Mathematical Modeling of Convective Heat Transfer:
Single Phase through Subcooled Boiling Flows
by
Matthew P. Wilcox
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
April 2013
i
© Copyright 2013
By
Matthew P. Wilcox
All Rights Reserved
ii
TABLE OF CONTENTS
TABLE OF CONTENTS ................................................................................................. iii
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES ......................................................................................................... vii
ABSTRACT ..................................................................................................................... ix
1. INTRODUCTION ....................................................................................................... 1
1.1
RESEARCH ....................................................................................................... 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 .............................................................................. 8
2.4
LAMINAR FLOW ............................................................................................. 9
2.5
TURBULENT FLOW ...................................................................................... 10
2.5.1
2.6
2.7
CALCULATING TURBULENCE PARAMETERS .......................... 13
TWO-PHASE FLOW ...................................................................................... 15
2.6.1
MODELING TWO-PHASE FLOWS .................................................. 17
2.6.2
POPULATION BALANCE MODEL.................................................. 18
BOILING HEAT TRANSFER ........................................................................ 19
3. HEAT TRANSFER AND FLUID FLOW: MODELING ......................................... 21
3.1
NATURAL CONVECTION ............................................................................ 21
3.1.1
HORIZONTAL CYLINDER ............................................................... 21
3.1.2
VERTICAL PLATE ............................................................................ 28
3.2
LAMINAR FLOW ........................................................................................... 34
3.3
TURBULENT FLOW ...................................................................................... 38
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER ..................... 38
3.3.2
TURBULENT FLOW WITH HEAT TRANSFER ............................. 43
iii
3.4
3.5
TWO-PHASE FLOW ...................................................................................... 46
3.4.1
GAS MIXING TANK .......................................................................... 46
3.4.2
BUBBLE COLUMN ............................................................................ 51
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL ... 56
BOILING FLOWS ........................................................................................... 60
3.5.1
POOL BOILING .................................................................................. 60
3.5.2
SUBCOOLED Flow BOILING ........................................................... 66
4. DISUSSION AND CONCLUSIONS ........................................................................ 81
REFERENCES ................................................................................................................ 83
iv
LIST OF TABLES
Table 2.5.1-1: Turbulent Flow Input ............................................................................... 14
Table 2.5.1-2: Turbulence Parameter Calculation ........................................................... 14
Table 3.1.1-1: Horizontal Cylinder Model Input ............................................................. 22
Table 3.1.1-2: Horizontal Cylinder Model Water Density .............................................. 22
Table 3.1.1-3: Mesh Validation for Horizontal Cylinder Model ..................................... 27
Table 3.1.2-1: Vertical Plate Model Input ....................................................................... 29
Table 3.1.2-2: Vertical Plate Model Water Density ........................................................ 29
Table 3.1.2-3: Mesh Validation for Vertical Plate Model ............................................... 33
Table 3.2-1: Laminar Flow Model Input ......................................................................... 35
Table 3.2-2: Laminar Flow Model Water Density .......................................................... 35
Table 3.2-3: Mesh Validation for Laminar Flow Model ................................................. 37
Table 3.3.1-1: Turbulent Flow Without Heat Transfer Model Input ............................... 39
Table 3.3.2-1: Turbulent Flow With Heat Transfer Model Input .................................... 44
Table 3.3.2-2: Turbulent Flow Model Water Density ..................................................... 44
Table 3.3.2-3: Mesh Validation for Turbulence With Heat Transfer Model ................... 46
Table 3.4.1-1: Gas Mixing Tank Model Input ................................................................. 48
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank Model ......................................... 50
Table 3.4.2-1: Bubble Column Model Input ................................................................... 52
Table 3.4.2-2: Mesh Validation for Bubble Column Model ........................................... 56
Table 3.4.3-1: Population Balance Model Input .............................................................. 57
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.072 N/m ..................... 59
Table 3.4.3-3: Bubble Size Distribution – Surface Tension of 0.0072 N/m ................... 60
Table 3.5.1-1: Pool Boiling Model Input......................................................................... 61
Table 3.5.1-2: Pool Boiling Model Water Density .......................................................... 62
Table 3.5.1-3: Mesh Validation for Pool Boiling Model................................................. 65
Table 3.5.2-1: Subcooled Flow Boiling Model Input ...................................................... 69
Table 3.5.2-2: Subcooled Flow Boiling Model Water Properties ................................... 70
Table 3.5.2-3: Boiling Model Study Case Input .............................................................. 71
Table 3.5.2-4: Boiling Model Study Case Results .......................................................... 74
Table 3.5.2-5: Initial Conditions Study Case Input ......................................................... 75
v
Table 3.5.2-6: Initial Condition Study Case Results ....................................................... 75
Table 3.5.2-7: Axial Location Liquid Volume Fraction .................................................. 76
Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction .......................................... 78
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction ........................................... 79
Table 3.5.2-10: Mesh Validation for Subcooled Boiling Model ..................................... 80
vi
LIST OF FIGURES
Figure 2.5-1: Transition from Laminar to Turbulent Flow.............................................. 11
Figure 2.6-1: Flow Regimes ............................................................................................ 15
Figure 2.6-2: Baker Flow Pattern .................................................................................... 16
Figure 2.7-1: Boiling Heat Transfer Regimes ................................................................. 19
Figure 3.1.1-1: Heated Cylinder Schematic .................................................................... 21
Figure 3.1.1-2: Temperature (K) ..................................................................................... 23
Figure 3.1.1-3: Density (kg/m3) ....................................................................................... 23
Figure 3.1.1-4: Velocity Vectors (m/s) ............................................................................ 24
Figure 3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder (K) ............ 25
Figure 3.1.1-6: Temperature at θ = 30° Vs. Radial Distance .......................................... 26
Figure 3.1.1-7: Temperature at θ = 90° Vs. Radial Distance .......................................... 26
Figure 3.1.1-8: Temperature at θ = 180° Vs. Radial Distance ........................................ 27
Figure 3.1.2-1: Vertical Plate Schematic ......................................................................... 28
Figure 3.1.2-2: Temperature (K) ..................................................................................... 30
Figure 3.1.2-3: Velocity Vectors (m/s) ............................................................................ 30
Figure 3.1.2-4: Interference Fringes Around a Heated Vertical Plate (K) ...................... 31
Figure 3.1.2-5: Dimensionless Temperature as a Function of Prandtl Number .............. 32
Figure 3.1.2-6: Dimensionless Temperature as a Function of Prandtl Number .............. 33
Figure 3.2-1: Laminar Flow Schematic ........................................................................... 34
Figure 3.2-2: Velocity Profile .......................................................................................... 34
Figure 3.2-3: Radial Velocity (m/s) ................................................................................. 36
Figure 3.2-4: Temperature (K) ........................................................................................ 36
Figure 3.2-5: Wall Shear Stress ....................................................................................... 37
Figure 3.3.1-1: Turbulent Flow Without Heat Transfer Schematic................................. 38
Figure 3.3.1-2: Velocity Magnitude ................................................................................ 38
Figure 3.3.1-3: Wall Shear Stress .................................................................................... 40
Figure 3.3.1-4: Radial Velocity (m/s) ............................................................................. 40
Figure 3.3.1-5: dAxial-Velocity/dx ................................................................................. 40
Figure 3.3.1-6: Flow Results for Mass Flow Rate of 0.5 kg/s ......................................... 41
Figure 3.3.1-7: Flow Results for Larger Mass Flow Rate of 1.5 kg/s ............................. 41
vii
Figure 3.3.1-8: Turbulent Kinetic Energy (m2/s2) ........................................................... 42
Figure 3.3.1-9: Production of Turbulent Kinetic Energy ................................................ 42
Figure 3.3.2-1: Turbulent Flow With Heat Transfer Schematic ...................................... 43
Figure 3.3.2-2: Temperature (K) ..................................................................................... 43
Figure 3.3.2-3: Radial Velocity (m/s) .............................................................................. 45
Figure 3.3.2-4: Velocity Magnitude ................................................................................ 45
Figure 3.3.2-5: Wall Shear Stress .................................................................................... 45
Figure 3.4.1-1: Gas Mixing Tank Schematic................................................................... 47
Figure 3.4.1-2: Gas Volume Fraction .............................................................................. 49
Figure 3.4.1-3: Liquid Velocity Vectors (m/s) ................................................................ 49
Figure 3.4.1-4: Gas Velocity Vectors (m/s)..................................................................... 50
Figure 3.4.2-1: Bubble Column Schematic ..................................................................... 51
Figure 3.4.2-2: Gas Volume Fraction .............................................................................. 53
Figure 3.4.2-3: Liquid Velocity Vectors (m/s) ................................................................ 54
Figure 3.4.2-4: Gas Velocity Vectors (m/s)..................................................................... 55
Figure 3.4.2-5: Gas Volume Fraction (0.10 m/s)............................................................. 55
Figure 3.4.3-1: Gas Volume Fraction with PBM ............................................................ 57
Figure 3.4.3-2: Liquid Velocity Vectors with PBM (m/s)............................................... 58
Figure 3.4.3-3: Liquid Velocity Vectors with PBM (m/s)............................................... 59
Figure 3.5.1-1: Pool Boiling Schematic .......................................................................... 61
Figure 3.5.1-2: Instantaneous Gas Volume Fraction ....................................................... 63
Figure 3.5.1-3: Liquid Velocity Vectors (m/s) ................................................................ 64
Figure 3.5.1-4: Gas Velocity Vectors (m/s)..................................................................... 64
Figure 3.5.1-5: Volume Fraction of Vapor on Heated Surface ....................................... 65
Figure 3.5.2-1: Subcooled Flow Boiling Model Schematic ............................................ 69
Figure 3.5.2-2: Case 1 - Temperature (K) ....................................................................... 71
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction ........................................................... 71
Figure 3.5.2-4: Case 1 - Mass Transfer Rate (kg/m3-s) ................................................... 72
Figure 3.5.2-5: Liquid Volume Faction for Cases 1-6..................................................... 73
Figure 3.5.2-6: Liquid Volume Faction for Cases 7-12................................................... 77
viii
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 each
regime was discussed in detail and followed by the development 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
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 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
initial conditions. The bubble departure diameter models 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 initial conditions impact the liquid volume fraction
during subcooled flow boiling was explored.
The initial conditions of heat flux, inlet
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 initial conditions with respect to liquid volume fraction were
developed. Overall, the inlet temperature had the greatest impact on liquid volume
fraction, the wall heat flux had the second greatest impact and mass flow rate had the
smallest impact.
ix
1. INTRODUCTION
Since the 19th century, the world’s standard of living has greatly increased
primarily due to the generation and distribution of electricity. Over 80% of the world’s
electricity production is generated by converting thermal energy, from a fuel source into
electrical energy. A common energy conversion process known as the Rankine Cycle
burns fuel to generate steam which is used to turn a turbine and spin an electric
generator.
Electricity production involves numerous 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 here will be uranium or
nuclear fuel. Nuclear power plants harness energy released during fission to heat water.
The energy transfer mechanisms within a nuclear reactor involve all three major forms
of heat transfer; conduction, convection and radiation. The fluid flow through the
reactor core is complex due to the intense energy transfer and phase change.
In
Pressurizer Water Reactors, the water surrounding the reactor core is prevented from
bulk boiling because it is highly pressurized; however, a small amount of localized
boiling does occur. This 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 in locations called nucleation sites. The number of bubbles that form is heavily
dependent on fluid inlet 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 steam is dispersed in a subcooled liquid
which is where the term subcooled boiling originates.
