Tube Side Nusselt Number Correlations for Cocurrent and

Countercurrent Flow Keywords: 1-1 Heat Exchanger, Correlation, Nusselt number, Model, Cocurrent,

Countercurrent, Heat Transfer Coefficient

Unit Operations ChE 433

Submitted to: Dr. David Thiessen

Unit Operations Professor

Date: 26 February, 2016

Members:

Abdi Ahmed

Andrew Wright

Paulo Dias

Shane Kelly

1

Abstract

In this experiment, cocurrent and countercurrent Nusselt number correlations for the 1:1 silver

heat exchanger were developed. Tube-side Nusselt Number correlations are required for a plant

which utilizes a heat exchanger with the same tube diameter and aspect ratio as one found in the

lab. The inlet and outlet temperatures were measured via thermocouples while varying tube side

flow rate while the shell side was kept constant. This data was used to determine the overall heat

transfer coefficient over a range of Reynolds numbers and subsequently to fit the parameters

required in the Nusselt number correlation. The parameters determined for cocurrent flow were c

= 0.0228 and n = 0.759 while they were found to be c=0.0230 and n=0.784, for countercurrent

flow. These correlations were then compared to three generalized Nusselt number correlations.

These correlations are the Seedier Tate correlation, Dittus-Boelter correlation, and Gnielinski

correlation. The experimental Nusselt number correlations were determined to be dependent on

the direction of flow and the error in the parameters was determined to be insignificant. The error

for both experimental Nusselt number correlations was determined to be ±0.1.

2

Table of Contents

Abstract……………………………………………………………………………………………1

Introduction………………………………………………………………….…………………….3

Theory…………………………………………………………………………….………….……4

Experimental Methods………………………………………………………………….………... 8

Results/Discussion……….……………………………………….…………………….……......11

Conclusion ……………….………………….…………………………………....…..................13

Recommendations………………………………………………………………………..………14

Nomenclature……………………………………………………………….………….…..…….15

References…………………………………………………………………….………………….16

Appendices…………………………………………………………………….…………….…...18

Appendix A: Error Analysis……………....................…………………………..……... 18

Appendix B: Operating Procedures………............………..………………………….... 21

Appendix C: Sample Calculations……………………………………………………….22

Appendix D: Process Flow Diagram....………………………………………………….25

3

Introduction

Heat exchangers are a necessary part of any industrial chemical process that necessitates a

change in temperature. They are used to heat or cool process streams in order to achieve a

desired stream property. These properties may include enthalpy, viscosity, or phase, to name a

few. A heat exchanger is a piece of equipment which transfers heat from one fluid to another [1].

This is done by allowing the two process fluids to flow on opposite sides of some heat transfer

medium such as steel. There are many ways to meet this requirement. One of the most common

methods of achieving heat transfer is to use coaxial piping [2]. Heat exchangers which employ

this method are called shell and tube heat exchangers. They consist of one large outer pipe and

many smaller pipes contained within the large pipe. Two fluids flow through the heat exchanger,

one outside of the small tubes and one inside. It is favorable to increase the area for which heat

transfer may occur, which is done by increasing the number of small pipes. In addition to the

increased surface area, this will increase the pressure drop through the heat exchanger, so these

two effects must be balanced economically. Other factors that affect the heat transfer are the

fluid flow rates and the temperature driving force. The heat transfer coefficient depends on the

Reynolds number, which in turn depends on the flow rate and design of the heat exchanger, for

example, tube diameter and cross-sectional area. The reciprocal of the heat transfer coefficient is

a measure of the resistance to heat transfer, which indicates how easily thermal energy may pass

through a medium [3]. If the heat transfer coefficient is large, the resistance to heat transfer is

minimized. For this reason, it is desirable to have a high fluid flow rate in order to increase

turbulence and decrease heat transfer resistance. The temperature driving force also affects the

heat transfer rate because the rate is proportional to the temperature difference between the hot

and cold fluids. This means the rate of heat transfer will be different for countercurrent flow and

cocurrent flow because the driving force changes more drastically for cocurrent flow. As a result,

the rate of heat transfer must be studied as a function of fluid flow rate and direction of flow.

It is important to know the heat transfer coefficients for the fluids used in a heat exchanger

because the rate of heat transfer at a given flow rate will determine how large a heat exchanger

must be. These heat transfer coefficients may be determined through the use of Nusselt number

correlations. Nusselt number correlations are primarily a function of the Reynolds number and

subsequently flow rate, and pipe diameter. Thus, it is important to know how the Nusselt number

will change as the flow rate varies because it will ultimately determine the temperature change of

the process stream. This is the premise of the current experiment in which the purpose is to find

Nusselt number correlations for cocurrent and countercurrent flow in a 1-1 shell and tube heat

exchanger. Proper temperature control can inhibit side reactions, prevent explosions, and

facilitate chemical separation. It is also common to use heat exchangers to perform the task of

heat integration for process streams. This allows a facility to save on energy costs by recycling

the energy back into their process. These are only a few instances in which temperature control is

necessary but they are illustrative of the importance of stream temperature control, which is

paramount to chemical engineering.

4

Theory

Heat exchangers take advantage of the physical phenomena of energy conservation while

moving from a hot substance to a colder one. In a heat exchanger, this process occurs through a

material that is thermally conductive. Hot and cold fluids are passed on either side of the heat

transfer medium to achieve heat transfer without mixing the two liquids. The rate of heat transfer

from the hot fluid to the cold fluid is determined by the hot fluid mass flow rate, the heat

capacity, and the temperature difference between the two fluids in the heat exchanger. The ideal

amount of heat a fluid transfers can be determined by Eq. (1).

