Wilcox Thesis FinalReport

8/10/2019 Wilcox Thesis FinalReport

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Copyright 2013

By

Matthew P. Wilcox

All Rights Reserved


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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. v

LIST OF FIGURES ......................................................................................................... vii

LIST OF SYMBOLS ......................................................................................................... x

ABSTRACT .................................................................................................................... xii

1. INTRODUCTION ....................................................................................................... 1

1.1 SUMMARY OF PRIOR WORK ....................................................................... 2

1.2 CONTENT ......................................................................................................... 3

2. HEAT TRANSFER AND FLUID FLOW: THEORY ................................................ 5

2.1 GOVERNING EQUATIONS ............................................................................ 5

2.2 NUMERICAL METHODS ................................................................................ 6

2.3 NATURAL CONVECTION .............................................................................. 9

2.4 LAMINAR FLOW ........................................................................................... 11

2.5 TURBULENT FLOW ...................................................................................... 12

2.5.1 CALCULATING TURBULENCE PARAMETERS .......................... 14

2.6 TWO-PHASE FLOW ...................................................................................... 16

2.6.1

MODELING TWO-PHASE FLOW .................................................... 18

2.6.2 POPULATION BALANCE MODEL.................................................. 19

2.7 BOILING HEAT TRANSFER ........................................................................ 20

2.7.1 SUBCOOLED BOILING .................................................................... 22

3.

HEAT TRANSFER AND FLUID FLOW: MODELING ......................................... 25

3.1

NATURAL CONVECTION ............................................................................ 25

3.1.1 HORIZONTAL CYLINDER ............................................................... 25

3.1.2 VERTICAL PLATE ............................................................................ 32

3.2 LAMINAR FLOW ........................................................................................... 38

3.3 TURBULENT FLOW ...................................................................................... 42

3.3.1 TURBULENT FLOW WITHOUT HEAT TRANSFER ..................... 42


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3.3.2 TURBULENT FLOW WITH HEAT TRANSFER ............................. 47

3.4 TWO-PHASE FLOW ...................................................................................... 50

3.4.1 GAS MIXING TANK .......................................................................... 50

3.4.2

BUBBLE COLUMN ............................................................................ 56

3.4.3 BUBBLE COLUMN WITH POPULATION BALANCE MODEL ... 61

3.5 BOILING HEAT TRANSFER ........................................................................ 65

3.5.1 POOL BOILING .................................................................................. 65

3.5.2 SUBCOOLED FLOW BOILING ........................................................ 71

4. DISUSSION AND CONCLUSIONS ........................................................................ 84

REFERENCES ................................................................................................................ 86


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LIST OF TABLES

Table 2.5.1-1: Turbulent Flow Input ............................................................................... 15

Table 2.5.1-2: Calculation of Turbulent Parameters ....................................................... 15

Table 3.1.1-1: Horizontal Cylinder Model Input ............................................................. 26

Table 3.1.1-2: Horizontal Cylinder Model Fluid Density ............................................... 26

Table 3.1.1-3: Mesh Validation for Horizontal Cylinder Model ..................................... 31

Table 3.1.2-1: Vertical Plate Model Input ....................................................................... 33

Table 3.1.2-2: Vertical Plate Model Fluid Density .......................................................... 33

Table 3.1.2-3: Mesh Validation for Vertical Plate Model ............................................... 37

Table 3.2-1: Laminar Flow Model Input ......................................................................... 39

Table 3.2-2: Laminar Flow Model Fluid Density ............................................................ 39

Table 3.2-3: Mesh Validation for Laminar Flow Model ................................................. 41

Table 3.3.1-1: Turbulent Flow Without Heat Transfer Model Input ............................... 43

Table 3.3.2-1: Turbulent Flow With Heat Transfer Model Input .................................... 48

Table 3.3.2-2: Turbulent Flow With Heat Transfer Model Fluid Density ...................... 48

Table 3.3.2-3: Mesh Validation for Turbulent Flow With Heat Transfer Model ............ 50

Table 3.4.1-1: Gas Mixing Tank Model Input ................................................................. 52

Table 3.4.1-2: Mesh Validation for Gas Mixing Tank Model ......................................... 55

Table 3.4.2-1: Bubble Column Model Input ................................................................... 57

Table 3.4.2-2: Mesh Validation for Bubble Column Model ........................................... 61

Table 3.4.3-1: Population Balance Model Input .............................................................. 62

Table 3.4.3-2: Bubble Size DistributionSurface Tension of 0.072 N/m ..................... 64

Table 3.4.3-3: Bubble Size DistributionSurface Tension of 0.0072 N/m ................... 65

Table 3.5.1-1: Pool Boiling Model Input ......................................................................... 66

Table 3.5.1-2: Pool Boiling Model Fluid Density ........................................................... 67

Table 3.5.1-3: Mesh Validation for Pool Boiling Model ................................................. 70

Table 3.5.2-1: Subcooled Flow Boiling Model Input ...................................................... 71

Table 3.5.2-2: Subcooled Flow Boiling Model Fluid Properties ..................................... 72

Table 3.5.2-3: Boiling Model Study Case Input .............................................................. 73

Table 3.5.2-4: Boiling Model Study Case Results .......................................................... 76

Table 3.5.2-5: Inlet Condition Study Case Input ............................................................. 78


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Table 3.5.2-6: Inlet Condition Study Case Results.......................................................... 79

Table 3.5.2-7: Axial Liquid Volume Fraction ................................................................. 79

Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction .......................................... 82

Table 3.5.2-9: Relative Impact on Liquid Volume Fraction ........................................... 82

Table 3.5.2-10: Mesh Validation for Subcooled Flow Boiling Model ............................ 83


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LIST OF FIGURES

Figure 2.2-1: Control Volume Schematic for Pressure Correction Equation .................... 7

Figure 2.2-2: Control Volume Schematic for Momentum Equation ................................. 8

Figure 2.2-3: Control Volume Schematic for Energy Equation ........................................ 8

Figure 2.5-1: Transition from Laminar to Turbulent Flow .............................................. 12

Figure 2.6-1: Two-Phase Flow Patterns .......................................................................... 16

Figure 2.6-2: Baker Flow Pattern .................................................................................... 17

Figure 2.7-1: Boiling Heat Transfer Regimes ................................................................. 20

Figure 3.1.1-1: Horizontal Cylinder Schematic ............................................................... 25

Figure 3.1.1-2: Temperature (K) ..................................................................................... 27

Figure 3.1.1-3: Density (kg/m3) ....................................................................................... 27

Figure 3.1.1-4: Velocity Vectors (m/s) ............................................................................ 28

Figure 3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder ................... 29

Figure 3.1.1-6: Dimensionless Temperature at = 30................................................... 30

Figure 3.1.1-7: Dimensionless Temperature at = 90................................................... 30

Figure 3.1.1-8: Dimensionless Temperature at = 180 ................................................. 31

Figure 3.1.2-1: Vertical Plate Schematic ......................................................................... 32


Figure 3.1.2-3: Velocity Vectors (m/s) ............................................................................ 34

Figure 3.1.2-4: Interference Fringes Around a Heated Vertical Plate ............................. 35

Figure 3.1.2-5: Dimensionless Temperature for Various Prandtl Numbers .................... 36

Figure 3.1.2-6: Dimensionless Temperature for Various Prandtl Numbers (Fluent) ...... 37

Figure 3.2-1: Laminar Flow Schematic ........................................................................... 38

Figure 3.2-2: Velocity Magnitude ................................................................................... 38

Figure 3.2-3: Radial Velocity (m/s) ................................................................................. 40

Figure 3.2-4: Temperature (K) ........................................................................................ 40

Figure 3.2-5: Wall Shear Stress ....................................................................................... 41

Figure 3.3.1-1: Turbulent Flow Without Heat Transfer Schematic ................................. 42

Figure 3.3.1-2: Velocity Magnitude ................................................................................ 42

Figure 3.3.1-3: Wall Shear Stress .................................................................................... 44

Figure 3.3.1-4: Radial Velocity (m/s) ............................................................................. 44


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Figure 3.3.1-5: ................................................................................................... 44Figure 3.3.1-6: Results for a Mass Flow Rate of 0.5 kg/s ............................................... 45

Figure 3.3.1-7: Results for a Mass Flow Rate of 1.5 kg/s ............................................... 45

