Modeling of Subcooled Flow Boiling and Other Heat
Transfer and Fluid Flow Phenomena
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
March 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 .......................................................................................................... vi
ABSTRACT .................................................................................................................... vii
1. INTRODUCTION ....................................................................................................... 1
1.1
RESEARCH ....................................................................................................... 2
1.2
CONTENT ......................................................................................................... 4
2. HEAT TRANSFER AND FLUID FLOW: THEORY ............................................... 5
2.1
GOVERNING EQUATIONS ............................................................................ 5
2.2
NUMERICAL METHODS................................................................................ 7
2.3
NATURAL CONVECTION .............................................................................. 9
2.4
LAMINAR FLOW ........................................................................................... 11
2.5
TURBULENT FLOW ...................................................................................... 12
2.5.1
CALCULATING TURBULENCE PARAMETERS .......................... 14
2.6
TWO-PHASE FLOW ...................................................................................... 16
2.7
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
3.4
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER ..................... 38
3.3.2
TURBULENT FLOW WITH HEAT TRANSFER ............................. 43
TWO-PHASE FLOW ...................................................................................... 47
3.4.1
GAS MIXING TANK .......................................................................... 47
iii
3.5
3.4.2
BUBBLE COLUMN ............................................................................ 51
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL ... 57
BOILING FLOWS ........................................................................................... 61
3.5.1
POOL BOILING .................................................................................. 61
3.5.2
SUBCOOLED BOILING .................................................................... 66
4. DISUSSION AND CONCLUSIONS ........................................................................ 79
REFERENCES ................................................................................................................ 81
APPENDIX A: MULTIPHASE FLOW MODELS ........................................................ 83
A.1 Volume of Fluid Model .................................................................................... 83
A.2 Mixture Model ................................................................................................. 83
APPENDIX B: POPULATION BALANCE MODEL................................................... 84
B.1 EQUATION FORMULATION ....................................................................... 85
B.1.1 Particle State Vector ......................................................................................... 85
B.1.2 Population Balance Equation ............................................................... 86
APPENDIX C: SUBCOOLED BOILING ..................................................................... 87
iv
LIST OF TABLES
Table 2.4.1-1: Turbulent Flow Input ............................................................................... 15
Table 2.5.1-2: Turbulence Parameter Calculation ........................................................... 15
Table 3.1.1-1: Horizontal Cylinder Input ........................................................................ 22
Table 3.1.1-2: Horizontal Cylinder Water Density ......................................................... 22
Table 3.1.1-1: Mesh Validation for Horizontal Cylinder ................................................ 27
Table 3.1.2-1: Vertical Plate Input .................................................................................. 28
Table 3.1.2-2: Vertical Plate Water Density .................................................................... 29
Table 3.1.2-1: Mesh Validation for Vertical Plate .......................................................... 33
Table 3.2-1: Laminar Flow Input..................................................................................... 34
Table 3.2-2: Laminar Flow Water Density ...................................................................... 34
Table 3.2-3: Mesh Validation for Laminar Flow ............................................................. 37
Table 3.3.1-1: Turbulent Flow Without Heat Transfer Input .......................................... 38
Table 3.3.2-1: Turbulent Flow With Heat Transfer Input ............................................... 43
Table 3.3.2-2: Turbulent Flow Water Density................................................................. 44
Table 3.3.2-3: Mesh Validation for Turbulence With Heat Transfer .............................. 46
Table 3.4.1-1: Gas Mixing Tank Input ............................................................................ 48
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank .................................................... 50
Table 3.4.2-1: Bubble Column Input ............................................................................... 52
Table 3.4.2-2: Mesh Validation for Bubble Column ....................................................... 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 ..................... 60
Table 3.4.3-3: Bubble Size Distribution – Surface Tension of 0.0072 N/m ................... 60
Table 3.5.1-1: Pool Boiling Input .................................................................................... 61
Table 3.5.1-2: Pool Boiling Water Density ..................................................................... 62
Table 3.5.1-3: Mesh Validation for Pool Boiling ............................................................ 65
Table 3.5.2-1: Subcooled Flow Boiling Input ................................................................. 68
Table 3.5.2-2: Subcooled Boiling Water Properties ........................................................ 69
Table 3.5.2-3: Boiling Model Case Input ........................................................................ 70
Table 3.5.2-4: Boiling Model Case Results ..................................................................... 72
Table 3.5.2-5: Input Parametric Case Matrix .................................................................. 74
v
Table 3.5.2-6: Input Parametric Case Results ................................................................. 74
Table 3.5.2-7: Axial Height Liquid Volume Fraction ..................................................... 75
Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction .......................................... 77
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction ........................................... 78
Table 3.5.2-10: Mesh Validation for Subcooled Boiling ................................................ 78
vi
LIST OF FIGURES
Figure 2.5-1: Example of Turbulent Flow ....................................................................... 12
Figure 2.6-1: Flow Regimes ............................................................................................ 16
Figure 2.6-2: Baker Flow Pattern .................................................................................... 17
Figure 2.7-1: Boiling Heat Transfer Regimes ................................................................. 19
Figure 3.1.1-1: Horizontal Cylinder Temperature ........................................................... 21
Figure 3.1.1-2: Horizontal Cylinder Density ................................................................... 23
Figure 3.1.1-3: Horizontal Cylinder Velocity Vector ...................................................... 24
Figure 3.1.1-4: Interference Fringes Around a Heated Horizontal Cylinder ................... 24
Figure 3.1.1-5: Temperature at θ = 30° Vs. Radial Distance .......................................... 25
Figure 3.1.1-6: Temperature at θ = 90° Vs. Radial Distance .......................................... 26
Figure 3.1.1-7: Temperature at θ = 180° Vs. Radial Distance ........................................ 26
Figure 3.1.2-1: Vertical Plate Temperature ..................................................................... 29
Figure 3.1.2-2: Vertical Plate Velocity Vector Plot ........................................................ 30
Figure 3.1.2-3: Interference Fringes Around a Heated Vertical Plate ............................. 31
Figure 3.1.2-4: Dimensionless Temperature as a Function of Prandtl Number .............. 32
Figure 3.1.2-5: Dimensionless Temperature as a Function of Prandtl Number .............. 32
Figure 3.2-1: Laminar Flow Velocity Profile Vs. Positon............................................... 35
Figure 3.3.1-1: Turbulent Flow Velocity Magnitude Vs. Position .................................. 39
Figure 3.3.1-2: Wall Shear Stress Vs. Position ............................................................... 40
Figure 3.3.1-3: Radial Velocity ...................................................................................... 40
Figure 3.3.1-4: dAxial-Velocity/dx Vs. Position ............................................................. 40
Figure 3.3.1-5: Flow Results for Mass Flow Rate of 0.5 kg/s ......................................... 41
Figure 3.3.1-6: Flow Results for Larger Mass Flow Rate of 1.5 kg/s ............................. 41
Figure 3.3.1-7: Turbulent Kinetic Energy ....................................................................... 42
Figure 3.3.1-8: Production of Turbulent Kinetic Energy ................................................ 42
Figure 3.3.2-1: Temperature ............................................................................................ 44
Figure 3.3.2-2: Radial Velocity ....................................................................................... 44
Figure 3.3.2-3: Velocity Magnitude Vs. Position ............................................................ 45
Figure 3.3.2-4: Wall Shear Stress Vs. Axial Position...................................................... 45
Figure 3.4.1-1: Gas Volume Fraction .............................................................................. 49
vii
Figure 3.4.1-2: Liquid Vector Velocity ........................................................................... 49
Figure 3.4.1-3: Gas Vector Velocity................................................................................ 50
Figure 3.4.2-1: Bubble Column Reactor.......................................................................... 51
Figure 3.4.2-2: Instantaneous Gas Volume Fraction ....................................................... 53
Figure 3.4.2-3: Instantaneous Liquid Velocity Vectors................................................... 54
Figure 3.4.2-4: Instantaneous Gas Velocity Vectors ....................................................... 55
Figure 3.4.2-5: Instantaneous Gas Volume Fraction (10 cm/s) ....................................... 56
Figure 3.4.3-1: Instantaneous Gas Volume Fraction with PBM ..................................... 58
Figure 3.4.3-2: Bubble Column Liquid Vector Velocity with PBM ............................... 59
Figure 3.4.3-3: Bubble Column Liquid Vector Velocity with PBM ............................... 60
Figure 3.5.1-1: Instantaneous Gas Volume Fraction ....................................................... 63
Figure 3.5.1-2: Instantaneous Liquid Velocity Vectors................................................... 64
Figure 3.5.1-3: Instantaneous Gas Velocity Vectors ....................................................... 64
Figure 3.5.1-4: Volume Fraction of Vapor on Heated Surface ....................................... 65
Figure 3.5.2-1: Case 1 - Temperature (K) ....................................................................... 70
Figure 3.5.2-2: Case 1 - Liquid Volume Fraction ........................................................... 71
Figure 3.5.2-3: Case 1 - Mass Transfer Rate (kg/m3-s) ................................................... 71
Figure 3.5.2-4: Liquid Volume Faction Vs. Position for Cases 1-6 ................................ 73
Figure 3.5.2-5: Liquid Volume Faction Vs. Position for Cases 7-12 .............................. 76
Figure 3.5.2-1: Void Fraction in Various Boiling Regimes ............................................ 87
viii
ABSTRACT
Investigations into various fluid flow and heat transfer regimes were modeled
numerically to better understand the phenomena that occur during subcooled flow
boiling. The theory of each fluid flow and heat transfer regime that occurs during
subcooled flow boiling is 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. 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 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.
