Modeling of Subcooled Flow Boiling and Other Heat
Transfer and Fluid Flow Scenarios
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 ......................................................................................................... vii
ABSTRACT ...................................................................................................................... x
1. INTRODUCTION ....................................................................................................... 1
1.1
Current Research ................................................................................................ 2
1.2
Work Herein ....................................................................................................... 3
2. HEAT TRANSFER AND FLUID FLOW: THEORY ................................................ 5
2.1
GOVERNING EQUATIONS ............................................................................ 5
2.2
NATURAL CONVECTION .............................................................................. 8
2.3
LAMINAR FLOW ........................................................................................... 10
2.4
TURBULENT FLOW ...................................................................................... 11
2.4.1
Calculating Turbulence Parameters ..................................................... 13
2.5
TWO-PHASE FLOW ...................................................................................... 15
2.6
BOILING HEAT TRANSFER ........................................................................ 18
2.7
POPULATION BALANCE ............................................................................. 20
2.7.1
Background .......................................................................................... 20
2.7.2
Equation Formulation .......................................................................... 21
3. HEAT TRANSFER AND FLUID FLOW: MODELING ......................................... 23
3.1
NATURAL CONVECTION ............................................................................ 23
3.1.1
HORIZONTAL CYLINDER ............................................................... 23
3.1.2
VERTICAL PLATE ............................................................................ 31
3.2
LAMINAR FLOW ........................................................................................... 37
3.3
TURBULENT FLOW ...................................................................................... 41
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER ..................... 41
3.3.2
TURBULENT FLOW WITH HEAT TRANSFER ............................. 46
iii
3.4
3.5
TWO-PHASE FLOW ...................................................................................... 50
3.4.1
GAS MIXING TANK .......................................................................... 50
3.4.2
BUBBLE COLUMN ............................................................................ 54
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL ... 61
BOILING FLOWS ........................................................................................... 66
3.5.1
POOL BOILING .................................................................................. 66
3.5.2
SUBCOOLED BOILING .................................................................... 72
4. DISUSSION AND CONCLUSIONS ........................................................................ 88
5. REFERENCES .......................................................................................................... 89
APPENDIX A: MULTIPHASE FLOW MODELS ........................................................ 92
A.1 Volume of Fluid Model .................................................................................... 92
A.2 Mixture Model ................................................................................................. 92
A.3 Eulerian Model ................................................................................................. 93
iv
LIST OF TABLES
Table 2.4.1-1: Input ......................................................................................................... 14
Table 2.4.1-2: Turbulence Parameter Calculation ........................................................... 14
Table 3.1.1-1: Horizontal Cylinder Input ........................................................................ 24
Table 3.1.1-2: Water Density........................................................................................... 24
Table 3.1.1-1: Mesh Validation for Horizontal Cylinder ................................................ 30
Table 3.1.2-1: Vertical Plate Input .................................................................................. 31
Table 3.1.2-2: Water Density........................................................................................... 32
Table 3.1.2-1: Mesh Validation for Vertical Plate .......................................................... 36
Table 3.2-1: Laminar Flow Input..................................................................................... 37
Table 3.2-2: Water Density.............................................................................................. 37
Table 3.3.2-1: Mesh Validation for Laminar Flow .......................................................... 40
Table 3.3.1-1: Turbulent Flow Without Heat Transfer Input .......................................... 41
Table 3.3.2-1: Turbulent Flow With Heat Transfer Input ............................................... 46
Table 3.3.2-2: Water Density........................................................................................... 47
