Modelo de Torres de Refrigeración

Methodologies for the Design and Control ofCentral Cooling Plants

by

James Edward Braun

A thesis submitted in partial fulfillment of therequirements for the degree of

DOCTOR OF PHILOSOPHY(Mechanical Engineering)

at the

UNIVERSITY OF WISCONSIN - MADISON

1988

Methodologies for the Design and Control ofCentral Cooling Plants

James Edward Braun

Under the supervision of Professors

Sanford A. Klein, John W. Mitchell, and William-A. Beckman

This work presents general methodologies useful to engineers and plant managers

for designing, retrofitting, and controlling the equipment in large central chilled water

systems. The methodologies are in the form of: 1) mathematical models for the

individual equipment, 2) optimal control algorithms, and 3) general guidelines for

design and control.

Mathematical models of varying complexity are developed for the purpose of: 1)

component design and retrofit analyses, 2) system simulation and control optimization

studies, and 3) on-line optimal controL The models developed in this study representimprovements over those that appear in the literature. Where possible, measurements

are used to validate the models.

Two methodologies are presented for determining optimal control of systems

without thermal storage. A component-based algorithm is developed for non-linear

optimization applied to system simulations. Results of a program implementation of

this optimization algorithm are used to develop a "simple" near-optimal control

methodology. This near-optimal methodology relies on an empirical model for the total

power consumption of the system that lends itself to rapid determination of optimal

go11

control variables and may be fit to measurements using linear regression techniques.

For systems with stratified thermal storage in parallel with the chillers and load, two

methodologies are also presented for optimal control. A methodology is developed for

determining the optimal control of stratified thermal storage systems using Dynamic

Programming. For systems with time-of-day utility rates, a simple strategy for near-

optimal control is also identified.

The methodologies developed for evaluating the optimal control of chilled water

systems are utilized to study the effects of alternative control strategies and system

configurations. In addition, control guidelines useful to plant engineers for improved

control practices are identified.

Sanford- A. Klein ________

400

Acknowledgements

I must admit that I never had a burning desire to undertake this task. I wouldn't

have done it anywhere but at the Solar Energy Laboratory. Jack, Bill, Sandy, and

John, youtve created a tremendous working environment that stimulates independence,

creative thought, and comraderie. Thanks for all the support, ideas, and friendship.

I can't honestly say that these have been the best years of my life. However, there

have been some very special people who've made difficult times alot better. Special

thanks to Sandy for always being supportive and understanding. You're always

moving forward and doing positive things, even though you're seldom satisfied.

Chris, Joe, Jill, David, Jim, Al, and Ruth: Thanks for listening to me when I needed it

most. Most importantly, thanks Liz for being you. You've added meaning and balance

to my life.

The financial support from ASHRAE is gratefully acknowledged. The TC 4.6

Research Subcommittee provided the right amount of independence and supervision for

this project.

iv

Table of Contents

Abstract ii

List of Figures ix

List of Tables xiv

Nomenclature xv

Chapter 1 Introduction 1

1.1 Background 11.2 Research Objectives and Approach 8

1.2.1 Model Development 91.2.2 Methodologies for Optimal Control 101.2.3 Applications 11

1.3 Description of the Dallas/Fort Worth Airport System 111.4 Organization 13

Chapter 2 Models for Centrifugal Chillers 14

2.1 A Mechanistic Model for Variable-Speed Chillers 142.1.1 Model Formulation 152.1.2 Solution of the Equations 302.1.3 Parameter Estimation and Comparison with Measurements 322.1.4 Surge Predictions 38

2.2 A Model for Correlating Performance Data 412.3 Performance Characteristics of the D/FW Chiller 482.4 Summary 52

Chapter 3 Condenser-Side Component Models 54

3.1 Effectiveness Models for Cooling Towers 54

v

3.1.1 Detailed Analysis 563.1.2 Merkel Analysis 603.1.3 Effectiveness Model 613.1.4 Estimating the Water Loss 643.1.5 Model Comparisons 663.1.6 Correlating Performance Data 683.1.7 Potential Improvements in the Model 723.1.8 Sump and Fan Power Analyses 75

3.2 Condenser-Loop Pumping Requirements 773.3 Summary 82

Chapter 4 Evaporator-Side Component Models 84

4.1 Development of an Effectiveness Model for Cooling Coils 844.1.1 Detailed Analysis 864.1.2 Dry Coil Effectiveness 894.1.3 Wet Coil Effectiveness 904.1.4 Combined Wet and Dry Analysis 934.1.5 Model Comparisons 944.1.6 Correlating Performance Data 97

4.2 Air Handler Analysis 994.3 Chilled Water Loop Pumping Requirements 1024.4 Summary 104

Chapter 5 Methodologies for Optimal Control of Systems without Storage 106

5.1 A Component-Based Optimization Algorithm 1085.1.1 Quadratic Costs and Linear Outputs 1115.1.2 Nonlinear Optimization 1155.1.3 Constraints 1195.1.4 Algorithm Summary and Program Implementation 122

5.2 A System-Based Algorithm for Near-Optimal Control 1245.2.1 System Cost Function 1245.2.2 Near-Optimal Control Algorithm 125

5.2.3 Parameter Estimation 1275.2.4 Application to Chilled Water Systems 128

vi

5.3 Comparisons 133

5.4 Summary 138

Chapter 6 Applications to Systems without Storage 140

6.1 System Description 1406.2 Control Guidelines for Multiple Components in Parallel 143

6.2.1 Multiple Cooling Tower Cells 1446.2.2 Multiple Chillers 1466.2.3 Multiple Air Handlers 1516.2.4 Multiple Pumps 154

6.3 Sensitivity Analyses and Control Characteristics of Subsystems 1566.3.1 Effects of Load and Ambient Conditions 1566.3.2 Condenser Water Loop 1576.3.3 Chilled Water Loop 159

6.4 Optimal versus "Alternative" Control Strategies 1606.4.1 Conventional Control Strategies 1616.4.2 Humidity Control 163

6.5 Comparisons of Alternative System Configurations underOptimal Control 1656.5.1 Variable versus Fixed-Speed Equipment 1656.5.2 Series versus Parallel Chillers 169

6.7 Summary 172

Chapter 7 Methodologies for Optimal Control of Systems with Storage 175

7.2 Optimization of Systems with Storage 1767.1.1 Dynamic Programming 1787.1.2 Application Systems with Fully-Stratified Storage 1817.1.3 Results for Fully-Stratified Storage 187

7.1 Models for Forecasting Building Cooling Requirements 1937.1.1 Time-Series Models 1947.1.2 Application to Forecasting Cooling Loads 197

7.3 Summary 202

vli

Chapter 8 Conclusions and Recommendations 204

8.1 Mathematical Models 204

8.2 Optimal Control Methodologies 205

8.3 Guidelines for Design and Control 208

Appendix A Refrigerant Property Data 212

Appendix B Method for Determining the Performance of Partially Wet 213and Dry Cooling Coils

Appendix C Component Parameters for Optimization Studies 216

References 220

vii

List of Figures

Figure 1.1.

Figure 2.1.

Figure 2.2.

Figure 2.3.

Figure 2.4.

Figure 2.5.

Figure 2.6.

Figure 2.7.

Figure 2.8.

Figure 2.9.

Figure 2.10.

Figure 2.11.

Schematic of a Typical Chilled Water System

Schematic and Pressure-Enthalpy Diagram for a Single-StageChiler

Velocity Components for Refrigerant Exiting a CompressorImpeller

Comparison of Modeled Evaporating Temperatures with D/FWData

Comparison of Modeled Condensing Temperatures with D/FWData

Comparison of Modeled Chiller Power with D/FW Data

Comparison of Modeled Compressor Speeds with D/FW Data

Modeled Compressor Speed Versus Load Requirement

Comparison of Modeled Surge Speed with D/FW Data

Effect of Water Flow Rates on Chiller Performance for Fixed

Leaving Water Temperatures

Dependence of Chiller Performance on Leaving Water

Temperatures for Fixed Temperature Differences

Comparison of Measured and Correlated Chiller Power

ix

2

16

24

35

36

36

37

39

41

42

43

47

Figure 2.12.

Figure 2.13.

Figure 2.14.

Figure 2.15.

Figure 3.1.

Figure 3.2.

Figure 3.3

Figure 3.4

Figure 3.5.

Figure 3.6.

Figure 3.7.

Figure 3.8.

D/FW Chiller COP for Variable-Speed Control

D/FW Chiller COP for Vane Control

Comparison of Chiller Power Consumptions under Vane andVariable-Speed Control

Effect of Refrigerant Type on Chiller Performance

Schematic of a Counterflow Cooling Tower

Saturation Air Enthalpy versus Temperature

Air Heat Transfer Effectiveness Comparisons (Dry Bulb,

Wet Bulb, and Water Inlet Temperatures of 70 F, 60 F, 90 F)

Water Temperature Effectiveness Comparisons (Dry Bulb,Wet Bulb, and Water Inlet Temperatures of 70 F, 60 F, 90 F)

Comparisons of Relative Water Loss (Dry Bulb, Wet Bulb,

and Water Inlet Temperatures of 70 F, 60 F, and 90 F)

Comparisons of Leaving Water Temperature Measurementsfrom Simpson and Sherwood [1946] with EffectivenessModel Results

Comparisons of Relative Water Loss Measurements fromSimpson and Sherwood [1946] with Effectiveness ModelResults

Comparisons of Leaving Water Temperature Measurementsfrom D/FW Airport with Effectiveness Model Results

x

49

50

51

52

56

62

67

68

70

71

72

Figure 3.9.

Figure 3.10.

Figure 3.11.

Figure 3.12.

Figure 4.1.

Figure 4.2.

Figure 4.3.

Figure 4.4.

Figure 5.1.

Figure 5.2.

Figure 5.3.

Figure 5.4.

Figure 5.5.

Comparisons of Detailed Model with Effectiveness ModelResults for Extreme Conditions ( Ambient Dry and Wet BulbTemperature 80 F and 60 F)

Quadratic Fit to Variable-Speed Fan Power ConsumptionI

Pump Pressure Rise and System Pressure Drop Characteristics

Quadratic Fit to Variable-Speed Pump Power Consumption

Schematic of a Counterflow Cooling Coil

Air-Side Heat Transfer Effectiveness for Detailed andEffectiveness Models (Ambient Dry and Wet Bulb of 95 F and68 F, Water Inlet of 41 F, Ntuo/Ntui = 2)

Air Temperature Effectiveness for Detailed and EffectivenessModels (Ambient Dry and Wet Bulb of 95 F and 68 F, WaterInlet of 41 F, Ntuo/Ntui = 2)

Correlation of Cooling Coil Model Results with Catalog Data

Schematic of the Modular Optimization Problem

Comparisons of Optimal Chilled Water Temperature forComponent and System-Based Methodologies

Comparisons of Optimal Supply Air Temperature forComponent and System-Based Methodologies

Comparisons of Optimal Tower Control for Componentand System-Based Methodologies

Comparisons of Optimal Condenser Pump Control for

Component and System-Based Methodologies

xi

75

77

78

82

86

95

96

99

109

134

135

136

137

Figure 5.6.

Figure 6.1.

Figure 6.2.

Figure 6.3.

Figure 6.4.

Figure 6.5.

Figure 6.6.

Figure 6.7.

Figure 6.8.

Figure 6.9.

Figure 6.10.

Figure 6.11.

Figure 6.12.

Comparisons of Optimal System Performance for Componentand System-Based Methodologies

Effect of Condenser Water Flow Distribution for Two Chillersin Parallel

Effect of Relative Loading for Two Identical Parallel Chillers

Comparison of Optimal System Performance for IndividualSupply Air Setpoints with that for Identical Values

Effect of Chiller and Pump Sequencing on Optimal SystemPerformance

Effect of Uncontrolled Variables on Optimal System

Performance

Power Contours for Condenser Loop Control Variables

Power Contours for Chilled Water and Supply Air Tempera-

tures

Comparisons of Optimal-Control with "Conventional" Control

Strategies

Comparison of "Free" Floating and "Fixed" Humidity Control

Optimal System Performance for Variable and Fixed-SpeedChillers

Comparison of One-Speed, Two-Speed, and Variable-Speed

Cooling Tower Fans (Four Cells)

Comparison of Variable and Fixed-Speed Pumps

xii

138

147

150

152

155

157

158

160

162

164

166

167

168

Figure 6.13.

Figure 6.14.

Figure 7.1.

Figure 7.2.

Figure 7.3.

Figure 7.4.

Figure 7.5.

Figure 7.6.

Figure 7.7.

Comparison of Chiller Performance for Parallel and Series

Configurations

Optimal System Performance for Series and Parallel Chillers

Dynamic Programming Network

Parallel Configuration for Thermal Storage

Optimal Storage Charge Level for One Day with Time-of-DayElectric Rates

Optimal Chiller Loading for One Day with Time-of-DayElectric Rates

One-Hour Forecasts of March D/FW Cooling Load Data forAR(4) Model

Five-Hour Forecasts of March D/FW Cooling Load Data forAR(4) Model

Five-Hour Forecasts of March D/FW Cooling Load Data forCombined Deterministic and Stochastic Model

so*xml

170

172

180

182

190

191

198

199

200

List of Tables

Page

Table 2.1 Known D/FW Chiller Parameters 33

Table 2.2 Compressor Parameters Determined from Regression 35

Table 2.3 Comparisons of Chiller Model with Controlled Tests 38

Table 2.4 Measurements of the D/FW Chiller Performance for Variable 46and Fixed-Speed Control

Table 3.1 Mass Transfer Correlation Coefficients for Data of Simpson 70

and Sherwood [1946]

Table 4.1 Effectiveness Model Comparisons with Cooling Coil Catalog Data 97

Table 6.1 Summary of System Component Characteristics 143

Table 6.2 Cooling Season Results for Optimal vs. Conventional Control 163

Table 6.3 Cooling Season Results for Variable vs. Fixed-Speed 169Equipment

Table 7.1 One-Day Operating Cost Comparisons for Systems with 193Thermal Storage

Table 7.2 AR Model Fit to March D/FW Data (1 hour sampling) 197

xiv

Nomenclature

Chapter 2

A

Ax

Cpw

h

LMMm

P

Pch

0

r

R

T

UA

ux

V

V

w

Wpol

xv

area

- exit flow area from compressor impeller

specific heat of water

specific enthalpy of refrigerant or heat transfer coefficient

- log-mean temperature difference

- mass flow rate

- refrigerant pressure

- power consumption of compressor drive motor

- rate of heattransfer

- ratio of outside (finned) tube area to inside area or impeller radius

- heat transfer resistance of tube, including fouling factor

- temperature

- overall heat transfer conductance

- velocity of tip of impeller

- refrigerant specific volume

- refrigerant velocity

- compressor work input per unit mass of refrigerant

- polytropic work per unit mass of refrigerant, defined as the compressor

work required for a reversible polytropic process occurring between the

actual inlet and outlet states

- blade angle of compressor impeller

- dimensionless flow coefficient, defined'as the ratio of the fluid

velocity normal to the impeller to the impeller tip speed

- work coefficient, defined as the ratio of the tangential component of

the fluid velocity to the impeller tip speed

- work coefficient for a polytropic process

- polytropic efficiency, defined as the ratio of the polytropic work to

the actual compressor work required

- overall efficiency associated with the motor and gearbox

peak compres polytropic efficiency

- angular velocity of impeller

Subscripts

- condenser conditions

- chilled water conditions

- condenser water conditions

- chiller water return from the load

chilled water supply to the load

- leaving water temperature from the condenser

- supply water to the condenser

- evaporator conditions

- inside

- normal component

- outside

xvi

Ppol

Ipol

Tim

Tiref

0)

Additional

c

chw

cw

chwr

chws

cwr

cws

e

i

n

0

r

R

t

1

2

3

4

- radial component

- refrigerant

- tangential component

- conditions at tip of compressor impeller

- refrigerant state at exit of evaporator

- refrigerant state at exit of compressor

- refrigerant state at exit of condenser

- refrigerant state at inlet to evaporator

Chapters3and4

A - surface area

Av - surface area of water droplets per volume of cooling tower

Cpa - constant pressure specific heat of dry air

Cpm - constant pressure specific heat of moist air

Cpv - constant pressure specific heat of water vapor

Caw - constant pressure specific heat of liquid water

Cs - derivative of saturation air enthalpy with respect to temperature

C* - ratio of air to water capacitance rate for dry analysis

ha - enthalpy of moist air per mass of dry air

hc - convection heat transfer coefficient

hD - mass transfer coefficient

hf - enthalpy of liquid water

hg - enthalpy of water vapor

hs - enthalpy of saturated air

00xvln

M - mass flow rate

xia - mass flow rate of dry air

m- ratio of air to water effective capacitance rate for wet analysis

Le - Lewis number

Ntu - overall number of transfer units

P - power consumption

Q - overall heat transfer rate

Ta - air temperatur

Td, - ambient air dewpoint temperature

Tref - reference temperatue for zero enthalpy of liquid water

Ts - surface temperature

Tw - water temperature

Twb - ambient air wet bulb temperature

UA - overall heat transfer conductance

V - volume

AP - pressure drop or rise

APO - static pressure drop.

ea - air-side heat transfer effectiveness

7 - pump speed relative to maximum design speed

Coa - air humidity ratio

COS - humidity of saturated air

xvii

Additional Subscripts

a - air stream conditions

ahu - air handler

cw - condenser water loop

chw - chilled water loop

des - design conditions

dry - dry surface

e - effective

i - inlet or inside conditions

max - maximum

o0- outlet or outside conditions

p - pump conditions

s- surface conditions

T -total

w - water stream conditions

wet - wet surface

-Chapter 5

J

f

g

h

M

- instantaneous operating cost

- vector of uncontrolled variables

- vector of equality constraints

- vector of inequality constraints

- vector of discrete control variables

xix

u - vector of continuous control variables

x- vector of component input stream variables

y vector of component output stream variables

Chapter 7

J - integrated operating cost

L - instantaneous or stage operating cost

P - system power consumption

R cost of electricity

t - ime

T - storage temperature

Twb - ambient wet bulb temperature

Qch - rate of cooling provided by the chillers

QL - rate of cooling provided by the cooling coils to meet the load

Qs - rate of energy transfer from storage to meet the load (positive) or

to storage as supplied by the chillers (negative)

v - velocity in the x direction

x - position measured from the storage inlet

Xt - output of a pure time-series model at time t

Yt - output of a combined deterministic and time-series model at time t

f - vector of uncontrolled variables

u - vector of continuous control variables

x - vector of state variables

v - velocity relative to tank height measured in the x direction

xx

x - position of front between storage temperature zones measured from

the bottom of storage relative

At - timestep associated with stage

xxi

Chapter 1Introduction

1.1 Background

The heating and cooling requirements of many building complexes in this country

are provided with centrally located facilities. The University of Wisconsin at Madison

has two such central plants that provide the bulk of the heating and cooling

requirements for the university. Total annual fuel costs for both plants are on the order

of ten million dollars. Even a small relative savings in the energy requirements of a

plant such as this can translate into significant reductions in operating costs. In large

buildings, a significant portion of the total energy requirements are associated with air

conditioning. The potential for reductions in operating costs associated with improved

design and control practices for central cooling systems provides the impetus for this

project.

A centralized cooling plant consists of one or more chillers, cooling towers,

pumps, and air handlers controlled so as to satisfy the cooling requirements of one or

more buildings. Figure 1.1 shows a simplified schematic of a typical centralized chilled

water system. Return air from the zones is mixed with an outside ventilation air stream

and is cooled and dehumidified through cooling coils. In a variable air volume system,

the air handler air flows are typically adjusted depending upon the supply air

temperatures to maintain fixed zone temperatures. The supply air temperatures are

controlled by modulating the water flows through each cooling coil with control valves.

Cool and relatively dry air is supplied to the zones where both the temperature and

humidity rise due to sensible and latent energy gains from people, lights, equipment,

Supply Air Fans

Ventilation Air Exhaust Air

Figure 1.1 Schematic of a Typical Chilled Water System

Air

solar radiation through windows, infiltration, and conduction through the envelope.

The heat extracted from the air streams across the cooling coils warms the water

returning to the chiller. The chilled water supply temperature is maintained by control

of the chiller refrigerant flow through modulation of the compressor and expansion

device. Cooling towers with multiple cells sharing common sumps are typically used

to reject heat from the condenser of a chiller to the environment.

Design retrofits to existing systems may provide significant savings in operating

costs. Many systems were designed using old technology, with little concern for

energy costs. For instance, most large cooling systems utilize pumps and cooling

tower fans that operate at fixed speeds. Modulation of flow rates in response to

changing load conditions is limited to the on/off cycling of individual equipment

operated in parallel or series. A more efficient method of operation is to continuously

vary flow rates using variable-speed equipment Recently, the cost and reliability of

variable-speed electric motors has improved to the point where their use is cost effective

in many applications. Treichler [1985] concluded that variable-speed pumping is

economically attractive for both chilled water distribution and condenser water systems.

Centrifugal chillers are the largest consumers of energy in a central cooling system.

Significant operational savings can be realized with certain chiller retrofits. Many

applications utilize chillers that are operated at fixed speeds. Control of the chilled

water setpoint temperature is maintained by varying the position of pre-rotation inlet

vanes to the centrifuigal compressor. An alternative method, which gives significantly

higher efficiencies at part-load conditions, utilizes variable-speed control of the

compressor.

Another possibility for improving the overall chiller performance is through a

change in refrigerant. The refrigerant that gives the best performance depends upon the

4

load requirement. As an existing plant improves its energy management practices and

applies conservation methods to reduce the building load, the load requirement of the

chiller is reduced. As a result, the optimal refrigerant choice may change from that of

the original design.

The addition of chilled water storage, if properly controlled, can significantly

reduce operating costs (ASHRAE Bulletin [1985]). For systems with electrically

driven chillers, storage can be utilized to take advantage of time-of-day charges and

limit demand charges by shifting the load requirements. In the absence of special rate

structures, storage may also reduce costs by shifting the load to times when

environmental conditions result in improved chiller performance.

In addition to design retrofits, plant operating costs can be reduced through better

control practices. A central cooling plant has many operating variables that may be

controlled in a manner that minimizes the operational costs. At any given time, it is

possible to meet the cooling needs with any number of different modes of operation and

setpoints. Optimal supervisory control of the equipment involves determination of the

control that minimizes the total operating cost. The optimal control depends upon time,

through changing cooling requirements and ambient conditions. Currently, the

operators of central cooling plants determine control practices that yield "reasonable"

operating costs by experience gained through trial and error operation over a long

period of time. Little research has been pefformed in developing general methodologies

that would be suitable for optimal control of large centralized cooling systems. The

major independent control variables in a plant without chilled water storage are the

chilled water and supply air setpoint temperatures, number of chillers, pumps, and

cooling tower cells in operation, and cooling tower cell air flow and condenser pump

water flow rates. These variables may interact strongly and with proper control it is

possible to significantly reduce operating costs.

5

Much of the literature related to cooling systems pertains to sizing of the equipment.

The ASHRAE Handbook of Fundamentals [1981] gives background necessary for

individual equipment sizing. In order to properly evaluate the economics associated

with design retrofits or improved control practices, it is necessary to perform annual

simulations of the complete system using long-term average weather data and load

conditions. However, with inexpensive energy costs, there has been little incentive for

the use of detailed simulations in defing designs and control strategies that minimize

operating costs. More often than not, systems have been designed based upon

minimizing the first cost, while ensuring that the capacity of the system was sufficient

to meet the worst possible conditions. This has often resulted in systems being

oversized and operating inefficiently.

A transient system simulation program, TRNSYS [1984] was developed over 10

years ago by the University of Wisconsin Solar Energy Laboratory for analyzing the

performance of solar energy systems. TRNSYS is a modular program in which

models of system components (e.g. pipes, heat exchangers, storage) are written as

FORTRAN subroutines. The user can formulate or modify existing models and add

them to a library of components. The models are connected together to form a complex

system simulation model, analogous to the way pipes and wires connect the physical

pieces of equipment. The modular approach is advantageous in that it provides a format

in which a number of individuals can work independently and it minimizes the

programming required to investigate alternative component models and plant

configurations. This feature makes TRNSYS a good choice as a tool for studying the

design and control characteristics of cooling plants as compared with other existing

simulation programs (DOE-2 [ 1980], BLAST [1981], TRACE). However, since

TRNSYS has been used primarily for solar energy systems, its standard component

library does not include models for all of the equipment found in central cooling

6

systems. It also does not have the ability to determine the optimal control associated

with simulation of a system.

Simulations involving cooling systems have primarily been used for equipment

selection and building design. Most of the control studies which have been performed

have been concerned with the local-loop control of an individual component or

subsystem needed to maintain a prescribed set point, rather than the global

determination of optimum set points that minimize operating costs. Global optimum

plant control has been studied by Marcev [1980], Sud [1984], Lau [1985], Hackner

[1984,1985], Johnson [1985], and Nugent[1988].

Lau studied the effect of control strategies on the overall energy costs of a large

facility located in Charlotte, North Carolina through the use of annual simulations. The

operating costs associated with the existing control strategy were compared with those

resulting from "near-optimal" control of the condenser flow rates, the tower fan flows,

and the number of chillers operating, along with the use of storage. The reduction in

the utility bill was about 5.2 % of the total. The 50,000 gallon storage capacity was

small relative to the 6000 ton plant capacity. In addition, no time-of-day or peak

demand electric rates were in effect.

Hackner investigated optimal control strategies for a cooling system without storage

at a large office building in Atlanta, Georgia. Optimal control of the chiller plant

resulted in a reduction in the utility bill of about 8.7% when compared with the existing

control strategy. If the plant had been operated with fixed setpoints, similar to

conventional practice, the optimal control would have resulted in a cost reduction ofabout 19%. Similar conclusions were reached by Arnold, Sud, and Johnson. These

studies demonstrated the potential savings through the use of optimal control in a plant

without storage.

7

The goal of these control studies was to quantify the cost savings associated with

optimal control rather than to produce general algorithms suitable for on-line optimal

plant control. Optimal control set points were identified in Hackner's study for the

specific plant through the use of performance maps. These maps were generated by

many simulations of the plant over the range of expected operating conditions. The use

of established performance maps for on-line plant control is advocated by Johnson.

This procedure lacks generality and is not useful when storage is utilized.

Very little work has been performed in developing general methodologies that

would be suitable for on-line optimal control of large cooling systems. Nizet [1984]

applied a state-space model to a building zone in order to minimize the energy

consumption with respect to a single control variable, the supply air flow rate to the

zone. This optimization was carried out over a single day with the known weather data

utzed for forecasts. The conjugate gradient method-was applied to an integral

objective function that accounted for the cost of energy associated with the boiler,

chiller, and the supply air fans. Constant efficiencies were assumed for the boiler and

chiller. Thermal comfort constraints associated with upper and lower temperature

setpoints for the zone were considered through the introduction of penalty terms in the

cost function. Since component efficiencies were constant, the opportunity for savings

associated with optimal control of this hypothetical system resulted from operation

closer to the limits of comfort. Depending upon the comfort constraints, optimal

control of the supply air flow rate resulted in energy savings of between 12 and 30 %

when compared with the existing control. The authors concluded that there weredifficulties associated with the use of penalty functions for handing constraints. The

proper choice of the penalty coefficients is difficult and sometimes leads to a badly

conditioned problem. They also concluded that it was not practical to apply this

methodology to more complicated systems in an on-line application.

8

Gunewardana [1979] applied dynamic programming techniques for minimizing the

costs associated with a cooling system for power plants. The cooling System consisted

of a cooling tower and spray pond that could be operated in series or parallel through

eight possible modes in order to reject heat from the condenser of the power plant. The

overall cost to be minimized included the cost of operating the power plant and the cost

associated with pumping water through the cooling system. The dynamics of both the

cooling sump and spray pond were considered. A sampling interval of 15 minutes and

an optimization period of 2 hours were found to be adequate for this problem.

Forecasts of the ambient conditions and load were made by assuming that their rates of

change were constant. Depending upon the weather conditions, substantial cost

savings could be realized.

1.2 Research Objectives and Approach

The goal of this work is to develop general methodologies useful to engineers and

plant managers for designing, retrofitting, and controlling the equipment in large central

chilled water systems. The methodologies would be in the form of:

1) mathematical models for the individual equipment

2) optimal control algorithms

3) general guidelines for design and control

The methodologies developed are of a general nature so as to be applicable to a wide

variety of systems.

9

1.2.1 Model Development

Appropriate model development and selection depends strongly upon the desired

goals. There are three distinct uses for models in this study that result in three different

levels of mathematical representation.

1) ComZt sie and Analysis: Detailed mechanistic models are necessary in

order to evaluate design retrofits of individual equipment. They are also useful

in identifying the important variables that affect a components' overall

performance and provide a basis for the development of simpler models. When

little or no performance data are available, mechanistic models are a means of

generating a complete performance map for a particular component.

2) System Simulation: It is possible to correlate the performance of individual

components with simple mathematical relationships. In this manner, the time

required to perform a simulation of a complete system for studying the effects

of different control strategies and system configurations on overall performance

is significantly reduced.

3) On-Line Optimal Control: In order to practically apply on-line optimal control

to cooling systems, it is advantageous to have a simple model of the overall

power consumption of the plant in terms of a minimum of inputs and with

which optimal control setpoints can be readily determined. If the characteristics

of the system or measuring equipment is changing with time, it may necessary

to update parameters of the model using recursive identification techniques.

This is most easily carried out for models that are linear with respect to the

10

empirical coefficients. For systems with thermal storage, it is necessary to

forecast cooling requirements and ambient conditions in order to perform

optimal control.

In this study, models are developed for all three of the above purposes. The models

developed in this study represent improvements over those that appear in the literature.

Where possible, measurements from the cooling system at the Dallas/Fort Worth

(DF/W) airport and from the literature are used to validate the models.

1.2.2 Methodologies for Optimal Control

Methodologies for determining the minimum of a given cost function are well

established. For systems without significant thermal storage, the dynamics of the

cooling equipment can be neglected (Hackner [19851) and the plant optimization

involves minimizing the total instantaneous energy consumption of the chillers, cooling

tower fans, pumps, and cooling coil fans in terms of the current load and ambient

conditions. In this study, two methodologies are presented for determining optimal

control. First of all, a component-based algorithm is developed for non-linear

optimization applied to system simulation. In this study, this methodology is used

primarily as a simulation tool for analyzing the design and control characteristics of

chilled water systems under optimal control. Theoretically, the algorithm could be used

for on-line optimization of the simulation of an operating system using "simple"

component models. The simulation would proceed in parallel with the actual system

with the possibility of updating parameters of the component models using on-line

measurements. However, in this study, results of the optimization program are used to

develop a "simple" near-optimal control methodology. An overall empirical cost

function for the total power consumption of the cooling plant is inferred from the cost

11

functions associated with the components utilized in chilled water systems. This cost

function lends itself to rapid determination of optimal control variables and may be fit to

measurements using linear regression techniques.

Optimal control of a system with thermal storage involves minimizing the integral

of the instantaneous operating costs, while satisfying required constraints. In this

study, a methodology is developed based upon dynamic programming for detemiing

the optimal control of systems with stratified thermal storage. Optimization results are

then used to develop a "simple" near-optimal control strategy when time-of-day electric

utility rates are present.

For systems with storage, it is necessary to forecast cooling requirements and

ambient conditions in order to perform optimal control. A combined deterministic and

time series model is developed for forecasting cooling loads and is evaluated using data

from the D/FW airporL

1.2.3 Applications

In order to make any general conclusions concerning design and control of central

cooling plants, it is necessary to simulate the performance of a variety of system

configurations under optimal control. The methodologies developed for evaluating the

optimal control of chilled water systems are utilized in studying the effects of alternative

control strategies and system configurations. In addition, control guidelines useful to

plant engineers for improved control practices are identified.

1.3 Description of the D/FW Airport System

The central cooling plant at the Dallas/Fort Worth (D/FW) airport is used as the

primary test facility for this study. This particular system is of general interest because

12

of the large data acquisition system and the unique retrofits that the plant personnel have

undertaken.

The D/FW central facility provides heating and cooling for approximately 3 million

square feet of floor area for airport terminals, hotels, and office space. This system has

three centrifugal chillers originally rated at 8700 tons each. There are two sets of

cooling towers, each having four cells originally with two-speed 125 hp fans. Each

bank of tower cells shares a common sump and has two 500 hp condenser water

pumps in parallel that draw off the towers sumps. One set of cells has an additional

250 hp pump that is utilized at low load conditions. The chilled water pumping is

provided with two 500 hp pumps and one 250 hp pump. An additional unique feature

of the D/FW system is that there are several hours of thermal storage available within

the chilled water distribution system that is used to increase the overall plant cooling

capacity and improve part-load operation.

The data acquisition system records a wide range of conditions including

temperatures, pressures, flow rates, and power consumptions on magnetic tape each

minute. This information is utilized by the plant operators for the purpose of billing the

individual energy users and for energy management of the system as a whole. For this

study, this data is invaluable in developing and validating computer simulation models

of the equipment that are used in evaluating both improved control strategies and

retrofits for the plant.

Foremost among the retrofits implemented in the D/FW plant to reduce energy

consumption was the conversion of the drive for the primary centrifugal chiller. Each

of the three 8700 ton chillers as initially installed was driven with a steam turbine. As a

result of poor turbine efficiencies at low speeds, these chillers were primarily operated

at fixed compressor speeds. The capacity modulation was provided by control of inlet

pre-rotation and outlet diffuser vanes. Through goodl energy management practices, the

13

energy consumption at the D/FW airport has been reduced to the point where a single

chiller provides the necessary cooling requirements. To further reduce energy

consumption at part-load operation, the primary chiller was retrofitted with a 5000 hp

variable-speed electric motor. At the same time, the refrigerant was changed from R-22

to R-500 and the chiller capacity was derated to 5500 tons. This cooling capacity in

conjunction with the use of thermal storage is sufficient to satisfy the system demand

most of the time. Additional retrofits at the D/FW system include conversion of chilled

water distribution pumps and cooling tower fans with variable-speed motors. All of

these retrofits improve the part-load performance of the cooling plant.

1.4 Organization

The main body of the thesis is presented in Chapters 2 - 7. Both detailed and

simplified models for the individual system equipment are developed and compared in

Chapters 2,3, and 4. Results of these models are also compared with measurements.

Both the component-based and system-based optimization algorithms are developed in

Chapter 5. Results of applications of these methodologies to systems without storage

are given in Chapter 6. Included in these results are control simplifications that are

useful as guidelines for plant operators and that simplify the methodology for near-

optimal control. Typical results for the savings associated with optimal control and

comparisons between different system configurations under optimal control are also

presented in Chapter 6. In Chapter 7, dynamic programming is applied to fully-

stratified thermal storage systems in order to identify control simplifications for near-optimal control. In addition, a model for forecasting cooling loads is developed based

upon time-series methods. Computer programs developed in this study are listed in a

separate document (Braun [19881).

14

Chapter 2Models for Centrifugal Chillers

In this chapter, mathematical models are developed for centrifugal chillers. A

mechanistic model is presented for chillers operated with variable-speed control.

Results of this model are used to identify a simpler empirical model more suitable for

system simulation and optimal control studies. The empirical model correlates data for

both variable and fixed-speed chillers.

The models described in this chapter are compared with performance data for the

5500 ton centrifugal chiller at the Dallas/Fort Worth (D/FW) airport under both

variable-speed and fixed-speed control They are also used to study the performance

characteristics of the D/FW chiller.

2.1 A Mechanistic Model for Variable-Speed Chillers

A common method for modeling the performance of chillers for use in the

simulation of central cooling plants is to fit empirical relationships to manufacturers'

data. Stoecker [1971] and Bullock [1984] give examples of functional forms that are

adequate for this purpose. One limitation of this approach is that the model can only be

trusted within the range of conditions for which it was fit. Often the available data are

too limited to provide a complete performance map. In addition, empirical models are

not useful for investigating design retrofits associated with the chiller, such as changes

in refrigerant type or conversion from fixed-speed with vane control to variable-speed

capacity modulation. Such models are also limited in studying the importance of certain

control variables, such as chilled and condenser water flow rates. Although

manufactutrers' have developed mechanistic models of centrifugal chillers, their

15

descriptions are not available in the open literature.

In this section, a mechanistic model of a centrifugal chiller with variable-speed

control is described. The model utilizes mass, momentum, and energy balances on the

compressor, evaporator, condenser, and expansion device. Given a chilled water set

temperature and entering chilled and condenser water temperatures and flow rates, the

model determines both the required compressor speed and power consumption. In

addition, it may be used to estimate, for any given set of conditions, the chiller capacity

at a specific speed or power consumption or the compressor speed at which compressor

surge develops.

When little or no performance data are available, the mechanistic model described in

this section provides a tool for generating a complete chiller performance map. A

simpler empirical model appropriate for system simulation is then fit to the generated

data.

2.1.1 Model Formulation

Figure 2.1 shows a schematic and associated pressure-enthalpy diagram of a

centrifugal chiller with single-stage compression. The refrigerant entering the

compressor at state 1 is assumed to be a saturated vapor. Both the enthalpy and

pressure rise as the refrigerant passes through the compressor to a superheated state 2.

In the condenser, the refrigerant is cooled and condensed at a relatively constant

pressure and is assumed to exit at state 3 as a saturated liquid. It is then expanded at

constant enthalpy to the evaporator pressure (state 4). In the development that follows,

the chilled water supply and return temperatures, Tchws and TChw, refer to supply and

return to the load (i.e. from and to the evaporator), while the condenser water supply

and return, Tcw and To r are to and from the condenser. Each of the components are

modeled as follows.

16

14-" Ph

Tchwr Qe T chws

2

Enthalpy

Figure 2.1. Schematic and Pressure-Enthalpy Diagram for a Single-Stage Chiller

17

Evaporator

The evaporator is assumed to be a flooded shell and tube type design. Refrigerant

boils at the outside of horizontal tubes and rises out the top. Heat transfer in the

evaporator is modeled using an overall conductance and log-mean temperature

difference. Three expressions for evaporator heat flow that result from this model and

from energy balances on the two fluid streams are

Qe = UAeLMDe= rR(hj-h4)

= IMdiCpwCrI dwr- Tdw

where,o

UAe

LMD e

thR

h,

h4

Cpw

(2.1.1)

(2.1.2)

(2.1.3)

= rate of heat transfer to evaporator

= overall evaporator conductance

= evaporator log-mean temperature difference

= refrigerant mass flow rate

= specific enthalpy of refrigerant exiting evaporator

= specific enthalpy of refrigerant entering evaporator

= chilled water flow rate

- specific heat of water

The log-mean temperature difference and overall conductance are determined as

18

Tdmw- TchwsLMTe Td r- Te] (2.1.4)

'[T&iW, -Te]

Ae,iUAe 1 1 (2.1.5)+- 1 + Re

he,i+ reheo

where,

Te = refrigerant evaporation temperature

Aei = total inside surface area of evaporator tubes

heJ = heat transfer coefficient for water flow through evaporator tubes

he,o = boiling refrigerant heat transfer coefficient

re = ratio of effective outside evaporator tube area (product of overall fin

efficiency and total surface area) to inside area

Re = the resistance to heat transfer associated with the tube material, including

the fouling factor.

Nucleate boiling is assumed to take place from the evaporator tubes to the pool of

refrigerant. Bubbles nucleate and grow from spots on the surface in a thin layer of

superheated liquid formed adjacent to the tubes. There is much data available for

boiling heat transfer coefficients, but there is no universally accepted correlation.

Generally, the heat transfer coefficient for a particular application may be correlated in

the form

19

he,o = a (Tes'T )b (2.1.6)

where a and b are empirical constants that depend upon the properties of the refrigerant

and the nucleate characteristics of the surface and T, is the average tube outside

surface temperature. Myers (1952) gives typical results for the heat transfer coefficient

of Refrigerant 12 with finned tubes that are used in this study. An expression for the

tube surface temperature obtained by considering the heat transfer resistance between

the water and the outside tube surface is

r Re (2.1.7)= Tchw+Aeh ,

where T w is the average of the entering and leaving chilled water temperatures.

The chilled water flow through the evaporator tubes is assumed to be turbulent such

that the heat transfer coefficient is given as (ASHRAE [1985])

rk 0 8 0.4he, i == 0.023 dwRe ePhe., (2.1.8)

where kw is the thermal conductivity of water, 4 is the inside tube diameter, Ree is the

Reynolds number associated with water flow in an individual evaporator tube, and Pr is

the Prantl number for water.

Condenser

The condenser is also considered to be a horizontal shell and tube design.

Refrigerant condenses on the outside of the tubes and drains out the bottom.

20

Analogous to the evaporator, the three equations for heat transfer are

=QC- UACLIVM C (2.1.9)

= MR (h2 -h 3) (2.1.10)

= rCwCpw (Tcw- Tw) (2.1.11)

where,

h2 = specific enthalpy of refrigerant entering condenser

h3 = specific enthalpy of refrigerant exiting condenser

cw= condenser water flow rate

and

LAM C . TCW$

Ti- T] (2.1.12)

UAC+R 1 (2.1.13)hCji +rchc,o R

Tc = refrigerant condensing temperature

AC~ = total inside surface area of condenser tubes

hci = heat transfer coefficient for water flow through condenser tubes

hco = condensing refrigerant heat transfer coefficient

rc = ratio of effective outside condenser tube area (product of overall fin

efficiency and total surface area) to inside area

21

R = the resistance to heat transfer associated with the tube material, including

the fouling factor.

It is not strictly correct to use the refrigerant condensing temperature, Tc, in

determining the log-mean temperature difference for the entire condenser. Refrigerant

entering the condenser is superheated and is cooled sensibly to the saturation

temperature at a relatively constant pressure prior to condensation. During this process,

the heat transfer coefficient is lower and the temperatuN difference higher than during

condensation. These tradeoffs justify the use of a single condenser temperature and a

single condensing heat transfer coefficient in the model.

Theoretical expressions for determining the heat transfer coefficients for laminar

film condensation of pure vapors on plates and tubes were first developed by Nusselt.

The average heat transfer coefficient associated with a vapor condensing on N

horizontal tubes is estimated from

0.253 2

he9 =0.725 kf(pf-pv) ghfg (2.1.14)N dcLf (Tc - Tcs)

where,

kf = conductivity of the liquid refrigerant

hfg = heat of vaporization of the refrigerant

g = gravitational acceleration

pf = density of saturated liquid refrigerant

Pv= density of saturated vapor refrigerant

= condenser tube diameter

22

viscosity of saturated liquid refrigerant

TCOs = average tube outside surface temperaure

The vapor density is usually small compared to the liquid density and may be

neglected. Analogous to the evaporator analysis, the tube surface temperature is

0 1 1 (2.1.15)Ts T+Ri +

where TC, is the average condenser water temperature.

The correlation for turbulent flow in tubes, equation (2.1.8), is applicable to acondenser tube and is used for evaluating the water side heat transfer coefficient for the

condenser, he i

Compressor

One approach to modeling the performance of the compressor is to use performance

curves from the manufacturer. Davis (1974) presents a method of correlating data that

reduces the family of compressor head characteristics to a single curve of dimensionless

head versus a dimensionless flow. A limitation associatedt with this approach is that

complete performance data is not always readily available or it is presented in such a

way as to be specific to the refrigerant employed.The model developed in this study relies on relationships that are commonly used in

the design of centrifugal compressors. It employs fundamental mass, momentum, and

energy balances and empirical correlations that are representative of well-designed

centrifugal compressors.

23

Ferguson [1963] provides a good background for much of the presentation to

follow concerning the compressor. Figure 2.2 shows a cross-section of the impeller of

a centrifugal compressor showing a single blade with pertinent dimensions and

velocities.

The impeller rotates with an angular velocity co having a tip speed equal to u,. The

refrigerant vapor exits the impeller with a relative velocity V,, and an absolute velocity

V. The components of velocity tangential and normal to the impeller wheel are

denoted as Vx t and Vx,n.

Neglecting the angular momentum of the incoming refrigerant, a momentum

balance on the impeller gives an expression for the required work input per unit mass of

refrigerant.

w =COVXtr = uXVX 9 = 2 XU (2.1.16)

where the work coefficient, gx, is defined as the ratio of the tangential component of

the fluid velocity to the impeller tip speed.

Vx~t

UX (2.1.17)

If the velocity of the fluid relative to the impeller, VX, exits tangential to the blade

(i.e. no slippage), then the theoretical work coefficient determined from the vector

diagram of Figure 2.2 is

24

VX~

{ . Blade

Compressor Impeller

Velocity Vectors

Vx,n

Ux

Figure 2.2. Velocity Components for Refrigerant Exiting

a Compressor Impeller

25

Uxo VxncOt(o3)

= 1- CxCO() (2.1.18)

The dimensionless flow coefficient, is the ratio of the fluid velocity normal to

the impeller to the impeller tip speed.

Vxn MrgVxUx Aux (2..19

where Ax is the exit flow area of the impeller and vx is the exiting vapor specific

volume.

In reality, slippage and non-uniform velocity profiles at the impeller exit limit the

accuracy of this formulation. Wiesner [1959, 1960] has correlated the real performance

of centrifuigal compressors with vaneless diffusers. His results are presented as curves

of polytropic efficiency, rpoI, and polytropic work coefficient, , versus the

dimensionless flow coefficient, x where,

wpOl (2.1.20)1p1P Ww

2 (2.1.21)

The polytropic work, wpo is the quantity of work required for a reversible

polytropic process occurring between the actual inlet and outlet states. A polytropic

26

process satisfies

Pv = constant (2.1.22)

The polytropic coefficient, n, is determined by the actual initial and end states.

v2 PIP21V1 1-PJ (2.1.23)

Since there is an increase of entropy in the actual irreversible compressor, there

must be a reversible heat input to the reversible compressor for the end states to be the

same. The polytropic work is given by

P2 n P 2 n (..4WpoI f - vdP=Piv nf- 1(2.1.24)

p n - 1 PI]

The stage work coefficient, , is determined from the polytropic coefficient and

efficiency as

g-tx = (2.1.25)

Wiesner's results for are a series of straight lines that can be represented by

9 = 0.69 (1- 4xcot([3)) = 0.69 (2.1.26)

27

The polytropic efficiency results of Wiesner are presented as curves of relative

efficiency versus the dimensionless flow coefficient, x, for different rotational Mach

Numbers, M., where

M O = __

ao (2.1.27)

and a. is the sonic velocity in the refrigerant at the impeller inlet conditions. The

following equation provides a good fit to the graphical data of Wiesner.

ip! = [1 +a(1.1.Mo)] 1 -exp( x(b x + cox + d))(2.1.28)

1Tref

The reference polytropic efficiency, Trf, is the peak value associated with a

reference rotational Mach Number of 1.1. It is typically in the range of 0.80 to 0.85.

The empirical constants (a, b, c, d) that provide a good match to Wiesner's data are

0.109, 58.5, -6.0, and -18.8, respectively. Wiesner also presents a correction factor

for polytropic efficiencies due to differences in Reynolds numbers associated with the

use of different refrigerants. This effect is relatively small and is negligible for the

refrigerants considered in this study (R-500, R-22, and R-12).

In order to evaluate the flow coefficient using equation (2.1.19), it is necessary to

determine the specific volume of the refrigerant at the exit of the impeller. Most of the

entropy rise associated with the compression process occurs within the diffuser. For

this reason, the entropy at the impeller exit is assumed to be equal to the entropy at the

inlet.

28

sx = S1 (2.1.29)

The additional property necessary to define the state at the impeller outlet is

determined from an energy balance on the diffuser. Assuming that the kinetic energy

exiting the diffuser is small compared to that at the diffuser inlet, the incoming enthalpy

is

2

hx = h 2 - -2 (2.1.30)

From an energy balance on the impeller and equation (2.1.16)

2h2 = h1 + xux

So,

2

= , u y2 Vxhx-h+ g x'2

(2.1.31)

(2.1.32)

The absolute refrigerant velocity at the impeller exit, Vx, is determined from the

normal and tangential components (Figure 2.2), such that

2 2 2 2(2 2V = Vn+ V~ x t ) (..3

29

Thus the exit impeller enthalpy is

2 2]

hX =thi+u x .- (2.1.34)

The thermal expansion device is assumed to modulate the refrigerant flow such that

a saturated vapor state is maintained at the compressor inlet. The entering and exiting

enthalpy of the expansion device are assumed to be equal.

h4= h3 (2.1.35)

Finally, the power input to the motor driving the compressor is calculated as

11R(h 2 - h1)Pch =

T1m (2.1.36)

where im is the overall efficiency associated with the motor and gearbox if present.

The model, as defined through equations (2.1.1) - (2.1.36), requires properties of

the refrigerant at various states. A computer program developed from the equations

given by Downing (1981) was used to evaluate thermodynamic properties of the

refrigerants at any state. Additionally, curve-fits were developed for viscosity

and conductivity at saturated liquid conditions for refrigerants considered in this study.

The sonic velocity associated with the vapor refrigerant exhibits very little variation

over a wide range of temperatures. It was assumed to be constant evaluated at 50

degrees Fahrenheit. Appendix A gives the viscosity and conductivity curve-fits and

30

sonic velocity data. A listing of the refrigerant properties program is given by Braun

[1988].

2.1.2 Solution of the Equations

For a given chiller design, there are five independent input variables that define the

chiller performance through the equations presented in the previous section. It is

possible to solve these equations in different ways depending upon the desired inputs

and outputs. Most commonly, the independent variables controlling the compressor

performance would be the entering chilled and condenser water temperatures and flow

rates and the chilled water setpoint. In this case, the primary outputs would be the

compressor power requirement and speed. Alternatively, it is possible to specify the

compressor speed as an input, in which case the leaving chilled water temperature and

power requirement are outputs. Other possibilities include specifying the leaving in

place of entering condenser water temperature or the power consumption. In any case,

the solution of the equations is not as complicated as it might first appear.

For a single-stage compressor with a specified chilled water setpoint and entering or

leaving condenser water temperature, the equations are solved in two separate steps.

1) Given the chilled water entering flow rate and temperature and the setpoint,

determine the evaporator refrigerant temperature, Te, by iteratively solving

equations (2.1.1) and (2.1.3)-(2.1.8). This can be accomplished with

Newton's method applied to the function.

F(Te) = UAeLMT4"De - mehwCpw(T ehwr 'Thws) (2.1.37)

2) The solution of the remaining set of equations can be reduced to finding the

31

zeros of three functions with three unknowns using Newton's method. The

three iteration variables are T,, u., and x, while the three functions are

F1 = UAcLMIDc - rncwCp(T,- -Tcw) (2.1.38)

F2 = rX ARVx (2.1.39)

F3 = (h2 - h1) - wPOI (2.1.40)1 lpol

For given values of the iteration values, each of the terms in the above equations are

uniquely defined with equations (2.1.2), (2.1.9) - (2.1.35) and property data.

The analysis is complicated a bit further if two-stage compression is considered. In

this case, the energy and momentum balances are applied to both compression stages.

In the absence of an economizer, the outet from the first stage is the inlet the second..

Most multi-stage centrifugal chillers utilize an economizer. For a two-stage

compressor, refrigerant exiting the condenser is expanded to the intermediate pressure

between compression stages and enters a flash tank. Saturated refrigerant vapor is

removed from the flash tank, mixed with the outlet stream from the first stage and fed

to the second-stage compressor. Liquid refrigerant from the flash tank is expanded to

the evaporator pressure. In order to include an economizer in the analysis, mass and

energy balances are applied to the economizer to determine the additional states and

refrigerant flow rates.

In the solution of equations for two-stage compression, two additional iteration

variables are the intermediate pressure and the flow coefficient for the second stage.

32

Equations (2.1.39) and (2.1.40) apply to the first stage and an analogous two

additional equations are used for the second stage.

The maximum cooling capacity of a chiller is limited by several factors. For

instance, it may be controlled by the maximum allowable power input to the motor or

the maximum rotational speed. Alternatively, there may be a lower limit on the

refrigerant temperature in order to avoid localized ice formation within the evaporator or

an upper limit on the condenser pressure. In any of these situations, the model can be

adapted to determine the maximum capacity and associated power input and

compressor speed. The D/FW chiller capacity is primarily limited by the power input to

the motor. In this case, the equations are solved such that power is an input and

cooling capacity is output. Equations (2.1.37) - (2.1.40) are solved concurrently,

rather than separately in this situation. Listings of computer programs for evaluating

the chiller power consumption, operating speed, and cooling capacity for single and

two-stage compression (with or without an economizer) appear in a separate document

(Braun [1988]).

2.1.3 Parameter Estimation and Comparison with Measurements

The D/FW chiller has two-stage compression with an economizer. Many of the

parameters characterizing this design were available from the manufacturer and are

presented in Table 2.1. Additional parameters necessary for evaluating the chiller

performance were determined by regression using measurements from the D/FW

airport. The data used in the regression was randomly selected from two different time

periods to give a range of conditions.

The ratios of the outside finned tube area to the inside area for both the evaporator

and condenser were unknown. Sufficient data were available to estimate these ratios

33

from a regression analysis. The saturation temperatures were estimated from

compressor suction and discharge pressure measurements using refrigerant property

data for R-500. Good agreement between the model and the saturation temperatures is

obtained for values of r. and r, of 3.1 and 3.2.

Table 2.1Known D/FW Chiller Parameters

Description Value Units

Effective evaporator internal tube 11,300 sq. ft.surface areaNumber of evaporator tubes 3560Number evaporator tube passes 3Evaporator tube length 22 feetTube inside diameter (evaporator 0.75 inchesand condenser)Effective condenser internal tube 14,800 sq. ft.Number of condenser tubes 3349Number of condenser tube passes 1Condenser tube length 30 feetDiameter of compressor impellers 2.33 feet

The efficiency of the electric motor driving the compressor is approximately 95%.

Additionally, there is significant energy loss in the gearbox between the motor and the

compressor. At maximum loading of 5000 hp, the energy loss is approximately 200

hp. This gives an overall efficiency of about 91%.

There are three additional unknown parameters concerning the compressor that are

necessary in order to analyze the chiller performance: 1) the impeller blade angle, f3, 2)

the impeller exit flow area, Ax, 3) the reference polytropic efficiency, Tlrf-

Estimates of these parameters were obtained from the D/FW plant personnel and the

34

literature as follows.

From a photograph of a centrifugal compressor impeller available from the D/FW

plant personnel, the blade angle appears to be approximately 30 degrees. The impeller

width at the exit is between 2 and 3 inches. This gives an impeller exit area of between

1.2 and 1.8 square feet. Wiesner's [1960] curves of polytropic efficiency derived from

measurements of several centrifugal compressors operating with R-500, R-22, or R-12

give a relative polytropic efficiency of about 0.82.

In order to fine-tune these estimates, a regression analysis was applied to the

centrifugal compressor. Measurements included the motor electrical consumption and

the compressor suction and discharge pressures. Table 2.2 gives the parameter values

determined from the regression analysis that yield the best agreement with the data.

They are surprisingly close to the original estimates.

Figures 2.3 - 2.6 show comparisons between measurements and the overall model

predictions of refrigerant temperatures in the evaporator and condenser and the

compressor power and speed. Overall, the agreement is very good. The best

predictions are of the power consumption and the evaporator temperature. The

estimates of the condenser temperature and compressor rpm are not quite as good.

There appears to be a slight bias in the comparisons. The model tends to underestimate

power consumption and speed at low values. One possibility is that the motor and

gearbox efficiencies are lower at lower speeds and loadings. The model assumes a

fixed overall efficiency for these components at all conditions. Another possibility is

that the compressor polytropic efficiency may fall-off more significantly at low loads

than the Wiesner data exhibits.

35

Table 2.2Compressor Parameters Determinedfrom Regression

Parameter Value Units

f3 27.2 degreesAx 1.53 square feet

'Iref 0.814

50

45

40

35

30 35 40 45 50Measured Evaporator Temperature (F)

Figure 2.3. Comparison of Modeled Evaporating Temperatures with D/FW Data

30

I-

09tS.'

0:

36

92 5

I 80 x95 x x

S90"

• 80.-,,

.:. 75

: 7070 75 80 85 90 95 100

Measured Condensing Temperature (F)

Figure 2.4. Comparison of Modeled Condensing Temperatums with D/FW Data

4000 .1,

po

0

3-

I-

3000

2000

1000

0

0 1000 2000 3000 4000

Measured Chiller Power (kW)

Figure 2.5. Comparison of Modeled Chiller Power with D/FW Data

37 •

4500-

& 4000

3500

S3000

xx

~2500'

x

20002000 2500 3000 3500 4000 4500

Measured Compressor Speed (rpm)

Figure 2.6. Comparison of Modeled Compressor Speeds with D/FW Data

The D/FW measurements were originally recorded on magnetic tape at one minute

intervals. For the comparisons of Figures 2.3 - 2.6, the data were selected randomly.

Part of the variability in these results may be a result of unsteady conditions. As

another test of the accuracy of the model, controlled tests were performed on the chiller

and compared with model predictions for a range of conditions. The conditions for all

measurements were stabilized for at least 15 minutes. Both the chilled and condenser

water flow rates were held relatively constant for all tests. The results of the

comparisons as summarized in Table 2.3 show that the model agrees well with the data.

Once again, the estimates of power consumption are better than for compressor speed.

The root-mean-square of the differences is 84 kW for power and 140 rpm for

compressor speed. The relative error of the power consumption estimates is larger at

low loads. This may be due in part to the much larger uncertainty in the load evaluation

38

at this condition. Errors in measurements of chilled water temperature differences used

to determine the chiller load, have a much more significant effect when the differences

are small.

Table 2.3Comparisons of Chiller Model with Controlled Tests

Power (kW)Data Model

364650

10101411805126

241627363580

415610

1299940

1316

302755

10821421

867134

224827743519

461689

1154864

1391

Speed (rpm)Data Model

235026003100340014001400

36003700390027002700320026503000

21772695299232581563156235723755397124982497295526243004

2.1.4 Surge Predictions

The operating temperature and pressure of the refrigerant in the evaporator are

uniquely determined by the chilled water load, the water flow rate, and the chilled water

setpoint. Similarly, the condensing pressure depends upon the total heat rejection and

the condenser water stream conditions. A surge condition occurs when the compressor

is unable to develop a discharge pressure sufficient to satisfy the condenser heat

rejection requirements. This results in an unstable mode of operation in which the total

Load(tons)

13752475280027502710135554205460542026902750273040654065

Tchws(F)

4041404040504040405050505050

Tcwr(F)*

5764697964576976866269826475

39

flow in the compressor oscillates.

A mathematical characteristic of the surge condition is that it first develops at a point

of zero slope of the compressor discharge pressure versus flow relationship. This also

corresponds to a point of zero slope of the compressor speed versus chiller loading

characteristic. Figure 2.7 shows results of the model for required compressor

3500=

3300

3100

CL

2900 Required Speed

Q 2700

2500 - • • -0 500 1000 1500 2000 2500 3000

Chiller Load (tons)

Figure 2.7. Modeled Compressor Speed Versus Load Requirement

speed versus loading. The modeled surge point occurs at the minimum speed. To the

left of this point the model predicts that the compressor speed increases with decreasing

load. The minimum modeled compressor speed is 2700 rpm at a load of about 500

tons. It is difficult to predict the load associated with the development of surge. As

evident in Figure 2.7, the chiller cooling capacity is more sensitive to compressor speed

40

near the surge point. The determination of the onset of surge requires a subjective

decision and cannot be measured directly. In this work, the approach used in modeling

surge is to first determine the minimum possible compressor speed for any given set of

conditions. The point at which surge develops is then assumed to occur at 50 rpm's

greater than this minimum. At this point, Figure 2.7 indicates that there is essentially a

linear relationship between speed and chiller capacity.

The chiller model is easily adapted to determine the compressor speed at which

surge first occurs for a given set of chilled and condenser water conditions. An

optimization routine, such as golden section search, is used to calculate the minimum

compressor speed as a function of chiller loading. A program listing for determining

the surge point is given by Braun [1988].

Figure 2.8 shows a comparison of compressor speeds at which surge develops for

the model and D/FW data as a function of the temperature difference between the

leaving condenser and evaporator water flow streams. The agreement is surprisingly

good.

41

3500

Sm.

~3000

Data0 Modeix

U)2500

2 0 0 0 W ' - , -

10 20 30 40 50 60

Leaving Water Temperature Difference (F)

Figure 2.8. Comparison of Modeled Surge Speed with D/FW Data

2.2 A Model for Correlating Performance Data

The mechanistic model described in the previous section is useful for investigating

the performance of variable-speed chillers in detail. However, it requires too much

computation to be used in system simulation or optimization studies. In this section, a

simpler empirical model for correlating the performance of variable or fixed-speed

chillers is presented.

There are five independent input variables that uniquely define the performance of a

chiller. One possible set of independent variables is 1) chilled water load, 2) chilled

water supply temperature, 3) chilled water flow rate, 4) leaving condenser water

temperature, 5) condenser water flow rate. Alternatively, entering chilled water and

condenser water temperatures could be utilized in place of leaving conditions.

By correlating chiller performance in terms of leaving rather than entering

42

temperatures, the dependence on either the chilled water or condenser water flow rate is

not significant. Figure 2.9 shows the effect of chilled and condenser water flow rates

on the coefficient of performance (COP) of the D/FW variable-speed chiller determned

with the mechanistic model described in Section 2.1. In the normal operating range of

8.0

7.5-

7.0-

6.5-

6.0

5.5

5.0 -

1000

Figure 2.9.

I U U - I - U - U

2000 3000 4000 5000 6000 7000

Chilled Water Load (tons)

Effect of Water Flow Rates on Chiller Performance

for Fixed Leaving Water Temperatures

this chiller, the effect of variations in either flow rate on the overall chiller performance

is relatively small (< 2%) when the results are presented in terms of leaving water

temperatures. It is interesting to note that reducing the evaporator flow shows an

improvement in the performance. This results from the characterization of the

performance in terms of the load and leaving chilled water temperature. For a given

load and chilled water setpoint, a lower flow gives a higher chilled water return

temperature. Assuming that this dominates over the reduced heat transfer coefficient

x 50% greater evaporator water flowa 50% greater condenser water flow

m Normal water flow

*0.'

43

effect, the evaporation temperature rises and the performance improves. In practice, the

chilled water return temperature is constrained by comfort considerations at the

distribution points to the load.