𝑄 = �̇�𝐶𝑝∆𝑇 (1)

Where 𝑄 is the rate of heat transferred, �̇� is the fluid's mass flow rate, 𝐶𝑝 is the fluid's specific

heat capacity, and ∆𝑇 is the change in temperature of the fluid. To calculate the heat transfer rate with respect to the temperature driving force and physical geometry of a heat exchanger Eq. (2)

is used.

𝑄 = 𝑈𝐴∆𝑇𝐿𝑀𝐹 (2) Where 𝑄 is the rate heat transferred, 𝑈 is the overall heat transfer coefficient, 𝐴 is the surface area of the heat transfer area, and ∆𝑇𝐿𝑀 is the log mean temperature difference shown in Eq. (3).

∆𝑇𝐿𝑀 =(𝑇𝐻𝑖−𝑇𝐶𝑜)−(𝑇𝐻𝑜−𝑇𝐶𝑖)

ln((𝑇𝐻𝑖−𝑇𝐶𝑜)

(𝑇𝐻𝑜−𝑇𝐶𝑖))

(3)

Where 𝑇𝐻𝑖 is the temperature of the hot fluid entering the heat exchanger, 𝑇𝐶𝑜 is the temperature of the cold fluid leaving the heat exchanger, 𝑇𝐻𝑜 is the temperature of the hot fluid leaving the heat exchanger and 𝑇𝐶𝑖 is the temperature of the cold fluid entering the heat exchanger. An efficiency factor, 𝐹, is included; however for a 1-1 heat exchanger, 𝐹 is 1. The log mean temperature difference, ∆𝑇𝐿𝑀, represents the average temperature driving force for given inlet and outlet temperatures for both cocurrent and countercurrent flow.

The overall heat transfer coefficient, 𝑈, is unique to the flow conditions, geometry of the heat exchanger, thermal conductivity of the heat exchanger, and fluid properties. This value can be

calculated based on the inside or outside tube surface area of the heat exchanger. It is the

proportionality constant between the heat transfer rate and temperature driving force. The

reciprocal of this value, 1

𝑈, is a measure of the resistance to heat transfer by analogy to Ohm’s

Law, Eq. (4).

𝑉 = 𝐼𝑅 (4)

Where resistances in series can be added.

𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3 (5) The relationship between total heat transfer resistance represented by the inverse overall heat

transfer coefficient, 1

𝑈, and the heat transfer characteristics at the surfaces of the inner and outer

tube wall, as well as the material properties of the tube are shown in Eq. (6). 1

𝑈𝑜𝐴0=

1

ℎ𝑖𝐴𝑖+

1

𝑆𝑘𝑤+

1

ℎ𝑜𝐴𝑜 (6)

Where ℎ𝑖 and ℎ𝑜 are the heat transfer coefficients at the inner and outer tube surfaces, 𝐴𝑖 and 𝐴𝑜 are the inner and outer tube surface areas of the heat exchanger, 𝑆 is the shape factor which accounts for the surface geometry of the heat exchanger, and 𝑘𝑤 is the thermal conductivity of the heat exchanger tube walls.

5

The Reynolds number, Eq. (7), is a dimensionless number that is a ratio of inertial forces to

viscous forces.

𝑅𝑒 =𝜌�̅�𝐷

𝜇𝑁𝑇 (7)

Where 𝜌 is the fluid density, �̅� is the velocity, 𝐷 is the inner diameter of the tube side, 𝜇 is the dynamic viscosity, and 𝑁𝑇 is the total number of tubes. The Reynolds number is used to predict the characteristics of fluid flow. When viscous forces

outweigh inertial forces, the flow is considered laminar and smooth. This occurs when the

Reynolds number is low (2100) the inertial

forces overcome the viscous forces which causes the flow to become turbulent and cause eddies

to form in the fluid. Temperature influences the Reynolds number by affecting the density and

viscosity. However, while temperature will change the Reynolds number, a change in flow rate

will far outweigh a change in temperature with respect to the Reynolds number.

The Prandtl number, Eq. (8), is a dimensionless number that is the ratio of viscous diffusion to

thermal diffusion.

𝑃𝑟 =𝐶𝑝𝜇

𝑘𝑤 (8)

The Prandtl number is a ratio of the momentum diffusion rate to the thermal diffusion rate.

Momentum diffusion is the movement of particles in a fluid with respect to the cohesive forces

between neighboring atoms. Thermal diffusion is the movement of particles in a fluid with

respect to thermal energy. While all three variables related to the Prandtl number vary with

temperature, in this work it is assumed to be constant. This is acceptable because the Prandtl

number changes very little with temperature and this project utilizes a relatively narrow

temperature range.

The Nusselt number, Eq. (9), is the ratio of convective heat transfer to conductive heat transfer.

When the Nusselt number is less than 1, conduction plays a larger role than convection. When

the Nusselt number is greater than 1, the opposite is true.

𝑁𝑢 =ℎ𝑖𝐷𝑖

𝑘𝑤 (9)

Where ℎ𝑖 is the heat transfer coefficient for the tube-side fluid, 𝐷𝑖 is the diameter of the inside tube wall and 𝑘𝑤 is the thermal conductivity of the tube-side fluid. Many correlations exist that relate the Reynolds and Prandtl numbers to the Nusselt number.

These correlations depend on the type of heat exchanger, its geometry, and fluidic properties

(laminar or turbulent flow). Commonly used correlations include the Donohue equation [4], Eq.

(10), for the shell-side, Sieder-Tate equation [5], Eq. (11), Dittus-Boelter equation [6], Eq. (12),

and Gnielinski equation [7], Eq. (13) for turbulent flow on the tube-side. It should be noted that

these models assume smooth pipes and it is unlikely the tubes in the heat exchanger used in this

project are completely smooth.