1
1.1
RESEARCH
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. A
number 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. This 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 the energy from the wall is transferred to the
liquid adjacent to the heated wall. Then, a fraction of the energy is transferred to vapor
bubbles by evaporation while the remainder goes into the bulk liquid. [1]
Additionally, focus has been placed on accurately modeling the three most
impactful parameters in subcooled flow boiling.
These parameters are the active
nucleation site density (Na), departing bubble 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.
Recently, the use of population balance equations (PBEs) has been used to improve
the modeling of subcooled flow boiling to better determine how swarms of bubbles
2
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. Population balance equations help
define how particle size populations develop in specific properties over time. Population
balance equations are available in Fluent but 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 Section 1.1 impact the liquid volume fraction at different
axial locations. The third objective was to evaluate how heat flux, inlet 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 development of each model, a more complicated fluid
flow or heat transfer scenario was analyzed. 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 Section 3.1. 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 Section 3.2. The third
model developed was for turbulent flow. The theory of turbulent flow is described in
Section 2.5 and the analytical modeling results are displayed in Section 3.3. Section 3.3
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
3
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 model created includes a phase transformation (vaporization and condensation).
The theory of boiling heat transfer is described in Section 2.7 and the analytical
modeling results are presented in Section 3.5. Two models were created, the first for
pool boiling and the second for subcooled flow boiling. 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.
4
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 in
subcooled flow boiling.
2.1
GOVERNING EQUATIONS
Conservation equations are a local form of conservation 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 in Vector Form (continuity equation):
𝜕𝜌
⃑ ∙ 𝜌v
+ (∇
⃑)= 0
𝜕𝑡
Conservation of Mass in Cartesian Form:
𝜕𝜌 𝜕
𝜕
𝜕
(𝜌𝑣𝑥 ) +
(𝜌𝑣𝑌 ) + (𝜌𝑣𝑧 ) = 0
+
𝜕𝑡 𝜕𝑥
𝜕𝑦
𝜕𝑧
Conservation of Momentum in Vector Form:
𝜌
𝐷v
⃑
⃑ 𝜌 + 𝜇∇
⃑ 2v
= −∇
⃑ + 𝜌𝑔
𝐷𝑡
Conservation of Momentum in Cartesian Form:
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑝
𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥
𝜌(
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑥
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜌(
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦
𝜕𝑝
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑝
𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧
𝜌(
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
5
In subcooled flow boiling, as in many other instances of fluid dynamics, energy
is added or removed from the system. In this situation, the conservation of energy
equation is important.
Conservation of Energy in Vector Form:
𝜌𝐶̂𝑝
𝐷𝑇
𝜕 ln 𝜌 𝐷𝑝
⃑ ∙ 𝑞) − (
= −(∇
)
𝐷𝑡
𝜕 ln 𝑇 𝑝 𝐷𝑡
Conservation of Energy in Cartesian Form:
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑞𝑥 𝜕𝑞𝑦 𝜕𝑞𝑧
𝜕 ln 𝜌 𝐷𝑝
𝜌𝐶̂𝑝 ( + 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧 ) = − (
+
+
)−(
)
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕 ln 𝑇 𝑝 𝐷𝑡
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 various
circumstances 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 equation for 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 only focusing on the solution of the
differential equations at discrete locations, the need to find an exact solution to the
differential equation is not necessary.
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
6
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
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 for deriving discretization equations
include variational formulation, method of weighted residuals and control volume
formulation. 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,
7
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, it is generally referred to as forced
convection. Natural convection is a transport mechanism in which the fluid motion is
not generated by any external source (like a pump, fan, suction device, etc.) but driven
by buoyancy-induced motion resulting from internal body forces produced by density
gradients. 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 submersed
in 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 moves to take its place. The cooler fluid is then heated and the
process continues forming a convection current 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 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,
Gr, is a dimensionless parameter that represents the ratio of buoyancy to viscous forces
acting on a fluid; and is defined as:
𝐺𝑟 =
𝑔𝛽(𝑇𝑠 − 𝑇∞ )𝐿3
(𝜇 ⁄𝜌)2
where β is the thermal expansion coefficient:
1 𝜕𝑃
𝛽=− ( )
𝜌 𝜕𝑇 𝑝
8
The Prandtl number, Pr, is a dimensionless parameter that represents the ratio of
momentum diffusivity to thermal diffusivity; and is defined as:
Pr =
Cp μ
k
The Rayleigh number, Ra, is a dimensionless parameter that represents the ratio
of buoyancy and viscosity forces times the ratio of momentum and thermal diffusivities;
and is defined as:
Ra = GrPr
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, natural
convection can either be laminar or turbulent. Rayleigh numbers less than 108 indicate 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:
Gr
gβΔTL
=
Re2
V2
When this number approaches or exceeds unity, there are strong buoyancy
contributions to the flow. Conversely, if the ratio is very small, buoyancy forces may be
ignored.
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.
Laminar flow occurs at very low velocities where there are only small disturbances and
little to no local velocity variations. In laminar flow, the motion of the fluid particles is
very orderly and can be characterized by high momentum diffusion and low momentum
convection.
9
The Reynolds number is used to characterize the flow regime. The Reynolds
number, Re, is a dimensionless number that represents the ratio of inertial forces to
viscous forces; and is defined as:
Re =
ρVA
μ
The Reynolds number helps to 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 of laminar flow in a pipe is can be calculated by [5]:
𝑢=
𝑟𝑠2
𝑑𝑃
𝑟2
(− ) (1 − 2 )
4𝜇
𝑑𝑥
𝑟𝑠
Or, in terms of the mean velocity, V:
𝑟2
𝑢 = 2𝑉 (1 − 2 )
𝑟𝑠
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]:
𝜕𝑇
1𝜕
𝜕𝑇
𝜕 2𝑇
𝑢
= 𝛼[
(𝑟 ) + 2 ]
𝜕𝑥
𝑟 𝜕𝑟 𝜕𝑟
𝜕𝑥
2.5
TURBULENT FLOW
In fluid dynamics, turbulence is a flow regime characterized by chaotic and
stochastic changes. Turbulent flows exist everywhere in nature from the jet stream to
the oceanic currents. Turbulent flows are highly irregular and random which makes a
deterministic approach to turbulence problems impossible. They have high diffusivity,
meaning there is rapid mixing and increased rates of momentum, heat and mass transfer.
Because of these properties, turbulent flows are very important to many engineering
applications.
Turbulent flows involve large Reynolds numbers and contain three10
dimensional vorticity fluctuations. The unsteady vortices appear on many scales and
interact with each other generating high levels of mixing. Also, like laminar flows,
turbulent flows are dissipative. Because turbulence cannot maintain itself, it depends on
its 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
mechanisms, the turbulence will decay 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].
Figure 2.5-1: Transition from Laminar to Turbulent Flow
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
11
happens, the flow of the smoke becomes more random and rapidly mixes with the air
causing it to dissipate.
Perfect 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 NavierStokes equation 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 (timeaveraged) and fluctuating components.
The RANS equations can be used with
approximations based on knowledge of the turbulent flow to give approximate timeaveraged solutions to the Navier–Stokes equations. [9]
For the velocity terms:
𝑢𝑖 = 𝑢̅𝑖 + 𝑢𝑖′
where 𝑢̅𝑖 and 𝑢𝑖′ are the mean and fluctuating velocity components respectively.
Similarly, for scalar quantities:
𝜙 = 𝜙̅ + 𝜙 ′
where 𝜙 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:
𝛿𝑝
𝛿
(𝜌𝑢̅𝑖 ) = 0
+
𝛿𝑡 𝛿𝑥𝑖
𝛿
𝛿
𝛿𝑃
𝛿
𝛿𝑢𝑖 𝛿𝑢𝑗 2 𝛿𝑢𝑙
𝛿
′ ′
̅̅̅̅̅̅
(𝜌𝑢̅𝑖 ) +
(𝜌𝑢̅𝑖 𝑢̅𝑗 ) = −
+
[𝜇 (
+
− 𝜁𝑖𝑗
)] +
(−𝜌𝑢
𝑖 𝑢𝑗 )
𝛿𝑡
𝛿𝑥𝑗
𝛿𝑥𝑖 𝛿𝑥𝑗
𝛿𝑥𝑗 𝛿𝑥𝑖 3 𝛿𝑥𝑙
𝛿𝑥𝑗
The two above equations are called the RANS equations. They have the same
general form as the instantaneous Navier-Stokes equations, with the velocities and other
solution variables now representing time-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 Navier–Stokes equations. An additional term,(−𝜌𝑢
𝑖 𝑢𝑗 ),
12
known as the Reynolds stress 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-ϵ
turbulence model. The k-ϵ model was first introduced by Harlow and Nakayama in
1968 [10]. The k-ϵ model has become the most widely used model for industrial
applications because of its overall accuracy and small computational demand. In the k-ϵ
model, k represents the turbulent kinetic energy and ϵ represents its dissipation rate.
Turbulent kinetic energy is the average kinetic energy per unit mass associated with
eddies in the turbulent flow while epsilon (ϵ) is the rate of dissipation of the turbulent
energy per unit mass.
In the derivation of the k-ϵ 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-ϵ model became known, modifications were introduced to improve its
performance.
These improvements have helped create many, new, more accurate
models, among them, the realizable k-ϵ model which differs from the standard k-ϵ 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, ϵ,
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 flows. [9]
2.5.1
CALCULATING TURBULENCE PARAMETERS
All of the computational fluid dynamic models discussed in this thesis use the
k-ϵ turbulence model when applicable. In Fluent, turbulence models require certain
parameters to be established prior to initialization to properly set the initial and boundary
conditions for the flow. For instance, based on the conditions in Table 2.5.1-1, the
equations in Table 2.5.1-2 [9] were used to determine the boundary and initial condition
inputs for the turbulent flow models presented in Section 3.3.
13
Table 2.5.1-1: Turbulent Flow Input
Input Parameter
Mass Flow Rate (𝑚̇)
Pipe Diameter (D)
Viscosity (μ)
Density (ρ)
Turbulence Empirical Constant (Cμ)
Numerical Value
1.0 kg/s
0.03 m
0.001003 kg/m-s
998.2 kg/m3
0.09 [9]
Table 2.5.1-2: Turbulence Parameter Calculation
Variable
Hydraulic Diameter (Dh)
Flow Area (A)
Average Flow Velocity (uavg)
Reynolds Number (ReDh)
Turbulence Length Scale (l)
Equation
Numerical Value
4∗𝐴
𝐷ℎ =
𝑃
𝐷 2
𝜋 ∗ (2 )
=
=𝐷
4∗𝜋∗𝐷
𝐷 2
𝐴 =𝜋∗( )
2
0.03 𝑚 2
=𝜋∗(
)
2
𝑚̇
𝑢𝑎𝑣𝑔 =
𝜌∗𝐴
0.5 𝑘𝑔/𝑠
=
𝑘𝑔
998.2 3 ∗ 0.00070686 𝑚2
𝑚
𝑚̇𝐷ℎ
𝑅𝑒𝐷ℎ =
𝜇𝐴
𝑘𝑔
0.5 𝑠 ∗ 0.03 m
=
𝑘𝑔
0.001003 𝑚 − 𝑠 ∗ 0.00070686 𝑚2
𝑙 = 0.07 ∗ 𝐷ℎ
= 0.07 ∗ 0.03 𝑚
−
Turbulent Intensity (I)
Turbulent Kinetic Energy (k)
Dissipation Rate (ϵ)
0.00070686 m2
1.41726 m/s
42314
0.0021 m
1
𝐼 = 0.16 ∗ 𝑅𝑒𝐷 8
ℎ
=
0.03 m
4.22483 %
1
0.16 ∗ 42314−8
3
2
𝑘 = (𝑢𝑎𝑣𝑔 ∗ 𝐼)
2
2
3
𝑚
= (1.41726 ∗ 0.0422483)
2
𝑠
3/2
3/4 k
ε = C𝜇
𝑙
0.00537853/2
= 0.093/4
0.0021
14
0.0053785 m2/s2
0.030859 m2/s3
2.6
TWO-PHASE FLOW
Fluid flows that contain two or more components are 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 the
same 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 flows involving
water and steam.