The number of independent variables effecting the chiller performance is reduced to

three, when utilizing leaving evaporator and condenser water temperatures and

neglecting the effect of variations in flow rate. Correlating chiller performance may be

further simplified by utilizing the difference between the leaving condenser and

evaporator water temperaturs as an independent variable rather than the individual

temperatures. Figure 2.10 shows a comparison between modeled chiller COP as a

function load for a leaving water temperature difference of 40 F for different chilled

water supply and leaving condensing water temperatures. The performance is

7.0

6.5-

o 6.0

5-5-- -Tcw s = 40 F,Tewr = 80 F

x Tchws = 50 F, Towr= 90 F

5.0 . , w . , . ,

1000 2000 3000 4000 5000 6000 7000Chilled Water Load

Figure 2.10. Dependence of Chiller Performance on Leaving Water Temperaturesfor Fixed Temperature Differences

44

nearly independent of the individual leaving evaporator and condenser water

temperatures, depending primarily upon their difference. Similar results were obtained

for a range of other conditions.

Results of the mechanistic chiller model indicate that chiller power consumption is

primarily a function of only two variables, the load and the temperature difference

between the leaving condenser and chilled water flows. The following functional form

correlates the chiller power in terms of these variables.

Pch22Pdcs = a0 + ajX + a2X 2 + a3Y + a4Y2 + a5XY (2.2.1)

where X is the ratio of the chiller load to a design load, Y is the leaving water

temperature difference divided by a design value, P&c is the chiller power consumption,

and Pd, is the power consumption associated with the design conditions. The

empirical coefficients of the above equation (a0, a1, a2, a3, a4, and a5) are determined

with linear least-squares curve-fitting applied to measured or modeled performance

data.

It is also necessary to model the limits of chiller operation associated with maximum

chiller capacity and compressor surge. The following relationships were found to

work well for estimating the maximum and minimum capacities.

Xmx = bo + blY (2.2.2)

Xmin = cO + clY + c2y 2 (2.2.3)

45

where X.. and X.- are the maximum and minimum possible chilled water loads

associated with the capacity and surge limits divided by the'design load. Again, the

empirical coefficients of the above equations are determined with linear regression

applied to measurements or model generated data.

In order to estimate chiller power consumption using the above relationships, it is

necessary to know the leaving water temperatures. The chiller is controlled to give a

specified leaving chilled water temperature. The leaving condenser water temperature,

on the other hand, is not known. It depends upon the entering conditions, load, and

power consumption. An implicit relationship for the dimensionless leaving water

temperature difference results from the use of an overall energy balance on the chiller.

e + lPe(a0+alX +a 2 X2+a 3Y + a4Y2+ a5 XY)

= micwCPW[Y(Tcwr" Tchw)des - (Tcws Tchws)] (2.2.4)

Equation (2.2.4) is quadratic in Y and may solved explicitly. If the design conditions

are appropriately chosen, then Y will typically vary between about 0.2 and 1. This may

be used as a criteria for selecting between the two solutions of equation (2.2.4).

If the specified chiller load falls outside of the limits imposed by equations (2.2.2)

and (2.2.3), then it is necessary to adjust the setpoint such that operation occurs at the

limit. In this case, the load is given by equations (2.1.3) and (2.2.2) or (2.2.3) and Y

is determined with (2.2.4). These equations are solved to find the load, chilled water

setpoint, and value of Y that give performance at the operation limit. Equation (2.2.1)

is then used to evaluate the chiller power consumption.

The empirical relationships for estimating chiller power consumption and the

capacity and surge limits were tested over a wide range of operating conditions using

data derived from the mechanistic model of a variable-speed chiller. In all cases, these

46

models provided an excellent fit to the generated data.

The model for power consumption was also fit to measurements from the D/FW

airport system for both variable and fixed-speed control. Tests were performed for

both types of control at nearly identical conditions. Table 2.4 summarizes the results of

these tests. The test conditions are given in sets of two. The first of each set is for the

variable-speed measurements, the second for fixed-speed control. The chilled and

condenser water flow rates were held constant for these tests. The compressor speed

for the fixed-speed, vane control tests was also held constant at 4000 rpm and the inlet

pre-rotation and outlet diffuser vanes were operated using the automatic control

implemented by the manufacturer.

Table 2.4Measurements of the D/FW Chiller Performance

for Variable and Fixed-Speed Control

Test Load(tons) Tchws(F) Tcwr(F) Power Consumption (kW)Variable-Speed Fixed-Speed

1 1375, 1355 40, 40 57, 58 364 8602 2475, 2625 41, 40 64, 62 650 14103 2800, 2710 40, 40 69, 69 1010 18004 2750, 2670 40, 40 79, 80 1411 19305 2710, 2710 40, 40 64, 64 805 14106 1355, 1625 50, 49 57, 57 126 10367 5420, 5420 40, 40 69, 70 2416 27808 5460, 5420 40, 40 76, 76 2736 27369 5420, 5420 40, 40 86, 86 3580 362710 2690, 2710 50, 50 62, 62 415 156011 2750, 2750 50, 50 69, 69 610 132612 2730, 2730 50, 50 82, 82 1299 183013 4065, 4065 50, 50 64, 64 940 244614 4065, 4065 50, 50 75, 75 1316 2480

47

Figure 2.11 shows a comparison between measured and correlated chiller power

consumption. Separate curve-fits were performed for the D/FW chiller operated with

both variable and fixed-speed control. The design conditions for equation (2.2.1) were

taken to be those associated with the maximum measured power consumption. The

root-mean-square of the error in the modeled power is 64 kW for the variable-speed

and 152 kW for the fixed-speed control. The larger errors for fixed-speed

4000

now

noON"

3000

2000

1000

00 1000 2000 3000

Measured Chiller Power (kW)

4000

Figure 2.11. Comparison of Measured and Correlated Chiller Power

operation may be due to a more unstable control characteristic. In examining the test

results in Table 2.4, there is more inconsistency in the results of the fixed-speed tests.

For instance, in comparing tests 7 and 8 (or 13 and 14), the only condition that changed

appreciably was the leaving condenser water temperature. However, the power

48

consumption associated with vane control did not increase as would be expected. In

this mode of operation, both the inlet and outlet vanes are adjusted in some automatic

fashion to meet the desired conditions. Therefore, it is possible to realize the same

conditions with different power consumptions. The inconsistencies in the fixed-speed

measurements and resulting errors in the empirical fit to the data may be due to

inconsistent control of the inlet and outlet compressor vanes.

2.3 Performance Characteristics of the D/FW Chiller

The performance of the D/FW chiller operated with both variable and fixed-speed

control may be compared directly through the test results of Table 2.4. At all part-load

conditions, the performance associated with the variable-speed control is superior.

However, the power requirements come together at loads approaching the capacity of

the chiller. This is expected, since at this condition the vanes are wide open and the

speed under variable-speed control approaches that of the fixed-speed operation.

In order to see the differences between variable and fixed-speed operation more

clearly, their performance was correlated using the form of equation (2.2.1). Figures

2.12 and 2.13 show chiller performance for both controls in terms of COP as a

function of load for different condenser to evaporator leaving water temperature

differences. For the variable-speed control, the COP reaches a maximum at part-load

conditions. This peak occurs as a result of tradeoffs between increasing heat transfer

efficiencies due to decreased water to refrigerant temperature differences in the

evaporator and condenser and the decreased polytropic efficiencies of the compressor

that occur at low refrigerant flow rates. The peak COP moves to lower loads at lower

temperature differences.

It is typical for centrifugal chillers with either variable or fixed-speed control to

exhibit a peak in perfonnance at part-load conditions of between 40% to 70% of the

49

chiller design capacity. However, the fixed-speed performance data shown in Figure

2.13 does not exhibit such a peak. The best performance occurs at the capacity of the

chiller. The capacity of the D/FW chiller was derated when retrofit with a different

refrigerant, so that the evaporator and condenser are oversized at the current capacity

relative to the original design capacity. As a result, the performance is more sensitive to

penalties associated with part-load operation of the compressor than to heat exchange

improvements. Part of the improvement with variable-speed chiller control may result

from the unique characteristics of the D/FW chiller.

10

0NMI

30 F

8

6

4-

A -1V-

1000I - U - U - U

2000 3000 4000 5000


6000

Figure 2.12. D/FW Chiller COP for Variable-Speed Control

50

10

8-

3.30F

•" 4 -

2

01 V I W i " I ' '' I " "

1000 2000 3000 4000 5000 6000


Figure 2.13. D/FW Chiller COP for Vane Control

Figure 2.14 shows a direct comparison of the results of Figures 2.12 and 2.13.

The ratio of power under variable-speed control to that with vane control is plotted as a

function of load and leaving water temperature differences. At typical summer

conditions of 5500 tons and temperature differences of between 40 and 50 F, the

performance ratio is near unity. As the load decreases, the COP of the variable-speed

control ineases, while that associated with fixed-speed operation is reduced. At part-

load conditions of about 3000 tons and temperature differences between 30 and 40 F,

the variable-speed control uses about 30% less power.

51

0.8

0.7 30F

- 0.60C

i: 0.4-

0.3-

0.2 i

2000 3000 4000 5000 6000


Figure 2.14. Comparison of Chiller Power Consumptions under Vane andVariable-Speed Control

The mechanistic chiller model is useful for studying the effect of different

refrigerants on the overall chiller performance. Figure 2.15 shows COP as a function

of load for R-500, R-22, and R- 12 at a leaving water temperature difference of 30 F.

The D/FW chiller as installed was charged with R-22. Upon retrofit with a variable-

speed drive, the refrigerant was changed to R-500. Since the maximum chiller load is

generally less than about 5500 tons, Figure 2.15 indicates that this was a relatively

good choice. Overall, R-22 does well at very high loads, both R-12 and R-500 are

good at loads near 5500 tons, and R-12 is a clear choice at lower loads. It appears that

R- 12 may be a better overall choice than R-500 for the D/FW system.

52

10

R-500a R-22

8-

7

60 2000 4000 6000 8000


Figure 2.15. Effect of Refrigerant Type on Chiller Performance

2.4 Summary

A detailed mechanistic model for variable-speed centrifugal chillers was developed.

The model requires only design parameters and the operating conditions in order to

estimate the power requirement. The model is also capable of estimating the

compressor speed at which surge develops or the maximum chiller cooling capacity at a

given power input or speed. Results of the model compare favorably with

measurements from the D/FW airport for both power requirement and the speed

associated with the onset of surge.

Using results of the mechanistic model, a simpler empirical model for correlating

performance data was also identified. Chiller power consumption is correlated as a

quadratic function of only two variables, the load and temperature difference between

the leaving condenser and chilled water flows. This model fits data for both variable-

53

speed and fixed-speed with vane control chillers. Models for correlating the limits of

chiller operation associated with surge and capacity were also presented.

Results of the models were used to study the performance of the D/FW chiller.

Data for both variable-speed with wide-open vanes and fixed-speed with vane control

were correlated using the empirical model and compared over a wide range of

conditions. At design conditions, the power consumption associated with the two

controls is essentially equal. At 60% of the design load, the variable-speed operation

requires 50% to 80% of the power requirements for fixed-speed, vane control. The

magnitude of the improvement depends upon the temperature difference between the

leaving condenser and chilled water streams. The use of different refrigerants was also

investigated. The peak performance associated with using R-500 shows on the order

of 5 - 10% improvement over the original charge of R-22.

54

Chapter 3Condenser-Side Component Models

In this chapter, models are presented for cooling towers and condenser water

pumping requirements. These are the prinary components associated with rejecting

heat from chiller condensers to the environment.

3.1 Effectiveness Models for Cooling Towers

This section presents a simple, yet mechanistic method for modeling the

performance of cooling towers. Through the introduction of an air saturation specific

heat, effectiveness relationships are developed. The advantages of this approach are its

simplicity, accuracy, and consistency With the methods for analyzing sensible heat

exchangers. The accuracy of the effectiveness model is as good or better than that

associated with standard methods presented in the literature and has significantly less

computational requirements.

The first practical theory of cooling tower operation was developed by Merkel

[1925]. Merkel's method has been the basis of most cooling tower analyses (Nottage

[1941], Lichtenstein[1943], Mickley [1949], Carey [1950], Webb [1984]) and is

outlined in the ASHRAE Equipment Guide [1983].

In the Merkel analysis, the water loss due to evaporation is neglected and a Lewis

number of unity is assumed. The determination of outlet states requires an iterative

numerical integration of two differential equations. A more rigorous analysis of a

cooling tower that did not utilize the assumptions of Merkel was performed by

Sutherland [1983]. He found that counterfiow cooling towers can be undersized

between 5% and 15% through the use of the Merkel method if "true" mass transfer

55

coefficients are used. In practice, however, the errors are not nearly as large, because

the mass transfer coefficients utilized in the Merkel method are generally determined by

matching results of the model to measurements from small-scale tests. Another

approach for modeling cooling towers was presented by Whillier [1976]. Whillier

developed a simple method for correlating performance data. However, this method is

not useful for design purposes.

In this section, an effectiveness model is developed for cooling towers by utilizing

the assumption of a linearized air saturation enthalpy. On its own, the linearization is

not a unique contribution. It was utilized for cooling coils by Threlkeld [1970] and

suggested for, but not evaluated for cooling towers by Nizet [1985]. Berman [1961]

described a log-mean enthalpy method for analyzing cooling towers that implicitly

assumes a linear saturation curve. Maclaine-Cross [1985] also developed a model for

wet surface heat exchangers that incorporated the assumption of a linearized saturation

humidity ratio. Recently, an effectiveness model for cooling towers was also presented

by Jaber and Webb [1987]. The contributions of the work described in this chapter are

primarily: 1) development of the basic equations leading to effectiveness relationships

for cooling towers analogous to those for sensible heat exchangers, 2) development of

a simple method for estimating the water loss in cooling towers, and 3) validation of the

methodologies over wide ranges of conditions. Results of the methods are compared

with both numerical solutions to the detailed heat and mass transfer modeling equations

and with measurements. In Chapter 4, a similar analysis is applied to modeling the

performance of cooling coils.

56

3.1.1 Detailed Analysis

The assumptions and basic equations employed here closely follow those of

Sutherland [1983]. A schematic of a counterflow cooling tower showing pertinent

states and dimensions is given in Figure 3.1.

ma; 0 a,o;ha,o

4

ma; oa,i; ha

Air

Watermw4 ; Tw

rnw; TW9

Figure 3.1. Schematic of a Counterflow Cooling Tower

(Oa + da m + dmwha +dha T ~

OD a mw; Tw

dV

TIt

57

For negligible heat transfer from the tower walls, a steady-state energy balance on

the incremental volume, dV, gives the following relation between the water and air

enthalpies.

adh a = d(whf,w)

= MW dh f,w + h f,wdw(31)

where,

ma = mass flow rate of dry air

ha = enthalpy of moist air per mass of dry air

m, = mass flow rate of water

hfow = enthalpy of liquid water

Relationships for the incremental water loss, dm,, and the water flow rate at any point

within the tower, m.x, are determined from steady-state mass balances on the water.

d w =adcoa

w= Mw,i - ma(cOao- coa)

(3.1.2)

(3.1.3)

where,

wa = local air humidity ratio

w~i = inlet water flow rate

0a, o = outlet air humidity ratio

From equations (3.1.1), (3.1.2), and (3.1.3) and assuming a constant water

58

specific heat, the change in water temperature across the incremental control volume

(water inlet minus outlet) is

dh a -Cpw(Tw "Trf)doadTw *a oa(314[ .w (Oao '-a)]CPw

Ma

where Tw is the local water temperature, Trf is the reference temperature for zero

enthalpy of liquid water, and Cp is the constant pressure specific heat of liquid water.

The incremental increase in enthalpy of the air stream is equal to the rate of energy

transfer from the water droplets due to both heat and mass transfer or

madha= hcAvdV(Tw-Ta) + hgw Iadcoa (3.1.5)

where,

hc = convection heat transfer coefficient

Av = surface area of water droplets per unit volume of cooling tower

hgVw = enthalpy of water vapor at the local water temperature.

Assuming that the mass fraction of water vapor in the mixture of air and vapor is

approximately equal to the humidity ratio, the rate of mass transfer of water vapor to the

air stream may be expressed as

rnadoa = hDAvdV(COs,w- Co)a) (3.1.6)

59

where

hD = mass transfer coefficient

oslw = saturated air humidity ratio at the local water temperature.

Substituting equation (3.1.6) into (3.1.5) and making use of the Lewis number

definition (Le = hc/(hDCpm)) and the definitions for the enthalpy of water vapor, the

result may be written as

xlaha-hDAVdVLeCPm(rw-Ta)+ (cos,w- oa)hgw]

= LehDAvdF(hs,w " ha) + (osw" .cia)(1/Le - 1)hg,w] (3.1.7)

where,

C -Cpm i

Cpa

CpV=

hg o -

constant pressure specific heat of moist air = Cpa + C(aCpv

constant pressure specific heat of dry air

constant pressure specific heat of water vapor

enthalpy of water vapor at 0 degree reference level

The overall number of transfer units for mass transfer is defined as

Ntu hDAVVTMa

(3.1.8)

where VT is the total tower volume. Utilizing the Ntu definition, equations (3.1.6) and

(3.1.7) may be reduced to

60

do a = Ntu

dV VT (c a s (3.1.9)

dh = -L Ntu (h - h0) + (Oa -CO )(1/Le "-1)hg, (3.1.10)

- VT ~

For given Ntu, Le, and inlet conditions, equations (3.1.9), (3.1.10), and (3.1.4)

may be solved numerically for the exit conditions of both the air and water stream. The

solution is iterative with respect to two variables, o° and Tw.o. At each iteration,

equations (3.1.9), (3.1.10), and (3.1.4) are integrated numerically over the entire tower

volume from air inlet to outlet.

3.1.2 Merkel AnalysisIn order to simplify the analysis, Merkel [1925] made two assumptions. Most

significantly, he neglected the effect of the water loss due to evaporation on the right-

hand side of equation (3.1.1). In equation (3.1.1), this eliminates the last term and

implies that the water flow rate at each point in the tower in the first term on the right-

hand side is equal to the inlet flow. The second assumption is a Lewis number of

unity. With these approximations, the equations for the cooling tower may be reduced

to

dha Ntu(ha - (3.1.11dV - VT

61

dTw m a(dha/dV)dV thWC pw

The primary computational savings associated with the Merkel assumptions is that

the solution of equations (3.1.11) and (3.1.12) is iterative with respect to a single

variable, Tw, o, instead of two variables. However, the solution of equations (3.1.11)

and (3.1.12) gives only the exit temperature of the water stream and exit enthalpy of the

air stream. In order to obtain the exit humidity ratio, it is still necessary to numerically

integrate equation (3.1.9) after the solution of equations (3.1.11) and (3.1.12) has been

obtained.

3.1.3 Effectiveness Model

Equation (3.1.12) may be rewritten in terms of only air enthalpies by introducing

the derivative of the saturated air enthalpy with respect to temperature evaluated at the

water temperature.

dhS,W -aCs(dha/dV (3.1.13)dV flMCpw

where,

C s - T - T w(3 .1.14 )

C5 has the units of specific heat and will be termed the saturation specific heat. If the

saturation enthalpy were linear with respect to temperature, (i.e constant C5 ), then

62

equations (3.1.11) and (3.1.13) could be solved analytically for the exit conditions.

These equations are analogous to the differential equations that result for a sensible heat

exchanger with C,, ha, and h,,w replaced by Cp7, Ta, and Tw. Figure 3.2 shows the

variation of saturation enthalpy with temperature, along with a straight line connecting

two typical water inlet and outlet states. It is apparent that the saturation enthalpy is not

linear with respect to temperature. However, by choosing an appropriate average slope

between inlet and outlet water conditions, as depicted in Figure 3.2, an effectiveness

relationship may be derived in terms of C.

100

90

80

70

60

50

40

30 Water'

20

100 '- -

40 50I - UIa = -f = = UN=

----- --vI a I - II - I

60 70 80 90

Saturation Temperature (F)

Figure 3.2. Saturation Air Enthalpy versus Temperature

Effectiveness is usually defined as the ratio of the actual to maximum possible heat

transfer. However, when C, is taken as the slope of the straight line between the water

inlet and outlet conditions, then an air-side effectiveness rather than an overall

.0

0

I'

dI-.

0n

Twj,- Tw,o

1

100 110

63

effectiveness relationship should be used. The air-side effectiveness is defimed as the

ratio of the actual heat transfer to the maximum possible air-side heat transfer that

would occur if the exiting air stream were saturated at the temperature of the incoming

water (i.e. ha,o = hs,'wj). The actual heat transfer is then given in terms of this

effectiveness as

a ea a(hs w,i -ha, ) (3.1.15)

where, analogous to a dry counterflow heat exchanger, the effectiveness is evaluated as

1 - exp(-Ntu(1 - m))F a = ,-,(3.1.16)

1 - m exp(-Ntu(1 - m))

where,

M ma (3.1.17)m=--

)mwji (CpVXS)

The exit air enthalpy and water temperature are determined from overall energy balances

on the flow streams.

ha,o ha,i+ a(hs,wi -'hai) (3.1.18)

mnw,i(Tw i - Tref)Cpw - mna(hao - ha,) (31.9Tw~~o -- rhw,o pw (..9

64

Equation (3.1.19) is written in terms of the mass flow rates of water entering and

leaving the cooling tower. The simplest strategy is to completely neglect the water loss

as is done with the Merkel method. In this case, equations (3.1.15) - (3.1.19), along

with psychrometric data, are sufficient for determining the overall heat transfer and exit

conditions. The enthalpies associated with the inlet air and saturated air at the water

inlet temperature are evaluated with a psychrometric chart or using empirical

correlations as given in the ASHRAE Handbook of Fundamentals [1985]. The average

saturation specific heat, Cs, is estimated as the average slope between the inlet and

outlet water conditions.

Cs= Twj- Twp (3.1.20)

Since C, depends upon T,, the solution of equations (3.1.15) - (3.1.20) is iterative.

However, C,5 is only weakly dependent upon the exit temperature, so that any

reasonable initial guess of Tw.0 (such as the entering air wet bulb temperature) usually

results in convergence in only 2 iterations.

3.1.4 Estimating the Water Loss

Due to water loss, the water flow rate exiting the cooling tower is on the order of

1% to 4% less than the entering flow rate. Neglecting this loss may result in up to a 2

degree Celcius error in the exit water temperature. This error can effect the calculated

performance of other system equipment such as a chiller. The computation of the exit

water temperature given by equation (3.1.19) is improved if the water loss is

considered. It is also necessary to know the water loss in order to perform mass and

65

energy balances on the cooling tower sump. From an overall mass balance, the exit

water flow rate is

w9= m ,- ra( a,o" a) (3.1.21)

The exit air humidity ratio could be determined by numerically integrating equation

(3.1.9) over the tower volume, An alternative approach, which allows an analytic

solution, is to assume that equation (3.1.9) is approximately satisfied over the entire

tower volume with the local co, replaced with a constant appropriately averaged value.

Assuming a Lewis number of unity, integration of equation (3.1.9) gives

coa,o = o's,w,e + ( oai-oswe)exp(-Ntu) (3.1.22)

By integrating equation (3.1.11) for constant l an effective saturation enthalpy is

determined as

=h+ "(ha,o- h adi) (3.1.23)

hsw' ha'i +1 - exp(-Ntu)

The effective saturation humidity ratio, cos,w,., associated with hs,we is found from

psychrometric data.

66

3.1.5 Model Comparisons

Figures 3.3 - 3.5 show comparisons between cooling tower results obtained with:

1) a detailed analysis defined by the numerical solution of equations (3.1.4), (3.1.9),

and (3.1.10) with a Lewis number of unity; 2) the Merkel analysis; 3) the

effectiveness model.

In Figure 3.3, cooling tower air heat transfer effectiveness is plotted versus Ntu for

different ratios of water to air flow rate for entering conditions typical of those

associated with heat rejection from the condenser of a chiller. Neglecting the water loss

in the Merkel approach causes a slight underprediction in the cooling tower

effectiveness resulting from a reduced mass transfer. Errors associated with the

effectiveness method are primarily a result of the assumption of a linear saturation

enthalpy relationship. Overall, both the Merkel and effectiveness models agree closely

with the detailed analysis for these conditions.

Figure 3.4 shows cooling tower water temperature effectiveness results for the

same conditions as for Figure 3.3. The water temperature effectiveness is defined as

the ratio of the temperature difference between the inlet and outlet water to the

maximum possible temperat difference if the leaving water were at the entering air

wet bulb temperature. As a result of the water loss, this temperature effectiveness does

not correspond to a heat transfer effectiveness. In Figure 3.4, the Merkel model

somewhat overestimates the water temperatue effectiveness due to the assumption of

no water loss. The effectiveness model gives results that are closer to the more exact

analysis.

67

1.0

0.8MW 2

0.8 Ma mw=1

ma

0.6* =0.5

0.4,a

-"Detailed Analysis

0.2 + Merkel Methodn Effectiveness Model

0.0'I i iI V

0 1 2 3 4 5

Number of Transfer Units (Ntu)

1.0m

0.8 M a

0.6

V 0.4 Ma

Detailed Analysis

0.2 Merkel Method

* Effectiveness Model0 .0 * • , • , .

0 1 2 3 4 5


Figures 3.3 & 3.4. Air Heat Transfer and Water Temperature EffectivenessComparisons (Dry Bulb, Wet Bulb, and Water Inlet Temperatures of 70 F, 60 F, 90 F)

68

Figure 3.5 gives comparisons between tower water loss evaluated with a detailed

analysis and the loss determined using the simplified approach presented in Section

3.1.4 (The Merkel method neglects the water loss). Water loss as a percentage of the

inlet water flow rate is plotted versus Ntu for different ratios of water to air flow rates.

No significant differences are evident in these comparisons.

3'

-0.5ma = 2

.. " 1 ma =

-Detailed Analysis* Effectiveness Model

2 M 11 a *a 1111110000olm

0 1 2 3 45


Figure 3.5. Comparisons of Relative Water Loss (Dry Bulb, Wet Bulb, and Water

Inlet Temperatures of 70 F, 60 F, and 90 F)

3.1.6 Correlating Performance Data

General correlations for heat and mass transfer in cooling towers in terms of the

physical tower characteristics do not exist. It is usually necessary to correlate data for

specific tower designs. Mass transfer data are typically correlated with the following

form (ASHRAE [1983]).

69

I. n

hDAVVT = c MW(3.1.24)

where c and n are empirical constants specific to a particular tower design. Multiplying

both sides of the above equation by Mw/ma. and utilizing the definition for Ntu gives

0 1 w I + n

Ntu = cm- (3.1.25)

According to the above equation, data should correlate as a straight line on a log-log

plot of Ntu versus the flow rate ratio. The slope and intercept of this line are (1+n) and

log(c), respectively. Linear regression may be utilized to determine the "best" fit

straight line through the data. For given entering conditions and heat transfer rate, the

tower Ntu is estimated from equation (3.1.16) with the air-side effectiveness computed

using equation (3.1.15).

Simpson and Sherwood [1946] give data for a number of different tower designs.

The coefficients of equation (3.1.25) were fit to the measurements of Simpson and

Sherwood for four different tower designs over a range of performance conditions.

Table 3.1 gives the coefficients of the mass transfer correlation determined from the

regressions for each tower. The data include measurements of outlet states for both the

water and air flow streams. A comparison between results of model estimates for

leaving water temperature and measurements are shown in Figure 3.6. The model

matches these data quite closely over a wide range of conditions. Water loss was

computed from the measurements using a mass balance and compared with model

70

predictions as shown in Figure 3.7. The model tends to underpredict the water loss as

compared with the data. This bias may be associated with the assumption of Lewis

number of unity or may be due to errors in the measurements of the air outlet state.

Energy balances on the data close only to within 10 percent.

Table 3.1Mass Transfer Correlation Coefficients for Data of

Simpson and Sherwood [1946]

Tower c n

R-1R-2M-1

M-2

110

a

'0i

100

90

1.6841.4050.9841.130

-0.391-0.727-0.852-0.617

80

70 , "70 80 90 100 110

Measured Two (F)

Figure 3.6. Comparisons of Leaving Water Temperature Measurements fromSimpson and Sherwood [1946] with Effectiveness Model Results

71

3,

20

2

2 1'

0 1 2 3

Measured W mW,0 x 100mwA

Figure 3.7. Comparisons of Relative Water Loss Measurements from Simpson andSherwood [1946] with Effectiveness Model Results

Results of the cooling tower model were also to fit to data from the D/FW airport

for a 3 day period in October. This tower has four cells, each with two-speed fans that

share a common sump. D/FW measurements included the entering tower water

temperature and supply water temperature to the chillers from the sump. Ambientwet

bulb temperatures were available from the National Weather Service for this time

period. The maximum tower air flow rates and coefficients of the Ntu correlation

determined from nonlinear regression analysis were 635,000 cfm, c = 3.76, and

n = -0.63. As exhibited in Figure 3.8, the model does a relatively good job of

estimating the leaving cooling tower water temperature for this data. The root-mean-

square of the error is approximately 1.4 degrees F. Since the differences between

72

entering and leaving water temperatures were in the range of 4 to 15 degrees, not nearly

as good agreement was realized in terms of tower heat rejection rates. However, the

agreement is within the accuracy of the measurements.

80

.+++

' 70"+ ++ +4. +

+

'i

N 60 +

4. .D/FW Measurements-Model Predictions

50 , • •0 20 40 60 80 100

Time (hours)

Figure 3.8. Comparisons of Leaving Water Temperature Measurements from D/FW

Airport with Effectiveness Model Results

3.1.8 Potential Improvements in the Model

Theoretically, it is possible to improve the effectiveness model by accounting for

the effect of water loss on the change in the water stream enthalpy in an approximate

manner. Rather than totally neglecting the water loss due to evaporation in the last term

of equation (3.1.1), a slightly better assumption is a constant specific enthalpy of this

evaporation term throughout the tower. A corrected enthalpy is then defined as

h'= h - WCpw (Twi- Tref) (3.1.26)

73

With this definition, equations (3.1.11) and (3.1.13) may be replaced with

I

dha = Nm (27dV " VT (ha"hsv) (3.2.27)

dh _w maaCs(dha/dV)S = (3.1.28)

d Vw C

where,

CS = ].dT JT Tw (3.1.29)

Equations (3.1.26) - (3.1.29) may be solved analytically for outlet conditions given

inlet conditions for a linearized Cr Overall, the accuracy of this analysis is no better

and in many cases worse than the simpler effectiveness method. Neglecting the effect

of the water loss on the enthalpy change of the water stream is a conservative

assumption. On the other hand, the linearization of the saturation enthalpy between the

inlet and outlet water temperatures tends to overpredict the tower heat transfer. These

are relatively small, but compensating effects that result in a model with good accuracy.

By including the water loss, the model becomes slightly optimistic.

The accuracy of the effectiveness model presented in this chapter depends primarily

upon the temperature difference between the water stream inlet and outlet. As the inlet-

to-outlet water temperature difference increases, the accuracy associated with using a

linear relationship for the saturation enthalpy is reduced. Comparisons with both

74

detailed numerical solutions and measurements indicate that this model works well over

a wide range of typical conditions. However, for more extreme conditions, it is

possible to improve the accuracy of the model.

One approach for reducing the errors associated with the linearization involves

dividing the cooling tower into two or more increments. The effectiveness model is

then applied to each increment and the set of equations is solved for the intermediate

and outlet states. With a sufficient number of increments, this model will always yield

satisfactory results. This procedure is advocated by Jaber and Webb [1987]. Another

approach involves choosing a better linearization than a straight line between the inlet

and outlet conditions. Berman [ 1961] developed a correction factor to account for the

curvature of the saturated air enthalpy versus temperature relationship.

Figure 3.9 shows a comparison between the detailed analysis, the effectiveness

model as described in Section 3.1.3 , and the effectiveness model using the correction

of Berman [1961]. Cooling tower air effectiveness is plotted as a function of the water

inlet temperature for different water-to-air mass flow rates with constant ambient dry

and wet bulb temperatures (80 F and 60 F). The Ntu's were determined with equation

(3.1.25) using coefficients determined from the regression to the D/FW airport data.

Depending upon the flow rate ratio, the accuracy of the effectiveness model is degraded

with increasing temperature differences between water inlet and ambient wet bulb. As

the water-to-air flow rate ratio increases, the water temperature difference decreases and

the linearization is more accurate. At the low flow rate ratios, the water loss becomes

more significant and compensates for greater inaccuracies in the linearization. Overall,

the effectiveness model appears to give satisfactory results for temperature differences

up to 50 degrees F between the water inlet and ambient wet bulb. The correction of

Berman improves the results for the higher flow rate ratios, but has negative effect at

75

low flow rate ratios where the water loss becomes important. At low flow rate ratios

and large water inlet and wet bulb temperature differences, the combination of the

Berman correction and the approximate method for including water loss described

above may be justified. However, the accuracy of the simpler effectiveness model is

most likely sufficient over the practical operating range when considering the other

uncertainties in the analysis such as inaccurate input data or the assumption of a Lewis

number of unity.

1n0.8-

0.8"

0.4-

60 70

I I ilIiI w I w l " I "

80 90 100 110

Water Inlet Temperature (F)

Figure 3.9. Comparisons of Detailed Model with Effectiveness Model Results forExtreme Conditions ( Ambient Dry and Wet Bulb Temperature 80 F and 60 F)

3.1.8 Sump and Fan Power Analyses

In analyzing the performance of a system, it is necessary to include effects of a

tower sump. Water enters a sump from each of the operating tower cells and from a

water make-up source. The level of the sump is assumed to be constant, so that the

U'

II

OW

=w 2

1

* Effectivness Model++ Effectiveness with Berman [ 1961] Correction

0.2120

a - - 9 N

76

flow of water make-up is equal to the total water loss from the cells. The water volume

of the sump is further assumed to be fully-mixed, so that a steady-state energy balance

yields

Ncell N 11

X ( Tw'o-Ts + tw~i- mwSo) (Tmain-Ts) = 0 (3.1.30)k=1 k=1

where Nn is the number of tower cells that share a common sump, Tmin is

temperature of the water make-up and T. is the fully-mixed sump temperature. This

equation is solved for the sump temperature, Ts, which is generally the supply

temperature to the condenser of a centrifugal chiller in a central chilled water facility.

The cooling tower fans are assumed to obey the fan laws. Given the power

requirement at maximum fan speed, the power consumption for a cooling tower

consisting of N, 1 tower cells is calculated as

Ncell 3

Pt= Yt,kPt,k,des (3.1.31)k=1

where Yk and Pt~kd are the relative fan speed and design power consumption at

maximum speed for the kth tower cell.

As discussed in Chapter 5, the deter ation of optimal control points that minimize

operating costs is simplified when the individual component operating costs are

expressed as quadratic functions of the continuous controls and/or outputs. In the

optimization process, the power consumption given by equation (3.1.31I) may be

approximated with a quadratic function using a second-order Taylor series expansion

77

about the most recent iteration point. Alternatively, it is possible to accurately correlate

the power consumption of a variable-speed fan as a quadratic function of speed over its

operating range. Figure 3.10 shows a comparison of the relative fan power

consumption as determined with equation (3.1.31) for one cell and a quadratic

function. The coefficients of the quadratic function were determined analytically by

matching the power consumption determined with equation (3.1.3 1) at relative fan

speeds of one-third, two-thirds, and full speed. The quadratic function works

extremely well in correlating variable-speed fan power consumption.

u

.2

0m

Lo

0J

0.0 " I I " l "

0.0 0.2 0.4 0.6 0.8

Relative Fan Speed

1.0

Figure 3.10. Quadratic Fit to Variable-Speed Fan Power Consumption

3.2 Condenser-Loop Pumping Requirements

The performance of a centrifugal pump is characterized by a rating curve that gives

pressure rise developed by the pump and efficiency versus pump flow. The maximum

78

pressure rise is produced at zero flow rate, while the maximum efficiency occurs at

what is considered to be the design flow. At the operating point of a pump, the

pressure rise developed is equal to the total pressure drop through the system. The

primary pressure losses in the condenser loop are to due to the elevation difference

between the tower inlet stream and the sump and the flows through the condenser and

tower nozzles. Figure 3.11 shows typical pump pressure rise and system pressure

drop characteristics as a function of flow rate. The operating point of the pump occurs

where the two curves intersect.

0I.i

f-

APpm

Al'.

Pump Characteristic

mcw mp,max

Mass Flow Rate

Figure 3.11. Pump Pressure Rise and System Pressure Drop Characteristics

For turbulent flow, the overall condenser water loop pressure drop takes the

following form.

79

r 2APcw = AP 0 + AP cw,des 1 (3.2.1)

where AP. is the static pressure drop associated with the elevation change in the tower,

rncw is the condenser loop flow rate, and APwd. is a design pressure drop for the flow

components at a design condenser loop flow rate of Mew,&s. Considering only the

pressure losses within the chillers and cooling tower cells, the design condenser loop

pressure drop may be expressed as

2 2AP cwdes--APc md] + AP ImCWtS](3.2.2)

lch,d csJ [mtde 5 ]A~hdSMchdesJ Lmrdes

where mCh,des and APchde are the flow rate and pressure drop through the chillers at

specified design conditions and mr, des and AP t,des are the design flow and pressure

drop for the cooling tower nozzles. The condenser loop pressure drop also depends

upon the number of chillers and cooling tower cells in operation. Cooling tower cells

operate in parallel and the total flow is generally divided equally between the cells. For

identical chillers, the condenser flow would also be divided equally between the

chillers. For parallel flow paths with identical pressure drop characteristics, the design

flow rates utilized in equation (3.2.2) are simply the design values for an individual

device multiplied by the number operating in parallel. The design pressure drop for the

parallel components is equal to the pressure drop for an individual component at its

design flow rate.

The pressure rise developed by a pump may be correlated with a simple parabolic

relationship (White [1986]). In tenns of the maximum pressure developed at zero

.80

flow, AP pmaW and the maximum flow at zero pressure rise, mp,ma , the pump

pressure rise is

21Ic

APp AP pmax 1 - ( j(3.2.3)

Multiple pumps operating in parallel will produce a total flow capacity equal to the

sum of the individual capacities at the same pressure rise. The combined pump

characteristic is obtained by adding together flow capacities at fixed pressure rises. For

identical parallel pumps, the maximum flow rate of equation (3.2.3) is equal to the

product of the individual maximum flow rate for an individual pump and the number of

pumps operating in parallel.

For a variable-speed pump, the pump characteristic changes with the pump speed.

From similarity rules for centrifugal pumps, the maximum pressure rise and mass flow

rate depend upon the relative pump speed according to the following relationships.

2APp,max = 7p APp,maxcdes (3.2.4)

mp,max = Yp mp,max,des (3.2.5)

whereyp is the pump speed relative the maximum design speed and APp,maxdes andmp,made are the maximum pump pressure rise and flow rate associated with the

design speed.

Equating the condenser loop pressure drop to the pump pressure rise and solving

81

for the total flow rate gives the following relation.

S= Y pmaxdcs (3.2.6)

mpPmax,des APp,max,des

For fixed-speed pumps, the relative speed in the above equation is equal to unity.

For a given mass flow rate and pressure rise, the overall pump power requirement

for the condenser water loop is determined as

thCW APCWPP - (3.2.7)rip Pcw

where pcw is the density of the condenser water and 1p is the overall efficiency of the

pump and motor. The overall pump efficiency is primarily a function of the flow rate

relative to the speed of the pump (White [1986])and may be correlated with

0r 2P ao+ [al] + a2 [+ades](3.2.8)

The determination of optimal control points that minimize operating costs is

simplified when the individual component operating costs are expressed as quadratic

functions of the controls and/or outputs. It is possible to accurately correlate the power

consumption of a variable-speed pump as a quadratic function of pump speed over its

82

operating range. Figure 3.12 demonstrates that a quadratic function works extremely

well in correlating pump power consumption. The coefficients of the quadratic

function were determined analytically by matching the power consumption detenined

with equation (3.2.7) at relative speeds of one-half, three-quarters, and full speed.

0

0.5 0.6 0.7 0.8 0.9

Relative Pump Speed

1.0

Figure 3.12. Quadratic Fit to Variable-Speed Pump Power Consumption

3.3 Summary

An effectiveness model has been presented for cooling towers. The model utilizes

existing effectiveness relationships developed for sensible heat exchangers with

modified definitions for number of transfer units and the capacitance rate ratio. A

simple method was also developed for estimating the water loss in cooling towers.

Results of the model compare well with the results of more detailed numerical solutions

to the basic heat and mass transfer equations and with experimental data. The

simplicity and accuracy of the effectiveness model is such that it is useful for both

83

design and system simulations. This modeling approach is also appropriate for other

wet surface heat exchangers and is applied to*cooling coils in Chapter 4.

A model was also presented for determining the condenser pump flow rate and

power consumption based upon a hydraulic analysis. It was shown that the variable-

speed pump power consumption may be correlated as a simple quadratic function of the

relative pump speed. This was also found to be the case for variable-speed cooling

tower fans. The use of quadratic functions for the component operating costs

simplifies the determination of the optimal control points as described in Chapter 5.

84

Chapter 4Evaporator-Side Component Models

In this chapter, models are presented for air handling and chilled water pumping

equipment. An air handler consists of a cooling coil, supply air fan, and valve for

modulating chilled water flow through the coil. In Section 4.1, an effectiveness model

is developed for coiling coils that is similar to the cooling tower model presented in

Chapter 3. The complete air handler analysis is presented in Section 4.2, while the

chilled water pumping power requirements are developed in Section 4.3.

4.1 Development of an Effectiveness Model for Cooling Coils

A cooling coil is similar to a cooling tower in that energy is transferred between air

and water streams due to both sensible and latent effects. The ASHRAE Equipment

Guide [1983] presents a method for the design and analysis of cooling coils in which

the overall heat transfer is evaluated by using log-mean temperature differences between

the coolant and the surface and log-mean enthalpy differences between the surface

conditions and the air stream. The use of log-mean enthalpy differences requires the

assumption that the enthalpy associated with the saturated air at the surface interface is a

linear function of temperature. For a given coil and entering conditions, the solution is

iterative with respect to the leaving condition. For system simulation purposes,

Stoecker [1975] presents an empirical model for cooling coils. This model is not useful

for design and requires the determination of fifteen empirical constants from

performance data.

A simple, yet fundamental model for cooling coils was proposed by Threlkeld

[1970]. By assuming a linear relationship for saturation air enthalpy with respect to

85

both surface and water coolant temperature conditions, Threlkeld obtained an analytic

solution to the heat and mass transfer equations. Threlkeld did not compare his

methodology extensively either with more detailed analyses or experimental results.

Elmahdy [1977] compared results of this model with experimental measurements for

two different coil designs and found good agreement.

Threlkeld presented his solution for cooling coils in terms of log-mean enthalpy

differences between the entering and leaving flow streams. This form is particularly

useful for design purposes in which entering and leaving conditions are specified and

the area requirements are to be computed. For a given coil design and entering

conditions, a more convenient expression for determining the cooling capacity is based

upon the coil effectiveness.

The pr#mary difference between the analysis of cooling coils and cooling towers is

associated with the fact that the air and water streams are not in direct contact. As a

result, there is an additional heat transfer resistance associated with the material

separating the streams and there is no loss of mass associated with the water flow. The

use of multi-pass crossflow geometries also complicates the analysis of cooling coils.

However, as the number of passes increases beyond about four, the performance of a

crossflow heat exchanger approaches that of a counterflow.

In this section, the basic theory of a counterflow cooling coil is presented, leading

to the development of an effectiveness model. A simple method for analyzing the

performance of cooling coils that have both wet and dry sections is also presented. The

resulting effectiveness model is compared with results of finite-difference solutions to

the basic heat and mass transfer equations and manufacturrs' data. In addition,

functional forms for correlating performance data are presented.

86

4.1.1 Detailed Analysis

A schematic representation of a counterflow cooling coil in which air is cooled as a

result of a flow of chilled water is shown in Figure 4.1. Assuming that the local

surface temperaure is less than the dewpoint of the air, then the air is both cooled and

dehumidified.

Air hc all..]

Water Film

&ha +dh 0 a+ do)a

Surface_ ___________

Tw4A

-IV Tw + dTw Water4- dA

Figure 4.1. Schematic of a Counterflow Cooling Coil

Following the assumptions and development of Section 3.1.1, the equations modeling

the local heat and mass transfer from the air to the chilled water for a counterflow coil

without fins may be written

do)a Ntuo(0 )adA - LoAo

dlh NtUo ,---at A (h a - hs ts) + ( co- os1)(I/ eo - l)hg l

dA A0 S- I ~s

(4.1.1)

(4.1.2)

87

dha do)adTw dA Cw(s r)dAdT (4.1.3)

dA faw. pw

ma

=Ntui (Tw- Ts) (4.1.4)

where,

Ao = total cooling coil outside surface area

Cpw = constant pressure specific heat of liquid water

ha = enthalpy of moist air pernmass of dry air

hg,s = enthalpy of water vapor at the coil. surface temperature

hS's = enthalpy of saturated air at the surface temperature

Mna = mass flow rate of dry air

MW = mass flow rate of water

Leo = Lewis number for air streamNtu o = overall air-side number of heat transfer unitsNtui = overall water-side number of heat transfer units

Tref = reference temperature for zero enthalpy of liquid water

TS = coil surface temperature

Tw = water temperature

Oa = ir humidity ratio

COs s = humidity of saturated air at the coil surface temperature

In contrast to the use of mass transfer Nt's for the cooling tower analysis, heat

transfer Ntu's are utilized for the inside and outside flow streams. The Ntu's and

88

Lewis number are defined as

(UA)iNtu-i hC,A w(4.1.5)

0 maC pm(416

Leo0- hDC 0 (4.1.7)

where,

Cpm =constant pressure specific heat of moist air(UA)i = overall heat transfer conductance between water stream and outside

surface

h9o f= convection heat transfer coefficient for air stream

hD0o =f mass transfer coefficient for air stream

Equations (4.1.1) - (4.1.4) may be solved numerically for the air and water exit

states. For given entering conditions, the solution is iterative with respect to the exit

water temperature, Tw~o. Fr a given value of Tw~o, the equations are numerically

integrated from air inlet to outlet. For sections of the coil where the local surface

temperature is greater than the dewpoint of the air, no condensation occurs and

equations (4.1.1) - (4.1.4) apply with cos~s = 0a*a In the following two sections,

effectiveness relations are presented for both completely dry and completely wet coils.

89

4.1.2 Dry Coil Effectiveness

If the coil surface temperature at the air outlet is greater than the dewpoint of the

incoming air, then the coil is completely dry throughout and standard heat exchanger

effectiveness relationships apply. In terms of the air-side heat transfer effectiveness,

the dry coil heat transfer is

Qdry = Edry,araCpm (Tai Twj) (4.1.8)

where, analogous to a dry counterflow heat exchanger, the effectiveness is evaluated as

1 - exp(-Ntudry(1 - C))8 dry,a -- C)),(4.1.9)

1 - Cexp(-NtUdry(1-C

where,

C* rnaCPm (4.1.10)mwCpw

The overall number of transfer units for the dry coil is

Ntu &yNtu oNty = Ntu, (4.1.7)

1 + C (Ntuo/Ntui)

For finned surfaces, the air-side number of transfer units, Ntu0 , defined by equation

(4.1.6) is multiplied by an overall fin efficiency factor. For air flow over finned coil

90

surfaces, general correlations for detemining the overall heat transfer coefficient have

been developed by Elmahdy [1979] for dry heat exchangers.

The exit air humidity ratio is equal to the inlet value, while the exit air and water

temperatures are determined from energy balances on the flow streams as

Ta,o =Ta,i" -Sdrya(Tai" Tw) (4.1.8)

, (4.1.9)Tw,0 =Twi + C (Ta,i - Ta,o)

The coil surface temperature at the air outlet is determined by equating the rate

equation for heat transfer between the water and air streams with that between the water

and the outside surface.

•NtuTS 0 TWi + C N-ry(Ta'o Tw'o) (4.1.10)

If the surface temperature evaluated with the above equation is less than the inlet air

dewpoint, then at least a portion of the coil is wet and the analysis in the following

section must be applied.

4.1.3 Wet Coil Effectiveness

If the coil surface temperature at the air inlet is less than the dewpoint of the

incoming air, then the coil is completely wet and dehumidification occurs throughout.

Introducing the definition for the air saturation specific heat, Cs, from Chapter 3,

assuming that Cs evaluated at the local surface temperature is equal to the value at the

91

local water temperature, assuming a Lewis number of unity, and neglecting the energy

flow associated with condensate draining from the coil surface, equations (4.1.1) -

(4.1.4) may be written

dh a Ntuwet -h~)(..1dV = - A--- (h a h" W

dhs.w maCs(dha/dA)- a(4.1.12)

dwCpw

where the overall number of transfer units for a wet coil is defined as

Ntu NtuoNtuwet Ntu, (4.1.13)1 + m (NtuO/Ntu-)

and

, ma (4.1.14)m=

iiw(CPgC )

C dTT =] (4.1.15)

For finned surfaces, Ntuo as defined by equation (4.1.6)-should be multiplied by an

overall fin efficiency. The air-side transfer units, Ntu, will differ for wet and dry coils

due to different heat transfer coefficients and fin efficiencies for finned coils. There are

no general correlations for heat transfer coefficients for wet coils, however ASHRA

[1985] presents data that show a relatively small difference between wet and dry

92

coefficients. The fin efficiency for a wet coil must account for the effects of both heat

and mass transfer. Threlkeld [1970] gives a method for computing fin efficiencies for

wet coils using the relationships available for dry coils.

For a constant C, equations (4.1.11) and (4.1.12) may be solved analytically. For

a completely wet coil, the heat transfer is

Owet = ewetaa(hai - hs wi) (4.1.16)

where,

1 - exp(-Ntuwet(1 - in))ea, 1 - mexp(-Ntuwet(-m.1.17)

Analogous to the dry analysis, the exit air enthalpy and water temperature are

hao = ha.i - ewet~a(hai - hs,w, ) (4.1.18)

W = Tw.+ a a, o) (4.1.19)

The exit air temperature is determined in a manner analogous to that for determining

the exit humidity ratio for the cooling tower described in Chapter 3 and as described in

the ASHRAE Equipment Guide [1983].

Ta, o = Ts,e + (Ta,i - Ts,e) exp(-Ntuo) (4.1.20)

93

where, the effective surface temperature is determined from its corresponding saturation

enthalpy.

ha,o - ha ihs,s,e = hai + 1 - exp(-N) (4.1.21)

From the rate equations, the surface temperature at the air inlet is computed as

mfa [Ntuwet1Tsj i Tw,o + N -- pa't j (ha,i- 'hs'wo (4.1.22)

mIwC 1 tuiJ

If the surface temperatur evaluated with the above equation is greater than the inlet air

dewpoint, then a portion of the coil beginning at the air inlet is dry, while the remainder

is wet.

4.1.4 Combined Wet and Dry Analysis

Depending upon the entering conditions and flow rates, only part of the coil may be

wet. A detailed analysis would involve determining the point in the coil at which the

surface temperature equals the dewpoint of the entering air. A method for determining

the heat transfer and outlet states in this manner for a coil with dry and wet sections is

given in Appendix B.

A simpler approach is to assume that the coil is either completely wet or dry. Either

assumed condition will tend to underpredict the actual heat transfer. With the

completely dry assumption, the latent heat transfer is neglected and the predicted heat

transfer is low. With the assumption of a completely wet coil, the model predicts that

94

the air is humidified during the portion of the coil in which the dewpoint of the air is

less than the surface temperature. The latent heat transfer to the air associated with this

"artificial" mass transfer reduces the overall calculated cooling capacity as compared

with the actual situation. Since both the completely dry and wet analyses underpredict

the heat transfer, a simple approach is to utilize the results of the analysis that gives the

largest heat transfer. The error associated with this method is generally less than 5

percent.

The steps for determining the heat transfer and oudet conditions for a cooling coil

are summarized as follows:

1) Assume that the coil is completely dry and apply the analysis of Section 4.1.2.

2) If the surface temperature at the air outlet determined with the dry analysis is less

than the dewpoint of the entering air, then assume that the coil is completely wet

and apply the analysis of Section 4.1.3.

3) If the surface temperature at the air inlet determined with the wet analysis is

greater than the entering dewpoint temperature, then a portion of the coil is dry.

At this point, the analysis given in Appendix B could be used to determine the

fractions of the coil that are wet and dry, and the corresponding heat transfers

and outlet states. More simply, choose the result of steps 1) or 2) that yields the

largest heat transfer.

4.1.5 Model Comparisons

Figures 4.2 and 4.3 show typical comparisons between results of the numerical

solutions of equations (4.1.1) - (4.1.4) with a Lewis number of unity and those of the

95

effectiveness model described in the previous section. Air heat transfer and

temperature effectivenesses are plotted versus the air-side transfer units, Ntuo, for

different water-to-air flow rate ratios. The air heat transfer effectiveness is the ratio of

the actual heat transfer to the heat transfer that would occur if the exit air were saturated

at temperature of the inlet water. The air temperature effectiveness is the ratio of the

actual air temperature difference across the coil to the temperature difference between

the inlet air and inlet water.

1.0

0.8 M ama

4C 0.6, mw

0.

. 0.4II

Detailed Analysis0.2 * Effectiveness: Wet and Dry

+ Effectiveness: Wet or Dry

0 .0 •1 - 1 • •

0 1 2 3 4 5Air-Side Transfer Units (Ntuo0)

Figure 4.2. Air-Side Heat Transfer Effectiveness for Detailed and EffectivenessModels (Ambient Dry & Wet Bulb of 95 F & 68 F, Water Inlet of 41 F, NtuO/Ntu i = 2)

As evident in Figures 4.2 and 4.3, the effectiveness model agrees very closely with

the more detailed analysis. The accuracy is better than that for cooling tower analyses

because the saturation enthalpy is more nearly linear in the temperature range associated

with cooling coil operation. Similar agreement was found over a wide range of

96

conditions and coil parameters. For situations in which there were wet and dry

portions of the coil, both the method of Appendix B and the simpler method of

assuming the coil is either all wet or dry were utilized for the effectiveness model

results. The results include conditions in which there are significant wet and dry coil

sections. The differences in both the air heat transfer and temperature effectivenesses

between the two methods are at most 5 percent.

1.0mw 2 Mw

0.8M Ima

MW 0.60.6 "Ma

E NOE 0.4

- Detailed Analysis

0.2 -Effectiveness: Wet and Dry

I+ Effectiveness: Wet or Dry

0.0 • , • , I I ,

0 1 2 3 4 5

Air-Side Transfer Units (Ntu 0)

Figure 4.3. Air Temperature Effectiveness for Detailed and Effectiveness Models

(Ambient Dry and Wet Bulb of 95 F and 68 F, Water Inlet of 41 F, Ntuo/Ntui = 2)

Catalog data for the performance of a cooling coil were utilized in order to further

test the accuracy of the effectiveness model. The physical characteristics of the coil and

flow rates were used in order to determine heat transfer coefficients and fin efficiencies

from general relationships available in the literature. For air flow over finned coil

surfaces, correlations developed by Elmalidy [1979] for dry heat exchangers were

utilized. Fin efficiencies for wet surfaces were determined as outlined in Threlkeld

97

[1970] for straight fms. Turbulent flow was assumed for the water-side heat transfer.

Table 4.1 shows representative comparisons between the model and data for a 6

row coil with 8 fins per inch of tubing for a range of conditions. The agreement

between the model and the manufacturers' data is excellent.

Table 4.1

Effectiveness Model Comparisons with Cooling Coil Catalog Data

VelocityWater (fps) Air (fpm)

4424244668

500700400500600500500600500700

Tw,0 (F)Data Model

47.952.055.349.759.053.853.755.453.753.1

47.851.955.049.758.454.154.055.654.053.7

Ta,i (F)Data Model

48.452.752.450.556.654.355.158.657.260.4

48.953.053.150.856.653.855.158.656.559.3

4.1.6 Correlating Performance Data

The heat and mass transfer characteristics of a cooling coil depend primarily upon

its geometry and the flow rates. In order to simulate the performance of a cooling coil

in a system simulation, it is necessary to estimate the air-side and water-side transfer

units as a function of the flows. General correlations exist for the air-side and water-

side heat transfer coefficients and overall fin efficiencies. However, it is necessary to

know specific details concerning the dimensions and configuration of the tubes and

fins, which is not always readily available. It is possible to correlate the transfer units

Tai(F)

7575758080809090

100100

Twb(F)

60606064647272728080

Twji

(F)

40444640444040464040

98

when both inlet and outlet coil conditions are provided for a range of flows. The

following forms have been found to work well.

ma

Ntui-k, mW (4.1.023)

Ntuo = k3 ,,,(4.1.24)

where k 1, k2, k3, and k4 are empirical constants that may be determined with nonlinear

regression applied to differences between measurements and cooling coil model

predictions of the water and air outlet temperatures. Although, the air-side transfer

units differ for dry and wet coils due to different heat transfer coefficients and fin

efficiencies, a single correlating function appears to work well for both cases.

Coefficients of the transfer unit equations were determined with nonlinear

regression applied to the manufacturers' data presented in Table 4.1. These data cover

a fairly wide range of inlet conditions and flow rates and include conditions for which

the coil is completely wet and almost completely dry. The assumption that the coil was

either all wet or all dry was utilized for the coil model. Figure 4.4 shows a comparison

between the data and model results for the difference between outlet temperatures (both

water and air) and inlet conditions. The agreement between the model results and the

data is excellent.

99

m Water Temperature Difference

40 x Air Temperature Difference

0

S20L10

0

0 10 20 30 40 50

Temperature Differences from Data (F)

Figure 4.4. Correlation of Cooling Coil Model Results with Catalog Data

4.2 Air Handler Analysis

Local loop control of a chilled water valve modulates the flow of chilled water

through the coil in order to maintain a specified supply air temperature to the zones.

For a variable-air-volume (VAV) system, a local loop controller also adjusts the supply

air flow in order to maintain the zone temperature (and possibly humidity) at prescribed

conditions. For the purposes of this study, local loop control of the air handlers is

assumed to be ideal such that specified supply air and zone temperatures (and humidity

in some cases) are maintained exactly. These controls are handled as constraints in the

overall optimization of the system as described in Chapter 5.

There are two possibilities for modulating the air flow in a VAV system that are

considered in this study. The most efficient method involves the use of variable-speed

100

fan motors. The speed of the fan is adjusted to give the required air flow. The power

consumption for a variable-speed air handler fan is assumed to obey the fan laws and is

computed as

r3P u = P ahu, des (4.2.1)

where rna is the supply air flow rate from the air handler and P ahudes is the air

handler fan power at a design flow of mhu,&d Analogous to the cooling tower

analysis, the air handler fan power may also be expressed as a quadratic function of the

relative flow with a second-order Taylor series expansion of equation (4.2.1) or a

correlation over its operating range.

A more common method for modulating supply air flow utilizes variable-pitch fan

blades with fixed-speed motors. In this study, the power consumption is determined

for variable-pitch fan control using a correlation from the BLAST (1981) simulation

program.

Pahu - Pah.,de. 0.517 - 0.784mahu s + 1.26 1'ahud-1 (4.2.2)~aiiu Pahudes\h5 l' s Imahu.sJ)

At part-load conditions, the power consumption is always greater for fixed-speed,

variable-pitch than for variable-speed control.

In order to analyze the performance of the cooling coil, it is necessary to know the

inlet enthalpy and humidity. Return air from the zones serviced by an air handler is

101

mixed with ambient air so that

h a,i = Xambharab + (1 - Xamb)har (4.2.3)

oa, i = X amtO~amb + (1 - Xamb)a,r (4.2.4)

where Xamb is the fraction of the total air handler air flow that is drawn from the

ambient, hnb and cOab are the enthalpy and humidity ratio associated with ambient

conditions, and har and (oar are the enthalpy and humidity ratio of the return air from

the zones. The return air conditions are determined with overall energy and mass

balances.

h' h + gain

ha,r = ha,o + • (4.2.5)Mahu

00gainCja,r = aa,o + . (4.2.6)

liahu

where Qgain and (0 gain are the total rate of energy and humidity gains to the ventilation

stream from zones serviced by the air handler. The energy and moisture gains in a

building are due to many factors that include conduction through walls, solar radiation

through windows, people, lights, and equipment. In order to model these gains in

detail, it is necessary to include the dynamics associated with walls and possibly the

furnishings. Simulation programs such as TRNSYS [1984] contain zone models that

determine energy and moisture gains in a detailed fashion. Many of the results in this

102

work are presented in terms of the gains to the ventilation stream. However, for the

purpose of performing simulations over a cooling season, the energy and moisture

gains are determined with a very simple model. The internal gains (both energy and

moisture) due to people, lights, and equipment are considered to be constants that do

not depend upon time. The overall conduction through walls is computed assuming an

overall conductance for heat gain driven by an ambient sol-air temperature. The overall

energy gains are

Ogain= (UA)z(Tsol-air-" Tz) + Oint (4.2.7)

where (UA) z is an overall conductance, (int is the overall internal gains due to people,

lights, equipment, etc., Tz is the zone temperature, and Tso1.k is an ambient sol-air

temperatur determined as

Tsol'air -=Tamb + ho I1(4.2.8)

where Tamb is the ambient dry bulb temperature, o and ho are the solar absorptivity

and convection coefficient for the outside of the walls, and I is the instantaneous

horizontal radiation.

4.3 Chilled Water Pumping Requirements

The analysis of the pumping requirements for the chilled water loop is similar to

that for the condenser loop presented in Chapter 3. The primary pressure losses in the

chilled water loop are to due to the flows through the chiller and cooling coils.

Commonly, the pressure drop between the chilled water supply to and retur from the

103

air handlers is maintained at a minimum value necessary to ensure adequate flow to all

air handlers in the system. For variable-speed pumping, the pump speed controller

responds to changes in the air handler control valves in order to maintain the specified

pressure drop. In this manner, the chilled water flow requirements of the air handlers

may be met exactly (i.e. no bypass) by the pumps. For fixed-speed pumps, a bypass

valve between the supply and return lines maintains the pressure drop. As the supply

air temperature is increased for a given chilled water supply temperature, the water flow

requirements of the air handlers are reduced and the air handler pressure drops increase.

Under these circumstances, the bypass valve would open to bypass flow and reduce the

pressure drop. However, for best system performance, the supply air temperatures

should always be chosen to minimize the bypass flow. For systems with fixed-speed

chilled water pumps, the bypass flow is considered to be zero in this study.