𝑁𝑢 = 0.2𝑅𝑒0.6Pr0.33 (10) 𝑁𝑢 = 0.023𝑅𝑒0.8𝑃𝑟0.33 (11) 𝑁𝑢 = 0.023𝑅𝑒0.8𝑃𝑟𝑛 (12)

Where 𝑛 = 0.4 for heating, 𝑛 = 0.3 for cooling. This is due to variation in viscosity as the temperature of the fluid changes. In a hot liquid, the viscosity at the wall is reduced allowing

more uniform flow. In this project hot water was cooled thus 𝑛 of 0.3 was used.

6

𝑁𝑢 =(

𝑓2)

(𝑅𝑒 − 1000)𝑃𝑟

1 + 12.7 (𝑓2)

12

(𝑃𝑟23 − 1)

(13)

Where,

𝑓 = 𝐴 +𝐵

𝑅𝑒−

1𝑚

(14)

And the constants A, B, and m are as follows,

Reynolds Number 𝐴 𝐵 𝑚

2100 − 4000 0.0054 2.3 ∙ 10−8 − 2 3⁄

4000 − 107 1.28 ∙ 10−3 0.1143 3.2154

*The Gnielinski Nusselt number correlation is only valid for 2300< Re < 5 ∗ 106

Model

The Nusselt number is defined by the heat transfer coefficient of the inside tube wall multiplied

by the inside diameter of the tube wall over the thermal conductivity of the fluid inside the tube

as seen in Eq. (15). It can also be found with a generalized correlation such as Sieder-Tate or

Dittus-Boelter. To obtain a model of the Nusselt number for the silver heat exchanger, a similar

correlation must be found. This model will be of the same form as the Sieder-Tate but containing

different constants.

𝑁𝑢 =ℎ𝑖𝐷𝑖

𝑘𝑤= 𝐶𝑅𝑒𝑛𝑃𝑟

13⁄ (15)

Where ℎ𝑖, 𝐶, and 𝑛 are unknown parameters. In order to solve for the unknowns, Eq. (6), will be put in terms of the outside overall heat

transfer coefficient resulting in Eq. (16). 1

𝑈𝑜=

1

ℎ𝑜+ 𝑅𝑤 +

𝐷𝑜

ℎ𝑖𝐷𝑖 (16)

In Eq. (16), the outside heat transfer coefficient, ℎ𝑜, and the wall resistance, 𝑅𝑤, are assumed to

be constant because the shell side fluid was kept at a constant flow rate. Thus 1

ℎ𝑜+ 𝑅𝑤 can be

combined into a new constant, 𝐴. The heat transfer coefficient, ℎ𝑖, is unknown and a relationship with a measurable value must be found. This is done by taking Eq. (15) and solving for ℎ𝑖.

ℎ𝑖 =𝐶𝑅𝑒𝑛𝑃𝑟

13⁄ 𝐾𝑤

𝐷𝑖 (17)

Next, the right side of Eq. (17) is plugging into Eq. (16) and the outside heat transfer coefficient

and wall resistance terms are also combined into the constant 𝐴. 1

𝑈𝑜= 𝐴 +

𝐷𝑜

𝐶𝑅𝑒𝑛𝑃𝑟1

3⁄ 𝐾𝑤𝐷𝑖

𝐷𝑖

(18)

The constants in the fraction are then combined into a new constant, 𝐵. 1

𝑈𝑜= 𝐴 +

𝐵

𝑅𝑒𝑛 (19)

Where,

7

𝐵 =𝐷𝑜

𝐶𝑃𝑟1 3⁄ 𝐾𝑤 (20)

Then, a best-fit line for the plot of 1

𝑈𝑜 vs 𝑅𝑒 in the form of Eq. (19) is required to solve for the

constants. These are found using excel solver to minimize the error between the points and the

line through a least squares analysis. Figure 1 shows these results.

Figure 1. Plot of 1/U0 versus Reynolds number using the experimental data obtained from

cocurrent flow. A model was fit to the experimental values to return the parameters A, B, and n

which were further used in determining the Nusselt number correlation.

From there in order to solve for C, a relationship must be found by taking advantage of the

relationship between ℎ𝑖 and the Nusselt number correlation. ℎ𝑖𝐷𝑖

𝑘𝑤= 𝐶𝑅𝑒𝑛𝑃𝑟

13⁄ (21)

Next, the fraction containing ℎ𝑖 in Eq. (16) is set equal to the fraction containing B in Eq. (19) to give Eq. 22.

𝐷𝑜

ℎ𝑖𝐷𝑖=

𝐵

𝑅𝑒𝑛 (22)

Eq. 22 is then solved for ℎ𝑖 to yield Eq. 23.

ℎ𝑖 =𝑅𝑒𝑛𝐷𝑜

𝐵𝐷𝑖 (23)

Then Eq. (23) is plugged into in Eq. (21), which results in Eq. (24)

8

𝑅𝑒𝑛𝐷𝑜𝐷𝑖

𝐵𝐷𝑖𝑘𝑤= 𝐶𝑅𝑒𝑛𝑃𝑟

13⁄ (24)

Solving Eq. (24) for 𝐶 results in Eq. (25).

𝐶 =𝐷𝑜

𝐵𝑃𝑟1

3⁄ 𝐾𝑤

(25)

The constant 𝐶, is then found using the value of 𝐵 that was obtained through the excel solver method and the Nusselt number correlation is complete. The form of the final correlation is given

in Eq. 26.

𝑁𝑢 = 𝐶𝑅𝑒𝑛𝑃𝑟1

3⁄ (26)

Experimental Method

Figure 2. Process flow diagram detailing the equipment used.