Table 3.3.2-1: Mesh Validation for Turbulence With Heat Transfer .............................. 49
Table 3.4.1-1: Gas Mixing Tank Input ............................................................................ 50
Table 3.4.1-1: Mesh Validation for Gas Mixing Tank .................................................... 53
Table 3.4.2-1: Bubble Column Input ............................................................................... 55
Table 3.4.2-1: Mesh Validation for Bubble Column ....................................................... 60
Table 3.4.3-1: Population Balance Model Input .............................................................. 61
Table 3.4.3-1: Bubble Size Distribution – Surface Tension of 0.072 N/m ..................... 65
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.0072 N/m ................... 65
Table 3.5.1-1: Pool Boiling Input .................................................................................... 66
Table 3.5.1-2: Water Density........................................................................................... 67
Table 3.5.1-1: Mesh Validation for Pool Boiling ............................................................ 71
Table 3.5.2-1: Subcooled Flow Boiling Input ................................................................. 76
Table 3.5.2-2: Liquid Properties ...................................................................................... 77
Table 3.5.2-3: Boiling Model Case Input ........................................................................ 77
Table 3.5.2-4: Boiling Model Case Input ........................................................................ 79
Table 3.5.2-5: Subcooled Boiling Case Matrix ............................................................... 82
v
Table 3.5.2-6: Axial Height Liquid Volume Fraction ..................................................... 82
Table 3.5.2-4: Subcooled Boiling Case Results .............................................................. 87
Table 3.5.2-5: Mesh Validation for Subcooled Boiling .................................................. 87
vi
LIST OF FIGURES
Figure 2.3-1: Example of Turbulent Flow ....................................................................... 11
Figure 2.5-1: Flow Regimes ............................................................................................ 15
Figure 2.5-2: Baker Flow Pattern .................................................................................... 16
Figure 2.6-1: Boiling Heat Transfer Regimes ................................................................. 18
Figure 3.1.1-1: Horizontal Cylinder Temperature ........................................................... 25
Figure 3.1.1-2: Horizontal Cylinder Density ................................................................... 25
Figure 3.1.1-3: Horizontal Cylinder Velocity Vector ...................................................... 26
Figure 3.1.1-4: Interference Around a Horizontal Cylinder in Free Convection ............ 27
Figure 3.1.1-5: Temperature at θ = 30° Vs. Radial Distance .......................................... 28
Figure 3.1.1-6: Temperature at θ = 90° Vs. Radial Distance .......................................... 28
Figure 3.1.1-7: Temperature at θ = 180° Vs. Radial Distance ........................................ 29
Figure 3.1.2-1: Vertical Plate Temperature Plot .............................................................. 32
Figure 3.1.2-2: Vertical Plate Velocity Vector Plot ........................................................ 33
Figure 3.1.2-3: Interference Around a Vertical Plate in Free Convection Flow ............. 34
Figure 3.1.2-4: Dimensionless Temperature as a Function of Prandtl Number .............. 35
Figure 3.1.2-5: Dimensionless Temperature as a Function of Prandtl Number (Fluent) 35
Figure 3.2-1: Laminar Flow Velocity Profile Vs. Positon............................................... 38
Figure 3.3.1-1: Velocity Magnitude Vs. Position ............................................................ 42
Figure 3.3.1-2: Wall Shear Stress Vs. Position ............................................................... 43
Figure 3.3.1-3: Radial Velocity ...................................................................................... 43
Figure 3.3.1-4: dAxial-Velocity/dx Vs. Position ............................................................. 43
Figure 3.3.1-5: Flow Results for Smaller Mass Flow Rate (0.5 kg/s) ............................. 44
Figure 3.3.1-6: Flow Results for Larger Mass Flow Rate (1.5 kg/s) ............................... 44
Figure 3.3.1-7: Turbulent Kinetic Energy ....................................................................... 45
Figure 3.3.1-8: Production of Turbulent Kinetic Energy ................................................ 45
Figure 3.3.2-1: Temperature ............................................................................................ 47
Figure 3.3.2-2: Turbulent Kinetic Energy ....................................................................... 47
Figure 3.3.2-3: Radial Velocity ....................................................................................... 47
Figure 3.3.2-4: Velocity Magnitude Vs. Position ............................................................ 48
Figure 3.3.2-5: Wall Shear Stress Vs. Axial Position...................................................... 48
vii
Figure 3.4.1-1: Gas Volume Fraction .............................................................................. 51
Figure 3.4.1-2: Liquid Vector Velocity ........................................................................... 52