Depending on the volume fraction of each component in the two-phase flow,
different flow patterns can exist. Understanding the flow pattern of the two-phase flow
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 Figure 2.6-1 [11].
Figure 2.6-1: Flow Regimes
The flow patterns shown in Figure 2.6-1 can be 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 distributed
15
in the liquid phase in the form of bubbles. This flow pattern occurs at low gas volume
fractions. Subcooled 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, 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). Flow
film boiling occurs when the heat flux is relatively large compared to the mass flux.
Film boiling is discussed further in Section 2.7.
As stated previously, knowing the flow pattern is important to determine the
pressure drop and heat transfer rate within a system. The flow pattern changes as a
function of gas volume fraction and flow velocity. The flow pattern of a system can be
determined using the Baker flow criteria shown in Figure 2.6-2 [11].
Figure 2.6-2: Baker Flow Pattern
16
2.6.1
MODELING TWO-PHASE FLOWS
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 demand.
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 the marker-and-cell method and quickly became popular due to its low
computer storage requirements. Typical applications of VOF include stratified or freesurface 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 in between the VOF and Eulerian models both in
complexity and computational expense. The Mixture multiphase 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 singlefluid 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 velocities.
The mixture model is a good substitute for the full Eulerian multiphase 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.
17
Typical applications include
sedimentation, cyclone separators, 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 momentum and continuity equations for
each of the phases, and the equations are coupled through pressure and exchange
coefficients. With the Eulerian multiphase model, the number of secondary phases is
limited only by memory requirements and convergence behavior.
The Eulerian
multiphase 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
Many industrial fluid flow applications including subcooled boiling involve a
secondary phase with a size distribution. The size distribution of particles may include
solid particles, bubbles, or droplets that evolve in a multiphase system. Thus, in
multiphase flows involving a size distribution, a balance equation is required to describe
the changes in the particle size distribution, in addition to momentum, mass, and energy
balances. This balance is generally referred to as the population balance.
To make use of this modeling concept, a number density function is introduced
to account for the different sizes in the particle population. With the aid of particle
properties (for example, particle size, 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 boiling model
created in Section 3.5.2.
However, the population balance model is utilized in
Section 3.4.3 to track bubble size distribution within a bubble column.
18
2.7
BOILING HEAT TRANSFER
Boiling is defined as 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 temperature which is made possible due to the large
amount of energy required to cause a phase change. This is essential for industrial
cooling applications, such as nuclear reactors and fossil boilers. Due to 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 pool boiling.
If heat addition causes a phase change in a moving fluid then it is
flow boiling. Both types of boiling heat transfer can be separated into four regimes
which are shown in Figure 2.7-1 [12].
Figure 2.7-1: 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
5/4
bulk fluid by natural convection. The heat transfer rate is proportional to 𝛥𝑇𝑠𝑎𝑡 [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 forming
bubbles at the nucleation sites. When the liquid evaporates, a significant amount of
19
energy is removed from the heated surface due to the latent heat of the vaporization.
Vaporization also increases the convective heat transfer by mixing the liquid water near
the heated surface. There are two subregimes of nucleate boiling. The first subregime is
when local boiling occurs in a subcooled liquid (subcooled boiling). In this situation,
bubbles form on the 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, the
bubbles do not condense after detaching from the heated surface since the liquid is the
same temperature as the steam. It is possible for both subregimes to take place between
points A and C. 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 heat transfer boiling regime. [11]
As the heated surface increases in temperature, more and more nucleation sites
become active. As more bubbles form at the nucleation sites, they begin to merge
together and form columns or slugs of gas, 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 Figure 2.7-1 to decrease until a maximum is reached (point C). Point C is
referred to as the critical heat flux. When the critical heat flux is reached, the vapor
begins to form an insulating blanket around the heated surface which dramatically
increases the surface temperature. 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]
20
3. HEAT TRANSFER AND FLUID FLOW: MODELING
3.1
NATURAL CONVECTION
Two natural convection scenarios were examined in this section. 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 an elevated constant surface temperature submerged in an
infinite pool of liquid was analyzed in this section. 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
important boundary conditions used to model the horizontal cylinder. The top and
bottom walls of the rectangle represent inlet and outlet pressure boundaries with pressure
conditions set such that the fluid is stagnant until heated by the cylinder. The left and
right walls of the rectangle are slip walls to more accurately model an infinite pool. See
Table 3.1.1-1 for a more detailed list of input parameters used in this section.
Figure 3.1.1-1: Heated Cylinder Schematic
21
Table 3.1.1-1: Horizontal Cylinder Model Input
Input
Geometry
Cylinder Diameter
Pool Height
Pool Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Density
Initial Conditions
Cylinder Surface Temperature
Initial Fluid Temperature
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
Transient Formulation
Value
0.02 m
0.28 m
0.24 m
Planar
Transient
0.05 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Boussineq
310 K
300 K
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.1.1-2
PISO
Least Square Cell Based
PRESTO!
Second Order Upwind
Second Order Upwind
Second Order Implicit
Table 3.1.1-2: Horizontal Cylinder Model Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
22
Figure 3.1.1-2 shows the liquid temperature field after 20 seconds of heating. As
the temperature increases, the fluid begins to rise due to buoyancy forces.
Figure 3.1.1-2: Temperature (K)
Figure 3.1.1-3 indicates 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).
Figure 3.1.1-3: Density (kg/m3)
23
As the warm fluid rises, it loses energy to the surrounding bulk fluid which
causes the buoyancy driving head to diminish and the warm fluid climbs more slowly
until it eventually stops. When it reaches its maximum elevation, it is pushed to the left
or right by the fluid travelling upwards below it and fluid recently pushed aside 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 indefinitely 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 cylinder.
Figure 3.1.1-4 is a velocity vector plot that displays how the liquid moves within
the control volume. The cycle of energy absorption and replacement around the cylinder
and the two convection cells above the cylinder are visible in this figure.
Figure 3.1.1-4: 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 constant temperature.
24
(a)
(b)
Figure 3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder (K)
(a) From Eckert [13] (b) Isotherms From Fluent
The model of a horizontal cylinder submerged in an infinite pool was in
qualitative agreement to experimental data. 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 away from one another as they get farther from the heated surface.
Quantitative experimental data from Ingham [14] was compared to the Fluent
results to provide model validation. Figure 3.1.1-6, Figure 3.1.1-7 and Figure 3.1.1-8
show 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 = (T’ – T0) / (Twall – T0) where T’ is the actual
fluid temperature, T0 is the bulk fluid temperature and Twall is the wall temperature.
Dimensionless time is t = t’ * (βgΔT/a)1/2 where t’ is real time, ΔT is (Twall – T0), β is the
coefficient of thermal expansion and a is the diameter of the cylinder.
The heated horizontal cylinder model developed in Fluent showed good
agreement compared to experimental data at the three different radial locations. This
comparison provided confidence that the information obtained from the Fluent model
was accurate.
25
(a)
(b)
Figure 3.1.1-6: Temperature at θ = 30° Vs. Radial Distance
(a) From Ingham [14] and (b) From Fluent
(a)
(b)
Figure 3.1.1-7: Temperature at θ = 90° Vs. Radial Distance
(a) From Ingham [14] and (b) From Fluent
26
(a)
(b)
Figure 3.1.1-8: Temperature at θ = 180° Vs. Radial Distance
(a) From Ingham [14] and (b) From Fluent
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
(“Analysis Value” in Table 3.1.1-3) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.1.1-3). The results from the mesh validation
shown in Table 3.1.1-3 prove that the results are mesh independent.
Table 3.1.1-3: Mesh Validation for Horizontal Cylinder Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (°F)
Min Density (kg/m3)
Analysis Value
19716
38688
0.01627
309.9239
993.1765
27
Mesh Validation
23636
46400
0.01621
309.9531
993.1625
Difference
19.88 %
19.93 %
-0.37 %
0.01 %
0.00 %
3.1.2
VERTICAL PLATE
A vertical plate with an elevated constant surface temperature submerged in an
infinite pool of liquid was analyzed in this section. Energy passed from the slightly
warmer plate to the fluid causing its temperature to increase and the fluid to rise. Figure
3.1.2-1 shows a schematic representation of the geometry and important boundary
conditions used to model the vertical plate. Although the top and bottom walls of the
rectangle represent inlet and outlet pressure boundaries, the fluid is stagnant until heated
by the plate. The left and right walls of the rectangle are slip walls to more accurately
model an infinite pool. See Table 3.1.2-1 for a more detailed list of input parameters
used in this section.
Figure 3.1.2-1: Vertical Plate Schematic
Figure 3.1.2-2 presents the liquid temperature field after 20 seconds. 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 Figure 3.1.2-2 shows the growth of the
thermal boundary layer. The thermal boundary layer is relatively small at the bottom of
the plate because there has been little heat addition but it grows (teal color expands away
from the plate) as the fluid reaches the top of the plate.
28
Table 3.1.2-1: Vertical Plate Model Input
Input
Geometry
Plate Height
Plate Width
Pool Height
Pool Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Density
Initial Conditions
Plate Surface Temperature
Initial Fluid Temperature
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
Transient Formulation
Value
0.18 m
0.01 m
0.20 m
0.13 m
Planar
Transient
0.05 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Boussineq
310 K
300 K
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.1.2-2
PISO
Least Square Cell Based
PRESTO!
Second Order Upwind
Second Order Upwind
Second Order Implicit
Table 3.1.2-2: Vertical Plate Model Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
29
Figure 3.1.2-2: Temperature (K)
Figure 3.1.2-3 shows the fluid velocity in vector form.
The growth of the
momentum boundary layer is more visible in this figure (the teal colored arrows expand
away from the plate). The figure shows that the velocity is primarily vertical with a
magnitude that increases with elevation. The increase in fluid velocity is caused by
longer contact time with the heated surface which causes a greater temperature gradient
and therefore a larger buoyancy force.
Figure 3.1.2-3: Velocity Vectors (m/s)
30
Comparing Figure 3.1.2-3 (vertical plate velocity vectors) with Figure 3.1.1-4
(horizontal cylinder 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 was giving 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
constant temperature.
(a)
(b)
Figure 3.1.2-4: Interference Fringes Around a Heated Vertical Plate (K)
(a) From Eckert [13] and (b) From Fluent
31
The model of a vertical plate submerged in an infinite pool was in qualitative
agreement to experimental data. 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
away from one another as they get farther from the heated surface.