9

In this experiment, a stream of water is initially pumped into the tube side (P13) of a 1-1 heat

exchanger with 76 tubes and a transfer surface area of 2.76 square meters. The outgoing tube

side stream (P7) is then sent into a steam heat exchanger (HX-2) where it is heated by the steam

in the shell side. The shell side steam then enters steam trap 2 where it is condensed and sent to

the drain. The heated water from the steam heat exchanger (HX-2) is sent to a holding tank

where it is then sent back to the hot water tank which then processed to the heat exchanger of

interest (HX-1), thus furthering the loop.

The program, LabVIEW, was used to collect data on the four thermocouples installed on the heat

exchanger. The program was started at the beginning of each lab period and the data collected

was saved at the end. Next, the required system valves (V6, V7 and V18) would be opened to

ensure that the water can circulate. The hot water pump (1) was then turned on and the desired

flowrate was regulated by a variable frequency drive. The steam valves (V10 and V17) were

opened and the steam regulator was slowly tightened until the steam began to enter the steam

heat exchanger (HX-2). The steam regulator is left only slightly opened until the cold water

stream is turned on. Once the hot water tank reaches the desired temperature for that day, the

cold water valves (V1, V2 and V4) would be opened and the cold water pump (2) would be

turned on. The cold water was pumped from a 300-gallon tank that was constantly replenished

with a hose. The cold water pump (2) was set to 13.75 gallons per minute to ensure that the water

level in the tank remained constant. This flowrate was kept constant to ensure that the outside

heat transfer coefficient remained constant throughout this project. The cold water would pass

from the tank, to the heat exchanger of interest (HX-1), and finally to the drain. Steady state was

achieved when all the thermocouple readings on LabVIEW were not changing significantly (± 0.5 C), which took about 15 minutes. At this point, all the temperatures were recorded. These

temperatures along with the tube and shell-side flowrate were inputted into Microsoft Excel for

further use in calculations towards the Nusselt number correlation. The tube side temperature

was then changed by increasing/decreasing the steam pressure. This resulted in an

increase/decrease in the steady state temperatures. This project required data for both

countercurrent and cocurrent flow through the silver heat exchanger (HX-1). To obtain this data

the configuration of the heat exchanger was changed during the experiment to change the flow

pattern. This was done by opening all of the tube side valves (V5, V6, V7 and V8) then closing

the desired valves to operate in the desired flow directions. For example, countercurrent flow

required valves 5 and 8 to be open and valves 6 and 7 to be closed, cocurrent required the

opposite. The experimental design and the variables held constant are shown in Tables 1 and 2,

respectively. The heat exchanger (HX-1) specifications that were used are shown in Table 3.

The thermocouples were calibrated using ice water and boiling water. This was done by

removing a thermocouple from heat exchanger and placing it into ice water and boiling water.

The temperatures measured by the thermocouple and a thermometer were recorded and the

difference was found. This difference was utilized during the error analysis of the system.

In order to shut down the heat exchanger, the steam valves (V10 and V17) were turned off, the

steam regulator was loosened, the pumps (1 and 2) were shut off and finally any open valves

(V5, V6, V7, V8, and V18) were closed. The LabVIEW data was saved and the program was

closed.

10

Table 1. Detailing the experimental design followed in this project. Five different temperatures

were obtained for each flowrate and flow regime.

Manipulated variables

Flowrate for tube-side fluid

(GPM)

Temperature of inlet tube-side (°C) increment

of 5 degrees

Flow regime

5 35-60 Cocurrent

Countercurrent

10 35-60 Cocurrent

Countercurrent

15 35-60 Cocurrent

Countercurrent

20 35-60 Cocurrent

Countercurrent

25 35-60 Cocurrent

Countercurrent

30 35-60 Cocurrent

Countercurrent

Table 2. Variables held constant throughout the project.

Constant variables

Flow rate for cold water 13.75 GPM

Incoming cold water

temp

13.2-16.8 °C

11

Table 3. Specifications of the heat exchanger used in this project. The length, the tube diameters,

and the number of tubes were used.

Parameter Value

L tube 47.75 inch

N tube 76

Tube OD 0.375 inch

Tube ID 0.304 inch

Results and Discussion

Figure 3. Experimental Nusselt number correlation comparison between cocurrent and

countercurrent flow.

12

Figure 4. Experimental Nusselt number correlation comparison between generalized

correlations.

In the current experiment, Nusselt number correlations have been developed for both cocurrent

and countercurrent flow. These correlations were determined using flowrates ranging from 5-30

GPM and with temperature roughly ranging from 30-60°C. These are the largest possible ranges

for safe data collection in this system. The two parameters which were used to fit the two

correlations are c and n. These values are provided in Table 4. It can be seen from Figure 3 that

the two correlations are indeed dependent on the direction of fluid flow. The countercurrent

Nusselt numbers are consistently higher than the cocurrent values. The difference between the

two correlations becomes more drastic as the Reynolds number increases. However, in the

laminar flow regime they are reasonably close and exhibit a difference of roughly 3 near a

Reynolds number of 2100. In the regime of fully developed turbulence, the Nusselt numbers

differ by as much as 10. This indicates a stronger dependence on flow direction at higher flow

rates, for a given heat exchanger.

Both experimental correlations are consistently lower than those predicted by the Sieder-Tate

Equation and the Dittus-Boelter Equation as shown in Figure 4. A third generalized correlation

was used to compare the data collected. This correlation is the Gnielinski correlation, which

contains separate parameters for the Reynolds number regions 2300-4000 and 4000-10,000,000.