Figure 3.4.1-3: Gas Vector Velocity................................................................................ 52
Figure 3.4.2-1: Bubble Column Reactor.......................................................................... 54
Figure 3.4.2-2: Instantaneous Gas Volume Fraction ....................................................... 56
Figure 3.4.2-3: Instantaneous Liquid Velocity Vectors................................................... 57
Figure 3.4.2-4: Instantaneous Gas Velocity Vectors ....................................................... 58
Figure 3.4.2-5: Instantaneous Gas Volume Fraction (10 cm/s) ....................................... 59
Figure 3.4.3-1: Instantaneous Gas Volume Fraction with PBM ..................................... 62
Figure 3.4.3-2: Bubble Column Liquid Vector Velocity with PBM ............................... 63
Figure 3.4.3-3: Bubble Column Liquid Vector Velocity with PBM ............................... 64
Figure 3.5.1-1: Instantaneous Gas Volume Fraction ....................................................... 68
Figure 3.5.1-2: Instantaneous Liquid Velocity Vectors................................................... 69
Figure 3.5.1-3: Instantaneous Gas Velocity Vectors ....................................................... 69
Figure 3.5.1-4: Volume Fraction of Vapor on Heated Surface ....................................... 70
Figure 3.5.2-1: Void Fraction in Various Boiling Regimes ............................................ 72
Figure 3.5.2-2: Case 1 - Temperature (K) ....................................................................... 78
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction ........................................................... 78
Figure 3.5.2-4: Base Case - Mass Transfer Rate (kg/m3-s) ............................................. 79
Figure 3.5.2-4: Case 1 – Liquid Volume Faction Vs. Position........................................ 79
Figure 3.5.2-4: Case 2 – Liquid Volume Faction Vs. Position........................................ 80
Figure 3.5.2-4: Case 3 – Liquid Volume Faction Vs. Position........................................ 80
Figure 3.5.2-5: Case 4 – Liquid Volume Faction Vs. Position........................................ 81
Figure 3.5.2-7: Case 5 – Liquid Volume Faction Vs. Position........................................ 81
Figure 3.5.2-7: Case 6 – Liquid Volume Faction Vs. Position........................................ 81
Figure 3.5.2-8: Base Case – Liquid Volume Faction Vs. Position .................................. 83
Figure 3.5.2-9: Case 1 - Liquid Volume Faction Vs. Position ........................................ 83
Figure 3.5.2-10: Case 2 - Liquid Volume Faction Vs. Position ...................................... 84
Figure 3.5.2-11: Case 3 - Liquid Volume Faction Vs. Position ...................................... 84
Figure 3.5.2-12: Case 4 - Liquid Volume Faction Vs. Position ...................................... 85
Figure 3.5.2-13: Case 5 - Liquid Volume Faction Vs. Position ...................................... 85
viii
Figure 3.5.2-14: Case 6 - Liquid Volume Faction Vs. Position ...................................... 86
ix
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 curs during
subcooled boiling is discussed in detail and followed up by a numerical model.
Numerical models to analyze natural circulation, laminar flow, turbulent flow with and
without heat transfer, two-phase flow, pool boiling and subcooled flow boiling were
developed. The commercial software Fluent was used to create and analyze the models.
Different modeling techniques and numerical solvers were employed depending on the
scenario analyzed. The results of each model were compared to experimental data when
available to prove its validity.
especially in the natural convection and laminar flow regimes. The bubble column
results were similar to those found experimentally. The impact the different boiling
model options has on voiding was investigated. It was found that all the models behave
similarly but xxxxx tends to predict more voiding than xxxxxx. Relationships between
mass flow, inlet temperature and heat flux with respect to voiding were also developed.
Overall, Fluent showed good agreement in many areas with experimental data.
x
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
(coal, oil, natural gas and uranium), 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 the energy released during the fission
process to heat the surrounding water called the Reactor Coolant System (RCS). 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 is also complex due to the extreme energy transfer and
phase change. In Pressurizer Water Reactors (PWRs), the RCS 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 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 amount 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
CURRENT RESEARCH
Subcooled boiling is a complex phenomenon involving heat transfer, fluid flow
and phase change characterized by the combination of numerous phenomena such as
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 nucleate flow boiling was developed by Del Valle and Kenning. Their
model accounts for bubble dynamics at the heated wall using concepts developed
initially by Graham and Hendricks for wall heat flux partitioning during nucleate pool
boiling. More recently, a new approach to the partitioning of the wall heat flux has been
proposed by Basu et al. The fundamental idea of the 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. [19]
Additionally, focus has been put towards accurately modeling the three most
impactful parameters in subcooled 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 lift diameter.