Experimental data from Ostrach [15] was compared to the Fluent results to assess
the quantitative accuracy of the model.
Figure 3.1.2-5 and Figure 3.1.2-6 show a
comparison of dimensionless temperature versus dimensionless distance for five
different Prandtl numbers. Figure 3.1.2-5a shows theoretical values and Figure 3.1.2-5b
compares some of the theoretical values to experimental data.
Dimensionless
temperature is H(η) = (T – T∞) / (T0 – T∞) where T is the actual fluid temperature, T∞ is
the bulk fluid temperature and T0 is the wall temperature. Dimensionless distance is
η = (Y / X) * (Grx / 4)1/4 where Grx is the Grashof number, Y is the vertical height and X
is the distance from the plate.
The information contained in Figure 3.1.2-6 was
calculated by Fluent.
(a)
(b)
Figure 3.1.2-5: Dimensionless Temperature as a Function of Prandtl Number
(a) Theoretical Values and (b) Experimental Values [15]
32
Figure 3.1.2-6: Dimensionless Temperature as a Function of Prandtl Number
The heated vertical plate model developed in Fluent produced similar
temperature results to the experimental data for five different Prandtl numbers. This
comparison provided confidence that the information obtained from the Fluent 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
(“Analysis Value” in Table 3.1.2-3) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.1.2-3). The results from the mesh validation
shown in Table 3.1.2-3 prove that the results are mesh independent.
Table 3.1.2-3: Mesh Validation for Vertical Plate Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (°F)
Min Density (kg/m3)
Analysis Value
12310
23572
0.01376
309.8089
993.2319
33
Mesh Validation
18081
35168
0.01380
309.7991
993.2365
Difference
46.88 %
49.19 %
0.29 %
0.00 %
0.00 %
3.2
LAMINAR FLOW
A simple axisymmetric laminar flow model was developed in this section.
Figure 3.2-1 shows a schematic representation of the geometry and important 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 wall 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 Table 3.1.1-1 for a more detailed
list of input parameters used in this section.
Figure 3.2-1: Laminar Flow Schematic
Based upon the selected initial conditions, the Reynolds number is 352, which is
well within the laminar regime. A noteworthy characteristics of laminar flow is the
parabolic shape of its velocity profile. Figure 3.2-2 displays the velocity magnitude
versus position (distance from the pipe centerline) at different lengths from the pipe
entrance.
Figure 3.2-2: Velocity Profile
34
For example, “line-10cm” is the velocity profile 10 cm from the pipe entrance. 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.
Table 3.2-1: Laminar Flow Model Input
Input
Value
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Initial Conditions
Pipe Wall Surface Temperature
Fluid Inlet Temperature
Fluid Inlet Velocity
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
0.03 m
0.50 m
Axisymmetric
Steady
Pressure Based
Relative
-9.8 m/s2 (X-direction)
Active
Laminar
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.2-2
305 K
300 K
0.05 m/s
Coupled
Least Square Cell Based
Second Order
Second Order Upwind
Second Order Upwind
Table 3.2-2: Laminar Flow Model Water Density
Density (kg/m3)
999.9
994.1
Temperature (K)
273
308
35
Another characteristic of laminar flow is the lack of mixing that occurs within the
fluid as it travels through the pipe. The radial velocity within the pipe is basically zero
and each fluid molecule or atom remains about the same distance from the centerline as
it travels through the pipe. Figure 3.2-3 shows 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 pipe
boundary conditions and entrance effects but these have a negligible impact on system as
a whole.
Figure 3.2-3: Radial Velocity (m/s)
Figure 3.2-4 shows the temperature profile for the laminar flow analyzed.
Because there is little to no radial velocity, diffusion and conduction are the primary
forms of heat transfer which causes the growth of the thermal boundary layer to be very
slow. The growth of the thermal boundary layer is shown in Figure 3.2-4 by the
expansion of the teal colored region.
Figure 3.2-4: Temperature (K)
As in natural convection, laminar flow generates a momentum boundary layer
but its development is not visible pictorially. The momentum boundary layer is created
by drag, or shear, forces created by the wall. Figure 3.2-5 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.
36
Figure 3.2-5: Wall Shear Stress
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
(“Analysis Value” in Table 3.2-3) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.2-3). The results from the mesh validation
shown in Table 3.2-3 prove that the results are mesh independent.
Table 3.2-3: Mesh Validation for Laminar Flow Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Min Radial Velocity (m/s)
Max Dynamic Pressure (Pa)
Max Temperature (K)
Analysis
Value
26320
25353
0.079561
-0.003293
3.15925
304.6503
37
Mesh
Validation
31000
29970
0.079507
-0.003528
3.155022
304.6855
Difference
17.78 %
18.21 %
-0.07 %
7.12 %
-0.13 %
0.01 %
3.3
TURBULENT FLOW
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER
A simple axisymmetric turbulent flow model was developed in this section.
Figure 3.3.1-1 shows a schematic representation of the geometry and important
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
wall 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 Table 3.3.1-1 for a more detailed list of input parameters used in this section.
Figure 3.3.1-1: Turbulent Flow Without Heat Transfer Schematic
Based upon the selected initial conditions, the Reynolds number is 42314, which
is well within the turbulent regime. Figure 3.3.1-2 shows the velocity magnitude versus
position (distance from the pipe centerline) at different distances from the pipe entrance.
Figure 3.3.1-2: Velocity Magnitude
38
The velocity profile of turbulent flow differs significantly in two ways compared
to the velocity profile of laminar flow (Section 3.2). First, turbulent flow velocity
profiles are much flatter. Therefore, the fluid velocity doesn’t 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. Figure 3.3.1-2 (turbulent flow) shows that flow reached a steady profile about
10 cm from the pipe entrance. Figure 3.2-2 (laminar flow) shows that flow reached a
steady profile about 45 cm from the pipe entrance.
This qualitatively matches
experimental data well.
Table 3.3.1-1: 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
Velocity Formulation
Relative
Gravity
-9.8 m/s2 (X-direction)
Models
Energy
Inactive
Viscous
Realizable k-ε
Turbulence Model
Near Wall Treatment
Enhanced
Turbulent Intensity
4.22483 % *
Initial Conditions
Fluid Mass Flow Rate
1.0 kg/s
Material Properties (Water)
Density
998.2 kg/m3
Viscosity
0.001003 kg/m-s
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Pressure
Second Order
Momentum
Second Order Upwind
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
* Calculation shown in Table 2.5.1-2.
39
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.
Figure 3.3.1-4 shows that that maximum radial
velocity occurs near the pipe entrance. Figure 3.3.1-5 reveals that the greatest reduction
in axial velocity occurs near the pipe entrance which is necessary to conserve
momentum. Since shear stress is related to change in velocity parallel to the wall (axial
velocity), the increase in wall shear stress is reasonable.
Figure 3.3.1-3: Wall Shear Stress
Figure 3.3.1-4: Radial Velocity (m/s)
Figure 3.3.1-5: dAxial-Velocity/dx
40
To further investigate the impact of entrance effects, two additional scenarios
were examined using a mass flow rate of 0.5 kg/s (Figure 3.3.1-6) and a mass flow rate
of 1.5 kg/s (Figure 3.3.1-7).
(a)
(b)
(c)
Figure 3.3.1-6: Flow Results for Mass Flow Rate of 0.5 kg/s
(a) Radial Velocity (m/s) (b) Wall Shear Stress (c) dAxial-Velocity/dx
(a)
(b)
(c)
Figure 3.3.1-7: Flow Results for Larger Mass Flow Rate of 1.5 kg/s
(a) Radial Velocity (m/s) (b) Wall Shear Stress (c) dAxial-Velocity/dx
41
Figures 3.3.1-6 and 3.3.1-7 show that wall shear stress and 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.
Figure 3.3.1-8 and Figure 3.3.1-9 show the turbulent kinetic energy and the
production of turbulent kinetic energy as a function of distance, respectively.
Figure 3.3.1-8: Turbulent Kinetic Energy (m2/s2)
Figure 3.3.1-9: Production of Turbulent Kinetic Energy
Most of the turbulent kinetic energy is located near the pipe wall due to shear
stress. The trend of Figure 3.3.1-9 is similar to that of Figure 3.3.1-3 because shear
stress, created by the wall, produces turbulent kinetic energy.
A mesh validation was not performed for this model directly.
The mesh
accuracy is proven adequate in Section 3.3.2 which utilizes the same model with the
addition of energy transfer from the pipe walls to the fluid.
42
3.3.2
TURBULENT FLOW WITH HEAT TRANSFER
The turbulent flow model described in Section 3.3.1 was modified to include heat
transfer from the pipe wall to the fluid. Figure 3.3.2-1 shows a schematic representation
of the geometry and important boundary conditions used to model turbulent flow within
a pipe with 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 wall with a constant heat flux 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 Table 3.3.2-1 for a more detailed
list of input parameters used in this section.
Figure 3.3.2-1: Turbulent Flow With Heat Transfer Schematic
Figure 3.3.2-2 displays the fluid temperature change caused by energy addition
from the pipe walls. The radial temperature distribution in Figure 3.3.2-2 is more evenly
distributed than the radial temperature distribution in Figure 3.2-4 (laminar flow).
Uniform temperature distribution is a characteristic of turbulent flow and made possible
by the chaotic nature of the flow regime.
Figure 3.3.2-2: Temperature (K)
The radial velocity in Figure 3.3.2-3 is very similar to that in Figure 3.3.1-4
which means that the heat addition has a negligible impact on fluid velocity. If the heat
transfer rate to the fluid was increased sufficiently such that flow velocity was impacted,
then the radial velocity between the two scenarios would also differ.
43
Table 3.3.2-1: Turbulent Flow With 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
Velocity Formulation
Relative
Gravity
-9.8 m/s2 (X-direction)
Models
Energy
Active
Viscous
Realizable k-ε
Turbulence Model
Near Wall Treatment
Enhanced
Turbulent Intensity
4.22483 % *
Initial Conditions
Fluid Mass Flow Rate
1.0 kg/s
Fluid Inlet Temperature
300 K
Wall Heat Flux
450 kW/m2
Material Properties (Water)
Specific Heat
4182 J/kg-K
Thermal Conductivity
0.6 W/m-K
Viscosity
0.001003 kg/m-s
Density
See Table 3.3.2-2
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Pressure
Second Order
Momentum
Second Order Upwind
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
Energy
Second Order Upwind
* Calculation shown in Table 2.5.1-2.
Table 3.3.2-2: Turbulent Flow Model Water Density
Density (kg/m3)
999.9
994.1
974.9
Temperature (K)
273
308
348
44
Figure 3.3.2-3: Radial Velocity (m/s)
Closely comparing the velocity profiles for the two turbulent flow models
(Figure 3.3.1-2 and Figure 3.3.2-4) reveals that the velocity magnitude is slightly larger
for the case with heat transfer. The energy addition causes the density of the fluid to
decrease and the velocity increases slightly in order to maintain a constant mass flow
through the pipe.
Figure 3.3.2-4: Velocity Magnitude
As expected, the wall shear stress shown in Figure 3.3.2-5 is similar to the wall
shear stress shown in Figure 3.3.1-3.
Figure 3.3.2-5: Wall Shear Stress
45
Comparing the velocity magnitude, radial velocity and wall shear stress from
Section 3.3.1 to Section 3.3.2 proves that the addition of heat transfer has a negligible
impact the turbulent flow. This is reasonable since the heat flux is relatively small and
does not create any localized phase change.