The experimental correlations were both above the lower Gnielinski correlation, but were

consistently below the predictions for Reynolds numbers above 4000. The shapes of the

experimental correlations match those of the Sieder-Tate and the Dittus-Boelter Equations fairly

well. However, the Gnielinski correlation predicts steeper response to the Reynolds number,

particularly in the 2300-4000 range.

13

Table 4. Modeling parameters along with their associated uncertainties for cocurrent and

countercurrent.

Cocurrent Countercurrent

A 0.000504 ± 7.43E-8 0.000501 ± 4.46E-10

B 0.413 ± 4.02E-5 0.411 ± 2.69E-11

c 0.0228 ± 5.76E-6 0.0230 ± 7.54E-7

n 0.759 ± 2.17E-4 0.784 ± 2.03E-9

The data for inlet and outlet temperatures of the hot and cold streams were collected using

thermocouples and the LabVIEW program. The calibration of the thermocouples were performed

in order to determine the systematic error. This was done using water at 0 ᴼC and 100 ᴼC. The

deviation is presented in Table 5. The deviation of the temperature was accounted for following

data collection.

Table 5. Calibration results for the thermocouples used in the heat exchanger.

Thermocouples Ice water

Temperature

(°C)

Thermometer

Reading for ice

water (°C)

Boiling water

Temperature (°C)

Thermometer

Reading (°C) for

boiling water

Tube (north) -0.5 0 95.5 93.9

Shell (north) -0.900 0 95.5 94.0

Shell (south) 0.180 0 95.0 93.0

Tube (south) Broken Broken Broken Broken

Conclusion

The Nusselt number correlation for the silver heat exchanger was found to be 0.0228Re0.759

Pr1/3

and 0.0230Re0.784

Pr1/3

for cocurrent and countercurrent flow, respectively. The Nusselt number

obtained from countercurrent flow is greater than cocurrent flow. This shows that heat transfer is

enhanced in countercurrent flow than cocurrent flow which is due to the larger temperature

driving force resulting from countercurrent flow. This can be explained due to a greater

temperature driving force resulting in higher heat transfer. The uncertainties in the fitting

parameters were insignificant, the error in Reynolds number was ±55, and the error in the Nusselt number was ±0.1 for both cocurrent and countercurrent flows. The model obtained from this project can be used as a useful tool in predicting trends in the Nusselt number as a function

of Reynolds number for heat exchangers with a similar aspect ratio.

14

Recommendations

There are a few things that can be done to increase the efficiency of the heat exchanger, these

include adding insulation, adding a lid to the hot water tank, and pipe cleaning by using an

industrial cleaner to remove any fouling caused by Pullman water [8]. Currently the hot water

tank is filled and never emptied, the top is exposed and a very noticeable amount of heat is being

lost. Adding a lid and applying insulation will ensure that only the silver heat exchanger is

removing heat from the water and should improve the stability of the hot water temperature. This

will also reduce the amount of steam heating required, reducing some of the energy cost of the

heat exchanger. Pullman water is high in heavy metals and minerals, it could be possible for

some to build up inside the heat exchanger. This fouling may be removed with an industrial

cleaner. The water from the hose should be analyzed and the proper cleaner researched and

pumped through the heat exchanger. This will reduce any error due to build up hampering heat

transfer.

In order to improve the system itself, it is recommended to calibrate the flowmeters. With the

current system set up, it is incredibly difficulty to calibrate these reliably. The cold water stream

splits in two, half a foot above the ground making sampling very difficult. They would need to

be removed from the system to properly test them.

A suggestion to another group using the system would be to always start with a small amount of

steam pressure and slowly increase the temperature. This will ensure that the steam will not build

up in the steam heat exchanger and activate the pressure relief due to inadequate heat transfer to

the warm water stream. Slowly increasing the steam pressure would result in the water slowly

heating up. This eliminates the requirement of steady-state data collection because the

temperatures will change very slowly. This allows constant data recording over a range of inlet

temperatures.

15

Nomenclature

Symbol Description Units

𝑄ℎ Hot side heat duty W

𝑄𝑐 Cold side heat duty W

𝐶𝑝 Heat capacity of water J/kg K

𝐷𝑖 Inside diameter of tube m

𝐷𝑜 Outer diameter of tube m

𝑁𝑇 Number of tubes None

𝑚𝑐 Mass flow rate of water shell

side kg/s

A

Inner surface area of tubes

(𝐴𝑖) and outer surface area of tubes (𝐴𝑜)

m2

f fraction factor None

hi Inside heat transfer

coefficient W/(m2 ∗ K)

ho Outside heat transfer

coefficient W/(m2 ∗ K)

k Thermal conductivity of

water W/m K

mh Mass flow of water in the

tube side kg/s

Nu Nusselt Number None

Pr Prandtl Number None

Re Reynolds Number None

U Overall Heat Transfer

(o=outer, i=inner) W/(m2 ∗ K)

𝜇 Viscosity of water Pa s

𝜌 Density of water kg m3⁄

16

References

[1] McCabe, Warren L., Julian C. Smith, Peter Harriott. “Unit operations of Chemical

Engineering: seventh edition”. McGraw-Hill’s Chemical Engineering Series. New York, NY.

2005.

[2] Michael Frankel, CIPE, CPD,: Facilities Site Piping Systems Handbook. VOLATILE

LIQUIDS: TREATMENT AND DISPOSAL, Chapter (McGraw-Hill Professional, 2012),

AccessEngineering < http://accessengineeringlibrary.com/browse/facilities-site-piping-systems-

handbook/c9780071760270ch15>

[3] Thomson, William. Introduction to Transport Phenomena. 1st ed. 1. Upper Saddle River.

Prentice Hall PTR, 2000. 335-342, 345-350. Print.