Figure 3.3.1-8: Turbulent Kinetic Energy (m2/s

2) ........................................................... 46

Figure 3.3.1-9: Production of Turbulent Kinetic Energy ................................................ 46

Figure 3.3.2-1: Turbulent Flow With Heat Transfer Schematic ...................................... 47


Figure 3.3.2-3: Radial Velocity (m/s) .............................................................................. 49

Figure 3.3.2-4: Velocity Magnitude ................................................................................ 49

Figure 3.3.2-5: Wall Shear Stress .................................................................................... 49

Figure 3.4.1-1: Gas Mixing Tank Schematic ................................................................... 51

Figure 3.4.1-2: Gas Volume Fraction .............................................................................. 53

Figure 3.4.1-3: Gas Volume Fraction at Jet Centerline ................................................... 53

Figure 3.4.1-4: Liquid Velocity Vectors (m/s) ................................................................ 54

Figure 3.4.1-5: Gas Velocity Vectors (m/s) ..................................................................... 55

Figure 3.4.2-1: Bubble Column Schematic ..................................................................... 56

Figure 3.4.2-2: Gas Volume Fraction .............................................................................. 58


Figure 3.4.2-4: Gas Velocity Vectors (m/s) ..................................................................... 60

Figure 3.4.2-5: Gas Volume Fraction (0.10 m/s)............................................................. 60

Figure 3.4.3-1: Gas Volume Fraction with PBM ............................................................ 62

Figure 3.4.3-2: Liquid Velocity Vectors with PBM (m/s) ............................................... 63

Figure 3.4.3-3: Gas Velocity Vectors with PBM (m/s) ................................................... 64

Figure 3.5.1-1: Pool Boiling Schematic .......................................................................... 66

Figure 3.5.1-2: Vapor Volume Fraction .......................................................................... 68


Figure 3.5.1-4: Vapor Velocity Vectors (m/s) ................................................................. 69

Figure 3.5.1-5: Volume Fraction of Vapor on Heated Surface ....................................... 70

Figure 3.5.2-1: Subcooled Flow Boiling Model Schematic ............................................ 71

Figure 3.5.2-2: Case 1 - Temperature (K) ....................................................................... 73

Figure 3.5.2-3: Case 1 - Liquid Volume Fraction ........................................................... 73


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Figure 3.5.2-4: Case 1 - Mass Transfer Rate (kg/m3-s) ................................................... 74

Figure 3.5.2-5: Case 1 - Vapor Generation Rate ............................................................. 75

Figure 3.5.2-6: Liquid Volume Faction for Cases 1-6 ..................................................... 77

Figure 3.5.2-7: Liquid Volume Faction for Cases 7-12................................................... 80


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LIST OF SYMBOLS

A flow area (m2)

a cylinder diameter (m)

thermal diffusivity (m

2

/s) coefficient of thermal expansion (K

-1)

Cp specific heat at constant pressure (J/kg-K) partial differentialD/Dt substantial differential with respect to time

D pipe diameter (m)

Dh hydraulic diameter (m)

dbw bubble departure diameter (m)

turbulent dissipation rate (m2/s

3)

f bubble departure frequency (s-1

)

g acceleration due to gravity (m/s2)

g subscript referring to gas/vapor

h interfacial heat transfer coefficient (W/m2-K)

hfg latent heat of vaporization (J/kgmol)

I turbulent intensity

k thermal conductivity (W/m-K) turbulent kinetic energy (m2/s2)l turbulence length scale (m)

l subscript referring to liquid

L length (m) mass flow rate (kg/s)Na nucleation site density (m

-2)

P perimeter (m)p pressure (Pa)

density (kg/m3) heat flux in vector form (W/m2)

Qw wall heat flux (W/m2)

r radial distance in cylindrical coordinates (m)


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rs radius of circular pipe (m)

surface tension (N/m)

S suppression factor

t time (s)

T temperature (K)

Twall wall temperature (K) bulk fluid temperature (K)Tsat fluid saturation temperature (K)

Tsub liquid subcooling temperature (K)

contact angle (radians)

generalized velocity (m/s)

axial velocity (m/s) velocity in x-direction (m/s) velocity in y-direction (m/s) time-mean velocity (m/s) fluctuating component of velocity (m/s) viscosity (kg/m-s) average mass velocity in vector form (m/s)V mean velocity (m/s) del operator scalar quantityxi distance in x-direction (m)

xj distance in y-direction (m)

x spatial coordinate in a Cartesian or cylindrical system (m)

y spatial coordinate in a Cartesian system (m)


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ABSTRACT

Various fluid flow and heat transfer regimes were investigated to provide insight

into the phenomena that occur during subcooled flow boiling. The theory of eachregime was discussed in detail and followed by the development of a numerical model.

Numerical models to analyze natural convection, laminar flow, turbulent flow with and

without heat transfer, two-phase flow, pool boiling and subcooled flow boiling were

created. The commercial software Fluent was used to produce the models and analyze

the results. Different modeling techniques and numerical solvers were employed in

Fluent depending on the scenario to generate acceptable results. The results of each

model were compared to experimental data when available to prove its validity.

Although numerous heat transfer and fluid flow phenomena were analyzed, the

primary focus of this research was subcooled flow boiling. The impact that different

boiling model options have on liquid volume fraction was examined. Three bubble

departure diameter models and two nucleation site density models were studied using the

same inlet conditions. The bubble departure diameter models examined did not show

any relationship with liquid volume fraction; however, the Kocamustafaogullari-Ishii

nucleation site density model tended to predict a greater liquid volume fraction, meaning

less vapor production, than the Lemmert-Chawla nucleation site density model.

A second study on how inlet conditions impact the liquid volume fraction during

subcooled flow boiling was explored. The inlet conditions of heat flux, fluid

temperature and mass flow rate were increased or decreased relative to a base case value.

The difference in liquid volume fraction between scenarios was compared and

relationships relating the inlet conditions with respect to liquid volume fraction were

developed. Overall, the fluid temperature had the greatest impact on liquid volume

fraction, the wall heat flux had the second greatest impact and the mass flow rate had the

smallest impact.


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1. INTRODUCTION

Since the 19th

century, the worlds standard of living has greatly increased

primarily due to the generation and distribution of electricity. Over 80% of the worldselectricity production is generated by converting thermal energy, from a fuel source, into

electrical energy. The Rankine Cycle is a common energy conversion process that burns

fuel and generates steam which is used to spin an electric generator. Electricity

production involves several engineering processes but is primarily based around heat

transfer and fluid flow.

Coal, oil, natural gas and uranium are some of the different fuel sources

available to electrical power plants. The fuel source in focus in this research is uranium

or nuclear fuel. Nuclear power plants harness energy released during fission to heat the

water that flows over the uranium fuel rods. The energy transfer mechanisms within a

nuclear reactor involve the three major forms of heat transfer; conduction, convection

and radiation. The fluid flow through the reactor is complex because of intense energy

transfer and phase change. In Pressurizer Water Reactors, the water flowing through the

reactor is prevented from bulk boiling because it is highly pressurized; however, a small

amount of localized boiling does occur which is known as subcooled flow boiling. This

research focuses on the convective heat transfer and fluid flow phenomena that occur

during subcooled flow boiling. Specifically, topics on turbulence, two-phase flow and

phase change are discussed.

Subcooled boiling occurs when an under-saturated fluid comes in contact with a

surface that is hotter than its saturation temperature. Small bubbles form on the heated

surface at preferential locations called nucleation sites. The number of bubbles that form

is heavily dependent on fluid temperature, pressure, mass flow, heat flux and

microscopic features of the surface. After the bubbles form on the heated surface, they

detach and enter the bulk fluid. When this occurs, saturated vapor is dispersed in a

subcooled liquid which is where the term subcooled boiling originates.