Thus, the relationships developed in
Section 3.3.1 (impact mass flow has on shear stress and radial velocity) are applicable to
turbulent flows with heat transfer as long as the heat transfer rate is small.
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
(“Analysis Value” in Table 3.3.2-3) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.3.2-3). The results from the mesh validation
shown in Table 3.3.2-3 prove that the results are mesh independent.
Table 3.3.2-3: Mesh Validation for Turbulence With Heat Transfer Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Temperature (°F)
Min Density (kg/m3)
Max Dynamic Pressure (Pa)
3.4
TWO-PHASE FLOW
3.4.1
GAS MIXING TANK
Analysis
Value
31031
31000
1.502045
317.6659
989.4604
1122.853
Mesh
Validation
35739
34624
1.500343
318.1447
989.2305
1119.909
Difference
15.17 %
11.69 %
-0.11 %
0.15 %
-0.02 %
-0.26 %
In many branches of engineering, gas injection techniques have been extensively
utilized to enhance chemical reaction rates, homogenize temperature and chemical
compositions, and remove impurities. In the steel industry, the advancements made in
mixing have increased the level of control over the steelmaking process which has
improved the quality of steel produced. To mix the molten metal, gas is pumped through
a porous plug located at the bottom of the mixing tank. The porous plug controls the
velocity and bubble diameter of the gas. Buoyancy forces cause the injected gas to
move quickly through the molten metal and drag forces causes mixing. Figure 3.4.1-1
46
shows a schematic representation of the geometry and important boundary conditions
used to model the gas mixing tank. The top line of the rectangle is a pressure outlet and
the left, right and most of the bottom lines of the rectangle represent no slip walls. The
red line on the bottom of the rectangle represents a velocity inlet and is where the air jet
enters the tank to mix the liquid. See Table 3.4.1-1 for a more detailed list of input
parameters used in this section.
Figure 3.4.1-1: Gas Mixing Tank Schematic
Figure 3.4.1-2 shows the gas volume fraction, Figure 3.4.1-3 shows the liquid
vector velocity and Figure 3.4.1-4 shows the gas vector velocity after 5 seconds of gas
injection. Midway through the liquid volume in Figure 3.4.1-2, the air jet begins to
become wavy. The wavy behavior is explained by Rayleigh instability which states that
surface tension tends to minimize surface area between the two phases. Thus, after a
certain distance the air jet will transform into air bubbles with the same volume but less
surface area. The length required for the jet to breakup is dependent upon the gas
velocity and gas / liquid surface tension.
The liquid and gas velocities shown in Figure 3.4.1-3 and Figure 3.4.1-4,
respectively, are similar in trend and magnitude which indicates that the drag force
between the two phases is strong. The maximum gas velocity is much greater than the
inlet velocity (0.5 m/s); therefore, buoyancy forces are significant. Figure 3.4.1-3 shows
that there is a number of small eddies created by the injected gas which provide a
significant amount of mixing within the liquid.
47
Table 3.4.1-1: Gas Mixing Tank Model Input
Input
Geometry
Tank Width
Tank Height
Porous Plug Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Turbulence Model
Near Wall Treatment
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Initial Conditions
Water Level
Gas Velocity
Bubble Diameter
Material Properties (Water)
Density
Viscosity
Material Properties (Air)
Density
Viscosity
Surface Tension
Solution Methods
Scheme
Gradient
Momentum
Volume Fraction
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Transient Formulation
48
Value
0.30 m
0.60 m
0.02 m
Planar
Transient
0.001 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Inactive
Standard k-ε
Eulerian
Schiller-Nauman
Standard
0 m2/s2
0 m2/s3
0.40 m
0.5 m/s
0.001 m
998.2 kg/m3
0.001003 kg/m-s
1.225 kg/m3
1.7894E-05 kg/m-s
0.072 N/m
Phase Coupled SIMPLE
Least Square Cell Based
Second Order Upwind
QUICK
Second Order Upwind
Second Order Upwind
Second Order Implicit
Figure 3.4.1-2: Gas Volume Fraction
Figure 3.4.1-3: Liquid Velocity Vectors (m/s)
49
Figure 3.4.1-4: Gas Velocity Vectors (m/s)
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
(“Analysis Value” in Table 3.4.1-2) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.4.1-2). The results from the mesh validation
shown in Table 3.4.1-2 prove that the results are mesh independent.
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Max Static Pressure (psia)
Max Liquid Total Pressure (psia)
Max Liquid Volume Fraction
Analysis
Value
30625
30256
1.539086
2.046923
3925.424
4775.512
1.000000
50
Mesh
Validation
36045
35644
1.453488
2.086285
3894.616
4732.633
1.000000
Difference
17.70 %
17.81 %
-5.56 %
1.92 %
-0.78 %
-0.90 %
0.00 %
3.4.2
BUBBLE COLUMN
A bubble column reactor is an apparatus primarily used to study gas-
liquid reactions.
This apparatus is a vertical column of liquid with gas introduced
continuously at the bottom through a sparger. The bubble column contains gas dispersed
as bubbles in a continuous volume of liquid. Per Section 2.6, the flow is considered to
be bubbly. Bubbles form and travel upwards through the column due to the inlet gas
velocity and buoyancy.
The gas introduced through the sparger provides mixing,
similar to the gas mixing tank in Section 3.4.1 but much less intense. This method of
mixing is less invasive and requires less energy than mechanical stirring.
Bubble
column reactors are often used in industry to develop and produce chemicals and fuels
for use in chemical, biotechnology, and pharmaceutical processes. Figure 3.4.2-1 shows
a schematic representation of the geometry and important boundary conditions used to
model the bubble column. The top line of the rectangle is a pressure outlet and the left
and right lines of the rectangle represent no slip walls. The bottom line of the rectangle
signifies a velocity inlet and is where the air bubbles enter the column.
Table 3.4.2-1 for a more detailed list of input parameters used in this section.
Figure 3.4.2-1: Bubble Column Schematic
51
See
Table 3.4.2-1: Bubble Column Model Input
Input
Geometry
Column Width
Column Height
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Turbulence Model
Near Wall Treatment
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Initial Conditions
Water Level
Gas Flow Rate
Bubble Diameter
Material Properties (Water)
Density
Viscosity
Material Properties (Air)
Density
Viscosity
Surface Tension
Solution Methods
Scheme
Gradient
Momentum
Volume Fraction
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Transient Formulation
52
Value
0.10 m
0.75 m
Planar
Transient
0.001 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Inactive
Standard k-ε
Eulerian
Schiller-Nauman
Standard
0 m2/s2
0 m2/s3
0.50 m
0.05 m/s
0.005 m
998.2 kg/m3
0.001003 kg/m-s
1.225 kg/m3
1.7894E-05 kg/m-s
0.072 N/m
Phase Coupled SIMPLE
Least Square Cell Based
Second Order Upwind
QUICK
Second Order Upwind
Second Order Upwind
Second Order Implicit
Figure 3.4.2-2 is a comparison between the gas volume fraction 1 second and
5 seconds after gas has begun flowing through the bubble column. At both time points
the gas flows in slugs. After 5 seconds, the gas reaches the top of the liquid and caused
the surface to change shape.
Compared to the initial liquid level, the level after
5 seconds is about 5 cm higher. The level increase is known as gas holdup and is caused
by phase drag forces and displacement. Figure 3.4.2-2b reveals that most of the gas
travels along the wall in a quasi-annular flow type regime.
(a)
(b)
Figure 3.4.2-2: Gas Volume Fraction
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-3 is a comparison between the liquid velocity vectors 1 second and
5 seconds after the gas has begun flowing through the bubble column. Distinct paths of
liquid movement, primarily along the walls, can be seen at both time points. Due to
buoyancy and phase drag forces, the largest liquid velocities coincide with the regions of
greatest gas volume fraction.
53
(a)
(b)
Figure 3.4.2-3: Liquid Velocity Vectors (m/s)
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-4 is a comparison between the gas velocity vectors 1 second and
5 seconds after gas has begun flowing through the bubble column. The white region
two-thirds up the bubble column in Figure 3.4.2-4a is a region where the gas has not
reached. It is noteworthy that the original gas-liquid interface is not flat but consists of
two parabolas. The two parabolas are created because most of the gas travels close to
the wall (Figure 3.4.2-2). Figure 3.4.2-4b reveals that the greatest gas velocities occur
near the walls which are also the areas of greatest gas volume fraction. Higher gas
volume fractions lead to greater buoyancy forces which cause greater gas velocities.
A second scenario was analyzed to compare the impact that gas inlet velocity has
on gas holdup. This case is the same as the case described in Table 3.4.2-1 except that
the gas inlet velocity is increased to 0.10 m/s. Figure 3.4.2-5 illustrates the gas volume
fraction 1 second and 5 seconds after gas has begun flowing through the bubble column.
Figure 3.4.2-5b reveals that the injected gas causes the water level to rise about 15 cm.
This is a much larger increase than the gas holdup shown in Figure 3.4.2-2b, which
employs a gas inlet velocity of 0.05 m/s and proves that gas holdup is not proportional to
inlet velocity.
54
(a)
(b)
Figure 3.4.2-4: Gas Velocity Vectors (m/s)
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-5: Gas Volume Fraction (0.10 m/s)
After (a) 1 Second and (b) 5 Seconds
55
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed on the scenario with a gas velocity of 0.05 m/s. The mesh validation
compared the results shown in this section (“Analysis Value” in Table 3.4.2-2) to a
second case with an increased number of finite volumes (“Mesh Validation” in
Table 3.4.2-2). The results from the mesh validation shown in Table 3.4.2-2 prove that
the results are mesh independent.
Table 3.4.2-2: Mesh Validation for Bubble Column Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Average Gas Velocity (m/s)
Max Liquid Volume Fraction
Max Static Pressure (Pa)
3.4.3
Analysis
Value
7006
6750
0.625945
0.313947
0.998733
4929.094
Mesh
Validation
8785
8500
0.63157
0.308535
1.00000
4920.58
Difference
25.39 %
25.93 %
0.90 %
1.72 %
0.13 %
-0.17 %
BUBBLE COLUMN WITH POPULATION BALANCE MODEL
The bubble column model created in Section 3.4.2 was expanded to include a
population balance model (PBM) with three discrete bubble sizes so that bubble swarm
could be tracked. In all gas-liquid flows, the bubbles can increase or decrease in size
due to coalescence or breakup. Coalescence occurs when two or more bubbles collide
and the liquid barrier between them ruptures to form a larger bubble. Bubbles breakup
occurs when a bubble collides with a turbulent eddy approximately equal to its size.
Table 3.4.3-1 lists the input used to create the PBM implemented in this section.
Figure 3.4.3-1 is a comparison between the gas volume fraction at 1 second and
5 seconds after gas has begun flowing through the bubble column. When comparing
Figure 3.4.3-1 to Figure 3.4.2-2, there are noticeable differences. One of the obvious
differences between the two figures is the distribution of the gas phase at the two time
points. With the population balance model implemented (Figure 3.4.3-1), the gas phase
distribution is more uniform and does not contain any areas with large gas volume
fractions. This is most noticeable at the bottom of the bubble column after 5 seconds.