Tolubinsky and Kostanchuk proposed the most simplistic which evaluates bubble
departure as a function of subcooling temperature.
Kocamustafaogullari and Ishii
improved this model by including the contact angle of the bubble.
Finally, Unal
produced a more comprehensive correlation which includes the effect of subcooling and
the convection velocity and heater wall properties. All three of these correlations are
available in Fluent. The most common bubble departure frequency correlation for CFD
was developed by Cole. It is derived from the bubble departure diameter and a balance
between buoyancy and drag forces and is available in Fluent.
2
Another improvement to the modeling of subcooled flow boiling in recent years is
the additional use of population balance equations (PBEs) to better determine how the
swarm of bubbles interact after detaching from the heated surface. This is a relatively
new way of investigating subcooled boiling and is recommended by Krepper et. al. [13]
and investigated by Yeoh and Tu [19].
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.
1.2
WORK HEREIN
This thesis produced an investigation on subcooled flow boiling using Fluent.
Fluent is a widely accepted commercial CFD code that can simulate complex heat
transfer and fluid flow regimes. The objective of this thesis was to understand how
modeling options and initial conditions have on the level of subcooled boiling that
occurs at different axial locations. The impact that initial conditions such as mass flow
rate, inlet temperature and heat flux have on voiding were evaluated and understood.
Also, the impact that the models discussed in Section 1.1 have on voiding was
investigated
Due to its complexity, the development of a subcooled 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 is natural
convection.
The theory of natural convection is described in Section 2.2 and the
analytical modeling results are discussed in Section 3.1.
Two natural convection
geometries are analyzed. The first is a horizontal cylinder suspended in a pool and the
second is a vertical plate in a pool. The second model incorporated a turbulence model.
The theory of laminar flow is described in Section 2.3 and the analytical modeling
results are discussed in Section 3.2. The theory of turbulence is described in Section 2.4
and the analytical modeling results are discussed in Section 3.3. Section 3.3 discusses
two scenarios, turbulent flow without heat transfer and turbulent flow with heat transfer.
The fourth model developed contains a two-phase flow model. The theory of two-phase
3
flow is described in Section 2.5 and the analytical modeling results are discussed in
Section 3.4. Two scenarios of two-phase flow are discussed. The first is a gas mixing
tank and the second is a bubble column. Both scenarios use water and air as the primary
and secondary phases. 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 0 and the analytical modeling results are discussed in
Section 0. Two different models are created, the first is a pool boiling and the second is
for subcooled boiling. After each model is developed, a mesh validation is performed
and the results are compared to known experimental data whenever 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 for the various phenomena that are
involved in subcooled nucleate 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 [5]. 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
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦
𝜕𝑝
𝜌(
+ 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
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
scenarios 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. [6]
The goal in CFD 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)
has to be finite. By only focusing on the solution of the differential equations at
disscrete locations, the need to find an exact solution to the differential equation has
been replaced.
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. [6]
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
6
equation is derived from the differential equation governing the dependent variable and
thus expresses the same physical information as the differential equation. The piecewise
nature of the profile (or mesh) is created by the finite number of grid points that
participate in a given discretization equation. The value of the dependent variable at a
grid point thereby influences the value of the dependent variable in its immediate area.
As the number of grid points becomes very large, the solution of the discretization
equations is expected to approach the exact solution of the corresponding differential
equation. This is true because as the grid points get closer together, the change in value
between neighboring grid points becomes small and the actual details of the profile
assumption become less important.
This is where the term “mesh independent”
originates. If there are too few grid points (coarse mesh), the profile assumptions can
impact the solution results and the discretization equation solution will not match the
differential equation solution. To ensure that the discretization equation results are not
dependent on the profile assumptions, the solution should be checked for mesh
independence. [6]
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 [6]. Fluent allows for
manipulation of the relaxation constants for may 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.
7
2.2
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
[3]. 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 [3]. For example, in a system where a heated surfaces
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 that use fins to distribute heat and in 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 the 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 =
8
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 be
either laminar or turbulent. Rayleigh numbers less than 108 indicate a buoyancy-induced
laminar flow, with transition to turbulence occurring at about Ra ≈ 109. [4]
In many engineering applications, convection is primarily 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.