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1.1 SUMMARY OF PRIOR WORK

Subcooled flow boiling is characterized by the combination of convection,

turbulence, boiling and two-phase flow. Determining the amount of voiding that occurs

during subcooled flow boiling has become a topic of great interest in recent years. Anumber of mechanistic models for the prediction of wall heat flux and partitioning have

been developed. One of the most commonly used mechanistic models for subcooled

flow boiling was developed by Del Valle and Kenning. Their model accounts for bubble

dynamics at the heated wall using concepts developed initially by Graham and

Hendricks for wall heat flux partitioning during nucleate pool boiling. Recently, a new

approach to the partitioning of the wall heat flux has been proposed by Basu et al. The

fundamental idea of this model is that all of the energy from the wall is transferred to the

adjacent liquid. A fraction of the energy is absorbed by vapor bubbles through

evaporation while the remainder goes into the bulk liquid. [1]

In addition to the development of mechanistic heat transfer and partitioning

models, focus has been placed on accurately modeling three of the most impactful

parameters in subcooled flow boiling. These parameters are the active nucleation site

density (Na), bubble departure diameter (dbw) and bubble departure frequency (f). The

two most common nucleation site density models were developed by Lemmert and

Chwala and Kocamustafaogullari and Ishii. Both of these models are available in

Fluent.

Many correlations have been developed to determine the bubble departure

diameter. Tolubinsky and Kostanchuk proposed the most simplistic correlation which

evaluates bubble departure diameter as a function of subcooling temperature.

Kocamustafaogullari and Ishii improved this model by including the contact angle of the

bubble. Finally, Unal produced a comprehensive correlation which includes the effect of

subcooling, the convection velocity and the heater wall properties. All three of these

bubble departure diameter correlations are available in Fluent.

The most common bubble departure frequency correlation for computational

fluid dynamics was developed by Cole. It is based on a bubble departure diameter

model and a balance between buoyancy and drag forces. The Cole bubble departure

frequency model is available in Fluent.


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Recently, the use of population balance equations have been used to improve the

modeling of subcooled flow boiling by determining how swarms of bubbles interact

after detaching from the heated surface. This technique was recommended by Krepper

et. al. [2]and investigated by Yeoh and Tu [1]. Population balance equations have been

introduced in several branches of modern science, mainly areas with particulate entities

such as chemistry and materials because they help define how particle populations

develop in specific properties over time. Population balance equations are available in

Fluent; however, not in combination with the boiling model.

1.2

CONTENT

This research produced an investigation on subcooled flow boiling using Fluent.

Fluent is a widely accepted commercial computational fluid dynamics code that can

simulate complex heat transfer and fluid flow regimes. This thesis had three major

objectives. The first objective was to gain an understanding of the phenomena that occur

during subcooled flow boiling. The second objective was to determine how the boiling

model options described in Section1.1 impact the liquid volume fraction at different

axial locations. The third objective was to evaluate how heat flux, fluid temperature and

mass flow rate impact the liquid volume fraction at different axial locations.Due to its complexity, development of the subcooled flow boiling model was

performed in stages. With the expansion of each model, a more complicated fluid flow

or heat transfer scenario was analyzed. After each model was created, a mesh validation

was performed and the results were compared to known experimental data when

possible to validate the information generated by Fluent.

The first and simplest model created was for natural convection. The theory of

natural convection is described in Section 2.3 and the analytical modeling results are

presented in Section3.1. Two natural convection geometries were analyzed. The first

was a horizontal cylinder suspended in an infinite pool and the second was a vertical

plate suspended in an infinite pool. The second model developed was for laminar flow.

The theory of laminar flow is described in Section 2.4 and the analytical modeling

results are discussed in Section3.2. The third model developed was for turbulent flow.


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The theory of turbulent flow is described in Section 2.5 and the analytical modeling

results are displayed in Section3.3. Section3.3 contains two turbulent flow scenarios;

turbulent flow without heat transfer and turbulent flow with heat transfer. The fourth

model developed was for two-phase flow with water and air. The theory of two-phase

flow is described in Section 2.6 and the analytical modeling results for the scenarios

analyzed are shown in Section 3.4. The first scenario is a gas mixing tank and the

second scenario is a bubble column. The final and most complex models created include

phase transformation (vaporization and condensation). Section2.7 contains the theory

of boiling heat transfer with a subsection specific to subcooled boiling. Section 3.5

presents the analytical results for the two models created; the first for pool boiling and

the second for subcooled flow boiling. A summary of the results and the conclusions

reached from the models developed herein is documented in Section 4.


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2. HEAT TRANSFER AND FLUID FLOW: THEORY

This section discusses basic theory behind some common heat transfer and fluid

flow scenarios. It is meant to provide a brief introduction to the phenomena involved insubcooled flow boiling.

2.1

GOVERNING EQUATIONS

Conservation equations are a local form ofconservation laws which state that

mass, energy and momentum as well as other natural quantities must be conserved. A

number of physical phenomena may be described using these equations [3]. In fluid

dynamics, the two key conservation equations are the conservation of mass and the

conservation of momentum.

Conservation of Mass (continuity equation):

( ) Conservation of Momentum:

In subcooled flow boiling, as in many other instances of fluid dynamics, energy

is added or removed from the system. When this occurs, the conservation of energy

equation is important.

Conservation of Energy:

( )
http://en.wikipedia.org/wiki/Conservation_lawhttp://en.wikipedia.org/wiki/Conservation_law


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2.2 NUMERICAL METHODS

After the conservation laws governing heat transfer, fluid flow and other related

processes are expressed in differential form (Section 2.1), they can solved using

numerical methods to determine pressure, temperature, mass flux, etc. for variouscircumstances and boundary conditions. Each differential equation represents a

conservation principle and employs a physical quantity as its dependent variable that is

balanced by the factors that influence it. Some examples of differential equations that

may be solved through numerical methods are conservation of energy, conservation of

momentum and time-averaged turbulent flow. [4]

The goal of computational fluid dynamics is to calculate the temperature,

velocity, pressure, etc. of a fluid at particular locations within a system. Thus, the

independent variable in the differential equations is a physical location (and time in the

case of unsteady flows). Due to computational limitations, the number of locations (also

known as grid points or nodes) must be finite. By concentrating on a solution to the

differential equations at discrete locations, the requirement to find an exact solution is

avoided. The algebraic equations (also known as discretization equations) involving the

unknown values of the independent variable at chosen locations (grid points) are derived

from the differential equations governing the independent variable. In this derivation,

assumptions about the value of the independent variable between grid points must be

made. This concept is known as discretization. [4]

A discretization equation is an algebraic relationship that connects the values of

the dependent variable for a group of grid points within a control volume. This type of

equation is derived from the differential equation governing the dependent variable and

thus expresses the same physical information as the differential equation. The piecewise

nature of the profile (or mesh) is created by the finite number of grid points that

participate in a given discretization equation. The value of the dependent variable at a

grid point thereby influences the value of the dependent variable in its immediate area.

As the number of grid points becomes very large, the solution of the discretization

equations is expected to approach the exact solution of the corresponding differential

equation. This is true because as the grid points get closer together, the change in value

between neighboring grid points becomes small and the actual details of the profile


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assumption become less important. This is where the term mesh independent

originates. If there are too few grid points (coarse mesh), the profile assumptions can

impact the solution results and the discretization equation solution will not match the

differential equation solution. To ensure that the discretization equation results are not

dependent on the profile assumptions, the solution should be checked for mesh

independence. [4]

One of the more common procedures for deriving discretization equations is

using a truncated Taylor series. Other methods include variational formulation, method

of weighted residuals and control volume formulation. The conservation equations in

Section 2.1 in discretized form are shown below:

Pressure Correction Equation (continuity equation) [4]:

Figure 2.2-1: Control Volume Schematic for Pressure Correction Equation


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Conservation of Momentum in Discretized Form [4]:

(a) (b)

Figure 2.2-2: Control Volume Schematic for Momentum Equation

Conservation of Energy in Discretized Form [4]:

|| || || ||

Figure 2.2-3: Control Volume Schematic for Energy Equation


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In the iterative process for solving a discretization equation, it is often desirable

to speed up or to slow down the changes, from iteration to iteration, in the values of the

dependent variable. The process of accelerating the rate of change between iterations is

called over-relaxation while the process of slowing down the rate of change between

iterations is called under-relaxation. To avoid divergence in the iterative solution of

strongly nonlinear equations, under-relaxation is a very useful tool [4].

Fluent allows for manipulation of the relaxation constants for many independent

variables to improve convergence ability. It also offers numerous spatial discretization

solvers for the various independent variables such as pressure, flow, momentum,

turbulence, and energy. Fluent implements the control volume formulation with

upwinding which was first proposed by Courant, Isaacson, and Rees in 1952. Other

options include QUICK, power law and third-order MUSCL.