56
Table 3.4.3-1: Population Balance Model Input
Input
Method
Number of Bins
Bin-0
Bin-1
Bin-2
Bin Distribution
Bin-0
Bin-1
Bin-2
Aggregation Kernel
Model
Surface Tension
Breakage Kernel
Model
Surface Tension
Formulation
Value
Discrete
3
0.0075595 m
0.0047622 m
0.0030000 m
25 %
50 %
25 %
Luo
0.072 N/m
Luo
0.072 N/m
Hagesather
(a)
(b)
Figure 3.4.3-1: Gas Volume Fraction with PBM
After (a) 1 Second and (b) 5 Seconds
57
Figure 3.4.3-2 is a comparison between the liquid velocity vectors at 1 second
and 5 seconds after gas has begun flowing through the bubble column. Figure 3.4.3-2b
reveals that the liquid velocity increases as elevation increases. This is less noticeable in
Figure 3.4.2-3 which displays a more uniform overall liquid velocity.
Table 3.4.3-1
shows that there is an increase in the number of large bubbles at the outlet compared to
the inlet in Figure 3.4.3-2b. The larger bubbles attain higher velocities due to greater
buoyancy forces which in turn increases the liquid velocity due to drag between the two
phases. The velocity gradient in Figure 3.4.2-3 is more uniform because the bubbles do
not coalesce and therefore remain at a constant diameter.
(a)
(b)
Figure 3.4.3-2: Liquid Velocity Vectors with PBM (m/s)
After (a) 1 Second and (b) 5 Seconds
The population balance model calculates the bubble size distribution at each axial
height using the Luo breakup and coalescence model. Table 3.4.3-2 shows the bubble
size population distribution at the inlet and outlet of the bubble column. This table
shows that there is a strong bias for the smaller bubbles to coalesce into larger bubbles;
thus, surface tension is a strong driver to reduce surface area and there is very little
turbulence within the column to cause the bubbles to break apart.
58
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.072 N/m
Inlet
(Fraction)
0.250
0.500
0.250
Bin-0 (0.0076 m)
Bin-1 (0.0048 m)
Bin-2 (0.0030 m)
Outlet
(Fraction)
0.865
0.117
0.018
Net
(Fraction)
+0.615
-0.383
-0.232
Figure 3.4.3-3 is a comparison between the gas velocity vectors at 1 second and
5 seconds after gas has begun flowing through the bubble column.
Similar to
Figure 3.4.2-4, the shape of the gas as it initially climbs the bubble column is made up of
two adjacent parabolas; however, it is much more severe in Figure 3.4.3-3a.
Figure 3.4.3-3b shows a uniform gas velocity distribution throughout the entire bubble
column. This is different from Figure 3.4.2-4b which illustrated large gas velocities
along the wall and smaller gas velocities in the center.
(a)
(b)
Figure 3.4.3-3: Liquid Velocity Vectors with PBM (m/s)
After (a) 1 Second and (b) 5 Seconds
59
The impact that surface tension has on bubble size distribution was evaluated by
reducing the surface tension by a factor of ten to 0.0072 N/m. Table 3.4.3-3 displays the
bubble size distribution at the inlet and outlet of the bubble column with the reduced
surface tension. The smaller surface tension decreases the driving force for bubbles to
coalesce and significantly reduces the average bubble diameter.
Table 3.4.3-3: Bubble Size Distribution – Surface Tension of 0.0072 N/m
Bin-0 (0.0076 m)
Bin-1 (0.0048 m)
Bin-2 (0.0030 m)
3.5
BOILING FLOWS
3.5.1
POOL BOILING
Inlet
(Fraction)
0.250
0.500
0.250
Outlet
(Fraction)
0.495
0.335
0.170
Net
(Fraction)
+0.245
-0.165
-0.080
Pool boiling occurs when a liquid transforms to a vapor due to energy absorption
in a fluid that is stagnant.
When the surface temperature of the heated surface
sufficiently exceeds the saturation temperature of the liquid, vapor bubbles nucleate on
the heated surface. The bubbles grow on the surface until they detach and move out into
the bulk liquid. While rising is the result of buoyancy, the bubbles either collapse or
continue to grow depending upon whether the liquid is locally subcooled or saturated.
Pool boiling involves complex fluid motions initiated and maintained by the nucleation,
growth, departure and collapse of bubbles, and by natural convection. [11]
Figure 3.5.1-1 shows a schematic representation of the geometry and important
boundary conditions used to model pool boiling. The top line of the rectangle is a
pressure outlet and the bottom wall of the rectangle is the heated surface. The left and
right lines of the rectangle represent no slip walls. See Table 3.5.1-1 for a more detailed
list of input parameters used in this section.
60
Figure 3.5.1-1: Pool Boiling Schematic
Table 3.5.1-1: Pool Boiling Model Input
Input
Value
Geometry
Column Width
Column Height
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Slip
Mass Transfer
Initial Conditions
Bubble Diameter
Initial Fluid Temperature
Heater Temperature (Bottom)
Backflow Temperature (Top)
Backflow Volume Fraction (Top)
61
0.01 m
0.05 m
Planar
Transient
0.002 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Mixture
Schiller-Nauman
Manninen et al.
Evaporation-Condensation
0.0002 m
372 K
383 K
373 K
0
Material Properties (Water) [16]
Density
Specific Heat
Thermal Conductivity
Viscosity
Heat of Vaporization
Material Properties (Vapor) [16]
Density
Specific Heat
Viscosity
Thermal Conductivity
Surface Tension
Solution Methods
Scheme
Gradient
Pressure
Momentum
Volume Fraction
Energy
Transient Formulation
See Table 3.5.1-2
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
2.418379E+08 J/kgmol
0.5542 kg/m3
2014 J/kg-K
1.34E-05 kg/m-s
0.0261 W/m-K
0.072 N/m
PISO
Least Square Cell Based
Body Force Weighted
Second Order Upwind
QUICK
Second Order Upwind
Second Order Implicit
Table 3.5.1-2: Pool Boiling Model Water Density
Density (kg/m3)
974.9
958.4
Temperature (K)
348
373.15
Figure 3.5.1-2 displays the instantaneous gas volume fraction after 0.9 seconds
and 1.7 seconds of heating. The first time point was chosen because it shows steam
releasing from the heated surface and entering the bulk fluid which is the driving force
behind all fluid motion. The second time point was chosen because it reveals the
interaction between the liquid and vapor at a high level.
The evolution of the steam generation, upward movement (due to buoyancy) and
liquid refill is illustrated in Figure 3.5.1-2 through Figure 3.5.1-4. Figure 3.5.1-2a
indicates that the bottom of the control volume is heated and some vapor has formed
(two areas of significant steam generation are in green). Figure 3.5.1-2b shows that the
vapor has moved upward (teal region) and that the liquid has moved downward to take
its place (blue area at the bottom).
62
(a)
(b)
Figure 3.5.1-2: Instantaneous Gas Volume Fraction
After (a) 0.9 Seconds and (b) 1.7 Seconds
Figure 3.5.1-3 and Figure 3.5.1-4 display the liquid and gas velocities,
respectively, at the two time points. Comparing these two figures indicates that the
largest upward liquid and vapor velocities occur in generally the same regions. These
regions also coincide with the areas of largest gas volume fraction (Figure 3.5.1-2). As
vapor forms on the heated surface, it eventually detaches and enters the liquid above.
Due to buoyancy, the vapor travels upward through the liquid. Drag forces between the
two phases cause the liquid to also travel upwards but at a slower rate due to slip. Other
areas of high liquid velocity occur between the two swells of upward moving vapor and
along the walls. The liquid being of greater density flows downward to refill the void
created by the steam.
63
(a)
(b)
Figure 3.5.1-3: Liquid Velocity Vectors (m/s)
After (a) 0.9 Seconds and (b) 1.7 Seconds
(a)
(b)
Figure 3.5.1-4: Gas Velocity Vectors (m/s)
After (a) 0.9 Seconds and (b) 1.7 Seconds
64
Figure 3.5.1-5 shows the volume fraction of vapor on the heated surface after
2 seconds. The figure illustrates that vapor is produced significantly at two locations
(vapor volume fraction is at a maximum), 0.0008 m and 0.0095 m. In this situation 0.00
m is the left wall and 0.01 m is the right wall. The vapor volume fraction is at a
minimum at approximately 0.005 m which is where liquid is taking the place of the
recently created vapor.
Figure 3.5.1-5: Volume Fraction of Vapor on Heated Surface
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
(“Analysis Value” in Table 3.5.1-3) to a second case with an increased number of finite
volumes (“Mesh Validation” in Table 3.5.1-3). The results from the mesh validation
shown in Table 3.5.1-3 prove that the results are mesh independent.
Table 3.5.1-3: Mesh Validation for Pool Boiling Model
Number of Nodes
Number of Elements
Min Mixture Density (kg/m3)
Max Mixture Velocity (m/s)
Min Liquid Volume Fraction
Max Static Pressure (Pa)
Max Phase Transfer (kg/m3-s)
Analysis
Value
26645
26208
754.389
0.059396
0.787011
452.2354
2.169675
65
Mesh
Validation
32481
32000
742.115
0.062788
0.774197
452.2388
2.190905
Difference
21.90%
22.10%
-1.63%
5.71%
-1.63%
0.00%
0.98%
3.5.2
SUBCOOLED FLOW BOILING
Subcooled flow boiling occurs when a moving, under-saturated fluid comes in
contact with a surface that is hotter than its saturation temperature. It involves intense
interactions between the liquid and vapor phases and therefore modeling can be a
challenge. The Eulerian multiphase model is most appropriate for subcooled boiling
because it is capable of modeling multiple separate, yet interacting phases.
When
modeling subcooled boiling, there are three parameters of great importance. These
parameters are active nucleation site density (Na), departing bubble diameter (dbw) and
bubble departure frequency (f) [1].