9
2.3
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 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 flow with a Reynolds number less than 2300. The velocity of laminar
flow in a pipe is can be calculated by [3]:
𝑢=
𝑟𝑠2
𝑑𝑃
𝑟2
(− ) (1 − 2 )
4𝜇
𝑑𝑥
𝑟𝑠
Or, in terms of the mean velocity, V:
𝑢 = 2𝑉 (1 −
𝑟2
)
𝑟𝑠2
The energy equation for flow through a circular pipe assuming symmetric heat
transfer,, fully developed flow and constant fluid properties is [3]:
𝑢
𝛿𝑇
1𝛿
𝛿𝑇
𝛿 2𝑇
= 𝛼[
(𝑟 ) + 2 ]
𝛿𝑥
𝑟 𝛿𝑟 𝛿𝑟
𝛿𝑥
10
2.4
TURBULENT FLOW
In fluid dynamics, turbulence is a flow regime characterized by chaotic and
stochastic property changes. They exist everywhere in nature from the jet stream to the
oceanic currents.
Turbulent flows are highly irregular or 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. [15]
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. [15]
A common example of the transition of laminar flow to turbulent flow is smoke
rising from a cigarette.
Figure 2.3-1: Example of Turbulent Flow
11
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 because
of 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. [1]
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 [1]. 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 equations above 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
12
′ ′
̅̅̅̅̅̅
time-averaged solutions to the Navier–Stokes equations. An additional term,(−𝜌𝑢
𝑖 𝑢𝑗 ),
known as the Reynolds stress now appear in the equation as a results of using the RANS
method. [1]
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 [18]. 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, the assumption is 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 mean-square vorticity fluctuation. The term “realizable” means that the
model satisfies certain mathematical constraints on the Reynolds stresses, consistent
with the physics of turbulent flows. [1]
2.4.1
CALCULATING TURBULENCE PARAMETERS
All of the CFD 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 regime. For instance, based on the conditions in Table 0-1, the equations in
Table 0 were used to determine the boundary and initial condition inputs for the
turbulent flow models discussed in Section 0.
13
Table 2.4.1-1: 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 [1]
Table 2.4.1-2: 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
4.22483 %
ℎ
−
1
8
= 0.16 ∗ 42314
3
2
𝑘 = (𝑢𝑎𝑣𝑔 ∗ 𝐼)
2
2
3
𝑚
= (1.41726 ∗ 0.0422483)
2
𝑠
3/2
3/4 k
ε = C𝜇
𝑙
3/2
0.0053785
3/4
= 0.09
0.0021
14
0.03 m
0.0053785 m2/s2
0.030859 m2/s3
2.5
TWO-PHASE FLOW
Fluid flows that contain two or more components is referred to as multi-phase 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 will primarily focus on two-phase
flow involving water and air while Section 0 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, wave flow,
slug flow, annular flow, dispersed flow and fog or mist flow.
A schematic
representation of each of these flow patterns is shown in Figure 2.5-1 [2, Figure 3-2].
Figure 2.5-1: Flow Regimes
The flow patterns shown in Figure 3.3-1 can be further classified into three
categories, bubbly flow, slug flow and annular flow. Bubbly flow is when the liquid
phase is continuous and the vapor phase is discontinuous and the vapor phase is
distributed in the liquid in the form of bubbles. This flow pattern occurs at low gas
15
volume fractions. Subcooled boiling can be 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. In
subcooled film boiling, due the heated walls, there is a vapor annulus with a liquid core
(i.e., inverse annular flow). [2]
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 the gas volume fraction and the flow velocity. The flow pattern of a system
can be determined using the Baker flow criteria shown in Figure 2.5-2 [2, Figure 3-4].
Figure 2.5-2: Baker Flow Pattern
Two-phase flows obey all of the basic laws of fluid mechanics. However, the
equations are more complicated and more numerous than those for single-phase flow
because there are more equations to solve (second set of conservation equations for the
16
secondary phase) and more equations are introduced (mass transfer, etc.). Additionally,
phenomena like phase-interface interactions and slip must be considered.