Cases were analyzed that increased or
decreased the heat flux, the inlet temperature or the mass flow compared to a base case.
The differences in liquid volume fraction between the cases were compared and
relationships between heat flux, inlet temperature and mass flow rate 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. The Rankine Cycle is an energy conversion process where fuel is
burned to heat water and form steam. The steam is used to turn a turbine which spins an
electric generator. Electricity production involves numerous engineering processes but
is primarily based around heat transfer and fluid flow.
There are many different fuel sources available to electrical power plants such as
coal, oil, natural gas and uranium. The fuel source in focus here will be uranium or
nuclear fuel. Nuclear power plants harness energy released during fission to heat water.
This water is then pumped through a heat exchanger to produce steam. The heat transfer
mechanisms at work within a nuclear reactor core are extremely complex. All three
major forms of heat transfer are at work; conduction, convection and radiation. The
fluid flow through the reactor core is also 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 also 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 subcooled boiling
that occurs is challenging and 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. More 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 put towards 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.
Another improvement that has been made in recent years to the modeling of
subcooled flow boiling is the use of population balance equations (PBEs) to better
2
determine how swarms of bubbles interact after detaching from the heated surface. This
is a relatively new way of investigating subcooled flow boiling that 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. These 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.
3
1.2
CONTENT
This thesis 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 complex 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 a laminar flow model. 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 a turbulent flow model. 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 a water / air twophase flow model. The theory of two-phase flow is described in Section 2.6 and the
analytical modeling results for the two scenarios analyzed are shown in Section 3.4. The
first is a gas mixing tank and the second is a bubble column. The final and most
complex model created includes a phase transformation (vaporization and condensation)
model. 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 is for
pool boiling and the second is 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 (CFD).
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 background on 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
6
2.2
NUMERICAL METHODS
After the conservation laws governing heat transfer, fluid flow and other related
processes are expressed in differential form (shown above), they can solved using
numerical methods to determine pressure, temperature, mass flux, etc. for various
situations 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 the 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
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
7
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 the 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.
Fluent offers numerous spatial discretization solvers for the various independent
variables such as pressure, flow, momentum, turbulence, and energy. Fluent implements
the control volume formulation with upwinding which was first proposed by Courant,
Isaacson, and Rees in 1952. Other options include QUICK, power law and third-order
MUSCL.
8
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 gradient 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 will absorb energy from the heated
surface and become less dense. Buoyancy effects due to body forces will cause the
heated fluid to rise. At this point, the surrounding, cooler fluid will move in 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.
Most weather patterns are created by natural
convection. Natural convection also takes place in many engineering applications such
as home heating radiators and cooling computer chips.
The amount of heat transfer occurring 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
The Prandtl number, Pr, is a dimensionless parameter that represents the ratio of
momentum diffusivity to thermal diffusivity; and is defined as:
Pr =
9
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 engineering applications, convection is mixed meaning that both natural
and forced convection occurs 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.
10
2.4
LAMINAR FLOW
Single-phase 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 is characterized by high momentum diffusion and low momentum
convection.
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
This quantity helps to quantify the relative importance of these two types of
forces for given flow conditions. For internal flow, such as within a pipe, laminar flow
is characterized by 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 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
11
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 three-
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 of laminar flow to turbulent flow is smoke
rising from a cigarette.
Figure 2.5-1: Example of Turbulent Flow
12
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 into the turbulent regime. When this
happens, the flow of the smoke becomes more random and rapidly mixes with the air
causing the smoke to dissipate.
Exact 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. [8]
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 [8]. 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
13
′ ′
̅̅̅̅̅̅
time-averaged solutions to the Navier–Stokes equations. An additional term,(−𝜌𝑢
𝑖 𝑢𝑗 ),
known as the Reynolds stress appears in the equation as a results of using the RANS
method. [8]
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 [9].
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 was 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 have become known, modifications have been introduced to
improve its performance. These improvements have helped create many, new, more
accurate models, among these, 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 for the turbulent viscosity. Second, a modified transport equation for the
dissipation rate, ϵ, has been derived from an exact equation for the transport of the meansquare vorticity fluctuation. The term “realizable” means that the model satisfies certain
mathematical constraints on the Reynolds stresses, consistent with the physics of
turbulent flows. [8]
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 the Fluent code, 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 [8] were used to determine the boundary and initial
condition inputs for the turbulent flow models presented in Section 3.3.
14
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 [8]
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
15
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 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 a 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 2.5-1 [10].
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
16
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 occurs when there
are relatively large liquid slugs surrounded by vapor. This flow 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). 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 [10, Figure 3-4].
Figure 2.6-2: Baker Flow Pattern
17
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 complicated because there are more equations to solve due to
the secondary phase and additional phenomena to account for such as mass transfer,
phase-interface interactions (slip and drag). Three common multiphase flow models
available in Fluent are Volume of Fluid, Mixture and Eulerian, each with varying
strengths and computational demand.
The Volume of Fluid model solves a single set of momentum equations for two
or more fluids and tracks the volume fraction of each fluid throughout the domain. The
Mixture model solves for the momentum equation of the mixture and prescribes relative
velocities to describe the dispersed phase. The Eulerian model solves momentum and
continuity equations for each of the phases, and the equations are coupled through
pressure and exchange coefficients. These are discussed in detail in Appendix A.
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. Pool boiling occurs when heat is added to a stagnant fluid
while flow boiling occurs when heat is added to a moving fluid. Both types of boiling
heat transfer can be separated into four regimes which are shown in Figure 2.7-1 [11].
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 [10].
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
19
bubbles at the nucleation sites. When the liquid evaporates, a significant amount of
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, the
bubbles form on the heated surface but tend to condense after leaving the heated surface.
The second subregime is when local boiling occurs in a saturated liquid. In this case, the
bubbles do not collapse. 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. [10]
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 and 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 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. [11]
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. [10]
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. [11]
20
3. HEAT TRANSFER AND FLUID FLOW: MODELING
3.1
NATURAL CONVECTION
Two examples of natural convection 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
The results from modeling a cylinder with a constant surface temperature
submerged in an infinite pool of liquid were analyzed in this section. The cylinder was
slightly warmer than the surrounding fluid and therefore energy passed from the cylinder
to the nearby fluid causing its temperature to increase. Table 3.1.1-1 lists the important
input needed to replicate the results shown in this section. The liquid temperature field
after 20 seconds is shown in Figure 3.1.1-1.