As discussed previously, nucleation sites are preferential locations where vapor
tends to form. They are usually cavities or irregularities in a heated surface. However,
not all sites are active and the number of 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]:
𝑁𝑎 = [𝑚(𝑇𝑠𝑎𝑡 − 𝑇𝑤 )]𝑛
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 system 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 and can be
written as [1]:
𝑁𝑎 =
1
[
−4.4
2𝜎𝑇𝑠𝑎𝑡
2
𝑑𝑏𝑤
∆𝑇𝑒𝑓𝑓 𝜌𝑔 ℎ𝑓𝑔
]
𝑓(𝜌∗ )
𝑓(𝜌∗ ) = 2.157 ∗ 10−7 ∗ 𝜌∗−3.2 ∗ (1 + 0.0049𝜌∗ )
𝜌∗ = (
𝜌𝑙 −𝜌𝑔
66
𝜌𝑔
)
where
dbw is the lift off bubble diameter
σ is the surface tension
Tsat is the fluid saturation temperature
ΔTeff = SΔTw
ΔTw = Tsat - Twall
S is the suppression factor
ρl is the liquid density
ρg is the gas density
hfg is the latent heat of vaporization
The departing bubble diameter is the bubble size when it leaves the heated
surface and 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]:
𝑑𝑏𝑤 = 𝑚𝑖𝑛 [0.0006 ∗ exp (−
𝑇𝑠𝑢𝑏
45
) ; 0.00014]
The second correlation was created on the basis of the balance between the
buoyancy and surface tension forces at the heated surface. Kocamustafaogullari and
Ishii modified an expression by Fritz that involved the contact angle of the bubble [1]:
𝑑𝑏𝑤 = 2.5 ∗ 10−5 (
𝜌𝑙 − 𝜌𝑔
𝜎
) 𝜃√
𝜌𝑔
𝑔 ∗ (𝜌𝑙 − 𝜌𝑔 )
A third, more comprehensive correlation was proposed by Unal which includes
the effect of subcooling, the convection velocity and the heater wall properties [1]:
𝑑𝑏𝑤 =
2.42 ∗ 10−5 ∗ 𝑝0.709 ∗ 𝑎
√𝑏𝛷
67
Where
𝑎=
(𝑄𝑤 − ℎ𝑇𝑠𝑢𝑏 )1/3 𝑘𝑙
2𝐶 1/3 ℎ𝑓𝑔 √𝜋𝑘𝑙 ⁄𝜌𝑙 𝑐𝑝𝑙 𝜌𝑔
𝐶=
√
𝑘𝑤 𝜌𝑤 𝑐𝑝𝑤
𝑇𝑠𝑢𝑏
;𝑏 =
𝑘𝑙 𝜌𝑙 𝑐𝑝𝑙
2[1 − (𝜌𝑔 − 𝜌𝑙 )]
ℎ𝑓𝑔 𝜇𝑙 [𝑐𝑝𝑙 ⁄(0.013ℎ𝑓𝑔 𝑃𝑟 1.7 )]
3
𝜎
√𝑔(𝜌 − 𝜌 )
𝑙
𝑔
(𝑢𝑙 ) 0.47
𝑓𝑜𝑟 𝑢𝑙 ≥ 0.61 𝑚/𝑠
Φ = {0.61
1.0
𝑓𝑜𝑟 𝑢𝑙 < 0.61 𝑚/𝑠
The bubble departure frequency is the rate at which bubbles are generated and
detach at an active nucleation site and it is dependent on heat flux and a combination of
buoyancy and drag forces. The most common bubble departure frequency correlation
for computational fluid dynamics was developed by Cole. It is derived from the bubble
departure diameter and a balance between buoyancy and drag forces [1]:
𝑓=√
4𝑔(𝜌𝑙 − 𝜌𝑔 )
3𝜌𝑙 𝑑𝑏𝑤
The heat transfer rate from the wall to the fluid greatly impacts the number of
active nucleation sites, bubble diameter and bubble departure frequency. The amount of
energy transferred 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 can be utilized in Fluent.
Figure 3.5.2-1 shows a schematic representation of the geometry and important
boundary conditions used to model subcooled flow boiling. 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 the heated surface and after the rotation
becomes the pipe wall. The left and right lines of the rectangle are the inlet and outlet
68
areas respectively, which when revolved, are circular. See Table 3.5.2-1 for a more
detailed list of input parameters used in this section.
Figure 3.5.2-1: Subcooled Flow Boiling Model Schematic
Table 3.5.2-1: Subcooled Flow Boiling Model Input
Input
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Near Wall Treatment
Turbulent Intensity
Multiphase
Drag
Lift
Heat Transfer
Mass Transfer
Correlations
Interfacial Area
Bubble Diameter
Initial Conditions
Mass Flow Rate
Inlet Fluid Temperature
Wall Heat Flux
69
Value
0.03 m
0.50 m
Axisymmetric
Steady
Pressure Based
Relative
-9.8 m/s2 (X-direction)
Active
Realizable k-ϵ
Enhanced
4.2079 % *
Eulerian
Schiller-Nauman
Boiling-Moraga
Ranz-Marshall
RPI Boiling
See Table 3.5.2-3
Ia-Symmetric
Sauter-Mean
0.3 kg/s
370 K
90000 W/m2
Material Properties (Water) [16]
Density
See Table 3.5.2-2
Specific Heat
See Table 3.5.2-2
Thermal Conductivity
See Table 3.5.2-2
Viscosity
See Table 3.5.2-2
Heat of Vaporization
See Table 3.5.2-2
Material Properties (Vapor) [16]
Density
0.5542 kg/m3
Viscosity
1.34E-05 kg/m-s
Thermal Conductivity
0.0261 W/m-K
Surface Tension
0.072 N/m
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Momentum
Second Order Upwind
Volume Fraction
QUICK
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
Energy
Second Order Upwind
* Calculated using equations from Table 2.5.1-2.
Table 3.5.2-2: Subcooled Flow Boiling Model Water Properties
368 K
370 K
373.15 K*
Density (kg/m )
961.99
960.59
958.46
Specific Heat (J/kg-K)
4210.0
4212.1
4215.5
Viscosity (kg/m-s)
0.0002978
0.0002914
0.0002822
Conductivity (W/m-K)
0.6773
0.6780
0.6790
Heat of Vaporization (J/kgmol)
N/A
N/A
40622346
Surface Tension (N/m)
N/A
N/A
0.0589
* Saturation temperature at atmospheric pressure (14.7 psia).
3
Boiling Model Study
The impact that each boiling model has on liquid volume fraction was
investigated by analyzing a set of cases that implemented the inputs listed in
Table 3.5.2-1 and Table 3.5.2-3.
Based on the modeling options in Fluent, six
combinations were possible. The liquid volume fraction at different axial locations and
the values of average liquid volume fraction among cases were compared.
70
Table 3.5.2-3: Boiling Model Study Case Input
Case
Number
1
2
3
4
5
6
Bubble Departure
Diameter Model
Tolubinski-Kostanchuk
KocamustafaogullariIshii
Unal
Tolubinski-Kostanchuk
KocamustafaogullariIshii
Unal
Nucleation Site
Density Model
Lemmert-Chawla
Lemmert-Chawla
Frequency of Bubble
Departure Model
Cole
Cole
Lemmert-Chawla
KocamustafaogullariIshii
KocamustafaogullariIshii
KocamustafaogullariIshii
Cole
Cole
Cole
Cole
Plots of temperature, liquid volume fraction and mass transfer rate for Case 1 are
shown in Figures 3.5.2-2, 3.5.2-3 and 3.5.2-4, respectively. Although these figures are
specific to Case 1, their trends can be applied to all of the subcooled flow boiling cases
analyzed. Figure 3.5.2-2 displays how the liquid temperature increases as the fluid
travels down the pipe. The maximum bulk liquid temperature is about 373 K which is
also the fluid saturation temperature.
Figure 3.5.2-2: Case 1 - Temperature (K)
Figure 3.5.2-3 reveals how that the liquid volume fraction decreases as the fluid
travels down the pipe. The large reduction in liquid volume fraction is caused by energy
transfer from the walls and the small amount of liquid subcooling at the pipe entrance.
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction
71
Figure 3.5.2-4 is of particular interest because it shows both the generation and
destruction of steam bubbles. The light blue and teal areas next to the heated wall
depicts that steam is being generated. After the bubbles grow in size they detach and
join the bulk fluid. A small distance towards the pipe centerline away from the heated
wall is a dark blue region.
In this region, the steam bubbles lose energy to the
surrounding subcooled liquid and condense back into liquid.
The generation and
destruction of steam bubbles is characteristic of subcooled flow boiling.
Figure 3.5.2-4: Case 1 - Mass Transfer Rate (kg/m3-s)
The volume-weighted average liquid volume fraction of the entire control
volume for the six cases described in Table 3.5.2-3 is shown in Table 3.5.2-4. Case 4
predicted the largest liquid volume fraction while Case 2 predicted the smallest liquid
volume fraction; however, the difference between the two cases is only about 1.6%.
Therefore, the choice of boiling model seems to have only a small impact on the overall
liquid volume fraction for the conditions examined. The results also show that the
Kocamustafaogullari-Ishii nucleation site density model tends to predict a greater liquid
volume fraction, meaning less vapor production, than the Lemmert-Chawla nucleation
site density model. Cases 4 through 6 have a smaller liquid volume fraction range
(0.9124 to 0.9165) than Cases 1 through 3 (0.9003 to 0.9108). This means that when the
Kocamustafaogullari-Ishii nucleation site density model is employed, the choice of the
bubble departure diameter model has less of an impact on liquid volume fraction than if
the Lemmert-Chawla nucleation site density model is employed. Comparing the results
from a bubble departure diameter model perspective reveals that there is no tendency
any of the three models examined to predict a larger or smaller liquid volume fraction.
Thus, the nucleation site density model has a greater impact on liquid volume fraction
than the bubble departure diameter model.
72
(a) Case 1
(b) Case 2
(c) Case 3
(d) Case 4
(e) Case 5
(f) Case 6
Figure 3.5.2-5: Liquid Volume Faction for Cases 1-6
73
Table 3.5.2-4: Boiling Model Study Case Results
Case Number
1
2
3
4
5
6
Volume-Weighted Liquid Volume Fraction
0.91078539
0.90031346
0.90856631
0.91649488
0.91612881
0.91241595
Figure 3.5.2-5 shows the liquid volume fraction at nine axial heights as a
function of distance from the pipe center for the six cases described in Table 3.5.2-3.
The x-axis is position, or distance from the centerline and the pipe wall is located at
0.015 m. Although Table 3.5.2-4 indicates that the models predict similar liquid volume
fractions within the entire control volume, Figure 3.5.2-5 illustrates that there are
noticeable differences. First, there are significantly higher liquid volume fraction near
the pipe inlet (0 to 10 cm) in Cases 4 through 6 compared to Cases 1 through 3.
Therefore, vapor formation using the Kocamustafaogullari-Ishii nucleation site density
model requires more energy addition. Second, the liquid volume fraction 0.008 m from
the pipe centerline is significantly less in Cases 1 through 3 than in Cases 4 through 6.
This is due to the smaller vapor production rate at the pipe wall in Cases 1 through 3.
Initial Conditions Study
A second parametric study using the subcooled boiling model described in
Table 3.5.2-1 was used to determine how inlet temperature, mass flow and heat flux
impact liquid volume fraction. Six additional cases were analyzed in total as part of this
parametric study. For this set of cases, the active nucleation site density is determined
using the Lemmert and Chwala correlation, the bubble departure diameter is determined
using the Tolubinsky and Kostanchuk correlation and the bubble departure frequency is
determined using the Cole correlation. Case 1 from the boiling model study is used as
the nominal case to which the other six cases are compared. Cases 7 through 12 increase
or decrease the heat flux, the inlet temperature or the mass flow rate relative to the Case
1 value. The input for the cases analyzed is documented in Table 3.5.2-5.
74
Table 3.5.2-5: Initial Conditions Study Case Input
Case Number
Inlet Temperature
(K)
370
370
370
372
368
370
370
1 (base)
7
8
9
10
11
12
Mass Flow
(kg/s)
0.30
0.30
0.30
0.30
0.30
0.33
0.27
Heat Flux
(kW/m2)
90
100
80
90
90
90
90
The volume-weighted average liquid volume fraction of the entire control
volume for the seven cases described in Table 3.5.2-5 is displayed in Table 3.5.2-6.
Table 3.5.2-6 shows that the maximum and minimum liquid volume fractions occur in
Case 9 and Case 10 (inlet temperature variation cases), respectively. The significant
impact that inlet temperature has on liquid volume fraction can be attributed to the large
specific heat of water (4212 J/kg-K). If the specific heat was smaller, the difference in
liquid volume fraction between these two cases and the base case would be less. The
large specific heat value of water is one reason why it is commonly used in energy
conversion cycles.