Three
common multiphase flow models are Volume of Fluid, Mixture and Eulerian, each with
varying strengths and computational demand. These are implemented in the Fluent
software and are discussed further in Appendix A.
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.
This thesis uses the Eulerian model for the
simulation of two phase flows.
17
2.6
BOILING HEAT TRANSFER
Boiling heat transfer is defined as a mode of heat transfer that occurs when
saturated liquid changes to gas. 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 requiring high heat transfer capacities, 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 boiling heat transfer capacity and mechanism. There are two
basic types of boiling, pool boiling and flow boiling. Flow boiling is boiling in a
flowing stream of fluid, where the heating surface may be the channel wall confining the
flow. Both types of boiling heat transfer can be broken down into four regimes which
are shown in Figure 0-1 [16].
Figure 0-1: 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 [2].
The second regime of boiling, from point A to point C, is called nucleate boiling.
During this stage vapor bubbles are generated at certain preferred locations on the heated
surface called nucleation sites. Nucleation sites are often microscopic cavities or cracks
in the surface. When the liquid near the wall superheats, it evaporates forming bubbles
18
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. In this scenario, the bubbles form on the
heated surface but tend to condense after leaving the heated surface.
The second
subregime is when bulk boiling occurs in a saturated liquid. In this case, the bubbles do
not collapse. Note that both subregimes may take place between points A and C.
Nucleate boiling has very high heat transfer rates for only small temperature difference
between the bulk fluid and the heated surface. For this reason it is considered the most
efficient boiling regime for heat transfer. [2]
As the heated surface increases in temperature, more and more nucleation sites
become active. The bubbles 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. The vapor begins to
form an insulating blanket around the heated surface and thereby dramatically increases
the surface temperature. This is called the boiling crisis or departure from nucleate
boiling. [16]
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. [2]
If the temperature difference between the surface and the fluid continues to increase,
stable film boiling is achieved. When this occurs, there is a continuous vapor blanket
surrounding the heated surface and phase change occurs at the liquid-vapor interface
instead of the heated surface. During this regime, most heat transfer is carried out by
radiation. [16]
19
2.7
POPULATION BALANCE
2.7.1
BACKGROUND
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. [1]
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 number of
20
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 [2]. It can be seen that
the population of active sites is a strong function of wall temperature and therefore heat
flux. [2]
2.7.2
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. [1]
2.7.2.1 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
21
The total volume fraction of all particles is given by
𝛼(𝑥, 𝑡) = ∫ 𝑛 𝑉(𝜑) 𝑑𝑉𝜑
𝛺𝜑
Where 𝑉(𝜑) is the volume of a particle in state φ.
2.7.2.2 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.
22
3. HEAT TRANSFER AND FLUID FLOW: MODELING
3.1
NATURAL CONVECTION
Two examples of natural convection are examined in the following subsections: a
heated horizontal cylinder and a heated vertical plate submerged in an infinite pool.
These examples were chosen because of their simplicity, the fact they are commonly
found in nature and because they have been previously studied and results are available
for the validation of the numerical computations.
3.1.1
HORIZONTAL CYLINDER
In this scenario, a cylinder with a constant surface temperature is submerged in
an infinite pool of liquid. The cylinder is slightly warmer than the surrounding fluid and
therefore energy passes from the cylinder to the nearby fluid causing its temperature to
increase. Table 3.1.1-1 lists the important case information needed to replicate the
results shown in this section.
23
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: Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
24
The fluid temperature field after 20 seconds is shown in Figure 3.1.1-1.
Figure 3.1.1-1: Horizontal Cylinder Temperature
As the temperature increases, the fluid expands and its density decreases. As the
fluid density decreases, buoyancy forces take affect and the warmer, less dense fluid
rises. The density change is shown in Figure 3.1.1-2. Notice that even some distance
away from the cylinder there is a density change. This is caused by energy transfer by
conduction through the fluid which causes a small density changes in the fluid that is not
in direct contact with the cylinder. This density gradient is shown by the color transition
surrounding the cylinder from orange to yellow to green to blue.
Figure 3.1.1-2: Horizontal Cylinder Density
25
As the fluid rises, it separates from the cylinder and new, colder fluid takes its
place. When the warm fluid rises, it loses energy to the surrounding, cooler bulk fluid.