In the analysis of the chilled water loop, the controlled pressure drop is treated as a

static (constant) pressure loss, so that the overall loop pressure drop is

2

APchw = APahu + APevappdS[. chw ] (4.3.1)[chw,desJ

where APahu is the static pressure drop associated with the air handlers, mchw is the

chilled water flow rate, and AP,.p,des is a design pressure drop for the chiller

evaporator at a chilled water flow rate of mchwdes.

The chilled water loop pressure drop also depends upon the number of chillers in

operation. For identical parallel chillers, the chilled water flow is divided equally

between the chillers. In this case, the design flow rate utilized in equation (4.3.1) is

simply the design value for an individual chiller multiplied by the number operating.

104

The design pressure drop for the parallel chillers is equal to the pressure drop for an

individual component at its design flow rate.

Equating the chilled water loop pressure drop to the pump pressure rise and solving

for the total flow rate gives the following relation.

mchw = mchw,des A m.c(4.3.2)

[mp,max,des p,max,des

where Yp is the relative pump.speed, APp maxdcs is the maximum pump pressure

developed at zero flow, and mp,ma,des the maximum flow at zero pressure rise. For

fixed-speed pumps, the relative speed in the above equation is equal to unity.

For a given mass flow rate and pressure rise, the overall pump power requirement

for the chilled water loop is

minch w AP chwP p =(4.3.3)p ip P chw

where phw is the density of the chilled water and 11p is the overall efficiency of the

pump and motor determined with a correlation of the form given by equation (3.2.8).

4.4 Summary

An effectiveness model analogous to that presented for cooling towers in Chapter 3

has been presented for cooling coils. A simple method was also developed for

estimating the performance of cooling coils having both wet and dry portions. The

105

effectiveness relationships given in this chapter were for counterflow operation. Most

cooling coils use multi-pass crossflow geometries. However, if the number of coil

rows is greater than about four, then the performance of a crossflow coil approaches

that of a counterflow device. The counterflow effectiveness relationship is

recommended and was utilized throughout this study. However, standard effectiveness

relationships for other geometries could also be applied using the definitions for Ntu

and m* given in this chapter.

Models were also presented for the air handlers and chilled water pumps. Again, it

is important to note that quadratic functions work well in correlating the power

consumptions of these auxiliary devices.

106

Chapter 5Methodologies for Optimal Control

of Systems without Storage

The optimal control problem associated with a central chilled water system may be

thought of as having a two-level hierarchical structure. The first level involves local

loop control in response to prescribed setpoints. An example of a first level control

variable would be the compressor speed for a variable-speed chiller. The second level

controls are independent variables that-may be adjusted to minimize the operating costs,

while still satisfying the load requirements. In the previous example, the chilled water

supply temperature is a second level variable that may be adjusted independently.

The dynamics of the first level (local loop) controls must be considered in order to

maintain prescribed setpoints in an efficient manner. However, for systems without

significant thermal storage, the system dynamics may be neglected (Hackner [1985]) in

the determination of the optimal second level control setpoints. In this study, local loop

control is not considered and the first level (local loop) control is considered to be

entirely dependent upon the second level setpoints.

Optimal control of a system involves minimizing the total power consumption of the

chillers, cooling tower fans, condenser water pumps, chilled water pumps, and the air

handling fans at each instant of time with respect to the independent continuous and

discrete control while maintaining the desired zone conditions and ensuring that the

control variables are within acceptable bounds. Discrete control variables are not

continuously adjustable, but have discrete settings, such as the number of operating

chillers, cooling tower cells, condenser water pumps, and chilled water pumps and the

relative speeds for multi-speed fans or pumps. Independent continuous control

107

variables might include the chilled water and supply air set temperatures, relative water

flow rates to the chillers (evaporators and condensers), cooling tower cells, and cooling

coils, and the speeds for variable-speed fans or pumps.

In this chapter, two methodologies are presented for determining optimal values of

the independent control variables that minimize the instantaneous cost of operating

chilled water systems. In section 5.1, a modular component-based optimization

algorithm is presented. The algorithm is implemented in a computer program that

simulates the operation of a system through time, while minimizing the cost at each time

increment in response to the uncontrolled (e.g. weather) variables. Each hardware

component is represented as a separate subroutine in the simulation of a system.

Information concerning the cost of operation of individual components and the manner

in which the components are interconnected are used to perform the optimization in an

efficient manner. Each component may also have constraints associated with its

operation. The modular nature of this program is similar to that of the widely used

TRNSYS [1984] simulation program. However, TRNSYS has no capability for

performing control optimization and could not be utilized for this study.

There are three intended uses for the component-based optimization program in this

study:

1) Analysis of control and design options: The program may be used as a

simulation tool for comparing conventional and optimal control strategies.

Conventional control strategies, such as fixed temperature setpoints, are

implemented through the imposition of constraint equations. Design

comparisons, such as variable versus fixed-speed equipment, may be

performed for systems that are optimally controlled.

108

2) On4-line control optimization: The algorithm may be used for on-line

optimization of the simulation of an operating system using "simple" component

models. The simulation could proceed in parallel with the actual system

operation with the possibility of updating parameters of the component models

using on-fine measurements.

3) Nkar-otimal control algoithms: Results of the optimization program are useful

for developing "simple" near-optimal control algorithms.

In Chapter 6, the component-based algorithm is applied to typical systems to study both

design and control issues. Existing optimization packages proved to be extremely

inefficient for these systems and had difficulties handling the nonlinear equality

constraints that arose.

In section 5.2, a "simple" methodology is presented for near-optimal control. An

overall empirical cost function for the total power consumption of the cooling plant is

inferred from the cost functions associated with the components utilized in chilled water

systems. This cost function lends itself to rapid determination of optimal control

variables and may be fit to measurements using linear regression techniques. In section

5.3, results of the system-based and component based algorithms are compared.

5.1 A Component-Based Optimization Algorithm

Figure 5.1 depicts the general nature of the modular optimization problem. Each

component in a system is represented as a separate set of mathematical relationships

organized into a computer model. Its output variables and operating cost are functions

of parameter, input, output, uncontrolled, and controlled variables. The structure of the

109

complete set of equations to be solved for the entire system is dictated by the manner in

which the components are interconnected together.

fn, Mn, Un

xl

fl, M1 , Ul

Jn

J2

X2 Y2

f 2 , M 2 , U2

Figure 5.1. Schematic of the Modular Optimization Problem

The optimization problem is formally stated as the minimization of the sum of the

operating costs of each component, Ji, with respect to all discrete and continuous

controls or

Minimize

nJ(f,M,u) = XJi(xi, yi, fi, Mi, ui (5.1.1)

i=l

Yl

110

with respect to M and u, subject to equality constraints of the form

g(f, M, u) =

gl(fi, M1 , ui, X1, Yl)1

g 2(fbM2,u , x 2, Y2) = 0

gn(fn, Mn, Un, xn, Yn)

and inequality constraints of the form

h(f, M,u) =

h1(f 1, M1 , ui, xi, Y)1h2(f 2 ,M 2 , u2 x2 ,y 2 ) j 0

h(fn, Mn, u n, X asY )(5.1.3)

where, for any component i,

x= vector of input stream variables

y = vector of output stream variables

f. = vector of uncontrolled variables1

M. = vector of discrete control variables

u . = vector of continuous control variables

i operating cost

g = vector of equality constraints

hi = vector of inequality constraints

Typical input and output stream variables for chilled water systems are temperature

and mass flow rate. The uncontrolled variables are measurable quantities that may not

be controlled, but that affect the component outputs and/or costs, such as ambient dry

(5.1.2)

111

bulb and wet bulb temperature.

Both equality and inequality constraints arise in the optimization of chilled water

systems. One example of an equality constraint that arises when two or more chillers

are in operation is that the sum of their relative loadings must be one. The simplest type

of inequality constraints to handle are bounds on control variables. For example, lower

and upper limits are necessary for the chilled water set temperature, in order to avoid

freezing in the evaporator and to provide adequate dehumidification for the zones. Any

equality constraint may be rewritten in the form of equation (5.1.2) such that when it is

satisfied, the constraint equation is equal to zero. Similarly, inequality constraints may

be expressed as equation (5.1.3), so that the constraint equation is greater than or equal

to zero to avoid violation.

The mathematics associated with the optimization algorithm utilized in this study are

well known. Special advantage is taken of the characteristic that the operating costs

associated with each component in a chilled water system may be modeled with

quadratic functions. A background for the development that follows is presented by

Gill [ 1981]. In the next section, an algorithm is presented for determining optimal

values of the continuous control variables for the special case where all component

costs are quadratic functions and output stream variables are linear functions of the

controls. This algorithm is the basis for the more general nonlinear method presented

in section 5.1.2. The procedure for handling constraints is given in section 5.1.3 and

the complete algorithm including the determination of the optimal discrete controls and

implementation in a computer program is summarized in section 5.1.4.

5.1.1 Quadratic Costs and Linear Outputs

A simple function for which an optimum exists and may be determined analytically

112

is a quadratic function. In Chapter 2, it was shown that the power consumption of a

chiller may be adequately represented as a quadratic function of the load and the

temperature difference between leaving condenser and evaporator water temperatures.

The other energy consuming components in a chilled water system are pumps and fans.

As shown in Chapters 3 and 4, the power requirement of a continuously adjustable

pump or fan (variable-speed or variable-pitch) may be accurately represented with a

quadratic function of its control variable through either a second-order Taylor series

approximation or a single quadratic correlating function. As a result, the cost of

operating any of these components (chillers or auxiliary equipment) may be expressed

in a general form as a quadratic function of its continuous control and/or output stream

variables or

Ji= uTAiui + yiTBiYi + yiTCiui + pITui + %Tyi + ri (5.1.4)

where Ai, Bi , and Ci are coefficient matrices, pi and qi are vectors, and ri is a scalar

constant. The coefficients of this cost function may depend upon the component

operating modes (discrete controls) and uncontrolled variables.

The optimization problem is simplified if the output stream variables for each

component are linear functions of the input stream and continuous control variables or

Yi =i 0iui +- + i -+4i (5.1.5)

where Oi and Oi are coefficient matrices and . is a coefficient vector that may depend

upon the discrete control and uncontrolled variables. The inputs to component i are

outputs from other components, so that the solution for all output stream variables is of

the form

113-

y(f, M, u) = ou + (5.1.6)

where the coefficient matrix 0 and vector depend upon the coefficients for the

individual components and the interconnections between components.

The total cost of operation at any time is the sum of the individual component

operating costs. With individual component costs and outputs represented by equations

(5.1.4) and (5.1.5), the total cost may be reduced to

J(f,M,u) = u Au + b u + c (5.1.7)

where,

A= A + 0 [B+C]

b= p +0 T[2 BT +q]+CT4

= r + 1B + qj c

A2

B1

B =[B2

Anj

C=l C2

Bn C

114

Pl q]

P2 1 q q2

p LI r=r

The first-order condition for a minimizng or maximing point requires that the

Jacobian of the cost function be equal to zero. The Jacobian is a vector containing the

partial derivatives of the cost function with respect to each of the control variables. For

the cost function of equation (5.1.7)

r ~ ~ ~ o . .,T

=u(A + A) + b =0 (5.1.8)

where u* represents the optimal control vector. Solving for u* gives

u -[A + A b (5.1.9)

In general, the cost functions that arise with chilled water systems are globally

convex, so that a single global minimum exists. However, it is relatively simple to

determine convexity of a-quadratic equation. In order to determine whether u*

represents a minimum, maximum, or saddle point, it necessary to evaluate second

derivatives. The second-order condition for a minimum requires that the Hessian of

the cost function be positive semi-definite. The Hessian is a matrix containing partial

derivatives of the transpose of the Jacobian with respect to all control variables. For the

cost function of equation (5.1.7), the Hessian is simply

115

Ou u =(A + A) (5.1.10)

au DU

Positive semi-defmiteness of the above Hessian implies that

T Tu(A+A)u 0 (5.1.11)

for all u not equal to zero. Simple methods exist for determining the positive semi-

defimiteness of symmetric Hessian matrices such as that given by equation (5.1.10).

5.1.2 Nonlinear Optimization

Some component outputs depend non-linearly upon controls or input variables and

some component costs are only quadratic locally, so that an iterative technique is

required to determine the optimal control values. At each iteration, an overall quadratic

for the system expressed as equation (5.1.7) is forfed from individual quadratic

relationships for each component (eq. (5.1.4)) and a linearization of the output

variables (eq. (5.1.6)).

All output variables are linearized with respect to the continuous controls using a

first-order Taylor series expansion about the last iteration point, U0.

116

y(u) =y(u0) + r](u - Uo) (5.1.11)

This equation may be written in the form of equation (5.1.6) with

0 Y(Uo) - [ uO -- (5.1.12)

The Jacobian of the outputs with respect to control variables is determined numerically

using forward differences.

A simple estimate of the minimum point may be determined from equation (5.1.9).

However, for points that are "far away" from the minimum, this may give a point that

has a greater cost than the last iteration. A common procedure, that is utilized in this

study, is to perform a one-dimensional search between the previous iteration and the

point defined by equation (5.1.9). At the ith iteration, a new estimate of the optimal

control point is

- ui '1 + s(u' - u1) (5.1.14)

where u' is from equation (5.1.9) and the step length, s, is determined by minimizing

J(u i ' l + s(u' - ui'l)) with respect to s.

The optimal step length is approximated with polynomial interpolation. The costs at

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step lengths of zero, one-half, and one are used to construct a quadratic function for the

cost as a function of the step length. The optimal value of s is estimated as the

minimum of this quadratic function, constrained between zero and one. In some cases,

the polynomial may be a poor approximation to the real function and the estimated

optimal step length may result in a cost greater than that associated with a step length of

zero. Under this circumstance, the interpolation is repeated with the last computed

optimal step length becoming the new step length of unity.

It is necessary to iteratively solve for the outputs of each component at each iteration

of the optimization procedure with the most recent controls. A simple method that is

employed in the TRNSYS [1984] program is successive substitution. Outputs are

successively fed as inputs to connecting components until the values are not changing

significantly. However, this method can be extremely inefficient for solving systems

of equations, even if they are linear. The solution efficiency is important because the

equations must be solved at each iteration of the optimization procedure and a high

degree of accuracy is required for determining good numerical derivatives.

A better approach is to utilize the Newton-Raphson method applied to a set of

equations that measure the residual error for the independent variables. These residual

equations could be defined as differences between input values and output values that

feed those inputs for one component in each recyclic loop of components. At each

iteration of the Newton-Raphson procedure, new estimates of the independent variables

are determined by assuming that the residual equations are globally linear using

coefficients determined with a local linearization. As a result, this method converges in.

one iteration for linear equations. However, it is necessary at each iteration to solve a

system of linear equations involving a Jacobian of residuals with respect to the

independent iteration variables. The Jacobian contains the partial derivatives of each of

the residual equations with respect to the independent variables and must be determined

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numerically.

In the program developed in this study, an alternative method is employed that is a

compromise between the methods of successive substitution and Newton-Raphson. As

with Newton-Raphson, a set of residual equations are defined such that at the solution,

they are identically zero. However, these equations are solved using a series of one-

dimensional applications of the secant method. Consider a series of components

organized in a recyclic loop. Denoting the input to one component in the loop as x and

the output that feeds that input as y(x), then a residual function at the ith iteration may

be defined as

Gi = y(x i) = xi (5.1.15)

A new estimate of x is determined by approximating the function with a straight line

connected between the last two function values or

i+1 i Gx x - a i-i (5.1.16)

(G-G )/(x =x)

Initial values of x, y, and G are determined through successive substitution.

The advantage of this method is that it is not necessary to compute a Jacobian, so

that the computation associated with updating the independent variables is much less

than that for Newton-Raphson. However, since the residual equations are not coupled,

convergence is slower than for Newton-Raphson. For the chilled water systems

considered in this study, the coupling between recyclic loops is relatively small and the

solution algorithm works well.

119

5.1.3 Constraints

Linear Eaualitv Constraints

For constraints that are linear with respect to the control variables, the constraint

equations may be written in the form

g(u) = a + Ou=0 (5.1.17)

where g is a vector of constraint equations, 3 is a coefficient matrix, and a is a

coefficient vector.

A common method for solving optimization problems with linear constraints is the

method of Lagrange multipliers. This method involves redefining the cost function so

that at the minimum, the constraint function is automatically satisfied. The modified

cost function, termed the Lagrangian, is given as

T

J(u,,X) =-J(u) + X g(u) (5.1.18)

where X is a vector of Lagrange multipliers. The modified optimization problem

involves minimizing the Lagrangian with respect to both u and X. The first-order

conditions for a minimum require that

[ [ + T2!]= (5.1.19)

120

- -g(u) 0(5.1.20)

For the quadratic cost function and linear constraints of equations (5.1.7) and (5.1.17),

these conditions yield

u= [A]+ A ] (5.1.21)

A A) 0 (A +A TII T . (5.1.22)

Nonlinear Equality Constraints

Nonlinear constraints are handled through linearization and the use of Lagrange

multipliers. A fist-order Taylor series expansion for the constraints about the last

iteration point, u0 , gives

[gg(u)= g(uo) + -(U-Uo) (5.1.23)

The above equation may be written in the form of equation (5.1.17) with

a g(u9) - .1uuo (5.1.24)

13= )- f]u. (5.1.25)

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The Jacobian of the constraints with respect to control variables is determined

numerically using forward differences coincidently with the computation of the

Jacobian for the output stream variables with respect to controls.

With linearization applied to nonlinear constraints, the first-order condition applied

to the Lagrangian cost function does not guarantee that the constraints will be satisfied

at any point except the solution. More importantly, during the determination of the

optimal step length, the Lagrangian does not provide a good measure of the degree to

which the constraints are violated. To alleviate this problem, the optimal step length is

computed using a cost function that is the sum of the original cost function and a

quadratic penalty function.

TJ(u) = J(u) + g(u) g(u) (5.1.26)

At the ith iteration, a new estimate of the optimal control point is found with equation

(5.1.14), with u' determined from equations (5.1.21) and (5.1.22) and the step length,

s, determined by minimizing J(u i1 + s(u' - u with respect to s.

Inequality Constraints

The only inequality constraints considered in this study are simple bounds on the

control variables. These become linear equality constraints when violated and are

handled with Lagrange multipliers. In order to determ e the optimal control points

subject to inequality constraints, the optimal control values are first determined at each

iteration assuming that no inequality constraints are violated. If some control bounds

are exceeded, then linear equality constraints representing these limits are added and the

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optimal controls are recomputed. Additional constraints are added if violated and the

process is repeated until the number of constraints is not changing or equals the number

of control variables. The constrained control values represent the next iterates in the

overall nonlinear optimization. It is not possible to solve a problem in which the

number of constraints exceeds the number of control variables. In this situation, some

of the constraints are not satisfied.

5.1.4 Algorithm Summary and Program Implementation

The steps associated with the constrained optimization of the continuous controls

are summarized below.

1) Solve for component outputs with current controls.

2) Linearize outputs with respect to controls to get equation (5.1.6).

3) Determine the coefficients of the quadratic equation (5.1.7) from the component

quadratic equations (5.1.4) and the linearized output equation (5.1.6).

4) Determine the control point associated with a step length of unity with equations

(5.1.21) and (5.1.22).

5) Estimate optimal step length with polynomial interpolation applied to

minimizing J(u i 1 + s(u' - U') with respect to s, where the augmented cost

function is defined by equation (5.1.26).

6) Determine next estimate of control point, ui, with equation (5.1.14).

7) If J(u5> J(u'-) then set u' -ui and go to step 5.

8) If no controls exceed their bounds or the number of constraints equals the

number of controls then go to step 11.

9) Add eqtuality constraints for any controls that exceed bounds unless the number

of constraints equals the number controls.

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10) Determine a constrained optimum with equations (5.1.21) and (5.1.22) and go

to step 8.

11) If the change in cost from the last iteration is greater than a specified tolerance,

then go to step 1.

The complete optimization algorithm is implemented in'a computer program that can

simulate the optimal operation of a system through time (Braun [1988]). The system is

described through input data that specifies the components, their parameters, and their

interconnections in a manner similar to that for the TRNSYS [1984] simulation

program. At each simulation timestep, data for the uncontrolled variables (e.g.,

weather, schedules) are read and the constrained nonlinear optimization of the

continuous control variables is performed for each feasible combination of discrete

controls with the combination giving the minimum being the optimal control. Not all

possible combinations of discrete controls are feasible. For instance, the operation of

more than one condenser or chilled water pump might be non-optimal under all

conditions when only one chiller is on, so that these combinations would not be worth

considering. In the implementation of the optimization algorithm for this study, the

feasible combinations of discrete modes are specified as input data. In the event that the

optimization algorithm were implemented for on-line optimal control, then a better

approach for determining the optimal discrete control modes at each time interval would

be to order the feasible combinations of modes and only allow a single change (up or

down) between combinations within the list.The complete mathematical description of a specific optimization problem depends

upon the choice of the independent controls and the constraints imposed upon the

system. Generally the supply air temperature for each air handler is considered to be an

124

independent (adjustable) control variable. In this case, the local loop controller varies

the air and water flow through the cooling coil in order to maintain that setpoint, along

with the desired temperature in the zone. However, in terms of the optimization

algorithm, it is more straightforward to consider the relative air and water flow rates to

the cooling coil as control variables, with an equality constraint that forces the zone

temperature to be maintained. One advantage of this formulation is that the power

consumption of the air handler may be represented as a quadratic function of the speed

in a simple manner. It is also easier to handle bounds on the fan speed as compared

with the supply air setpoint, since the fan speed has a natural upper and lower bound,

while physical constraints on the setpoint vary according to the coil entering air and

water conditions and design.

5.2 A System-Based Algorithm for Near-Optimal Control

The methodology for determining optimal values of control variables described in

the previous section could be used for on-line optimal control of chilled water systems.

However, in order to calibrate the models for a specific plant, measurements would be

required for outputs and power consumptions for each component in the system. Also,

depending upon the number of control variables, the computational requirements may

be restrictive. An alternative approach for near-optimal control described in this section

involves correlating the overall system power consumption with a single function that

allows for a rapid determination of optimal control variables and requires measurements

of only total power over a range of conditions.

5.2.1 System Cost Function

The concept of utilizing quadratic functions for the power consumptions of

individual components can be extended to the system as a whole. In the vicinity of any

125

optimal control point, the system power consumption may be approximated with a

quadratic function of the continuous control variables according to equation (5.1.7).

However, the quadratic relationship changes with changes in the operating modes (i.e.

discrete controls) and uncontrolled variables. It will be shown that a quadratic function

also correlates power consumption in terms of the uncontrolled variables over a wide

range of conditions, so that the following cost function may be applied for determining

optimal control points.

T T T f oT Tn

J(f,M,u) =uAu +b +fT+f +df + fEu + g (5.2.1)

where A, C, and, E are coefficient matrices, b and d are coefficient vectors, and g is a

scalar. The empirical coefficients of the above cost function depend upon the operating

modes, so that it is necessary to determine these constants for feasible combinations of

discrete control modes. Once again, many mode combinations may be unfeasible or

clearly non-optimal under all conditions and therefore need not be considered. Some

advantage may also be taken of the symmetry in the quadratic terms of equation

(5.2.1). Both A and C may be expressed as symmetric matrices, so that only the upper

(or lower) triangular coefficients need be detenined.

5.2.2 Near-Optimal Control Algorithm

One advantage of the cost function of equation (5.2.1) is that a solution for the

optimal control vector that minimizes the cost may be determined analytically by

applying the first-order condition for a minimum. Equating the Jacobian of equation

(5.2.1) with respect to the control vector to zero and solving for the optimal control,

(with symmetric A), gives

126

u =k+Kf

where,

1 -1%k =--A b

K=--A E

The cost associated with the unconstrained control of equation (5.2.2) is

T Tf f [+ f + '

where,

E=KTK+EK+C

a -- 2KAk + Kb + Ek + d

kTI:-kT~k + b k + g

(5.2.2)

(5.2.3)

The control defined by equation (5.2.2) results in a minimum power consumption if

and only if the Hessian of the cost function is a positive-definite matrix or in the case of

equation (5.2.1), if A is positive definite. If this condition holds and if the system

power consumption correlates with equation (5.2.1), then equation (5.2.2) dictates that

127

the optimal continuous control variables vary as a near linear function of the

uncontrolled variables. However, a different linear relationship applies to each feasible

combination of discrete control modes. In the implementation of the algorithm, the

minimum cost associated with each mode combination is computed from equation

(5.2.3). The costs for each combination are compared in order to identify the

minimum. Simple bounds on the continuous control variables may be handled as

outlined in section 5.1.3.

5.2.3 Parameter Estimation

The total number of empirical coefficients in equation (5.2.1) that need to be

determined for each feasible set of modes is

2 Nu(N u - 1) 2 Nf(Nf- 1)Ncoef= N2- 2 + NU + Nf + 2 + Nf + NfNU+ 1 (5.2.4)

where Nu is the number of continuous control variables and Nf is the number

uncontrolled variables.

One approach for determining these constants would be to apply regression

techniques directly to measured total power consumption. Since the cost function is

linear with respect to the empirical coefficients, linear regression techniques may be

utilized. A set of experiments could be performed on the system over the expected

range of operating conditions. In some cases, the quadratic cost function may only be

an accurate index near the optimal control points, so that it would be necessary to repeat

the experiments in the vicinity of the control defined by equation (5.2.2). Possibly, the

regression could be performd on-line using least-squares recursive parameter updating

(Ljung [1983]).

128

Another approach for estimating coefficients of the empirical system model would

involve regression to results of a simulation of the system. By using mechanistic

models for the individual components, data over a limited range of conditions would be

sufficient to calibrate the coefficients of the models. The use of the component-based

optimization algorithm as a simulation tool would ensure a good fit near the optimal

control points.

Rather than fitting empirical coefficients of the system cost function of equation

(5.2.1), the coefficients of the optimal control equation (5.2.2) and the minimum cost

function of equation (5.2.3) could be determined directly with regression applied to

optimal control results. At a given set of conditions, optimal values of the continuous

control variables could be estimated through trial-and-error variations in the system or

with the component-based optimization algorithm. Only (Nf + 1) independent

conditions would be necessary to determine coefficients of the linear control law given

by equation (5.2.2). The coefficients of minimum cost function could then be

determined from system measurements with the linear control law in effect. The total

number of coefficients to determine with this approach is less than that for direct

regression to power measurements.

Nof = Nu(Nf+ 1) + N2 Nf(Nf- 1)

Nf 2 + Nf + 1 (5.2.5)

The disadvantage of this approach is that there is no direct way to handle constraints on

the controls.

5.2.4 Application to Chilled-Water Systems

In order to apply the system-based optimization technique, it is necessary to identify

129

both the control variables for which the optimization is to be performed and the

uncontrolled variables that effect the system performance through time. Using the

component-based optimization algorithm described in this chapter, it is shown in

Chapter 6 that the important uncontrolled variables are the total chilled water load and

ambient wet bulb temperature. Additional secondary uncontrolled variables that could

be important if varied over a wide range would be the individual zone latent to sensible

load ratios and the ratios of individual sensible zone loads to the total sensible loads for

all zones.

Chapter 6 also identifies control simplifications that reduce the number of

independent control variables and simplify the optimization. These simplifications and

their implications are summarized as follows.

1) VaOpaber-Saeed Tower Fans: Opete al1 tower cells at identical fan speeds.

The only tower control variable is fan-speed which is equivalent to air flow

relative to the maximum possible flow.

2) Multi-Speed Tower Fans: Increment lowest tower fans first when adding tower

capacity. Reverse for removing capacity. With this sequencing, a single

independent tower control variable is the relative tower air flow.

3) Variable-Speed Pumps: The sequencing of variable-speed pumps should be

directly coupled to the sequencing of chillers, to give peak pump efficiencies for

each possible combination of operating chillers. Multiple variable-speed pumps

should be controlled to operate at equal fractions of their maximum speed. With

this sequencing arrangement, a single independent control variable for the

condenser pump is the flow relative to the maximum possible flow.

130

4) Chillers: Multiple chillers should have identical chilled water set temperatures

and the evaporator and condenser water flows for multiple chillers should be

divided according to the chillers relative cooling capacities. The independent

chiller control variables are a single chilled water temperature and the number of

chillers operating.

5) Air Handlers: All parallel air handlers should have identical supply air setpoint

temperatures. As a result, only a single setpoint control variable applies to all

air handlers.

Using these general results, a reduced set of independent control variables are: 1)

supply air set temperature, 2) chilled water set temperature, 3) relative tower air flow,

4) relative condenser water flow, and 5) the number of operating chillers.

The supply air and chilled water setpoints are continuously adjustable control

variables. However, since the chilled water flow requirements are dependent upon

these controls, there may be discrete changes in power consumption associated with

varying these controls, ff there are discrete control changes in the pump operation. For

the same total flow rate, the overall pumping efficiency changes with the number of

operating pumps. However, this has a relatively small effect upon the overall power

consumption, so that the discontinuity may be neglected in fitting the overall cost

function to changes in the control variables.

For variable-speed cooling tower fans and condenser water pumps, the relative

tower air and condenser water flows are continuous control variables. Analogous to

the chilled water flow, the overall condenser pumping efficiency changes with the

number of operating pumps, so that there may be a discontinuity in the power

131

consumption associated with continuous changes in the overall relative condenser water

flow. This discontinuity may also be neglected in fitting the overall cost function to

changes in this control variable.

With variable-speed pumps and fans, the only significant discrete control variable is

the number of chillers operating. A chiller mode defines which of the available chillers

are to be on-line. The optimization problem involves determining optimal values of

only four continuous control variables for each of the feasible chiller modes. The

chiller mode giving the minimum overall power consumption represents the optimum.

In order for a chiller mode to be feasible, it must be possible to operate the specified

chillers safely within their capacity and surge limits. In practice, abrupt changes in the

chiller modes should also be avoided. Large chillers should not be cycled on or off,

except when the savings in associated costs with the change is significant.

For fixed-speed cooling tower fans and condenser water pumps, there are only

discrete possibilities for the relative flows. One method of handling these variables is

to consider each of the discrete combinations as separate modes. However, for

multiple cooling tower cells with multiple fan speeds, the number of possible

combinations may be large. A simpler approach, that works well in this case, is to treat

the relative flows as continuous control variables and to select the discrete relative flow

that is closest to that determined with the continuous optimization. At least three

relative flows (discrete flow modes) are necessary for each chiller mode in order to fit

the quadratic cost function. The number of possible sequencing modes for fixed-speed

pumps is generally much more limited than that for cooling tower fans, with at most

two or three possibilities for each chiller mode. In fact, with many current designs,

individual pumps are physically coupled with chillers and it is impossible to operate

greater or fewer pumps than the number of chillers operating. Thus, it is generally best

132

to treat the control of fixed-speed condenser water pumps with a set of discrete control

possibilities, rather than using a continuous control approximation.

The methodology for near-optimal control of a chilled water system may be

summarized as follows.

1) Change the chiller operating mode if at the limits of chiller operation (surge or

capacity).

2) For the current set of conditions (load and wet bulb), estimate the feasible

modes of operation that would avoid limits on the chiller and condenser pump

operation.

3) For the current operating mode, determine optimal values of the continuous

controls with equation (5.2.2).

4) Determine a constrained optimum, if controls exceed their bounds.

5) Repeat steps 3 and 4 for each feasible operating mode.

6) Change the operating mode if the optimal cost associated with the new mode is

significantly less than that associated with the current mode.

7) Change the values of the continuous control variables. When treating multiple-

speed fan control with a continuous variable, use the discrete control closest to

the optimal continuous value.

If the linear optimal control equation (5.2.2) is directly fit to optimal control results,

then there is no direct way of handling the constraints. A simple solution is to constrain

the individual control variables as necessary and neglect the effects of the constraints on

the optimal values of the other controls and the minimum cost function. The variables

of primary concern with regard to constraints are the chilled water and supply air set

temperatures. These controls must be bounded for proper comfort and safe operation

133

of the equipment. On the other hand, the cooling tower fans and condenser water

pumps should be sized so that the system performs efficiently at design loads and

constraints on control of this equipment should only occur under extreme conditions.

There is a strong coupling between optimal values of the chilled water and supply

air temperature, so that decoupling these variables in evaluating constraints is generally

not justified. However, when either control is operating at a bound, optimization

results indicate that the optimal value of the other "free" control is approximately

bounded at a value that depends only upon the ambient wet bulb temperature. The

optimal value of this "free" control (either chilled water or supply air setpoint) may be

estimated at the load at which the other control reaches its limit. Coupling between

optimal values of the chilled water and condenser water loop controls is not as strong,

so that interactions between constraints on these variables may be neglected.

5.3 Comparisons

Braun [1987] correlated the power consumption of the Dallas/Fort Worth airport

chiller, condenser pumps, and cooling tower fans with the quadratic cost function given

by equation (5.2.1) and showed good agreement. Since the chilled water loop control

was not considered, the chilled water setpoint was treated as a known uncontrolled

variable. The discrete control variables associated with the four tower cells with two-

speed fans and the three condenser pumps were treated as continuous control variables.

The optimal control determined by the near optimal equation (5.2.2) also agreed well

with that determined using a non-linear optimization applied to a detailed simulation of

the system.

In order to evaluate results of the system-based methodology for a complete system

that includes the air handlers, the component-based optimization was applied to an

example system, as described in Chapter 6. Coefficients of the optimal control and

134

minimum system cost function were fit to results of the component-based optimization

over a range of conditions. Figures 5.2 - 5.5 show comparisons between the controls

as determined with the component-based and system-based methods for a range of

loads, for a relatively low and high ambient wet bulb temperature (60 F and 80 F).

In Figures 5.2 and 5.3, optimal values of the chilled water and supply air

temperatures are compared. The near-optimal control equation provides a good fit to

the optimization results for all conditions considered. The chilled water temperature

was constrained between 38 F and 55 F, while the supply air setpoint was allowed to

float freely. Figures 5.2 and 5.3 show that for the conditions where the chilled water

temperature is constrained, the optimal supply air temperature is also nearly bounded at

a value that depends upon the ambient wet bulb.

60

55High Wet Bulb (80 F)

E- 50

5

Low Wet Bulb (60 F)

40 Optimal- Near-Optimal

35, - , - , • , • *

0.0 0.2 0.4 0.6 0.8 1.0

Relative Chilled Water Load

Figure 5.2. Comparisons of Optimal Chilled Water Temperature

135

65

Low WeWetBBulb((80FF

'~60cc

wo" 55

Low Wet Bulb (60 F)

50

- Near-Optimal45 , ,

0.0 0.2 0.4 0.6 0.8 1.0


Figure 5.3. Comparisons of Optimal Supply Air Temperature

Optimal relative cooling tower air and condenser water flow rates are compared in

Figures 5.4 and 5.5. Although the optimal controls are not exactly linear functions of

the load, the linear control equation provides an adequate fit. The differences in these

controls result in insignificant differences in overall system power consumption, since

as shown in Chapter 6, the optimum is extremely flat with respect to these variables.

The nonlinearity of the condenser loop controls is partly due to the constraints imposed

upon the chilled water set temperature. However, this effect is not very significant.

136

Figures 5.4 and 5.5 also suggest that the optimal condenser loop control is not very

sensitive to the ambient wet bulb temperature. However, the cooling tower considered

is representative of that at D/FW airport and operates at high effectiveness. For a lower

effectiveness tower, the sensitivity of the control to ambient wet bulb is greater.

* Optimal

- Near-Optimal

High Wet Bulb (80

0.2 0.4 0.6


I v 1.

0.8 1.0

Figure 5.4. Comparisons of Optimal Tower Control

0.8-

0.7-

0.6

0.5-

0.4-

OWN

cc

0

B

(60F)

0.310.1

w 5 w 0 w 0 v

0

137

1.0U Optimal- Near-Optimal

.2 U

7. 0.8High Wet Bulb (80 F)

- 0.6.

Low Wet Bulb (60 F)

1 0.4

0 .2- - , -

0.0 0.2 0.4 0.6 0.8 1.0


Figure 5.5. Comparisons of Optimal Condenser Pump Control

In order to determine the optimal discrete mode of operation, it is necessary to have

a reasonably accurate model of the minimum cost of operation for each mode. Figure

5.6 shows a comparison between the optimal system coefficient of performance (COP)

determined with the component-based optimization algorithm and the near-optimal

quadratic cost function of equation (5.2.3). The differences between the results are

very small over a wide range of chilled water loads and ambient wet bulbs. This model

works well, even though it does not explicitly consider the constraints on the chilled

water temperature that are exhibited at low and high loads in Figure 5.2.

138

8

*Optimal- Near-Optimal

7

Low Wet Bulb (60 F)nowE 6

High Wet Bulb (80 F)3

0.0 0.2 0.4 0.6 0.8 1.0


Figure 5.6. Comparisons of Optimal System Performance

5.4. Summary

Two methodologies have been presented for determining optimal control points of

chilled water systems. A component-based non-linear optimization algorithm was

developed as a simulation tool for investigating optimal system performance. Results

of this algorithm implemented in a computer program, led to the development of a

simpler system-based methodology for near-optimal control.

The advantage of the component-based algorithm over the system-based approach

olution to the optimization problem, including any nonlinear

constraints. Each of the components in the system is represented as a separate

subroutine with its own parameters, controls, inputs, and outputs. Models of

139

components may be either mechanistic or empirical in nature, so that the methodology

is useful for evaluating both system design or control characteristics. In Chapter 6, this

methodology is applied to typical chilled water systems to study both design and

control issues.

The component-based algorithm takes advantage of the quadratic cost behavior of

the components found in chilled water systems in order to solve the optimization

problem in an efficient manner. However, in order to utilize this methodology as a tool

for on-line optimization, it is necessary to have detailed performance data for each of

the individual system components. The system-based near-optimal control

methodology presented in this chapter utilizes an overall system cost function in terms

of the total chilled water load and ambient wet bulb temperature. This cost function

leads to a set of linear control laws for the continuous control variables in terms of these

uncontrolled variables. Separate control laws are required for each feasible

combination of discrete controls and the costs associated with each combination are

compared to identify the optimum. The overall procedure is simple enough so as to be

implementable either manually or on-line using microcomputers. For manual control

applications, charts such as those that appear in Figures 5.2 - 5.6 could be used to

determine optimal control as a function of load and wet bulb.

140

Chapter 6Applications to Systems without Storage

In this chapter, the modeling and optimization techniques described in previous

chapters are applied to chilled water systems without storage. The results that are

presented arise from both specified controls applied to individual component and

subsystem modeling and optimal control results determined with the component-based

algorithm described in Chapter 5.

6.1 System Description

Figure 1.1 showed a general schematic of the variable air volume (VAV) system

considered in this study. The central cooling facility, which consists of multiple

centrifugal chillers, cooling towers, and pumps, provides chilled water to a number of

air handling units in order to cool air that is supplied to building zones. General

descriptions of the components and modeling assumptions were given in Chapters 2, 3,

and 4. All energy consuming components in the system are assumed to be electrically

driven.

At any given time, it is possible to meet the cooling needs with any number of

different modes of operation and setpoints. Optimal control of a system involves

minimizing the total power consumption of the chillers, cooling tower fans, condenser

water pumps, chilled water pumps, and the air handling fans at each instant of time

with respect to the independent continuous and discrete control variables. Chapter 5

describes a method for determining optimal values of these control variables. The

discrete control variables considered in this chapter include the number of operating

chillers, cooling tower cells, condenser water pumps, and chilled water pumps and the

141

relative speeds for multi-speed fans or pumps. The independent continuous control

variables considered include the chilled water and supply air set temperatures, relative

water flow rates to multiple chillers (both evaporators and condensers) and multiple

cooling tower cells, and the speeds for variable-speed cooling tower fans and chilled

and condenser water pumps.

In addition to the independent optimization control variables, there are also local

loop (dependent) controls associated with the chillers, air handlers, and chilled water

pumps. All local loop controls are assumed to be ideal, such that their dynamics are not

considered. Each chiller is considered to be controlled such that the specified chilled

water set temperature is maintained. The air handler local loop control involves control

of both the coil water flow and fan air flow in order to maintain the prescribed supply

air setpoint and zonetemperature. The total requirement for the chilled water flow to

the air handlers is dictated by the chilled water and supply air setpoints and the load.

Control of the chilled water pumps is implemented through a local loop control that

maintains a constant pressure difference between the main supply and return pipes for

the air handlers. The setpoint for this pressure difference is chosen to ensure adequate

distribution of flow to all air handlers and is not considered an optimization variable.

The optimal control variables change through time in response to uncontrolled

variables. The uncontrolled variables are measurable quantities that may not be

controlled, but that affect the component outputs and/or costs, such as the load and

ambient dry and wet bulb temperature.

The results presented in this study are primarily representative of the Dallas/Fort

Worth (DIFW) airport cooling system. However, characteristics of the systems studied

by Lau [1985] and Hackner [1985] are also utilized. Table 6.1 summarizes the

different component characteristics considered in this study. The loads employed

throughout are representative of the D/FW system, so that components studied by Lau

142

and Hackner are scaled for D/FW conditions. Appendix C gives the parameter values

associated with the individual component models. The ventilation air flow is taken to

be 10% of the design air flow for the air handler under all circumstances. The use of an

economizer or "free" cooling cycles were not considered in this study. These modes of

operation would occur at low ambient wet bulb temperatures, such that the cooling

loads could be met without operating the chillers.

The base system makes use of the component characteristics associated with the

first choice for each component type in Table 6.1. For the most part, this corresponds

to the current D/FW system. However, detailed data were not available for the

performance of the air handlers. As a result, the air handler performance characteristics

from manufacturers data used by Hackner [1985] were utilized in the base system.

Except where otherwise noted, all pumps and fans are considered to be operated with

variable-speed motors. Although combinations of components from Table 6.1 other

than that associated with the base system were considered, results are only presented

for the base system, except when alternative systems yield different conclusions.

Many of the results presented in this chapter are for steady-state conditions and are

given as a function of load at ambient dry and wet bulb temperatures of 80 F and 70 F.

For the purpose of performing simulations over a cooling season, optimal costs of

operation are correlated in terms of the load and ambient conditions using the form

given by equation (5.2.3). The result is integrated over time in response to time-

varying load and weather conditions. In those cases where cooling season results are

presented, weather data for May through October in Dallas, Texas and Miami Florida

were utilized. Two different constant internal gains to the zones were considered: one-

third and one-half of the maximum chiller cooling capacity. Only conditions where the

ambient wet bulb temperature was greater than 60 F were considered in the

143

determination of plant operating costs.

Table 6.1Summary of System Component Characteristics

Component Component Characteristics

Chillers 1) D/FW variable-speed chiller2) D/FW fixed-speed chiller3) Lau (1985) chiller scaled to D/FW loads4) Hackner (1985) chiller scaled to D/FW loads

Cooling Towers 1) D/FW tower characeristics2) Lau (1985) tower scaled to D/FW loads3) Hackner (1985) tower scaled to D/FW loads

Pumping 1) D/FW system and pump characteristics2) 20% greater system and pumping head at design

conditions than 1)3) 20% lower system and pumping head at design

conditions than 1)

Air Handlers 1) Hackner (1985) scaled to D/FW loads2) 20% greater fan power requirements at design

conditions than 1)3) 20% lower fan power requirements at design

conditions than 1)

Loads 1) 20% of zone loads are latent2) 15% of zone loads are latent3) 25% of zone loads are latent

6.2 Control Guidelines for Multiple Components in Parallel

For some of the independent control variables, it is possible to determine control

guidelines that when implemented yield near-optimal performance. These guidelines

simplify the optimization process, in that these independent control variables may be

reduced to dependent variables. They are also useful to plant engineers as "rules of

144

thumb" for improved control practices. In this section, both component modeling and

optimization techniques are used to identify control guidelines associated with multiple

components arranged in parallel.

6.2.1 Multiple Cooling Tower Cells

The power consumption of a chiller is sensitive to the condensing water

temperature, which is, in turn, affected by both the condenser water and cooling tower

air flow rates. Increasing either of these flows reduces the chiller power requirement,

but at the expense of an increase in the pump or fan power consumption.

Braun [1987] and Nugent [1988] have shown that for variable-speed fans, the

minimum power consumption results from operating all cooling tower cells under all

conditions. The power consumption of the fans depends upon the cube of the fan

speed. Thus, for the same total air flow, operating more cells in parallel allows for

lower individual fan speeds and overall fan power consumption. An additional benefit

associated with full-cell operation is lower water pressure drops across the spray

nozzles, which results in lower pumping power requirements. However, at very low

pressure drops, inadequate spray distribution may adversely affect the thermal

performance of the cooling tower. Another economic consideration is the greater water

loss associated with full-cell operation.

Most current cooling tower designs utilize multiple-speed, rather than continuously

adjustable variable-speed fans. In this case, it is not optimal to operate all tower cells

under all conditions. The optimal number of cells operating and individual fan speeds

will depend upon the system characteristics and ambient conditions. However, simple

relationships exist for the best sequencing of cooling tower fans as capacity is added or

removed. When additional tower capacity is required, then in almost all practical cases,

145

the tower fan operating at the lowest speed (including fans that are off) should be

increased first. Similarly, for removing tower capacity, the highest fan speeds are the

first to be reduced.

These guidelines are derived from evaluating the incremental power changes

associated with fan sequencing. For two-speed fans, the incremental power increase

associated with adding a low-speed fan is less than that for increasing one to high speed

if the following condition is satisfied.

3 3Yt,low < (1- tlow)(6.2.1)

or

yt,low < 0.79 (6.2.2)

where Ytlow is the relative fan speed at low speed. If the low speed is less than 79% of

the the high fan speed, then the incremental power increase is less for adding a low-

speed as opposed to a high-speed fan. In addition, if the low speed is greater than 50%

of the high speed, then the incremental increase in air flow is greater (and therefore

better thermal performance) for adding the low-speed fan. Most commonly, the low

speed of a two-speed cooling tower fan is between one-half and three-quarters of full

speed. In this case, tower cells should be brought on-line at low speed before any

operating cells are set to high speed. Similarly, the fan speeds should be reduced to

low speed before any cells are brought off-line.

For three-speed fans, the sequencing logic is not as obvious. However, for the

special case where low speed is greater than or equal to one-third of full speed and the

146

difference between the high and intermediate speeds is equal to the difference between

the intermediate and low, then the best strategy is to increment the lowest fan speeds

first when adding tower capacity and decrement the highest fan speeds when removing

capacity. Typical three-speed combinations that satisfy this criteria are 1) one-third,

two-thirds, and full speed or 2) one-half, three-quarters, and full speed.

Another issue related to control of multiple cooling tower cells having multiple-

speed fans concerns the distribution of water flow to the individual cells. Typically, the

water flow is divided equally among the operating cells. However, the overall thermal

performance of the cooling tower is best when the flow is divided such that the ratio of

water to air flow rates is identical for all cooling tower cells. In comparing equal flow

rates to equal flow rate ratios, at worst, a five percent difference between the heat

transfer effectivenesses for a combination of two tower cells, one operating a one-half

and the other at full speed was found. Depending upon the conditions, these

differences generally result in less than a one percent change in the chiller power.

These differences should also be contrasted with the lower water pressure drop across

the spray nozzles (lower pumping power) associated with equally divided flow. In

addition, the performance differences are smaller for greater than two-cell operation,

when a majority of cells are operating at the same speed. Overall, equal water flow

distribution between cooling tower cells is near-optimal.

6.2.2 Multiple Chillers

Multiple chillers are normally configured in a parallel manner and typically

controlled to give identical chilled water supply temperatures. For the parallel chiller

combinations considered in this study, controlling for identical set temperatures was

found to be either optimal or near-optimal. Besides the chilled water setpoint,

additional control variables are the relative chilled and condenser water flow rates.

147

Simple guidelines may be established for distributing these flows.

In general, the relative condenser water flows to each chiller should be controlled to

give identical leaving condenser water temperatures for all chillers. This condition

approximately corresponds to relative condenser flow rates equal to the relative loads

on the chillers. Figure 6.1 shows results for four different sets of two chillers operated

7.0

6.5

6.0D

5.5

5.0

4.5-15

Figure 6.1.

I * I I * I - I -I l " 1 " a • I " I - "

-10 -5 0 5 10

Condenser Return Water AT (F)

Effect of Condenser Water Flow Distribution forTwo Chillers in Parallel

in parallel. The overall chiller coefficient of performance (COP) is plotted versus the

difference between the condenser water return temperatures for equal loadings on the

chillers. For the identical D/FW variable-speed chillers, the optimal temperature

difference is almost exactly zero. This was found to be the case for all identical chillers

considered in this study. For the non-identical chillers of Figure 6.1, equal leaving

D/FW Variable-Speed & Hackneres

0OWN

0

15

148

condenser water temperatures result in chiller performance that is close to the optimum.

Even for the D/FW variable and fixed-speed chiller combination, which have very

different performance characteristics, the penalty associated with the use of identical

condenser leaving water temperatures is insignificant. Similar results were obtained for

unequal loadings on the chillers.

For given chilled water return and supply temperatures, the relative chilled water

flow to each chiller is equal to its relative loading. Consider the problem of determining

the optimal relative loadings for Nh chillers in parallel. The relative condenser water

flows are assumed to be controlled to give identical return water temperatures. The

optimization problem is one of minimizing the total chiller power consumption or

Nch

Pch = X Pchi (6.2.3)i=1

with respect to the relative loadings, fji with the constraint

Nch

fL = 1 (6.2.4)i=l

By forming the Lagrangian and applying the first-order condition for a minimum or

maximum, it can be shown that the point of minimum or maximum overall power

occurs where the derivatives of the individual chiller power consumptions with respect

to their relative loadings are equal.

149

= for all i, j (6.2.5)

afLoi fLj

This condition, along with the constraint of equation (6.2.4) are sufficient to determine

the relative loadings. For identical chillers, these equations are satisfied for all chillers

loaded equally or

1fL, for i = ltoNch (6.2.6)

N~ tNch

For chillers with different cooling capacities, but identical part-load characteristics, the

constrained optimality conditions are satisfied when each chiller is loaded according to

the ratio of its capacity to the sum total capacity of all operating chillers. For the it

chiller,

fLi Qcap,iNch for i = 1to Nch (6.2.7)

XQcap, (i-=1

where Qap, is the cooling capacity of the ith chiller.

The relative loadings determined with equations (6.2.6) or (6.2.7) could result in

either minimum or maximum power consumptions. With the second-order necessary

condition for a minimizing point, it is possible to show that these points represent aminimum when the derivative of the COP with respect to relative loading is less than

zero. In other words, the chillers are operating at loads greater than the point at which

the maximum COP occurs. Typically, but not necessarily, the maximum COP occurs

150

at loads that are about 40 - 60% of a chillers cooling capacity. In this case, and with

loads greater than about 50% of cooling capacity, the control defined by equations

(6.2.6) and (6.2.7) results in a minimum power consumption.

Figure 6.2 shows the effect of the relative loading on chiller COP for different sets

of identical chillers loaded at approximately 70% of their overall capacities. Three of

the chillers have maximum COP's when evenly loaded, while the fourth (D/FW fixed-

speed) obtains a minimum at that point. The part-load characteristic of the D/FW fixed-

speed chiller is unusual in that the maximum COP occurs at its capacity (see Chapter 2).

This chiller was retrofit with a different refrigerant and drive motor which caused its

capacity to be derated from 8700 tons to 5500 tons. As a result, the evaporators and

condensers are oversized for its current capacity.

7.0

D/RW Variable-Speed Chiles

6.5

Q 6.0-D/FW Fixed-Speed Chillers

0.5" Hawlner's Chiers

5.0 Lau's Chillers

0.3 0.4 0.5 0.6 0.7

Relative Load on First Chiller

Figure 6.2. Effect of Relative Loading for Two Identical Parallel Chillers

151

The effect of loading on non-identical chillers was also investigated. The minimum

power consumption was realized with near-even loading for all combinations

considered, except for those involving the D/FW fixed-speed chiller. The best strategy

for this particular fixed-speed chiller in combination with other chillers is to load it as

heavily as possible, since its performance is best at ful load.

One of the important issues concerning control of multiple chillers is chiller

sequencing. Sequencing involves determining the conditions at which specific chillers

are brought on-line or off-line. The optimal sequencing of chillers depends primarily

upon their part-load characteristics. Chillers should be brought on-line at conditions

where the total power of operating with the additional chiller would be less than without

it. Optimization results indicate that the optimal sequencing of chillers may not be

decoupled from the optimization of the rest of the system. The characteristics of the

system change when a chiller is brought on-line or off-line due to changes in the system

pressure drops and overall part-load performance. The optimal point for switching

chiller operation may differ significantly from switch points determined if only c1iller

performance were considered at the conditions before the switch takes place.

6.2.3 Multiple Air Handlers

In a large system, a central chilled water facility may provide cooling to several

buildings, each of which may have a number of air handling units in parallel. If the

supply air setpoints of each of the air handlers were considered to be a unique control

variable, the optimization problem would become quite complicated. However, the

error associated with using identical supply air temperatures for all air handlers is

relatively small as compared with the optimal solution, even when the loading on the

various cooling coils differs significantly. As a result, the number of control variables

in the optimization process may be reduced by one less than the number of air handlers.

152

Figure 6.3 shows a comparison between individual and identical setpoint control

values for a system with two identical air handlers. The system coefficient of

performance (COP) associated with optimal control is plotted versus the relative loading

on one air handler. The difference between individual and identical setpoint control is

not significant over the practical range of relative loadings. This result is most easily

explained by considering the limits on the relative loadings. For the case of equal

loadings on the air handlers, the optimal control setpoints are equal for identical air

handlers. In the other extreme, where all of the load is applied to one air handler, then

the supply air temperature of the unloaded air handler has no importance, since its

power consumption is zero. Between the two extremes, the error associated with

assuming identical setpoints is relatively small. This result also extends to many air

handlers in parallel and to non-identical designs.

4.5.

4.4m4IdenicaSupply Air Set Temperatures

4.3

4.2

4.1

4.0

3.9 OptimalS 3.9

CL 3.8

3.7

3.6

3.5 • -0.5 0.6 0.7 0.8 0.9

Relative Load on First Air Handler

Figure 6.3. Comparison of Optimal System Performance for IndividualSupply Air Setpoints with that for Identical Values

153

Although the number of control variables is reduced by considering only a single

supply air set temperature, the overall operating cost still depends upon the performance

and loadings on the individual air handlers. However, for the purpose of determining

optimal control, air handlers may be combined into a single effective air handler under

the conditions that all zones are maintained at the same air temperature and the heat

transfer characteristics of the coils are similar. The required air flow as a fraction of the

design air flow for the ith air handler in parallel, #y , may be expressed in terms of an

overall air flow ratio as

fah,i

Y ahu,i fahudes, ahu (6.2.8)

where yahu is the ratio of the total required air flow to the total design air flow, fahu,i is

the ratio of air flow for the ith air handler to the total air flow for all air handlers, and

fmh,,,i is the ratio of design air flow for the idh air handler to the total design air flow

for all air handlers. If all air handlers have identical supply air and zone air temperature

setpoints, then f., is also equal to ratio the sensible loading on the zones supplied by

ith air handler to the total sensible zone loads, f In this case, the total air handler

power consumption for variable-speed fans is

3 Nahu e F fs,

Pahu = ahu XP Pahudes,i f-s,1 1(6.2.9)i = 1 ~ Lahudes,i

where Pahu,des,i1i the power consumption for the ith air handler at it's design air flow.

As a result, all the air handlers may be combined into one air handler having the

sum total area, water flow, air flow, and loading, with the power computed according

154

to equation (6.2.9). For the near-optimal control algorithm described in Chapter 5, it is

necessary to include the relative zone sensible loads (fs,,) as uncontrolled variables in

the empirical system cost function, if they change significantly.

6.2.4 Multiple Pumps

A common control strategy for sequencing both condenser and chilled water pumps

is to bring pumps on-line or off-line with chilers. In this case, there is a condenser and

chilled water pump associated with each chiller. For fixed-speed pumps, this strategy

is not optimal. At the point at which a chiller is brought on-line in parallel and

assuming that the pump control does not change, there is a reduction in the pressure

drop and subsequent increase in flow rate for both the condenser and chilled water

loops. The increased flow rates tend to improve the overall chiller performance.

However, if the pumps are operating near their peak efficiency before the additional

chiller is brought on-line, then there is a drop in the pump efficiency when adding the

additional chiller while holding the pump control constant. Most often, the

improvements in chiller performance offset the degradation in pump performance, so

that there is no need for an additional pump at the chiller switch point.

Figure 6.4 shows the optimal system performance for different combinations of

chillers and fixed-speed pumps in parallel as a function of load for a given wet bulb.

The optimal switch point for a second pump occurs at a much higher load (-40.62) than

the switch point for adding or removing a chiller (~0.38). If the second pump were

sequenced with the second chiller, then the optimal switch would occur at the maximum

chiller capacity and approximately a 10% penalty in performance would result at this

condition. The optimal control for sequencing fixed-speed pumps depends upon the

load and the ambient wet bulb temperature and should not be directly coupled to the

155

chiller sequencing.

6

One Chiller, One Pump

064 5 Optimal Chiller Switch Point

Two Chillers, One Pump

rj 4Two Chillers,

ego Two Pumps

Optimal Pump Switch Point0 3

2 '"1 i"Slm i

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 6.4. Effect of Chiller and Pump Sequencing on Optimal System Performance

The sequencing of variable-speed pumps is more straightforward than that for

fixed-speed pumps. For a given set of operating chillers and tower cells (i.e. given

system pressure drop characteristics), variable-speed pumps have relatively constant

efficiencies over their range of operation. As a result, the best sequencing strategy is to

select pumps so as to operate near their peak efficiencies for each possible combination

of operating chillers. Since the system pressure drop characteristics change when

chillers are added or removed, then the sequencing of variable-speed pumps should be

directly coupled to the sequencing of chillers. For identical variable-speed pumps

oriented in parallel, the best overall efficiency is obtained if they operate at identical

speeds. For non-identical pumps, near-optimal efficiency is realized if they operate at

equal fractions of their maximum speed.

156

6.3 Sensitivity Analyses and Control Characteristics of Subsystems

In this section, the sensitivity of the optimal system performance to the uncontrolled

and controlled variables is studied.

6.3.1 Effects of Load and Ambient Conditions

For a given system in which the relative loadings on each zone are relatively

constant, the optimal control variables are primarily a function of the total sensible and

latent gains to the zones, along with the ambient dry and wet bulb temperatures. Figure

6.5 shows the effect of these uncontrolled variables on the total operating costs. For a

given load and wet bulb temperature, the effect of the ambient dry bulb temperature is

insignificant, since air enthalpies depend primarily upon wet bulb temperatures and the

performance of wet surface heat exchangers are driven primarily by enthalpy

differences. Typically, the zone latent gains are on the order of 15-25% of the total

zone gains. In this range, Figure 6.5 shows that the effect of changes in latent gains

has a relatively small effect upon the system performance for given total load.

Consequently, results for overall system performance and optimal control may be

correlated in terms of only the ambient wet bulb temperature and total chilled water

load. In the event that the load distribution between zones changes significantly

through time, then this must also be included as a correlating variable.

157

11• ~Base Cs

10 Ambient Dry Bulb = 80 FAmbient Wet Bulb = 70 F

9 . Latent-Sensible Ratio = 0.20

O 8

%.Low Ambient Wet Bulb (55 F)

n 6

, 5

4+ High Ambient Dry Bulb (95 F)

3 x High Zone Latent-Sensible Ratio (0.25)

2 1 " 1 =M I "I I"M

0.0 0.2 0.4 0.6 0.8 1.0


Figure 6.5. Effect of Uncontrolled Variables on Optimal System Performance

6.3.2 Condenser Water Loop

The primary controllable variables associated with heat rejection to the environment

are the condenser water and tower air flow rates. Figures 5.4 and 5.5 showed how

optimal values of these flow rates vary with both total chiller load and ambient wet bulb

temperature. Both optimal air and water flows increase with load and wet bulb

temperature. Higher condensing temperatures and reduced chiller performance result

from either increasing loads or wet bulb temperatures for a given control. Increasing

the air and water flow under these circumstances reduces the chiller power consumption

at a faster rate than the increases in fan and pump power.

Figures 6.6 shows the sensitivity of the total power consumption to the tower fan

and condenser pump speed. Contours of constant power consumption are plotted

158

versus fan and pump speeds. Near the optimum, power consumption is not sensitive

to either of these control variables, but increases more quickly away from the optimum.

The rate of increase in power consumption is particularly large at low condenser pump

speeds. There is a minimum pump speed necessary to overcome the static head

associated with the height of the water discharge in the cooling tower above the takeup

from the sump. As the pump speed approaches this value, the condenser flow

approaches zero and the chiller power increases dramatically. It is generally better to

have too high rather than too low a pump speed. The "flatness" near the optimum

indicates that it is not necessary for an extremely accurate determination of the optimal

control.

1.0 J 5% > Minimum

0.9

. 0.8- 1% > Minimum

0.77Minimum

xW 0.6

0.5:.2

0.4

oom 0.3C1 10% > Minimum

0.2 • 25% > Minimum0.11 - , - , - -

0.5 0.6 0.7 0.8 0.9 1.0

Relative Condenser Pump Speed

Figure 6.6. Power Contours for Condenser Loop Control Variables

159

6.3.2 Chilled Water Loop

Figures 5.2 and 5.3 showed the dependence of'the optimal chilled water and supply

air set temperatures on the load and wet bulb. Both the optimal chilled water and

supply air temperatures decrease with increasing load for a fixed wet bulb temperature.

This behavior occurs because the rate of change in air handler fan power with respect to

load changes is larger than that for the chiller at the optimal control points. As the wet

bulb temperature increases for a fixed load, the optimal set temperatures also increase.

There are two pnmary reasons for this result: 1) For a given load, the chiller power

depends primarily upon the temperature difference between the leaving condenser and

chilled water temperatures. The condenser temperature and chiller power consumption

increase with increasng wet bulb temperature and the optimal chilled water temperature

increases in order to reduce the temperature difference across the chiller. 2) In the

absence of humidity control, optimal supply air and chilled water temperatures increase

with decreasing sensible zone loads for constant total chiller load. In addition, the

sensible to total load ratio decreases with increasing wet bulb temperature for constant

total load.

Figure 6.7 shows the sensitivity of the system power consumption to the chilled

water and supply air set temperatures for a given load and wet bulb temperature.

Within about 3 degrees F of the optimum, the power consumption is within 1% of the

minimum. Outside of this range, the sensitivity to the setpoints increases significantly.