Comparing the three cases that cause a decrease in liquid volume
fraction from the base case (Cases 7, 9 and 12) to the three cases that cause an increase
in liquid volume fraction from the base case (Cases 8, 10 and 11) demonstrates that the
liquid volume fraction decreases more than it increases for the same delta change in
initial conditions. Changes in initial condition near the saturation point will have a
larger impact on liquid volume fraction than changes in initial conditions farther away
from the saturation point. Therefore, the initial conditions do not linearly impact liquid
volume fraction.
Table 3.5.2-6: Initial Condition Study Case Results
Case Number
1
7
8
9
10
11
12
Volume-Weighted Liquid Volume Fraction
0.91078539
0.87799626
0.93408281
0.57124303
0.96969908
0.92067945
0.89072032
75
Table 3.5.2-7 shows the liquid volume fraction at nine axial locations for the
cases described in Table 3.5.2-5. This table allows for a finer comparison of the liquid
volume fraction between the cases. Table 3.5.2-7 does not show any irregular trends in
liquid volume fraction and the same relationships between initial conditions and liquid
volume fraction developed using Table 3.5.2-6 can be drawn using Table 3.5.2-7. Thus
making observations based on overall liquid volume fraction is acceptable.
Table 3.5.2-7: Axial Location Liquid Volume Fraction
Location* Case 1
Case 7
Case 8
0 cm
1.00000 1.00000 1.00000
5 cm
0.99168 0.98880 0.99397
10 cm
0.97680 0.97050 0.98231
15 cm
0.96151 0.95036 0.97201
20 cm
0.93987 0.92220 0.95598
25 cm
0.91595 0.89812 0.93589
30 cm
0.89784 0.87830 0.91644
35 cm
0.88190 0.85540 0.90250
40 cm
0.85984 0.80840 0.89019
* Distance from the pipe inlet.
Case 9
1.00000
0.95129
0.87624
0.80165
0.71266
0.57694
0.42719
0.31823
0.25132
Case 10
1.00000
0.99785
0.99390
0.98885
0.98427
0.97895
0.96784
0.95471
0.93927
Case 11
1.00000
0.99348
0.97938
0.96537
0.94748
0.92264
0.90098
0.88448
0.86812
Case 12
1.00000
0.98907
0.97482
0.95578
0.93222
0.91180
0.89572
0.87680
0.83296
Figure 3.5.2-6 illustrates the liquid volume fraction at different axial locations in
Table 3.5.2-6 with respect to distance from the centerline. The x-axis is position, or
distance from the centerline and the pipe wall is located at 0.015 m. The impact that
inlet temperature (Case 9 and Case 10) has on liquid volume fraction is extremely visible
in Figure 3.5.2-6. Case 9 shows significant voiding in the centerline after 25 cm from
the pipe inlet due to the high inlet temperature (subcooling of about 1 K). Case 10
reveals the opposite where 40 cm from the pipe inlet there is no voiding even at 0.010 m
from the pipe centerline.
76
(a) Case 7
(b) Case 8
(c) Case 9
(d) Case 10
(e) Case 11
(f) Case 12
Figure 3.5.2-6: Liquid Volume Faction for Cases 7-12
77
The liquid volume fraction at the nine axial locations from Cases 7 through 12
were compared to the liquid volume fraction of the base case (Case 1) using the
following three equations for heat flux, inlet temperature and mass flow, respectively,
where i stands for the axial location.
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥)
𝐶𝑎𝑠𝑒 𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥𝑖 − 𝐵𝑎𝑠𝑒 𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥𝑖
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝐼𝑛𝑙𝑒𝑡 𝑇𝑒𝑚𝑝. )
𝐶𝑎𝑠𝑒 𝐼𝑛𝑙𝑒𝑡 𝑇𝑒𝑚𝑝.𝑖 − 𝐵𝑎𝑠𝑒 𝐼𝑛𝑙𝑒𝑡 𝑇𝑒𝑚𝑝.𝑖
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤)
𝐶𝑎𝑠𝑒 𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤𝑖 − 𝐵𝑎𝑠𝑒 𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤𝑖
The results of comparing the values from Table 3.5.2-7 using the three above
equations are shown in Table 3.5.2-8. For example, at an axial height of 10 cm, by
increasing the heat flux from 90 kW/m2 to 100 kW/m2 (Case 1 to Case 7) the liquid
volume fraction decreased by 0.0063 or 0.00063 per kW/m2. Similar calculations were
carried out for the remaining axial locations and initial conditions. The change in liquid
volume fraction at each axial location was averaged to produce an overall impact that
each initial conditions on liquid volume fraction.
Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction
Height
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
Average
Case 7
Case 8
0.00000
0.00000
-0.00029
-0.00023
-0.00063
-0.00055
-0.00112
-0.00105
-0.00177
-0.00161
-0.00178
-0.00199
-0.00195
-0.00186
-0.00265
-0.00206
-0.00514
-0.00304
-0.00154 (kW/m2)-1
Case 9
Case 10
0.00000
0.00000
-0.02020
-0.00309
-0.05028
-0.00855
-0.07993
-0.01367
-0.11361
-0.02220
-0.16951
-0.03150
-0.23533
-0.03500
-0.28184
-0.03641
-0.30426
-0.03972
-0.08028 (K)-1
78
Case 11
Case 12
0.00000
0.00000
0.06000
0.08700
0.08600
0.06600
0.12867
0.19100
0.25367
0.25500
0.22300
0.13833
0.10467
0.07067
0.08600
0.17000
0.27600
0.89600
0.17178 (kg/s)-1
Table 3.5.2-8 reveals the average impact that changing the heat flux, inlet
temperature and mass flow rate have on the overall liquid volume fraction. Evaluating
which of the three inputs is most impactful on liquid volume fraction is difficult to do in
absolute terms (a 1 kg/s increase in mass flow rate is a larger percentage increase than a
10 kW/m2 increase in heat flux). Therefore, the values in Table 3.5.2-8 were compared
on a percentage basis to provide further insight. Table 3.5.2-9 shows the liquid volume
fraction change expected for a 1% change in each initial condition. The second column
of Table 3.5.2-9 repeats the initial conditions used in Case 1 (from Table 3.5.2-1), the
third column calculates 1% of the Case 1 input value (for example, 90 kW/m2 * 0.01 =
0.9 kW/m2), the fourth column replicates the results from Table 3.5.2-8, and the fifth
column shows the outcome when columns three and four are multiplied together.
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction
Initial
Condition
Heat Flux
Temperature
Mass Flow
Case 1
Input
90 kW/m2
370 K
0.3 kg/s
1% of Case 1
Table 3.5.2-8
Input
Results
2
0.9 kW/m
-0.00154 (kW/m2)-1
3.70 K
-0.08028 (K)-1
0.003 kg/s
0.17178 (kg/s)-1
Equivalent Liquid
Volume Fraction
-0.00139
-0.29704
0.00052
Table 3.5.2-9 illustrates that a 1% increase in heat flux causes the average liquid
void fraction to decrease by 0.00139, a 1% increase in temperature causes the average
liquid void fraction to decrease by 0.29704 and a 1% increase in mass flow rate causes
the average liquid void fraction to increase by 0.00052. It is understood that a 1%
increase in the inlet temperature from the Case 1 condition would be greater than the
saturation temperature at atmospheric pressure and therefore impossible; however, this
exercise was performed to show the impact of the initial conditions in a more revealing
manner. Table 3.5.2-9 indicates that inlet temperature has the greatest impact on liquid
volume fraction, the wall heat flux has the second greatest impact and mass flow rate has
the smallest impact.
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
(“Analysis Value” in Table 3.5.2-10) to a second case with an increased number of finite
79
volumes (“Mesh Validation” in Table 3.5.2-10). The results from the mesh validation
shown in Table 3.5.2-10 prove that the results are mesh independent.
Table 3.5.2-10: Mesh Validation for Subcooled Boiling Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Min Liquid Volume Fraction
Max Phase Transfer (kg/m3-s)
Analysis Value
25000
23976
0.8181624
0.9972627
0.4876771
24.87638
80
Mesh Validation
31000
29970
0.8199201
0.9982293
0.4853158
26.22442
Difference
24.00 %
25.00 %
0.21 %
0.10 %
-0.48 %
5.42 %
4. DISUSSION AND CONCLUSIONS
This thesis provided theoretical background and development of computational
fluid dynamic models for various fluid flow and heat transfer phenomena including
natural convection, laminar flow, turbulent flow with and without heat transfer, twophase flow, pool boiling and subcooled flow boiling.
Natural convection models of a heated horizontal cylinder and a heated vertical
plate were presented in Section 3.1.
These models implemented the Boussineq
approximation to calculate temperature induced density gradients and buoyancy forces.
The heated horizontal cylinder model predicted a greater maximum velocity compared to
the heated vertical plate even though the two models used the same surface and bulk
fluid temperatures. The heated vertical plate had a lower maximum velocity due to drag
forces invoked by the heated surface. Both natural convection models showed good
agreement qualitatively and quantitatively with experimental data.
Laminar flow within a pipe was investigated in Section 3.2. The parabolic
velocity profile that is characteristic of laminar flow matched well qualitatively with
experimental data. Also, the radial velocity for most of the pipe was near zero and was
less than 10-3 times the average axial velocity.
Two models involving turbulent flow within a pipe were created as part of
Section 3.3. As expected, the velocity profiles calculated where flat and the velocity
magnitude didn’t decrease until very close to the pipe wall which matched well
qualitatively with experimental data. The wall shear stress reached a maximum at a
short distance from the pipe inlet due to entrance effects causing a surge in radial
velocity which led to a dramatic reduction in axial velocity. The turbulent flow model
with the energy equation was compared to the turbulent flow model without energy
addition and it was determined that there was a small increase in the fluid velocity
magnitude for the scenario with heat addition. The velocity increase was due to the
constant mass flow rate boundary condition and the reduction in density caused by
energy addition.
Two-phase flow involving water and air was examined as part of Section 3.4.
The first model was a mixing tank that used an air jet to stir the liquid. Effects of
81
Rayleigh instability were observed. Before the jet broke the surface of the water, it
became wavy and surface tension started to transform the jet into bubbles to reduce
surface area. The second model created was a bubble column reactor. XXXXXXX was
observed to occur. Gas holdup due to phase drag forces and displacement was noted.
The amount of gas holdup was found to be related to inlet gas velocity however the
relationship was not linear. A population balance model was employed for two bubble
column cases. The model predicted that the air bubbles would coalesce and grow in size
as they traveled up the bubble column due to surface tension. When the surface tension
was reduced, the number of bubbles that grew in size dramatically reduced.
Section 3.5 discussed phase transformation due to heat addition in both stagnant
and flowing liquids. The pool boiling model showed the progression of vapor formation
on the heated surface, detachment and liquid refill. Drag forces between the two phases
caused the liquid to travel upwards with the rising vapor but at a slower rate due to slip.
The second phase transformation model developed and the focus of this research
was a subcooled flow boiling model. The impact that different boiling model options
have on liquid volume fraction was investigated. Three bubble departure diameter
models and two nucleation site density models were analyzed using the same initial
conditions. The bubble departure diameter models 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 initial conditions impact the liquid volume fraction
during subcooled flow boiling was explored.
The initial conditions of heat flux, inlet
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 initial conditions with respect to liquid volume fraction were
developed. Overall, the inlet temperature had the greatest impact on liquid volume
fraction, the wall heat flux had the second greatest impact and mass flow rate had the
smallest impact.
82
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