As this heat transfer process occurs the buoyancy driving head diminishes causing the
fluid to climb more slowly until it eventually stops. At this point it is pushed to the side
by the fluid travelling upwards below it and begins to sink. This motion creates two
small convection cells to the left and right of the rising plume about two diameters above
the heated cylinder.
This process continues ad infinitum as long as there is a
temperature gradient (i.e., buoyancy driving head). The convection cells are clearly
shown in the velocity vector plot, Figure 3.1.1-3.
Figure 3.1.1-3: Horizontal Cylinder Velocity Vector
26
To verify that the model produced realistic results, the solution was compared to
experimental data. Figure 3.1.1-4 shows isotherms surrounding a horizontal tube in
natural convection flow as revealed by an interference photograph.
(a)
(b)
Figure 3.1.1-4: Interference Around a Horizontal Cylinder in Free Convection
(a) is from [9] and (b) shows isotherms from Fluent
The Fluent model of a horizontal cylinder submerged in an infinite pool is in
qualitative agreement to experimental data. Figure 3.1.1-4 shows comparable results.
Both have isotherms that extend away from the plate and grow in distance away from
one another as they get farther from the plate.
Quantitative experimental data from Ingham [10] was compared to the Fluent
results to validate the model. 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 various
dimensionless times at 30°, 90° and 180°, respectively. 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.
27
(a)
(b)
Figure 3.1.1-5: Temperature at θ = 30° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
(a)
(b)
Figure 3.1.1-6: Temperature at θ = 90° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
28
(a)
(b)
Figure 3.1.1-7: Temperature at θ = 180° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
29
The heated horizontal cylinder model developed in Fluent shows good agreement
compared to experimental data at three different locations. This helps give confidence in
the information that is gathered from the model. 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 given in this section are mesh independent.
Table 3.1.1-1: Mesh Validation for Horizontal Cylinder
Analysis Value Mesh Validation
Difference
Number of Nodes
19716
23636
19.88 %
Number of Elements
38688
46400
19.93 %
Max Velocity (m/s)
0.01627
0.01621
-0.37 %
Max Total Temperature (°F)
309.9239
309.9531
0.01 %
Min Density (kg/m3)
993.1765
993.1625
0.00 %
Because the model for a uniformly heated horizontal cylinder submerged in a pool as
calculated by Fluent produce results that are similar to those measured experimentally
and are mesh independent, it can confidently be stated that the results are reliable.
30
3.1.2
VERTICAL PLATE
The second scenario of natural convection involves a heated vertical plate with a
constant surface temperature submerged in an infinite pool of liquid. Like the cylinder,
the plate is also slightly warmer than the surrounding fluid and therefore energy passes
from the plate to the fluid causing its temperature to increase. Table 0-1 lists the
important case information needed to replicate the results shown in this section.
Table 0-1: 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
31
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 0-2
PISO
Least Square Cell Based
PRESTO!
Second Order Upwind
Second Order Upwind
Second Order Implicit
Table 0-2: Water Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
The fluid temperature field after 20 seconds is shown in Figure 0-1.
Figure 0-1: Vertical Plate Temperature Plot
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. Notice how the thermal boundary layer is small at the bottom
of the plate and much larger at the top. The thermal boundary layer expands as the
momentum boundary layer expands which helps pull warm fluid away from the hot
plate.
For more information on thermal and momentum boundary layers, see
Reference 3.
32
Figure 0-2: Vertical Plate Velocity Vector Plot
Figure 0-2 shows the fluid velocity in vector form. 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.
Comparing Figure 0-2 (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 when 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 different is small, it is
counterintuitive. The horizontal cylinder produced a larger maximum velocity because
the buoyancy driving head does not compete with a drag force generated by the heated
surface. Although the plate continued to heat the fluid as it travels upward, the velocity
is limited by friction which is why the plate scenario had a smaller maximum velocity.
To ensure that the model is giving realistic results, the solution was again compared
to experimental data. Figure 0-3 shows isotherms surrounding a vertical plate in natural
convection flow as revealed by an interference photograph.