Figure 3.1.1-1: Horizontal Cylinder Temperature
21
Table 3.1.1-1: Horizontal Cylinder 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
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
On
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 Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
22
As the temperature increases, the fluid expands and its density decreases causing
the fluid to rise due to buoyancy forces. Even fluid that is not in direct contact with the
heated cylinder experiences a density change, shown in Figure 3.1.1-2. The density
gradient is shown by the color transition surrounding the cylinder from least dense (blue)
to most dense (red). This is caused by energy transfer via conduction to the bulk fluid.
Figure 3.1.1-2: Horizontal Cylinder Density
As the fluid rises, it separates from the cylinder allowing new, cooler fluid takes
its place. When the warm fluid rises, it loses energy to the surrounding bulk fluid which
causes the buoyancy driving head to diminish which causes the fluid to climb more
slowly until it eventually stops. At this point, the fluid 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 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. Figure 3.1.1-3 is a liquid
velocity vector plot that shows how the liquid moves within the control volume. The
cycle of fluid energy absorption and replacement around the cylinder and the two
convection cells above the cylinder are more visible in this figure.
23
Figure 3.1.1-3: Horizontal Cylinder Velocity Vector
To verify that the model produced realistic results, the solution was compared to
experimental data.
Figure 3.1.1-4 shows interference fringes surrounding a heated
horizontal cylinder in natural convection. Each interference fringe can be interpreted as
a band constant temperature.
(a)
(b)
Figure 3.1.1-4: Interference Fringes Around a Heated Horizontal Cylinder
(a) is from [12] and (b) shows isotherms from Fluent
24
Figure 3.1.1-4 shows comparable results between experimental data and the
results determined by Fluent. Both 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 [13] was compared to the Fluent
results to provide model validation. Figure 3.1.1-5, Figure 3.1.1-6 and Figure 3.1.1-7
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.
(a)
(b)
Figure 3.1.1-5: Temperature at θ = 30° Vs. Radial Distance
(a) is from [13] and (b) is from Fluent
25
(a)
(b)
Figure 3.1.1-6: Temperature at θ = 90° Vs. Radial Distance
(a) is from [13] and (b) is from Fluent
(a)
(b)
Figure 3.1.1-7: Temperature at θ = 180° Vs. Radial Distance
(a) is from [13] and (b) is from Fluent
26
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 gathered from the Fluent model
was accurate. To ensure that the mesh had no impact on the results, a mesh validation
was performed. The mesh validation was performed by comparing the results shown in
this section (“Analysis Value” in Table 3.1.1-1) to a second case with an increased
number of nodes and elements (“Mesh Validation” in Table 3.1.1-1). The second case
was exactly the same as the case used in this section except that its mesh was refined.
The results from the mesh validation are shown in Table 3.1.1-1, and prove that the
results are mesh independent.
Table 3.1.1-1: Mesh Validation for Horizontal Cylinder
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
The results from modeling a vertical plate with a constant surface temperature
submerged in an infinite pool of liquid were analyzed in this section. Like the cylinder,
the plate was also slightly warmer than the surrounding fluid and therefore energy
passed from the plate to the fluid causing its temperature to increase. Table 3.1.2-1 lists
the important input needed to replicate the results shown in this section.
Table 3.1.2-1: Vertical Plate Input
Input
Value
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
28
0.18 m
0.01 m
0.20 m
0.13 m
Planar
Transient
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
On
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 Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
The liquid temperature field after 20 seconds is shown in Figure 3.1.2-1. 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 is known as the thermal boundary
layer. The teal color in Figure 3.1.2-1 shows the growth of the thermal boundary layer.
It is relatively small at the bottom of the plate because there has been little heat addition
but the thermal boundary layer grows (teal color expands away from the plate) as the
fluid reaches the top of the plate. Although not visible in Figure 3.1.2-1, as the thermal
boundary layer expands, so does the momentum boundary layer which means that there
is more fluid motion due to heat addition from the hot plate.
Figure 3.1.2-1: Vertical Plate Temperature
29
Figure 3.1.2-2 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 creating a greater temperature gradient and
therefore a larger buoyancy force.
Figure 3.1.2-2: Vertical Plate Velocity Vector Plot
Comparing Figure 3.1.2-2 (vertical plate velocity vectors) with Figure 3.1.1-3
(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 produced a
maximum fluid velocity of 0.0149 m/s while the horizontal cylinder produced a
maximum fluid velocity of 0.0177 m/s. Although the difference is small, it is notable.
The horizontal cylinder produced a larger maximum velocity because the buoyancy
driving head does not fight against the drag force generated by the heated surface.
Although the plate continued to heat the fluid as it traveled upward, the velocity is
limited by friction which caused 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 3.1.2-3 shows interference fringes surrounding a heated
30
vertical plate in natural convection. Each interference fringe can be interpreted as a band
constant temperature.
(a)
(b)
Figure 3.1.2-3: Interference Fringes Around a Heated Vertical Plate
(a) is from [12] and (b) is from Fluent
The model of a vertical plate submerged in an infinite pool was in qualitative
agreement to experimental data. Figure 3.1.2-3 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.
Quantitative experimental data from Ostrach [14] was compared to the Fluent
results to assess the accuracy of the model. Figure 3.1.2-4 and Figure 3.1.2-5 show a
comparison of dimensionless temperature versus dimensionless distance for five
different Prandtl numbers. Figure 3.1.2-4a shows theoretical values and Figure 3.1.2-4b
compares some of the theoretical values to experimental data.
The information
contained in Figure 3.1.2-5 was calculated by Fluent. Dimensionless temperature is
T = (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 = (Grx / 4)1/4 * (Y / X) where Grx is the Grashof number, Y is the vertical height and X
is the distance from the plate.
31
(a)
(b)
Figure 3.1.2-4: Dimensionless Temperature as a Function of Prandtl Number
(a) Theoretical Values and (b) Experimental Values [14]
Figure 3.1.2-5: Dimensionless Temperature as a Function of Prandtl Number
32
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 gathered from the Fluent model
was accurate. To ensure that the mesh had no impact on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Analysis Value” in Table 3.1.2-1) to a second case with an increased number of nodes
and elements (“Mesh Validation” in Table 3.1.2-1). The second case was exactly the
same as the case used in this section except that its mesh was refined. The results from
the mesh validation are shown in Table 3.1.2-1, and prove that the results are mesh
independent.
Table 3.1.2-1: Mesh Validation for Vertical Plate
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 flow model was developed to gain a better understanding
of laminar flow in a pipe. The Reynolds number was 352, based upon the selected initial
conditions, which is well within the laminar regime. Table 3.2-1 lists the important
input needed to replicate the results shown in this section.
Table 3.2-1: Laminar Flow 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)
On
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 Water Density
Density (kg/m3)
999.9
994.1
Temperature (K)
273
308
34
A noteworthy characteristics of laminar flow is the parabolic shape of the
velocity profile. Fluid velocity within the pipe slowly decreases as distance from the
pipe centerline increases. This is vastly different from turbulent flow which has a very
flat velocity profile and is described in more detail in Section 3.3. Figure 3.2-1 shows
the velocity magnitude versus position (distance from the pipe centerline) at various
distances from the pipe entrance. For example, “line-10cm” shows the velocity profile
10 cm from the pipe entrance. As the flow develops, the entrance effects dissipate, the
velocity profile becomes more and more parabolic until it reaches a steady state at 45 cm
from the entrance.
Figure 3.2-1: Laminar Flow Velocity Profile Vs. Positon
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 3.2-2 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 spikes near the entrance of the pipe due to pipe boundary
conditions and entrance effects but these have little impact on system as a whole.
35
Figure 3.2-2: Laminar Flow Radial Velocity
Figure 3.2-3 shows the temperature profile of the laminar flow analyzed.