2.3

NATURAL CONVECTION

Convection is the transport of mass and energy by bulk fluid motion. If the fluid

motion is induced by some external force, like a pump, fan, or suction device, it is

generally referred to as forced convection. If the fluid motion is induced by an internal

force such as buoyancy produced by density gradients, it is generally referred to as

natural convection. The density gradients can arise from mass concentration and or

temperature gradients in the fluid [5]. For example, in a system where a heated surface

is in contact with a cooler fluid, the cooler fluid absorbs energy from the heated surface

and becomes less dense. Buoyancy effects due to body forces cause the heated fluid to

rise and the surrounding, cooler fluid takes its place. The cooler fluid is then heated and

the process continues forming a convection cell that continuously removes energy from

the heated surface.

In nature, natural convection cells occur everywhere from oceanic currents to air

rising above sunlight-warmed land. Natural convection also takes place in many

engineering applications such as home heating radiators and cooling of computer chips.

The amount of heat transfer that occurs due to natural convection in a system is

characterized by the Grashof, Prandtl and Rayleigh numbers. The Grashof number,


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Gr, is a dimensionless parameter that represents the ratio ofbuoyancy toviscous forces

acting on a fluid and is defined as:

where is the thermal expansion coefficient:

The Prandtl number, Pr, is a dimensionless parameter that represents the ratio of

momentum diffusivity to thermal diffusivity; and is defined as:

The Rayleigh number, Ra, is a dimensionless parameter that represents the ratio

of buoyancy to viscosity forces times the ratio of momentum diffusivity to thermal

diffusivity; and is defined as: When the Rayleigh number is below a critical value for a particular fluid, heat

transfer is primarily in the form of conduction; when it exceeds the critical value, heat

transfer is primarily in the form of convection. Like forced convection, naturalconvection can either be laminar or turbulent. Rayleigh numbers less than 10

8indicate a

buoyancy-induced laminar flow, with transition to turbulence occurring at about 109. [6]

In many situations, convection is mixed meaning that both natural and forced

convection occur simultaneously. The importance of buoyancy forces in a mixed

convection flow can be measured by the ratio of the Grashof and Reynolds numbers:

When this ratio approaches or exceeds unity, there are strong buoyancy

contributions to the flow. Conversely, if the ratio is very small, buoyancy forces may be

ignored.
http://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Buoyancy


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2.4 LAMINAR FLOW

Fluid flow can be grouped into two categories, laminar or turbulent flow.

Laminar flow implies that the fluid moves in sheets that slip relative to each other and it

occurs at very low velocities where there are only small disturbances and little to nolocal velocity variations. In laminar flow, the motion of the fluid particles is very

orderly and can be characterized by highmomentum diffusion and low momentum

convection.

The Reynolds number is used to characterize the flow regime. The Reynolds

number, Re, is adimensionless number that represents theratio of inertial forces to

viscous forces; and is defined as:

The Reynolds number helps quantify the relative importance of inertial and

viscous forces for given flow conditions. For internal flow, such as within a pipe,

laminar flow occurs at a Reynolds number less than 2300.

The velocity profile of a laminar flow in a pipe can be calculated by [5]:

Or, in terms of the mean velocity, V:

The above two equations indicate that the velocity for laminar flow is related to the

square of the pipe radius and thus the flow profile is parabolic.

The energy equation for flow through a circular pipe assuming symmetric heat

transfer, fully developed flow and constant fluid properties is [5]:

This equation shows that convection due to flow is balanced by diffusion in the radial

and axial directions.
http://en.wikipedia.org/wiki/Momentum_diffusionhttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Dimensionless_numberhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Dimensionless_numberhttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Momentum_diffusion


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2.5 TURBULENT FLOW

Influid dynamics,turbulence is a flow regime characterized by chaotic and

stochastic changes. Turbulent flows involve large Reynolds numbers and contain three-

dimensional vorticity fluctuations. The unsteady vortices appear on many scales andinteract with each other generating high levels of mixing and increased rates of

momentum, heat and mass transfer. Like laminar flows, turbulent flows are dissipative

and therefore depend on their environment to obtain energy. A common source of

energy for turbulent velocity fluctuations is shear in the mean flow; other sources, such

as buoyancy, exist too. If turbulence arrives in an environment where there is no shear

or other maintenance mechanism, the turbulence decays and the flow tends to become

laminar. [7]

In flows that are originally laminar, turbulence arises from instabilities at large

Reynolds numbers. For internal flows, such as within a pipe, turbulent flow is

characterized by a Reynolds number greater than 4000. For flows with a Reynolds

number between 2300 and 4000, both laminar and turbulent flows are possible. This is

called transition flow. [7]

A common example of the transition from laminar flow to turbulent flow is

smoke rising from a cigarette [8].

Figure2.5-1: Transition from Laminar to Turbulent Flow
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As the smoke leaves the cigarette, it travels upward in a laminar fashion as

shown by the single stream of smoke. At a certain distance, the Reynolds number

becomes too large and the flow begins to transition to the turbulent regime. When this

happens, the flow of the smoke becomes more random and rapidly mixes with the air

causing it to dissipate.

Modeling of turbulent flow requires the exact solution of the Continuity and

Navier-Stokes equations which can be extremely difficult and time consuming due to the

many scales involved. To reduce the complexity, an approximation to the Navier-Stokes

equations was developed by Osborne Reynolds called the Reynolds-averaged Navier

Stokes equations (or RANS equations). This method decomposes the instantaneous

fluid flow quantities of the Navier-Stokes equations into mean (time-averaged) and

fluctuating components. The RANS equations can be used with approximations based

on knowledge of the turbulent flow to give approximate time-averaged solutions to

the NavierStokes equations. [9]

For the velocity terms:

where and are the mean and fluctuating velocity components respectively.Similarly, for scalar quantities:

where denotes a scalar such as energy, pressure, or species concentration.

Substituting expressions of this form for the flow variables into the instantaneous

continuity and momentum equations and taking a time-average yields the time-averaged

continuity and momentum equations [9]. These are written in Cartesian tensor form as:

( ) ( )The two above equations are the Cartesian RANS equations for a two-

dimensional system. They have the same general form as the instantaneous Navier-

Stokes equations, with the velocities and other solution variables now representing time-


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averaged values. The RANS equations can be used with approximations based on

knowledge of the turbulent flow to give approximate time-averaged solutions to

the NavierStokes equations. An additional term( ), known as the Reynoldsstress, appears in the equation as a results of using the RANS method. [9]

One way that the Reynolds stress is evaluated in practice is through the k-

turbulence model. The k- model was first introduced by Harlow and Nakayama in

1968 [10]. The k- model has become the most widely used model for industrial

applications because of its overall accuracy and small computational demand. In the k-

model, k represents the turbulent kinetic energy and represents its dissipation rate.

Turbulent kinetic energy is the average kinetic energy per unit mass associated with

eddies in the turbulent flow while epsilon () is the rate of dissipation of the turbulent

energy per unit mass.

In the derivation of the k-model, it is assumed that the flow is fully turbulent,

and the effects of molecular viscosity are negligible. As the strengths and weaknesses of

the standard k-model have become known, modifications were introduced to improve

its performance. These improvements have helped create many, new, more accurate

models, among them, the realizable k- modelwhich differs from the standard k-model

in two important ways. First, the realizable model contains an alternative formulation of

the turbulent viscosity. Second, a modified transport equation for the dissipation rate, ,is derived from an exact equation for the transport of the mean-square vorticity

fluctuation. The term realizable means that the model satisfies certain mathematical

constraints on the Reynolds stresses, consistent with the physics of turbulent flow. [9]

2.5.1 CALCULATING TURBULENCE PARAMETERS

All of the computational fluid dynamic models discussed in this thesis use the

k-turbulence model when applicable. In Fluent, turbulence models require certain

parameters to be established prior to initialization to properly set the boundary

conditions for the flow. Based on the conditions specified in Table 2.5.1-1, the

equations in Table2.5.1-2 [9]were used to determine the boundary condition inputs for

the turbulent flow models presented in Section3.3.