[4] McCabe, Warren L., Julian C. Smith, and Peter Harriot. Unit Operations of Chemical

Engineering. New York: McGraw-Hill, 1993. Print.

[5] Don W. Green; Robert H. Perry: Perry's Chemical Engineers' Handbook, Eighth Edition.

HEAT TRANSFER BY CONVECTION, Chapter (McGraw-Hill Professional, 2008 1997 1984

1973 1963 1950 1941 1934), AccessEngineering <

http://accessengineeringlibrary.com/browse/perrys-chemical-engineers-handbook-eighth-

edition/p200139d899705_7001>

[6] Myer Kutz: Heat-Transfer Calculations. Determination of Heat-Transfer Film Coefficients

by the Wilson Analysis, Chapter (McGraw-Hill Professional, 2006), AccessEngineering <

http://accessengineeringlibrary.com/browse/heat-transfer-calculations/p2000de7499722_1001>

[7] Warren M. Rohsenow; James P. Hartnett; Young I. Cho: Handbook of Heat Transfer. HEAT

EXCHANGERS, Chapter (McGraw-Hill Professional, 1998), AccessEngineeringEXPORT

[8] "2013 Annual Drinking Water Quality Report." (n.d.): n. pag. Web. 25 Feb. 2016.

.

[9] "A Summary of Error Propagation." (n.d.): 1-5. Harvard University. Web. 17 Feb. 2016.

.

[10] "A Summary of Error Propagation." (n.d.): n. pag. Harvard University. Web. 17 Feb. 2016.

.

[11] "The Error of the Natural Logarithm." - Physics Stack Exchange. N.p., 26 Jan. 2014. Web.

16 Feb. 2016. .

[12] "Water Density Calculator." N.p., n.d. Web. 15 Feb. 2016.

.

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17

[13] "Water - Thermal Properties." Water - Thermal Properties. N.p., n.d. Web. 15 Feb. 2016.

.

[14] "Liquid Dynamic Viscosity." Calculation by Vogel Equation (Water). N.p., n.d. Web. 15

Feb. 2016.

.

[15] Ramires, M., C. A. Nieto De Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W.

A. Wakeham. "Standard Reference Data for the Thermal Conductivity of Water." °ûN (n.d.): n.

pag. Web. 15 Feb. 2016. .

18

Appendix A. Error Analysis

Measuring error and propagation of error both need to be accounted for to fully understand the

effects of error on this project. Measuring error can be broken down into two different types of

error that can contribute to recording a value that is different from the actual value. These are

random and systematic errors. Random errors would be an unaccountable change in a

measurement over a certain range. Systematic error is a constant difference in measurement from

a source. Systematic errors can be removed from a measurement while random errors must be

accounted for in using the measurement. Random errors can be accounted for by taking multiple

measurements. By taking multiple measurements and averaging, the random error above or

below the actual value will cancel out. Systematic errors will be offset by the same amount thus

to account for these errors equipment must be calibrated. By calibrating them with known values

the difference in the measured and actual value can be found.

Due to the setup of the heat exchanger, measuring error plays a large part of the known error

found in the results of this experiment. Measuring error in this project includes the error in the

pump flow rate and thermocouple measurements. The error obtained from the pump flow rate is

attributed to random error, as the flow rate would continue to fluctuate once the flow rate was

set. This could be due to the electrical current to the pumps, dirt building up in the filter, error in

the propeller flowmeter or temperature changes in the water. Due to the inability to calibrate the

pumps, the error observed must be assigned to random error. The error found in the

thermocouple can be explained by both random and systematic error. The systematic error was

determined when the thermocouples were calibrated, the random error was estimated by the

amount that the measurements fluctuated by while data was collected.

Propagation of error involves determining an error in a variable and carrying it through

successive calculations [9, 10]. This error is the error that each mathematical operation adds to

the original uncertainty. This error is found by using a mathematical analysis of each calculation

done on the experimental data. These equations were used to determine the uncertainty caused

by the propagation of error when calculating the Nusselt number correlation. For addition and

subtraction calculations, Eq. (1) was used.

𝜎𝑥 = √𝜎𝑎2 + 𝜎𝑏2 + 𝜎𝑐2 (A.1)

Application of Eq. (1) is shown using the average hot side temperature as an example:

𝑇𝑎𝑣𝑔 =𝑇ℎ𝑖 + 𝑇ℎ𝑜

2

The error in the average temperature is then,

𝜎𝑇𝑎𝑣𝑔 = √𝜎𝑇ℎ𝑖2 + 𝜎𝑇ℎ𝑜

2

For multiplication and division calculations, Eq. (2) was used.

𝜎𝑥

𝑥= √(

𝜎𝑎

𝑎)

2

+ (𝜎𝑏

𝑏)

2

+ (𝜎𝑐

𝑐)

2

(A.2)

For example, calculating the error in U0 is shown below:

𝑈0 =𝑄𝑎𝑣𝑔

𝐴0∆𝑇𝐿𝑀

Since this involves multiplication/division calculations, Eq. (2) is used.