Diffusion and conduction are the primary forms of heat transfer within the fluid. The
growth of the thermal boundary layer as the fluid travels down the pipe is shown in
Figure 3.2-3 by the expansion of the teal colored region.
Figure 3.2-3: Laminar Flow Temperature
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 forces produced by the wall. Figure 3.2-4 shows the wall shear stress as a
function of distance from the pipe entrance.
Figure 3.2-4: Laminar Flow Wall Shear Stress
36
Figure 3.2-4 shows that the wall stress is much larger in the first 10 cm which is
caused by entrance effects. Once the entrance effects dissipate, the wall shear stress
slowly decreases as the flow reaches a steady state. To ensure that the mesh had no
impact on the results, a mesh validation was performed. The mesh validation was
performed by comparing the results shown in this section (“Analysis Value” in
Table 3.3.2-1) to a second case with an increased number of nodes and elements (“Mesh
Validation” in Table 3.3.2-1). The second case was exactly the same as the case used in
this section except that its mesh was refined. The results from the mesh validation are
shown in Table 3.3.2-1, and prove that the results are mesh independent.
Table 3.2-3: Mesh Validation for Laminar Flow
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 flow model was developed to gain a better understanding
of turbulent flow in a pipe. The Reynolds number was 42314, based upon the selected
initial conditions, which is well within the turbulent regime. Table 3.3.1-1 lists the
important input needed to replicate the results shown in this section.
Table 3.3.1-1: Turbulent Flow Without Heat Transfer 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
Off
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.
38
Figure 3.3.1-1 shows the velocity magnitude versus position (distance from the
pipe centerline) at various distances from the pipe entrance. 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 3.3.1-1
(turbulent flow) shows that flow reached a steady profile about 10 cm from the pipe
entrance. Figure 3.2-1 (laminar flow) shows that flow reached a steady profile about
45 cm from the pipe entrance. This qualitatively matches experimental data well.
Figure 3.3.1-1: Turbulent Flow Velocity Magnitude Vs. Position
Figure 3.3.1-2 shows 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 the entrance effects. Figure 3.3.1-3 shows that that maximum absolute radial
velocity occurs near the pipe entrance. Conservation of momentum requires that the
axial velocity decrease near the entrance due to the increase in radial velocity.
Figure 3.3.1-4 shows that the greatest reduction in axial velocity occurs near the pipe
39
entrance which matches expectations. Since shear stress is related to change in velocity
perpendicular to the wall (axial velocity), the increase in wall shear stress is reasonable.
Figure 3.3.1-2: Wall Shear Stress Vs. Position
Figure 3.3.1-3: Radial Velocity
Figure 3.3.1-4: dAxial-Velocity/dx Vs. Position
40
To further investigate the impact of entrance effects, two additional cases were
run using a mass flow rate of 0.5 kg/s (Figure 3.3.1-5) and 1.5 kg/s (Figure 3.3.1-6).
(a)
(b)
(c)
Figure 3.3.1-5: Flow Results for Mass Flow Rate of 0.5 kg/s
(a) radial velocity (b) wall shear stress vs. position (c) daxial-velocity/dx vs. position
(a)
(b)
(c)
Figure 3.3.1-6: Flow Results for Larger Mass Flow Rate of 1.5 kg/s
(a) radial velocity (b) wall shear stress vs. position (c) daxial-velocity/dx vs. position
41
Figures 3.3.1-5 and 3.3.1-6 show that wall shear stress and maximum radial
velocity are directly related to mass flow rate. At a certain distance from the 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 directly related to the mass flow rate. A larger mass flow rate requires a greater
length.
Figure 3.3.1-7 and Figure 3.3.1-8 show the turbulent kinetic energy and the
production of turbulent kinetic energy as a function of distance.
Figure 3.3.1-7: Turbulent Kinetic Energy
Figure 3.3.1-8: 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 3.3.1-8 is similar to that of Figure 3.3.1-2 because shear
stress, created by the wall, produces turbulent kinetic energy.
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. Table 3.3.2-1 lists the important input needed to
replicate the results shown in this section.
Table 3.3.2-1: Turbulent Flow With Heat Transfer 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
On
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.
43
Table 3.3.2-2: Turbulent Flow Water Density
Density (kg/m3)
999.9
994.1
974.9
Temperature (K)
273
308
348
Figure 3.3.2-1 shows the fluid temperature change caused by energy addition
from the pipe walls. The radial temperature distribution in Figure 3.3.2-1 is significantly
more uniform than the radial temperature distribution in Figure 3.2-3 (laminar flow).
Uniform temperature distribution is a characteristic of turbulent flow and made possible
due to the chaotic nature of the flow regime.
Figure 3.3.2-1: Temperature
The radial velocity shown in Figure 3.3.2-2 is very similar to that shown in
Figure 3.3.1-3 which is expected since 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 significantly impacted, then the radial velocity between the two scenarios
would also differ.
Figure 3.3.2-2: Radial Velocity
Comparing the velocity profiles for the two turbulent flow scenarios
(Figure 3.3.1-1 and Figure 3.3.2-3) reveals that the velocity magnitude is slightly larger
for the case with heat transfer. The heat transfer caused the density of the fluid to
decrease and therefore the velocity increased slightly to maintain a constant mass flow
through the pipe.
44
Figure 3.3.2-3: Velocity Magnitude Vs. Position
As expected, the wall shear stress shown in Figure 3.3.2-4 is similar to the wall
shear stress shown in Figure 3.3.1-2.
Figure 3.3.2-4: Wall Shear Stress Vs. Axial Position
To ensure that the mesh had no impact on the results, a mesh validation was
performed. The mesh validation was performed by comparing the results shown in this
section (“Analysis Value” in Table 3.3.2-1) to a second case with an increased number
of nodes and elements (“Mesh Validation” in Table 3.3.2-1). The second case was
exactly the same as the case used in this section except that its mesh was refined. The
45
results from the mesh validation are shown in Table 3.3.2-1, and prove that the results
are mesh independent.
Table 3.3.2-3: Mesh Validation for Turbulence With Heat Transfer
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Temperature (°F)
Min Density (kg/m3)
Max Dynamic Pressure (Pa)
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 %
Comparing the velocity magnitude plots, radial velocity contours and wall shear
stress plots from Section 3.3.1 and Section 3.3.2 shows that the addition of heat transfer
has a negligible impact the fluid flow profile. 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 are applicable to turbulent flows with heat transfer as long as
the impact due to heat transfer is small.
46
3.4
TWO-PHASE FLOW
3.4.1
GAS MIXING TANK
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 available 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.
Table 3.4.1-1 lists the important input needed to replicate the results shown in this
section.
After 5 seconds of gas injection, Figure 3.4.1-1 shows the gas volume fraction,
Figure 3.4.1-2 shows the liquid vector velocity and Figure 3.4.1-3 shows the gas vector
velocity. Midway through the liquid volume in Figure 3.4.1-1, 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. 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 air velocity and gas / liquid surface
tension.
The liquid and gas velocities shown in Figure 3.4.1-2 and Figure 3.4.1-3,
respectively, are similar which indicates that the drag force between the two phases is
strong. The maximum gas velocity is greater than the inlet velocity (0.5 m/s); therefore,
buoyancy forces are significant. Figure 3.4.1-2 shows that there is a number of small
eddies, created by the injected gas, that provide a significant amount of mixing within
the liquid.
47
Table 3.4.1-1: Gas Mixing Tank 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)
Off
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 3.4.1-1: Gas Volume Fraction
Figure 3.4.1-2: Liquid Vector Velocity
49
Figure 3.4.1-3: Gas Vector Velocity
To ensure that the mesh had no impact on the results, a mesh validation was
performed. The mesh validation was performed by comparing the results shown in this
section (“Analysis Value” in Table 3.4.1-2) to a second case with an increased number
of nodes and elements (“Mesh Validation” in Table 3.4.1-2). The second case was
exactly the same as the case used in this section except that its mesh was refined. The
results from the mesh validation are shown in Table 3.4.1-2, and prove that the results
are mesh independent.