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Table2.5.1-1: Turbulent Flow Input

Input Parameter Numerical Value

Mass Flow Rate () 1.0 kg/sPipe Diameter (D) 0.03 m

Viscosity () 0.001003 kg/m-sDensity () 998.2 kg/m

Turbulence Empirical Constant (C) 0.09 [9]

Table2.5.1-2: Calculation of Turbulent Parameters

Variable Equation Numerical Value

Hydraulic Diameter (Dh)

0.03 m

Flow Area (A)

0.00070686 m2

Average Flow Velocity (V)

1.41726 m/s

Reynolds Number (ReDh) 42314

Turbulent Length Scale (l) 0.0021 m

Turbulent Intensity (I) 0.0422483

Turbulent Kinetic Energy (k) ( ) 0.0053785 m2/s2

Dissipation Rate ()

0.030859 m2/s

3


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2.6 TWO-PHASE FLOW

Fluid flow that contains two or more components is referred to as multiphase

flow. The flow components can be of the same chemical substance but in different

states of matter such as water and steam, be of different chemical substances but thesame state of matter such as water and oil or finally be of different chemical substance

and different states of matter such as water and air. This section focuses on two-phase

flow involving water and air while Section 2.7 focuses on two-phase flow involving

water and steam.

Depending on the volume fraction of each component in the two-phase flow,

different flow patterns can exist. Understanding the two-phase flow pattern is important

because pressure drops and heat transfer rates are heavily impacted by the flow type.

The characteristic flow patterns for two-phase flow, in order of increasing gas volume

fraction from liquid to gas, are bubbly flow, plug flow, stratification flow, wavy flow,

slug flow, annular flow and spray flow. A schematic representation of each of these

flow patterns is shown in Figure2.6-1 [11].

Figure2.6-1: Two-Phase Flow Patterns

The flow patterns shown in Figure 2.6-1 can be further classified into three

categories, bubbly flow, slug flow and annular flow. Bubbly flow is when the liquid

phase is continuous and the vapor phase is discontinuous such that the vapor phase is


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distributed in the liquid phase in the form of bubbles. This flow pattern occurs at low

gas volume fractions. Subcooled flow boiling is classified as bubbly flow. Slug flow is

when there are relatively large liquid slugs surrounded by vapor. This flow pattern

occurs at moderate gas volume fractions and relatively low flow velocities. Annular

flow is when the liquid phase is continuous along the wall and the vapor phase is

continuous in the core. This flow pattern occurs at high gas volume fractions and high

flow velocities. Although not considered to be a flow regime, flow film boiling is the

opposite of annular flow (the vapor phase is continuous along the wall and the liquid

phase is continuous in the core) and occurs when the heat flux is relatively large

compared to the mass flux. Film boiling is discussed further in Section2.7.

The flow pattern of a system can be determined using the Baker flow criteria

shown in Figure2.6-2 [11]if the gas volume fraction and mass velocity are known. For

example, if a two-phase flow consisting of air and water has a total mass velocity (air

plus water) of 0.10 x 106lbm/hr-ft

2and a gas quality of 0.4, then flow will be annular.

Figure2.6-2: Baker Flow Pattern


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2.6.1 MODELING TWO-PHASE FLOW

Two-phase flows obey the same basic laws of fluid mechanics that apply to

single phase flows; however, the equations are more complicated and more numerous.

Two-phase flows are more difficult to solve due to the secondary phase and additional

phenomena that must be accounted for such as mass transfer and phase-interface

interactions (slip and drag). Three common multiphase flow models available in Fluent

are Volume of Fluid (VOF), Mixture and Eulerian, each with varying strengths and

computational demands.

The VOF model is the simplest and least computationally expensive of the three

multiphase models offered in Fluent. The VOF model can analyze two or more

immiscible fluids by solving a single set of momentum equations and tracking the

volume fraction of each fluid throughout the domain. All control volumes must be filled

with either a single fluid phase or a combination of phases. The VOF model does not

allow for void regions where no fluid of any type is present. The VOF method was

based on themarker-and-cell method and quickly became popular due to its low

computer storage requirements. Typical applications of VOF include stratified or free-

surface flows such as the prediction of jet breakup, the motion of large bubbles in a

liquid, the motion of liquid after a dam break, and the steady or transient tracking of a

liquid-gas interface. [9]

The Mixture model is between the VOF and Eulerian multiphase models both in

complexity and computational expense. The Mixture model can analyze multiple phases

(fluid or particulate) by solving the momentum, continuity, and energy equations for the

mixture, the volume fraction equations for the secondary phases, and algebraic

expressions for the relative velocities. Like the VOF model, it uses a single-fluid

approach but has two major differences. First, the Mixture model allows for the phases

to be interpenetrating and therefore the volume fraction of a fluid in a control volume

can be equal to any value between zero and one. Second, the Mixture model allows for

the phases to move at different velocities, using the concept of slip. The Mixture model

is a good substitute for the full Eulerian model in several cases where a full multiphase

model may not be feasible or when the interphase laws are unknown or their reliability

can be questioned. Typical applications include sedimentation, cyclone separators,
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particle-laden flows with low loading, and bubbly flows where the gas volume fraction

remains low. [9]

The Eulerian model is the most complex and most computationally expensive

multiphase model offered in Fluent. It solves the momentum and the continuity

equations for each phase, and couples the equations through pressure and exchange

coefficients. With the Eulerian model, the number of secondary phases is limited only

by memory requirements and convergence behavior. The Eulerian model allows for the

modeling of multiple separate, yet interacting phases. The interacting phases can be

liquids, gases, or solids in nearly any combination. Due to its ability to model

interacting phases, typical applications of the Eulerian model are bubble columns, risers,

particle suspension, fluidized beds and boiling including subcooled boiling. [9]

2.6.2 POPULATION BALANCE MODEL

In many two-phase flow applications, including subcooled flow boiling, it is

helpful to know how the secondary phase (solids, bubbles, droplets, etc.) evolves over

time. Thus, a balance equation is required to describe the changes in the particle size

distribution over time, in addition to the momentum, mass, and energy balances already

employed. The additional balance equation is generally referred to as the population

balance equation.

The population balance model in Fluent implements a number density function to

account for the different sizes of the particle population. With the aid of particle

properties (i.e., size, density, porosity, composition, etc.), different particles in the

population can be distinguished and their behavior can be described. [9]

The link between the population balance and boiling models has not been fully

developed in Fluent and is therefore not employed in the subcooled flow boiling model

discussed in Section 3.5.2. However, the population balance model is utilized to track

bubble size distribution within a bubble column (Section 3.4.3).


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2.7 BOILING HEAT TRANSFER

Boiling is a mode of heat transfer that occurs when saturated liquid changes to

saturated vapor due to heat addition. It is normally characterized by a high heat transfer

capacity and a low wall-fluid temperature delta which is made possible by the generallylarge energy absorption required to cause a phase change. These heat transfer properties

are essential in industrial cooling applications such as nuclear reactors and fossil boilers.

Because of its importance in industry, a significant amount of research has been carried

out to study the capacity and the mechanism of boiling heat transfer. There are two

basic types of boiling, pool boiling and flow boiling. If heat addition causes a phase

change in a stagnant fluid, then it is called pool boiling. If heat addition causes a phase

change in a moving fluid, then it is called flow boiling. Both types of boiling heat

transfer can be separated into four regimes, which are shown in Figure2.7-1 [12].

Figure2.7-1: Boiling Heat Transfer Regimes

The first regime of boiling, up to point A, is known as natural convection boiling.

During this regime, no bubbles form; instead, heat is transferred from the surface to the

bulk fluid by natural convection. The heat transfer rate is proportional to[11].The second regime of boiling, from point A to point C, is called nucleate boiling.

During this stage, vapor bubbles are generated at certain preferred locations on the

heated surface called nucleation sites. Nucleation sites are often microscopic cavities or

cracks in the surface. When the liquid near the wall superheats, it evaporates and a


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significant amount of energy is removed from the heated surface due to the latent heat of

the vaporization which also increases the convective heat transfer by mixing the liquid

water near the heated surface. There are two subregimes of nucleate boiling that can

take place between points A and C. The first subregime is when local boiling occurs in a

subcooled liquid (subcooled boiling). In this situation, bubbles form on a heated surface

but tend to condense after detaching from it. The second subregime is when local

boiling occurs in a saturated liquid. In this case, bubbles do not condense after detaching

from the heated surface since the liquid is at the same temperature as the vapor.