The penalty associated with operation away from the optimum is greater in the direction

of smaller differences between the supply air and chilled water setpoints. As this

temperature difference is reduced, the required flow of chilled water to the coil

increases and the chilled water pumping power is larger. For a given chilled water or

supply air temperature,the temperature difference is limited by the heat transfer

160

characteristics of the coil. Below this limit, the required water flow and pumping

power would approach infinity if the pump output were not conswained. It is generally.

better to have too large rather than too small a temperatur difference between the

supply air and chilled water setpoints.

65

CuI

s

sm

60

55-

50

-F.., U - U - U - U -.9

35 40 45 50 55

Chilled Water Supply Temperature (F)

Figure 6.7. Power Contours for Chilled Water and Supply Air Temperatures

6.4 Optimal versus "Alternative" Control Strategies

There is no general strategy that has been established for controlling chilled water

systems. Most commonly, the chilled water and supply air set temperatures are

constants that do not vary with time. In some applications, these setpoints vary

according to the ambient dry bulb temperature. Generally, there is an attempt to control

the cooling tower and condenser water flow in response to changes in the load and

10% >25% >

A

161

ambient wet bulb. One strategy for controlling these flow rates is to maintain constant

temperature differences between the cooling tower outlet and the ambient wet bulb

(approach) and between the cooling tower inlet and outlet (range), regardless of the

load and wet bulb. In some applications, humidity along with temperature is controlled

within the zones. In this section, the performance of some of these control strategies is

compared with that of an optimally controlled system.

6.4.1 Conventional Control Strategies

Fixed values of chilled water and supply air setpoints and tower approach and range

that are optimal or near-optimal over a wide range of conditions do not exist. In

addition, it is not obvious how to choose values that work best overall. One simple,

yet reasonable approach is to determine fixed values that result in near-optimal

performance at design conditions. Figure 6.8 shows a comparison between the

performance associated with optimal control and 1) fixed chilled water and supply air

temperature setpoints (40 and 52 F) with optimal condenser loop control, 2) fixed

tower approach and range (5 and 12 F) with optimal chilled water loop control, and 3)

fixed setpoints, approach, and range. The results are given as a function of load for a

fixed wet bulb temperature. Since the fixed values were chosen to be appropriate at

design conditions, the differences in performance as compared with optimal control are

minimal at high loads. However, at part-load conditions, Figure 6.8 shows that the

savings associated with the use of optimal control become very significant. Optimal

control of the chilled water loop results in greater savings than that for the condenser

loop for part-load ratios less than about 50%.

162

7.

6 Optimal Fixed Approach and Range

Fixed Chilled Water andE 5 Supply Air Setpoints

0Fixed Setpoints, Approach, and Range

3

2 1

0.2 0.4 0.6 0.8 1.0


Figure 6.8. Comparisons of Optimal Control with "Conventional" Control Strategies

The overall savings over a cooling season for optimal control depends upon the

time variation of the load. If the cooling load were relatively constant, then fixed values

of temperature setpoints, approach, and range could be chosen to give near-optimal

performance. Table 6.2 summarizes cooling season operating costs for the

conventional control strategies relative to optimal control for two different load

characteristics in both Dallas and Miami. The low and high internal gains are

approximately one-third and one-half of the maximum cooling requirement of the

system. In Dallas, the maximum total operating cost difference of approximately 17%occurs for fixed setpoints, approach, and range with low internal gains. The difference

is reduced to 7% for the high internal gains, since the system operates at a more

uniform load near the design conditions. There is approximately twice the penalty

163

associated with the use of fixed chilled water and supply air setpoints as compared with

fixed tower approach and range for low internal gains, but the penalties are equal for

the high internal gains. The results for the Miami climate are similar to those for Dallas.

Table 6.2Cooling Season Results for Optimal vs. Conventional Control

Cost Relative to Optimal Control

Control Description Low Internal Gains High Internal GainsDallas Miami Dallas Miami

fixed temperauresetpoints 1.09 1.10 1.03 1.03fixed approach and range 1.05 1.06 1.03 1.03fixed setpoints, approach, range 1.17 1.19 1.07 1.07

6.4.2 Humidity Control

In a variable air volume system, it is generally possible to adjust the chilled water

temperature, supply air temperature, and air flow rate in order to maintain both

temperature and humidity. In constraining the room humidity, the number of "free"

control variables in the optimization is reduced by one. For a given chilled water

temperature, there is at most one combination of the supply air temperature, air flow

rate, and water flow rate that will maintain both the room temperature and humidity.

ASHRAE [1981] defines acceptable bounds on the room temperature and humidity

for human comfort. For a zone that is being cooled, the equipment operating costs are

minimized when the zone temperature is at the upper bound of the comfort region.

However, operation at the humidity upper limit does not minimize costs. Figure 6.9

compares costs and humidities associated with fixed and free floating zone humidities

as a function of the load. Over the range of loads for this system, the free floating

humidity operates within the comfort zone at lower costs and humidities with the largest

164

differences occurring at the high loads. Operation at the upper humidity bound results

in lower latent loads, but the addition of thishumidity control constraint requires higher

supply air temperatures than that associated with free floating humidities. In turn, the

higher supply air temperature results in greater air handler power consumption. In

effect, the addition of any constraint reduces the number of free control variables by

one and results in operation at a higher cost. In the determination of optimal control

points, the humidity should be allowed to float freely, unless it fals outside the bounds

of human comfort.

6 o.o14

"fixed" humidity 0.013

-0.012

,5"firee" flaig humidity -0.011CIO-0.010

0.009 l

S 4 0.008

0.007

0.0061*06 COP for "free"

2 ,... ..I' - . , - 0.005

o 3 ~~floating humidity 0.5 N

COP for "fixed" humidity 0.004

0.003

0.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 6.9. Comparison of "Free" Floating and "Fixed" Humidity Control

165

6.5 Comparisons of Alternative System Configurations

under Optimal Control

In order to compare the operating costs associated with different system designs, it

is most appropriate that each be optimally controlled. In this section, alternative system

configurations are compared in terms of their optimal system performance.

6.5.1 Variable versus Fixed-Speed Equipment

The part-load performance of a centrifugal chiller depends upon the method by

which its capacity is modulated. As noted earlier, the Dallas/Fort Worth primary chiller

was originally operated at a constant speed and the cooling capacity was modulated

with the use of pre-rotation inlet vanes and outlet diffuser vanes. The chiller was

subsequently retrofit with a variable-speed electric motor and the vane control was

disconnected.

In general, the part-load performance of a chiller is better for variable-speed as

compared with vane control. In this study, the performance of the D/FW chiller was

measured for both types of capacity modulation for nearly identical conditions as

described in Chapter 2. Figures 2.13 and 2.14 showed the performance for the

variable and fixed-speed control, while Figure 2.15 gave a direct comparison of their

performance. Figure 6.10 gives a comparison between the overall optimal system

performance for both types of control. At part-load conditions, the performance

associated with the variable-speed control is significantly better. However, the power

requirements are similar at conditions associated with the peak loads. This is expected,

since at this condition, the vanes are wide open and the speed under variable-speed

control approaches that of the fixed-speed operation.

166

7.

6

o D/FW Variable-Speed Chiller

E 5

D/RF ixd-Speed Chiller

3

0.2 0.4 0.6 0.8 1.0


Figure 6.10. Optimal System Performance for Variable and Fixed-Speed Chillers

Part of the improvement with variable-speed chiller control may result from the

unique characteristics of the D/FW chiller. The capacity of this chiller was derated, so

that the evaporator and condenser are oversized at the current capacity relative to the

original design capacity. As a result, the performance is more sensitive to penalties

associated with part-load operation of the compressor than to heat exchange

improvements that occur with lower loads.

The most common design for cooling towers utilizes multiple tower cells in parallel

that share a common sump. Each tower cell has a fan that may have one, two, or

possibly three operating speeds. Although multiple cells, having multiple fan settings

offers a wide flexibility in the control, the use of variable-speed tower fans can provide

additional improvements in the overall system performance.

Figure 6.11 compares optimal system performance for single-speed, two-speed,

167

and variable-speed tower fans as a function of load for a given wet bulb. There are

four tower cells for this system. All cells operate for the variable-speed fan results

under all conditions, while cells are isolated for discrete fan control results when their

fans are off. The discrete changes in the control of the multi-speed fans causes the

discontinuities in the slopes of the curves in Figure 6.11. The flexibility in the control

with one or two-speed fans is most limited at low loads. Below about 70% of full-load

conditions, the difference between one-speed and variable-speed fans becomes very

significant. With two-speed fans, the differences are on the order of 3 - 5% over the

entire range.

7,

Variable-Speed Fans6

Two-Speed Fans

" 5

3.0.2 0.4 0.6 0.8 1.0


Figure 6.11. Comparison of One-Speed, Two-Speed, and Variable-SpeedCooling Tower Fans (Four Cells)

If a fixed-speed pump is sized to give proper flow to a chiller at design conditions,

then it is oversized for part-load conditions and the system will have higher operating

168

costs than with a variable-speed pump having the same design capacity. The use of a

smaller fixed-speed pump for low load conditions improves the flexibility in control

and can reduce the overall power consumption. Figure 6.12 gives the optimal system

performance for variable-speed and fixed-speed pumps applied to both the condenser

and chilled water flow loops. The "large" fixed-speed pumps are sized for design

conditions, while the "small" pump has one-half the flow capacity of the large. Below

about 60% of full-load conditions, the use of variable-speed pumps shows a very

significant improvement over single fixed-speed pumps. With the addition of "small"

fixed-speed pumps, the improvements with variable-speed become significant at about

40% of the maximum load.

7.

6 Variable-Speed Pumps

~.) 5

No 4 Large Fixed-Speed Pump

S3

o,-

0.2 0.4 0.6 0.8 1.0


Figure 6.12. Comparison of Variable and Fixed-Speed Pumps

The overall savings over a cooling season associated with the use of variable-speed

equipment depends upon the time variation of the load. Table 6.3 summarizes cooling

169

season operating costs for the fixed-speed equipment relative to all variable-speed for

low and high internal gains in both Dallas and Miami. Included in this table are results

for fixed-speed air handler fans with variable-pitch blades to control the air flow. The

results do-not differ significantly between the Dallas and Miami climates. The largest

differences for all situations are seen in comparing the fixed-speed to the variable-speed

chillers (13% to 18%). The use of individual fixed-speed pumps or fans in the system

carries a penalty of about 3 - 10% depending upon the load characteristics. Overall, the

use of all fixed-speed equipment results in operating costs that are 26 - 43% higher than

for all variable-speed drives.

Table 6.3Cooling Season Results for Variable vs. Fixed-Speed Equipment

Cost Relative to All Variable-Speed EquipmentConfiguration Low Internal Gains High Internal Gains

Dallas Miami Dallas Miami

fixed-speed chiller 1.16 1.18 1.14 1.13fixed-speed tower fans 1.06 1.06 1.05 1.05fixed-speed pumps 1.10 1.09 1.05 1.04fixed-speed air handler fans 1.07 1.07 1.04 1.04all fixed-speed equipment 1.42 1.43 1.29 1.26

6.5.2 Series versus Parallel Chillers

Multiple chillers are typically arranged in parallel and the chilled and condenser

water flows are divided between the chillers according to their loading. Alternatively, it

is possible to arrange chillers in series, so that the total chilled and condenser water

flows pass through each evaporator and condenser. Two possible arrangements for

series chillers are series-parallel and series-counterflow. In the series-parallel

170

arrangement, the chilled and condenser water flows are in parallel, in that these streams

enter the same chiller. For the series-counterflow configuration, the streams enter at

opposing chillers.

For the same total flow, multiple chillers operate more efficiently in series rather

than in parallel. Figure 6.13 gives a comparison between the chiller coefficients of

performance for parallel and series arrangements of two identical chillers as a function

of the relative loading on the first chiller for identical entering temperatures and flow

rates. Both series arrangements require significantly less power than the parallel

chillers, the best arrangement being series counterflow. For the same total flow, the

heat transfer coefficients are higher and leaving water temperature differences are lower

for the individual chillers in series orientations than for parallel.

8.5

8Series-Counterflow8.0-

~7.5Iowian"

=" 7.0

oParallel Chillers

6.5

0.3 0.4 0.5 0.6 0.7

Relative Load on First Chiller

Figure 6.13. Comparison of Chiller Performance forParallel and Series Configurations

171

Although the chillers perform more efficiently in series rather than parallel, there are

significant increases in water stream pressure drops across both the evaporators and

condensers for the series arrangement. For two identical chillers, the ratio of the

pressure drop across either the evaporator or condenser for a series arrangement as

compared with that for parallel is approximately 8-to-1 for identical flow rates. The

difference in the overall system performance for series versus parallel depends upon the

magnitude of evaporator and condenser pressure drops as compared with the other

pressure losses in the chilled and condenser water loops. Figure 6.14 compares the

optimal system performance for series and parallel chillers as a function of load for a

given wet bulb. For this system, the tradeoffs between improved chiller performance

and increased pressure drops balance such that overall system performance is similar

for both configurations. Reducing the evaporator and condenser pressure drops by

20% did not have a significant effect upon this result.

172

5.0-

4.5"Sees-CounterbowChillers

4.0

Parallel Chiller

' 3.5

3.0

2.5

2.0' - • " ,

0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 6.14. Optimal System Performance for Series and Parallel Chillers

6.6 Summary

Optimization techniques were applied to the control of chilled water systems. The

important uncontrolled variables that effect system performance and optimal control

variable settings were identified as the total chilled water load and ambient wet bulb

temperature. Additional secondary uncontrolled variables that could be important if

varied over a wide range would be the individual zone latent to sensible load ratios and

the ratios of individual sensible zone loads to the total sensible loads for all zones.

Control guidelines that reduce the number of independent control variables and

simplify the optimization were also identified. These general results were utilized in

Chapter 5 to develop a "simple" methodology for near-optimal control of chilled water

systems without storage. The guidelines are also useful to plant engineers for

173

improved control practices and are summarized as follows.

1) Variable-Sed Toweran: Operate all tower cells at identical fan speeds.

2) Multi-Speed Tower Fans: Increment lowest tower fans first when adding tower

capacity. Reverse for removing capacity.

3) Variable-Speed Pumps: The sequencing of variable-speed pumps should be

directly coupled to the sequencing of chillers, to give peak pump efficiencies for

each possible combination of operating chillers. Multiple variable-speed pumps

should be controlled to operate at equal fractions of their maximum speed.

4) Chiller: Multiple chillers should have identical chilled water set temperatures


divided according to the chillers relative cooling capacities.


temperatures.

No general simplifications could be found for the optimal sequencing of chillers and

fixed-speed pumps. It is necessary to evaluate the overall system performance in order

to determine the optimal points for adding or removing chillers. In general, it is not

optimal to sequence fixed-speed pumps with chillers.

Additional results and conclusions concerning both control and design under

optimal control of chilled water systems are summarized as follows.

174

1) Depending upon the load characteristics, fixed values of chilled water and

supply air setpoints and cooling tower approach and range resulted in

approximately 7 - 19% greater cooling season operating costs than that for

optimal control in Dallas and Miami.

2) In the determination of optimal control points, the humidity should be allowed

to float freely, unless it falls outside the bounds of human comfort. In effect,

the addition of any constraint reduces the number of free control variables by

one and results in operation at a higher cost.

3) Depending upon the load characteristics, the cooling season operating costs

were approximately 26 - 43% greater for all fixed-speed equipment as compared

with all variable-speed in Dallas and Miami. The most significant difference

was attributed to the chiller.

4) The performance of multiple chillers is enhanced by orientation in series rather

than parallel. However, the increase in pumping power requirements for series

chillers offsets the chiller improvements and the overall performance for the two

configurations is similar.

175

Chapter 7Methodologies for Optimal Control of Systems with Storage

There are several advantages associated with the use of thermal storage in large

cooling systems. From a design point of view, the use of storage can increase the peak

cooling capacity of a system, thereby reducing the size and initial cost of chillers,

cooling towers, and air handlers. For existing systems, introducing storage can allow

operation at more favorable part-load conditions. However, the most important

applications of storage are for 1) shifting the load from times of high to times of low

utility electric rates and 2) limiting the peak system power consumption in the presence

of utility demand charges.

There are two types of storage that are presently being installed for chilled water

systems: water and ice. Typically, both storage types are arranged in parallel between

the chillers and the air handlers in the chilled water loop. These systems generally

operate in different modes that allow for charging (cool-down) and discharging of

storage and for direct cooling in which the storage is isolated and the chillers meet the

load directly. In an ice system, a chiller produces ice during the charging mode and a

brine solution is pumped between the storage and air handlers during discharge. A

water storage system relies on stratification in order to achieve good performance.

Cold water from the chillers is fed to the bottom of storage during the charging cycle.

During discharge, this chilled water is pumped to the air handlers and warm return

water from the air handlers is added to the top of storage. The stratification of storage

is produced by the cold supply to the bottom and warm return to the top.

In this chapter, a methodology is outlined for determining the optimal control of a

fully-stratified chilled water storage system. This methodology is utilized as tool for

176

analyzing the control of these systems. A simpler near-optimal control strategy is

developed from the results of the detailed analysis. In order to implement either the

optimal or near-optimal control, it is necessary to forecast the cooling load

requirements. Time-series techniques are applied to the development of a model for

forecasting the cooling requirements of buildings.

7.1 Optimization of Systems with Storage

Optimal control of a system with thermal storage involves minimizing the integral of

the operating costs, while satisfying required constraints. There are capacity limits

associated with each piece of equipment in the plant. If sensible energy storage is

utilized, then there are upper and lower limits on the temperature of storage in order to

provide proper comfort and avoid freezing. The general form of the optimization

problem is

Minimize

J -fL (x(t), u(t), f(t),t)dt (7.1.1)

to

subject to

dx-= (x, u, f) (7.1.2)

x(t) C X

u(t) C U

177

where J is the total cost of the process between the times to and tf, L is the

instantaneous cost at time t, x is a vector of state variables, u is the vector of control

variables, f is a vector of uncontrolled variables, and X and U are constraint sets

associated with the state and control variables. The instantaneous operating cost at any

time is the product of the total system power consumption and the cost of electricity or

L(x(t), u(t), f(t), t) = R(t) P(x(t), u(t), f(t)) (7.1.3)

where P is power consumption and R is the cost of electricity.

The solution of the optimal control problem gives a vector of control functions for a

continuous representation or sequences of values for the discrete case. This solution

can either be expressed as a function of time, u*(t), in open-loop control, or

alternatively as a function of the state variables, u*(x), for closed-loop control.

One approach for solving the optimal control problem expressed by equations 7.1.1

and 7.1.2 that evolved from the calculus of variations was introduced by Pontryagin

[1962]. The minimum principle provides a set of necessary conditions which the

optimal control, u*, must satisfy if it exists. One limitation of this approach is that

constraints on the state and control variables are often difficult to handle. It is also

possible to determine a local minimum as a solution.

Alternatively, the optimization problem may be solved in a discrete manner using

dynamic programming (Bellman [ 1957]). This method handles constraints on both

state and control variables in a straightforward manner. This method also guarantees a

global minimum The major limitations are associated with large memory and

computation requirements when many state variables are involved. In this study,

dynamic programming is utilized to study the optimal control of chilled water systems

with storage.

178

7.1.1 Dynamic Programming

Dynamic programming may be applied as a numerical technique for determining

global solutions to dynamic optimization problems. The method involves discretization

of the problem with respect to both time and state variables. The discrete-time

representation of the optimization problem becomes

Minimize

K

= XL(x(k), u(k), f(k),k) (7.1.4)k=O

subject to

x(k) = (x(k-1), u(k-1), f(k-1)) (7.1.5)

x(k) e X

u(k) e U

The integral cost function has been converted to a sum of individual stage costs from 0

to K, while the state equations are represented in difference form.

The heart of dynamic programming is Bellman's principle of optimality. The

principle can be stated as follows:

Suppose the sequence of optimal controls, u*(k), k = 0, 1,..., K, optimizes J(u).

Define a subproblem:

179

KJm = XL (x(k), u (k), f(k), k)

k--m

with the initial state at stage m equal to that associated with the original optimal

control, x(m) = x *(m). Then the solution to the original optimal control problem

from stage m to K, u*(k), k = m, m+l,..., K also optimizes Jm(u).

This fairly simple principle leads to an iterative functional equation for determining

optimal control termed the dynamic programming equation.

I(x(m), m) = min{L(x(m), u(m), f(m)) + I(x(m+l), m+1)} (7.1.6)

I(x(m), m) is defined as the minimum cost-to-go from state x at stage m to K. The

dynamic proamming equation is related to the principle of optimality in that the

minimum cost-to-go from any state x(m) is found by minimizig the sum of the stage

cost for the present stage, m and the minimum cost in going to the end of the process

from the resulting state at the next stage, m+l.

Equation 7.1.6 may be used in a numerical scheme for determining optimal control.

The state variables are discretized within the constraint set establishing a network of

paths between successive stages and possible states. Figure 7.1 shows a possible

network for a single state variable with specified initial and final conditions for 5

stages. For clarity in Figure 7.1, the change in state from one stage to the next is

limited to one state increment. The problem is solved for all stages by beginning at the

end of the process and working towards the beginning. For the network of Figure 7.1,

an initial cost-to-go of zero is assumed at the final state, x(K), K = 5. At stage K-i,

180

the minimum cost-to-go for each possible state is simply the stage cost associated with

the path from that state to the final state. At any stage k, the minimum cost-to-go for

any state, according to equation 7.1.6, is associated with the path having the smallest

sum of the stage cost and cost-to-go of the next state. At the Oth stage, the minimum

cost- to-go is the cost associated with the optimal path through the network.

Xmax

x(O)

Xmm L.Ik=0

Stage

Figure 7.1. Dynamic Programming Network

There are many possible controls that will take the system from any state i to any

other state j, one stage ahead. The minimization of equation 7.1.6 dictates that the stage

cost be a minimum associated with optimal control between the two states. For any

path from i to j, this stage cost optimization problem may be formally stated as

Minimize

=5

Jij(k) = L(x(k), u(k), f(k), k) (7.1.7)

181

subject to

xj(k+l) = 4(xi(k), u(k), f(k)) (7.1.8)

Depending upon the number of stages and state variables, the computational and

memory requirements associated with the use of dynamic programming may be

excessive. The number of nodes in a dynamic programming network is equal to the

number stages times the number of state variable increments together raised to a power

equal to the number state variables. For a problem with 10 stages having 3 state

variables, each discretized with 10 increments, the number of network nodes would be

10,000. Morin (1979) provides a review of improvements in the the dynamic

programming procedures that address this problem.

One such improvement, utilized in this study, involves application of dynamic

programming in an iterative scheme in which bounds on the state variables are reduced

at each iteration in the vicinity of the last computed optimal solution. Initially, the

network is coarse and encompasses the total range of acceptable state variables. At

each iteration, an optimal path is determined through the network and the size of the

network is reduced for the next iteration. At each iteration and at each stage, the

number of state variables is constant and bounds on the network are defined equidistant

above and below the states along the last computed optimal path. In the limit, the size

of the network approaches zero along the optimal control path.

7.1.2 Application to Systems with Fully-Stratified Water Storage

Figure 7.2 shows a schematic of a chilled water storage oriented in parallel between

the chillers and air handler equipment. There are five primary modes of operation

182 •

associated with control of storage that are considered in this study:

Return toChillers

Supply fromChillers

Return fromLoad

Supply toLoad

Figure 7.2. Parallel Configuration for Thermal Storage

1) Direct Cooling: The cooling load is met completely by chilled water supplied

from the chillers and there is no flow to or from storage. All independent control

variables are "free" in the control optimization.

2) Partial Charge: Chilled water from the chillers is provided to meet the load and

to lower the temperature of storage. In this mode, the chilled water supply

temperature is controlled to be a constant. All other independent control

variables are "free" in the control optimization.

4) Partial Discharge: Chilled water is supplied from both the chillers and storage to

183

meet the load. In this mode, the return water from the load is controlled to be

constant. All other independent control variables are "free".

4) Full Charge: There is no cooling load requirement (i.e. air handlers are off) and

the chillers are operating solely to charge the storage. The chilled water supply

temperature is controlled to be constant and all other independent control

variables are "free".

5) Full Discharge: The cooling load is met completely by water supplied from

storage and the chillers and condenser loop equipment is off. The return water

from the load is controlled to be a constant.

The use of constant inlet temperatures to storage during charging and discharging is

common practice. The storage capacity is proportional to the difference between the

load water return temperature for discharging and the chilled water supply temperature

for charging. However, there are practical limits on these temperatures. The limit on

the chilled water supply in order to avoid freezing problems in the evaporator is

approximately 40 F. A load water return temperature of 60 F during discharge is

approximately the upper limit necessary to insure proper dehumidification at the cooling

coils for the load. The constant values of chilled water supply temperature for charging

and load water return temperature for discharging were chosen as 40 and 60 F,

respectively.

Neglecting heat gains from the environment, conduction, and mixing at the inlets,

the energy equation for charging or discharging storage is

184

aT aT- = -(7.1.9)at ax

where T is temperature, t is time, x is position measured from the inlet, and v is the

velocity of the fluid. The fully-stratified model represented by equation (7.1.9) has a

known analytic solution. Given an initial temperature distribution at time t, the solution

of this hyperbolic partial differential equation for the distribution at time t + At and

position x is

T(t + At, x) = T(t, x - vAt) for x > vAt (7.1.10)

T(t + At, x) = Ti(t + At - x/v) for x < vAt (7.1.11)

where Ti(r) is the inlet temperature (i.e. chiller supply or load return) at any time r. For

fluid that was within the storage at time t, equation (7.1.10) states that the distribution

at time t + At is a copy of the original distribution moved by an amount vAt. For fluid

that enters the storage during the time period, equation (7.1.11) shows that the

temperature distribution downstream of the inlet is a history of the inlet conditions.

The use of constant inlet temperatures to the storage during charging and

discharging simplifies the storage model and the overall optimization problem. In this

case, the storage has two distinct temperature zones and the state of storage is

represented by the position of the front between cold fluid supplied from the chillers

and warm fluid returned from the load. Ifx represents the distance of this front above

the bottom of storage relative to the height of storage, then it follows from equation

('7.1.11) that a discrete state equation for storage at stage k may be expressed as

185

x (k+1) = x (k) + v At (7.1.12)

where v is the velocity of the fluid relative to the tank height measured in the x

direction. The relative position of the front may also be thought of as the relative

storage charge. At a value of 0, the storage is empty in terms of its ability to provide

cooling to the load. At a value of 1, the storage is fully charged. In terms of the rate at

which energy is removed from storage, the relative fluid velocity is

OsV (Tcharge _ Tdischarg) Caps (7.1.13)

where s is the energy discharge (positive) or charge (negative) rate of storage, Tchge

is the chilled water supply temperature in the charging mode (40 F), Tdishn is the

load water return temperature in the discharging mode (60 F), and Caps is the thermal

capacitance of storage. From an energy balance, the required chiller cooling rate is

Qeh Q0L- Qs (7.1.14)

where QLis the total building cooling load rate.

In the dynamic programming algorithm, the state variable (i.e. relative charge level)

is discretized between 0 and 1. Between any two states on the dynamic programming

network, it is necessary to determine the optimal stage cost. For a given mode of

operation at a given stage, the minimum power consumption of the equipment depends

primarily upon the total cooling load, ambient wet bulb temperature, and the storage

charging or discharging rate. The load and wet bulb are considered to be known

(forecasted) uncontrolled variables. The required storage charge or discharge rate is

186

computed from equations (7.1.12) and (7.1.13) using the known states. Rather than

perform the stage optimization within the dynamic programming algorithm, minimum

power requirements from steady-state optimizations were correlated for each mode of

operation using the form of equation (5.2.3). The optimal stage cost between any two

states, for stage k is

Ji4k) = R(k) Pi. j(QL(k), Twb(k), Qsijk))j At (7.1.15)

*

where R is the cost of electricity and Pi* is the minimum average power consumption

between states i and j evaluated at a load, wet bulb, and storage discharge (or charge)

rate of QL, Twb, and 0,if

There are limits associated with operation of the equipment that must be considered.

If the required chiller cooling rate is greater than the capacity of the chiller, then it is not

possible to make the transition between the two states and an infinite stage cost is

assigned. There is also a minimum cooling rate required to avoid compressor surge.

However, in the full discharge mode, the chillers are off and the required chiller cooling

is zero. If storage is being discharged and the required cooling rate determ ed with

equation (7.1.14) is less than the minimum chiller cooling rate, then the chiller is

assumed to operate a portion of the time during that stage at its minimum capacity in the

partial discharge mode and is considered to be off the rest of the time with the system in

the full discharge mode. The average rate of power consumption for the stage is then

P* =P (Q L, Twb, Q L- Qch,min) At~ + p*( T r At0~ (.116At At]J(..

187

where Qch,min is the minimum chiller cooling capacity and Aton is the time period for

operation of chillers at their minimum capacity determined as

Aton= - L"s i-j At (7.1.17)Och,min

Similarly, if the system is operating in the charge (partial or full) or direct mode and the

required chiller load is less than the minimum allowable, then the chillers are considered

to operate at their minimum capacity for the time necessary to meet the total load

requirement and are off the remainder of the stage.

There are also considered to be additional limits associated with minimum and

maximum rates.of charging and discharging storage. The maximum charging or

discharging rate is limited by the maximum chilled water pump flow and the supply

temperatures to storage during charging or discharging. The minimum rates are due to

limits on the control valves that distribute the flow between storage and the air handlers

during charging and between storage and the chillers during discharging. If the

required charging or discharging rate is greater than the maximum, then it is not

possible to make the transition between the two states on the dynamic programming

network and an infinite stage cost is assigned. If the required charging or discharging

rate is less than the minimum rate, then the system is assumed to operate a period of the

time during that stage in the partial charging or discharging mode at the minimum rate

and the remainder in the direct cooling mode.

7.1.3 Results for Fully-Stratified Storage

The methodology described in the previous section was applied to a fully-stratified

chilled water storage system in order to answer two questions: 1) What is the

188

magnitude of cost savings associated with the use of optimally controlled storage in the

absence of time-of-day or peak demand utility rates? and 2) Is it necessary to perform

"true" optimal control for systems having time-of-day rates or do simpler control

strategies work "well enough"? In order to study these issues, one-day (24-hour)

simulations were performed for known load distributions and ambient wet bulb

conditions. Only steady-periodic solutions were considered so that the final state

equalled the initial state. Both the load and wet bulb were varied according to a

sinusoidal function as

fmax + fmin fmax- fmin 2f(t)- 2 + mx msi 2 (t - 12) (7.1.18)

where f(t) is the value of the uncontrolled variable (load or wet bulb) at time t and fmin

and fmax are the minimum and maximum values that occur at 6 a.m. and 6 p.m.,

respectively. The effects of different minimum and maximum values were considered.

In some cases, the cooling system was assumed to be off at night (6 p.m. to 6 a.m.),

such that the load during this time was zero. The system considered was the base

system described in Chapter 6.

In the absence of time-of-day electric utility rates and if the peak load is less than the

capacity of the chiller, then the optimal control strategy for all 24 hour load and wet

bulb patterns considered was to operate in the direct mode for all times. In other

words, storage provided no benefit for this system. For no night-time load, thermal

storage provided at most a 5% reduction in the overall power consumption for the

system and loads considered.

One of the reasons that storage provides little or no operating cost savings when

189

there are no special time-of-day utilities rates is due to the fact that this system operates

most efficiently in the direct mode. For charging of storage, the chillers operate at a

fixed chilled water supply, while for discharging, the return temperature from the load

is constant. In the direct mode, these variables are optimized as a function of the load

and wet bulb temperature. Undoubtably, storage would provide some additional

benefit if the chilled water supply and load water return temperatures during charging

and discharging were also "free" control variables in the optimization. However, this

complicates the dynamic optimal control problem considerably by introducing many

state variables and would be difficult to implement in practice.

Figures 7.3 and 7.4 show how the optimal charge level of storage and chiller

loading vary over a day for different storage sizes with time-of-day electric rates. The

cost of electricity is assumed to be $0.15/kW-hr for the hours from 8 a.m. to 8 p.m.

and $0.05/kW-hr for the rest of the day. The ambient wet bulb varied between 65 and

75 F, while the minimum and maximum loads were 1500 and 4000 tons. The storage

charge level is defined as the fraction of the tank that is filled with chilled water at 40 F,

with the remainder of the tank at 60 F. Two different storage sizes were considered for

these results: 1500 and 2000 ton-hours/deg. F. The no storage results shown in Figure

7.4 represent the total building load and would be the load on the chiller if there were

no storage or if the system always operated in the direct cooling mode.

In the presence of time-of-day rates, Figure 7.3 shows that the optimal charge level

reaches a maximum at the time at which the rates increase and realizes a minimum when

the rates decrease. The smaller storage capacity is insufficient to meet the load during

the period of high utility rates. As a result, the storage is completely discharged when

the utility rates are lowered. Between the hours of about 8 a.m. to 1 p.m., this system

operates primarily in the direct cooling mode and the storage remains near its full

charge. Since the storage is undersized, then the chiller must operate sometime during

190

the high rate period. The optimal time to operate the chiller is during the first part of the

high rate period when the ambient wet bulb is lower and the system performs more

efficiently.

no

I-

Oo0'U

Figu

8 a.m. 12 4 8 p.m. 12 4 8

Time of Day

re 7.3. Optimal Storage Charge Level for One Day with

Time-of-Day Electric Rates

a.m.

The system with the larger storage has excess capacity and operates in the full

discharge mode for the entire high rate period and reaches a minimum charge level

when the rates are reduced. As evident from Figure 7.4, there is no load on the chillers

(i.e. chillers off) for this system between 8 a.m. and 8 p.m. After the rates are

reduced, both the small and large storage systems operate in the direct cooling mode for

a period of time before charging the storage. During the charging period, the chillers

are loaded at a fairly constant rate until the maximum storage charge is reached at the

time when the rates increase.

191

* No Storage

O 1500 ton-hours/deg. F

E3 2000 ton-hours/deg. F

ii II II

60001

5000-0O0

04000

o* 30000

2000

1000

0

~- ri - r

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

Hour of Day (from 8 a.m.)

Figure 7.4. Optimal Chiller Loading for One Day withTime-of-Day Electric Rates

From the results of dynamic programming applied to systems with storage, some

simple guidelines may be established for "near-optimal" control when time-of-day

utility rates are in place.

1) Begin the period of high rates with the storage fully charged.

2) During high-rate hours, the system should operate in the full discharge mode for a

period of time necessary for the storage to completely discharge, if possible. If

the storage capacity is not sufficient to meet the load when rates are high, then the

system should operate in the direct mode for the remaining time. The direct mode

on-peak operation should occur during times when the wet bulb temperature is

I-- ~ - - ! - - - ~ - - -. - - Pm y60yboyV pra-.,I"Pa"IMAJ40509

192

lowest, generally during the morning and early afternoon hours.

3) During low-rate hours, the chillers may operate at their capacity to charge storage

during the time when the wet bulb temperature is lowest for near-optimal

performance. For the remainder of the time, the system should operate in the

direct cooling mode.

Table 7.1 compares the cost of operation associated with optimal control, the near-

optimal control strategy outlined above and a non-optimal, more conventional control

strategy. In the non-optimal strategy, the system operates in the full discharge mode

from the start of the on-peak utility rate period. If the storage is completely discharged

during this period, then the system operates in the direct mode to meet the load until the

electric rates decrease. The storage is then charged at full capacity until fully charged.

The advantage of this control strategy is that it does not require forecasting of the

cooling loads or ambient wet bulb. The disadvantage is that the chillers may operate at

high cooling loads during times when the wet bulb is higher.

Table 7.1 shows that the near-optimal control strategy results in operating costs that

are close to those for optimal control. However, depending upon the storage capacity

in relation to the load, the non-optimal strategy gives costs that can be significantly

greater. All three controls give similar results at large storage capacities, since the

system always operates in the discharge mode when rates are high and operates in the

charging mode at full capacity for a significant portion of the time when rates are low.

The conventional strategy also works well when the variation in the wet bulb

temperature is small.

193

Table 7.1 One-Day Operating Cost Comparisonsfor Systems with Thermal Storage

Daily Operating Costs ($)System Description Optimal Near-Optimal Conventional

24 hour load 2686 2727 3095

ran. and max loads: 1500, 4000 tonsstorage capacity: 1500 ton-hours/deg. F

24 hour load 2489 2497 2564

min. and max. loads: 1500, 4000 tons

storage capacity: 2000 ton-hours/deg. F

12 hour load 2167 2301 2625

min. and max loads: 1000, 5000 tons

storage capacity: 1500 ton-hours/deg. F

12 hour load 1943 2066 2095

min. and max loads: 1000, 5000 tonsstorage capacity: 2000 ton-hours/deg. F

7.3 Models for Forecasting Building Cooling Requirements

In order to implement optimal or near-optimal control of thermal storage, it is

necessary to forecast cooling load requirements. In this section, measurements from

the Dallas/Fort Worth airport are utilized to develop a time-series model for forecasting

cooling loads.

Building cooling loads are functions of many variables such as ambient

temperature, solar radiation, occupancy of people, lighting, computers, etc. All of

these heat gains are periodic functions having a dominant period of 24 hours. There are

also random components in the loads resulting from the stochastic nature of the weather

and the way that the buildings and the distribution system are utilized. Therefore, a

194

combined deterministic plus stochastic model is appropriate for load forecasting.

The procedure for fitting a combined model to a data set is a three step process. A

purely deterministic model is defined through regression to the data. Next, a stochastic

time-series model is fit to the residual errors associated with the deterministic model. In

this step, the proper time-series model order is defined. Finally, parameters of the

combined deterministic plus stochastic model are estimated simultaneously.

For on-line optimal control of the equipment, it would be advantageous to utilize

on-line recursive parameter estimation for the forecasting model. Recursive parameter

estimation is most easily carried out with linear models (i.e. the model is linear in the

unknown parameters). For that reason, this study was restricted to linear time-series

models termed AR (auto-regressive) models. A good background to material presented

in this section is found in Pandit and Wu [1983].

7.2.1 Time-Series Models

A time series is a sequence of observed data ordered in time. The statistical

methodology dealing with the analysis of such a data sequence is termed time-series

analysis. A time-series model expresses the dependence of the present observation as a

sum of two independent parts: one dependent upon preceding observations and the

other an independent random sequence. The simple auto-regressive model of order n

(AR(n)) is given by

n

Xt= OiXt- + et (7.2.1)i=1

where Xt is the current (zero mean) output of the system, Xt~ is the output i steps

previous, 4)i is i parameter of the model, and et can be thought of either as a random

195

input to the system or the one-step ahead prediction error of the model. At time t-1, the

best forecast of an observation for time t is obtained by evaluating the expected value of

the right-hand side. The expected value of a random variable with a zero mean is zero,

thus the one-step ahead prediction for the model defined by equation (7.2.1) at time t-1

is

nXt = iXt-1(7.2.2)

i=1

From equations (7.2. 1) and (7.2.2), it should be evident that et is the error associated

with the one-step ahead forecast of the model. This is true for all time series models.

In order to estimate parameters of the model for a given set of data, the sum of the

squares of the one-step ahead prediction errors (e's) is minimized with respect to the

unknown O's using linear regression. For a given model, greater than one-step ahead

forecasts are determined with equation (7.2.2) applied recursively to known and

forecasted values.

A more general time-series model that utilizes both previous observations and errors

is termed an auto-regressive moving average model (ARMA). Pandit and Wu [1983]

have defined a rational method for determining the order of an ARMA model that

provides an adequate fit to the data. However, a nonlinear regression is required in

order to determine the coefficients of the model. Theoretically, an ARMA model can be

represented by an AR model of infinite order. However, in practice, only a finite AR

model is necessary. The advantage of using AR models is that linear regression

techniques may be utilized. This is especially important for load forecasting in the

context of on-line optimal control, when coefficients of the model are updated using

196

recursive parameter estimation techniques.

Periodic trends in the data may be removed by the use of a trigonometric

polynomial of order m, having the form

m mP(t) = Yaj sin(jox) + I bj cos(jon) (7.2.3)

j=1 j=1

where o) is the frequency and the a's and b's are unknown coefficients that are fit with a

linear least squares method applied to the data.

A combined model can also be expressed in a linear form. The combined model for

a zero mean output, Yt, is the sum of the deterministic and stochastic models.

m m nYt = Xajsinjox) + X bjcos(jcot) + Oi Xt., + et (7.2.4)

j=1 j=1 i=l

But, by definition

= XYt"- aj sin(joct) + X:bjcos(jot) (7.2.5)j=1 j=1

Substituting equation (7.2.5) into (7.2.4) gives

m m n

Yt= a a'j sin(jct) + X b 'jcos(jcot) + X 4)i Yt- 1 + et (7.2.6)j=1 j=l i=1

where a 'j and b 'j are different coefficients than those appearing in equation (7.2.4).

197

All the empirical coefficients of equation (7.2.6) can be fit to data with linear least-

squares methods. For one-time parameter estimation from a batch of measurements,

the data is typically averaged and the average is subtracted from the data before the

fitting process. Parameters of this model may also be fit using on-line recursive

parameter estimation as outlined by Ljung [1983].

7.2.2 Application to Forecasting Cooling Loads

Simple AR models given by equation (7.2.1) were fit with approximately three

days of cooling load data in March from the D/FW airport for a sampling interval of 1

hour. Results of this analysis are summarized in Table 7.2.

Table 7.2

AR Model Fit to March D/FW Data (1 hour sampling)

Parameter AR(1) AR(2) AR(3) AR(4) AR(5)

01 0.881 0.870 0.886 0.8496 0.8140

02 0.009 0.060 0.0748 0.0930

03 -0.059 0.1390 0.1510

04 -0.2290 -0.1010

05 -0.1500

rms(tons) 334 334 334 329 327

At first glance, it might appear that an AR(1) model is adequate for this data set. The

reduction in the root-mean sum of squares of the 1-step prediction errors is insignificant

when going from AR(1) to AR(2). However, the AR(4) model does show some

additional improvement. This behavior results from the large degree of randomness

198

exhibited in the data on a small time scale. A four-hour history is useful in overcoming

the short-term fluctuations and to incorporate the larger time scale variations in the data.

Comparisons of the one-step and five-step ahead predictions with the data are shown in

Figures 7.5 and 7.6. The AR(4) model does reasonably well for the one hour

prediction, but falters badly with five hour forecasts. Similar results were obtained

with an ARMA(3,2) model fit to this data.

300002

0

I-

2000

1000

0 ww

0 10 20 30 40 50 60 70 80

Time (hours)

Figure 7.5. One-Hour Forecasts of March D/FW Cooling Load Datafor AR(4) Model

199

WIo+

2000 +o

+ +

- Data+ 5-hour Forecasts

0 -II V

0 20 40 60 80

Time (hours)

Figure 7.6. Five-Hour Forecasts of March D/FW Cooling Load Datafor AR(4) Model

An improved model results if some of the determinism is removed by fitting a

trigonometric polynomial to the data. If a combined model is fit to the data then the

adequate model has 4 auto-regressive parameters and two periods (24 and 12 hours).

In this case, the root-mean square (rms) error of one-step ahead forecasts is 287. This

is a significantly improved model over the pure AR(4). The combined model does an

extremely good job of forecasting the cooling loads 1 hour ahead. In addition to the

improved 1-step predictions, the ability of the model to perform 5-hour forecasts is

vastly improved. Figure 7.7 shows comparisons between five-step ahead predictions

of the combined model with the March data. The rms of the errors for 5-step

predictions of the combined model is 398, as compared with 625 for the pure AR(4).

200

3000

02 4.

C+

- 2000-+

+2 444

4. 410004.+4.

+-Daa+ 5-hour Forecasts

0I " I

0 20 40 60 80

Time (hours)

Figure 7.7. Five-Hour Forecasts of March D/FW Cooling Load Datafor Combined Deterministic and Stochastic Model

Even better long-term predictions can be realized by using a larger sampling interval

for the data. The model determined with a sampling interval of 2.5 hours gives

significantly better results for the long-term predictions (rms of 296 versus 398). The 1

hour sampling interval gives a model that utilizes only the most recent information

concerning the load. This occurs because the model is fit using the errors of the 1-step

ahead prediction and there is a large degree of randomness in the data at 1 hour

samples. The larger sampling period more appropriately captures the larger scale

variations in the data. The forecasts associated with a 5 hour sampling interval are

better still. It might be advantageous to have separate models for short and long-term

forecasting that utilize different sampling intervals.

A good test of the model determined with the March data is to compare it with

201

another data set. A comparison with October data gave good results for one-hour

forecasts, but the five-hour predictions were very poor. There is a seasonal effect that

alters the characteristics of the deterministic part of the data, The cooling requirement is

coupled closely to the ambient temperature. Generally, the peak cooling load occurs at

about the same time of the day as the maximum ambient temperature. During October,

this peak occurs later in the day than in March. In fitting a combined deterministic and

stochastic model to the October data, the adequate linear model was also found to be

AR(4) with a deterministic part having two periods. The accuracy of the forecasts was

similar to that for the model fit to the March data.

In an attempt to further improve the long-term forecasts of the model, the ambient

temperature was used as a deterministic input to the model. At the D/FW airport, the

cooling requirement is strongly coupled to the ambient conditions. However, since the

ambient temperature and cooling requirement are essentially in phase, there was little

improvement in the results with the additional information. In other words, the history

of the ambient temperature is almost completely reflected in the cooling load history.

It would be advantageous to use recursive on-line identification for this forecasting

model so that the parameters could be adjusted to account for changes that occur on a

seasonal basis, such as those discussed for the October and March data.. This can be

accomplished in two ways. First of all, the incoming measured data may be weighted

more heavily than past data (e.g. exponential weighting) such that the parameter

estimates more correctly reflect the current state of the system. Secondly, periodicities

that reflect seasonal changes in the climate may be included.

202

7.3 Summary

Dynamic Programming was applied as a numerical tool for determining optimal

control of systems with stratified thermal storage. In the absence of time-varying utility

rates, thermal storage provided little or no operating cost savings as compared with no

storage for the systems considered. For systems with time-of-day utility rates, a simple

strategy for near-optimal control was identified.

The advantage of the Dynamic Programming methodology over the simpler strategy

is that it provides a "true" solution to the optimization problem and can adapt to

changing circumstances. The algorithm is simple enough so as to be implementable on

a microcomputer. Although demand charges were not considered in this study, peak

power demands could be reduced by assigning large stage costs for "high" power

consumptions.

The near-optimal control strategy involves maximizing the operation of the system

in the full discharge mode whenever the utility rates are high and minimizing the

operation in the charge mode when the utility rates are low. If the storage capacity is

insufficient to meet the load during the high rate period, then the system operates in the

direct mode during the hours when the wet bulb is low. During the low rate period, the

storage is charged at the full capacity of the chiller during the hours of lowest wet bulb.

For the remainder of this period, the system operates in the direct mode. The advantage

of this near-optimal control strategy is that it is easily implemented and gives operating

costs similar to that for optimal control.

In order to apply either the optimal or near-optimal strategy for controlling storage,

it is necessary to forecast the total cooling requirements. Pure time-series and

combined deterministic plus time-series models were fit to cooling load data for the

DIFW airport. In all cases, the models worked well for making forecasts of one hour.

203

In order to make longer term predictions, it was necessary to include deterministic

components in the model. The results of 5-hour predictions with the combined model

were good. The resulting model is simple enough to be fit with linear least-square

methods.

204

Chapter 8Conclusions and Recommendations

In this study, general methodologies were developed for design and control of central

chilled water systems. These methodologies are in form of mathematical models,

optimization algorithms, and guidelines for design and control synthesized from results

of optimal control applied to chilled water systems. The computer programs developed

in this study are listed in a separate document (Braun [1988]). Specific conclusions

and recommendations concerning this work follow.

8.1 Mathematical Models

In Chapter 2, a detailed mechanistic model for variable-speed centrifugal chillers

was developed. The model requires only design parameters and the operating

conditions in order to estimate the power requirement. The model is also capable of

estimating the compressor speed at which surge develops or the maximum chiller

cooling capacity at a given power input or speed. Results of the model were compared

with measurements from the D/FW airport. The mechanistic model works well in

estimating both the power requirement and the speed associated with the onset of surge

for variable-speed centrifugal chillers. Additional work is necessary to develop

mechanistic models for analyzing the performance offixed-speed, variable-vane

controlled chillers.

Using results of the mechanistic model, a simpler empirical model for correlating

performance data was also identified. Chiller power consumption correlates as a

quadratic function of only two variables: the load and temperature difference between

the leaving condenser and chilled water flows. The empirically-based model fits data

205

for both variable-speed and ftxed-speed with vane control chillers. Models for

correlating the limits of chiller operation associated with surge and capacity were also

presented.

In Chapters 3 and 4, effectiveness models were developed for cooling towers and

cooling coils. These models utilize existing effectiveness relationships for sensible heat

exchangers with modified definitions for number of transfer units and the capacitance

rate ratio. Simple methods were also developed for estimating the water loss in cooling

towers and the performance of cooling coils having both wet and dry portions.

Results of the effectiveness models for cooling towers and cooling coils compare well

with the results of more detailed numerical solutions to the basic heat and mass transfer

equations and with experimental data. The advantages of this approach are its

simplicity, accuracy, and consistency with the methods for analyzing sensible heat

exchangers. Future work should involve application of this modeling approach to

other wet surface heat exchangers.

Models were also presented for the power requirements of cooling tower and air

handler fans and condenser water and chilled water pumps. It was shown that

quadratic functions work well in correlating the power consumptions of the auxiliary

devices in central chilled water facilities. The use of quadratic functions for the chiller

and auxiliary equipment was an important result in the development of an efficient

method for determining optimal control.

8.2 Optimal Control MethodologiesTwo methodologies were presented for determining optimal control points for

chilled water systems without storage. A component-based non-linear optimization

algorithm was developed as a simulation tool for investigating optimal system

performance. Results of this algorithm implemented in a computer program, led to the

206

development of a simpler system-based methodology for near-optimal control.

The advantage of the component-based algorithm over the system-based approach

is that it provides a "true" solution to the optimization problem, including any nonlinear

constraints. Each of the components in the system is represented as a separate

subroutine with its own parameters, controls, inputs, and outputs. Models of

components may be either mechanistic or empirical in nature, so that the component-

based methodology is useful for evaluating both system design or control

characteristics.

The component-based algorithm takes advantage of the quadratic cost behavior of

the components found in chilled water systems in order to solve the optimization

problem in an efficient manner. However, in order to utilize this methodology as a tool

for on-line optimization, it is necessary to have detailed performance data for each of

the individual system components. Results of detailed optimizations identified

simplifications that reduced the number of control variables to five and uncontrolled

variables to two. The system-based near-optimal control methodology utilizes an

overall system cost function in terms of these variables. This cost function leads to a

set of linear control laws for the continuous control variables in terms the total chilled

water load and ambient wet bulb temperature. Separate control laws are required for

each feasible combination of discrete controls and the costs associated with each

combination are compared to identify the optimum. Results of the system-based

optimization methodology agree well with those of a more detailed non-linear

optimization analysis. The system-based procedure is simple enough so as to be

implementable either manually or on-line using microcomputers. For manual control

applications, charts could be used to determine optimal control as a function of load and

wet bulb.

207

Additional work is necessary in order to apply either the component-based or

system-based methodologiy to on-line optimal control. In particular, methods for

determining parameters of the models need to be investigated. The performance

characteristics of the system may change over time, so that it could be necessary to

update the model parameters. It is also important to iden)fy an appropriate time

interval for making control decisions. There may be inefficiencies associated with

changing controls too frequently in response to small changes in the uncontrolled

variables. The next step is to test these methodologies as part of an energy

management system for controlling an actual system.

A methodology was presented for determining the optimal control of stratified

thermal storage systems using Dynamic Programming that is simple enough so as to

be implementable on a microcomputer. Although demand charges were not considered

in this study, peak power demands could be reduced by assigning large stage costs for

"high" power consumptions. Future work should consider the best strategy for

including peak demand charges in the optimization algorithm.

For systems with time-of-day utility rates, a simple strategy for near-optimal

control was identified. The advantage of this near-optimal control strategy is that it is

easily implemented and gives operating costs similar to thatfor optimal control.

Both the optimal and near-optimal strategies should be adaptable to systems that

incorporate ice storage, however these concepts need to be tested. Systems that

incorporate storage in-line with the chilled water distribution system, rather than in

parallel should also be considered. It is also necessary to develop strategies for near-

optimal control of systems that utilize more than one day of storage.

In order to apply either the optimal or near-optimal strategy for controlling storage,

it is necessary to forecast the total cooling requirements. Pure time-series and

combined deterministic plus time-series models were fit to cooling load data for the

208

D/FW airport. Pure time-series models work wellfor making forecasts of one hour.

In order to make longer term predictions, it was necessary to include deterministic

components in the forecasting model. The results of 5-hour predictions with the

combined model were good. The resulting deterministic and time-series model is

simple enough so that its coefficients may befit with linear least-square methods.

More data is necessary to determine whether the model works well under all

circumstances. Methods for recursive parameter estimation should also be tested for

this forecasting model.

In determining the optimal control of systems with storage, both the load and

ambient wet bulb temperature were assumed to be knowns. Future work should

address the effect of inaccuracies in forecasts on the accuracy of the optimal control

solution.

8.3 Guidelines for Design and Control

Optimization techniques were applied to the control of chilled water systems. The

important uncontrolled variables that effect system performance and optimal control

variable settings were identfied as the total chilled water load and ambient wet bulb

temperature. Additional secondary uncontrolled variables that could be important if

varied over a wide range would be the individual zone latent to sensible load ratios and

the ratios of individual sensible zone loads to the total sensible loads for all zones.

Control guidelines that reduce the number of independent control variables and

simplify the optimization were also identified. These general results were utilized to

develop near-optimal control strategies for chilled water systems with and without

storage. The guidelines are also useful to plant engineers for improved control

practices and are summarized as follows.

209

1) Variable-Speed Tower Fans: Operate all cooling tower cell fans at identical fan

speeds.

2) Multi-Speed Tower Fans: Increment lowest cooling towerfans first when

adding tower capacity. Reverse for removing capacity.

3) Variable-Speed Pum:The sequencing of variable-speed punps should be

directly coupled to the sequencing of chillers, to give peak pwnp efficiencies

for each possible combination of operating chillers. Multiple variable-speed

pumps should be controlled to operate at equal fractions of their maximum

speed.

4) hill: Multiple chillers should have identical chilled water set temperatures


divided according to the chillers relative cooling capacities.


tenperatures.

6) Straified Thermal Storage: A near-optimal control strategy involves

maximizing operation of the system in the full discharge mode whenever the

utility rates are high and minimizing operation in the charge mode when the

utility rates are low. Otherwise, the system should operate in the direct mode,

if necessary, during times of lowest wet bulb temperature. This control

strategy should also be tested for systems with ice storage.

210

No general simplifications could be found for the optimal sequencing of chillers and

fixed-speed pumps. It is necessary to evaluate the overall system performance in order

to determine the optimal points for adding or removing chillers. In general, it is not

optimal to sequence fixed-speed pumps with chillers.

Additional results and conclusions concerning both control and design under

optimal control of chilled water systems are summarized as follows.

1) There is a strong incentive for the use of optimal or near-optimal control of

chilled water systems. Depending upon the load characteristics, fixed values

of chilled water and supply air setpoints and cooling tower approach and range

resulted in approximately 7 - 19% greater cooling season operating costs than

that for optimal control in Dallas and Miami.

2) In the determination of optimal control points, the humidty should be allowed

tofloatfreely, unless it falls outside the bounds of human comfort. In effect,

the addition of any constraint reduces the number of free control variables by

one and results in operation at a higher cost.

3) There are significant operational cost savings associated with the use of

variable-speed equipment for chilled water systems. Depending upon the load

characteristics, the cooling season operating costs were approximately 26- 43%

greater for all fixed-speed equipment as compared with all variable-speed in

Dallas and Miami. The most significant difference was attributed to the chiller.

4) The current practice of orienting multiple chillers in parallel is near-optimal and

211

should be continued. The performance of multiple chillers is enhanced by

orientation in series rather than parallel. However, the increase in pumping

power requirements for series chillers offsets the chiller improvements and the

overall performance for the two configurations is similar.

5) The choice of chiller refrigerant can have a significant effect upon operating

costs. For the D/FW chiller, the peak performance associated with the use of

R-500 shows on the order of 5 - 10% improvement over the orginal charge of

R-22. This particular chiller was derated from its original capacity due to

changes in the load requirements.

6) In the absence of time-varying utility rates, thermal storage provided little or no

operating cost savings as compared with no storage for the systems

considered. Part of the reason for this is due to the assumption of a constant

chilled water supply temperature during charging of storage and a constant load

return water temperature during discharge. Future work should be performed

to compare the performance of optimally controlled storage systems with the

charging and discharging temperatures as 'free" variables in the optimization.

Analysis of systems with poorer part-load characteristics could also change this

conclusion. In addition, systems that incorporate ice storage should be

compared with water storage systems both with and without time-of-day

electic rates.

212

Appendix ARefrigerant Property Data

Liquid Refrigerant Thermal Conductivity Correlations

The following correlations require temperature, T, in degrees Fahrenheit and give

thermal conductivity, kf, in Btu/hr-ft-F.

R-500: kf = 0.05300 - 0.000127 T

R-22: kf = 0.06298 - 0.000159 T

R-12: kf- 0.04902 - 0.000117 T

Lgi _ " " Viseosit Correlation

Refrigerant viscosities, A, are in ibm/ft-hr and temperatures, T, are degrees

Fahrenheit.

R-500: lf = 0.6851 - 3.2943e-03 T + 6.4430e-06 T2

R-22: f 0.6517 - 2.5943e-03 T + 5.610le-06T

R-12: If = 0.7630 - 3.8905e-03 T + 9.88lOe-06 T2

Refrigerant Sonic Velocity Data at 50 degrees Fahrenheit

Refrigerant a- fft/sec)R-500 490

R-22 534R-12 446

213

Appendix BMethod for Determining the Performance of

Partially Wet and Dry Cooling Coils

Water will begin to condense on the surface of a cooling coil at the point where

surface temperature equals the dewpoint of the entering air. The process of determining

the relative areas associated with the wet and dry portions of the coil is iterative. In

terms of the exit water temperature, the dry coil surface area and the flow stream

conditions at the point where condensation occurs are found by solving the flow stream

energy balance and rate equations for the dry section and setting the surface temperature

at the condensation point equal to the dewpoint point temperature, Tdp. The resulting

equation for the fraction of the total coil surface area that is dry is

1 (Tdp- Tw,) + C (Ta,-Tdp (B. 1)dry = ""K (1 - K/Ntuo)(Ta,i -'Tdp)

where,

*

K NtUdry (1 - C) (B.2)

The effectivenesses for the wet and dry portions of the coil, eCwet and ady are

8 a~wt - 1 - exp(- (1 - fdj,) Ntu weL( 1 - in))(B)

1 - in exp(- (1 - fdry)Ntuwet(1 - in))

214

1 - exp(-fdy NmrY(1 - C)) (B.4)8dry =1 - C exp(-f&YNtu &Y(1- C

The water temperature at the point where condensation begins is

Tw i + CCwet, a(hai- hs,w,,i)/Cpm - C wetaedy,aTa,iTwwxe tadrya a i(B.5)

(1 - C ewetaedrya)

and a new estimate of the exit water temp is

Tw,0 = C eyaTa,i + (1- C ecirya)Tw,x (B.6)

An excellent initial estimate of the outlet temperature is the larger of the temperatures

obtained from the completely wet and dry analyses (i.e. fry = 0 and fdy = 1). As

shown in Figures 4.2 and 4.3, the all wet or dry assumption gives results that are close

to those of the partially wet and dry analysis. As a result, the above iterative process

converges very quickly, typically within two iterations.

The outlet air state from the coil is determined from

Tao Ts~e + (Tax- Tse)exp(- (1 - fdr)Ntuo) (B.7)

where, the effective surface temperature in the wet coil section is the saturation

temperature corresponding to an effective enthalpy condition determined as

ha,o - ha,xh ~ =h hai +1 - exp(- (1 - fdy)Ntuo)

The air temperature and enthalpy at the point at which condensation occurs are

Ta,x = Ta,i c-dry,a(T a,i- Twx)

ha,x = ha,i-dryaCpm(Ta, i-wx)

215

(B.8)

(B.9)

(B.10)

216

Appendix CComponent Data for Base System of Chapter 6

I. Chiller Power Consumption (see Chapter 2)

Pch =a 0 + aX + a2X2+ a3Y + a4Y2+ a5 XYPdes

where,

X =Oe

o-des

S(T -.Thw)

ATdes

Parameter Variable-Speed. Fixed-Speed Lau Hackner

ao 0.07336 0.0516 0.1107 0.2642

a, -0.3259 1.2199 0.3198 0.0207

a2 0.5744 -0.2517 0.4662 0.2643

a3 -0.03888 -0.6448 -0.0956 -0.4460

a4 0.3321 0.8963 0.2152 0.3555

a5 0.3684 -0.3119 -0.0656 0.5176

Qdes 5421. 5421. 1250. 550.

AT des 46. 46. 50. 50.

Pdes 3580. 3627. 961. 453.

217

I. Cooling Tower Cells (see Chapter 3)

1 +n

where,

c =2.0

n = -0.63

Design Air Flow: 635,000 cfm

Design Fan Power. 100 kW

Sump Make-Up Water Temperature: 50 F

I. Cooling Coil Data (see Chapter 4)

Ntu i = klmW:::ek]

Ntuo = mk

3 a]k4

I ma.des]

where,

k, = 2.25

k3 = 1.70

k2 = -0.20

k4 = -0.38

218

Total Design Air Flow: 1,800,000 cfm

Total Design Fan Power. 1000 kW

IV. Pump Analysis (see Chapters 3 and 4)

Press=urer s

Component Desien Flow (epm) Desin .AP (psi)

Chiller Condenser 10,000 10

Chiller Evaporator 10,000 25

Cooling Tower Nozzle 5,000 5

Condenser Water Loop Static 30

Chilled Water Loop Static 20

Pump Pressure Rise

2

APp AP Ppmx1 -(mcwP pmax

2Appmax = yAP pmaxdes

mp,max = yp mp,maxdes

219

where,

APp,maxdes = 100 psi

mp,maxdes = 10,000,000ibm/hr

PmL_ Efficiency

1p-= a0 + a,1 -. 2+ a2

'p p.maxdes p pmaxdes

where,

ao = 0.0

a1 = 2.93

a2 = -2.64

220

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Programs for Modelling andOptimizing the Performance

of Central Chilled Water Systems

James Edward BraunSolar Energy Laboratory

University of Wisconsin-Madison1500 Johnson Drive

Madison, WI 53706-1687 USA

Programs for Modelling and Optimizing the

Performance of Central Chilled Water Systems

James Edward Braun

Solar Energy Laboratory

UNIVERSITY OF WISCONSIN- MADISON

1988

This document contains computer listings for algorithms that are described in the Ph.D.

thesis "Methodologies for the Design and Control of Central Cooling Plants",

by James E. Braun, University of Wisconsin - Madison, 1988

Section 1 Chiller Models

Section 2 Cooling Tower and Coil Models

Section 3 Component-Based Optimization Program

Section 4 Dynamic Optimization Program

Section 5 Utility Routines

Table of Contents

1.0

2.0

3.0

4.0

5.0

Section 1 Chiller Models

Chiller Performance 1.1This program computes the performance (e.g., motor power consumption

and compressor speed) of variable-speed centrifugal chillers given the flow

stream conditions and cooling load requirement as outlined in Chapter 2. In

addition to the routines listed in this section, this program utilizes two utility

subroutines (SOLVER and FREON) that are listed in Section 5.