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank
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%
17.70%
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 a bubble column reactor.
Figure 3.4.2-1: Bubble Column Reactor
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
thin 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. The
method to calculate the change in bubble size due to turbulent eddies is discussed in
Section 3.4.3. Table 3.4.2-1 lists the important input needed to replicate the results
shown in this section.
51
Table 3.4.2-1: Bubble Column 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)
Off
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 shows a comparison between gas volume fraction 1 second and
5 seconds after gas has begun flowing through the bubble column. At both time points
the gas tends to flow in slugs. After 5 seconds, the gas has reached the top of the liquid
and caused the surface to change shape. The liquid level is also higher after 5 seconds
by about 5 cm. The level increase is known as gas holdup and caused by phase drag
forces and displacement. Figure 3.4.2-2b shows that most of the gas travels along the
wall in a quasi-annular flow type regime.
(a)
(b)
Figure 3.4.2-2: Instantaneous Gas Volume Fraction
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-3 shows 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: Instantaneous Liquid Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-4 shows 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. This is occurs because most of the gas travels close to the wall (shown in
Figure 3.4.2-2). Figure 3.4.2-4b shows that the greatest gas velocities occur near the
walls which are also areas of greatest gas volume fraction. Higher gas volume fractions
lead to greater buoyancy forces which cause greater gas velocities.
A second case was completed to better understand the impact that gas inlet
velocity has on gas holdup. This case is the same as the case described in Table 3.5.1-1
except that the gas inlet velocity was increased to 10 cm/s. Figure 3.4.2-5 shows the gas
volume fraction 1 second and 5 seconds after gas has begun flowing through the bubble
column. Comparing Figure 3.4.2-5a and Figure 3.4.2-5b reveals that the injected gas
caused the water level to rise about 15 cm due to gas holdup. This is a much larger
54
increase than the gas hold shown in Figure 3.4.2-2, which employed a gas inlet velocity
of 5 cm/s.
(a)
(b)
Figure 3.4.2-4: Instantaneous Gas Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
To ensure that the mesh has no impact on the results, a mesh validation was
performed on the original case (gas velocity of 5 cm/s). The mesh validation was
performed by comparing the results shown in this section (“Analysis Value” in
Table 3.4.2-2) to a second case with an increased number of nodes and elements (“Mesh
Validation” in Table 3.4.2-2). The second case was exactly the same as the case used in
this section except that its mesh was refined. The results from the mesh validation are
shown in Table 3.4.2-2, and prove that the results are mesh independent.
55
Figure 3.4.2-5: Instantaneous Gas Volume Fraction (10 cm/s)
After (a) 1 Second and (b) 5 Seconds
Table 3.4.2-2: Mesh Validation for Bubble Column
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Average Gas Velocity (m/s)
Max Liquid Volume Fraction
Max Static Pressure (Pa)
Analysis
Value
7006
6750
0.625945
0.313947
0.998733
4929.094
56
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 %
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL
The bubble column model discussed in Section 3.4.2 was expanded to include a
population balance model (PBM) so that the growth, coalescence, and breakup of the
bubble swarm within the column can be tracked.
For additional information on
population balance and the model implemented within Fluent, see Appendix B.
A population balance model with three discrete bubble sizes was added to the
Section 3.4.2 bubble column model. Table 3.4.3-1 lists the input used to create the
population balance model implemented in this section.
Table 3.4.3-1: Population Balance Model Input
Input
Value
Discrete
3
0.0075595 m
0.0047622 m
0.0030000 m
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
25 %
50 %
25 %
Luo
0.072 N/m
Luo
0.072 N/m
Hagesather
Figure 3.4.3-1 shows 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 significant differences. One of the more
obvious differences 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
much more uniform without any large areas of high gas volume. This is most noticeable
at the bottom of the bubble column.
57
(a)
(b)
Figure 3.4.3-1: Instantaneous Gas Volume Fraction with PBM
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.3-2 shows 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 shows that the top of the bubble column has the largest liquid velocities.
This is not as noticeable in Figure 3.4.2-3 where the liquid velocity is more uniform
from top to bottom because the bubbles remain at a constant diameter. Greater liquid
velocities are achieved at the top of the bubble column with the population balance
model because of bubble coalescence. Table 3.4.3-1 shows that there are more, larger
bubbles at the top of the column than at the bottom. The larger bubbles have more
surface area which cause more drag between the liquid and gas phases. The larger
bubbles also attain higher velocities because the buoyancy forces are larger.
The
combination of higher gas velocities and larger drag forces cause the liquid velocity to
be greater.
Figure 3.4.3-3 shows a comparison between the gas velocity vectors at 1 second
and 5 seconds after gas has begun flowing through the bubble column. Similar to
58
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 bubble column
where there are no sections of little or no movement.
This is different from
Figure 3.4.2-4b where areas of no movement (in the center of the column) are prevalent.
(a)
(b)
Figure 3.4.3-2: Bubble Column Liquid Vector Velocity with PBM
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 fraction 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
to cause the bubbles to break apart.
The impact that surface tension has on bubble size distribution was tested by
reducing the surface tension by a factor of ten to 0.0072 N/m. Table 3.4.3-3 shows the
59
bubble size distribution at the inlet and outlet of the bubble column with the reduced
surface tension. The smaller surface tension reduces the driving force for bubbles to
coalesce and significantly reduces the average bubble diameter.
Figure 3.4.3-3: Bubble Column Liquid Vector Velocity with PBM
After (a) 1 Second and (b) 5 Seconds
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.072 N/m
Bin-0 (0.76 cm)
Bin-1 (0.48 cm)
Bin-2 (0.30 cm)
Inlet
(Fraction)
0.250
0.500
0.250
Outlet
(Fraction)
0.865
0.117
0.018
Net
(Fraction)
+0.615
-0.383
-0.232
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)
Inlet
(Fraction)
0.250
0.500
0.250
Outlet
(Fraction)
0.495
0.335
0.170
60
Net
(Fraction)
+0.245
-0.165
-0.080
3.5
BOILING FLOWS
3.5.1
POOL BOILING
Pool boiling occurs when a liquid transforms to 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. [10]
Table 3.5.1-1 lists the important input needed to replicate the results shown in
this section.
Table 3.5.1-1: Pool Boiling Input
Input
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
Value
0.01 m
0.05 m
Planar
Transient
0.002 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
On
Laminar
Mixture
Schiller-Nauman
Manninen et al.
Evaporation-Condensation
0.0002 m
372 K
383 K
373 K
0
Material Properties (Water) [15]
Density
Specific Heat
Thermal Conductivity
Viscosity
Heat of Vaporization
Material Properties (Vapor) [15]
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 Water Density
Density (kg/m3)
974.9
958.4
Temperature (K)
348
373.15
Figure 3.5.1-1 shows the instantaneous gas volume fraction after 0.9 seconds and
1.7 seconds of heating. These two time points were chosen because the first time point
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
shows how the fluid and vapor interact at a high level.
The evolution of steam generation, upward movement (due to buoyancy) and
liquid refill is shown in Figure 3.5.1-1 through Figure 3.5.1-3. Figure 3.5.1-1a shows
that the entire bottom of the control volume is heated and some steam has formed (two
areas of significant steam generation are shown in green). Figure 3.5.1-1b shows the
vapor moving upward (teal region) and liquid taking its place (blue area at the bottom).
62
(a)
(b)
Figure 3.5.1-1: Instantaneous Gas Volume Fraction
After (a) 0.9 Seconds and (b) 1.7 Seconds
Figure 3.5.1-2 and Figure 3.5.1-3 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 3.5.1-1). As
vapor is formed 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. This causes large velocity gradients and mixing.
63
(a)
(b)
Figure 3.5.1-2: Instantaneous Liquid Velocity Vectors
After (a) 0.9 Seconds and (b) 1.7 Seconds
(a)
(b)
Figure 3.5.1-3: Instantaneous Gas Velocity Vectors
After (a) 0.9 Seconds and (b) 1.7 Seconds
64
Figure 3.5.1-4 shows the volume fraction of vapor on the heated surface after
2 seconds.