Nucleate boiling is characterized by a very high heat transfer rate and a small

temperature difference between the bulk fluid and the heated surface. For this reason, it

is considered to be the most efficient form of boiling heat transfer. [11]

As the heated surface increases in temperature, more and more nucleation sites

become active. As more bubbles form at these sites, they begin to merge together and

form columns or slugs of vapor, thus decreasing the contact area between the bulk fluid

and the heated surface. The decrease in contact area causes the slope of the line in

Figure2.7-1 to decrease until a maximum is reached (point C). Point C is referred to as

the critical heat flux and the vapor begins to form an insulating blanket around the

heated surface which dramatically increases the surface temperature when reached. This

is called the boiling crisis or departure from nucleate boiling. [12]

As the temperature delta increases past the critical heat flux, the rate of bubble

generation exceeds the rate of bubble separation. Bubbles at the different nucleation

sites begin to merge together and boiling becomes unstable. The surface is alternately

covered with a vapor blanket and a liquid layer, resulting in oscillating surface

temperatures. This regime of boiling is known as partial film boiling or transition

boiling and takes place between points C and D. [11]

If the temperature difference between the surface and the fluid continues to

increase, stable film boiling is achieved. During stable film boiling, there is a

continuous vapor blanket surrounding the heated surface and phase change occurs at the

liquid-vapor interface instead of at the heated surface. During this regime, most heat

transfer is carried out by radiation. [12]


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2.7.1 SUBCOOLED BOILING

Subcooled flow boiling occurs when a moving, under-saturated fluid comes in

contact with a surface that is hotter than its saturation temperature. Intense interaction

between the liquid and vapor phases occur and therefore the Eulerian multiphase model

is most appropriate for subcooled boiling because it is capable of modeling multiple,

separate, yet interacting phases.

The heat transfer rate from the wall to the fluid changes based on the amount of

vapor on the heated surface. Since the vapor area is constantly changing due to the

formation, growth and departure of bubbles, the use of a correlation is necessary. Del

Valle and Kenning created a mechanistic model to determine the area of the heated

surface influenced by vapor during flow boiling which is utilized by Fluent. When

modeling subcooled boiling, there are three parameters of importance that greatly impact

the liquid volume fraction; they are active nucleation site density (Na), bubble departure

diameter (dbw) and bubble departure frequency (f) [1].

As discussed previously, nucleation sites are preferential locations where vapor

tends to form and are usually cavities or irregularities in a heated surface. The number

of active nucleation sites per unit area is dependent on fluid and surface conditions. The

most common active nucleation site density relationship was developed by Lemmert and

Chwala. It is based on the heat flux partitioning data generated by Del Valle and

Kenning [1]:

According to Kurul and Podowski, the values of m and n are 210 and 1.805,

respectively. Another popular correlation for nucleation site density was created by

Kocamustafaogullari and Ishii. They assumed that the active nucleation site density

correlation developed for pool boiling could be used in forced convective systems if the

effective superheat was used rather than the actual wall superheat. This correlation

accounts for both the heated surface conditions and the fluid properties [1]:


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The bubble departure diameter is the bubble size when it leaves the heated

surface and it depends in a complex manner on the amount of subcooling, the flow rate,

and a balance of surface tension and buoyancy forces. Determining the lift off bubble

diameter is crucial because the bubble size influences the interphase heat and mass

transfer through the interfacial area and the momentum drag terms. Many correlations

have been proposed for this purpose; however, the following three are applicable for low

pressure, subcooled flow boiling. The first correlation was proposed by Tolubinsky and

Kostanchuk. It establishes the bubble departure diameter as a function of the subcooling

temperature [1]:

The second correlation was created by Kocamustafaogullari and Ishii who

modified an expression by Fritz that involved the contact angle of the bubble. Its basic

premise is to balance the buoyancy and surface tension forces at the heated surface [1]:

( )A third, more comprehensive correlation was proposed by Unal which includes

the effect of subcooling, the convection velocity, and the heated wall properties [1]:

where

[()]


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[ () ] ( )

{

The bubble departure frequency is the rate at which bubbles are generated and

detach from an active nucleation site. The most common bubble departure frequency

correlation for computational fluid dynamics was developed by Cole who derived it

based on the bubble departure diameter and a balance between buoyancy and drag

forces [1]:

( ) The use of a mechanistic heat transfer model with individual correlations to

calculate the number of active nucleation sites, the bubble departure diameter and the

bubble departure frequency assist in the accurate determination of liquid volume fraction

during subcooled flow boiling. Each of these correlations are tested and compared in

Section 3.5.2.


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3. HEAT TRANSFER AND FLUID FLOW: MODELING

3.1

NATURAL CONVECTION

Two natural convection scenarios were examined. The first was a heated

horizontal cylinder and the second was a heated vertical plate, both were submerged in

an infinite pool of liquid. These examples were chosen because of their simplicity,

because they are commonly found in nature and because they have been previously

studied and results are available for validation of the numerical computations.

3.1.1

HORIZONTAL CYLINDER

A cylinder with a constant surface temperature submerged in an infinite pool of

liquid at a lower temperature was analyzed. Energy passed from the slightly warmer

cylinder to the nearby fluid causing its temperature to increase and convection cells to

form. Figure 3.1.1-1 shows a schematic representation of the geometry and boundary

conditions used to model the horizontal cylinder. The top and bottom walls of the

rectangle represent inlet and outlet pressure boundaries respectively, with pressure

conditions set such that the fluid is stagnant until heated by the cylinder. The left andright walls of the rectangle are slip boundaries to more accurately model an infinite pool.

See Table3.1.1-1 for a detailed list of input parameters used.

Figure3.1.1-1: Horizontal Cylinder Schematic


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Table3.1.1-1: Horizontal Cylinder Model Input

Input Value

Geometry

Cylinder Diameter 0.02 m

Pool Height 0.28 m

Pool Width 0.24 m

2D Space Planar

Solver

Time Transient

Time Step Size 0.05 s

Type Pressure Based

Velocity Formulation Relative

Gravity -9.8 m/s (Y-direction)

Models

Energy Active

Viscous LaminarDensity Boussinesq

Initial Conditions

Cylinder Surface Temperature 310 K

Fluid Temperature 300 K

Material Properties (Water)

Specific Heat 4182 J/kg-K

Thermal Conductivity 0.6 W/m-K

Viscosity 0.001003 kg/m-s

Density See Table3.1.1-2

Solution Methods

Scheme PISO

Gradient Least Square Cell Based

Pressure PRESTO!

Momentum Second Order Upwind

Energy Second Order Upwind

Transient Formulation Second Order Implicit

Table3.1.1-2: Horizontal Cylinder Model Fluid Density

Temperature (K) Density (kg/m )273 999.9

308 994.1

348 974.9

373 958.4


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Figure3.1.1-2 presents the liquid temperature field after 20 seconds of heating.

As the temperature increases, the fluid begins to rise due to buoyancy forces.

Figure3.1.1-2: Temperature (K)

Figure3.1.1-3 shows that even the fluid not in direct contact with the heated

cylinder experiences a density change. The density gradient which is caused by energy

transfer via conduction to the bulk fluid is illustrated by the color transition surrounding

the cylinder from least dense (blue) to most dense (red).

Figure3.1.1-3: Density (kg/m3)


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As the warm fluid rises, it loses energy to the surrounding bulk fluid which

reduces its buoyancy driving head until the rising fluid eventually stops. When the fluid

reaches its maximum elevation, it is pushed aside by the fluid travelling upwards below

it and begins to sink. This motion creates a small convection cell to the left and to the

right of the rising plume about 3 cm above the heated cylinder. This process continues

as long as there is a temperature gradient between the cylinder and the bulk fluid. If the

bulk fluid temperature increases, the buoyancy driving head will be smaller and the

convection cells will develop closer to the heated surface.

Figure3.1.1-4 is a velocity vector plot that displays how the liquid moves within

the control volume. The two convection cells above the cylinder are clearly visible in

this figure which also reveals how the rising fluid is replaced by the cooler fluid

surrounding the cylinder.

Figure3.1.1-4: Velocity Vectors (m/s)

To verify that the model produced realistic results, the solution was compared to

experimental data. Figure 3.1.1-5 shows interference fringes surrounding a heated

horizontal cylinder in natural convection. Each interference fringe can be interpreted as

a band of constant density and therefore temperature.