Chiller Capacity 1.15This program computes the cooling capacity based upon the compressor

power input for variable-speed centrifugal chillers as outlined in Chapter 2.In addition to the routines listed in this section, this program utilizes two

utility subroutines (SOLVER and FREON) that are listed in Section 5.

Chiller Surge Point 1.28This program estimates the minimum speed and cooling load associatedwith the onset of surge for variable-speed centrifugal chillers as outlined in

Chapter 2 In addition to the routines listed in this section, this programutilizes two utility subroutines (SOLVER and FREON) that are listed in

Section 5.

-1.0-

C PROGRAM FOR CALCULATING THE PERFORMANCE OF A SINGLE OR DOUBLE-STAGE

C CENTRIFUGAL CHILLER. FOR A TWO-STAGE MACHINE, THERE IS ALSO AN OPTION FOR *

C A FLASH GAS ECONOMIZER. THE INPUTS ARE FLOW STREAM CONDITIONS AND COOLING *

C LOAD REQUI.REMENTS, WHILE OUTPUTS INCLUDE POWER CONSUMPTION, COMPRESSOR

C SPEED, ETC. THE UNITS UTILIZED WITHIN THIS PROGRAM ARE TYPICAL ENGLISH

C UNITS THAT ARE COMMON WITHIN THE HVAC INDUSTRY (BTU'S, TONS, KW). THE

C ANALYSIS IS DESCRIBED IN CHAPTER 2 OF "METHODOLOGIES FOR DESIGN AND CONTROL*

C OF CENTRAL COOLING PLANTS", J.E. BRAUN, PH.D THESIS, UNIVERSITY OF

C WISCONSIN - MADISON, 1988 *

C

C

EXTERNAL EVAPCHILLER

DOUBLE PRECISION X(5),F(5),W(5,5),DIFF(5),TOL

COMMON /DATA/ PAR(10),XIN(7),OUT(19),ISTAGE, IECON, IREF

DATA TOL/I.E-05/,DIFF/5*0.001/,PI/3.1415/

C

PARAMETER(IREF=500, ISTAGE=2, IECON=2)

PARAMETER(CPW=I., DENS=8.3333)

PARAMETER(AEI=11300., ACI=14800., REO=3.1, RCO=3.1, DIAE=.75/12.,

DIAC=.75/12., NEPASS=3, NETUBE=3560, NCPASS=1, NCTUBE=3349,

C1=103., C2=1.1, NVERT=1, RPE=2.E-04, RPC=2.E-04, AX=1.53,

BETA=27.2, RIMP=1.1667, EFFREF=0.814, EFFMOT=0.91)

C

C PROGRAM DATA DEFINED WITHIN THE ABOVE PARAMETER STATEMENTS:

C

C IREF = REFRIGERANT TYPE (12, 22, 500, ETC)

C ISTAGE = NUMBER OF COMPRESSION STAGES (1 OR 2)

C IECON = 1 FOR NO ECONOMIZER, 2 FOR ECONOMIZER

C CPW = SPECIFIC HEAT OF WATER (BTU/LBM-F)

C DENS = DENSITY OF WATER (LBM/GAL)

C AEI = EFFECTIVE INSIDE SURFACE AREA OF EVAPORATOR TUBES (FT**2)

C ACI = EFFECTIVE INSIDE SURFACE AREA OF CONDENSOR TUBES (FT**2)

C REO = RATIO OF EFFECTIVE OUTSIDE EVAPORATOR TUBE SURFACE AREA

C INCLUDING FINS AND FIN EFFICIENCY TO INSIDE AREA

C RCO = RATIO OF EFFECTIVE OUTSIDE CONDENSER TUBE SURFACE AREA

-1.1-

(Chillpr Pprf rmn~nepcnlbf;tin I

Q~rftin1 ihrP~frruae L~ I

INCLUDING FINS AND FIN EFFICIENCY TO INSIDE AREA

C DIAE = EVAPORATOR TUBE DIAMETER (FEET)

C DIAC = CONDENSER TUBE DIAMETER (FEET)

C NETUBE = NUMBER OF EVAPORATOR TUBES

C NEPASS = NUMBER OF EVAPORATOR TUBE PASSES

C NCTUBE = NUMBER OF CONDENSER TUBES

C NCPASS = NUMBER OF CONDENSER TUBE PASSES

C CIC2 = EMPIRICAL CONSTANTS FOR BOILING HEAT TRANSFER

C COEFFICIENT

C NVERT = NUMBER CONDENSER TUBES IN A VERTICAL LINE

C RPERPC = EVAPORATOR AND CONDENSER TUBE WALL RESISTANCE

C (FT**2-F/BTU)

C AX = EXIT AREA OF COMPRESSOR IMPELLER (FT**2)

C BETA = IMPELLER BLADE ANGLE (DEGREES)

C RIMP = COMPRESSOR IMPELLER RADIUS (FEET)

C EFFREF = REFERENCE POLYTROPIC EFFICIENCY

C EFFMOT = COMPRESSOR DRIVE MOTOR EFFICIENCY

C

C THE VALUES DEFINED ABOVE ARE REPRESENTATIVE OF THE PRIMARY CHILLER AT

C THE DALLAS/FORT WORTH AIRPORT.

C

C ** OPEN FILES FOR DATA INPUT AND OUTPUT **

C

OPEN (10, FILE='CHILLER.DAT' , STATUS='OLD')

OPEN(11,FILE='CHILLER.OUT' , STATUS='NEW')

WRITE (11, 101)

C

C ** LOOP FOR VARYING INPUT DATA **

C

10 READ (10, *,END=100) QEVAPGPMCHWGPMCWTCHWSTCWR

C

C INPUT DATA IS:

C QEVAP = CHILLER EVAPORATOR LOAD (TONS)

C GPMCHW = CHILLED WATER FLOW RATE (GAL/MIN)

C GPMCW = CONDENSOR WATER FLOW RATE (GAL/MIN)

C TCHWS = CHILLED WATER SETPOINT (DEGREES F)

C TCWR =CONDENSOR WATER RETURN (DEGREES F)

C

- 1.2-

a a a a a%-&-& %-a a %-V a a a a La a l%_lw_t IV I I LChiller Perfnrmanep

I

Section 1 Chiller Performance

C ** PRELIMINARIES **

C

C CALCULATE THE MASS FLOW RATE FROM THE GPM AND THE LOAD RETURN

C TEMPERATURE FROM THE CHILLER LOADING

C FLCHW = CHILLED WATER FLOW RATE (LBM/HR)

C FLCW = CONDENSER WATER FLOW RATE(LBM/HR)

C TCHWR = CHILLED WATER RETURN TEMPERATURE (DEGREES F)

C

FLCHW=60. *DENS*GPMCHW

FLCW=60. *DENS*GPMCW

TCHWR=TCHWS+12000 . *QEVAP/FLCHW/CPW

C

C ** EVAPORATOR ANALYSIS **

C

C GIVEN THE CHILLER LOAD AND WATER CONDITIONS, THE EQUATION SOLVER

C ITERATIVELY DETERMINES THE EVAPORATOR TEMPERATURE, TE, THROUGH THE

C USE OF THE SUBROUTINE EVAP. PARAMETERS, INPUTS, AND OUTPUTS ARE

C COMMUNICATED THROUGH THE COMMON BLOCK "DATA". HEI IS THE INSIDE

C EVAPORATOR HEAT TRANSFER COEFFICIENT (TURBULENT FLOW).

C

PAR(1) =AEI

PAR(2) =REO*AEI

HEI=1 .37* (NEPASS*GPMCHW/NETUBE) **0.8*DIAE**-1 .8

PAR(3) =HEI

PAR(4) =RPE

PAR (5) =CI

PAR(6)=C2

XIN (1) =FLCHW

XIN (2) =TCHWS

XIN (3) =TCHWR

XIN (4) =12000. *QEVAP

TE=TCHWS-5.

X ( 1) =TE

CALL SOLVER (I,DIFF, X,F, W,EVAP, TOL)

TE=X (1) )

C

C ** COMPRESSOR AND CONDENSOR ANALYSIS **

C

C THE EQUATION SOLVER SOLVES THE SYSTEM OF EQUATIONS MODELING THE

- 1.3 -

Crj%19tn I3 l94:1|ull 1N'uIta r i I 1 IVJI Iialu

C THE COMPRESSOR AND CONDENSER BY CALLING SUBROUTINE CHILLER. EITHER

C ONE OR TWO-STAGE COMPRESSION IS POSSIBLE. FOR SINGLE-STAGE

C COMPRESSION, THERE ARE THREE ITERATIVE VARIABLES: THE REFRIGERANT

C CONDENSING TEMPERATURE, THE COMPRESSOR DIMENSIONLESS FLOW COEFFICIENT,

C AND THE COMPRESSOR IMPELLER TIP SPEED. FOR TWO-STAGE COMPRESSION TWO

C ADDITIONAL UNKNOWNS ARE: THE SECOND STAGE DIMENSIONLESS FLOW COEFFI-

C CIENT AND THE SECOND STAGE IMPELLER INLET PRESSURE. THE INLET STATE

C TO THE COMPRESSOR IS KNOWN FROM THE EVAPORATOR ANALYSIS. THE FREON

C PROPERTIES AT THIS STATE ARE EVALUATED PRIOR TO THE ITERATIVE

C SOLUTION. PARAMETERS, INPUTS, AND OUTPUTS ARE COMMUNICATED THROUGH

C THE COMMON BLOCK "DATA".

C

C SET-UP PARAMETERS FOR ANALYSIS:

C C3 IS A CONSTANT FOR EVALUATING THE HEAT TRANSFER COEFFICIENT FOR

C CONDENSATION ON THE CONDENSER TUBES.

C

PAR(1) =ACI

PAR (2) =RCO*ACI

HCI=1.788*(NCPASS*GPMCW/NCTUBE) **0.8*DIAC**-I.8

PAR (3) =HCI

PAR(4) =RPC

C3=0.725*(32.2*3600.**2/NVERT/DIAC)**0.25

PAR(5) =C3

PAR (6) =AX

PAR(7) =0.

IF(ABS(BETA-90.).GT.1.E-06) PAR(7)=./TAN(PI*BETA/180.)

PAR (8) =EFFREF

PAR(9) =EFFMOT

PAR(10)=RIMP

C

C EVALUATE PROPERTIES AT COMPRESSOR INLET FROM RESULTS OF EVAPORATOR

C ANALYSIS

C

CALL FREON(TE,PI,HS1,1I.,V1,UIREF,15)

OUT ( 8) =P 1

C

C INPUTS AND INITIAL GUESSES FOR ITERATION VARIABLES

C

- 1.4 -

Chillarr r,m .... I

C7~.LIII L A I I .a I

XIN (1) =FLCW

XIN (2) =TCWR

XIN (3) =12000. *QEVAP

XIN(4) =P1

XIN(5) =VI

XIN(6)=HI

XIN(7)=Sl

C

X (1) =TCWR+10.

X(2) =0.25

X(3)=350.

C

C EQUATION SOLVER: NUMBER OF VARIABLES DEPENDS UPON WHETHER SINGLE OR

C TWO-STAGE COMPRESSION.

C

IF(ISTAGE.EQ.1) THEN

CALL SOLVER(3,DIFF, X,F,W, CHILLER, TOL)

ELSE

OUT (18) =OUT (15)

X(4)=0.25

CALL FREON(TCWR+10.,P2,H3,S3,0.,V3,U3, IREF, 15)

X(5)=(PI+P2) /2.

CALL SOLVER (5, DIFF, X, F, W, CHILLER, TOL)

ENDIF

C

C ** OUTPUT DETAILED RESULTS AND COMPARISONS TO FILE **

C

C COMPUTE THE CONDENSOR WATER SUPPLY TEMPERATURE (TCWS)

C

TCWS=TCWR-OUT (2) *12000 ./FLCW/CPW

WRITE(11,102) GPMCHW, GPMCW, TCHWSTCHWR, TCWSTCWR

IF(ISTAGE.EQ.1) THENWRITE(11,103) (OUT (I) ,1I=,16)

ELSE

WRITE (11,104) (OUT (I) ,I=l, 19)

END IF

GO TO 10

100 CONTINUE

STOP

- 1.5 -

C hiller Pprfnrm _nepCait;nn I -

Qao~nnI1!.t,.tlj)ll I , a l £ l IV Illilll,

C

101 FORMAT(///10X,'********** CHILLER MODEL **********')

102 FORMAT(//2X, '****************************** INPUT DATA *****'

'*************************',//2X,'FLCHW, FLCW ='

2(1XF8.2)/2X,'TCHWS, TCHWR, TCWS, TCWR = '4(1XF6.2))

103 FORMAT(//2X,'** ENERGY QUANTITIES **',//2X,'QEVAP = ',F8.2/2X,

'QCOND = 'F8.2/2X,'QCOMP-= ',F8.2/2X, 'ELEC - 'F8.2

* //2X,'** DETAILED RESULTS **'l,//2X,

'TE, TC, T2, P1, P2, RPM = ',6(IX,F7.2)/2X,

'COP, PHIX, NPOL, EFF = ',4(IX,F6.3)/2X,

'UAE, UAC = ',2(IX,1PE11.3))

104 FORMAT(//2X,'** ENERGY QUANTITIES **',//2X,'QEVAP = ',F8.2/2X,

'QCOND = ',F8.2/2X,'QCOMP - 'F8.2/2X, 'ELEC = ',F8.2

. //2X,'** DETAILED RESULTS **',//2X,

'TE, TC, T2, P1, P2, UX = ',6(IXF7.2)/2X,

'COP, PHIXI, NPOL1, EFF1 = ',4(IXF6.3)/2X,

'PHIX2, NPOL2, EFF2 -- '3(1XF6.3)/2X,

'UAE, UAC = ',2(1X,1PE11.3))

END

C

SUBROUTINE EVAP (NEQ, X, F)

C

C *

C THIS SUBROUTINE MODELS THE PERFORMANCE OF AN EVAPORATOR. THE EQUATION *

C SOLVER THAT CALLS THIS ROUTINE IS SEARCHING FOR THE EVAPORATOR TEMPERATURE *

C (TE) THAT YIELDS NO DIFFERENCE BETWEEN THE CHILLER LOAD CALCULATED WITH A *

C UA-LOG MEAN TEMPERATURE DIFFERENCE AND THE ENTHALPY DIFFERENCE OF THE *

C CHILLED WATER STREAM. *

C *

DOUBLE PRECISION X,F


AEI=PAR ( 1)

AEO=PAR (2 )

HEI=PAR (3)

-1.6 -

C~hillpr Pp rfnr .....

.,,VUI -- -hir P .fnrminc.

RPE=PAR (4)

C1=PAR (5)

C2=PAR (6)

TCHWS=XIN (2)

TCHWR=XIN (3)

QEVAP=XIN (4)

TE=X

UAI IS THE CONDUCTANCE ASSOCIATED WITH

MATERIAL (PLUS FOULING FACTOR). UAE IS

CONDUCTANCE, INCLUDING THAT ASSOCIATED

HEAT TRANSFER COEFFICIENT (HEO). TFILM

THE WATER FLOW AND PIPE

THE OVERALL EVAPORATOR

WITH THE REFRIGERANT BOILING

IS THE REFRIGERANT FILM

TEMPERATURE AT THE SURFACE OF THE TUBE.

UAI=AEI/ (1. /HEI+RPE)

TW= (TCHWS+TCHWR) /2.

TFILM=TW-QEVAP/UAI

DT=AMAXl (TFILM-TE, 0 .1)

HEO=C* (DT) **C2

UAE=1./ (1./UAI+1./HEO/AEO)

IF(TE.LT.TCHWS) THEN

XLMTD= ( (TCHWR-TE) - (TCHWS-TE) ) /ALOG ( (TCHWR-TE) / (TCHWS-TE))

ELSE

XLMTD=TW-TE

ENDIF

** FUNCTION FOR EQUATION SOLVER **

F=QEVAP -UAE *XLMTD

** OUTPUTS **

OUT (5) =TE

OUT (15) =UAE

RETURN

END

-1.7-

C

C

C

C

C

C

CC

C

C

C

C

Chiller Perfnrminep I.,p,-titin I

C'~L!... 17*.ectioni U i- pu I-p I al I'J

SUBROUTINE CHILLER (NEQ, X, F)

C

c*C *

C THIS SUBROUTINE IS CALLED AT EACH ITERATION OF THE EQUATION SOLVER TO *

C EVALUATE THE COMPRESSOR AND CONDENSER PERFORMANCE. THROUGHOUT THE PROGRAM *

C T, P, V, H, AND S REFER TO REFRIGERANT TEMPERATURE, PRESSURE, SPECIFIC, *

C VOLUME, SPECIFIC ENTHALPY, AND SPECIFIC ENTROPY FOR A GIVEN STATE. *

C STATES ARE: *

C

C STATE 1: ENTERING COMPRESSOR FIRST STAGE

C S.TATE Xl: EXIT FROM FIRST STAGE COMPRESSOR IMPELLER

C STATE 1A: EXIT FROM FIRST STAGE COMPRESSOR DIFFUSER

C STATE 2A: ENTERING COMPRESSOR SECOND STAGE

C STATE X2: EXIT FROM SECOND. STAGE COMPRESSOR IMPELLER

C STATE 2: EXIT FROM SECOND STAGE COMPRESSOR DIFFUSER

C STATE 3: EXIT FROM CONDENSER

C STATE 3A: ENTERING ECONOMIZER AFTER EXPANSION

C STATE 3L: LIQUID STATE WITHIN ECONOMIZER

C STATE 3V: VAPOR STATE WITHIN ECONOMIZER

C STATE 4: ENTERING EVAPORATOR

C

C

REAL M0,MU1,MU2,NEXP1,NEXP2

DOUBLE PRECISION X(NEQ) ,F (NEQ)

COMMON /DATA/ PAR(10),XIN(7),OUT(19),ISTAGEIECONIREF

DATA CPW/l./,TMIN/-150./,TMAX/220./,PI/3.1415/

C

C ** PARAMETERS AND INPUTS **

C

ACI=PAR(l)

ACO=PAR (2)

HCI=PAR(3)

RPC=PAR (4)

C3=PAR (5 )

AX=PAR (6)

CBETA=PAR (7)

- 1.8 -

Ci~hllr ar fnr.......v

owi.tli f n 1 x IlIVII'LuMI I I -./ IIUlaItII.

EFFREF=PAR (8)

EFFMOT=PAR (9)

RIMP=PAR (10)

c

FLCW=XIN(1)

TCWR=XIN (2)

QEVAP=XIN (3)

P1=XIN(4)

VI=XIN (5)

H1=XIN (6)

SI=XIN (7)

C

C ** ITERATION VARIABLES **

C

C THE THREE UNKNOWNS FOR SINGLE-STAGE COMPRESSION ARE:

C TCND = THE CONDENSING REFRIGERANT TEMPERATURE

C PHIXi = THE COMPRESSOR DIMENSIONLESS FLOW COEFFICIENT

C UX = THE IMPELLER TIP SPEED

C FOR TWO-STAGE COMPRESSION, TWO ADDITIONAL UNKNOWNS ARE:

C PHIX2 = SECOND STAGE DIMENSIONLESS FLOW COEFFICIENT

C PlA - INLET PRESSURE TO THE SECOND STAGE.

C

TCND=X (1)

PHIX1=X (2)

UX=X(3)

TCND=AMAX1 (TMIN, AMINI (TCND, TMAX))

PHIX1=AMAX1 (0.,AMIN1 (PHIXi, 1.0))

UX=AMAX1 (UX, 1.)


PHIX2=X(4)

PHIX2=AMAX1 (0. , AMIN1 (PHIX2, 1. 0))

PIA=X(5)

ENDIF

C

C ** EXPANSION VALVE AND ECONOMIZER **

C

C DETERMINE PROPERTIES FOR EXPANSION VALVE(S) AND ECONOMIZER, IF

C PRESENT. ALSO DETERMINE REFRIGERANT FLOW RATES FOR EVAPORATOR (FLi)

- 1.9 -

.qpvtinn II Chillpr Parfnrinnn"i-ja

~.qtin 1

C AND CONDENSER (FL2).

C

CALL FREON(TCNDP2,H3,S3,O.,V3,U3,IREF,15)

IF(ISTAGE.EQ.2 .AND. IECON.EQ.2) THEN

H3A=H3

CALL FREON(T3A, P1A, H3A, S3A, X3A, V3A, UDUM, IREF, 23)

CALL FREON(T3A,1PIA, H3L, S3L,0.,V3L, UDUM, IREF, 25)

CALL FREON(T3AP1A,H3V, S3V, 1.,V3VtUDUM, IREF, 25)

H4=H3L

FL1=QEVAP/ (HI-H4)

FL2=FL1/(1.-X3A)

ELSE

H4=H3

FL1=QEVAP/ (HI-H4)

FL2=FL1

ENDIF

C

C ** COMPRESSOR STAGE(S) **

C

C DETERMINE INLET AND OUTLET CONDITIONS OF IMPELLERS AND DIFFUSERS

C FOR BOTH STAGES.

C

CALL PROP (IREF, TCNDAOCONDVISC)

MO=UX/AO

EFFN1=(I.+0.109*(1.1-MO))*(i.-EXP(PHIX1*(58.45*PHIX1**2

-5.99*PHIXI-18.81)))

EFF1=EFFREF*EFFN1

MU1=0.69* (1. -PHIX1*CBETA) /EFF1

HX1=H1+(MU-0.5*MU1*MU-0.5*PHIX1*PHIX1) *UX*UX/32•2/778.

HX1=AMAX1 (HX1,H1)

CALL FREON(TX1,PX1,HX1,S1,XX1,VX1,UDUM, IREF, 34)

IF(ISTAGE.EQ.1) THENH2=HI +MUI*UX*UX/32 .2 /778.

ELSE

EFFN2= (I.+0. 109"*(I. -M0) ) *(I.-EXP (PHIX2* (58.45*PHIX2**2

• -5.99*PHIX2-.18.81)) )

EFF2 =EFFREF *EFFN2

MU2=0 .69* (1.-PHIX2*CBETA) /EFF2

HIA=HI1+MUI1*UX*UX/32 .2/778.

- 1.10-

irh illpr 1P,,...vP .......

C'i _..... INecuion ii -"~

CALL FREON(T1A, P1A, HIA, SIA, X1A, V1A, UDUM, IREF, 23)

IF(IECON.EQ.2) THEN

H2A=X3A*H3V+(1.-X3A) *H1A

CALL FREON (T2A, PlA, H2A, S2A, X2A, V2A, UDUM, IREF, 23)

ELSE

T2A=TlA

H2A=HIA

S2A=SlA

V2A=V1A

ENDIF

HX2=H2A+(MU2-0.5*MU2*MU2-0.5*PHIX2*PHIX2) *UX*UX/32.2/778.

HX2=AMAX1 (HX2, H2A)

CALL FREON(TX2,PX2,HX2,S2A, XX2,VX2,UDUM, IREF, 34)

H2=H2A+MU2*UX*UX/32.2/778.

ENDIF

CALL FREON(T2,P2,H2,S2,X2,V2,UDUM, IREF, 23)

C

C EVALUATE POLYTROPIC COEFFICIENTS FOR EACH STAGE

C


NEXP 1=ALOG (P1/P2) /ALOG (V2/VI)

XPON1= (NEXPi-1.) /NEXP1

ELSE

NEXP1=ALOG (P1/PlA) /ALOG (VIA/V1)

NEXP2=ALOG (PIA/P2) /ALOG (V2/V2A)

XPON1= (NEXPI-1.) /NEXP1

XPON2= (NEXP2-1.) /NEXP2

END IF

C

C ** CONDENSER **

C

C QCOND = CONDENSER HEAT REJECTION

C UAI = CONDUCTANCE ASSOCIATED WITH THE WATER FLOW AND PIPE

C MATERIAL (PLUS FOULING FACTOR),

C TFILM = CONDENSING FILM TEMPERATURE AT THE TUBE SURFACES,

C UAC = OVERALL CONDUCTANCE INCLUDING THAT ASSOCIATED WITH THE

C CONDENSING HEAT TRANSFER COEFFICIENT, HCO.

C

- 1.11 -

C~h~ pIIlaeD-.g %......a

01 .:,W)CcntIu tI I .- .uu i 1 ,l lla lllu "

CALL FREON(TCNDP2,H2V, S2V, 1.,V2V,UDUM, IREF, 15)

QCOND=FL2 * (H2-H3)

TCWS=TCWR-QCOND / FLCW/CPW

UAI=ACI/ (I./HCI+RPC)

TW= (TCWS+TCWR)/2.

TF I LM=TW+QCOND /UAI

DT=AMAX1 (TCND-TFILM, 0 .1)

DH=H2V-H3

HCO=C3*(COND**3*DH/VISC/DT/V3**2) **0.25

UAC=1./ (I./UAI+1./HCO/ACO)

IF (TCND. GT. TCWR) THEN

XLMTD=( (TCND-TCWS)- (TCND-TCWR) ) /ALOG ( (TCND-TCWS) / (TCND-TCWR))

ELSE

XLMTD=TCND-TW

END IF

C

C ** FUNCTIONS FOR EQUATION SOLVER **

C

C THE EQUATION SOLVER ITERATIVELY SOLVES FOR THE INDEPENDENT VARIABLES

C IN THE VECTOR X THAT YIELD ZEROS OF THE FOLLOWING FUNCTIONS.

C

F (1) =FLCW*CPW* (TCWR-TCWS) -UAC*XLMTD

F (2) =X (2) -FL1*VX1/AX/UX/3600.


F (3) =H2- (144./778. *PI*V1/EFF1/XPON1* ((P2/PI) **XPON1-. ) +H1)

ELSE

F (3) =HIA- (144./778. *P1*Vl/EFF1/XPON1* ((PlA/PI) **XPON1-. ) +HI)

F (4) =X (4) -FL2*VX2/AX/UX/3600.

F (5)=H2- (144./778. *PIA*V2A/EFF2/XPON2* ((P2/PlA) **XPON2-1. ) +H2A)

ENDIF

C

C ** OUTPUTS **

C

OUT (1) =QEVAP/12000.

OUT ( 2) =QCOND /12000 .

OUT ( 3) = (QCOND-QEVAP) / 12000.

OUT (4) =3 .515 *OUT (3) /EFFMOT

OUT ( 6) =TCND

OUT ( 7) =T2

-1.12 -

('h;llpr Pprfnrrnnn,,n a

-1.13-

Chiller PerformnnepQa,-tinn IvI I,|l;~n 1 .h..l.r P '1'rIfI irn lllnc%.

OUT (9) =P2

OUT (10) =60. *UX/RIMP/2./Pi

OUT (11) =EFFMOT*OUT (1)/OUT (3)

OUT (12) =PHIX1

OUT (13) =NEXP1

OUT (14) =EFF1

OUT (16) =UAC


OUT (15) =PHIX2

OUT (16) =NEXP2

OUT (17) =EFF2

OUT(19)=UAC

ENDIF

RETURN

END

C

SUBROUTINE PROP (IREF,T,A0,COND,VISC)

C

C *

C THIS SUBROUTINE DETERMINES PHYSICAL PROPERTIES FOR REFRIGERANTS R500, R22, *

C OR R12. THE ARGUMENTS ARE: *

C *

C IREF = REFRIGERANT TYPE (500, 22, OR 12) *

C T = REFRIGERANT TEMPERATURE (DEGREES F) *

C AO = SONIC VELOCITY (FT/SEC) *

C COND = THERMAL CONDUCTIVITY (BTU/HR-FT-F) *

C VISC = VISCOSITY (LBM/FT-HR) *

C *

C

IF(IREF.EQ.500) THENA0-490.

COND=0 .053-0. 000127"T

VISC=0 .6851-3. 2943E-03*T+6. 443E-06*T*T

END IF

IF(IREF.EQ.22) THEN

AO=534.

COND=0.063-0-000159*T

VISC=0.6507-2.5943E-03*T+5.6101E-06*T*T

ENDIF

3-ectionQ ^ 1^ 61 : dA rib I Chillpr Pprfhrm-an,,n b

cQae4;en I -aJcx.L5'JU I -I I I%-" " -dILI

C

C A PROGRAM FOR CALCULATING THE COOLING CAPACITY BASED UPON A SPECFIED *

C POWER INPUT REQUIREMENT FOR A SINGLE OR DOUBLE-STAGE CENTRIFUGAL CHILLER. *

C FOR A TWO-STAGE MACHINE, THERE IS ALSO AN OPTION FOR A FLASH GAS *

C ECONOMIZER. THE INPUTS ARE FLOW STREAM CONDITIONS AND POWER REQUIREMENT,

C WHILE OUTPUTS INCLUDE COOLING CAPACITY, COMPRESSOR SPEED, ETC. THE UNITS *

C UTILIZED WITHIN THIS PROGRAM ARE TYPICAL ENGLISH UNITS THAT ARE COMMON *

C WITHIN THE HVAC INDUSTRY (BTU'S, TONS, KW). THE ANALYSIS IS DESCRIBED IN *

C CHAPTER 2 OF "METHODOLOGIES FOR DESIGN AND CONTROL OF CENTRAL COOLING *

C PLANTS", J.E. BRAUN, PH.D THESIS, UNIVERSITY OF WISCONSIN - MADISON, 1988 *

C *

C

EXTERNAL CHILLER



DATA TOL/0.001/,DIFF/6*0.01/,PI/3.1415/

C

PARAMETER(CPW=1., DENS=8.3333)


DIAC=.75/12., NEPASS=3, NETUBE=3560, NCPASS=I, NCTUBE=3349,

C1=103., C2=1.1, NVERT=I, RPE=2.E-04, RPC=2.E-04, AX=1.53,


C

IREF = 500

ISTAGE = 2

IECON = 2

C

C PROGRAM DATA DEFINED ABOVE:

C








-1.15-

('hillpr C-anni,;tv I

cat ' 1nn




c INCLUDING FINS AND FIN EFFICIENCY TO INSIDE AREA







C CIC2 = EMPIRICAL CONSTANTS FOR BOILING HEAT TRANSFER

C COEFFICIENT


C RPE,RPC = EVAPORATOR AND CONDENSER TUBE WALL RESISTANCE

C (FT**2-F/BTU)

C AX = EXIT AREA OF COMPRESSOR IMPELLER (FT**2)C BETA - IMPELLER BLADE ANGLE (DEGREES)




C



C


C

OPEN (10, FILE= 'CAPACITY. DAT', STATUS= 'OLD')

OPEN(11,FILE='CAPACITY.OUT',STATUS='=NEW')

WRITE(11,101)

C


C

10 READ (10, ',END=100) POWER, GPMCHW, GPMCW, TCHWS, TCWR

c

c

C INPUT DATA IS:

C POWER = COMPRESSOR MOTOR POWER INPUT (KW)


- 1.16-

Chillpr Cnnq.v;tv

CA. Xt: -N" 1,I,3e 1 i o I - ' - .u u u ' . l ' , . l . ,

C GPMCW = CONDENSC -TER FLOW RATE (GAL/MIN)

C TCHWS = CHILLED :-ER SETPOINT (DEGREES F)

C TCWR = CONDENSOR WATER RETURN (DEGREES F)

C


C

C CALCULATE THE MASS FLOW RATE FROM THE GPM

C FLCHW - CHILLED WATER FLOW RATE (LBM/HR)


C

FLCHW=60. *DENS*GPMCHW

FLCW=60 . *DENS*GPMCW

C

C

C ** CHILLER ANALYSIS **

C


C THE CHILLER BY CALLING SUBROUTINE CHILLER. EITHER ONE OR TWO-STAGE

C COMPRESSION IS POSSIBLE. FOR SINGLE-STAGE COMPRESSION, THERE ARE FOUR

C ITERATIVE VARIABLES: THE REFRIGERANT EVAPORATOR TEMPERATURE, REFRIGERANT




C CIENT AND THE SECOND STAGE IMPELLER INLET PRESSURE. PARAMETERS, INPUTS,

C AND OUTPUTS ARE COMMUNICATED THROUGH THE COMMON BLOCK "DATA".

C


C HEI = INSIDE EVAPORATOR HEAT TRANSFER COEFFICIENT (TURBULENT FLOW)

C HCI = INSIDE CONDENSER HEAT TRANSFER COEFFICIENT (TURBULENT FLOW)

C C3 = CONSTANT FOR EVALUATING THE HEAT TRANSFER COEFFICIENT FOR


C

PAR(1) =AEI

PAR(2) =REO*AEIHEI=. *37* (NEPASS*GPMCHW/NETUBE) **Q0.8*DIAE**-I. . 8

PAR (3) =HEI

PAR (4) =RPE

PAR (5 ) =CI.

PAR( 6 )=C2

- 1.17 -

C'hillar 1 n . . :,x

IReton 1

PAR (7)=ACI

PAR (8) =RCO*ACI

HCI=1.788* (NCPASS*GPMCW/NCTUBE) **0.8*DIAC**-1 .8

PAR(9) =HCI

PAR(10) =RPC

C3=0 .725* (32.2*3600. **2/NVERT/DIAC) **0.25

PAR(11) =C3

PAR(12) =AX

PAR(13) =0.

IF(ABS(BETA-90.) .GT.1.E-06) PAR(13)=1./TAN(PI*BETA/180.)

PAR (14) =EFFREF

PAR(15)=EFFMOT

PAR(16) =RIMP

c


C

XIN(1) =FLCHW

XIN (2) =TCHWS

XIN(3) =FLCW

XIN (4) =TCWR

XIN (5) =12000. *EFFMOT*POWER/3.515

c

TE=TCHWS-5.

TCND=TCWR+ 10.

X(1) =TE

X (2) =TCND

X(3)=0.15

X(4)=400.

c



cIF(ISTAGE.EQ.1) THEN

CALL SOLVER(4,DIFF, X,F,W, CHILLER, TOL)

ELSE

X(5) =0.15

CALL FREON(TE,PI,HI,S1,1I.,V1,UI, IREF, 15)

CALL FREON(TCND,P2,H3,S3,0.,V3,U3, IREF, 15)

-1.18-

Chiller C nacitv

.... ,..., k1

-1.19-

rh;1lar

X(6)=(PI+P2)/2.

CALL SOLVER (6, DIFF, X,F, W,CHILLER, TOL)

ENDIF

C

c ** OUTPUT DETAILED RESULTS AND COMPARISONS TO FILE **

C

C COMPUTE THE CHILLED WATER RETURN TEMPERATURE (TCHWR) AND CONDENSOR

C WATER SUPPLY TEMPERATURE (TCWS)

C

TCHWR=TCHWS+OUT (1) *12000./FLCHW/CPW


WRITE (11, 102) GPMCHW, GPMCW, TCHWS, TCHWR, TCWS, TCWR


WRITE(11,103) (OUT(I),I=l,16)

ELSE

WRITE(II,104) (OUT(I),I=1,19)

ENDIF

GO TO 10

100 CONTINUE

STOP

C

101 FORMAT(///21X,'********** CHILLER MODEL **********')

102 FORMAT(//2X, '****************************** INPUT DATA *****'

•*************************I//2Xf'FLCHW, FLCW =

2(1xF8.2)/2X,'TCHWS, TCHWR, TCWS, TCWR = ',4(IXF6.2))

103 FORMAT(//2X,'** ENERGY QUANTITIES **I,//2X,'QEVAP= ',F8.2/2X,

'QCOND = 'F8.2/2X,'QCOMP-= 'fF8.2/2X,'ELEC = ',F8.2

1 I/2X,'** DETAILED RESULTS **',/12X,

e 'TE, TC, T2, P1, P2, RPM = ',6(1XF7.2)/2X,

'COP, PHIX, NPOL, EFF = ',4(IXF6.3)/2X,

'UAE, UAC = '2(1X,1PE11.3))

104 FORMAT(//2X,'** ENERGY QUANTITIES **I//2X,'QEVAP = ',F8.2/2Xf

* 'QCOND = ',F8.2/2X,'QCOMP = ',F8.2/2X,'ELEC = ',F8.2

* //2X, '** DETAILED RESULTS **',//2X,

* 'TE, TC, T2, P1, P2, RPM = ',6(IX, F7.2)/2X,

* 'COP, PHIXI, NPOL1, EFFI = ',4(IX, F6.3)/2X,

'PHIX2, NPOL2, EFF2 = ',3(1XF6.3)/2X,

* 'UAE, UAC = ',2(IXIPEII 3))

END

lqptiln I

SUBROUTINE CHILLER (NEQ, X, F)

C

C

C THIS SUBROUTINE IS CALLED AT EACH ITERATION OF THE EQUATION SOLVER TO

C EVALUATE THE CHILLER PERFORMANCE. THROUGHOUT THE PROGRAM T, P, V, H, AND

C REFER TO REFRIGERANT TEMPERATURE, PRESSURE, SPECIFIC, VOLUME, SPECIFIC

C ENTHALPY, AND SPECIFIC ENTROPY FOR A GIVEN STATE. STATES ARE:

C

C STATE 1: ENTERING COMPRESSOR FIRST STAGE

STATE

STATE

STATE

S

Xl: EXIT FROM FIRST STAGE COMPRESSOR IMPELLER

1A: EXIT FROM FIRST STAGE COMPRESSOR DIFFUSER

2A: ENTERING COMPRESSOR SECOND STAGE

STATE X2: EXIT FROM SECOND STAGE COMPRESSOR IMPELLER

STATE 2: EXIT FROM SECOND STAGE COMPRESSOR DIFFUSER

STATE 3: EXIT FROM CONDENSER

STATE 3A: ENTERING ECONOMIZER AFTER EXPANSION

STATE 3L: LIQUID STATE WITHIN ECONOMIZER

STATE 3V: VAPOR STATE WITHIN ECONOMIZER

STATE 4: ENTERING EVAPORATOR

REAL MOMU1,MU2,NEXP1,NEXP2

DOUBLE PRECISION X(NEQ),F(NEQ)


DATA CPW/l./,TMIN/-150./,TMAX/220./,PI/3.1415/

C

AEI=PAR (1)

AEO=PAR (2)

HEI=PAR(3)

RPE=PAR(4)

Cl=PAR(5)

C2=PAR(6)

AC I=PAR (7)

ACO=PAR (8)

- 1.20 -

C

C

C

C

C

C

C

C

C

C

C

Chillpr 'n n ,.,,t.

lpetinn 1

HCI=PAR(9)

RPC=PAR(10)

C3=PAR(11)

AX=PAR (12)

CBETA=PAR (2.3)

EFFREF=PAR (14)

EFFMOT=PAR (15)

RIMP=PAR(l.6)

C

FLCHW=XIN (1).

TCHWS=XIN (2)

FLCW=XIN (3)

TCWR=XIN (4)

QCOMP=XIN (5)

C


C

C THE FOUR UNKNOWNS FOR SINGLE-STAGE COMPRESSION ARE:

C TE = EVAPORATOR REFRIGERANT TEMPERATURE


C PHIXl = THE COMPRESSOR DIMENSIONLESS FLOW COEFFICIENT

C UX = THE IMPELLER TIP SPEED.



C P2IA = INLET PRESSURE TO THE SECOND STAGE.

C

TE=X (1)

TCND=X (2)

PHIX1=X(3)

UX=X(4)

TE=AMAXl (TMIN, AMINI (TE, TMAX))

TCND=AMAXl (TMIN, AMINI (TCND, TMAX))

PHIXI=AMAX1(0.001,AMINI (PHIXi, 1.0))


PHIX2=X(5)

PHIX2=AMAXl. (0 .001I, AMINI (PHIX2, 1. 0) )

P2.A=X (6)

END IF

- 1.21-

a AL 4 a Is w4w a %- Q Lt"-%., A V VC hillpr ("n ne~tv

Qatdfi nn I


C


C PRESENT.

C

CALL FREON(TE,P1,H1,S1,I.,V1,UlfIREF, 15)

CALL FREON(TCNDP2,H3,S3,0.,V3,U3,IREF, 15)

IF(ISTAGE.EQ•2 .AND. IECON.EQ.2) THEN

H3A=H3

CALL FREON(T3AP1A,H3A, S3AX3AV3AUDUM, IREF,23)

CALL FREON(T3AP1A,H3LS3LO.,V3LUDUMIREF,25)

CALL FREON (T3A, PIA, H3V, S3V, 1. ,V3V,UDUM, IREF, 25)

H4=H3L

ELSE

H4=H3

ENDIF

C


C


C FOR BOTH STAGES.

C

CALL PROP (IREFTCNDA0,COND,VISC)

MO=UX/AO

EFFN1=(. +0. 109* (1.1-MO)) * (1.-EXP (PHIX1* (58.45*PHIX1**2

• -5.99*PHIXI-18.81)) )

EFF 1=EFFREF*EFFN1

MU1=0.69* (1.-PHIX1*CBETA)/EFF1

HX1=HI+ (MUI-0.5*MU1*MUI-0.5*PHIX1*PHIX1) *UX*UX/32 .2/778.

HX1=AMAX1 (HX1, H1)

CALL FREON (TX1,PX1,HX1,S1, XX1,VX1,UDUM, IREF, 34)


H2=H1+MU1*UX*UX/32.2/778.

ELSE

EFFN2= (1.+0. 109"*(I.I-M0) ) *(I.-EXP (PHIX2* (58.45*PHIX2**2

• 5.99*PHIX2-18.81)) )

EFF 2=EFFREF * EFFN2

MU2=0 .69"* (1 .- PHIX2 *CBETA) /EFF2

- 1.22 -

(Nb.11 (nnar~t6MU.LVlI .1 " -, V. - ulI

Ipetion 1

H1A=H1+MU1*UX*UX/32.2/778.

CALL FREON(TIA, PlA, HIA, SlA, XlA, VIA, UDUM, IREF, 23)

IF(IECON.EQ.2) THEN

H2A=X3A*H3V+(1.-X3A) *HIA

CALL FREON(T2AP1A,H2A, S2AX2AV2AUDUM, IREF, 23)

ELSE

T2A=TlA

H2A=HlA

S2A=S1A

V2A=VIA

ENDIF

HX2=H2A+(MU2-0.5*MU2*MU2-0.5*PHIX2*PHIX2) *UX*UX/32.2/778.

HX2=AMAXI (HX2, H2A)

CALL FREON (TX2, PX2, HX2, S2A, XX2,VX2,UDUM, IREF, 34)

H2=H2A+MU2*UX*UX/32.2/778.

ENDIF


C


C


NEXP 1 =ALOG (P 1 / P 2) / ALOG (V2 / V1)

XPONI= (NEXP I-i.) /NEXP1

ELSE

NEXP I=ALOG (P1/P IA) / ALOG (V1A/Vl)

NEXP2=ALOG.(PIA/P2) /ALOG (V2/V2A)

XPONI= (NEXP 1-i.)/NEXP1


ENDIF

C


C

C UAI IS THE CONDUCTANCE ASSOCIATED WITH THE WATER FLOW AND PIPE

C MATERIAL (PLUS FOULING FACTOR). UAE IS THE OVERALL EVAPORATOR

C CONDUCTANCE, INCLUDING THAT ASSOCIATED WITH THE REFRIGERANT BOILING

C HEAT TRANSFER COEFFICIENT (HEO) . TFILM IS THE REFRIGERANT FILM

C TEMPERATURE AT THE SURFACE OF THE TUBE. ALSO DETERMINE REFRIGERANT

C FLOW RATES FOR EVAPORATOR (FLI) AND CONDENSER (FL2).

- 1.23 -

a Jkm, rmm v . , m',, m (4 ,.rq4Chillpr 'anneiltv

4Zpt;nn 1-7 . 1 %., a -a- gualwI

C


FL1=QCOMP/ ((HIA-HI) + (H2-H2A)/ (I.-X3A))

FL2=FL1/ (I.-X3A)

ELSE

FL1=QCOMP/ (H2-HI)

FL2=FL1

END IF

QEVAP=FL1* (HI-H4)

TCHWR=TCHWS+QEVAP /FLCHW/CPW

UAI=AEI/(1./HEI + RPE)


TFILM=TW-QEVAP/UAI

DT=AMAX1 (TFILM-TE, 0 .1)

HEO=C* (DT) **C2

UAE=I. / (1./UAI+1./HEO/AEO)


XLMTDE= ( (TCHWR-TE)- (TCHWS-TE) )/ALOG ( (TCHWR-TE) / (TCHWS-TE))

ELSE

XLMTDE=TW-TE

ENDIF

C

C ** CONDENSER **

C







C

CALL FREON (TCNDfP2,H2V, S2V, 1.,V2V, UDUM, IREF, 15)

QCOND=FL2 * (H2-H3)

TCWS=TCWR-QCOND / FLCW/CPWUAI=AC I/ ( 1. /HC I+RPC)

TW= (TCWS +TCWR) / 2.

TF I LM=TW+QCOND /UAI

DT=AMAXl (TCND-TFILM, 0 .1 )

DH=H2V-H3

- 1.24 -

Chiller "Cannit v

Rpetiln I

HCO=C3*(COND**3*DH/VISC/DT/V3**2)**0.25

UAC=1./ (./UAI+1./HCO/ACO)

IF(TCND.GT.TCWR) THEN

XLMTDC=( (TCND-TCWS)-(TCND-TCWR) ) /ALOG( (TCND-TCWS) / (TCND-TCWR))

ELSE

XLMTDC=TCND-TW

ENDIF

C


C



C

F (1) =FLCHW*CPW* (TCHWR-TCHWS) -UAE*XLMTDE

F (2) =FLCW*CPW* (TCWR-TCWS) -UAC*XLMTDC

F (3)-X (3) -FL1*VX1/AX/UX/3600.


F(4)=H2-(144./778.*P1*Vl/EFF1/XPON1*((P2/PI)**XPON1-1.)+H1)

ELSE

F (4)=HA-(144./778. *PI*Vl/EFF1/XPON1* ((PlA/PI) **XPON1-. ) +H1)

F (5) =X (5) -FL2*VX2/AX/UX/3600.

F (6) =H2- (144./778. *P1A*V2A/EFF2/XPON2* ((P2/PlA) **XPON2-1. ) +H2A)

ENDIF

C

C ** OUTPUTS **

C

OUT (1) =QEVAP/12000.

OUT (2)=QCOND/12000.

OUT (3) = (QCOND-QEVAP) /12000.

OUT(4)=3.515*OUT(3)/EFFMOT

OUT (5) =TE

OUT (6) =TCND

OUT (7) =T2

OUT (8) =P1OUT ( 9 ) =P2

OUT (10) =60. *UX/RIMP/2 ./2I

OUT (11) =EFFMOT*OUT (1) /OUT (3)

OUT (12) =PHIXl

- 1.25 -

I la JL a 9 %W A- t -TChiller iC nq-itv

IRctinn 1I L eIlI- ~L; 1 Jl vi t v

OUT (13) =NEXP 1

OUT (14) =EFF1

OUT (15) =UAE

OUT (16) =UAC


OUT (15) =PHIX2

OUT (16) =NEXP2

OUT (17) =EFF2

OUT (18) =UAE

OUT (19) =UAC

END IF

RETURN

END

C

SUBROUTINE PROP (IREF,T,A0,CONDVISC)

C*********** ****************** *********************************************

C *


C OR R12. THE ARGUMENTS ARE: *

C

C IREF = REFRIGERANT TYPE (500, 22, OR 12)

C T = REFRIGERANT TEMPERATURE (DEGREES F)

C AO = SONIC VELOCITY (FT/SEC)

C COND = THERMAL CONDUCTIVITY (BTU/HR-FT-F)

C VISC = VISCOSITY (LBM/FT-HR)

C

C

IF(IREF.EQ.500) THEN

AO=490.

COND=0.053-0. 000127*TVISC=0 .6851-3. 2943E-03*T+6 . 443E-06*T*T

END IF

IF(IREF.EQ.22) THEN

A0=534.

COND=0 .063-0. 000159"T

VISC=0. 6507-2. 5943E-03*T+5. 6101E-06*T*T

END IF

-1.26 -

Chillpr rnnneitv

3ectl()tj L '1,11111CI "kdLJdljtV('h;llpr Canan;fvCAMI^+;IImk I

IF (IREF. EQ. 12) THEN

AO=446.

COND=0.049-0.000117*T

VISC=0.763-3.8905E-03*T+9.881E-06*T*T

ENDIF

RETURN

END

IF (IREF. EQ. 12) THEN

AO=446.

COND=0.049-0.000117*T

VISC=0.763-3.8905E-03*T+9.881E-06*T*T

ENDIF

RETURN

END

q,,tinn I

C *

C PROGRAM FOR ESTIMATING THE SURGE POINT FOR A SINGLE OR DOUBLE-STAGE *

C CENTRIFUGAL CHILLER. FOR A TWO-STAGE MACHINE, THERE IS ALSO AN OPTION FOR *

C A FLASH GAS ECONOMIZER. THE INPUTS ARE FLOW STREAM CONDITIONS, WHILE *

C OUTPUTS INCLUDE THE COOLING LOAD, POWER CONSUMPTION, AND COMPRESSOR SPEED *

C ASSOCIATED WITH THE SURGE CONDITIONS. THE UNITS UTILIZED WITHIN THIS *

C PROGRAM ARE TYPICAL ENGLISH UNITS THAT ARE COMMON WITHIN THE HVAC INDUSTRY *

C (BTU'S, TONS, KW). THE ANALYSIS IS DESCRIBED IN CHAPTER 2 OF *

C "METHODOLOGIES FOR DESIGN AND CONTROL OF CENTRAL COOLING PLANTS", J.E. *

C BRAUN, PH.D THESIS, UNIVERSITY OF WISCONSIN - MADISON, 1988 *

C *

C


DATA PI/3.1415/,GOLD/.61803399/

C

PARAMETER(CPW=1., DENS=8.3333)

C

IREF = 500

ISTAGE = 2

IECON = 2

C

C PROGRAM DATA DEFINED ABOVE:

C






C


C

OPEN (10, FILE= 'SURGE. DAT ' ,STATUS =' OLD ')

OPEN (11,FILE=' SURGE .OUT' , STATUS='NEW')

WRITE (11, 101)

C


- 1.28 -

C'hillpr S,,rop

Qaritn 1LJ-V~lyf x."1,-0 tJLU UU rA

C

10 READ(10, *,END=100) QMINQMAXGPMCHWGPMCWTCHWSTCWR

C

C INPUT DATA IS:

C QMIN = LOWER LIMIT ON CHILLER EVAPORATOR LOAD (TONS)

C QMAX = UPPER LIMIT ON CHILLER EVAPORATOR LOAD (TONS)


C GPMCW = CONDENSOR WATER FLOW RATE (GAL/MIN)

C TCHWS = CHILLED WATER SETPOINT (DEGREES F)

C TCWR = CONDENSOR WATER RETURN (DEGREES F)

** DETERMINE MINIMUM COMPRESSOR SPEED **

THE MINIMUM COMPRESSOR SPEED IS DETERMINED USING AN OPTIMIZATION SCHEME

CALLED GOLDEN SECTION SEARCH.

STORE INITIAL GUESSES FOR ITERATION VARIABLES

OUT (5) =TCHWS-5.

OUT (6) =TCWR+10.

OUT (10) =450.

OUT (12) =0.15


OUT (15)=0.15

CALL FREON(OUT(5),P1,H1,S1,I.,V1,UIIREF,15)

CALL FREON(OUT(6),P2,H3,S3,O.,V3,U3,IREF,15)

OUT (20) = (P1+P2) /2.

END IF

INITIAL POINTS FOR GOLDEN SECTION SEARCH

STEP= (QMAX-QMIN) *GOLD

Q 1=QMAX-STEP

Q2=QMIN+STEP

QLAST=QMIN

UXm SPEED (GPMCHWGPMCW, TCHWS, TCWR,Q1, TCHWR)

UX2 = SPEED (GPMCHW, GPMCW, TCHWS, TCWR, Q2, TCHWR)

- 1.29 -

CC

C

Chillpr .'Riirap,

a a- h L-,A S .I

C LOOP TO FIND MINUMUM SPEED BY VARYING CHILLER COOLING LOAD

C

20 EPS-AMAX1(.05,QLAST*0.001)

IF(STEP.LT.EPS) GOTO 50

STEP=STEP*GOLD

IF(UX1.GT.UX2) THEN

TEMP=Q1

QI=Q2

UX1=UX2

Q2=TEMP+STEP


QLAST=Q2

ELSE

TEMP=Q2

Q2=QI

UX2=UX1

Q1=TEMP-STEP

UX1 = SPEED (GPMCHW, GPMCW, TCHWS, TCWR, QI, TCHWR)

QLAST=Q1

ENDIF

GO TO 20

50 CONTINUE

C

C INCREASE SPEED BY 50 RPM

C

UX1=OUT (10)

RIMP=PAR (10)

RPM=60.*UX1/RIMP/2./PI + 50.

UX=2. *P I *RIMP*RPM/60.

Q1=OUT (1)

Q2=1.5*QI

ITER=0

60 ITER=ITER+I


DUX= (UX2-UJXl) I (Q2-Q1)

QI=Q2

UX1=UX2

Q2=Q2 + (UX-UX2)/IDUx

IF(ABS(UX-UX2).GT.0.1 .AND. ITER.LT.10) GO TO 60

- 1.30 -

.qpction 1 Chiller Snrgp

Q e;,z 1 hiilr Rira_ll'JEIt ALIr5

C

C ** OUTPUT DETAILED RESULTS AND COMPARISONS TO FILE **

C

C COMPUTE THE CONDENSOR WATER SUPPLY FLOW AND TEMPERATURE (TCWS) AND

C COMPRESSOR RPM

C

FLCW=60. *DENS*GPMCW


RIMP=PAR (10)

OUT(10)=60.*OUT(10)/RIMP/2./PI

WRITE(11, 102) GPMCHW, GPMCW, TCHWSTCHWR, TCWSTCWR


WRITE(II,103) (OUT(I), I=1,16)

ELSE

WRITE(II,104) (OUT(T),I=1,19)

ENDIF

GO TO 10

100 CONTINUE

STOP

C

101 FORMAT(///21X, '********** CHILLER MODEL **********')

102 FORMAT(//2X, '****************************** INPUT DATA *

'*************************I//2X,'FLCHW, FLCW ='

0 2(1X,F8.2)/2X,'TCHWS, TCHWR, TCWS, TCWR = ',4(1X,F6.2))

103 FORMAT(//2X,'** ENERGY QUANTITIES **I,//2X,'QEVAP= ',F8.2/2X,

0 'QCOND = ',F8.2/2X,'QCOMP - ',F8.2/2X,'ELEC - ',F8.2

0 //2X,'** DETAILED RESULTS **',//2X,

a 'TE, TC, T2, P1, P2, RPM= ',6(1X,F7.2)/2X,

0 'COP, PHIX, NPOL, EFF = ',4(1X,F6.3)/2X,

0 'UAE, UAC = '2(1X,IPE11.3))

104 FORMAT(//2X,'** ENERGY QUANTITIES **',//2X,'QEVAP- ',F8.2/2X,

0 'QCOND = ',F8.2/2X,'QCOMP- ',F8.2/2X,'ELEC = ',F8.2

* //2X, '** DETAILED RESULTS **',//2X,

* 'TE, TC, T2, P1, P2, RPM = ',6(IX, F7.2)/2X,

* 'COP, PHIXl, NPOL1, EFFI = :',4(IX, FE.3)/2X,

* 'PHIX2, NPOL2, EFF2 = ',3(IX, F6.3)/2X,

* 'UA.E, UAC = ',2(IXIPE1I.3))

END

-1.31-

t I.UII !.,--,... I ,l

FUNCTION SPEED (GPMCHW, GPMCW, TCHWS, TCWR, QEVAP, TCHWR)

C

C

C ESTIMATE COMPRESSOR PERFORMANCE FOR GIVEN LOAD AND RETURN COMPRESSOR SPEED *

c*

C

EXTERNAL EVAP, CHILLER



DATA TOL/O.OO1/,DIFF/5*O.O1/,PI/3.1415/

C

PARAMETER(CPW=I., DENS-8.3333)


DIAC=.75/12., NEPASS=3, NETUBE=3560, NCPASS=I, NCTUBE=3349,

C1=103., C2=1.1, NVERT=I, RPE=2.E-04, RPC=2.E-04, AX=1.53,


C

C PROGRAM DATA DEFINED WITHIN THE ABOVE PARAMETER STATEMENTS:

C















C C1,C2 = EMPIRICAL CONSTANTS FOR BOILING HEAT TRANSFER

C COEFFICIENT


-1.32 -

Chiller ,,mopC!4..^J.:Af., 1

C RPERPC = EVAPORATOR AND CONDENSER TUBE WALL RESISTANCE

C (FT**2-F/BTU)

C AX = EXIT AREA OF COMPRESSOR IMPELLER (FT**2)

C BETA = IMPELLER BLADE ANGLE (DEGREES)




C



C


C

C CALCULATE THE MASS FLOW RATE FROM THE GPM AND LOAD RETURN TEMPERATURE

C FLCHW = CHILLED WATER FLOW RATE (LBM/HR)


C TCHWR = CHILLED WATER RETURN TEMPERATURE (DEGREES F)

C

C

FLCHW60. *DENS*GPMCHW

FLCW-60. *DENS*GPMCW

TCHWR=TCHWS+12000 . *QEVAP/FLCHW/CPW

C


C

C GIVEN THE CHILLER LOAD AND WATER CONDITIONS, THE EQUATION SOLVER

C ITERATIVELY DETERMINES THE EVAPORATOR TEMPERATURE, TE, THROUGH THE

C USE OF THE SUBROUTINE EVAP. PARAMETERS, INPUTS, AND OUTPUTS ARE

C COMMUNICATED THROUGH THE COMMON BLOCK "DATA". HEI IS THE INSIDE

C EVAPORATOR HEAT TRANSFER COEFFICIENT (TURBULENT FLOW).

C

XIN (1) =FLCHWXIN ( 2) --TCHWS

XIN ( 3) =TCHWR

XIN (4) =12000. *QEVAP

PAR (1) =AEI

PAR (2) =REO*AEI

- 1.33 -

Cart; nn I C~hillpr Rima

SPction I

HEI=1.37*(NEPASS*GPMCHW/NETUBE) **0.8*DIAE**-].8

PAR(3) =HEI

PAR(4) =RPE

PAR(5) =C1

PAR(6) =C2

C

X (1)=OUT (5)

CALL SOLVER(IDIFF,X,F,W,EVAP,TOL)

TE=X(I)

C

C ** COMPRESSOR AND CONDENSOR ANALYSIS **

C


C THE COMPRESSOR AND CONDENSER BY CALLING SUBROUTINE CHILLER. EITHER

C ONE OR TWO-STAGE COMPRESSION IS POSSIBLE. FOR SINGLE-STAGE

C COMPRESSION, THERE ARE THREE ITERATIVE VARIABLES: THE REFRIGERANT




C CIENT AND THE SECOND STAGE IMPELLER INLET PRESSURE. THE INLET STATE

C TO THE COMPRESSOR IS KNOWN FROM THE EVAPORATOR ANALYSIS. THE FREON

C PROPERTIES AT THIS STATE ARE EVALUATED PRIOR TO THE ITERATIVE

C SOLUTION. PARAMETERS, INPUTS, AND OUTPUTS ARE COMMUNICATED THROUGH

C THE COMMON BLOCK "DATA".

C


C HCI = INSIDE CONDENSER HEAT TRANSFER COEFFICIENT (TURBULENT FLOW)

C C3 = CONSTANT FOR EVALUATING THE HEAT TRANSFER COEFFICIENT FOR


C

PAR(1) =ACI

PAR (2) =RCO*ACIHCI=I. 788* (NCPASS*GPMCW/NCTUBE) **0 .8*DIAC**-I .8

PAR (3) =HCI

PAR (4) =RPC

C3=0 .725* (32.2*3600. **2/NVERT/DIAC) **0 .25

PAR (S5) =03

PAR (6) =AX

PAR (7) =0.

- 1.34-

C~hiller S,, op.

eOqc4tn1 Cllau eirr

IF(ABS(BETA-90.) .GT.l.E-06) PAR(7)=1./TAN(PI*BETA/180.)

PAR (8) =EFFREF

PAR (9) =EFFMOT

PAR(10) =RIMP

C

C EVALUATE PROPERTIES AT COMPRESSOR INLET FROM RESULTS OF EVAPORATOR

C ANALYSIS

C

CALL FREON(TE,Pl,H,S, 1.,V1,Ul, IREF, 15)

OUT (8) =P1

C


C

XIN(1)=FLCW

XIN(2) =TCWR

XIN (3) =12000. *QEVAP

XIN(4)=P1

XIN (5) =Vl

XIN(6)=HI

XIN(7)=S1

C

X (1) =OUT (6)

X(2) =OUT (12)

X(3)=OUT (10)

C



C


CALL SOLVER(3,DIFF,X,F,W, CHILLER, TOL)

ELSE

X(4)=OUT (15)

X(5) =OUT (20)

CALL SOLVER(5,DIFF,X,F,W,CHILLER,TOL)

END IF

SPEED = X(3)

-1.35 -

.Q.oet;nn I -- ('h;la i!,,* ra...

1J Lt IO ul I LJL.E

RETURN

END

SUBROUTINE EVAP (NEQ, X, F)

C

C *

THIS SUBROUTINE MODELS THE PERFORMANCE OF AN EVAPORATOR. THE EQUATION

SOLVER THAT CALLS THIS ROUTINE IS SEARCHING FOR THE EVAPORATOR TEMPERATURE

(TE) THAT YIELDS NO DIFFERENCE BETWEEN THE CHILLER LOAD CALCULATED WITH A

UA-LOG MEAN TEMPERATURE DIFFERENCE AND THE ENTHALPY DIFFERENCE OF THE

CHILLED WATER STREAM.