Vapor is being produced significantly at two locations (vapor volume
fraction is at a maximum), 0.0008 m and 0.0095 m. 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 3.5.1-4: Volume Fraction of Vapor on Heated Surface
To ensure that the mesh had no impact on the results, a mesh validation was
performed. The mesh validation was performed by comparing the results shown in this
section (“Analysis Value” in Table 3.5.1-3) to a second case with an increased number
of nodes and elements (“Mesh Validation” in Table 3.5.1-3). The second case was
exactly the same as the case used in this section except that its mesh was refined. The
results from the mesh validation are shown in Table 3.5.1-3, and prove that the results
are mesh independent.
Table 3.5.1-3: Mesh Validation for Pool Boiling
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 BOILING
Subcooled boiling involves intense interactions between the liquid and vapor
phases and therefore modeling can be a challenge. The Eulerian multiphase model is
most appropriate because it is the only Fluent option capable of modeling multiple
separate, yet interacting phases. When modeling subcooled boiling, there are three
parameters of great importance. These parameters are the 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 departing bubble diameter is the bubble size when it leaves the
heated surface and is dependent on the amount of subcooling and a balance of surface
tension and buoyance forces. The bubble departure frequency is the rate at which
bubbles are generated at an active nucleation site and it is dependent on heat flux and
buoyancy and drag forces.
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 formulation, 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.
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
66
effective superheat was used rather than the actual wall superheat. This correlation
accounts for both the heated surface conditions and fluid properties and can be written
as [1]:
−4.4
1
2𝜎𝑇𝑠𝑎𝑡
𝑏𝑤
𝑒𝑓𝑓 𝜌𝑔 ℎ𝑓𝑔
𝑁𝑎 = 𝑑2 [∆𝑇
]
𝑓(𝜌∗ )
𝑓(𝜌∗ ) = 2.157 ∗ 10−7 ∗ 𝜌∗−3.2 ∗ (1 + 0.0049𝜌∗ )
𝜌∗ = (
𝜌𝑙 −𝜌𝑔
𝜌𝑔
)
Determining the lift off bubble diameter is crucial because the bubble size
influences the interphase heat and mass transfer through the interfacial area
concentration and momentum drag terms. Many correlations have been determined;
however, the three discussed are applicable at 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]
On the basis of the balance between the buoyancy and surface tension forces at
the heating 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 more comprehensive correlation was proposed by Unal which included the
effect of subcooling and the convection velocity and heater wall properties [1]:
𝑑𝑏𝑤 =
2.42 ∗ 10−5 ∗ 𝑝0.709 ∗ 𝑎
√𝑏𝛷
where
𝑎=
(𝑄𝑤 − ℎ𝑇𝑠𝑢𝑏 )1/3 𝑘𝑙
2𝐶 1/3 ℎ𝑓𝑔 √𝜋𝑘𝑙 ⁄𝜌𝑙 𝑐𝑝𝑙 𝜌𝑔
√
𝑘𝑤 𝜌𝑤 𝑐𝑝𝑤
𝑇𝑠𝑢𝑏
;𝑏 =
𝑘𝑙 𝜌𝑙 𝑐𝑝𝑙
2[1 − (𝜌𝑔 − 𝜌𝑙 )]
67
𝐶=
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 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 subcooled flow boiling model developed in Fluent uses the inputs listed in
Table 3.5.2-1 to understand the impact that different boiling models and initial
conditions have on axial liquid volume fraction.
Table 3.5.2-1: Subcooled Flow Boiling Input
Input
Value
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Near Wall Treatment
Turbulent Intensity*
Multiphase
Drag
Lift
68
0.03 m
0.50 m
Axisymmetric
Steady
Pressure Based
Relative
-9.8 m/s2 (X-direction)
On
Realizable k-ϵ
Enhanced
4.2079 %
Eulerian
Schiller-Nauman
Boiling-Moraga
Heat Transfer
Mass Transfer
Correlations
Interfacial Area
Bubble Diameter
Initial Conditions
Mass Flow Rate
Inlet Fluid Temperature
Wall Heat Flux
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)
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)
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 Boiling Water Properties
368 K
370 K
Density (kg/m )
961.99
960.59
Specific Heat (J/kg-K)
4210.0
4212.1
Viscosity (kg/m-s)
0.0002978
0.0002914
Conductivity (W/m-K)
0.6773
0.6780
Heat of Vaporization (J/kgmol)
N/A
N/A
Surface Tension (N/m)
N/A
N/A
* Saturation temperature at atmospheric pressure.
3
69
373.15 K*
958.46
4215.5
0.0002822
0.6790
40622346
0.0589
To investigate the impact that the boiling models have on liquid volume fraction,
a set of cases using the inputs listed in Table 3.5.2-1 and Table 3.5.2-3 were analyzed.
Based on the modeling options in Fluent, six combinations were possible. The liquid
volume fraction at different axial heights and the average liquid volume fraction between
cases were compared.
Table 3.5.2-3: Boiling Model 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 3.5.2-1, 3.5.2-2 and 3.5.2-3, respectively. Although these figures are
specific to Case 1, their trends can be applied to all of the cases analyzed. Figure 3.5.2-1
shows how the liquid temperature increases.
Note that the maximum bulk liquid
temperature is about 373 K which is the fluid saturation temperature and therefore
maximum temperature achievable by the liquid.
Figure 3.5.2-1: Case 1 - Temperature (K)
Figure 3.5.2-2 shows that the liquid volume fraction decreases as the fluid travels
down the pipe. The phase change is caused by energy transferred from the walls to the
flowing liquid and the small amount of subcooling at the pipe entrance.
70
Figure 3.5.2-2: Case 1 - Liquid Volume Fraction
Figure 3.5.2-3 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 show
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 3.5.2-3: Case 1 - Mass Transfer Rate (kg/m3-s)
The volume weighted liquid volume fraction for the six cases described in
Table 3.5.2-3 are 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 in boiling
model has only a small impact on the overall liquid volume fraction for the conditions
examined. The results also shows 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 than Cases 1 through 3. 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 than if the
Lemmert-Chawla nucleation site density model was employed. Comparing the results
from a bubble departure diameter model perspective reveals that no one model has a
71
tendency to predict more or less of a liquid volume fraction. Thus, the nucleation site
density model has a greater impact on liquid volume fraction than the bubble departure
diameter model.
Table 3.5.2-4: Boiling Model 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 3.5.2-4 shows the liquid volume fraction at nine axial heights for the six
cases described in Table 3.5.2-3. Although Table 3.5.2-4 shows that the models predict
similar liquid volume fractions within the entire control volume, Figure 3.5.2-4 shows
that there are noticeable differences. First, there is a significantly higher liquid volume
fraction 5 to 10 cm from the pipe inlet in Case 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 8 mm
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.
A second parametric study using the subcooled boiling model described in
Table 3.5.2-1 was used to understand how inlet temperature, mass flow and heat flux
impact liquid volume fraction. For this set of cases, the active nucleation site density
was determined by the Lemmert and Chwala correlation, the bubble departure diameter
was determined by the Tolubinsky and Kostanchuk correlation and the bubble departure
frequency was determined by the Cole correlation.
The liquid properties at three
different temperatures are shown in Table 3.5.2-2 [15]. Six cases were analyzed in total
as part of this parametric study. Case 1 from the first study is used as the nominal case
to which the other six 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 six cases analyzed is documented in Table 3.5.2-5.