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(a) (b)

Figure3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder

(a) From Eckert [13](b) Isotherms From Fluent

Figure 3.1.1-5 shows that the experimental data and the model solution have

isotherms that extend away from the cylinder and grow in distance from one another as

they get farther from the heated surface. This indicates that the model is in qualitative

agreement with experimental data.

Quantitative experimental data from Ingham [14] was also compared to the

Fluent results to provide model validation. Figure3.1.1-6, Figure 3.1.1-7 and

Figure3.1.1-8 display a comparison of dimensionless temperature versus dimensionless

distance for four dimensionless times at an angle of 30, 90 and 180, respectively,

from the positive x-axis. Dimensionless temperature is T = (TT) / (TwallT) where

Tis the actual fluid temperature, Tis the bulk fluid temperature and Twall is the heated

wall temperature. Dimensionless distance is (r-a)/a * Gr1/4

where r is the radial

distance from the heated surface, a is the cylinder diameter and Gr is the Grashof

number. Dimensionless time is t = t * (gT/a)1/2

where t is real time, T is

(TwallT), is the coefficient of thermal expansion and a is the cylinder diameter.


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(a) (b)

Figure3.1.1-6: Dimensionless Temperature at =30

(a) From Ingham [14]and (b) From Fluent

(a) (b)

Figure3.1.1-7: Dimensionless Temperature at = 90



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(a) (b)

Figure3.1.1-8: Dimensionless Temperature at = 180


The heated horizontal cylinder model developed in Fluent showed good

agreement compared with experimental data at the three different radial locations. This

comparison provided confidence that the information obtained from the model was

accurate.

To ensure that the mesh had no significant effect on the results, a mesh validation

was performed. The mesh validation compared the results shown in this section

(Mesh 1 in Table3.1.1-3) to a second mesh with an increased number of finite

volumes (Mesh 2 in Table3.1.1-3). The results from the mesh validation displayed in

Table3.1.1-3 prove that the results are mesh independent.

Table3.1.1-3: Mesh Validation for Horizontal Cylinder Model

Mesh 1 Mesh 2 Difference

Number of Nodes 19716 23636 19.88 %

Number of Elements 38688 46400 19.93 %

Max Velocity (m/s) 0.01627 0.01621 -0.37 %

Max Total Temperature (K) 309.9239 309.9531 0.01 %

Min Density (kg/m ) 993.1765 993.1625 0.00 %


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3.1.2 VERTICAL PLATE

Single phase convection heat transfer around a vertical plate with a constant

surface temperature submerged in an infinite pool of liquid at a lower temperature was

also analyzed. Energy passed from the slightly warmer plate to the fluid causing its

temperature to increase and the fluid to rise. Figure3.1.2-1 shows a schematic

representation of the geometry and boundary conditions used to model the vertical plate.

The top and bottom walls of the rectangle represent inlet and outlet pressure boundaries

respectively, with pressure conditions set such that the fluid is stagnant until the plate is

heated. The left and right walls of the rectangle are slip boundaries to more accurately

model an infinite pool. See Table3.1.2-1 for a detailed list of input parameters used.

Figure3.1.2-1: Vertical Plate Schematic

Figure3.1.2-2 presents the liquid temperature field after 20 seconds of heating.

When energy is exchanged between the plate and the fluid, a thermal boundary layer is

created. Thermodynamic equilibrium demands that the plate, and the fluid in direct

contact with it, be at the same temperature. The region in which the fluid temperature

changes from the plate surface temperature to that of the bulk fluid temperature is known

as the thermal boundary layer. The teal color in Figure3.1.2-2 shows the growth of the

thermal boundary layer, which is relatively small at the bottom of the plate but grows

due to heat addition (teal color expands away from the plate) as the fluid climbs.


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Table3.1.2-1: Vertical Plate Model Input

Input Value

Geometry

Plate Height 0.18 m

Plate Width 0.01 m

Pool Height 0.20 m

Pool Width 0.13 m

2D Space Planar

Solver

Time Transient

Time Step Size 0.05 s

Type Pressure Based


Gravity -9.8 m/s2(Y-direction)

Models

Energy ActiveViscous Laminar

Density Boussinesq

Initial Conditions

Plate Surface Temperature 310 K






Density See Table3.1.2-2

Solution Methods

Scheme PISO


Pressure PRESTO!



Transient Formulation Second Order Implicit

Table3.1.2-2: Vertical Plate Model Fluid Density

Temperature (K) Density (kg/m3)

273 999.9

308 994.1

348 974.9

373 958.4


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Figure3.1.2-2: Temperature (K)

Figure3.1.2-3 shows the liquid velocity in vector form. The figure shows that

the velocity is primarily vertical and the magnitude increases with elevation. The

increase in fluid velocity with elevation is caused by an increase in energy absorption as

the fluid rises along the heated surface which causes a greater density gradient and

therefore a larger buoyancy force.

Figure3.1.2-3: Velocity Vectors (m/s)


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Comparing Figure 3.1.2-3 (vertical plate liquid velocity vectors) with

Figure3.1.1-4 (horizontal cylinder liquid velocity vectors) produces interesting results.

Because of the larger heated region, it was expected that the vertical plate would produce

a greater maximum fluid velocity compared to the horizontal cylinder. The vertical plate

has a maximum fluid velocity of 0.0149 m/s while the horizontal cylinder has a

maximum fluid velocity of 0.0177 m/s. Although the difference is small, it is notable.

The horizontal cylinder generates a larger maximum velocity because the buoyancy

driving force is not impeded by the drag force created by the heated surface. Although

the vertical plate continues to heat the fluid as it travels upward, the velocity is limited

by friction which causes the plate scenario to have a smaller maximum velocity.

To ensure that the model calculated realistic results, the solution was compared

to experimental data. Figure 3.1.2-4 shows interference fringes surrounding a heated

vertical plate in natural convection. Each interference fringe can be interpreted as a band

of constant density and therefore temperature.

(a) (b)

Figure3.1.2-4: Interference Fringes Around a Heated Vertical Plate

(a)From Eckert [13]and (b) Isotherms From Fluent


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Figure 3.1.2-4 shows that the experimental data and model solution have

isotherms that extend away from the plate and grow in distance from one another as they

get farther from the heated surface. This indicates that the model is in qualitative

agreement with experimental data.

Experimental data from Ostrach [15]was compared to the Fluent results to assess

the quantitative accuracy of the model. Figure3.1.2-5 and Figure 3.1.2-6 display a

comparison of dimensionless temperature versus dimensionless distance for five

different Prandtl numbers. Figure3.1.2-5a shows theoretical values and Figure3.1.2-5b

compares some of the theoretical values to experimental data. Dimensionless

temperature is H()= (TT) / (T0T) where T is the actual fluid temperature, Tis

the bulk fluid temperature and T0 is the wall temperature. Dimensionless distance is

= (Y / X) * (Grx/ 4)1/4where Grxis the Grashof number, Y is the vertical height and X

is the distance from the plate.

(a) (b)

Figure3.1.2-5: Dimensionless Temperature for Various Prandtl Numbers

(a) Theoretical Values and (b) Experimental Values [15]


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Figure3.1.2-6: Dimensionless Temperature for Various Prandtl Numbers (Fluent)

The heated vertical plate model developed in Fluent produced results that slightly

overestimate the thickness of the temperature profile when compared to experimental

data for five different Prandtl numbers. The slight over prediction is due to imperfect

extraction of the raw data from Fluent.To ensure that the mesh had no significant effect on the results, a mesh validation


(Mesh 1 in Table3.1.2-3) to a second mesh with an increased number of finite

volumes (Mesh 2 in Table3.1.2-3). The results from the mesh validation shown in

Table3.1.2-3 prove that the results are mesh independent.

Table3.1.2-3: Mesh Validation for Vertical Plate Model


Number of Nodes 12310 18081 46.88 %


Max Velocity (m/s) 0.01376 0.01380 0.29 %

Max Total Temperature (K) 309.8089 309.7991 0.00 %

Min Density (kg/m ) 993.2319 993.2365 0.00 %


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3.2

LAMINAR FLOW

A steady state, axisymmetric, laminar flow model was developed. Figure3.2-1

shows a schematic representation of the geometry and boundary conditions used to

model laminar flow within a pipe. The bottom line of the rectangle is an axis of rotation

which is used to simplify the geometry and represents the pipe centerline. The top line

of the rectangle is a no slip boundary and after the rotation, becomes the pipe wall. The

left and right lines of the rectangle are the inlet and outlet areas respectively, which

when revolved, are circular. See Table3.2-1 for a detailed list of input parameters used.