DOUBLE PRECISION X,F


AEI=PAR (1)

AEO=PAR (2)

HEI=PAR (3)

RPE=PAR (4)

C1=PAR (5)

C2=PAR (6)

TCHWS=XIN (2)

TCHWR=XIN (3)

QEVAP=XIN (4)

TE=X

UAI IS THE CONDUCTANCE ASSOCIATED WITH THE WATER FLOW AND PIPE

MATERIAL (PLUS FOULING FACTOR). UAE IS THE OVERALL EVAPORATOR

CONDUCTANCE, INCLUDING THAT ASSOCIATED WITH THE REFRIGERANT BOILING

HEAT TRANSFER COEFFICIENT (HEO). TFILM IS THE REFRIGERANT FILM

TEMPERATURE AT THE SURFACE OF THE TUBE.

UAI=AEI/ (1. /HEI+RPE)

- 1.36 -

CC

C

C

C

CC

C

C

C

C

C

(Chiilpr RiirapQ,aefJ;,n I ..I

Section 1 Chiller Surge


TF ILM=TW-QEVAP/UAI

DT=AMAX1 (TFILM-TE, C . 1)

HEO=CI* (DT) **C2

UAE=1./ (./UAI+1./HEO/AEO)


XLMTD= ( (TCHWR-TE) - (TCHWS-TE))/ALOG ((TCHWR-TE) / (TCHWS-TE))

ELSE

XLMTD=TW-TE

END IF

C

F=QEVAP -UAE *XLMTD

C

OUT (5) =TE


OUT (15) =UAE

ELSE

OUT (18) -UAE

ENDIF

C

RETURN

END

SUBROUTINE CHILLER (NEQX,F)

C

C

C THIS SUBROUTINE IS CALLED AT EACH ITERATION OF THE EQUATION SOLVER TO *

C EVALUATE THE COMPRESSOR AND CONDENSER PERFORMANCE. THROUGHOUT THE PROGRAM *

C T, P, V, H, AND S REFER TO REFRIGERANT TEMPERATURE, PRESSURE, SPECIFIC, *

C VOLUME, SPECIFIC ENTHALPY, AND SPECIFIC ENTROPY FOR A GIVEN STATE. *

C STATES ARE: *

C

C STATE 1: ENTERING COMPRESSOR FIRST STAGE*

C STATE XI: EXIT FROM FIRST STAGE COMPRESSOR IMPELLER*

C STATE IA: EXIT FROM FIRST STAGE COMPRESSOR DIFFUSER*

C STATE 2A: ENTERING COMPRESSOR SECOND STAGE*

C STATE X2: EXIT FROM SECOND STAGE COMPRESSOR IMPELLER*

C STATE 2: EXIT FROM SECOND STAGE COMPRESSOR DIFFUSER*

- 1.37 -

Ly Y616LLL 1 aCA.illpr u 16

C STATE 3: EXIT FROM CONDENSER

C STATE 3A: ENTERING ECONOMIZER AFTER EXPANSION

C STATE 3L: LIQUID STATE WITHIN ECONOMIZER

C STATE 3V: VAPOR STATE WITHIN ECONOMIZER

C STATE 4: ENTERING EVAPORATOR

C

C

REAL MOMU1,MU2,NEXP1,NEXP2



DATA CPW/1./,TMIN/-150./,TMAX/220./,PI/3.1415/

C

C ** PARAMETERS AND INPUTS **

C

ACI=PAR(1)

ACO=PAR (2)

HCI=PAR(3)

RPC=PAR(4)

C3=PAR (5)

AX=PAR (6)

CBETA=PAR (7)

EFFREF=PAR (8)

EFFMOT=PAR (9)

RIMP=PAR (10)

C

FLCW=XIN (1)

TCWR=XIN (2)

QEVAP=XIN (3)

P1=XIN (4)

V1=XIN (5)

H1=XIN (6)

SI=XIN (7)

C


C

C THE THREE UNKNOWNS FOR SINGLE-STAGE COMPRESSION ARE:


- 1.38 -

I qption 1 C'hilpr I,,irfy I

Lio Vi -IIn 1iCh1Iai li u u r_..

C PHIXi = THE COMPRESSOR DIMENSIONLESS FLOW COEFFICIENT

C UX = THE IMPELLER TIP SPEED



C PlA = INLET PRESSURE TO THE SECOND STAGE.

C

TCND=X (1)

PHIX1=X (2)

UX=X (3)

TCND=AMAX1 (TMIN, AMIN1 (TCND, TMAX))

PHIX1=AMAX1(0.,AMIN1(PHIX1, 1.0))

UX-AMAX1 (UX, 1.)


PHIX2=X(4)

PHIX2=AMAX1(0.,AMIN1(PHIX2, 1.0))

PIA=X(5)

ENDIF

C


C


C PRESENT. ALSO DETERMINE REFRIGERANT FLOW RATES FOR EVAPORATOR (FL1)

C AND CONDENSER (FL2).

C

CALL FREON(TCND, P2,H3,S3,0.,V3,U3, IREF, 15)


H3A=H3

CALL FREON (T3AP1A,H3A, S3A, X3AV3AtUDUM, IREF, 23)

CALL FREON (T3A, P1A, H3L, S3L,0. ,V3L, UDUM, IREF, 25)

CALL FREON (T3A, P1A, H3V, S3V, 1. ,V3V, UDUM, IREF, 25)

H4=H3L

FL1=QEVAP/ (H1-H4)

FL2=FLI/ (1.-X3A)

ELSE

H4=H3

FL1-QEVAP / (H1-H4)

FL2-FL1

END IF

- 1.39 -

Rpetion I (rhillpr ,, rup

. o , , fll V !i.r .uar--


C


C FOR BOTH STAGES.

C

CALL PROP (IREFTCNDA0,CONDVISC)

MO=UX/AO

EFFN1(I.+0.109*(I.I-MO))*(I.-EXP(PHIX1*(58.45*PHIX1**2

-5.99*PHIX1-18.81)))

EFF 1=EFFREF *EFFN1

MU1=0.69* (1.-PHIX1*CBETA) /EFF1

HX1=HI+(MUI-0.5*MU1*MUI-0.5*PHIX1*PHIX1) *UX*UX/32.2/778.

HX1=AMAX1 (HX1, Hi)

CALL FREON(TX1,PX1,HXI,S1,XXIVX1,UDUM, IREF, 34)


H2=HI +MU1*UX*UX/32.2/778.

ELSE

EFFN2=(I.+0.109*(1.1-MO))*(l.-EXP(PHIX2*(58.45*PHIX2**2

-5.99*PHIX2-18.81)))

EFF2 =EFFREF *EFFN2

MU2=0.69* (1. -PHIX2*CBETA)/EFF2

HlA=H+MU1*UX*UX/32.2/778.

CALL FREON(TA, PlA, HA, SlA, XlA,VlA,UDUM, IREF, 23)

IF(IECON.EQ.2) THEN

H2A=X3A*H3V+ (1 . -X3A) *HlA

CALL FREON (T2A, PlA, H2A, S2A, X2A, V2A, UDUM, IREF, 23)

ELSE

T2A=TlA

H2A=H1A

S2A=SlA

V2A=VlA

ENDIF

HX2=H2A+ (MU2-0.5*MU2 *MU2-0.5*PHIX2*PHIX2) *UX*UX/32.2/778.

HX2=AMAXI (HX2, H2A)

CALL FREON (TX2, PX2, HX2,5S2A, XX2, VX2, UDUM, IREF, 34 )

H2 =H2A+MU2 *UX*UX/32 .2 /77?8.

END IF


- 1.40-

.qpetion I tChillpr qitrap

c


C


NEXP 1 =ALOG(P 1 / P 2) / ALOG (V2 / Vl)


ELSE

NEXP1=ALOG (PI/PIA) /ALOG (VIA/Vl)

NEXP2=ALOG (P 1A/P2) /ALOG (V2/V2A)

XPON1- (NEXP 1-1.) /NEXP1


ENDIF

C

C ** CONDENSER **

C







C

CALL FREON (TCND, P2 ,H2VS2V, 1. ,V2V, UDUM, IREF, 15)

QCOND-FL2 * (H2 -H3)

TCWS=TCWR-QCOND/FLCW/CPW

UAI=ACI/ (I./HCI+RPC)

TW= (TCWS+TCWR)/2.

TFILM=TW+QCOND/UAI

DT=AMAX1 (TCND-TFILM, 0.1)

DH-H2V-H3

HCO=C3* (COND**3*DH/VISC/DT/V3**2) **0.25

UAC=1. / (1./UAI+1./HCO/ACO)IF (TCND.GT. TCWR) THEN

XLMTD= ( (TCND-TCWS) - (TCND-TCWR) ) /ALOG ( (TCND-TCWS) / (TCND-TCWR) )

ELSE

XLMTD=TCND-TW

END IF

C


- 1.41 -

C~hiller ,,itrvrZA,,.Pf 1

n%, tAn a L ,1 Chu11 r m u, i l. ra -

C



C

F (1) =FLCW*CPW* (TCWR-TCWS) -UAC*XLMTD

F (2)=X (2) -FL1*VX1/AX/UX/3600.


F (3) =H2- (144./778. *P1*Vl/EFF1/XPON1* ((P2/Pl) **XPON1-. ) +H1)

ELSE

F (3) =H1A- (144./778. *PI*VI/EFF1/XPON1* ((PIA/PI) **XPON-I. ) +HI)

F (4) =X (4) -FL2*VX2/AX/UX/3600.

F (5) =H2- (144./778. *PA*V2A/EFF2/XPON2* ((P2/PIA) **XPON2-1. ) +H2A)

END IF

C

C ** OUTPUTS **

C

OUT (1)=QEVAP/12000.

OUT (2) =QCOND/12000.

OUT (3) = (QCOND-QEVAP) / 12000.

OUT (4)=3. 515*OUT (3)/EFFMOT

OUT (6)=TCND

OUT (7)=T2

OUT (9) =P2

OUT (10) =UX

OUT (11) =EFFMOT*OUT (1) /OUT (3)

OUT (12) =PHIX1

OUT (13) =NEXP1

OUT (14) =EFF1

OUT (16) -UAC


OUT (15) =PHIX2

OUT (16) =NEXP2

OUT (17) =EFF2

OUT (19) =UAC

OUT (20) =PIA

END IF

RETURN

END

-1.42 -

.qpvtinn 1 Chillpr 'qeirap

Section 1- Chiller Surge

C

SUBROUTINE PROP (IREFrTAQ,CONDoVISC)

C


C OR R12. THE ARGUMENTS ARE. *

C

C IREF = REFRIGERANT TYPE (500, 22, OR 12)

C T = REFRIGERANT TEMPERATURE (DEGREES F)

C A0 = SONIC VELOCITY (FT/SEC)

C COND = THERMAL CONDUCTIVITY (BTU/HR-FT-F)

C VISC = VISCOSITY (LBM/FT-HR)

C

C

IF(IREF.EQ.500) THEN

A0-490.

COND-0.053-0. 000127*T

VISC=0.6851-3.2943E-03*T+6. 443E-06*T*T

END IF

IF(IREF.EQ.22) THEN

A0=534.

•COND=0.063-0.000159*T

VISC=0.6507-2. 5943E-03*T+5. 6101E-06*T*T

ENDIF

IF(IREF.EQ.12) THEN

A0=446.

COND=0. 049-0. 000117*T

VISC=0.763-3. 8905E-03*T+9. 881E-06*T*T

ENDIF

RETURN

END

- 1.43 -

Section 2 Cooling Tower and Coil Models

Cooling Tower Performance Comparisons 2.1This program compares three different methods for modeling the performanceof cooling towers as outlined in Chapter 3: 1) finite difference solution to thebasic heat and mass transfer equations, 2).Merkel method, and 3) theeffectiveness method. In addition to the routines listed in this section, thisprogram utilizes two utility subroutines (SOLVER and PSYCH) that are listed inSection 5.

Cooling Coil Performance Comparisons 2.10This program compares two methods for modeling the performance ofcooling coils as as outlined in Chapter 4: 1) finite difference solution tothe basic heat and mass transfer equations and 2) the effectiveness method. Inaddition to the routines listed in this section, this program utilizes two utilitysubroutines (SOLVER and PSYCH) that are listed in Section 5.

- 2.0 -

dahI ;na TvvrPgprfnrvmianea ( iia .. cnnt:LLlull &- 'AJ.JUi, A T ,J L I A I.. Ji ' iIlLii. ',. -UlJtil I )Ul3

C *

C PROGRAM FOR CALCULATING THE PERFORMANCE OF A COUNTERFLOW COOLING TOWER.

C THREE METHODS ARE COMPARED: DETAILED SOLUTION OF BASIC HEAT AND MASS *

C TRANSFER EQUATIONS, THE MERKEL METHOD, AND AN EFFECTIVENESS MODEL. THE *

C UNITS UTILIZED WITHIN THIS PROGRAM ARE TYPICAL ENGLISH. THE MODELS ARE *

C DESCRIBED IN CHAPTER 3 OF "METHODOLOGIES FOR DESIGN AND CONTROL OF CENTRAL *

C COOLING PLANTS", J.E. BRAUN, PH.D THESIS, UNIVERSITY OF WISCONSIN - *

C MADISON, 1988 *

C *

C

REAL NTUS(7),RAS(3),LE

COMMON /DATAS/ DATA(100,9),OUT(100,9),MODEIEXP

DATA NTUS/0.5,1.,1.5,2.,2.5,3., 4./

DATA RAS/0.5,1.,2./

C

C ** OPEN INPUT AND OUTPUT FILES **

C

OPEN (10, FILE= 'TOWERCOMPARE.DAT'fSTATUS = 'OLD')

OPEN(11,FILE='TOWERCOMPARE.OUT',STATUS='NEW')

C

C ** LOOP TO READ INPUT DATA **

C

5 READ(10,*,END=100) TWB, TWI, TDB, LE

c

C INPUT DATA:

C

C TDB = AMBIENT DRY BULB TEMPERATURE (DEGREES C)

C TWB = AMBIENT WET BULB TEMPERATURE (DEGREES C)

C TWI = INLET TEMPERATURE OF WATER TO COOLING TOWER (DEGREES C)

C LE = LEWIS NUMBER

c

c

C ** STORE DATA FOR A RANGE OF TOWER PARAMETERS **

c

C NTUS = NUMBER OF TRANSFER UNITS

C RAS = RATIO OF WATER FLOW TO AIR FLOW RATE

-2.1-

i

NEXP = 0

DO 10 NRA = 1,3

DO 10 NNTU - 1,7

IF(NRA .NE. 1 .OR. NNTU .LE. 5) THEN

NEXP = NEXP + 1

DATA (NEXP, 1) = NTUS (NNTU)

DATA (NEXP, 2) = RAS (NRA)

DATA(NEXP,3) = TWB

DATA(NEXP, 4) = TWI

DATA(NEXP, 5) = TDB

DATA(NEXP, 6) = LE

ENDIF

10 CONTINUE

C

C ** TOWER ANALYSIS **

C

C RESULTS ARE STORED IN THE OUT ARRAY FOR ALL SETS OF COIL DATA

C

CALL TOWER (NEXP)

C

C ** OUTPUT RESULTS TO FILE **

C

WRITE(11,101) TWB, TWI, TDB, LE

WRITE (11, 102)

DO 20 IEXP - 1,NEXP

WRITE(11,103) (DATA(IEXPJ),J=1,2), (OUT(IEXPJ),J=1,9)

20 CONTINUE

GOTO 5

C

100 CONTINUE

C101 FORAT(/1X'TWB, TWI, TDB, LE - ',4(iXFE.2))

102 FORMAT(//13X,'** FINITE DIFFERENCE ** ** MERKEL ** '

'*EFFECTIVENESS **'//

'NTU FLCW/FLA EPSA LOSS EPSW EPSA LOSS ',

'EPSW EPSA LOSS EPSW' )

103 FORMAT (T2, F3.,T9, F3.,'5, 3 (IX,3 (IX,F5.3) ))

- 2.2-

lqpfinn2 ,(Cnnlina Tnwpr Pprf'nrm~n = ... ..

t in la 'l''hw7Lr . prfnr ni ,a

C

END

C

SUBROUTINE TOWER (NEXP)

C

C *

C ROUTINE FOR CALCULATING COOLING TOWER PERFORMANCE. *

C *

C

EXTERNAL MODEL



DATA TOL/I.E-05/,DIFF/0.001,O.0001/

C

C***************** LOOP THROUGH ALL SETS OF COIL DATA *************************

C

C NOMENCLATURE:

C

C TWB = ENTERING AMBIENT WET BULB TEMPERATURE (DEGREES C)

C TW = WATER TEMPERATURE (DEGREES C)

C TA = AIR TEMPERATURE (DEGREES C)

C TS = SURFACE TEMPERATURE (DEGREES F)

C TDP = AIR DEWPOINT TEMPERATURE (DEGREES C)

C WA = AIR HUMIDITY RATIO

C HA = AIR SPECIFIC ENTHALPY (KJ/KG)

C WW = HUMIDITY RATIO ASSOCIATED WITH SATURATED AIR AT TW

C HW = SPECIFIC ENTHALPY ASSOCIATED WITH SATURATED AIR AT TW (KJ/KG)

C EPSW = WATER STREAM TEMPERATURE EFFECTIVENESS

C EPSA = AIR STREAM HEAT TRANSFER EFFECTIVENESS

C

C THE STATES 1 AND 2 REFER TO ENTERING AND EXITING CONDITIONS

C (1=ENTERING, 2 = EXITING) .

C

DO 100 IEXP = I,NEXP

RA = DATA (IEXP,2 )

TWB = DATA(IEXP, 3)

- 2.3 -

13 |1tl)11 4- t-UVuulu • u VV . •t u l uull I I u mlr i[I nllQ ^dl%+ ; &'%" I

~lInuII - ',. ninuuau • ivv%.n a a nun u Iuill, 'uuIIpIuUIsols

TWI = DATA(IEXP,4)

TDB -=DATA(IEXP,5)

CALL PSYCH (2,1,TDB,TWB, RHTDPWA1,HA)

DATA(IEXP,7) = HAl

DATA(IEXP,8) = WAl

CALL PSYCH(2,1,TWlTWlRHTDPWW1,HW1)

C

C ** EFFECTIVENESS-NTU MODEL **

C

CALL EFFEC (TW2,EPSAWA2, HA2)

EPSW - (TWI - TW2)/ (TW1 - TWB)

OUT(IEXP,7) = EPSA

OUT(IEXP,8) - 100.*(WA2 - WAI)/RA

OUT(IEXP,9) = EPSW

C

C ** DETAILED ANALYSIS **

C

C THE DETAILED COOLING COIL ANALYSIS INVOLVES AN ITERATIVE SOLUTION IN

C TERMS OF THE EXIT WATER TEMPERATURE (TW2) AND AIR HUMIDITY RATIO (WA2).

C THE EQUATION SOLVER CALLS THE SUBROUTINE MODEL AT EACH ITERATION. THE

C RESULTS OF THE EFFECTIVENESS ANALYSIS ARE USED AS AN INITIAL GUESS FOR

C THIS SOLUTION.

C

MODE = 1

X(I) = TW2

X(2) = WA2

CALL SOLVER (2,DIFF, X,F, W,MODEL, TOL)

C

TW2 = X(1)

WA2 = X(2)

HA2 = OUT(IEXP,I)

EPSA = (HA2 - HA1)/(HWI - HAI)

EPSW = (TWI - TW2) /(TWI - TWB)

CALL PSYCH (2, 6, TA2, TWB2, RH, TDP, WA2, HA2)

OUT(IEXP, I) = EPSA

OUT(IEXP,2) = 100.*(WA2 - WAI))/RA

OUT(IEXP,3) = EPSW

-2.4-

rnnlina Tnwar Parftirm4annaC adf:tnI

~cietmnn7 ('nnt in ~ Tnw~~r P~rfnrm~ n~ ~

C

C

C

C

C

C

C

OUT (I

OUT (I

- - - 1%- F - .. , - - -F . .• - - . .. • .. .

EXP,4) = EPSA

EXP,5) = Q100.*(WA2 - WAl) IRA

OUT(IEXP,6) = EPSW

C

100 CONTINUE

C

RETURN

END

C

SUBROUTINE MODEL (NEQ, X, F)

DETAILED MODEL OF A COOLING TOWER THAT SOLVES BASIC HEAT AND MASS TRANSFER *

EQUATIONS USING FINITE-DIFFERENCES. THIS ROUTINE IS CALLED AT EACH ITERA- *

TION OF AN EQUATION SOLVER THAT IS FINDING THE CORRECT LEAVING WATER *

TEMPERATURE. THERE ARE POSSIBLE MODES FOR THIS ROUTINE IN MODE 1 THE WATER*

TERMS ARE INCLUDED, WHILE IN MODE 2 THEY ARE NEGLECTED (MERKEL METHOD) *

C

REAL NTULE

DOUBLE PRECISION X (NEQ) ,F (NEQ)

- 2.5 -

** MERKEL METHOD **

SAME AS DETAILED ANALYSIS EXCEPT THAT WATER LOSS TERMS ARE NEGLECTED.

INVOLVES AN ITERATIVE SOLUTION IN TERMS OF THE EXIT WATER TEMPERATURE

(TW2). THE EQUATION SOLVER CALLS THE SUBROUTINE MODEL AT EACH ITERATION.

THE RESULTS OF THE DETAILED ANALYSIS ARE USED AS AN INITIAL GUESS FOR

THIS SOLUTION.

MODE = 2

CALL SOLVER (1, DIFF, X,F,W,MODEL, TOL)

TW2 = X(I)

HA2 = OUT(IEXP,4)

WA2 = OUT(IEXP,5)

EPSA = (HA2 - HA1) / (HW1 - HAl)

EPSW (TWI - TW2)/(TWl - TWB)

CALL PSYCH (2.6.TA2foTWB2.RH. TDP.WA2.HA2)

C

C

C

C

C

C

,Rpt inn2 Coolino Tnwpr Pprforrmn n ..... ...

I ASa dc- " Cn-Vmn v TnwraPprfnrmnn2 .gA v., ,¥lI


DATA CPW/1.0/,NVOL/30/,HGO/1075.1/,CPV/0.45/,TREF/32./

C

C ** RETRIEVE INPUT DATA **

c

NTU = DATA (IEXP, 1)

PA = DATA (IEXP, 2)

TWI = DATA(IEXP,4)

LE = DATA(IEXP,6)

HAl - DATA(IEXP,7)

WAl = DATA (IEXP, 8)

C

TW2 - X(I)

IF(MODE .EQ. 1) WA2 = X(2)

C

C ** INITIALIZATIONS **

C

DV = 1./NVOL

WA = WAl

HA = HAl

TW = TW2

C

C ** LOOP THROUGH TOWER SECTIONS **

C

C THE TOWER IS DIVIDED INTO NVOL SECTIONS. A SIMPLE FORWARD DIFFERENCING

C NUMERICAL INTEGRATION SCHEME IS UTILIZED. THE FIRST SECTION EVALUATED IS

C AT THE AIR INLET.

C

DO 100 I = ,NVOL

CALL PSYCH (2,1, TW, TWRH, TDP,WW, HW)

IF(MODE .EQ. 1) THEN

DWA = AMAX1(0.,NTU*(WW - WA)*DV)

CALL PSYCH (2, 6,TA, TDUM, RHTDPfWA, HA)

HGW= HGO + CPV*(TW - TREF)

DHA = LE*NTU*DV*((HW - HA) + (WW-WA)*(1./LE-I.)*RGW)

DHA = AMAXI(0.,DHA)

DTW = (DHA - CPW* (TW-TREF) *DWA) /(RA - (WA2 - WA) )/CPW

ELSE

-2.6 -

.qpetinn 2 Cooling Tower Performannep Cnnn ; ....

Section 2 Cooling Tower Performance Comparisons

DWA = 0.0

DHA = AMAX1(0.,NTU* (HW - HA) *DV)

DTW = AMAX1(0.,DHA/RA/CPW)

ENDIF

C

C UPDATE AIR AND WATER STATES FOR THIS SECTION

C

WA = WA + DWA

HA = HA + DHA

TW = TW + DTW

100 CONTINUE

C

C ** FUNCTION EVALUTION AND OUTPUTS **

C

C AT THE SOLUTION, THE WATER'TEMPERATURE FOR THE LAST SECTION SHOULD

C MATCH THE INLET WATER TEMPERATURE. IN MODE 2, THE HUMIDITY RATIO FOR

C THE LAST SECTION SHOULD BE EQUAL TO THE ITERATION VALUE.

C

F(1) = TWI -TW

C


F(2) = WA2 - WA

OUT(IEXP,I) = HA

ELSE

OUT(IEXP,4) = HA

OUT(IEXP,5) = WA

ENDIF

C

RETURN

END

C

o *

C EFFECTIVENESS MODEL FOR A COOLING TOWER. *

C *

C

SUBROUT INE EFFEC (TW2 , EP S, WA2 , HA2 )

- 2.7-

REAL NTU,MSTAR

COMMON /DATAS/ DATA (100,9) ,OUT (100, 9) ,MODE, IEXP

DATA CPW/I./,TREF/32./

C

C ** STATEMENT FUNCTION **

C

C CORRELATION FOR SATURATION TEMPERATURE IN TERMS OF SATURATION

C ENTHALPY: CORRELATION IS IN ENGLISH UNITS

C

TSAT(HSAT) = -0.8811 + 3.340*HSAT - 4.907E-02*HSAT**2 +

4.01E-04*HSAT**3 - 1.303E-06*HSAT**4

C


C

NTU = DATA( IEXP,1)

RAl = DATA (IEXP, 2)

TWB = DATA(:IEXP,3)

TWI = DATA(IEXP,4)

TAl = DATA (IEXP,5)

HA1 = DATA( IEXP,7)

WAI = DATA(I EXP,8)

C


C

CALL PSYCH(2,1,TW1,TW1,RHTDPWW1,HW1)

ITER = 0

TW2 = TWB

RA2 = RAI

C

C ** ITERATIVE LOOP **

C

10 ITER = ITER + 1

CALL PSYCH ( 2, 1,TW2, TW2, RH, TDP, WW2, HW2 )

TOLD = TW2

CS = (HWI - HW2) /(TWI - TW2)

MSTAR = CS/RAI/CPW

C = EXP (-NTU* (I.-MSTAR))

EPS = (1. - C)/(i. - MSTAR*C)

- 2.8-

,Qaetinn I Coonlingr Towpr Pprformfineo inm.n-; .... I

Vla .a.......,u, uw u reriurmance L omarlsons

HA2 = HAl + EPS*(HW1 - HAl)

HW = HA. + (HA2 - HA1)/(l. - EXP(-NTU))

TW = TSAT (HW)

CALL PSYCH (2, 1, TW, TW, RH, TDP, WW, HW)

WA2 = WW + (WAI - WW)*EXP(-NTU)

RA2 = RAl - (WA2 - WAl)

TW2 = TREF + (RA1*CPW*(TW1-TREF) - (HA2 - HA1))/RA2/CPW

IF(ABS(TW2 - TOLD) .GT. 1.E-03 .AND. ITER .LT. 20) GOTO 10

c

RETURN

END

c

- 2.9-

,Rpetinn I Cfd%];Imdv olrrwwy^se D^oetAmmem - - fN a

C~ifn I -Ol.J,.|I~JI - ~~'JL..7U~ :--_ 'uuv a l ''E lIillo,.Li ,tU hIIIJ4lI-IsoI)

C *

C PROGRAM FOR CALCULATING THE PERFORMANCE OF A COUNTERFLOW COOLING COIL. *

C TWO METHODS ARE COMPARED: DETAILED SOLUTION OF BASIC HEAT AND MASS TRANS- *

C FER EQUATIONS AND AN EFFECTIVENESS MODEL. THE UNITS UTILIZED WITHIN THIS *

C PROGRAM ARE MODIFIED SI. THE MODELS ARE DESCRIBED IN CHAPTER 4 OF *

C "METHODOLOGIES FOR DESIGN AND CONTROL OF CENTRAL COOLING PLANTS", *

C J.E. BRAUN, PH.D THESIS, UNIVERSITY OF WISCONSIN - MADISON, 1988 *

C *

C

REAL NTUO (6),RAS (3),LENTUR

COMMON /DATAS/ DATA(100,9),OUT(100,9),IEXP

DATA NTUO/0.5,1.,I1.5,2.,3.,4./

DATA RAS/0.5,1.,2./

C

C ** OPEN INPUT AND OUTPUT FILES **

C

OPEN(1OFILE='COILCOMPARE.DAT',STATUS='OLD')

OPEN(11,FILE='COILCOMPARE.OUT',STATUS='NEW')

C

C ** LOOP TO READ INPUT DATA **

C

5 READ(10, *,END=100) TDB, TWB, TWI, LE, NTUR

C

C INPUT DATA:

C

C TDB - AMBIENT DRY BULB TEMPERATURE (DEGREES C)

C TWB = AMBIENT WET BULB TEMPERATURE (DEGREES C)

C TWI = INLET TEMPERATURE OF WATER TO COOLING COIL (DEGREES C)

C LE = LEWIS NUMBER

C NTUR - RATIO OF NUMBER OF TRANSFER UNITS FOR WATER FLOW ON INSIDE

C SURFACE TO NUMBER OF TRANSFER UNITS FOR AIR FLOW ON OUTSIDE

C SURFACE

C

C

C ** STORE DATA FOR A RANGE OF COIL PARAMETERS **

C

- 2.10-

C nnl| inv Cill Iprfnrrn . P ... • ..

Cnnling Coil Performne Cnmnn rcnca .rr U ~'uU

1LIE3JI

C NTUO = NUMBER OF TRANSFER UNITS FOR AIR FLOW ON OUTSIDE SURFACE

C RAS = RATIO OF WATER FLOW TO AIR FLOW RATE

C

NEXP = 0

DO 10 NRA = 1,3

DO 10 NNTU = 1,6

IF(NRA .NE. 1 .OR. NNTU .NE. 6) THEN

NEXP = NEXP + 1

DATA (NEXP, 1) = NTUO (NNTU)

DATA(NEXP, 2) = RAS (NRA)

DATA(NEXP,3) = TWB

DATA(NEXP,4) = TWI

DATA(NEXP,5) = TDB

DATA(NEXP,6) = LE

DATA(NEXP,7) = NTUR

ENDIF

10 CONTINUE

C

C ** COIL ANALYSIS **

C

C RESULTS ARE STORED IN THE OUT ARRAY FOR ALL SETS OF COIL DATA

C

CALL COIL (NEXP)

C

C ** OUTPUT RESULTS TO FILE **

C

WRITE(II,101) TDB, TWB, TWI, LE, NTUR

WRITE (11, 102)

DO 20 IEXP = 1,NEXP

WRITE(11,103) (DATA(IEXPJ),J=1,2), (OUT(IEXPJ),J=I,6)

20 CONTINUE

GOTO 5

C

100 CONTINUE

C

101 FORMAT(///IX,'TDB, TWB, TWI, LE, NTUI/NTUO =f ',5(IX, F6.2)/)102 FOMA(/1X' ** FINITE DIFFERENCE **

'** EFFECTIVENESS **'//

.2.11 -

Qaef;nn I..I

Qenn7kjc; ,i J.v4,- dlu Ia I I Iv3 ILI5, .JIIU I-1uU !a t

5X,'NTU FLCW/FLA EPSA EPST FDRY',

EPSA EPST FDRY'/)

103 FORMAT (8 (3X,F6.3))

C

END

C

SUBROUTINE COIL (NEXP)

C

C *

C ROUTINE FOR CALCULATING COOLING COIL PERFORMANCE. *

C *

C

EXTERNAL MODEL

DOUBLE PRECISION X(1),F(1),W(II),DIFF(1),TOL


DATA TOL/I.E-05/,DIFF/0.001/

C

C***************** LOOP THROUGH ALL SETS OF COIL DATA *

C

C NOMENCLATURE:

C

C TWB - ENTERING AMBIENT WET BULB TEMPERATURE (DEGREES C)

C TW = WATER TEMPERATURE (DEGREES C)

C TA = AIR TEMPERATURE (DEGREES C)

C TS = SURFACE TEMPERATURE (DEGREES F)

C TDP = AIR DEWPOINT TEMPERATURE (DEGREES C)

C WA = AIR HUMIDITY RATIO

C HA = AIR SPECIFIC ENTHALPY (KJ/KG)

C WW = HUMIDITY RATIO ASSOCIATED WITH SATURATED AIR AT TW

C HW = SPECIFIC ENTHALPY ASSOCIATED WITH SATURATED AIR AT TW (KJ/KG)

C EPST = AIR STREAM TEMPERATURE EFFECTIVENESS

C EPSA = AIR STREAM HEAT TRANSFER EFFECTIVENESS

C

C THE STATES 1 AND 2 REFER TO ENTERING AND EXITING CONDITIONS

C (1=ENTERING, 2 = EXITING) .

C

DO 100 IEXP = I,NEXP

-2.12-

r~nnlin(a l Pprfnrmna . .. :..

13ec~.tIU L V~ '-e % IE I ulI I J iplue %,.

c

TWB = DATA(IEXP,3)

TWI = DATA(IEXP, 4)

TAl - DATA (IEXP ,5)

CALL PSYCH(1, ITA1,TWB, RHA1,TDPAIWAIHA1)

DATA(IEXP,8) = HAl

DATA(IEXP,9) = WAl

CALL PSYCH ( 1, 1,TW1, TW1,RHW1, TDPWI, WW1, HW1)

C

C ** EFFECTIVENESS-NTU MODEL **

C

CALL EFFEC (TW2, EPSA, TA2, HA2)

EPST = (TAl - TA2) /(TA1 - TWI)

OUT(IEXP,4) = EPSA

OUT(IEXP,5) = EPST

C

C ** DETAILED ANALYSIS **

C

C THE DETAILED COOLING COIL ANALYSIS INVOLVES AN ITERATIVE SOLUTION IN

C TERMS OF THE EXIT WATER TEMPERATURE (TW2). THE EQUATION SOLVER CALLS

C THE SUBROUTINE MODEL AT EACH ITERATION. THE RESULTS OF THE EFFECTIVENESS

C ANALYSIS ARE USED AS AN INITIAL GUESS FOR THIS SOLUTION.

C

X(1) = TW2

CALL SOLVER (IDIFF, X, F, W, MODEL, TOL)

C

TW2 = X(I)

HA2 = OUT(IEXP,I)

TA2 = OUT(IEXP,2)

EPSA = (HAl - HA2)/(HA1 - HW1)

EPST = (TAl - TA2)/(TAI - TWI)

OUT(IEXP,I) = EPSA

OUT(IEXP,2) = EPST

C10 0 CONTINUE

C

RETURN

END

- 2.13 -

Cnnl;na rn;l Parfhrrn*anrAm

.qpt;nfl )

C

SUBROUTINE MODEL (NEQ, X, F)

C* ** **** *** ** ** *** * ** ** * *** ***** ** * *** ****** *** * *** *** * ** ** * ******* *** ***** * * *

C *

C DETAILED MODEL OF A COOLING COIL THAT SOLVES BASIC HEAT AND MASS TRANSFER *

C EQUATIONS USING FINITE-DIFFERENCES. THIS ROUTINE IS CALLED AT EACH ITERA- *

C TION OF AN EQUATION SOLVER THAT IS FINDING THE CORRECT LEAVING WATER *

C TEMPERATURE. ,

C *

C

REAL NTUO, NTUI, NTU, LE



DATA CPW/4.19/,HGO/2501./,CPA/1.012/,CPV/I.805/

DATA TREF/0./,NAREA/30/

C


C

NTUO = DATA(IEXP,I)

RA = DATA (IEXP, 2)

TWI = DATA(IEXP,4)

TAl = DATA(IEXP,5)

LE = DATA(IEXP, 6)

NTUI = DATA (IEXP,7) *NTUO

HAl = DATA(IEXP,8)

WAl = DATA (IEXP, 9)

C

TW2 = X(1)

C


C

DA = I./NAREA

WA = WAl

HA = HA1

TW = TW2

TA =TAI

NDRY = 0

- 2.14 -

7%.. ,',,, l, - ,,...,,,J,,,r-, .,Xjt, t Iva '-veIllaillu " I, UIIUarIsollI Cnnl;na rni Pprfnrmnnr%,o ... ..

a3 |IUII 1 -,..,, I .I..- - ., I %JI. - i % %,,v Li I l I Ull1,v

C

c ** LOOP THROUGH COIL SECTIONS **

C

C THE COIL IS DIVIDED INTO NAREA SECTIONS. A SIMPLE FORWARD DIFFERENCING

C NUMERICAL INTEGRATION SCHEME IS UTILIZED. THE FIRST SECTION EVALUATED IS

C AT THE AIR INLET.

C

DO 100 I = 1,NAREA

C

C ** DRY ANALYSIS **

C

C ASSUME COIL SECTION IS COMPLETELY DRY

C

CPM = CPA + WA*CPV

CSTAR = CPM/RA/CPW

NTU = NTUO/(i. + NTUO/NTUI*CSTAR)

DWA = 0.

DHA = CPM*NTU*(TW - TA) *DA

DTW = DHA/RA/CPW

TSDRY = TW + CSTAR*NTU/NTUI*(TA - TW)

IF(I .EQ. 1) TS = TSDRY

CALL PSYCH(1, 4,TA, TWBSRHSTDPS,WA, HAS)

IF(TSDRY .GT. TDPS) THEN

TS = TSDRY

NDRY I

ELSE

C

C ** WET ANALYSIS **

C

C SURFACE TEMPERATURE IS LESS THAN DEWPOINT, SO ASSUME COMPLETELY WET

C

CALL PSYCH (I, I,1TW, TW, RHW, TDPW, WW, HW)

CALL PSYCH(1,1,TS,TS,RHS,TDPSWSHS)

HGS = CPV*(TS-TREF) + HGODWA = NTUO/LE* (WS - WA) *DA

DHA = NTUO*DA*( (HS-HA) + (WS-WA) *(1./LE-1.) *HGS)

DTW = (DHA - DWA*CPW* (TS-TREF))/RA/CPW

TS =TW - DTW/DA/NTUI

- 2.15 -

C~nalinaoinl Pprformqna n /mnr; ....

I3A lUll 4'.v ti r . A Us •a au *aa ui .. uIIIL ns II

END IF

C

C UPDATE AIR AND WATER STATES FOR THIS SECTION

C

WA = WA + DWA

HA = HA + DHA

TW = TW + DTW

CALL PSYCH (1, 6,TA, TWBA, RHA, TDPA, WA, HA)

C

100 CONTINUE

C

C ** FUNCTION EVALUTION **

C

C

C

C

C

C

C

AT THE SOLUTION, THE WATER TEMPERATURE FOR THE LAST SECTION SHOULD

MATCH THE INLET WATER TEMPERATURE.

F(1) = TWI - TW

** SET OUTPUTS **

OUT(IEXP,1) = HA

OUT(IEXP,2) = TA

OUT (IEXP, 3) = FLOAT (NDRY) /FLOAT (NAREA)

C

RETURN

END

C

C *

C EFFECTIVENESS MODEL FOR A COOLING COIL. *

C *

SUBROUTINE EFFEC (TW2, EPSA, TA2, HA2)

REAL NTUO, NTUI, NTU, NTUD, NTUW, MSTAR


DATA CPW/4.19/,CPA/1.012/,CPV/1.805/

C

** STATEMENT FUNCTIONS **

rnnl;na Cnil Pprfnrrnnnro

L, rvt.lvll A .IL .,% -e x.va ,m , ,a-. , i.%ll flI U I

C

C CORRELATION FOR SATURATION TEMPERATURE IN TERMS OF SATURATION

C ENTHALPY: CORRELATION IS IN SI UNITS

C

TSAT(HS) -- 5.75904E+00 + 6.58002E-01*HS

- 4.82511E-03*HS**2 + 2.39673E-05*HS**3

- 6.47307E-08*HS**4 + 7.08692E-11*HS**5

C

C EFFECTIVENESS RELATIONS FOR COUNTERFLOW HEAT EXCHANGERS

C

C(NTURATIO) = EXP(-NTU*(1.-RATIO))

EPS(NTU, RATIO) = (1. - C(NTU,RATIO))/(1. - RATIO*C(NTU,RATIO))

C


C

NTUO = DATA(IEXP,1)

RA = DATA(IEXP,2)

TWB = DATA (IEXP,3)

TWI = DATA(IEXP,4)

TAl = DATA(IEXP,5)

NTUI = DATA(IEXP,7)*NTUO

C

C ** PROPERTIES FOR INLET CONDITIONS **

C

CALL PSYCH(1,ITAITWB, RHTDPWAIfHA1)

CALL PSYCH(I,1,TWITW1,RH1,TDP1,WW1,HW1)

C

C ** DRY ANALYSIS **

C

C INITIALLY ASSUME THAT THE COIL IS COMPLETELY DRY

C

FDRY = 1.CPM = CPA + WA1*CPV

CSTAR = CPM/RA/CPW

NTUD - NTUJO/ (I. + NTUO/NTUI*CSTAR)

EPSD = EPS(NTUD,CSTAR)

TA2 = TAI - EPSD*(TAI - TWI)

HA2 = HA1. - CPM* (TAI - TA2)

-2.17 -

Cnnlina o i Pprformn~nep rtmnq.-;: ....C,,,,t;nnI

, tinnIa Aw %Vi 11-., v,, - , u.o X ., ,v,&--r ,'tL ll Ua L II U l ISonl5

WA2 = WAl

TW2 = TWI + (HAl - HA2)/RA/CPW

TS2 = TWI + CSTAR*NTUD/NTUI*(TA2 - TWI)

IF(TS2 .LT. TDP) THEN

C

C ** WET COIL **

C

C THE SURFACE TEMPERATURE AT THE WATER INLET IS LESS THAN THE DEWPONT OF

C THE AIR. ASSUME THAT THE COIL IS COMPLETELY WET.

C

FDRY = 0.

ITER = 0

10 ITER = ITER + 1

TOLD = TW2

CALL PSYCH(1, 1,TW2,TW2,RHW2,TDPW2,WW2,HW2)

CS = (HW2 - HW1)/(TW2 - TWI)

MSTAR = CS/RA/CPW

NTUW = NTUO/(1. + NTUO/NTUI*MSTAR)

EPSW = EPS (NTUW, MSTAR)

HA2 = HAl - EPSW*(HAl - HWl)

TW2 = TWI + (HAl - HA2)/RA/CPW

IF(ABS(TW2 - TOLD) .GT. 1.E-03 .AND. ITER .LT. 20) GOTO 10

TAX = TAl

TS1 = TW2 + CSTAR/CPM*NTUW/NTUI* (HAl - HW2)

IF(TS1 .GT. TDP) THEN

C

C ** PARTIALLY WET AND DRY **

C

C THE SURFACE TEMPERATURE AT THE AIR INLET (WATER OUTLET IS GREATER THAN

C THE DEWPOINT OF THE INLET AIR, SO THE COIL IS PARTIAL WET (EXIT SECTION)

C AND PARTIALLY DRY (AIR INLET SECTION). ITERATIVELY DETERMINE THE FRACTION

C OF THE COIL THAT IS DRY.

C

TWX = TW2

FDRY = 0.

ITER = 0

20 ITER - ITER + 1

TOLD = TW2

-2.18-

Coolina o il Prform~ne ... ..

De i !uII4-- n I i nn llticzu lkI parII l r s

EXPK = ((TDP-TW2) + CSTAR*(TA1-TDP))/

(1. - NTUD*(1. - CSTAR)/NTUO)/(TAI-TW2)

EXPK = AMAX1(1.E-06,EXPK)

FDRY = -1./NTUD/(1. - CSTAR)*ALOG(EXPK)

FDRY = AMAX1(0.,AMIN1(1.,FDRY))

EPSD = EPS(FDRY*NTUD,CSTAR)

CALL PSYCH (I 1,TWX,TWXRRHWXTDPWX, WWX, HWX)

CS - (HWX - HW1)/(TWX - TWI)

MSTAR = CS/RA/CPW

NTUW = NTUO/(1. + NTUO/NTUI*MSTAR)

EPSW = EPS((1.-FDRY)*NTUW,MSTAR)

TWX = (TWI + CSTAR/CPM*EPSW*(HA1 -EPSD*CPM*TA1 - HW1))/

(1. - CSTAR*EPSW*EPSD)

TW2 = CSTAR*EPSD*TAI + (1. - CSTAR*EPSD)*TWX

IF(ABS(TW2-TOLD) .GT. 1.E-03 .AND. ITER .LT. 20) GOTO 20

HA2 = HAl - RA*CPW* (TW2 - TWI)

TAX = TAl - EPSD*(TAI - TWX)

END IF

C

C ** DETERMINE EXIT AIR CONDITIONS **

C

HAX = HAl - CPM* (TAI -TAX)

HSW = HAX + (HA2 - HAX)/(I. - EXP(-(l.-FDRY)*NTUO))

TSW = TSAT(HSW)

TA2 = TSW + (TAX - TSW)*EXP(-(I.-FDRY)*NTUO)

ENDIF

C

C ** SET OUTPUTS **

C

EPSA = (HAl - HA2)/(HAl - HWl)

OUT(IEXP,6) = FDRY

C

C

RETURN

END

- 2.19-

t"qnnl;na n "t;I ,"Parfhrr a ^^ . .. ". .

Section 3 Component-Based Optimization Program

Main Program 3.1Primarily calls other routines for reading input system description and timedependent data, minimizing costs, and outputting results. Loops through timeand at each time evaluates every mode of operation for minimum cost.

I/O Routines 3.5Routines for inputting and outputting data and results.

Steady-State Optimization 3.14Routines for determining the values of continuous control variables thatminimize the instantaneous cost of operation for specified modes anduncontrolled variables.

System Model 3.28Routines for solving the equations that model the system that is specified.

Component Models 3.33Routines for modeling the individual component performance for chillers,air handlers, pumps, etc.

Sample Data 3.59Sample data for a system input description and time-dependent uncontrolledvariables.

- 3.0 -

Mnin Prnornm

C

C *

C Program for determining the optimal steady-state values of controls that *

C minimize instantaneous operating costs of chiller plants through time as *

C described in Chapter 6 of "Methodologies for the Design and Control of *

C Central Cooling systems", Jim Braun, PhD Thesis, UW-Madison, 1988.

C *

C

Implicit Real*8 (A-H,O-Z)

Integer OptMode

Logical Bounded

Parameter (MaxU = 10, MaxStr= 2, NUmax = 20, NYmax = 50,

0 NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,

0 NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension mapF (NFmax), mapM(NMVmax), mapX (NYmax), Par (NPmax),

& Modes(NMVmax), Info(MaxInfMaxU), U(NUmax),

0 Y(NYmax), Outs(NOmax), RData(NFmax) , F(NFmax),

0 Uopt(NUmax), Yopt(NYmax), Oopt(NOmax)

Common /ModeDat/ NModes, NModeV, ModeV(NMVmax,NMmax)

Common /Limits/ Umin (NUmax), Umax (NUmax)

Common /Tolerance/ CostTol, DerivTol, SolnTol

C

C********* Set Tolerances for optimization and solution of equations *

C

CostTol = l.e-06

DerivTol = 1.e-06

SolnTol = 1.e-08

C

C* ************************* Get Input Description *****************************

CC Descriptions of the data are given in the Input subroutine.

C

Call Input(Ntimes, *Nunits, NData, Par, mapF, mapM, mapX,

*Info, U, Y)

C

C** Total Number of Uncontrolled, Discrete Control, Continuous Control,

-3.1-

Qart;nn 'A-

C Stream Output, and Printed Output Variables **

IC

NFmap= Info(4, Nunits+1) - 1

NMmap = Info(5, Nunits+1) - 1

NU = Info(6, Nunits+1) - 1

NY = Info(8, Nunits+1) - 1

NO = Info(9, Nunits+1) - 1

C

*************************** Main Loop over Time ******************************

C

Do 200 itime 1,Ntimes

C

C ------- Read New Data and Copy into Sequential List for Components

C

If(Ndata .gt. 0 .and. NFmap .gt. 0) Then

Call Reader(Ndata, RData)

Call CopyF(NFmap, mapF, RData, F)

Endif

C

C ------------------ Costs for First Discrete Control Mode

C

OptMode = 1

C

C** Copy initial values of mode variables into sequential list for components *

C

If(NMmap .gt. 0) Then

Call Copy_Modes(OptMode, NMmap, mapM, Modes)

Endif

C

C** Determine costs associated with optimal continuous control variables **

C

Call Costs(Nunits, Par, F, Modes, mapX, Info, U, Y, Outs,• OptCost, Penalty, Bounded)

Do 10 i = I,NU

Uopt(i) = U(i)

10 Continue

Do 20 i = 1,NO

Oopt(i) = Outs(i)

-3.2-

Main Prorm1-Q ^41% 91% r1k I

~j~~.,LI~)uE .- ' &'iutii I uu~taaum

20 Continue

If (.not. (Bounded)) Then

OptCost = l.e+30

Endif

C

If(NModes .gt. 1) Then

C ------------------------- Evaluate All Mode Possibilities

C

Do 100 Mode = 2,NModes

Call CopyModes(Mode, NMmap, mapM, Modes)

Call Costs(Nunits, Par, F, Modes, mapX, Info, U, Y, Outs,

Cost, Pen, Bounded)

CC** Store controls associated with minimum cost **

C

If (Cost .lt. OptCost

OptMode = Mode

Do 40 i = 1,NU

Uopt(i) = U(i)

Continue

Do 50 i = 1,NY

Yopt(i) = Y(i)

Continue

Do 60 i - 1,NO

Oopt(i) = Outs(i)

Continue

OptCost = Cost

Penalty = Pen

Endif

Continue

.and. Bounded) Then

Endif

C

C ------------------------------- Print Output----------------------------------

C

Call Printer(itime, NU, NY, OptMode, Uopt, Y, Oopt,

OptCost, Penalty)

C

- 3.3-

40

50

60

100

C

UnI'o;n Pr,-n,-far"ICodf;nn I

Main ProgramSection 3

200 Continue

End

3.4.

jjC%;|IVI| - 0Ju \ U|I¢

C

C *

C Routine for reading input data for describing a system. *

C *

C

Subroutine Input(Ntimes, Nunits, NData, Par, mapF, mapM,

mapX, Info, U, Y)


Parameter (MaxU = 10, MaxStr = 2, NUmax = 20, NYmax = 50,

NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,

• NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension Par(NPmax), mapF (NFmax), mapM(NMVmax), mapX(NYmax),

• Info(MaxInfMaxU), U(NUmax), X(NYmax), Y(NYmax)

Dimension NTF (MaxU), NTP (MaxU), NTM(MaxU), NTU(MaxU), NTC (MaxU),

0 NTS (MaxU), NTO (NOmax)

Dimension InUnit (MaxStr,MaxU), InNum(MaxStr,MaxU),

• IUOut(NOmax), IVOut(NOmax)

Common /ModeDat/ NModes, NModeV, ModeV(NMVmax,NMmax)

Common /Limits/ Umin(NUmax), Umax(NUmax)

Common /Tears/ NTears, mapT(NYmax)

Common /Print/ NPrt, NOuts(5), ListO(10,5)

C

S********************* Get information concerning Types **********************

C

C NTypes number of total types available

C NStrV - number of stream variables within a stream

C NTP number of parameters for each type

C NTF - number of forcing functions for each type

C NTM number of mode variables for each type

C NTU - number of continuous control variables for each typeC NTC - number of equality constraints for each type

C NTS - number of streams for each type

C NTO - number of component outputs for printer (not stream variables)

C

NTypes = 6

NStrV = 2

-3.5 -

Qatt;nn I T/(") IDAllvt;nnc

C

C -------------------- Initial Call to Components Gets Info

C

Do 9 itype = 1,NTypes

Info(l, itype) - 0

goto (1, 2, 3, 4, 5, 6) ,itype

1. Call Typel(Info(l, itype))

goto 7

2 Call Type2(Info(l, itype))

goto 7

3 Call Type3(Info(l,itype))

goto 7

4 Call Type4(Info(l,itype))

goto 7

5 Call Type5 (Info (1, it-ype) )

goto 7

6 Call Type6(Info(litype))

7 Continue

NTP(itype) = Info(3,itype)

NTF(itype) = Info(4, itype)

NTM(itype) = Info(5,itype)

NTU(itype) = Info(6,itype)

NTC(itype) = Info(7,itype)

NTS(itype) = Info(8,itype)

NTO(itype) = Info(9,itype)

9 Continue

c

************************* Get System Descripton *

C

C---------------------------- General Simulation Data

C

C Ntimes - number simulation timestepsC Nunits - number of total components

C NData - number of forcing function data items to be read each time

C NModeV - number of discrete control (mode) variables

C NModes - number of possible combinations of mode variables

C ModeV - two-dimensional array of mode variables for each possible

C mode

- 3.6-

qIRptinn3 T/O" Rontine,

(atnIn

C

Read(10,*) Ntimes, Nunits, NData

Read(10,*) NModeV, NModes

If(NModeV .gt. 0 .and. NModes .gt. 0) Then

Do 11 j = 1,NModes

Read(10,*) (ModeV(ij),i=lNModeV)

11 Continue

Endif

C

C** Intialize pointers to data **

C

Do 12 i = 3,9

Info(i,i) = 1

12 Continue

C

C ----------------------- Data Concerning Each Component

C

C iu - unit number

C itype - type number

C

Do 30 iunit = 1, Nunits

Read(10,*) iu, itype

Info (liunit) = iu

Info(2,iunit) = itype

C

C** Component Parameters **

C

C ipP - pointer in Par array to parameters for iunit

C

ipP = Info(3,iunit)

Info(3, iunit+l) = ipP + NTP(itype)

If(NTP(itype) .gt. 0) Then

Read(10,*) (Par(i),i=ipP,ipP+NTP(itype)-l)

Endif

C** Forcing Function Data **

C

C ipF - pointer in F array to forcing function data for iunit

C mapF - mapping between F array and data items read each time

- 3.7 -

IUutlllcbI T/l') 12,.v,,1;,.,a,

Section 3

C mapF(i) gives the data item associated with the ith position

C in the F array

C

ipF = Info(4, iunit)

Info(4, iunit+1) = ipf + NTF(itype)

If(NTF(itype) .gt. 0) Then

Read(10,*) (mapF(i),i=ipf,ipf+NTF(itype)-l)

Endif

C

C** Discrete Control Mode Data **

C

C ipM - pointer in Modes array to mode data for iunit

C mapM - mapping between Modes array and list of mode variables

C mapM(i) gives the mode variable associated with the ith

C position in the Modes array

C

ipM - Info(5,iunit)

Info(5,iunit+1) = ipM + NTM(itype)

If(NTM(itype) .gt. 0) Then

Read(10,*) (mapM(i), i=ipMipM+NTM(itype)-1)

Endif

C

C** Continuous Control Data **

C

C ipU - pointer in U array to continuous control varibles for iunit

C U - initial values of continuous controls

C Umin - minimum value of continous controls

C Umax - maximum values of continous controls

C

ipU = Info(6,iunit)

Info(6,iunit+l) = ipU + NTU(itype)

If(NTU(itype) .gt. 0) ThenDo 15 i - ipU, ipU+NTU(itype)-I

Read(10,*) U(i), Umin(i), Umax(i)

15 Continue

Endif

Info(7, iunit+1) = Info(7,iunit) + NTC(itype)

- 3.8-

1/(0 Rmntinpo.

C

C** Stream Data **

C

C ipX - pointer in X array to output stream variables for iunit

C InUnit - unit number providing inputs to iunit

C InNum - number of output stream providing inputs

C X - initial value of input stream variables

C

ipX = Info(8,iunit)

Info(8,iunit+l) = ipX + NStrV*NTS(itype)

Do 20 i = l,NTS(itype)

ipX = ipX + (i-1)*NStrV

Read(10,*) InUnit(i,iunit), InNum(i, iunit)

Read(10,*) (X(j), j = ipX, ipX+NStrV-1)

20 Continue

C

C************************ Pointers to Component Outputs *

C

ipO = Info(9,iunit)

Info(9,iunit+l) = ipO + NTO(itype)

C

30 Continue

C

******************************** Tear Streams *********************************

C

Read(10,*) NTears

If(NTears .gt. 0) Then

Do 40 i = l,NTears

Read(10, *) iunit, istr,ivar

mapT(i) = Info(8,iunit)+ NStrV*(istr-1) + ivar - 1

40 Continue

EndifC

*** **********~~******** Printer Output List * ** * *** * * ********* ** ****** *** *

C

Read(10,*) NPrt

If(NPrt .gt. 0) Then

NPrt -- Min0(NPrt,5)

Do 55 iprt = l,NPrt

-3.9 -

Spetion _3 T/O Rmitinpr.

Read(10,*) NOuts(iprt)

If(NOuts(iprt) .gt. 0) Then

Read(10,*) ((IUOut(i), IVOut(i)), i = 1,NOuts(iprt))

Do 50 i =1,NOuts(iprt)

iu = IUOut(i)

num= 0

45 num = num + 1

If(iu .ne. Info(l,num) .and. num .1t. Nunits) goto 45

ipO -=Info(9,num)

ListO(i, iprt) = ipO + IVOut(i) - 1

50 Continue

Endif

55 Continue

Endif

C

C********************* Determine Pointers for Input Mapping *

C

C mapX - mapping between Y array and X array

C mapX(i) gives the position in Y associated with the ith

C position in the X array

C

Do 100 iunit = 1, Nunits

itype = Info(2,iunit)

Do 80 i 1, NTS(itype)

InU = InUnit(i,iunit)

num= 0

60 num = num + 1

If(InU .ne. Info(1,num) .and. num .1t. Nunits) goto 60

If(InU .eq. Info(1,num)) Then

ipX = Info(8,iunit) + (i-l)*NStrV

ipY = Info(8,num) + (InNum(i,iunit)-l)*NStrV

Do 70 j = 1,NStrVmapX(ipX+j-1) = ipY + j - 1

Y(ipY+j-1) = X(ipX+j-1)

70 Continue

Endif

80 Continue

100 Continue

- 3.10-

Spetion 3 - T/O Rntitinoc

C

Return

End

C

********************** Read Data as a Function of Time ************************

C

Subroutine Reader(Ndata, RData)


Dimension RData (NData)

C

C** Read Next Set of Data Values **

C

Read(11,*) (RData(i),i=1,Ndata)

C

Return

End

C

C************* Copy Data into Sequential List for Components *

C

Subroutine Copy_F(Nmap, mapF, RData, F)


Dimension RData(1), mapF(Nmap), F(Nmap)

C

Do 10 i = 1,Nmap

ipf = mapF(i)

F(i) = RData(ipf)

10 Continue

C

Return

End

C

C************************ Mode Array for All Components ***********************

C

Subroutine CopyModes(Mode, Nmap, mapM, Modes)Implicit Real*8 (A-H,O-Z)

Parameter (NMmax = 20, NMVmax = 10)

Dimension mapM (Nmap), Modes (Nmnap)

Common /ModeDat! NModes, NModeV, ModeV(NMVmax, NMmax)

- 3.11 -

63(eKtiull j__-* - IIL" LUutlllul- ,1i'l Dne..,m.cCae, f.; n n I - -

- 3.12 -

1/0'. Ron,cn.qpetinn ILrtinl a3-YI/O R L 411 %gn

Do 10 i = 1,Nmap

ipm = mapM (i)

Modes (i) = ModeV (ipm, Mode)

10 Continue

C

Return

End

C

********************** *** Optimization Output *****************************

C

Subroutine Printer(itime, NU, NY, Mode, U, Y, Outs, Cost, Penalty)

.Implicit Real*8 (A-H,O-Z)


.. NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,

NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension U(NUmax), Y(NYmax), Outs (NOmax), Output (NOmax)

Common /Print/ NPrt, NOuts(5), ListO(10,5)

C

C********* Print All Controls, Optimal Cost, and Constraint Penalty ***********

C

Write(12,1001) itime, Mode, (U(i) ,i=l,NU), Cost-Penalty, Penalty

C

C********************* Output User Specified Outputs *

C

If(Nprt .gt. 0) Then

Lu = 12

Do 20 iprt = 1, NPrt

Lu = Lu + 1

If(NOuts(iprt) .gt. 0) Then

Do 10 i = 1,NOuts(iprt)

ipO = ListO(iiprt)

Output(i) = Outs(ipO)10 Continue

Write(Lu, 1002) itime, (Output(i),i1l,NOuts(iprt))

Endif

20 Continue

Erndif

~otiuln 3 I/OI RnAVU LItInC3

1001 Format (lx,14,1lx,12,10 (lx,Ipell.3))

1002 Format(lx,14,10(lx,lpell.3))

1003 Format (lx,14, 5 (10 (lx,pell.3)/))

CEnd

-3.13 -

'Retinn3 1/0' R n,,tinp.I

Q-lf;n

C

C *

C This routine determines the costs associated with the current modes of *

C operation at optimal values of the continous control variables. *

C *

C

Subroutine Costs(Nunits, Par, F, Modes, mapX, Info, U, Y, Outs,

Cost, Penalty, Bounded)


Parameter (MaxU = 10, MaxStr= 2, NUmax = 20, NYmax = 50,

• NPmax = 50, NFmax = 10, NCmax = 10, NMVmax 1=0,

• NMmax = 20, NOmax = 50, MaxInf = 10)

Logical Debug, Converge, First, Bounded, UFixed(NUmax)

Dimension Par(NPmax), F (NFmax), Modes (NMVmax), mapX(NYmax),

* Info-(MaxInf,MaxU), U(NUmax), X(NYmax), Y(NYmax),

* Outs(NOmax) , UO(NUmax), UM(NUmax)o, GO(NCmax),

* dGdU (NCmax, NUmax)

Common /CostDat/ AA(NUmaxNUmax), BB(NYmax,NYmax),

CC(NYmax,NUmax), p(NUmax), q(NYmax), r(MaxU)

Common /ConDat/ alpha (NCmax), Beta (NCmax, NUmax)

Common /SolDat/ YO(NYmax), dYdU(NYmax, NUmax), Ahat(NUmax, NUmax),

bhat (NUmax), chat, Ainv(NUmax,NUmax)

Common /Constr/ GC(NCmax)

Common /Limits/ Umin (NUmax), Umax (NUmax)

Common /Tolerance/ CostTol, Deriv Tol, Soln Tol

Data imax/50/, First/.True./

C

diff = DerivTol

Tol = CostTol

Debug = .False.