19

𝜎𝑈0 = 𝑈0√(𝜎𝑄𝑎𝑣𝑔𝑄𝑎𝑣𝑔

)

2

+ (𝜎∆𝑇𝐿𝑀∆𝑇𝐿𝑀

)2

The error propagated in the log mean temperature difference calculation was found using an

approximation described by Eq. (3) [11],

𝜎 ln 𝑥 ≈𝜎𝑥

𝑥 (A.3)

Using the denominator of log mean temperature difference as the example,

ln ((𝑇ℎ𝑖 − 𝑇𝑐𝑜)

(𝑇ℎ𝑜 − 𝑇𝑐𝑖))

Using Eq. (3),

σln ((𝑇ℎ𝑖 − 𝑇𝑐𝑜)

(𝑇ℎ𝑜 − 𝑇𝑐𝑖)) =

σ ((𝑇ℎ𝑖 − 𝑇𝑐𝑜)(𝑇ℎ𝑜 − 𝑇𝑐𝑖)

)

(𝑇ℎ𝑖 − 𝑇𝑐𝑜)(𝑇ℎ𝑜 − 𝑇𝑐𝑖)

To find the error propagated when the inverse of a calculation was taken (Uo to 1

Uo for example),

Eq. (4) was used

𝜎𝑥 = 𝑥|𝑛|𝜎𝑎

|𝑎| (A.4)

Where

𝑥 = 𝑎𝑛 (A.5) With

𝑛 = −1

𝑥 =1

𝑈0

𝑎 = 𝑈0 Thus,

1

𝑈0= 𝑈0

−1

The error in 1/U0 is then,

𝜎 1𝑈0

=1

𝑈0|−1|

𝜎𝑈0|𝑈0|

The error in the Nusselt number was calculated using the theoretical propagation of error.

𝜎𝑥 = √(𝜕𝑥

𝜕𝑎)

2

𝜎𝑎2 + (𝜕𝑥

𝜕𝑏)

2

𝜎𝑏2 + (𝜕𝑥

𝜕𝑐)

2

𝜎𝑐2 (A.6)

The general form of the Nusselt number is,

𝑁𝑢 = 𝑐𝑅𝑒𝑛𝑃𝑟1/3 The error is calculated by

𝜎𝑁𝑢 = √(𝜕𝑁𝑢

𝜕𝑐)

2

𝜎𝑐2 + (𝜕𝑁𝑢

𝜕𝑛)

2

𝜎𝑛2 + (𝜕𝑁𝑢

𝜕𝑅𝑒)

2

𝜎𝑅𝑒2 + (𝜕𝑁𝑢

𝜕𝑃𝑟)

2

𝜎𝑃𝑟2

20

Table A.1. Uncertainties in the experimental values.

Pumps Error

Hot Water ±0.1

Cold Water ±0.1

Thermocouples

Shell side in ±1.1 °C

Shell side out ±1.2 °C

Left Tube ±1.1°C

Right Tube ±1.1 °C

𝟏

𝑼𝒐 ±9.3 ∗ 10−4

𝑚2𝐾

𝑊

Reynolds

Number

±60

Nusselt

Number

±0.1

c ≈ ±0

n ≈ ±0

21

Appendix B. Operating Procedures

Startup procedure:

1. Open LabVIEW program

2. Ensure E2 is closed

3. Open E5 and D to allow water to fill tank, TH1

4. Set valves

a. Close V1, and V2

b. Set V4 – V7 depending on concurrent or countercurrent flow

c. Open S2, E1, A, and B

d. Open V10 to fill TC1

5. Turn on power and make sure C1 and C2 are at the lowest setting

6. Turn on pump 1 by controlling C1

7. Open S1 to allow steam to flow

8. Adjust V3 to desired steam pressure

9. Once desired hot water temperature is reached, turn on pump 2

10. Collect data

Shutdown procedure:

1. Close S1 to shut off the steam

2. Turn off pumps 1 and 2, and turn off power

3. Set valves

a. Close V3, V10, S2, E1, E3, A, B, and D

b. Close V4 – V7

c. Open E2

4. Save LabVIEW data

Emergency shutdown:

1. Close S2

2. Turn off power

3. Evacuate to safest, nearest exit

22

Appendix C. Sample Calculation for a countercurrent run at 15 GPM

The goal in using the data is to plot 1

𝑈0 vs. Re so a model can be fit to the curve to obtain

parameters for the Nusselt number correlation.

Temperature data obtained from the experimental procedure,

Thi = 56.41°C

Tho = 41.55°C

Tci = 15.53°C

Tco = 32.20°C

Need to do a heat balance on hot and cold side. Theoretically these would be equal.

�̇�ℎ𝑜𝑡 = �̇�𝑐𝑜𝑙𝑑 Hot side

�̇�ℎ𝑜𝑡 = �̇�𝐶𝑝∆𝑇

Mass flowrate is found by multiplying the volumetric flowrate by the density.

�̇� = �̇�𝜌

�̇� = 15.00 𝐺𝑃𝑀 = 0.0009464𝑚3

𝑠

Density was found using a calculator online [12] where the temperature inputted was the average

hot side temperature.

𝑇𝑎𝑣𝑔 =𝑇ℎ𝑖 + 𝑇ℎ𝑜

2=

56.41℃ + 41.55℃

2= 49.98℃

From this,

𝜌 = 998.5𝑘𝑔

𝑚3

Now,

�̇� = 0.0009464𝑚3

𝑠∗ 998.5

𝑘𝑔

𝑚3= 0.9354

𝑘𝑔

𝑠

The heat capacity was also found using an online source [13]. This was not a calculator so the

value was interpolated at the average temperature.

𝐶𝑝 = 4182𝐽

𝑘𝑔 ∙ 𝐾

The heat on the hot side can now be calculated.

�̇�ℎ𝑜𝑡 = 0.9354𝑘𝑔

𝑠∗ 4182

𝐽

𝑘𝑔 ∙ 𝐾∗ (41.55℃ − 56.41℃) = −58,130 𝑊

The negative value means the stream is losing heat.

Cold side

The same procedure is used for the cold side so the values will just be stated. The flowrate on the

cold side was kept at a constant 13.75 GPM.