Figure3.2-1: Laminar Flow Schematic

Based on the selected inlet conditions, the Reynolds number is 352, which is well

within the laminar regime. Figure3.2-2 displays the velocity magnitude versus position

(distance from the pipe centerline) at different lengths from the pipe entrance. For

example, line-10cm is the velocity profile 10 cm from the pipe entrance. The

parabolic shape of the velocity profile is clearly visible which is characteristics of

laminar flow.

Figure3.2-2: Velocity Magnitude


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Fluid velocity within the pipe slowly decreases as distance from the pipe centerline

increases. Also, as the flow develops, the entrance effects dissipate, the velocity profile

becomes more parabolic until it reaches a steady state at about 45 cm from the entrance

which is in good agreement with well known entrance lengthcalculations [5].

Table3.2-1: Laminar Flow Model Input

Input Value

Geometry

Pipe Diameter 0.03 m

Pipe Length 0.50 m

2D Space Axisymmetric

Solver

Time Steady

Type Pressure BasedVelocity Formulation Relative

Gravity -9.8 m/s2(X-direction)

Models

Energy Active

Viscous Laminar





Density See Table3.2-2

Inlet Conditions

Pipe Wall Surface Temperature 305 K


Fluid Velocity 0.05 m/s

Solution Methods

Scheme Coupled


Pressure Second Order



Table3.2-2: Laminar Flow Model Fluid Density

Temperature (K) Density (kg/m )

273 999.9

308 994.1


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Another characteristic of laminar flow is the lack of mixing that occurs within the

fluid. The radial velocity within the pipe is basically zero and each fluid element

remains about the same distance from the centerline from entrance to exit. Figure3.2-3

displays the radial flow velocity. As expected, the radial velocity for most of the pipe is

near zero and is less than 10-3

times the average axial velocity. Radial velocity is at a

maximum near the entrance of the pipe due to inlet boundary conditions and entrance

effects but these have a negligible impact on system as a whole.

Figure3.2-3: Radial Velocity (m/s)

Figure 3.2-4 provides the temperature profile for the laminar flow analyzed.

Because there is little to no radial velocity, convection and conduction are the primary

forms of heat transfer which causes the thermal boundary layer to grow at a very slow

rate. The growth of the thermal boundary layer is shown in Figure 3.2-4 by the

expansion of the teal colored region.

Figure3.2-4: Temperature (K)

Figure3.2-5 shows the wall shear stress as a function of distance from the pipe

entrance. The wall stress is much larger in the first 10 cm due to entrance effects. Once

the entrance effects dissipate, the wall shear stress slowly decreases as the flow reaches a

steady state.


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Figure3.2-5: Wall Shear Stress

To ensure that the mesh had no significant effect on the results, a mesh validation


(Mesh 1 in Table3.2-3) to a second mesh with an increased number of finite volumes

(Mesh 2 in Table 3.2-3). The results from the mesh validation displayed in

Table3.2-3 prove that the results are mesh independent.

Table3.2-3: Mesh Validation for Laminar Flow Model


Number of Nodes 26320 31000 17.78 %


Max Velocity (m/s) 0.079561 0.079507 -0.07 %

Min Radial Velocity (m/s) -0.003293 -0.003528 7.12 %

Max Dynamic Pressure (Pa) 3.15925 3.155022 -0.13 %

Max Temperature (K) 304.6503 304.6855 0.01 %


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3.3

TURBULENT FLOW

3.3.1 TURBULENT FLOW WITHOUT HEAT TRANSFER

A steady state, axisymmetric, turbulent flow model was developed.

Figure3.3.1-1 shows a schematic representation of the geometry and boundary

conditions used to model turbulent flow within a pipe without heat transfer. The bottom

line of the rectangle is an axis of rotation which is used to simplify the geometry and

represents the pipe centerline. The top line of the rectangle is a no slip boundary and

after the rotation becomes the pipe wall. The left and right lines of the rectangle are the

inlet and outlet areas respectively, which when revolved, are circular. See Table3.3.1-1

for a detailed list of input parameters used.

Figure3.3.1-1: Turbulent Flow Without Heat Transfer Schematic

Based on the selected inlet conditions, the Reynolds number is 42314, which is

well within the turbulent regime. Figure3.3.1-2 displays the velocity magnitude versus

position (distance from the pipe centerline) at different distances from the pipe entrance.

Figure3.3.1-2: Velocity Magnitude


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The velocity profile of turbulent flow differs significantly in two ways compared

to the velocity profile of laminar flow (Figure 3.2-2). First, turbulent flow velocity

profiles are much flatter. Therefore, the fluid velocity doesnt decrease significantly

until close to the pipe wall. Second, entrance effects dissipate much quicker in turbulent

flow [5]and thus the fluid velocity reaches a steady state velocity profile in a shorter

distance. Figure3.3.1-2 (turbulent flow) shows that flow reaches a steady profile about

10 cm from the pipe entrance. Figure3.2-2 (laminar flow) shows that flow reaches a

steady profile about 45 cm from the pipe entrance. This qualitatively matches

experimental data well.

Table3.3.1-1: Turbulent Flow Without Heat Transfer Model Input

Input Value

Geometry

Pipe Diameter 0.03 m

Pipe Length 0.50 m

2D Space Axisymmetric

Solver

Time Steady

Type Pressure Based


Gravity -9.8 m/s (X-direction)

Models

Energy InactiveViscous Realizable k-

Turbulence Model

Near Wall Treatment Enhanced

Turbulent Intensity 0.0422483 *

Inlet Conditions

Fluid Mass Flow Rate 1.0 kg/s


Density 998.2 kg/m


Solution Methods

Scheme Coupled


Pressure Second Order


Turbulent Kinetic Energy Second Order Upwind

Turbulent Dissipation Rate Second Order Upwind

* Calculation shown in Table2.5.1-2.


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Figure 3.3.1-3 displays the wall shear stress versus distance from the pipe

entrance. The shear stress is very large at the pipe entrance and decays to the steady

state value after about 10 cm (same location where the velocity profile reaches steady

state). The large increase in shear stress at the beginning of the pipe (~1-2 cm from the

inlet) is caused by entrance effects. Figure3.3.1-4 shows that that maximum radial

velocity occurs near the pipe entrance. Figure3.3.1-5 reveals that the greatest reduction

in axial velocity occurs near the pipe entrance which is necessary to conserve

momentum when radial velocity increases. Since shear stress is related to change in

velocity parallel to the wall (axial velocity), the increase in wall shear stress near the

pipe entrance is reasonable.

Figure3.3.1-3: Wall Shear Stress

Figure3.3.1-4: Radial Velocity (m/s)

Figure3.3.1-5:


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To further investigate the impact of entrance effects, two additional scenarios

were examined using a mass flow rate of 0.5 kg/s (Figure3.3.1-6) and a mass flow rate

of 1.5 kg/s (Figure3.3.1-7).

(a)

(b) (c)

Figure3.3.1-6: Results for a Mass Flow Rate of 0.5 kg/s

(a) Radial Velocity (m/s) (b) Wall Shear Stress (c)

(a)

(b) (c)

Figure3.3.1-7: Results for a Mass Flow Rate of 1.5 kg/s

(a) Radial Velocity (m/s) (b) Wall Shear Stress (c)


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Figures 3.3.1-6 and 3.3.1-7 prove that the maximum wall shear stress and the

maximum radial velocity are directly related to mass flow rate. At a certain distance

from the pipe entrance, the change in axial velocity as a function of position reaches zero

and the wall shear stress reaches a constant value. The pipe length necessary to reach a

steady state shear stress is also related to the mass flow rate. A larger mass flow rate

requires a greater distance to reach a constant shear stress.

Figure3.3.1-8 shows that most of the turbulent kinetic energy is located near the

pipe wall due to shear stress.

Figure3.3.1-8: Turbulent Kinetic Energy (m2/s

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