C

C******** Total Number of controls, constraints, and stream variables *********

C

NU = Info (6,Nunits+1) - 1

NC = Info(7,Nunits+1) - 1

NY = Info (8,Nunits+1) - 1

-3.14-

qtead v-Sqta tp nti;75.1tinn

qpvwtinn -3kjv -,..Vl j - T.... _ 1, JLr LULItl IIJIUil

NO = Info(9,Nunits+l) - 1

C

********* Loops for Determining Optimal Control ******

C

C -------------- Iterative loop for updating quadratic cost models

C

iter = 0

Cost = 0.0

10 Continue

iter = iter + 1

CostL = Cost

C

C --------------------------- Determine Current Costs

C

If(NU .gt. 0) Then

C

C - ** Get Initial Cost On First Call **

C

If(First) Then

Call Solve(Nunits, Par, F, Modes, mapX, Info, U, Y,

Outs, Cost, Penalty)

Do 12 i = 1,NU

UO(i) = U(i)

12 Continue

First = .False.

C

C ** Use Polynomial Interpolation In Search Direction **

C

Else



iloop= 0

13 iloop = iloop + 1

Do 15 i = I,NUUM(i) = (U(i) + UO(i))/2.

15 Continue

Costl = CostL

Call Solve(Nunits, Par, F, Modes, mapX, Info, UM, Y,

- 3.15 -

Steadv-,lRt.itpOnf;m;,7-af;nn

.qpetion -13

Outs, Cost2, Pen2)

Cost3 = Cost

aO = Costi

al = -3.*Costl + 4.*Cost2 - Cost3

a2 = 2.*(Costl - 2.*Cost2 + Cost3)

step = Dminl(l.d0,Dmaxl(0.d0,-0.5d0*al/a2))

Do 20 i = INU

U(i) - UO(i) + step*(U(i) - U0(i))

20 Continue



If(Cost .gt. CostL .and. iloop .1t. 3) goto 13

Do 22 i = 1,NU

UO(i) = U(i)

22 Continue

Endif

C

C ** Output Results to Screen if Debugging **

C

If(Debug) Then

Write(*,*) 'U = ', (U(i),i=1,NU)

Write(*,*) 'GC = ',(GC(i),i=1,NC)

Write(*,*) 'CostCostL = ',Cost.CostL

Write(*,*) 'Y = ',(Y(i),i=1,NY)

Write(*,*) 'Outs = ', (Outs(i),i=1,NO)

Endif

C

C ** Convergence Check **

C

If(abs(Cost) .gt. 0) Then

Ck - abs(Cost-CostL)/Cost

Else

Ck = abs (Cost-CostL)

EndifConverge = Ck .lt. Tol

If(.not. (Converge) .and. iter .lt. imax) Then

CC ** Determine linearization for Outputs and Constraints **

- 3.16-

c~L~auy-~tiuLe uorlmlzaripnj - ...... teauy-atate UDIti mlzationl

C

Do 25 i = 1,NY

YO(i) = Y(i)

25 Continue

Do 30 i = 1,NC

G0(i) = GC(i)

30 Continue

Do 45 j = 1,NU

dU = Dmaxl(diff*U(i),diff)

U(j) = U(j) + dU


Outs, Costj, Penj)

U(j) = U(j) - dU

Do 35 i = 1,NY

dYdU(ifj) = (Yfi) - YO(i))/dU

Y(i) = Y0(i)

35 Continue

If(NC .gt. 0) Then

Do 40 i - 1,NC

dGdU(i, j) = (GC(i) - GO(i))/dU

GC(i) = GO(i)

40 Continue

Endif

45 Continue

C

Do 50 i = 1,NY

Do 50 j = 1,NU

YO(i) = Y0(i) - dYdU(i, j)*UO(j)

50 Continue

C

C ** Constraint Equation: G = alpha + Beta*u **

CIf(NC .gt. 0) Then

Do 60 i- = ,NC

alpha(i) = GO(i)

Do 55 j = I,NU

alpha(i) = alpha(i) - dGdU(i,j)*U0(j)

Beta(i, j) = dGdU(i, j)

- 3.17-

Steadv-,'tqtp ntiri7Tqtinn'Roetinn -1

L3eL lUi -LJQ L5.5- -JL L I UL IillfLIUil

55 Continue

60 Continue

Endif

C

C --------------------- Optimal Continuous Control Variables

C

C

C ** Quadratic Cost Terms**

C

Call Coef (Nunits, Info, NU, NC, NY)

C

C ** Terms for Quadratic Penalty Function from Linearized Constraints **

C

Do 210 i -- ,NU

Do 210 j = 1,NU

Do 210 k= 1,NC

Ahat(ij) - Ahat(ij) + Beta (k,i) *Beta (k,j)

210 Continue

Do 220 i = 1,NU

Do 220 j = 1,NC

bhat(i) = bhat(i) + 2.*Beta(ji)*alpha(j)

220 Continue

Do 230 i = 1,NC

chat - chat + alpha(i)*alpha(i)

230 Continue

C

C ** Get inverse of matrix of coefficients to quadratic terms **

C

Do 310 i = 1,NU

Do 310 j = INU

Ainv(i,j) = Ahat(i,j) + Ahat(ji)

310 Continue

Call Dinvert (NUmax, NU,, Ainv, if lag)

If(iflag .ne. 0) ThenWrite(*, *) 'Ahat not invertible'

Write(*,*) ' Ahat ='

Do 311 i = I,NU

Write (*, *) (Ahat (i, j) ,j=,NU)

311 Continue

- 3.18 -

C!,m^+;fn 'A

Ch2 ' ;9%i

Stop

Endif

C

C

C--------------------- Optimization with Equality Constraints

C

Call Optimize(NC, NU, U)

C

If(Debug) Then

Write(*,*) 'Unconstrained U = ',(U(i),i=1,NU)

Endif

C

C --------------------------- Inequality Constraints

C

C ** Add to Constraints if Controls Outside Limits **

C

NFixed = NC

Do 320 i = 1,NU

UFixed(i) = .False.

320 Continue

330 Nold = NFixed

Do 340 i = 1,NU

If(.not.(UFixed(i)) .and. U(i) .lt. Umin(i)) Then

U(i) = Umin(i)

UFixed(i) = .True.

NFixed = NFixed + 1

alpha(NFixed) = -Umin(i)

Beta(NFixed,i) = 1.0

Endif

If(.not.(UFixed(i)) .and. U(i) .gt. Umax(i)) Then

U(i) - Umax(i)

UFixed(i) = .True.

NFixed = NFixed + 1

alpha(NFixed) = Umax(i)Beta(NFixed, i) = -1.0

Endif

340 Continue

C

- 3.19-

3-ectJI.L Y 5 Ya7lULta rz Lt IIlplaijJII

C ** Can't have more constraints than controls **

C

If(NFixed .gt. NU) Then

Write(*,*) 'Warning: Constraints not Satisfied'

Else

C

If(NFixed .ne. Nold) Then

C

C ** Redo Constrained Optimization **

C

Call Optimize(NFixed, NU, U)

If(Debug) Then

Write(*,*) 'Constrained U = ' (U(i) i-i NU)

Endif

goto 330

Endif

C

Endif

goto 10

Endif

Endif

C

If (.not. (Converge)) Then

Write(*,*) ' Optimization Loop not Converged: Ck- ',Ck

Endif

C

C ----------------------------- Check Total Cost

C

If(Debug) Then

CostCk = chat

If(NU .gt. 0) Then

Do 410 i- 1,NUCostCk = CostCk + bhat (i) *UJ(i)

temp = 0.0

Do 400 j = I,NU

temp - temp + Ahat(i, j)*UJ(j)

400 Continue

CostCk = Cost Ck + U(i)*temp

410 Continue

- 3.20 -

Qad-fl*nn I -

I petonnI,.

Endif

Write(*,*) 'Cost Check = ',CostCk

Endif

C

* ** Check for Controls Within Bounds *

C

Bounded = .True.

Do 420 i = 1,NU

If( U(i) .lt. (Umin(i)-Tol) .or. U(i) .gt. (Umax(i)+Tol) ) Then

Bounded = .False.

Endif

420 Continue

C

Return

End

C

************************** Constrained Optimization *

C

Subroutine Optimize(NC, NU, U)



* NPmax = 50, NFmax = 10,NCmax = 10, NMVmax = 10,

* NMmax = 20, MaxInf = 10)

Real*8 U(NUmax), Y(NYmax), Lamdah(NCMax)

Common /SolDat/ YO(NYmax), dYdU(NYmax,NUmax), Ahat(NUmax, NUmax),

bhat(NUmax), chat, Ainv(NUmax,NUmax)

C

Call Lagrange(NC, NU, Lamdah, iflag)

If(iflag .eq. 0) Then

Do 10 i - 1,NU

U(i) = 0.0

Do 10 j = 1,NUU(i) = U(i) - Ainv(i, j)*(bhat(j) - Lamdah(j))

10 Continue

Endif

Return

End

- 3.21 -

I,Rtpndv..Iqtqtp Ont;vn;,vn+,oyr%

Determine Pointers for Input Mapping *******************

C

Subroutine Lagrange(NC, NU, Lamdah, iflag)

Implicit Real*8 (A-HO-Z) A


* NPmax = 50, NFmax = 10,NCmax = 10, NMVmax = 10,

* NMmax = 20, MaxInf = 10)

Real*8 Lamdah(1), Lamda (NCmax), Linv(NCmaxNCmax)

Dimension tempm(NCmaxNUmax), tempv(NCmax)

Common /ConDat/ alpha(NCmax), Beta(NCmaxNUmax)

Common /SolDat/ YO(NYmax), dYdU(NYmax,NNUmax), Ahat(NUmaxNUmax),

bhat(NUmax), chat, Ainv(NUmax,NUmax)

C

Do 5 i = INU

Lamdah(i) = 0.0

5 Continue

If(NC .gt. 0) Then

CC ---------------------------- Lagrange Multipliers-----------------------------

C

Do 20 i = 1,NC

tempv(i) = -alpha(i)

Do 20 j = INU

tempm(i,j) = 0.0

Do 15 k = 1,NU

tempm(i,j) = tempm(i,j) + Beta(ik)*Ainv(kj)

15 Continue

tempv(i) = tempv(i) + tempm(i,j)*bhat(j)

20 Continue

Do 30 i - 1,NC

Do 30 j - INC

Linv(i,j) = 0.0Do 30 k = 1,NUJ

BTkj - Beta(j,k)

Linv(i, j) = Linv(i,j) + tempm(i,k)*BTkj

30 Continue

Call Dinvert(NCmax, NC, Linv, if lag)

If(iflag .ne. 0) Then

- 3.22-

CtPI('fV..qtritp ()nf;rn;-sniP;iincnd,4.1*lnn I

Write(*,*) ' Linv not invertible'

Return

Endif

Do 40 i = 1,NC

Lamda(i) - 0.0

Do 40 j = 1,NC

Lamda(i) - Lamda(i) + Linv(i, j)*tempv(j)

40 Continue

C

C------------------------ Modified Lagrange Multipliers---------------

C

Do 50 i = 1,NUDo 50 j = 1,NC

BhTij = Beta(j,i)

Lamdah(i) = Lamdah(i) + BhTij*Lamda(j)

50 Continue

C

Endif

C

Return

End

C* *** ******* *** ** ***** ** *** ***** *** ** ** ** ** * *** *** ** ** * *** * ****** * *** * *** ** * *

C *

C Routine for determining coefficients of continuous cost function. *

C *

C

Subroutine Coef(Nunits, Info, NU, NC, NY)

Implicit Real*8 (A-H, O-Z)


. NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,* NMmax - 20, NOmax = 50, Maxlnf = 10)

Dimension Info (Maxlnf,MaxU), tempm(NYmax, Numax), tempv(NUmax)

Common /CostDat! AA (NUmax, NUmax), BB (NYmax, NYmax),

• ~CC (NYmax, NUmax), p (NUmax), q(NYmax) , r (MaxU)

Common /SolDat! Y0O(NYmax) , dYdU (NYmax, NUmax), Ahat (NUmax, NUmax) ,

• bhat (NUmax), chat, Amny (NUmax, NUmax)

Common /Con_Dat! alpha (NCmax) , Beta (NCmax, NUmax)

-3.23 -

.qtPnfiv..qfnta Ont;mv*-Ynf;nnfin I

C

*********************** Overall Quadratic Cost Function **********************

C

C** Quadratic term **

C

Do 20 i = INY

Do 20 j = 1,NU

tempm(i,j) = 0.0

Do 15 k = 1INY

tempm(i,j) = tempm(i, j) + BB (i, k) *dYdU (k,j)

15 Continue

tempm(i, j) - tempm(i,j) + CC(i, j)

20 Continue

Do 40 i = INU

Do 40 j = 1,NU

Ahat(i, j) = AA(ij)

Do 30 k = I,NY

dYdUT = dYdU(k,i)

Ahat(i,j) - Ahat(i,j) + dYdUT*tempm(k, j)

30 Continue

C If(NC .gt. 0) Then

C Do 35 k=INC

C BetaT = Beta(k,i)

C Ahat(ij) - Ahat(ij) + BetaT*Beta(k,j)

C35 Continue

C Endif

40 Continue

C

C** Linear term **

C

Do 50 i = INY

tempv(i) = 0.0

Do 45 j = 1,NY

BBTij = BB(ji)tempv(i) - tempv(i) + BBTij*Y0O(j)

45 Continue

tempv(i) = 2.0*tempv(i) + q(i)

50 Continue

- 3.24 -

L ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X- LJ L,|iVI

"'l m fl|,V ,|I I I IIt I LI iV 1IQ'-arf;inn Ii Sqteadv.,Stntp (Ontimi7-qtinn

I

£qPhehfl -I~.7'.. ~ LI 'J** ~..~IFI.tIE5IJ~.lLI~JII

55

CC

C

C60

c

70

cC**

c

Do 70 i = 1,NU

bhat(i) = p(i)

Do 55 j = 1,NY

dYdUT = dYdU(ji)

CTij = CC(ji)

bhat(i) = bhat(i) + dYdUT*tempv(j) + CTij*Y0(j)

Continue

If(NC .gt. 0) Then

Do 60 j= INC

bhat(i) = bhat(i) + 2.*alpha(j)*Beta(j,i)

Continue

Endif

Continue

Constant Term **

chat = 0.0

Do 80 iunit = l,Nunits

iu = Info(i,iunit)

chat = chat + r(iu)

80 Continue

Do 90 i INY

temp = 0.0

Do 85 j = 1,NY

temp = temp + Y0(j)*BB(ji)

85 Continue

chat = chat + (temp + q(i))*Y0(i)

90 Continue

C If(NC .gt. 0) Then

C Do 100 i= 1,NC

C chat = chat + alpha(i)*alpha(i)

ClO Continue

C Endif

C

Return

End

C

C******************* Initialize Coefficients of Cost Function *****************

C

- 3.25-

I 19teadv.-Statp Oi n;r;7 t ;nnI

Subroutine Initialize(Nunits, Info, NU, NC, NY)



NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,

NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension Info(MaxInfMaxU)

Common /CostDat/ AA(NUmax,NUmax), BB(NYmax,NYmax),

CC(NYmaxNUmax), p(NUmax), q(NYmax), r(MaxU)

Common /StrDat/ theta(NYmax,NUmax), phi(NYmax,NYmax), psi(NYmax)

Common /ConDat/ alpha(NCmax), Beta(NCmax,NUmax)

C

Do 10 i = INU

Do 10 j = 1,NU

AA(ij) = 0.0

10 Continue

C

Do 20 i = 1,NY

Do 20 j -1,NY

BB(ij) = 0.0

20 Continue

C

Do 30 i = INY

Do 30 j = 1,NU

CC(i,j) = 0.0

30 Continue

C

Do 40 i = 1,NU

p(i) - 0.0

40 Continue

C

Do 50 i = 1,NY

q(i) = 0.0

50 Continue

CDo 60 iunit = l,Nunits

iu -- Info(l,iunit)

r(iu) - 0.0

60 Continue

- 3.26 -

omaux.:QUA -L .C 'k-IntimizationQapf;nn 'A

qpeti~n -

C

Do 70 i - lNCmax

Do 70 j = 1,NU

Beta(i,j) = 0.0

70 Continue

C

Do 80 i = l,NCmax

alpha(i) = 0.0

80 Continue

C

Return

End

- 3.27 -

, a .K.,,lr a "" %0, " ILWT 716.7 tgiV| %./WtlII 11, aitI IIS I teadv-Stntp Ont;mi*i-Fat;nn

p,-tnnI71LJ . -I L I. 15 iIUUei

C

C *

C Routine for solving component stream outputs at a given iteration of the *

C optimization algorithm. *

C *

C

Subroutine Solve(Nunits, Par, F, Modes, mapX, Info, U, Y,



Logical Converge


* NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,

0 NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension Par(NPmax), F (NFmax), Modes (NMVmax), mapX(NYmax),

& Info (MaxInfMaxU), U(NUmax), X(NYmax), Y(NYmax),

0 Outs(NOmax), Xold(NYmax), Xnew(NYmax), Yold(NYmax),

0 Gnew(NYmax), Gold(NYmax)

Common /Tears/ NTears, mapT(NYmax)

Common /Tolerance/ CostTol, Deriv Tol, SolnTol

Data imax/50/

C

Tol = SolnTol

NY = Info(8,Nunits+1) - 1

C

C***** Secant Method Solution to Output Stream Variables: No Interactions *****

C

C

C** Call model of system to get initial results **

C

Do 2 i - INYYold(i) =- Y(i)

2 Continue

Call System(Nunits, Par, F, Modes, mapX, Info, U, Y, Outs,

* Cost, Penalty)

iter = 0

- 3.28-

.Rvctam 1%4nrial

C-----------------------------Iterative Loop

5 iter = iter + 1

C

C** Define residuals for specified tear stream variables **

C

Do 10 i = 1,NTears

ipT = mapT(i)

ipx --mapX(ipT)

Xold(i) = Yold(ipx)

Xnew(i) = Y(ipx)

Gold(i) -- Y(ipx) - Yold(ipx)

10 Continue

Do 15 i = INY

Yold(i) Y(i)

15 Continue

C

C** Call Model of the System **

C

Call System(Nunits, Par, F, Modes, mapX, Info, U, Y, Outs,

Cost, Penalty)

C

Apply One-Dimensional Secant Method to Each Tear Variable **

Do 20 i = l,NTears

ipT = mapT(i)

ipx = mapX(ipT)

Gnew(i) = Y(ipx) - Yold(ipx)

If (dabs(Xnew(i) - Xold(i)) .gt. Tol) Then

dGdX = (Gnew(i) - Gold(i))/(Xnew(i) - Xold(i))

If (dabs (dGdX) .gt. Tol) Then

Y(ipx) - Y(ipx) - Gnew(i)/dGdX

Endif

Endif

Continue

Check Convergence **

Converge = .True.

-3.29 -

C**

c

20

C

C**

c

lqpetinn -3 .qvctam lncil

Section 3-- System Model

Do 50 i = 1,NY

Converge = Converge .and.

dabs(Y(i) - Yold(i))/Y(i) .lt. Tol

50 Continue

C

If(.not.(Converge) .and. iter .1t. imax) goto 5

If (.not. (Converge)) Then

Write(*,*) ' Process equations not converged'

Endif

C

Return

End

C

C *

C System Model: Components and Connections *

C *

C

Subroutine System(Nunits, Par, F, Modes, mapX, Info, U, Y,




* NPmax = 50, NFmax = 10, NCmax = 10, NMVmax = 10,* NMmax = 20, NOmax = 50, MaxInf = 10)

Dimension Par(NPmax), F(NFmax), Modes(NMVmax), mapX(NYmax),

* Info(MaxInfMaxU) , U(NUmax) , X(NYmax)0, Y(NYmax),

0 Outs(NOmax), Costi(MaxU)


CC******** Total Number of controls, constraints, and stream variables *********

CNU-= Info(6,Nunits+l) - 1

NC-= Info(7,Nunits+l) - 1

NY = Info(8,Nunits+l) - 1

C

C----------------Initialize cost coefficients-------------

C

Call Initialize(Nunits, Info, NU, NC, NY)

- 3.30.

Setinn 3 .,yv .-- i vr_ p u

C

C ------------------------ Loop through all components

C

Do 250 iunit - 1,Nunits

itype = Info(2,iunit)

ipp = Info(3,iunit)

ipf = Info(4,iunit)

ipm = Info(5,iunit)

ipu = Info(6, iunit)

ipY = Info(8,iunit)

ipO = Info(9,iunit)

NYi = Info(8,iunit+l) - ipY

NUi = Info(6,iunit+l) - ipu

NOi = Info(9,iunit+l) - ipo

Do 50 i =1, NYi

ipx = mapX(ipY+i-1)

X(ipY+i-l) = Y(ipx)

50 Continue

C

C** Branch according to Type **

C

goto (110, 120, 130, 140, 150, 160) ,itype

C

110 Call Typel(Info(liunit), Par(ipp), F(ipf), Modes(ipm),

U(ipu), X(ipY), Y(ipY), Outs(ipO), Costi(iunit))

goto 200

C

120 Call Type2(Info(1, iunit), Par(ipp), F(ipf), Modes(ipm),


goto 200

C

130 Call Type3(Info(1,iunit), Par(ipp), F(ipf), Modes(ipm),U(ipu), X(ipY), Y(ipY), Outs(ipO), Costi(iunit))

goto 200

C140 Call Type4(Info(l, iunit), Par(ipp), F(ipf), Modes(ipm),

UJ(ipu), X(ipY), Y(ipY), Outs (ipO), Costi(iunit))

goto 200

-3.31 -

qetinn-1

C

150 Call Type5(Info(l,iunit), Par(ipp), F(ipf), Modes(ipm),


goto 200

C

160 Call Type6(Info(1,iunit), Par(ipp), F(ipf), Modes(ipm),


goto 200

C

200 Continue

C

250 Continue

C

C ---------------------------- Determine Total Costs

C

Cost = 0.0

Do 300 iunit = l,Nunits

Cost = Cost + Costi(iunit)

300 Continue

C

C -------Add Costs Associated with Penalty Function Applied to Constraints

C

Penalty - 0.0

Do 400 i - 1,NC

Penalty = Penalty + GC(i)*GC(i)

400 Continue

Cost = Cost + Penalty

C

Return

End

- 3.32 -

,Rvqtpm Mndpl

CQ. t; nI

C

******* * ************************* Type 1 ************************************

C

C ** COOLING TOWER **

C

Subroutine Typel(Info, Par, F, Modes, U, X, Y, Out, Power)


Real*8 mwi, mwo, mwl, ma, mamax

Parameter (MaxU = 10, NUmax = 20, NYmax = 50, NCmax = 10)

Parameter (mamax 2.85e06, Pmax = 100.)

Dimension Info(1), F(1), Modes(1), U(1), X(1), Y(1), Out(1)


. CC(NYmax,NUmax), p(NUmax) , q(NYmax) , r(MaxU)

C

C-------------------------- Type Info on First Call---------------------------

C

If(Info(1) .eq. 0) Then

Info (3) = 0

Info(4) = 2

Info (5) = 1

Info(6) = 1

Info (7) - 0

Info(8) = 1

Info (9) - 3

Return

Endif

Number

Number

Number

Number

Number

Number

Number

of

of

of

of

of

of

of

Parameters

Forcing Function Variables

Mode Variables

Continuous Control Variables

Constraints

Streams

Outputs

C ----------------------------------- CostsC

C** Fan Power Model **

C

C Power = Ncells*(a0 + al*gama + a2*gama*gama)*Pmax

C

Ncells = Modes (1)

gama = U (1)

aO = gama**3

al = -3.*gama**2

a2 = 3.*gama

Number of tower cells operating

Relative tower cell air flow rate

-3.33 -

acuti.l -Y % -VtlJiy;llt |VlIIeICnmnnnavi+ MeAalo

Setion 3

Power = Ncells*(aO + al*gama + a2*gama*gama)*Pmax

C

C** Modify appropriate locations in cost coefficient arrays **

C

iunit = Info(1)

ipu = Info(6) - 1 ! pointer to continuous controls

c

AA(ipu+l,ipu+l) = Ncells*a2*Pmax

p(ipu+l) = Ncells*al*Pmax

r(iunit) = Ncells*aO*Pmax

C

C-----------------------------------Inputs

C

SF (1)

= F(2)

- gama*mamax

=X (1)- X(2)

! inlet air dry bulb temperature

! inlet air wet bulb temperature

! individual tower air flow rate

! tower inlet water temperature

! tower (total) water flow rate

C----------------------------------- Outputs

C

Two = Y(i)

mwl = mwi/Ncells

C

C** All operating tower cells have equal water and air flow rates **

C

Call Tower(Tdb, Twb, Twi, mwl, ma, Two)

C

mwi

Two

mwo

tower

tower

(Twi - Two) / (Twi - Twb)

= Two

= Power

= epsw

outlet water temperature

outlet mass flow rate

tower water effectiveness

- 3.34-

Tdb

Twb

ma

Twi

mwi

mwo -

Y(1) -

Y(2) =

epsw =

Out (1)

Out (2)

Out (3)

Return

2., a f-t w l "lVIIil I UCA5 !r-nmnnna n t Uf l'e

2l~':lr

End

C

C------------------------ Cooling Tower Outlet Conditions

C

Subroutine Tower(Tdb, Twb, Twl, Flw, Fla, Two)


Real*8 Ntu

Parameter (Cl = 2.0, C2 = 0.37, Cpw = 1.0, Tmain = 55.,

Tol = 1.e-10, imax- 50)

Data Tw2/70./

C

C** Transfer Units: Correlation to D/FW Data **

C

Ral = Flw/Fla

Ntu = Cl*RaI**C2

C

C** Secant method solution for outlet water temperature **

C

Call DPsych(2, 1, Tdb, Twb, rh, Tdp, wal, hal)

Call DPsych(2, 1, Twl, Twl, rhwl, Tdpwl, wwl, hwl)

iter = 0

Told = Tw2

Call DPsych(2, 1, Tw2, Tw2, rhw2, Tdpw2, ww2, hw2)

Call Outlet(Ral, Ntu, Cpw, hal, wal, Twl, hwl, Tw2, hw2,

wa2, GI)

Tw2 = Told - G1

If(abs(Gl) .gt. Tol) Then

10 iter = iter + 1


Call Outlet(Ral, Ntu, Cpw, hal, wal, Twl, hwl, Tw2,

hw2, wa2, G2)

dGdT = (G2 - Gl)/(Tw2 - Told)

Told = Tw2

G1 = G2

Tw2 = Told - G2/dGdT

If(abs(G2) .gt. Tol .and. iter .lt. imax) goto 10

Endif

C

C** Steady-State Sump Analysis **

-3.35-

A rn F% 4-% " Am W% + lk if -,, Ar,, I r,

C

rloss = Fla*(wa2 - wal)/Flw

Two = (1. - rloss)*Tw2 + rloss*Tmain

C

Return

End

C

C -------------------------- Effectiveness Model

C

C As outlined in Chapter 3.

C

Subroutine Outlet(Ral, Ntu, Cpw, hal, wal, Twl, hwl, Tw2,

hw2, wa2, G)


Real*8 Ntu, mstar

Parameter (Tref = 32.)

C

C** Statement function for saturation temperature given enthalpy **

C

Tsat(hsat) =-0.8811 + 3.340*hsat - 4.907e-02*hsat**2 +

4.Ole-04*hsat**3 - 1.303e-06*hsat**4

C

C** Compute effectiveness for counterflow tower **

C

Cs = (hwl - hw2)/(Twl - Tw2)

mstar = Cs/Ral/Cpw

If(dabs(l. - mstar) .gt. l.e-06) Then

c = exp(-Ntu*l.-mstar))

eps = (1. - c)/(l. - mstar*c)

Else

eps = Ntu/(l. + Ntu)

EndifC

C** Determine outlet states **

C

ha2 - hal + eps*(hwl - hal)

hw = hal + (ha2 - hal)/(I. - exp(-Ntu))

Tw = Tsat (hw)

- 3.36-

Section 3 C-Omnnnipnt Unriale I

Section 3 Component Models

Call DPsych(2, 1, Tw, Tw, rhw, Tdpw, ww, hw)

wa2 = ww + (wal - ww)*exp(-Ntu)

Ra2 - Ral - (wa2 - wal)

Tw2new = Tref + (Ral*Cpw*(Twl-Tref) - (ha2 - hal))/Ra2/Cpw

C

C** Define residual for secant method solution **

C

G = Tw2 - Tw2new

C

Return

End

-3.37 -

O. lVI L ! VII 3 k tlUUI11 , I.UIo l

C

************************************ Type 2 ********************************

C

C ** CHILLER **

C

Subroutine Type2(Info, Par, F, Modes, U, X, Y, Out, Pch)


Real*8 mchws, mcws


Parameter (Cpw = 1.0)

Dimension Info(1), Par(1), F(1), Modes(1), U(1), X(1),

Y(1), Out(1)


CC(NYmax, NUmax), p(NUmax), q(NYmax), r(MaxU)

C

C-------------------------Type Info on First Call-----------------------------

C


Info (3) = 2

Info (4) = 0

Info (5) = 1

Info(6) = 1

Info (7) = 0

Info (8) = 2

Info (9) = 3

Return

Endif

Number

Number

Number

Number

Number

Number

Number

of

of

of

of

of

of

of

Parameters


Mode Variables


Constraints

Streams

Outputs

C ------------------------------------ Costs --------------------

C

C** Chiller Power Model **

C

C Power = Nch*Pdes*(aO + al*(Qload/Qdes) + a2*(Qload/Qdes)**2

C + a3*(Tcwr-Tchws)/DTDES + a4*((Tcwr-Tchws) /DTDES)**2

C + a5* (Qload/Qdes) * (Tcwr-Tchws) /DTDES

C

Ich

Qcap

= Par (1)

= Par (2)

Chiller choice (1 to 4)

Design chiller capacity

- 3.38-

Qadt;nn I t[ ,w r An, m, 4-

I/Ii'...4 ^ I .. ,

2

Nch = Modes (1) ! Number of chillers in parallel

C

Tchws = U(1) ! Chilled water supply setpoint

C

Tchwr = X(l) ! Chilled water return temperature

mchws = X(2) ! Chilled water flow rate

Tcws = X(3) ! Condenser water supply temperature

mcws = X(4) ! Condenser water flow rate

C

C** Determine individual chiller performance (loaded evenly) **

C

Qload = mchws*Cpw*(Tchwr - Tchws)/12000./Nch

Call Chiller(Ich, aO, al, a2, a3, a4, a5, Qcap, Qdes, DTdes,

Pdes, Qload, Tchws, Tcws, mcws/Nch, Tcwr)

C

C** Rewrite power as

C

C Power = bO + bl*Tchws + b2*Tchws**2 + b3*Tcwr +

C b4*Tcwr**2 + b5*Tchws*Tcwr

C

bO = Nch*Pdes*(aO + al*(Qload/Qdes) + a2*(Qload/Qdes)**2)

bl = -Nch*Pdes*(a3/DTdes + a5*(Qload/Qdes)/DTdes)

b2 = Nch*Pdes*a4/DTdes**2

b3 = Nch*Pdes*(a3/DTdes + a5*(Qload/Qdes)/DTdes)

b4 = Nch*Pdes*a4/DTdes**2

b5 = -2.*Nch*Pdes*a4/DTdes**2

Pch = bO + bl*Tchws + b2*Tchws**2 + b3*Tcwr +

b4*Tcwr**2 + b5*Tchws*Tcwr

C


C

iunit = Info(1)


ipy = Info(8) - 1 ! pointer to first output stream vector

r(iunit) = bO

AA (ipu+l, ipu+l) - b2

p (ipu+l) = bl

- 3.39-

2-*t'll'uuuutent ivtoaets

BB(ipy+3,ipy+3) -=b4

q(ipy+3) = b3

CC(ipy+3, ipu+l) = b5

C

C----------------------------------Outputs------------------------------------

C

Y(1) = Tchws

Y(2) = mchws

Y(3) = Tcwr

Y =(4) mcws

C

Out(1) = Tcwr

Out(2) = Nch*Qload

Out(3) = Pch

C

Return

End

C

C *

C Empirical model of a centrifugal chiller *

C *

C

Subroutine Chiller(Ich, aO, al, a2, a3, a4, a5, Qcap, Qdes, dTdes,

Pdes, Qevap, Tchws, Tcws, mcws, Tcwr)


Real*8 mcws

Dimension a0s(4), als(4), a2s(4), a3s(4), a4s(4), a5s(4),+

Qcaps(4),Qdess(4), dTdess(4), Pdess(4)

Data a0s/ 0.07336, 0.0516, 0.1107, 0.2642/

Data als/-0.32590, 1.2199, 0.3198, 0.0207/

Data a2s/ 0.57440,-0.2517, 0.4662, 0.2643/

Data a3s/-0.03888, -O.6448,-0.0956, -0.4460/Data a4s/ 0.33210, 0.8963, 0.2152, 0.3555/

Data a5s/ O.36840,-0.3119,-0.0656, 0.5176/

Data Qcaps /5500.,5500.,1250., 550.1

Data Qdess /5421.,5421.,1250., 550.!

Data dTdess/ 46., 46., 50., 50.!

- 3.40 -

Vn " A Pt r n r% d%,w%^ . 74 M .I - -

2~~+ nLJiiLI I "lJuua at [viU.II

Data Pdess /3580.,3627., 961., 453./

Data Cpw /1.0/

C

c------------------------- Chiller Parameters

C

C Ich chooses the chiller

C

C Ich = 1 for the D/FW variable-speed chiller

C Ich = 2 for the D/FW vane controlled chiller

C Ich = 3 for the chiller from Lau's study

C Ich = 4 for the chiller from Hackner's study

C

aO = a0s(Ich)

al = als(Ich)

a2 = a2s(Ich)

a3 = a3s(Ich)

a4 = a4s(Ich) -

a5 = ass (Ich)

C

C** Scale design conditions according to specified capacity **

C

fdes = Qcap/Qcaps(Ich)

Qdes = fdes*Qdess(Ich)

dTdes = dTdess (Ich)

Pdes = fdes*Pdess(Ich)

C "

C ----------- Determine dimensionless leaving temperature difference

C

X = Qevap/Qdes

C = Qevap + Pdes/3.515*(aO + al*X + a2*X*X) +

mcws*Cpw*(Tcws - Tchws)/12000.

B = Pdes/3.515*(a3 + a5*X) - mcws*Cpw*dTdes/12000.

A = a4*Pdes/3.515

Sqrd = B*B - 4.*A*CIf(Sqrd .gt. 0) Then

root = dsqrt (Sqrd)

Else

root = 0.0

- 3.41-

rllmnA,%r%,mlft+ NAd-%Anlc-


Endif

y = (-B - root)/2./A

If(Y .LT. 0) Y = (-B + root) /2./A

C

C -------------- Power consumption and condenser return temperature

C

Pch = Pdes*(aO + al*X + a2*X*X + a3*Y + a4*Y*Y + a5*X*Y)

Qcond= Qevap + Pch/3.515

Tcwr =Tcws + 12000.*Qcond/mcws/Cpw

C

Return

End

- 3.42-

C

~~ Type3

C

C ** CONDENSER PUMPS **

C

Subroutine Type3(Info, Par, F, Modes, U, X, Y, Out, Work)


Real*8 mwi, mwo, mmax, mpump, mchdes, mTdes, Ksys

Parameter (MaxU = 10, NUmax = 20, NYmax = 50, NCmax - 10)

Parameter (mmax = 1.e07, mchdes = 5.eO6, mTdes - 2.5e06,

dP0 - 30., dPTdes = 5., dPCdes = 10., dPmax = 100.,

rhow = 62.4, effmax- 0.80, cony = 144*3.765e-07)

Dimension Info(1), F(1), Modes(1), U(1), X(1), Y(1), Out(1)

Dimension a(3), W(3), Ginv(3,3)

Common /CostDat/ AA(NUMax,NUmax), BB(NYmax,NYmax),

CC(NYmax,NUmax) , p(NUmax), q(NYmax), r(MaxU)

C

C-------------------------- Type Info on First Call

C


Info(3) = 0

Info(4) = 0

Info(5) = 3

Info(6) - 1

Info(7) = 0

Info(8) = 1

Info(9) = 3

Return

Endif

Number

Number

Number

Number

Number

Number

Number

of

of

of

of

of

of

of

Parameters


Mode Variables


Constraints

Streams

Outputs

C

C------------------------------------ Inputs------------------

= Modes (1)

= Modes (2)

= Modes (3)

= u(1)

= X(1)

= x(2)

Number of pumps operating in parallel

Number of chillers operating in parallel

Number of tower cells operating in parallel

Relative pump speed

Pump water inlet temperature

Pump water inlet flow rate

-3.43-

Npumps

Nch

NCells

gama

Twi

mwi

iectloll , %fliUnlti voUUelsCN I&! AM W% Al

QapC n'a %,L11, 111V!!U IIl Lt IIUe IS

C

C -- Costs

C

C** Pump Power Model **

C

C dPPump = dPmax*(I - (mwo/Npumps/mmax)**2)*gama**2

C dPChiller = dPCdes*(mwo/Nch/mchdes)**2

C dPTower = dPTdes*(mwo/Ncells/mTdes)**2

C

C Work = mpump*dP*conv/rhow/effp

C

C** Fit quadratic to three points over a small range **

C

Ksys - dPCdes*( (Npumps*mmax)/(Nch*mchdes) )**2

+ dPTdes* ( (Npumps*rnmax) / (Ncells*mTdes) ) **2

dgama = 0.05

gammin = (dP0ldPmax)**0.5 + 0.005

gamal = Dmaxl (gammin, gama-dgama)

Do 50 i = 1,3

gamai = gamal + (i-l)*dgama

c = (gamai**2 - dPO/dPmax)/(I + Ksys/dPmax)

mpump = rmax*c**0.5

dP = dPO + Ksys* (mpump/nnax) **2

effpi = effmax* (3.66* (mpump/gamai/mmax) -3.30* (mpump/gamai/mmax) **2)

W(i) = mpump*dP*conv/rhow/effpi

Do 50 j = 1,3

Ginv(i,j) = gamai**(j-1)

Continue50

c

Call Dinvert(3,

Do 60 i - 1,3

a(i) = 0.0

Do 60 j - 1,3

a(i) = a(i)

Continue

3, Ginv, iflag)

+ Ginv(i,j)*W(j)

Modify appropriate locations in cost coefficient arrays **

- 3,44 -

60

CC**

rnmnn",m"t X4^Anic

2 ftAz3 Com rNz aan AnIt MvA y d bI'.

C

iunit

ipu

ipy

- Info (1)

= Info(6) - 1

= Info(8) - 1

AA(ipu+1, ipu+1)

p (ipu+l)

r (iunit)

pointer to continuous controls

pointer to output variables

= Npumps*a(3)

= Npumps*a (2)

= Npumps*a(i)

CC

C---------------------------------- Outputs------------------------------------C

Two = Twi

c = (gama**2 - dPO/dPmax)/(1 + Ksys/dPmax)

mpump = mmax*c**0.5

mwo = Npumps*mpump

Work = Npumps* (a (1) + a (2) *gama + a (3) *gama*gama)

effp = effmax*(3.66*(mpump/gama/mmax) -

3.30* (mpump/gama/mmax) **2)

Y(1)Y(2)

Out (1)

Out (2)

Out (3)

Return

End

= Two

= mwo

Outlet temperature

Outlet flow rate

= mwo

= Work

= effp

- 3.45-

C

onennt ModelsSIetinn3

L.IC71-LlNJE " Cnvii ai MI

C

*************************Type 4******************

C

C ** Air Handlers **

C

Subroutine Type4(Info, Par, F, Modes, U, X, Y, Out, Cost)


Real*8 mwi, mwo, ma, mamax, mamb

Logical Converge


Parameter (Xdes = 0.1, Tz = 75., Cpa = 0.24, Cpw = 1.0,

hfg = 1054., Penalty = 1.)

Dimension Info(1), Par(l), F(1), Modes(l), U(1), X(1),

Y(1), Out(1)


CC(NYmax,NNUmax), p(NUmax), q(NYmax), r(MaxU)


Common /Load/ Qcoil

C

C ------------------------- Type Info on First Call----------------------------

C


Info (3) = 2

Info(4) = 4

Info (5) = 0

Info(6) = 1

Info (7) = 1

Info (8) = 1

Info (9) = 4

Return

Endif

Number

Number

Number

Number

Number

Number

Number

of

of

of

of

of

of

of

Parameters


Mode Variables


Constraints

Streams

Outputs

C

C- --------------------------------- Parameters

C

= Par (1)

= Par (2)

= Xdes*mamax

! Maximum air handler flow rate

! Maximum air handler fan power

! Ambient ventilation air flow rate

- 3.46 -

mamax

Pmax

mamb

lqartinn -3 - -'nm nnn,)n t lMAndl-i

~J~.LIUII .2 -Vl'Ju'Ij'tIcIl £ viuucts

C - -IInputs u-

C

- U (1)

= X(2)

= F (2)

= F (3)

= F(4)

*12000.

*12000./hfg

DPsych (2, 1, Tamb,

= gama*mamax

= mamb/ma

Relative fan speed

Chilled water inlet temperature

Chilled water inlet flow rate

Load sensible energy requirement

Load humidity gains

Ambient dry bulb temperature

Ambient wet bulb temperature

Twb, rh, Tdp, wamb, hamb)

C

C** Call AHU to determine outlet air handler states **

C

Ts

Two

Call

Call

= Tz - Qsens/ma/Cpa

= Twi + 1.5*Qsens/mwi/Cpw

DPsych(2, 2, Ts, Twbs, 0.95, Tdps, was, has)

AHU(ma, Xamb, hamb, wamb, Tz, wgain, Twi, mwi, Two,

Ts2, was, haz, waz, wal)

C ----------------------------------- Costs

C

C** Fan Power Model **

C

C Pfan = (aO + al*gama + a2*gama*gama)*Pmax

C

gama = U(1)

aO = gama**3

al = -3.*gama**2

a2 = 3.*gama

Pfan = (aO + al*gama + a2*gama*gama)*Pmax

C


C

iunit = Info(1)


- 3.47 -

gama

Twi

mwi

Qsens

wgain

Tamb

Twb

C

Call

ma

Xamb

rnMnnnan+ IA^AmioC'm ̂ +; A,% " I

C'nm nnnpnt MXnic 1c_3C'LI lI --.- I V.l L.v I.

ipc = Info(7) - 1 ! pointer to equality constraint

C

AA(ipu+l,ipu+l) = a2*Pmax

p(ipu+l) = al*Pmax

r(iunit) = aO*Pmax

C

C** Penalty for not maintaining zone temperature **

C

g = Penalty*(Qsens - gama*mamax*Cpa*(Tz - Ts2))/12000.

GC(ipc+l) = g

Cost = Pfan

C

C ---------------------------------- Outputs

C

Qamb = mamb*(hamb - haz)/12000.

Qzone = (Qsens + wgain*hfg)/12000.

Qcoil = Qzone + Qamb

Qlat = ma*(wal - was)*hfg/12000.

C

Two =Twi + 12000.*Qcoil/mwi/Cpw

mwo = mwi

C

Y(1) = Two

Y(2) = mwo

C

Out(1) = Ts2

Out(2) = waz

Out(3) = Pfan

Out(4) = Qlat/Qcoil

C

Return

End

-3.48-

4z,pt;nn I

Cnmnnltn ,ai. AI. ^U.2

C

C *

C Model for air handler including mixing and cooling coil process *

C *

C

Subroutine AHU(ma, Xamb, hamb, wamb, Tz, wgain, Twl, mw, Tw2,

Ta2, was, haz, waz, wal)


Real*8 mw, ma

Data Tol/l.e-10/, imax/20/

C

C----------- Iterative solution air handler for exit humidity ratio

C

iter = 0

Call Mix(Tz, was, wgain, ma, Xamb, hamb, wamb, hal, wal,

haz, waz)

Call Coil(wal, hal, ma, Twl, mw, Ta2, wa2, Tw2, eps)

wold = was

G1 = wa2 - was

was = wa2


10 iter = iter + 1

Call Mix(Tz, was, wgain, ma, Xamb, hamb, wamb, hal, wal,

haz, waz)

Call Coil(wal, hal, ma, Twl, mw, Ta2, wa2, Tw2, eps)

G2 = wa2 - was

dGdw = (G2 - Gl)/(was - wold)

wold = was

GI = G2

was = wold - G2/dGdw

If(abs(G2) .gt. Tol .and. iter .lt. imax) goto 10

EndifReturn

End

- 3.49-

z5ectliln J uutlen t ---tvi weisi

C-lflrnnn nfnl la aIc,

C

C *

C Determine mixed air return to air handler *

C *

C

Subroutine Mix(Tz, wa2, wgain, ma, Xamb, hamb, wamb, hal, wal,

haz, waz)


Real*8 ma

C

waz = wa2 + wgain/ma

Call DPsych(2, 4, Tz, Twbz, rhz, Tdpz, waz, haz)

hal = (1. - Xamb)*haz + Xamb*hamb

wal = (1. - Xamb)*waz + Xamb*wamb

C

Return

End

- 3.50 -

,Rpetinn -3


C

C

C This routine determines the thermal performance of a cooling coil using an *

C effectiveness model as described in Chapter 4. *

C *

C

Subroutine Coil(wal, hal, ma, Twl, mw, Ta2, wa2, Tw2, eps)

Implicit Real*8 (A-HO-Z)

Real*8 ma, mw, mades, mwdes, NTUo, NTUi, NTUd, NTUw

Logical Dry, Wet

Data Cpa/0.240/, Cpw/l.000/, Cpv/0.45/

Data mades/8.e06/,mwdes/l.e07/

Data Tol/l.e-12/, imax/20/, Twx/40.01/

C

C** Statement functions for saturation temperature at saturation

C** enthalpy and air-side effectivness

C

Tsat(hsat) = 0.1905 + 3.184*hsat - 4.091E-02*hsat*hsat +

2.258E-04*hsat*hsat*hsat

C

****** *********************** Coil Analysis *******************************

C

C** Determine NTU's **

C

Cpm = Cpa + wal*Cpv

C

C** Correlation to Carrier Cooling Coil Data **

C

NTUo = 1.70*(ma/mades)**-0.38NTUi = 2.25* (mw/mwdes) **-0.2

Ra - mw/ma

Call DPsych(2, 6, Tal, Twbl, rh, Tdp, wal, hal)

Call DPsych(2, 1, Twl, Twi, rhwl, Tdpwl, wwl, hwl)

Twsav = Tw2

- 3.51 -

I I 2 C-AT- 1nnnil .nk Nitn,I .c

C

C** Dry analysis **

C Determines the air effectiveness (epsd) and outlet water temperature

C (Tw2) assuming the coil is dry

C If the surface temperature at the outlet is greater than the air

C entering dewpoint then coil is completely dry, otherwise do the wet

C analysis

C

Cstar = Cpm/Ra/Cpw

NTUd = NTUo/(l. + NTUo/NTUi*Cstar)

C = exp(-NTUd*(l. - Cstar))

epsd = (1. - C) / (. - Cstar*C)

Tw2 - Twl + epsd*Cstar*(Tal - Twl)

.Ta2 = Tal - epsd*(Tal - Twl)

Ts2 = Twl + Cstar*NTUd/NTUi*(Ta2 - Twl)

ha2 = hal - Cpm*(Tal - Ta2)

wa2 - wal

C

Dry = Ts2 .gt. Tdp

If(.not. Dry) Then

C

C** Wet Coil **

C Determines the air effectiveness (epsw) and outlet water temperature

C (Tw2) assuming the coil is wet

C The average saturation specific (Cs) heat depends upon the outlet

C conditions: Therefore an iterative solution

C If surface temperature at water outlet is less than the dewpoint of

C the entering air then the coil is completely wet

C

iter = 0

Tw2 = Twsav

Call WetCoil(Ra, NTUi, NTUo, Cpw, hal, Twl, hwl,• Tw2, Tsl, ha2, GI)

Told = Tw2

Tw2 = Told - Gi

If (dabs(Gl) .gt. Tol) Then

10 iter = iter + 1

Call wetCoil(Ra, NTUi, NTUo, Cpw, hal, Twl, hwl,

- 3.52-

C.Omnnnant Mnil ceC!^d%+2^lmk A

CA^;9%

Tw2, Tsl, ha2, G2)

If(dabs(Tw2 - Told) .gt. Tol) Then

dGdT = (G2 - GI)/(Tw2 - Told)

Told = Tw2

G1 = G2

If (dabs(dGdT) .gt. Tol) Then

Tw2 - Told - G2/dGdT

Endif

Endif

If(dabs(G2) .gt. Tol .and. iter .lt. imax) goto 10

Endif

Endif

C

C ** Determine if all Wet **

c

Wet = (.not. Dry) .and. (Tsl .lt. Tdp)

If(Wet) Then

hsavg = hal - (hal - ha2)/(l. - exp(-NTUo))

Tsavg = Tsat(hsavg)

Ta2 = Tsavg + (Tal - Tsavg)*exp(-NTUo)

Call DPsych(2, 5, Ta2, Twb2, rh2, Tdp2, wa2, ha2)

Endif

C

If(.not.(Dry) .and. .not.(Wet)) Then

C

C** Partially Wet and Dry **

C

Tw2 = Twsav

iter = 0

Call DryWet(NTUi, NTUo, NTUd, Ra, Cpm, Cpw, Cstar, Tal, Tdp,

hal, Twl, hwl, Tw2, Twx, fdry, epsd, Gl)

Told = Tw2

Tw2 = Told -Gi


20 iter = iter + 1

Call Dry_Wet(NTUi, NTto, NTtd, Ra, Cpm, Cpw, Cstar, Tal,

• Tdp, hal, Twi, hwl, Tw2, Twx, fdry, epsd, G2)

If (dabs(Tw2 - Told) .gt. Tol) Then

dGdT = (G2 - GI)/(Tw2 - Told)

- 3.53 -

c2nA :&%i

Told = Tw2

G1 = G2

If (dabs(dGdT) .gt. Tol) Then

Tw2 = Told - G2/dGdT

Endif

Endif

If(abs(G2) .gt. Tol .and. iter .1t. imax) goto 20

Endif

ha2 = hal - Ra*Cpw*(Tw2 - Twl)

Tax = Tal - epsd*(Tal - Twx)

hax = hal - Cpm*(Tal -Tax)

hsw = hax + (ha2 - hax)/(i. - EXP(-(l.-fdry)*NTUo))

Tsw = Tsat(hsw)

Ta2 = Tsw + (Tax - Tsw)*EXP(-(l.-fdry)*NTUo)

Call DPsych(2, 5, Ta2,. Twb2, rh2, Tdp2, wa2, ha2)

Endif

C

C** Air Effectiveness **

C

eps - (hal - ha2)/(hal - hwl)

C

Return

End

C

*************************** Wet Coil Analysis *** * *****

C

Subroutine WetCoil(Ra, NTUi, NTUo, Cpw, hal, Twl, hwl,

Tw2, Tsl, ha2, G)


Real*8 NTUo, NTUi, NTUw, mstar

C


Cs = (hw2 - hwl)/(Tw2 - Twl)

mstar = Cs/Ra/Cpw

NTUw = NTUo/(l. + NTUo/NTUi*mstar)

C - exp(-NTUw*(l. - mstar))

epsw = (1. - C)/(l. - mstar*C)

Tsl = Tw2 + NTUw/NTUi*(hal - hw2)/Ra/Cpw

- 3.54.-

3-ectlull -'I T--,IjlllLj licift Ivigursibrnmnnnant X4f%.rialc!

ha2 = hal - epsw*(hal - hwl)

G = Tw2 - (Twl + (hal - ha2)/Ra/Cpw)

Return

End

C

** * Dry & Wet Coil Analysis *

C

Subroutine DryWet(NTUi, NTUo, NTUd, Ra, Cpm, Cpw, Cstar, Tal,

Tdp, hal, Twl, hwl, Tw2, Twx, fdry, epsd, G)


Real*8 NTUo, NTUi, NTUd, NTUw, mstar

C

EXPK = ((Tdp - Tw2) + Cstar* (Tal - Tdp))/

(1. - NTUd*(1. - Cstar)/NTUo) /(Tal - Tw2)

EXPK = Dmaxl(1.d-06,EXPK)

fdry = -1./NTUd/ (1. - Cstar) *Dlog(EXPK)

fdry = Dmaxl(l.d-06,Dminl(l.dO - l.d-06,fdry))

C = exp(-fdry*NTUd*(l. - Cstar))

epsd = (1. - C)/(1. - Cstar*C)

Call DPsych(2, 1, Twx, Twx, rhwx, Tdpwx, wwx, hwx)

Cs = (hwx - hwl)/(Twx - Twl)

mstar = Cs/Ra/Cpw

NTUw = NTUo/(l. + NTUo/NTUi*mstar)

C = exp(-(l.-fdry)*NTUw*(l. - mstar))

epsw = (1. - C)/(1. - mstar*C)

Twx = (Twl + Cstar/Cpm*epsw*(hal -epsd*Cpm*Tal - hwl))/

(1. - Cstar*epsw*epsd)

G = Tw2- (Cstar*epsd*Tal + (1. - Cstar*epsd)*Twx)

C

Return

End

- 3.55-

¢ IIUI! a lVilUUIIlIIu ivioUUeI

Comnnnent Mndplc

CC*

C

C

********************************** Type 5 *

** CHILLED WATER PUMPS **

Subroutine Type5(Info, Par, F, Modes, U, X, Y, Out, Work)


Real*8 mwi, mwo, mmax, mpump, mchdes, Ksys


Parameter (mmax = l.e07, mchdes = 5.e06, dPO = 20., dPCdes = 25.,

dPmax - 100., rhow = 62.4, effmax - 0.80,

cony = 144*3.765e-07)

Dimension Info(1), F(1), Modes(1), U(1), X(1), Y(l), Out(1)

Dimension a(3), W(3), Ginv(3,3)

Common /CostDat/ AA(NUmaxNUmax), BB(NYmax,NYmax),

CC(NYmax,NUmax), p(NUmax), q(NYmax) , r(MaxU)

C

C-------------------------- Type Info on First Call --------------

C


Info (3) = 0

Info (4) = 0

Info (5) = 2

Info(6) = 1

Info (7) - 0

Info (8) = 1

Info (9) = 3

Return

Endif

Npumps = Modes(1)

Nch = Modes(2)

gama = U(1)

Twi = X(1)

mwi = X(2)

Number

Number

Number

Number

Number

Number

Number

of

of

of

of

of

of

of

Parameters


Mode Variables


Constraints

Streams

Outputs

SNumber of pumps operating in parallel! Number of chillers operating in parallel

! Relative pump speed

! Pump inlet temperature

! Pump inlet flow rate

c ----------------------------------- Costs

3.56 -

C

c

Cml.+; ein I

Cnmnflnlldt U1nfiatxec

C

C** Pump Power Model **

C

C dPPump = dPmax*(1 - (mwo/Npumps/nax)**2)*gama**2

C dPChiller = dPCdes*(mwo/Nch/mchdes)**2

C

C Work = mpump*dP*conv/rhow/effp

C

C** Fit quadratic to three points over a small range **

C

Ksys = dPCdes*( (Npumps*mmax)/(Nch*mchdes) )**2

dgama = 0.05

gammin = (dPO/dPmax)**0.5 + 0.005

gamal = Dmaxl(gammin, gama-dgama)

Do 50 i = 1,3

gamai = gamal + (i-l)*dgama

c = (gamai**2 - dPO/dPmax)/(I + Ksys/dPmax)

mpump = mmax*c**0.5

dP = dP0 + Ksys*(mpump/mmax)**2

effpi = effmax*(3.66*(mpump/gamai/mmax) -3.30* (mpump/gamai/mmax) **2)

W(i) = mpump*dP*conv/rhow/effpi

Do 50 j = 1,3

Ginv(i,j) = gamai**(j-l)

50 Continue

C

Call Dinvert(3, 3, Ginv, iflag)

Do 60 i = 1,3

a(i) = 0.0

Do 60 j = 1,3

a(i) = a(i) + Ginv(i,j)*W(j)

60 Continue

C

C** Modify appropriate locations in cost coefficient arrays **C

iunit = Info(1)

ipu = Info(6) - 1. ! pointer to continuous controls

ipy = Info(8) - 1 . pointer to output variables

- 3.57-

- r1 iV.i.1 k" VV %IIIIil L lviucill3.qpetion -I-

i

AA (ipu+ 1, ipu+1)

p (ipu+l)

r (iunit)

= Npumps*a(3)

= Npumps*a (2)

= Npumps*a(I)

C ---------------------------------- Outputs------------------------------------C

c = (gama**2 - dPO/dPmax) / (1 + Ksys/dPmax)

mpump = mmax*c**0.5

mwo = Npumps*mpump

Two = Twi

Y () = Two

Y(2) = mwo

Work =

effp =

C

Out (1)

Out (2)

Out (3)

Return

End

Pump outlet temperature

Pump outlet flow rate

Npumps*(a(1) + a(2)*gama + a(3)*gama*gama)

effmax* (3.66* (mpump/gama/mmax) -3.30* (mpump/gama/mmax) **2)

= mwo

= Work

= effp

- 3.58 -

Q Iaf fn A

C

C

O-S.tivli l 0 -VILLlUUmjl 'll l, oUUeIs

15, 5, 4

4 1

1 141

1, 4

8.e06, 1000.1 2 3 4

0.452, 0.1, 2.0

3, 1

40. 1.e07

2, 5

i, 2

0.52, 0.50, 2.0

i, 1

40. 1.e07

3, 2

1 5500.

2

55. 38. 55.

2, 1

50. l.e07

5, 1

70. l.e07

4, 1

3 4

3

0.428 0.2 2.

3, 2

85. 1.e07

5, 3

4 2 3

0.673, 0.57,

4, 1

70. 1.e07

2.0

Timesteps: Units: Data Values

4 Mode Variables: 1 Discrete Control Mode

Mode 1 Values:

Unit 1 Type 4 AHU

maximum air flow and power

Four Forcing Functions

Relative Fan Speed

Inputs from Unit 3 Stream 1

Initial Input Values

Unit 2 Type 5 Chilled Water Pumps

First and Second Mode Variables

Relative Pump Speed



Unit 3 Type 2 Chiller

D/FW Chiller, 5500 Tons

Second Mode Variable: Number of Chillers

Chilled Water Set Temperature





Unit 4 Type 1 Cooling Tower

Forcing Functions: Dry and wet Bulb Temp.

Mode Variable for'number of cells

Fan Control



Unit 5 Type 3 PumpThree Mode Variables

Relative Pump Speed



- 3.59 -

.qetilnnV .. m. m V mm

9L j & 'u" GLm qmmrI %P a-IF S& t "S amnle Data

1

411

3

5

1,3 2,2 3,3

4,2 5,2

5

4,1 4,3 3,1

5,1 5,3

6

3,2 2,1 2,3

, 1 I, 2 1, 4

Only 1 Tear Variable

Unit 4 Stream 1 Variable 1

3 Printers

Power Consumptions

Cooling Tower and Pump Outputs

Chiller, Pump, and AHU Outputs

! each line of data gives zone sensible

! and latent load and ambient dry bulb

! and wet bulb temperature

- 3.60-

;etinn 3

1000.

1750.

2500.

3250.

4000.

1000.

1750.

2500.

3250.

4000.

1000.

1750.

2500.

3250.

4000.

200.

350.

500.

650.

800.

200.

350.

500.

650.

800.

200.

350.

500.

650.

800.

80.

80.

80.

80.

80.

90.

90.

90.

90.

90.

70.

70.

70.

70.

70.

70.

70.

70.

70.

70.

80.

80.

80.

80.

80.

60.

60.

60.

60.

60.

60 TLIAAtl Vw -- LoFQ L- Q

Samnlo n1nt. I

Section 4 Dynamic Optimization Program

This program determines the optimal cost for a cooling plant with perfectly stratified storage

using dynamic programming as outlined in Chapter 7.

- 4.0 -

4

C Program for determining the optimal cost for a cooling plant with *

C storage using dynamic programming. Intially, a course grid is *

C utilized. Each successive iteration produces a refined grid. The *

C iteration is terminated when the total optimal cost does not change *

C significantly. *

C


Integer Stage, StageOn

Parameter (MStages = 24, MTimes = MStages + 1)

Dimension Xopt(MTimes), DX(MTimes)

Common /Ahead/ Qloads (MStages) , Twbs (MStages), Rates (MStages)

Common /Limits/ Qchmin, Qchmax, Qsmin

Common /Stor/ Delt, Cap, Ttop, Tbot

Common /ModelCoef/ Adir(2,2), bdir(2), cdir,

- Ach(3,3), bch(3), cch,

Adch(3,3), bdch(3), cdch,

Atch(2,2), btch(2), ctch,

Atdch(2,2), btdch(2), ctdch

Data pi/3.14159/

C

C ************************* Initializations ***************************

C

Nstages = 24

Nstates = 21

Delt = 1.

Cap = .1500.

Qchmin = 1000.

Qchmax = 5500.

Qsmin = 100.

Ttop = 60.

Tbot = 40.

Number of stages in time

Number of possible states at each time

Timestep associated with each stage

Capacitance of storage

Minimum allowable chiller load (surge)

Maximum allowable chiller load (capacity)

Minimum allowable storage charge & discharge

Controlled return temperature for discharge

Controlled supply temperature for charging

Read input data and generate load and wet bulb data ****

Time starts at 8 a.m., onpeak rates start at 8 a.m. and go for 12

hours, Minimum load and wet bulb are at 6 a.m.