72
(a) Case 1
(b) Case 2
(c) Case 3
(d) Case 4
(e) Case 5
(f) Case 6
Figure 3.5.2-4: Liquid Volume Faction Vs. Position for Cases 1-6
73
Table 3.5.2-5: Input Parametric Case Matrix
Case Number
1 (base)
7
8
9
10
11
12
Inlet Temperature
(K)
370
370
370
372
368
370
370
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 liquid volume fraction for the six cases described in
Table 3.5.2-5 are displayed in Table 3.5.2-6. Table 3.5.2-6 shows that the maximum and
minimum liquid volume fractions occur in Case 8 and Case 9, respectively. The large
impact that inlet temperature has on liquid volume fraction is 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. It is the
specific heat property of water that makes it so useful 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.
Therefore, the initial conditions do not proportionally impact liquid volume fraction.
Table 3.5.2-6: Input Parametric 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
Table 3.5.2-7 shows the liquid volume fraction at nine axial heights for the cases
described in Table 3.5.2-5. This table allows for a finer comparison of liquid volume
fraction between the cases analyzed. Table 3.5.2-7 verifies the relationships developed
74
using Table 3.5.2-6 and making observations based on overall liquid volume fraction is
acceptable.
Table 3.5.2-7: Axial Height Liquid Volume Fraction
Height
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
Case 1
1.00000
0.99168
0.97680
0.96151
0.93987
0.91595
0.89784
0.88190
0.85984
Case 7
1.00000
0.98880
0.97050
0.95036
0.92220
0.89812
0.87830
0.85540
0.80840
Case 8
1.00000
0.99397
0.98231
0.97201
0.95598
0.93589
0.91644
0.90250
0.89019
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 3.5.2-5 shows the liquid volume fraction of the axial heights in
Table 3.5.2-6 with respect to distance from the centerline. The x-axis is position, or
distance from the centerline. The pipe wall is located at 0.015 m. The impact that inlet
temperature (Case 8 and Case 9) has on liquid volume fraction is highly visible in
Figure 3.5.2-5. Case 9 shows significant voiding in the centerline after 0.25 m from the
pipe inlet due to the high inlet temperature (subcooling of about 1 K). Case 8 shows the
opposite where 0.400 m from the pipe inlet there is no voiding 0.010 m from the pipe
centerline.
The liquid volume fraction at the nine axial heights from Cases 7 through 12
were compared to the liquid volume fraction of the base case (Case 1) at the same axial
height using the following three equations for heat flux, inlet temperature and mass flow,
respectively, where i stands for the axial height 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
75
(a) Case 7
(b) Case 8
(c) Case 9
(d) Case 10
(e) Case 11
(f) Case 12
Figure 3.5.2-5: Liquid Volume Faction Vs. Position for Cases 7-12
76
The results of comparing the values from Table 3.5.2-7 using the three 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. This calculation was carried out for Cases 7
and 8 since they both alter heat flux. The change in liquid volume fraction for each axial
location was averaged to produce the overall impact that heat flux has on liquid volume
fraction. The same process was followed for inlet temperature (Cases 9 and 10) and
mass flow rate (Cases 11 and 12).
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
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 shows 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 in
absolute terms (i.e. 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 (heat
flux, inlet temperature and mass flow rate). 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 shows 1% of
the Case 1 value (for example, 90 kW/m2 * 0.01 = 0.9 kW/m2), the fourth column shows
the results from Table 3.5.2-8, and the fifth column shows the outcome when columns
three and four are multiplied together.
77
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction
Initial
Condition
Heat Flux
Temperature
Mass Flow
Case 1
Value
90 kW/m2
370 K
0.3 kg/s
1% of Case 1
Value
0.9 kW/m2
3.70 K
0.003 kg/s
Table 3.5.2-8
Results
-0.00154 (kW/m2)-1
-0.08028 (K)-1
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 reduce by 0.00139, a 1% increase in temperature causes the average
liquid void fraction to reduce 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 impact on the results, a mesh validation was
performed. The mesh validation was performed by comparing the results shown in this
section (“Analysis Value” in Table 3.5.2-10) to a second case with an increased number
of nodes and elements (“Mesh Validation” in Table 3.5.2-10). The second case was
exactly the same as the case used in this section except that its mesh was refined. The
results from the mesh validation are shown in Table 3.5.2-10, and prove that the results
are mesh independent.
Table 3.5.2-10: Mesh Validation for Subcooled Boiling
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
78
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. The areas
explored include natural convection, laminar flow, turbulent flow with and without heat
transfer, two-phase flow, pool boiling and subcooled flow boiling.
Natural convection models of a heated horizontal cylinder and a heated vertical
plate were developed in Section 3.1.
These models implemented the Boussineq
approximation to calculate density and thus temperature 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 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, as expected, 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 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 a slight
distance from the entrance due to entrance effects causing a surge in radial velocity
which caused a dramatic reduction in axial velocity. The turbulent flow model with the
energy equation employed showed a very small increase in the fluid velocity magnitude
due to the constant mass flow rate and the reduction in density due to the added energy.
Two-phase flow involving water and air was examined as part of Section 3.4.
The first model was a mixing tank using an air jet to stir the liquid. Effects of 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.
79
Gas holdup was observed due to phase drag forces and displacement. 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
causes 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 the 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 liquid volume fraction was
explored. Numerous cases were analyzed that increased or decreased the heat flux, the
inlet temperature or the mass flow compared to a base case. The differences in liquid
volume fraction between the cases were used to develop relationships between heat flux,
inlet temperature and mass flow rate with respect to liquid volume fraction. 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.
80
REFERENCES
1.
Yeoh, G. H; Tu, J. Y., “Modelling Subcooled Boiling Flows,” Nova Science
Publishers, Inc., 2009.
2.
Krepper, E.; Koncar, B.; Egorov, Y., “CFD Modeling of Subcooled Boiling –
Concept,
Validation
and
Application
to
Fuel
Assembly
Design,”
Elsevier B.V., 2006.
3.
Bird, R. B., Steward, W. E., Lightfoot, E. N., “Transport Phenomenon,” Wiley &
Sons Inc., 2nd Edition, 2007.
4.
Patankar, S. V., “Numerical Heat Transfer and Fluid Flow,” Hemisphere
Publishing Co., 1st Edition, 1980.
5.
Kays, William, Crawford, Michael, Bernhard, Weigand, “Convective Heat and
Mass Transfer,” McGraw-Hill, 4th Edition, 2005.
6.
Incropera, DeWitt, Bergman, Levine, “Introduction to Heat Transfer,” Wiley &
Sons Inc., 5th Edition, 2007.
7.
Tennekes, H; Lumley, J. L., “A First Course in Turbulence,” The MIT Press, 1972.
8.
ANSYS, Inc., “ANSYS Fluent 14.0 Theory Guide,” 2012.
9.
F. H. Harlow; P. I. Nakayama, “Transport of Turbulence Energy Decay Rate,” Los
Alamos Sci. Lab., LA-3854, 1968.
10.
Tong, L. S. “Boiling heat Transfer and Two-Phase Flow,” Wiley & Sons Inc., 2nd
Edition, 1965.
11.
Faghri, A.; Zhang, y.; Howell, J., “Advanced Heat and Mass Transfer,” Global
Digital Press, 2010.
81
12.
Eckert, E. R. G., “Introduction to the Transfer of Heat and Mass,” 1st Edition,
1950.
13.
Ingham, D. B., “Free-Convection Boundary Layer on an Isothermal Horizontal
Cylinder,” Journal of Applied Mathematics and Physics, Vol. 29, 1978.
14.
Ostrach, S., “An Analysis of Laminar Free-Convection Flow and Heat Transfer
About a Flat Plate Parallel to the Direction of the Generating Body Force,” Report
1111 – National Advisory Committee for Aeronautics.
15.
NIST/ASME Steam Properties, Database 10, Version 2.11, 1996.
16.
Krepper, E.; Rzehak, R., “CFD for Subcooled Flow Boiling: Simulation of
DEBORA Experiments,” Elsevier B.V., 2011.