�̇� = 13.75 𝐺𝑃𝑀 = 0.0008675𝑚3

𝑠

𝑇𝑎𝑣𝑔 =𝑇𝑐𝑖 + 𝑇𝑐𝑜

2=

15.53℃ + 32.20℃

2= 23.87℃

23

𝜌 = 997.3𝑘𝑔

𝑚3

�̇� = 0.0008675𝑚3

𝑠∗ 997.3

𝑘𝑔

𝑚3= 0.8652

𝑘𝑔

𝑠

𝐶𝑝 = 4181𝐽

𝑘𝑔 ∙ 𝐾

�̇�𝑐𝑜𝑙𝑑 = 0.8652𝑘𝑔

𝑠∗ 4181

𝐽

𝑘𝑔 ∙ 𝐾∗ (32.20℃ − 15.53℃) = 60,300 𝑊

This value is positive to show that it is gaining heat.

Since �̇�ℎ𝑜𝑡 ≠ �̇�𝑐𝑜𝑙𝑑, an average �̇� is taken for use in the empirical heat transfer rate equation.

𝑄𝑎𝑣𝑔 =|�̇�ℎ𝑜𝑡 + �̇�𝑐𝑜𝑙𝑑|

2=

|−58,130 𝑊 + 60,300 𝑊|

2= 59,220 𝑊

The empirical formula is defined as,

𝑄𝑎𝑣𝑔 = 𝑈0𝐴0∆𝑇𝐿𝑀𝐹

The objective is to obtain 1

𝑈0 so,

1

𝑈0=

𝐴0∆𝑇𝐿𝑀𝐹

𝑄𝑎𝑣𝑔

𝐴0 is the outside surface area of the tube bundle

𝐴0 = 𝜋𝐷0𝐿𝑁𝑇 The given heat exchanger specifications are

𝐷0 = 0.375 𝑖𝑛 = 0.00953 𝑚 𝐿 = 47.75 𝑖𝑛 = 1.213 𝑚

𝑁𝑇 = 76 tubes The outside surface area is then,

𝐴0 = 𝜋 ∗ 0.00953 𝑚 ∗ 1.213 𝑚 ∗ 76 tubes= 2.76 𝑚2

The log mean temperature difference is described as,

∆𝑇𝐿𝑀 =(𝑇ℎ𝑖 − 𝑇𝑐𝑜) − (𝑇ℎ𝑜 − 𝑇𝑐𝑖)

ln ((𝑇ℎ𝑖 − 𝑇𝑐𝑜)(𝑇ℎ𝑜 − 𝑇𝑐𝑖)

)=

(56.41°C − 32.20°C) − (41.55°C − 15.53°C)

ln (56.41°C − 32.20°C41.55°C − 15.53°C

)= 25.10℃

F is the correction factor for a shell and tube heat exchanger. For a 1-1 heat exchanger, as used in

this project, F = 1.

Now 1

𝑈0 is,

1

𝑈0=

2.76 𝑚2 ∗ 25.10℃ ∗ 1

59,220 𝑊= 0.00117

𝑚2 ∙ 𝐾

𝑊

This value is then plotted against the Reynolds number for the hot side.

𝑅𝑒 =𝜌�̅�𝐷

𝜇=

4�̇�

𝜋𝐷𝑖𝜇𝑁𝑇

The inside diameter is,

𝐷𝑖 = 0.304 𝑖𝑛 = 0.00772 𝑚 The dynamic viscosity is found from a calculator online [14].

𝜇 = 5.59 × 10−4 𝑃𝑎 ∙ 𝑠 The Re is then,

24

𝑅𝑒 =4 ∗ 0.9354

𝑘𝑔𝑠

𝜋 ∗ 0.00772 𝑚 ∗ 5.59 × 10−4 𝑃𝑎 ∙ 𝑠 ∗ 76 𝑡𝑢𝑏𝑒𝑠= 3630

This procedure of calculations was followed for each run. A plot of 1

𝑈0 vs. Re was then plotted. A

model with the form of 𝐴 +𝐵

𝑅𝑒𝑛 was fitted to the data. The parameters returned from Solver in

Microsoft Excel using a least squares analysis were,

A = 0.000501 𝑚2∙𝐾

𝑊

B = 0.411 𝑚2∙𝐾

𝑊

n = 0.784

The general form the of the Nusselt correlation used is,

𝑁𝑢 = 𝑐𝑅𝑒𝑛𝑃𝑟1/3 The Prandtl number is assumed to be constant in order to make the modeling portion easier. The

equation used was,

𝑃𝑟 =𝐶𝑝𝜇

𝑘

k is the thermal conductivity of water at the average temperature. The thermal conductivity was

obtained by referring to tabulated values [15] and then linearly interpolated to the desired

temperature. k for this run was,

𝑘 = 0.6412𝑊

𝑚 ∙ 𝐾

The Prandtl number is thus,

𝑃𝑟 =4182

𝐽𝑘𝑔 ∙ 𝐾

∗ 5.59 × 10−4 𝑃𝑎 ∙ 𝑠

0.6412𝑊

𝑚 ∙ 𝐾

= 3.65

The Prandtl number was calculated for each run and then averaged. The average value was

approximately 4. The parameter, c, is calculated by,

𝑐 =𝐷0

𝑘𝐵𝑃𝑟1/3=

0.00953 𝑚

0.634 𝑊

𝑚 ∙ 𝐾 ∗ 0.411𝑚2 ∙ 𝐾

𝑊 ∗ 41/3

= 0.0230

From here the Nusselt number was calculated by using the general form above,

𝑁𝑢 = 0.0230 ∗ 36300.784 ∗ 413 = 22.7

This procedure was followed for each run to obtain the Nusselt number. The Nusselt number was

then plotted against the Reynolds number for the final figure.

25

Appendix D. Process Flow Diagram