-4.1-

CC***

c

c

aec;tlou+I "p' T ' I -ulll.V / lllz, LiH lnvnnmir Onfi*m;-Ynfi-irw%

flvn~mr vm;wdr,

Read(10,*) CapI, Nstor, dstor, Xinit

Read(lO,*) Qld avg, Q Range, Icntr, Qadd

Read(10,*) Twbavg, TRange

Read(10,*) RateOn, RateOff

Do 10 i = 1, Nstages

Vary = sin(pi/12*(i - 4))

If(Icntr .eq. 1 .or. i .le. 10 .or. i .ge. 22) Then

Qloads(i) = Qld_avg + QRange*Vary

If(i .eq. 22) Then

Qloads(i) = Qloads(i) + Qadd

Endif

Else

Qloads(Stage) = 0.0

Endif

Twbs(i) = Twbavg + TRange*Vary

If(i .le. 12) Then

Rates(i) = RateOn

Else

Rates(i) = Rate Off

Endif

10 Continue

C

C***** Read Curve-Fit Coefficients for Minimum Power Consumptions *****

C

C The form of the curve-fits are: P = f'Af + b'f + c

C where f is a vector of uncontrolled (forcing function) variables and

C A, b, and c contain coefficients of the fit. The five possible modes

C of operation are: 1) direct (no storage), 2) partial storage charging

C (cool-down) at a fixed supply, while also supply the load, 3) partial

C storage discharge (warm-up) at fixed return, while also operating the

C chillers to meet the load, 4) full charge (no load), 5) full discharge

C (chillers off).

C

Read(11,*) ((Adir(i,j),i=1,2),j=1,2), (bdir(i),i=1,2), cdirRead(1l,*) (( Ach(i,j),i=1,3),j=1,3), ( bch(i),i=1,3), cch

Read(ll,*) (( Adch(i,j),i=l,3), j=l,3), ( bdch(i),i=l,3), cdch

Read(11,*) (( Atch(i, j),i=1,2), j=1,2), ( btch(i),i=l,2), ctch

Read(l1,*) ((Atdch(i,j),i=1,2), j=l,2), (btdch(i),i=1,2), ctdch

- 4.2 -

LYV%,LIUII -T a-PaAlAalliMS,_%JL)LtllllziltlulI.qpetnn 4 -I

C

* * ** Loop through storage sizes *

C

Cap = CapI - dstor

Do 200 istor = 1,Nstor

Cap = Cap + dstor

Write(12,999) Cap

C

**************** Initialized States and Bounds *********************

C

Xlow = 0.0

Xhigh = 1.0

Ntimes = Nstages + 1

Do 25 i = 1, Ntimes

Xopt(i) = Xinit

DX(i) = 1.0

25 Continue

C

C****************** Call Dynamic Programming Routine ******************

C

Call Opt(Nstages, Nstates, Xopt, Xlow, Xhigh, DX, Total)

C

C*********** Loop through optimal path and output results to file *****

C

Tot Dir = 0.0

Write (12,103)

Do 50 Stage = 1, Nstages

Qload - Qloads (Stage)

Twb - Twbs (Stage)

Rate = Rates (Stage)

Xi = Xopt (Stage)

Xj = Xopt(Stage+1)Cost_Opt = Cost(Stage, Xi, Xj, Qch, Power)

Cost Dir = Cost(Stage, Xinit, Xinit, Qdir, Pdir)

If (Cost__Dir .gt. l.e+10) Then

CostDir = 0.0

Endif

Tot Dir = TotDir + CostDir

Write(12,104) Stage, Xi, Xj, Rate, Twb, Qload, Qch,

- 4.3 -

0 " 0I)VnnMir Ontimr7nt;nn

Section 4- Dynamic Optimization

Power, CostOpt, CostDir

50 Continue

Write(12,105) Total, Tot Dir

C

200 Continue

C

101 Format (A20)

102 Format(10(1x, F6.1))

103 Format(//2x,'Stage, XI, XF, Rate, Twb, Qload, Qch, Power, ,

'Cost (Opt), Cost(Dir) 1/)

104 Format (2x, 12,3 (2x,F4.2), 2x,F5.1,5 (lx, F8.2))

105 Format(/2x,'Optimal Cost = ',F8.2f

/2x, ' Direct Cost = ',F8.2)

999 Format(//2x,'**** Storage Capacity = ',F8.2,' ****')

End

C

C *

C Dynamic Programming routine for minimizing the sum of costs over *

C time for a single state variable: Steady-Periodic Solution *

C *

C

Subroutine Opt(Nstages, Nstates, Xopt, Xlow, Xhigh, DX, Total)

C

C Nstages = number of total stages (between times)

C Nstates = number of states at each time

C Xopt = array of optimal state values

C Xlow = minimum allowable state variable

C Xhigh = maximum allowable state variable

C DX = initial bounds on dynamic programming gridC Total = minimum total cost

C

Implicit Real*8 (A-H, O-Z)Parameter (M_States = 51, M_Stages = 24, M_Times = MStages + 1)

integer First, Stage, Statei, Statej, FNode(M_States, MTimes)

Dimension Xopt (Nstages+1), DX(Nstages+1), Xmin(MTimes),

Xmax (MTimes) , Togo (M_States, M_Times )

-4.4.

Section 4 Dynamic Optimizato

Data Tol/l.e-4/, imax/100/

C

* ************* ***** One-time Initializations ********************

C

First = 1

Xmin(l) = Xopt(1)

Xmax(1) = Xopt(1)

Xmin(Nstages+1) = Xopt(I)

Xmax(Nstages+l) = Xopt (I)Total = 0.0 ! Initialize total cost to zero

iter- 0

C******************** Main loop for refining grid *****************

C

5 Continue

iter = iter + 1

C

C ---------------------------- Initializations

CC Togo(i,j) = Minimum cost-to-go from state i, stage j until the endC FNode(i,j) = pointer to state at next stage along optimal path fromC state i, stage j

C

Do 15 i = 1,Nstates

Do 15 j = 1,Nstages+l

Togo(i,j) = 1.e+20

FNode(i,j) = 0

15 Continue

Togo(FirstNstages+1) = 0.0

Do 20 i = 2,Nstages

Xmin (i) = Dmaxl (Xlow, Xopt(i) - DX(i))

Xmax(i) = Dminl (Xhigh, Xopt (i) + DX(i))20 Continue

CC---------------Dynamic Programming Loop

CC Loop through all stages and state combinations within current bounds

C

Do 50 Stage = Nstages,1l,-I

- 4.5 -

L V t &%Z _tDan-Y i L Ln I i a a n

Do 40 Statei - 1,Nstates

Do 30 Statej = 1,Nstates

C

C Compute optimal stage costs between states. If at first stage, then

C only allow paths from known initial to end states

C

If(Stage .gt. 1 .or. Statei .eq. First) Then

Xi = Xmin(Stage) +

(Statei-l) * (Xmax (Stage)-Xmin (Stage)) / (Nstates-1)

Xj = Xmin(Stage+1) +

(Statej-1) * (Xmax (Stage+1) -Xmin (Stage+1)) / (Nstates-1)

SCost = Cost(Stage, Xi, Xj, Qch, Power)

Else

SCost = I.e+20

Endif

CCost = Togo(Statej,Stage+1) + SCost

If(CCost .lt. Togo(StateiStage)) Then

Togo(Statei, Stage) = CCost

FNode(StateiStage) = Statej

Endif

30 Continue

40 Continue

50 Continue

C

C ---------- Refine the grid and recalculate optimum if necessary

C

If(Abs(Total - Togo(First,,1))/Togo(First,,1) .gt. Tol .and.

iter .lt. imax) Then

Total = Togo(First,l)

Statei = First

Do 100 Stage = 2,Nstages

Statei = FNode(StateiStage-1)Xopt(Stage) = Xmin(Stage) +

• (Statei-1) * (Xmax (Stage) -Xmin (Stage) ) / (Nstates-1)

If( dabs(Xopt(Stage) - Xmax(Stage)) .gt. Tol .and.

*dabs (Xopt (Stage) - Xmin(Stage)) .gt. Tol .or.

*dabs(Xopt (Stage) - Xlow) .lt. Tol .or.

*dabs(Xopt(Stage) - Xhigh) .lt. Tol ) Then

-4.6-

Dvnamie Ontimi7ntinn'Roction 4

QArf n3.-feulVI!J " -" " ,"i a, '.%j IIUL ILxu .P

DX(Stage) = 0.5*DX(Stage)

Endif

Continue100

goto 5

C

Endif

Return

End

C Function for determining minimum costs between states*

Function Cost(Stage, Xi, Xj, Qch, Power)


Integer Stage

Parameter (MStages - 24)

Common /Ahead/ Qloads(MStages), Twbs(MStages), Rates(MStages)

Common /Limits/ Qchmin, Qchmax, Qsmin

Common /Stor/ Delt, Cap, Ttop, Tbot

Qload

Twb

Qdisch

Qch

= Qloads (Stage)

= Twbs (Stage)

= Cap*(Xi - Xj)*(Ttop - Tbot)/Delt

= Qload - Qdisch

Chiller Capacity Check **********************

C

If(Qch .gt. Qchmax .or. Qch .1t. 0.0) Then

Cost = l.e+20

Return

Endif

C

C****************** Branch According to Mode of Operation *

C

4.7-

DvmmieOntimi-intinn

C 4 ^d: A

Qstor - abs(Qdisch)

C

C ---------------------- Direct Cooling: No Storage

C

If(Qstor .lt. l.e-06) Then

If(Qch .gt. Qchmin) Then

Call System(1, Qload, Twb, Qstor, Power)

ElseIf(Qload .1t. i.e-06) Then

Power = 0.0

Else

Cost = l.e+20

Return

Endif

C

C --------------------------- Charging Mode

C

ElseIf(Qch .gt. Qload) Then

If(Qch .gt. Qchmin .and. Qload .gt. l.e-06) Then

C

C** Partial Charge (i.e. meet the load)

C

If(Qstor .gt. Qsmin) Then


C

C** Limit on Charging Rate: assume part-time in direct mode and

C part-time at minimum charging rate

C

Else

deltC = Qstor/Qsmin*delt

Call System(l, Qload, Twb, 0., Pdirect)

Call System(2, Qload, Twb, Qsmin, Pcharge)

Power = (Pcharge*deltC + Pdirect*(delt-deltC))/deltEndif

C** Full Charge with N'~o Load

ElseIf(Qch .gt. Qchmin) Then


- 4.8 -

aI

k.1LJ /,!,Li J! "-It!" JLuII6 I|I II

C

C** Below Minimum Chiller Capacity

C

Else

Cost = l.e+20

Return

Endif

C

C --------------------------- Discharge Mode

C

Else

If(Qch .gt. Qchmin) Then

C

C** Partial Discharge: meet load with chillers and storage

C

If(Qstor .gt. Qsmin) Then


C

C** Limit on Discharge Rate: assume part-time in direct mode and

C part-time at minimum discharge rate

C

Else

deltD = Qstor/Qsmin*delt

Call System(l, Qload, Twb, 0., Pdirect)

Call System(3, Qload, Twb, Qsmin, Pdisch)

Power = (Pdisch*deltD + Pdirect*(delt-deltD))/delt

Endif

C

C** Below Chiller Capacity: assume part-time at minimum capacity and

C part-time in full charge mode (chillers off)

C

ElseIf (Qload-Qchmin .gt. Qsmin) ThenQson =Q load - Qchmin

Qsoff - Qload

delton - Qch/Qchmin*delt

Call System (3, Qload, Twb, Qson, Pon)

Call System(5, Qload, Twb, Qsoff, Poff)

Power = (Pon*delton + Poff*(delt-delton))/delt

- 4.9-

f)vnnTnie Ont;m;-7nf;nnQcietinn L

Catmfi'nn IA1cq t..l.Iu ll "I' . UlllLai i ll

C

c** Chiller and storage limits exceeded

C

Else-

Cost = l.e+20

Return

c

c

Endif

Endif

C

Cost = Rates (Stage) *Power*Delt

C

Return

EndC

C *

C Routine for determining minimum power consumption for equipment *

C *

C

Subroutine System(Icntr, Qload,


Dimension f(3)

Common /ModelCoef/ Adir(2,2),

Ach (3, 3),

Adch (3, 3),

Atch(2,2),

Atdch(2,2),

Twb, Qstor, Power)

bdir (2),

bch (3),

bdch (3),

btch (2),

btdch (2),

cdir,

cch,

cdch,

ctch,

ctdch

C******************** Set-Up Uncontrolled Variables *****

C

f(l) = Qload

f(2) = Twb

f(3) = Qstor

Goto (100, 200, 300, 400, 500) ,Icntr

4.1O-

D'vnnim; ie Iti m .7ainn A

qetinn #1

C** Direct Mode: All control variables are free!

C Compute optimal power based upon chiller load and wet bulb

C

Continue

Power = cdir

Do 10 i = 1,2

temp = 0.

Do 5 j = 1,2

temp = temp +

Continue

Power = Power +

Continue

goto 1000

Partial Charge Mode:

Compute optimal power

storage charging rate

Adir(i, j) *f(j)

(temp + bdir (i))*f(i)

fixed chiller supply!

based upon chiller load, wet bulb, and

Continue

Power = cch

Do 20 i- 1,3

temp = 0.

Do 15 j = 1,3

temp = temp + Ach (ij) *f (j)

Continue

Power = Power + (temp + bch(i))*f(i)

Continue

goto 1000

Partial Discharge Mode: fixed AHU water return!

Compute optimal power based upon chiller load, wet bulb, and

storage discharging rate **

Continue

Power = cdch

Do 30 i = 1,3

temp = 0.

-4.11 -

**

100

5

10

C

C

C

200

15

20

C

C

C**

C

CC

300

."JaLaIlILIL; kjUtImizatioll.Dvnnmioi% I

CrapMn d

Do 25 j = 1,3

temp = temp + Adch(i, j) *f (j)

25 Continue

Power = Power + (temp + bdch (i) ) *f (i)

30 Continue

goto 1000

CC** Full Charge Mode: fixed chiller supply and return


C

400 Continue

f(1) = Qstor

Power = ctch

Do 40 i = 1,2

temp = 0.

Do 35 j = 1,2

temp = temp + Atch(i, j)*f(j)

35 Continue

Power = Power + (temp + btch(i))*f (i)

40 Continue

goto 1000

C

C** Full Discharge Mode: fixed chiller supply and return


C

500 Continue

Power = ctdch

Do 50 i = 1,2

temp =0.

Do 45 j 1,2

temp = temp + Atdch(ij)*f(j)

45 ContinuePower = Power + (temp + btdch (i) )*f (i)

50 Continue

C

1000 Continue

C

Return

End

- 4.12 -

Dvnrl~nt rn;m-nt nni

Section 5 Utility Routines

NonLinear Equation Solver 5.1This routine solves N non-linear equations in N unknowns using a dampedNewton's method.

Matrix Inverse 5.4This routine determines the inverse of a NxN matrix.

Psychrometrics 5.7Routine for determining properties of moist air.

Freon Properties 5.12Routine for determining properties of various refrigerants.

- 5.0-

C

C*********** Double Precision version of Newtons method *******************

C

Subroutine Solver(N, Diff, Xnew, Fnew, Jinv, Func, Tol)


Real*8 Jinv(N,N)

External Func

Dimension Diff(N), Xnew(N), Fnew(N), X(50), F(50), dX(50)

Data imax/50/, jmax/5/, Nmax/50/

C

C------------------ Check for limit on number of equations

C

If (N.gt.Nmax) Then

Write(*,I) Nmax

Format(' ** Error - Only ',I2,' Equations Allowed **')

Stop

Endif

C

C ----------------- Iterate using Newton's method with damping

C

iter = 0

Call Func(N, Xnew, Fnew)

C

5 iter=iter + 1

C

C** Save information from last iteration **

C

Fnorm = 0.

Do 10 i = 1,N

X(i) = Xnew(i)

F(i) = Fnew(i)

Fnorm = Fnorm + F(i)*F(i)10 Continue

C

C** Determine numerical approximation to Jacobian using forward

C differences **

C

Do 20 j = 1,N

-5.1-

Qortinn ; - Non.l~inpnr Vli.;,nn , mw ...

DeltX = Dabs(Diff(j)*X(j))

Xnew(j) = X(j) + DeltX


Do 15 i = 1,N

Jinv(i,j) - (Fnew(i) - F(i))/Deltx

15 Continue

Xnew(j) = Xnew(j) - DeltX

20 Continue

C

C** Inverse of the Jacobian **

C

Call DInvert(N, N, Jinv, Iflag)

If(Iflag .ne. 0) Then

Write(*, 26)

26 Format(' ** Error - No Solution to Equations **')

Stop

Endif

C

C** Determine maximum change in the independent variables **

C

Do 30 i = 1,N

dX(i) = 0.

Do 25 j = 1,N

dX(i) = dX(i) + Jinv(i,j)*F(j)

25 Continue

dX(i) = 2.*dX(i)

30 Continue

C

C** Determine new guess but don't accept if there is an increase in

C the residual error---> "damping" **

C

j 0Fnlast = Fnorm

40 Continue

Do 45 i -- I,N

dX(i) = dX(i)/2.

Xnew(i) = X(i) - dX(i)

-5.2-

Non-Linear Jmntinn ,RnivprQmpt; n n 4

LJ1. I%,,,v I" "- " = l,,W l Ill L)VIY"I

45 Continue


Fnorm = 0.

Do 50 i = 1,N

Fnorm = Fnorm + Fnew(i)*Fnew(i)

50 Continue

If(Fnorm.gt.Fnlast .and. j.lt.jmax) go to 40

70 Continue

C

C** Check for convergence: Use both absolute and relative checks **

C

rerror = 0.

aerror = 0.

Do 80 i =1N

rerror Dmaxl(rerror, Dabs(dX(i) /Xnew(i)))

aerror = Dmaxl(aerror, Dabs(dX(i)))

80 Continue

C

If(rerror.gt.Tol .and. aerror.gt.l.e-06

.and. iter.lt.imax) go to 5

If(iter .eq. imax) Then

Write(*,101) (Xnew(i),i=l,N)

Write(*,102) (Fnew(i),i=l,N)

Endif

C

101 Format(' ** Warning - Lack of Convergence in Equation Solver *

/4x,'Last Values of X = ',5(lx,lpell.3)/

9(23x,5(1x,lpell.3)))

102 Format(4x,'Last Values of F = ',5(Ix,lpell.3)/

9(23x,5(lx,lpell.3)))

Return

End

- 5.3-

Q-,a',t;nn; Nonn-I,inpgqr F1ninn n1 ...

Rpetion -LI IVA~~lL1 IAI VI

C* ************************************************ *************************** *

C *

C Double precision subroutine for determining the inverse of a matrix. *

C *

C

Subroutine DInvert (Nrc,N,A, if lag)


Parameter (Nmax = 50)

Dimension A(Nrc,Nrc), y(Nmax), irow(Nmax), jcol(Nmax)

Data eps/l.d-15/

C

if lag=0

C

C************************ Check for too large a matrix ************************

C

If(N .gt. Nmax) Then

iflag=1

Return

Endif

C

• ******************** Loop for elimination ***************************

C

Do 50 k =1,N

kml = k - 1

C

C

C

** Search for the pivot element **

pivot = 0.0

Do 10 i - l,N

Do 10 j = I,N

C** Look for invalid pivot subscripts **

C

If(k .ne. 1) Then

Do 5 iscan = 1,kml

Do 5 jscan = 1,kml

5.4-

Ma[. * .. T,...-...o..I

L.7%,%,aMti ae 11" M,!"1litr&iy niVlg ,.D,

If(i .eq. irow(iscan)) go to 10

If(j .eq. jcol(jscan)) go to 10

5 Continue

Endif

If (dabs(A(ij)) .gt. dabs (pivot)) Then

pivot = A(i,j)

irow(k) = i

jcol(k) = j

Endif

10 Continue

C

C ** Check for too small a pivot **

C

If(dabs(pivot) .lt. eps) Then

iflag= 2

Return

Endif

C

C ** Normalize pivot row elements **

C

irowk = irow(k)

jcolk = jcol(k)

Do 20 j = 1,N

A(irowk, j) = A(irowk, j)/pivot

20 Continue

C

C ** Determine inverse **

C

A(irowk, jcolk) = 1.0/pivot

Do 40 i = 1,N

Aijck = A(i,jcolk)

If(i .ne. irowk) Then

A(i, jcolk)= -Aijck/pivot

Do 30 j = 1,N

If(j .ne. jcolk) Then

A(i,j) = A(i, j) - Aijck*A(irowk, j)

Endif

30 Continue

- 5.5-

Section 5 Matriy Tnvprcp

Section-S MatriY Tnvpr.qp

Endif

40 Continue

50 Continue

C

****************************** Unscramble inverse *****************************

C

Do 80 j - 1,N

Do 60 i = 1,N

irowi - irow(i)

jcoli = jcol(i)

y(jcoli) = A(irowi,j)

60 Continue

Do 70 i - 1,N

A(i, j)=y(i)

70 Continue

80 Continue

Do 110 i = 1,N

Do 90 j = I,N

irowj - irow(j)

jcolj - jcol(j)

y(irowj) = A(i,jcolj)

90 Continue

Do 100 j = 1,N

A(ij) = y(j)

100 Continue

110 Continue

C

Return

End

- 5.6-

Section 5 Psychrometrics

c

c *

C THIS ROUTINE TAKES AS INPUT A DRY BULB TEMP. AND ONE OTHER PROPERTY: WET *

C BULB TEMP., REL.HUMIDITY(FRACTION), DEW PT.TEMP., HUMIDITY RATIO, OR *

C ENTHALPY, DEPENDING ON MODE. OUTPUTS ARE HUMIDITY RATIO (OR REL.HUMIDITY *

C IN MODE 4), WET BULB TEMP., AND ENTHALPY (OR REL.HUMIDITY IN MODE 5). *

C TEMPERATURES ARE IN CELSIUS (IUNITS=1) OR FAHRENHEIT (IUNITS=2). ENTHALPY *

C IS IN KJ/KG (IUNITS=1) OR BTU/LBM (IUNITS=2). *

C *

C

SUBROUTINE DPSYCH(IUNITSMODETDBI,TWBI,RHTDPI,W,H)

IMPLICIT REAL*8 (A-H,O-Z)

DATA PATM/I./

C

C***************************** UNIT CONVERSIONS *

C

TDB =

TWB =

TDP =

TDBI

TWBI

TDP I

C

IF(IUNITS .EQ. 2) THEN

TDB = (TDB - 32.)/1.8

TWB

TDP

H

= (TWB - 32.)/1.8

= (TDP - 32.)/1.8

= (H - 7.68)/0.43002

END IF

C

C** SATURATION PRESSURE OF WATER AT WET BULB, DRY BULB, OR DEW POINT

C TEMPERATURE. **

C

IF(MODE .EQ. 6) TDB (H -

CALL SAT (TDB, PSATDB)

GOTO (1,2,3,2,2,2) ,MODE

CALL SAT (TWBPSAT)

GOTO 5

PSAT -=PSATDB

2501.*W)/(1.005 + 1.859*W)

-5.7-

£hPhflfl~ P~veh rnni~trie~L7~. ~ LI

GOTO 5

3 CALL SAT(TDPPSAT)

5 CONTINUE

C

C************** CALCULATE HUMIDITY RATIO AND WET BULB TEMPERATURE *************

C

GO TO (10,20,30,40,50,50), MODE

C

C ------------------- MODE 1: DRY BULB AND WET BULB SUPPLIED

C

10 IF (TWB .LE. 0.) THEN

P = PSAT - 5.704E-4*(TDB-TWB)*PATM

W = .62198 * P/(PATM-P)

ELSE

WSAT = .62198 * PSAT/(PATM-PSAT)

W = WSAT - (TDB-TWB)*(0.24 + .441*WSAT)/(597.31

+ 0.441*TDB - TWB)

END IF

H = 1.005*TDB + W*(2501. + 1.859*TDB)

GO TO 100

C

C ---------------- MODE 2: DRY BULB AND RELATIVE HUMIDITY SUPPLIED

C

20 W = .62198 * PSAT*RH/(PATM-PSAT*RH)

GO TO 40

C

C --------------------- MODE 3: DRY BULB AND DEW POINT SUPPLIED

C

30 W = .62198 * PSAT/(PATM-PSAT)

C

C ------------------------ FIND ENTHALPY FOR MODES 2 - 4

C

40 H = 1.005*TDB + W*(2501. + 1.859*TDB)

CC------------FIND WET BULB TEMPERATURE FOR MODES 2 - 6

C

50 DPRESS = ABS(I.-PATM)

IF (H .GT. 9.67 .AND. DPRESS .LT. 0.00001) THEN

-5.8 -

Q,,-tinn ; -- Ps vchrornptrics

l.tc%, , I P v % a ,rnVml itLA M a

Y = DLOG(H+17.68)

TWB = 26.7453 + Y*(-43.44 + Y*(13.909 - Y*.977))

ELSE

C

C** USE LINEAR APPROXIMATION WHEN H LT 9.67 **

C

IF (MODE .EQ. 3) CALL SAT(TDBPSAT)

WSAT = .62198 * PSAT/(PATM-PSAT)

T = (H - 2501.*WSAT)/(1.005 + 1.859*WSAT)

CALL SAT(TPSAT)

WT = .62198*PSAT/(PATM-PSAT)

C

C** FIND POINT ON LINE BETWEEN (TWT) AND (TDBWSAT) WITH ENTHALPY H **

C

IF (ABS(T-TDB) .LT. 0.01) THEN

TWB = (T+TDB)*0.5

ELSE

SLOPE = (WSAT-WT)/(TDB-T)

A = 1.859*SLOPE

B = 1.005 + WSAT*1.859 + SLOPE*(2501.-I.859*TDB)

C = -H + 2501.*(WSAT - SLOPE*TDB)

IF (SLOPE .LT. I.E-7) TWB--C/B

IF (SLOPE .GT. .99E-7) TWB = (-B + SQRT(B*B-4.*A*C))/(2.*A)

ENDIF

ENDIF

C

C--------------MODE 5 CONTINUED: DRY BULB AND ENTHALPY SUPPLIED

C


W = (H - 1.005*TDB)/(2501. + 1.859*TDB)

ENDIF

C

C********************* CONVERT OUTPUTS TO APPROPRIATE UNITS *******************

C100 CONTINUE

PV = PATM'*W/(.62198 + W)

IF (MODE .NE. 2) RH = PV/PSATDB

IF (MODE .NE. 3) THEN

Y = DLOG(1.013E05*PV)

-5.9-

cPetinnI Pqqvehrnm, trie qI

Pqvrh rnmptriv-€

TDP = -35.957 - 1.8726*Y + 1.1689*Y*Y

IF(TDP .LT. 0.) TDP = -60.45 + 7.0322*Y + 0.3700*Y*Y

ENDIF

IF(IUNITS .EQ. 2) THEN

H = H*0.43002+7.68

TDB = 1.8*TDB + 32.

TWB = 1.8*TWB + 32.

TDP = 1.8*TDP + 32.

ENDIF

C

IF (MODE .EQ. 6) TDBI = TDB

IF (MODE .NE. 2) TWBI = TWB

IF (MODE .NE. 3) TDP I = TDP

C

RETURN

END

C* ** ** ** **** * * *** * **** ** ** * ** * ** ** **** * ** ** * ***** ** * **** ** * *** * *** *** *** **** * *

C *

C SUBROUTINE FOR FINDING SATURATION PRESSURE OF WATER AT A GIVEN TEMPERATURE *

C *

C

SUBROUTINE SAT(T, PSAT)

IMPLICIT REAL*8 (A-H,O-Z)

C

C** STATEMENT FUNCTIONS FOR THE SATURATION PRESSURE OF WATER (IN ATMOSPHERES)

C AS FUNCTION OF TEMPERATURE **

C

P1(Z) = -7.90298*(Z - 1.0)

P2(Z) - 5.02808*DLOG10(Z)

P3(Z) = -1.3816E-07*(10.**(11.344*(1. - 1./Z)) -1.)

P4(Z) - 8.1328E-03*(10.**(-3.49149*(Z - 1.)) - 1.)

P5(Z) = -9.09718*(Z - 1.)P6(Z) = -3.56654*DLOGI0(Z)

P7(Z) - 0.876793"(I. - I./Z)

P8 = -2.2199

- 5.10-

3-ection 2 %, 1111 xi " I Ic L-t- I % apd-% in Z

CURVE-FITS FOR SATURATION PRESSURE *****************

C

IF (T .LE. 0.0) THEN

z = 273.16/(T + 273.16)

PSAT= 10.**(P5(Z) + P6(Z) + P7(Z) + P8)

ELSE

Z = 373.16/(T + 273.16)

PSAT = 10.**(PI(Z) + P2(Z) + P3(Z) + P4(Z))

END IF

RETURN

END

-5.11 -

Srtion 5 - P.sveh romptrics

__ . 'l1%; tI II ,,I,,AYIuL a u V I t I IC 3

SUBROUTINE FREON(TP,H,S,X,V,U,NREFITYPE)

C

C *

C SUBROUTINE FOR DETERMINING THERMODYNAMIC PROPERTIES OF REFRIGERANTS. THIS *

C ROUTINE IS DEVELOPED FROM EQUATIONS GIVEN IN "REFRIGERANT EQUATIONS", *

C R.C. DOWNING, DUPONT COMPANY. THE ARGUMENTS IN A CALL TO FREON ARE: *

C

C T = TEMPERATURE (DEGREES F) *

P = PRESSURE (PSI)

H = SPECIFIC ENTHALPY (BTU/LBM)

S = SPECIFIC ENTROPY (BTU/LBM-F)

X = QUALITY

V = SPECIFIC VOLUME (LBM/FT**3)

U - SPECIFIC INTERNAL ENERGY (BTU/LBM)

NREF REFRIGERANT TYPE (E.G., 11 FOR R12, 12 FOR

ITYPE = TWO-DIGIT INTEGER CODE FOR IDENTIFYING THE

SUPPLIED. THE CODE IS OF THE FORM IJ WHERE

TWO STATES AND ARE BETWEEN 1 AND 5 (1 =T,

5 = X, 6 = V, 7 = U)

R12, ETC.)

TWO STATES THAT

I AND J SPECIFY

2 = P, 3 = H, 4

THE REFRIGERANTS CURRENTLY SUPPORTED BY THIS PROGRAM ARE RII, R12, R13,

R14, R22, R114, R500, AND R502.

C

EXTERNAL THCON, TSCON, SVCON, XVCON, SXCON, HXCON, PSCON, PHCON

LOGICAL ERROR

D IMENS ION Q (44)

COMMON /CONST/ Q

DATA IMAX/100/,JMAX/1O/,TOL/O.O001/

C


C

C FILL INITIALIZES CORRELATION CONSTANTS ACCORDING TO REFRIGERANT TYPE

C

ERROR=. FALSE.

CALL FILL(NREF)

-5.12-

CC

C

C

C

C

C

C

C

C

C

C

C

C

C

ARE

THE

= S

Frpnn Prnnart;ac I

..- tinm,, aSg. Fr n Pi.

J=ITYPE/10

K=ITYPE-J*10

IF(J.EQ.1 .OR. K.EQ.1) THEN

T=T+Q (44)

END IF

C

C ** BRANCH ACCORDING TO INPUT STATES SUPPLIED **

C

C TX KNOWN

IF(ITYPE.EQ.51.OR.ITYPE.EQ.15)GO TO 111

C T,P KNOWN


C T,V KNOWN


C P,V KNOWN


C P,X KNOWN


C T,H GIVEN


C T,S KNOWN


C SV KNOWN


C V,X KNOWN


C S,X KNOWN


C H,X KNOWN


C P,S KNOWN


C P, H KNOWN

IF (ITYPE.EQ.23 .OR.ITYPE.EQ.32)GO TO 231

C H, S KNOWN

IF (ITYPE.EQ. 34 .OR. ITYPE.EQ. 43) GO TO 241

c

C T, X KNOWN FIND LIQUID SPECIFIC VOLUME

- 5.13-

Setion 5 Frenn Pronnrtips

-.. a, a t- L-n ' F Xn.s P V aVt 2

C

11 VL=1./DLIQ(T)

P=VPR (T)

CALL SPVOL(T,P,VV)

V=VL+X* (VV-VL)

DH=DHLAT (T, VV, VL, P)

CALL ENTHAL (HVfP,VV, T)

H=HV-(1.-X) *DH

DS=DH/T

CALL ENTROP (SVAPfT,VV)

S=SVAP- (1. -X) *DS

GO TO 1000

C

C TP KNOWN FIND VAPOR PRESSURE ASSUME SUPERHEAT

C

121 CALL SPVOL(TP,V)

CALL ENTHAL (HfP,V, T)

CALL ENTROP (SfT,V)

X=1. 5E+38

GO TO 1000

C

C TV KNOWN

C

131 IF(T.GT.Q(42))GO TO 22

PV=VPR(T)

CALL SPVOL (T, PV, VV)

20 IF (V-VV) 21,21,22

21 P=PV

VL=1./DLIQ (T)

X= (V-VL) / (VV-VL)DH-DHLAT (T, W, VL, P)

DS=DH/T

CALL ENTHAL (HV, P, VV, T)

H=HV-(1.-X)*DH

S CALL ENTROP (SV, T, VV)

S=SV- (1.-X) *DS

GO TO 1000

C IF SUPERHEAT

22 X=1. 5E+38

-5.14-

Section 5 F'rmrn Prnnert;pqI

lo166tMI.flJ FeonPrn- - IL-~~~

P=PR(T,V)

23 CALL ENTHAL(H,P,V,T)

CALL ENTROP (ST,V)

GO TO 1000

C

C P,V KNOWN

C

141 IF(P.GT.Q(41))GO TO 45

CALL TSAT(PTV)

CALL SPVOL (TV, FP, VV)

IF (V-VV) 44,44,45

C IF SATURATED

44 T=TV

VL-1./DLIQ (T)

X= (V-VL) / (VV-VL)


CALL ENTHAL(HV;P,VVT)

H=HV- (I.-X)*DH

DS=DH/T

CALL ENTROP (SVT,VV)

S=SV- (I.-X)*DS

GO TO 1000

C IF SUPERHEAT

45 X=1.5E+38

ITER=0

T1=Q(42)

42 XT=PR (TI, V) -P

ITER=ITER+I

DT=DPDT (TI, V)

T=TI- (XT/DT)

Z=ABS (T-TI)

T1=T

IF(ITER.GT.IMAX) THEN

ERROR=. TRUE.

GO TO 43

END IF

IF(Z-.001) 43, 43, 42

43 CONTINUE

- 5.15-

Treon Pronnrtieslqpctonn5

0,%,LJtnll " a a- ,n P, %,I." 1.

CALL ENTHAL(HP,V,T)

CALL ENTROP(ST,V)

GO TO 1000

C

C P,X KNOWN

C

151 CALL TSAT(PT)

VL=1./DLIQ (T)

CALL SPVOL(T,P,VV)

V=VV-(I.-X)*(VV-VL)

CALL ENTHAL(HV,P,VV,T)


H=HV - (1.-X)*DH

DS=DH/T

CALL ENTROP (SV, T,VV)

S=SV-(1.-X)*DS

GO TO 1000

C

C TH KNOWN

C

161 IF(T.GT.Q(42))GO TO 61

PV=VPR (T)


CALL ENTHAL (HV, PV, VV, T)

IF (H-HV) 65, 65,61

C FOR SUPERHEAT

61 X=1.5E+38

V=Q (43)

CALL SOLVE (T, H, V, THCON, TOL, IFLAG)

IF (IFLAG. EQ. 1) ERROR=. TRUE.

P=PR(T,V)

CALL ENTROP(ST,V)

GO TO 1000

C IF SATURATED

65 VL-1./DLIQ (T)

P=PV

DH=DHLAT (T,VV, VL, P)

X=I.- (HV-H) /DH

V-VV- (1 1.-X) * (VV-VL)

- 5.16-

Rpetnn- Freon PrnnortipqI

peL,7. t,,6Lnn F • r• a49IL; % I .IJljI I Ia

DS=DH/T

CALL ENTROP (SV, T,VV)

S=SV- (1.-X) *DS

GO TO 1000

C

C TS KNOWN

C

171 PV=VPR(T)



IF (S-SV) 75, 75, 71

C FOR SUPERHEAT

71 X=1.5E+38

V=VV

CALL SOLVE (T,S,V,TSCONTOL, IFLAG)

IF(IFLAG.EQ.1) ERROR=.TRUE.

P=PR(T,V)

CALL ENTHAL(HP,V,T)

GO TO 1000

75 VL=1./DLIQ(T)

P=PV

DH=DHLAT (T,VV,VL,P)

DS=DH/T

X=1.- (SV-S)/DS

V=VV- (1 .-X) * (VV-VL)

CALL ENTHAL (HV,P, VV, T)

H=HV- (1.-X)*DH

GO TO 1000

C

C SV KNOWN

C

181 T=500.

CALL SOLVE (V, S, T, SVCON, TOL, IFLAG)

IF (IFLAG. EQ. 1) ERROR=. TRUE.

P-PR (T,V)

CALL ENTHAL (H, P,V, T)

X=1. 5E+38

GO TO 1000

- 5.17 -

Section 5 Fripnn Prnnprt;iaQI

~ik onW& - -r -- nAK P ta uI ata ; a

c

C x, V KNOWN

C

191 T=500.

CALL SOLVE (X, V1 T, XVCON, TOL, IFLAG)


P=VPR (T)

CALL SPVOL(TP,VV)

VL=1./DLIQ (T)


CALL ENTHAL(HVP,VVT)

H-HV-(1.-X)*DH

DS=DH/T


S=SV-(1.-X) *DS

GO TO 1000

C

C SX KNOWN

C

201 T=500.

CALL SOLVE (X, S, T, SXCON, TOL, IFLAG)

IF (IFLAG.EQ. 1) ERROR=. TRUE.

P=VPR (T)

CALL SPVOL(TP,VV)

VL=1./DLIQ (T)

V=VV-(I.-X) * (VV-VL)



H=HV- (1. -X) *DH

GO TO 1000

C

C HX KNOWN

C

211 T=500.

S CALL SOLVE (X, H, T, HXCON, TOL, IFLAG)

IF(IFLAG.EQ.1) ERROR=. TRUE.

P=VPR (T)

CALL SPVOL (T, P, VV)

VL- . /DLIQ (T)

- 5.18-

'Section 5 Fre~rn )rnnprtipq

Section 5 Freon Prooerties

V=VV-(1.-X)*(VV-VL)

DS=DHLAT(TVVVLP)/T


S=SV-(i.-X)*DS

GO TO 1000

C

C PS KNOWN

C

221 IF(P.GT.Q(41))GO TO 15

CALL TSAT(PTS)

CALL SPVOL(TSP,VV)

-CALL ENTROP (SVTSVV)

IF(S-SV) 18,18,15

C FOR SATURATED

18 T=TS

P=VPR (T)

VL=1./DLIQ (T)


DS=DH/T

X=1.-(SV-S)/DS

V=VL+X* (VV-VL)


H=HV- (1. -X) *DH

GO TO -1000

15 X=11.5E+38

T=Q(42)

CALL SOLVE (P, S, T, PSCON, TOL, IFLAG)


CALL SPVOL(T,PV)

CALL ENTHAL(HP,V,T)

GO TO 1000

C

C PH KNOWN

C231 IF (P .GT.Q (41) )GO TO 12

CALL TSAT(P,TS)

CALL SPVOL (TS, P, VV)

CALL ENTHAL (HV, P, VV, TS)

- 5.19 -

OVJ%LIMII 1j .LqJl v I LzfJ I LI 3

IF (H-HV) 11, 11, 12

C IF SATURATED

11 T=TS

P=VPR (T)

VL=1./DLIQ (T)


DS=DH/T

X=1.-(HV-H)/DH

V=VL+X* (VV-VL)


S=SV- (1.-X)*DS

GO TO 1000

C IF SUPERHEAT

12 X=1.5E+38

T=Q(42)

CALL SOLVE (PH, T, PHCON, TOL, IFLAG)


CALL SPVOL(T,P,V)

CALL ENTROP (ST,V)

GO TO 1000

C

C HS KNOWN

C

241 T=460.

P=100.0

CALL HSCON(TP,H,S,F,G)

ENORM=F*F+G*G

ITER=0

95 ITER=ITER+1

TOLD=T

POLD=P

FOLD=F

GOLD=G

T=TOLD+AMAX1 (0.001,0. 001*TOLD)P--POLD+AMAX1 (0 .001, 0. 001I*POLD)

CALL HSCON (T, POLD, H,S, F,G)

DFDT= (F-FOLD) / CT-TOLD)

DGDT= (G-GOLD) / CT-TOLD)

- 5.20-

Qpetinn -; Frpnn Prnnart;ac I

L3 CLl U a a -I ' cult i IU-gi

CALL HSCON(TOLDP,H,S,F,G)

DFDP- (F-FOLD) / (P-POLD)

DGDP-(G-GOLD) / (P-POLD)

IF(ABS(DFDT*DGDP) .GT.1.E-06 .OR. ABS(DGDT*DFDP) .GT.I.E-06) THEN

DP=2. * (FOLD*DGDT-GOLD*DFDT) / (DFDT*DGDP-DGDT*DFDP)

DT=2. * (GOLD*DFDP-FOLD*DGDP) / (DFDT*DGDP-DGDT*DFDP)

ELSE

DP=0.

DT=0.

ENDIF

J=0

ELAST=ENORM

98 J=J+l

DP=DP/2.

DT=DT/2.

P=AMAX1 (AMINI (POLD+DPQ(41) ) , 1.E-06)

T=AMAX1 (AMINI (TOLD+DT, Q (42) ) , 1.E-06)

CALL HSCON(T,P,H,S,F,G)

ENORM=F*F+G*G

IF(ENORM.GT.ELAST .AND. J.LT.JMAX) GO TO 98

Z1=ABS (P-POLD)

Z2=ABS (T-TOLD)


ERROR=. TRUE.

GO TO 96

END IF

IF (Zl-. 001) 96,96,95

96 IF (Z2-.001) 97,97, 95

97 CONTINUE

CALL SPVOL(T,P,V)

X=1. 5E+38

GO TO 1000

1000 CONTINUE

U=H-P*V* (144./778.)

T-T-Q (44)

IF(ERROR) WRITE(*,1001) ITYPE,T,P,H,S,X,V

1001 FORMAT.(' ** WARNING - SOLUTION DID NOT CONVERGE **'/4X,

*'ITYPE = ',I2, ' T, 2, H, 5, X, V -',

*3 (IX,IPEII. 3)/33X,3 (1X,IPEII. 3) )

- 5.21 -

qpetinn ;-

q.etiAfl; urn ~'iuu ri upertues

RETURN

END

C

C SUBROUTINE FOR FINDING SPECIFIC VOLUME

C

SUBROUTINE SPVOL (TA, PA, VA)

D IMENS ION Q (44)

COMMON /CONST/ Q

DATA IMAX/50/

R=Q (14)

V1= (R*TA)/PA

ITER=0

5 X=PR(TA,Vl)-PA

ITER=ITER+1

DX=DPDV (TA, Vi)

VA=Vl- (X/DX)

Z=ABS (VA-Vl)

Vl=VA


WRITE(*,1001) TAPAVAZ

RETURN

END IF

IF (Z-.0001) 10,10,5

10 CONTINUE

1001 FORMAT(' ** WARNING - SUBROUTINE SPVOL, SOLUTION DID NOT CON',

'VERGE **1/4X,'T, P, V, ABS(V-VLAST) = ',4(IX,1PE11.3))

RETURN

END

C

C SUBROUTINE FOR FINDING ENTHALPY

C

SUBROUTINE ENTHAL (H, P, V, T)DIMENSION Q(44)

COMMON /CONST/ Q

XJ-0 .185053:

T2=T**2/2 .0

T3=T**3/3 .0

T4=T**4/4. 0

-5.22-

VB=AMAX1 (AMINI (V-Q (15) , 1 .E+05) , 1.E-05)

VB2=VB**2*2.0

VB3=VB**3*3.0

VB4=VB**4* 4.0

XKT=Q(31) *T/Q(42)

EKT=EXP (-XKT)

AV=Q(32) *V

C AVOID DIVIDING BY ZERO

IF(AV.EQ.0.0 .OR. ABS(AV).GT.30.) GO TO 105

EAV=EXP (AV)

IF(Q(33) .EQ.0.0)GO TO 100

CLN=Q(33) *(ALOG(1.0+(1.0/ (Q(33) *EAV))))

GO TO 110

100 CLN=0.0

GO TO 115

105 EAV=0.0

RX=0.0

RZ=0.0

CLN=0.0

GO TO 115

110 RX=(Q(28)/Q(32))*(1.0/EAV-CLN)

RZ=Q(30) / (Q(32) *EAV) -Q(30) *CLN/Q(32)

115 HI=Q(34)*T+Q(35)*T2+Q(36)*T3+Q(37)*T4-Q(38)/(4.0*T2)+XJ*P*V

H2=XJ* (Q(16)/VB+Q(19)/VB2+Q(22)/VB3+Q (25)/VB4+RX)

H3=XJ* (Q(18)/VB+Q (21)/VB2+Q (24)/VB3+Q(27)/VB4+RZ) * (1.0+XKT) *EKT

H=H1+H2+H3+Q (39)

RETURN

END

C

C SUBROUTINE FOR FINDING ENTROPY

C

SUBROUTINE ENTROP (S, T, V)DIMENSION Q(44)

COMMON /CONST/ Q

xJ-0 .185053

R=Q (14)

T2=T**2/2 .0

T3=T**3/3.0

VB=AMAXl (AMINI (V-Q (15) , 1 .E+05) , 1I.E-05)

- 5.23-

c.,petinn -5 Freon Prnnprt;pqI

VB2=2.0"*'2vB3=3.0*VB**3

VB4=4.0*VB**4

XKT=Q (31) *T/Q (42)

EKT=EXP (-XKT)

AV=Q(32) *V

IF(AV.EQ.0.0 .OR. ABS(AV).GT.30..OR. Q(33).EQ.0.0)GO TO 100

EAV=EXP (AV)

CLN=Q(33) *ALOG(1.0+(1.0/ (Q(33) *EAV)))

RX= (Q (29) /Q (32)) * (1. 0/EAV-CLN)

IF(CLN.GT.1.E-20) THEN

RZ=(Q(30)/Q(32))*EAV-Q(30)/(Q(32)*CLN)

ELSE

RZ=0.

ENDIF

GO TO 110

100 RX=0.0

RZ=0.0

EAV=0.0

CLN-0.0

110 G=Q (18)/VB+Q (21)/VB2+Q (24)/VB3+Q (27)/VB4+RZ

SI=Q(34)*ALOG(T)+Q(35)*T+Q(36)*T2+Q(37)*T3-Q(38)/(2.0*T**2)

S2=XJ*R*ALOG (VB)

S3=-XJ* (Q (17) /VB+Q (20) /VB2+Q (23) /VB3+Q (26) /VB4+RX)

S4=( (XJ*Q (31) *EKT) /Q (42) ) *G

S=S1+S2+S3+S4+Q (40)

RETURN

END

C

C SUBROUTINE FOR FINDING SATURATION TEMP AT A PRESSURE

C

SUBROUTINE TSAT (P, TS)EXTERNAL PCON

DATA TOL/0.0001/

TS=500.

CALL SOLVE (0. ,P,TS,PCON, TOL, IFLAG)

IF (IFLAG.EQ.1I) WRITE (*,1i001)

- 5.24 -

Spetonn 5- lPrpnn Prfnav-+i*,,ac

L-1 tioNr 5 FZ aa .f n Prtt V nVIVItiv

1001 FORMAT(' ** WARNING - SUBROUTINE TSAT, SOLUTION DID '

NOT CONVERGE **'/4X, 'P, T =',2(1X,IPE11.3))

RETURN

END

C

C

C FUNCTION FOR FINDING LIQUID DENSITY AT A TEMPERATURE

C

FUNCTION DLIQ(T)

DIMENSION Q(44)

COMMON/CONST/Q

TTC=AMAX1(1.-T/Q(42),0.)

DLIQ=Q(1)+Q(2)*TTC**(I./3.)+Q(3)*TTC**(2./3.)+Q(4)*TTC

@ +Q(5) *TTC** (4./3.) +Q (6) *TTC**0.5+Q (7) *TTC**2.

RETURN

END

C

C FUNCTION FOR FINDING VAPOR PRESSURE

C

FUNCTION VPR(T)

DIMENSION Q(44)

COMMON/CONST/Q

IF (Q (13) -T) 701,701,702

701 PL=Q (8) +Q(9)/T+Q (10) *LOG10(T) +Q (11) *T

GO TO 703

702 PL=Q (8) +Q (9)/T+Q (10) *LOG10 (T) +Q (11) *T+Q (12)*((Q(13) -T)/T)

@*LOG10 (Q (13) -T)

703 VPR=10**PL

RETURN

END

C

C FUNCTION FOR FINDING LATENT HEAT OF VAPORIZATION

C

FUNCTION DHLAT(TVG, VFP)

DIMENSION Q(44)

COMMON/CONST/Q

XJ=0 .185053

XLN1O0-2 . 302585093

XLOGE=0 . 4342944819

- 5.25-

'Section 5 Frpnn Prnnprtipc.I

,R1cinn -5L1~' ~ p U ~~JUj p i

IF (Q (13) -T) 711,711, 712

711 E=Q (12) * (XLOGE/T)

GO TO 713

712 E=Q (12) * (XLOGE/T+Q (13) *LOG10 (Q (13)-T)/T**2)

713 DHLAT=XJ*T*(VG-VF)*(P*XLNI0*(-Q(9)/T**2+Q(10)/(T*XLNI0)+Q(11)-E))

RETURN

END

C

C FUNCTION FINDING PRESSURE

C

FUNCTION PR(TV)

DIMENSION Q(44)

COMMON/CONST/Q

RI=Q (14)

EKT=EXP (-Q (31) *T/Q (42))

VB=AMAX1 (AMINI (V-Q (15), 1.E+05), 1.E-05)

AV=Q(32) *V

IF(AV.EQ.0..OR. ABS(AV).GT.30.) THEN

P5=0.

ELSE

EAV=EXP (AV)

P5=(Q (28) +Q (29) *T+Q (30) *EKT) / (EAV* (1.+Q (33)*EAV))

ENDIF

P1= (RI*T) /VB+ (Q (16) +Q (17) *T+Q (18) *EKT) /VB**2

P2= (Q (19) +Q (20) *T+Q (21) *EKT) /VB**3

P3= (Q (22) +Q (23) *T+Q (24) *EKT)/VB**4

P4= (Q (25) +Q (26) *T+Q (27) *EKT) /VB**5

PR=PI+P2+P3+P4+P5

RETURN

END

C

C FUNCTION FOR FINDING DP/DV AT CONST T

C

FUNCTION DPDV(TV)

D IMENS ION Q (44 )

COMMON/CONST/Q

R=Q (14)

EKT=EXP (-Q (31) *T/Q (42))

- 5.26 -

Frpnn Pvwfnav-*;AmcI

VB=AMAX1 (AMINI (V-Q (15), 1.E+05), 1.E-05)

AV=Q(32) *V


RR=0.

ELSE

EAV=EXP (AV)

EAV2=EXP (2. 0*AV)

RR=- (Q (32) *EAV+2.0Q*Q (32) *Q (33) *EAV2) / (EAV+Q (33) *EAV2) **2

ENDIF

DX1=-R*T/VB**2-2.0* (Q(16)+Q(17) *T+Q (18) *EKT)/VB**3

DX2=-3 .0* (Q (19) +Q (20) *T+Q (21) *EKT) /VB**4

DX3=-4. 0* (Q (22) +Q (23) *T+Q (24) *EKT) /VB**5

DX4--5 .0* (Q (25) +Q (26) *T+Q (27) *EKT) /VB**6

DX5=(Q(28)+Q(29) *T+Q(30) *EKT) *RR

DPDV=DX1 +DX2 +DX3 +DX4+DX5

RETURN

END

C

C FUNCTION FINDING DP/DT AT CONST V

C

FUNCTION DPDT(T,V)

DIMENSION Q(44)

COMMON/CONST/Q

R=Q (14)

EKT=EXP (-Q (31) *T/Q (42))

VB=AMAX1 (AMINi (V-Q(15), 1.E+05), 1 .E-05)

AV=Q(32) *V


TERM=0.

ELSE

EAV=EXP (AV)

TERM-(Q(29)-TTC*Q(30)) / (EAV* (1.+Q(33) *EAV))END IF

TTC-=(Q (31) *EKT) /Q (42)

DTI=R/VB+ (Q (17)-TTC*Q (18) )/VB**2

DT2-(Q(20)-TTC*Q(21) ) /VB**3

DT3= (0(22)-TTC*Q (24) )/VB**4

DT4= (Q (26) -TTC*Q (27) )/VB**5+TERM

DPDT=DT 1+DT2 +DT3 +DT 4

- 5.27 -

S I ,tinn5 Freon Pronnrties

A a

RETURN

END

C

C SUBROUTINE TO INTIALIZING CORRELATION CONSTANTS FOR SPECIFIED REFRIGERANT

c

SUBROUTINE FILL (NREF)

INTEGER RTYPE (8)

DIMENSION Rll(44),,Rl2(44)fRl3(44)rRl4(44)tR22(44)

DIMENSION R114(44)rR500(44)rR502(44)rQ(44)

COMMON /CONST/ Q

DATA RTYPE/11fl2ri3fl4r22fll4r5OOr5O2/,NRTYPE/8/

DATA NCOEF/44/jLAST/0/

C

DATA Rll/34.57,57.638llf43.6322f-42.82356,36.70663fO.rO.r

0 42.147028651-4344.343807f-12.84596753f4.0083725E-03rO.O3l3605356f

0 862.0710.078ll7fO.OOl9Of-3.126759fl.318523E-03f-35.76999f

a -0.025341,4.875121E-05,1.220367,1.687277E-031-1.805062E-06t

0 O.r-2.358930E-05r2.448303E-08r-1.478379E-04fl.057504EO8f

0 -9.472103EO4rO.f4.5Or580.fO.rO.023815f2.798823E-04r-2.123734E-07f

0 5.999018E-llf-336.80703f5O.5418t-0.0918395r639.5r848.0710.028927f

0 459.67/

C

DATA R12/34.84,53.34118710.118.6913710.121.983961-3-150994I

0 39.883817271-3436.6322281-12.4715222814.73044244E-03,0.fO.f

0 0.088734fO.OO65093886r-3.40972713rl.59434848E-031-56.7627671t

0 0.0602394465r-1.87961843E-05rl.31139908f-5.4873701E-04fO.fO.rO.r

a 3.468834E-09f-2.54390678E-05tO.fO.rO.r5.47510.,O.r8.0945E-03f

0 3.32662E-04r-2.413896E-O7r6.72363E-llrO.f39.556551t

0 -0.016537936,596.9,r693.3rO.O287r459.7/

C

DATA R13/36.06996f54.39512410.18.512776,0.125.879906r9.589006t

0 25.967975t-2709.538217f-7.1723439lr2.545154E-03fO.28030109lr

.-5.28-

,;Pction 5 - Freon PrnnprtipQI

DATA R14/39.06,69.56848914.5866114,36.1716662t-8.05898610.fO.f

20.715453891-2467.505285f-4.69017025r6.4798076E-04fO.770707795F

424.rO.1219336rO.OOl5r-2.16295912.135114E-03,-18.941131I

4.404057E-03,1.282818E-05fO.539776fl.921072E-04f-3.918263E-07f

0.,-4.481049E-06r9.062318E-091-4.836678E-05f5.838823EO71

-9.263923EO410.14.1661.19999710.10.0300559282f2.3704335E-04r

-2.85660077E-08,-2.95338805E-llfO.186.102162fO.3617 2528,543.16F

409.5fO.O256f459.69/

c

DATA R22/32.76,54.634409f36.74892f-22.2925657,20.473288610.fO.r

29.35754453r-3845.193152r-7.861032212.1909390E-031

0 .445746703f 686.11 0 .124098f 0 .002, -4 .35354712 .407252E-03f

-44.066868,-0.017464,7.62789E-05,1.483763,2.310142E-031

-3.605723E-06,0.r-3.724044E-0515.355465E-081-1.845051E-04I

1.363387EO8f-1.672612EO,5fO.l4.2f548.2fO.rO.O2812836r2.255408E-04t

-6.509607E-08,0.,257.341,62.4009r-0.0453335,721.9lr664.51

0 .030525, 45 9. 6 9/

c

DATA R114/36.32f6l.146414fO.tl6.418015rO.fl7.476838rl.119828I

0 27.071306f-5113.7021f-6.308676lf6.91003E-04fO.78142lllf768.351

0 0.0627808O7fO.OO5914907t-2.3856704rl.0801207E-03r-6.5643648I

9 0.0340556871-5.3336494E-0610.163660571-3.857481E-04,0.10.1

0 1.6017659E-O6r6.2632341E-10f-1.0165314E-05,0.rO.fO.r3.fO.fO.r

0 0.0175r3.49E-04f-1.67E-07fO.fO.l25.3396621f-0.11513718f477.2f

753. 97, 0. 027531, 459. 69/

c

DATA R500/3l.Of43.562f74.709f-87.583,56.483tO.rO.fl7.780935r

-3422.69717,-3.6369lr5.0272207E-0410.462940lr695.57FO.10805r

0.0060342291-4.54988812.308415E-03,-92.90748rO.O8660634I

-3.141665E-05,2.742282,-8.726016E-04,0.tO.f-1.375958E-06F

9.149570E-09f-2.102661E-04rO.fO.fO.r5.475fO.fO.fO.026803537f

2.8373408E-04f-9.7167893E-0810.rO.r46.4734f-0.09012707564,646.3f

-5.29-

Section 5 Freon Pronerties

Qpetnn ;Jl;Ut~lJU t' l" *I Wll I IJIL 1

* 7.0240549E-07,0.022412368,8.8368967E-06,-7.9168095E-09,

* -3.7167231E-04,-3.8257766E07,5.5816094E04,1.5378377E09,4.2,609.,

* 7.E-07,0.020419,2.996802E-04,-1.409043E-07,2.210861E-11,0.,

* 35.308,-0.07444,591.,639.56,0.028571,459.67/

c

IGO=0

DO 5 I=INRTYPE

IGO=IGO+I

IF(NREF.EQ.RTYPE(IGO)) GO TO 100

5 CONTINUE

C

WRITE (*, 11)

11 FORMAT(///' ** ERROR - NO PROPERTIES FOR THIS REFRIGERANT **')

STOP

C

100 IF(IGO.EQ.LAST) RETURN

LAST= IGO

GO TO (l10,120,130,140,150,160,170,180) ,IGO

C

110 DO 115 I-1,NCOEF

Q(I)=R1I (I)

115 CONTINUE

GO TO 500

c

120 DO 125 I-1,NCOEF

Q (I) =R12 (I)

125 CONTINUE

GO TO 500

c

130 DO 135 I=1,NCOEF

Q(I)=R13 (I)135 CONTINUE

GO TO 500

C

140 DO 145 I-I,NCOEF

Q (I) =R14 (I)

145 CONT INUE

GO TO 500

- 5.30 -

Preinni

3.,mL IUI !-7C - Freon. ..oner....

C


Q(I)=R22 (I)

155 CONTINUE

GO TO 500

C


Q(I)=R14 (I)

165 CONTINUE

GO TO 500

C


Q(I)=R500(I)

175 CONTINUE

GO TO 500

C


Q(I)=R502 (I)

185 CONTINUE

C

500 CONTINUE

RETURN

END

C

C

SUBROUTINE SOLVE (Y1, Y2, XNEW, FUNC, TOL, IFLAG)

EXTERNAL FUNC

DATA IMAX/100/,JMAX/10/

C

IFLAG0

C

C

C Iterate using Newton's method with damping

CITER=0

CALL FUNC (Y1, Y2, XNEW, FNEW)

C

5 ITER-ITER+I

- 5.31 -

reon PronertiesCa^+; nn C -

C

C Determine numerical approximation to derivative

C

X=XNEW

F=FNEW

C

XNEW=X+AMAX1 (TOL, X*TOL)


DFDX-(FNEW-F) / (XNEW-X)

DX=2. *F/DFDX

C

C Determine new guess but don't accept if there is an increase in

C the residual error---> "damping"

C

J=0

40 CONTINUE

J=J+1

DX=DX/2.

XNEW=X-DX


IF(ABS(FNEW).GT.ABS(F) .AND. J.LT.JMAX) GO TO 40

70 CONTINUE

C

C Check for convergence

C

ERROR=ABS (DX/XNEW)

IF(ERROR.GT.TOL .AND. ABS(DX) .GT.l.E-06

.AND. ITER.LT.IMAX) GO TO 5

C

IF (ITER. EQ. IMAX) IFLAG=1

RETURN

END

SUBROUTINE THCON (T, H,V, F)

P=PR(T, V)

CALL ENTHAL ( HNEW, 2, V, T)

F=HNEW-H

RETURN

END

-5.32-

Freon PrnnortipqSetionn5

.,petiAnn-5 -LJJL.-. ... '-.-A 'uVa F I UUJI tIt!s

c

SUBROUTINE TSCON (T, S, V, F)

CALL ENTROP (SNEWT,V)

F=SNEW-S

RETURN

END

C

SUBROUTINE SVCON (V, S, T, F)

CALL ENTROP (SNEW, T, V)

F=SNEW-S

RETURN

END

C

SUBROUTINE XVCON(X, V,T,F)

P=VPR(T)

CALL SPVOL(TP,VV)

VL=1./DLIQ (T)

F=VV-(I.-X)*(VV-VL)-V

RETURN

END

C

SUBROUTINE SXCON (X, S, T, F)

P=VPR (T)

CALL SPVOL (TfP,VV)

VL=1./DLIQ (T)

CALL ENTROP (SVAPfT,VV)

DS=DHLAT (T,VV,VL, P) /T

F=SVAP- (1 . -X) *DS-S

RETURN

END

C

SUBROUTINE HXCON (X, H, T, F)

P=VPR (T)

CALL SPVOL(TP,VV)VL=1. /DLIQ (T)

CALL ENTHAL (HV, P, VV, T)

F=HV- ( 1.-X) *DHLAT (T, VV, VL, P) -H

RETURN

-5.33 -

Frpnn Prfnar-+;,nv

J- triNn Prn-L- iwp urp LIU")

END

C

SUBROUTINE PSCON(P,S,T,F)

CALL SPVOL(TP,V)

CALL ENTROP (SNEW, T,V)

F=SNEW-S

RETURN

END

C

SUBROUTINE PHCON(PH,T,F)

CALL SPVOL(TP,V)

CALL ENTHAL (HNEW, P, V, T)

F=HNEW-H

RETURN

END

C

SUBROUTINE PCON (DUM, P, T, F)

F=VPR (T) -P

RETURN

END

c

SUBROUTINE HSCON(TP,H,S,F,G)

CALL SPVOL(TP,V)

CALL ENTROP (SGUESST,V)

CALL ENTHAL (HGUESS, P, V, T)

F=SGUESS-S

G=HGUESS-H

RETURN

END

-5.34-

Qartinn -;- ]Freon Prnnart;ac I

Modelo de Torres de Refrigeración

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