82
APPENDIX A: MULTIPHASE FLOW MODELS
A.1
VOLUME OF FLUID MODEL
The VOF model can model two or more immiscible fluids by solving a single set
of momentum equations and tracking the volume fraction of each of the fluids
throughout the domain. Typical applications include 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 any liquid-gas interface. [8]
A.2
MIXTURE MODEL
The mixture model is a simplified multiphase model that can be used in different
ways. It can be used to model multiphase flows where the phases move at different
velocities, but assume local equilibrium over short spatial length scales. It can be used to
model homogeneous multiphase flows with very strong coupling and phases moving at
the same velocity and lastly, the mixture models are used to calculate non-Newtonian
viscosity.
The mixture model can model 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.
Typical applications include sedimentation, cyclone separators, particle-laden flows with
low loading, and bubbly flows where the gas volume fraction remains low.
The mixture model is a good substitute for the full Eulerian multiphase model in
several cases. A full multiphase model may not be feasible when there is a wide
distribution of the particulate phase or when the interphase laws are unknown or their
reliability can be questioned. A simpler model like the mixture model can perform as
well as a full multiphase model while solving a smaller number of variables than the full
multiphase model. [8]
83
APPENDIX B: 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, and so on), different
particles in the population can be distinguished and their behavior can be described. [8]
The population balance model gives the ability to track steam bubbles on a
particle size basis after they have detached from a heated wall. The fate of a steam
bubble traveling in a subcooled bulk fluid is not well understood. There are a number of
possibilities that can occur which include breakup into smaller steam bubbles due to
turbulent eddies, coalescence of multiple bubbles into one larger bubble or shrinkage due
to transfer of energy from the bubble to the surrounding fluid.
The growth rate is based on particle volume and therefore surface area. In
nucleate boiling, the bulk fluid is subcooled. When steam bubbles form on the heated
surface and eventually detach, they travel within the subcooled bulk fluid loosing energy
through the steam-liquid interface causing the bubbles to shrink.
The birth and death of particles can occur due to breakage and aggregation
processes. In the case of subcooled nucleate boiling, mixing caused by turbulence plays
an important role. Particle birth (and death) is caused by the breakage of a single large
bubble into multiple smaller bubbles due to liquid turbulence eddies. Particle death (and
birth) is due to the coalescence of multiple small bubbles into one larger bubble.
In boiling applications, another way that bubbles are born is through phase
change. Bubbles form on the heated wall at preferential locations called nucleation sites.
The number of potential nucleation sites is dependent on the surface condition of the
heated wall. A very smooth surface has a low number of cavities and therefore a low
84
number of potential nucleation sites. A rough surface has a large number of cavities and
therefore a large number of potential nucleation sites. However, just because a heated
surface has a high number of potential nucleation sites it does not mean that they are all
active. An empirical formula governing the population of active sites is:
̅ = 𝑁0 exp (−
𝑁
𝐾
3 )
𝑇𝑤𝑎𝑙𝑙
Where N0 and K represent the liquid and surface conditions [10]. It can be seen
that the population of active sites is a strong function of wall temperature and therefore
heat flux. [10]
B.1
EQUATION FORMULATION
The goal of this section is to present an overview of the theory and governing
equations used to calculate particle growth and nucleation. [8]
B.1.1 PARTICLE STATE VECTOR
The particle state vector is characterized by a set of external coordinates (𝑥),
which denote the spatial position of the particle and “internal coordinates” (φ), which
could include particle size, composition, and temperature. From these coordinates, a
number density function 𝑛(𝑥, φ, t) can be postulated where φ ϵ 𝛺𝑉 , 𝑥 𝜖 𝛺𝑥 . Therefore,
the average number of particles in the infinitesimal volume 𝑑𝑉𝑥 𝑑𝑉𝜑 is 𝑛(𝑥, φ, t) 𝑑𝑉𝑥 𝑑𝑉𝜑 .
The total number of particles in the entire system is
∫ ∫ 𝑛𝑑𝑉𝑥 𝑑𝑉𝜑
𝑉𝑥
⃑ 𝑉𝜑
The local average number density in physical space (that is, the total number of
particles per unit volume is given by
𝑁(𝑥, 𝑡) = ∫ 𝑛𝑑𝑉𝜑
𝛺𝜑
The total volume fraction of all particles is given by
85
𝛼(𝑥, 𝑡) = ∫ 𝑛 𝑉(𝜑) 𝑑𝑉𝜑
𝛺𝜑
Where 𝑉(𝜑) is the volume of a particle in state φ.
B.1.2 POPULATION BALANCE EQUATION
Assuming that φ is the particle volume, the transport equation for the number
density function is given as:
𝜕
𝜕𝑡
[𝑛(𝑉, 𝑡)] + ∇ ∙ [𝑢
⃑ 𝑛(𝑉, 𝑡)] + ∇𝑉 ∙ [𝐺𝑉 𝑛(𝑉, 𝑡)] =
𝑉
∫ 𝑎
2 0
1
(𝑉 − 𝑉 ′ , 𝑉 ′ ) 𝑛 (𝑉 − 𝑉 ′ , 𝑡) 𝑛 (𝑉 ′ , 𝑡) 𝑑𝑉 ′
∞
Birth due to Aggregation
− ∫0 𝑎 (𝑉, 𝑉 ′ ) 𝑛 (𝑉, 𝑡) 𝑛 (𝑉 ′ , 𝑡) 𝑑𝑉 ′
Death due to Aggregation
+ ∫𝛺 𝜌𝑔 (𝑉 ′ ) 𝛽 (𝑉|𝑉 ′ ) 𝑛 (𝑉 ′ , 𝑡) 𝑑𝑉 ′
Birth due to Breakage
−𝑔 (𝑉) 𝑛 (𝑉, 𝑡)
Death due to Breakage
𝑉
The boundary and initial conditions are given by
𝑛 (𝑉, 𝑡 = 0) = 𝑛𝑉 ; 𝑛(𝑉 = 0, 𝑡) 𝐺𝑉 = 𝑛̇ 0
Where 𝑛̇ 0 is the nucleation rate in particles / m3-s.
86
APPENDIX C: SUBCOOLED BOILING
Subcooled flow boiling is a very efficient form of heat transfer that is described
as having high heat transfer rates and low levels of wall superheat. Figure 3.5.2-1 [10]
shows the various boiling regimes as a function of void fraction as the fluid travels along
the heated surface. The void fraction in Region I is small and the level of voiding is
mainly dependent on surface flux conditions. This region is known as wall voidage.
Region II is known as the bubble detachment region and is mainly dependent upon the
bulk flow characteristics. Eventually bulk boiling begins to occur and the MartinelliNelson curve can be used to determine void fraction.
Figure 3.5.2-1: Void Fraction in Various Boiling Regimes
If the heat flux from a heated wall into a subcooled fluid is slowly increased for a
set of initial conditions, a point will be reached, known as the onset of nucleate boiling,
where the transition from single-phase convection to subcooled flow boiling occurs.
During nucleate boiling, heat transfer rates increase dramatically due to bubbles
formation on the heated surface. As the bubble generation rate increases, heat carried by
bubbles becomes a larger portion of the total energy transferred. If the wall heat flux is
87
allowed to increase further, the transition from subcooled flow boiling to saturated flow
boiling will occur when the bulk fluid temperature reaches the saturation point.
Although saturated flow boiling is an important form of heat transfer, the primary topic
of this section is subcooled flow boiling.
The efficient heat transfer mechanism provided by vapor generation in subcooled
flow boiling is limited to the point where vapor generation exceeds the rate at which the
liquid can replace it on the heated surface which leads to a greater portion of the heated
surface being covered by vapor. This is known as the critical heat flux where the heat
transfer coefficient begins to decrease with increasing temperature leading to an unstable
situation. In this event, the temperature of the heated surface increases rapidly which can
lead to melting or destruction of the heater. The critical heat flux is dependent upon the
working fluid, the mass flux, the inlet temperature and the saturation pressure. The
verification of design improvements and their influence on the critical heat flux requires
expensive experiments. Therefore, the supplementation of experiments by numerical
analyses is of high interest in industrial applications. [16]
88