33
(a)
(b)
Figure 0-3: Interference Around a Vertical Plate in Free Convection Flow
(a) is from [9] and (b) is from Fluent
The model of a vertical plate submerged in an infinite pool is in qualitative
agreement to experimental data. Figure 4.2-3 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 [11] was compared to the Fluent
results to assess the accuracy of the model.
Figure 0-4 and Figure 0-5 show a
comparison of dimensionless temperature versus dimensionless distance for various
Prandtl numbers. Figure 0-4a shows theoretical values and Figure 0-4b compares some
of the theoretical values to experimental data. The information contained in Figure 4.2-5
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.
34
(a)
(b)
Figure 0-4: Dimensionless Temperature as a Function of Prandtl Number
(a) Theoretical Values and (b) Experimental Values [11]
Figure 0-5: Dimensionless Temperature as a Function of Prandtl Number (Fluent)
The heated vertical plate model developed in Fluent produced very similar
temperature results to the experimental data for five different Prandtl numbers. This
helps give confidence in the information that is gathered from the model. To ensure that
the mesh had no impact on the results, a mesh validation was performed. The mesh
35
validation compared the results shown in this section (“Analysis Value” in Table 0-1) to
a second case with an increased number of nodes and elements (“Mesh Validation” in
Table 0-1). 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 0-1,
and prove that the results given in this section are mesh independent.
Table 0-1: 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
36
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
Using Fluent, a simple axisymmetric flow model was developed to gain a better
understanding of laminar flow in a pipe. The Reynolds number for the scenario was
selected as 352 which is well within the laminar regime. Table 3.2-1 lists the important
case information 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 0-2
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: Water Density
Density (kg/m3)
999.9
994.1
Temperature (K)
273
308
37
One of the most notable characteristics of laminar flow is the parabolic shape of
its velocity profile. Figure 3.2-1 shows the velocity magnitude versus position (distance
from the pipe centerline) for various distances from the pipe entrance. The distance
from the pipe entrance is given in the legend. For example, “line-10cm” shows the
velocity profile 10 cm from the pipe entrance. As the flow develops, i.e., 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 tends to stay about the same distance from the
centerline as it travels through the pipe. Figure 3.2-2 shows the temperature profile of
the laminar flow analyzed. Diffusion and conduction are the primary forms of heat
transfer. The growth of the fluid thermal boundary layer as it travels down the pipe is
also visible in Figure 3.2-2.
Figure 3.2-2: Laminar Flow Temperature
38
Figure 3.2-3 shows the radial flow velocity. As expected, the radial velocity for
most of the pipe is near zero and is less than 10-3 times the average axial velocity.
Radial velocity spikes near the entrance of the pipe due to pipe boundary conditions and
entrance effects but this has little impact on system as a whole.
Figure 3.2-4: Laminar Flow Radial Velocity
Laminar flow also tends to create momentum boundary layers which cause
frictional force on the wall. Figure 3.2-5 shows the computed drag force on the wall.
Figure 3.2-5: Laminar Flow Wall Shear Stress
The wall stress is much larger in the first 5 cm due to entrance effects. Once the
entrance effects dissipate, the wall shear stress slowly decreases as the flow becomes
more and more parabolic. To ensure that the mesh had no impact on the results, a mesh
validation was performed.
The results from the mesh validation are shown in
Table 3.3.2-1, and prove that the results given in this section are mesh independent. 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
39
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 given in this
section are mesh independent.
Table 3.3.2-1: 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
40
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
Due to its random and chaotic nature, turbulent flows are considered to be
homogenized very quickly in terms of energy and solute concentration.
A simple
axisymmetric flow model was developed in Fluent to gain a better understanding of
turbulent flow in a pipe. The Reynolds number for the scenario was selected as 42314
which is well within the turbulent regime.
Table 3.3.1-1 lists the important case
information 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 0-2.
41
Figure 3.3.1-1 shows the velocity magnitude versus position (distance from the
pipe centerline) at various distances from the pipe entrance. The distance from the pipe
entrance is given in the legend and for example, “line-10cm” shows the velocity profile
10 cm from the pipe entrance. The velocity profile of turbulent flow differs significantly
in two ways when compared to the velocity profile of laminar flow (Section 0). First,
the turbulent flow velocity profiles are much flatter. This means that the fluid velocity
doesn’t decrease significantly until close to the pipe wall. Second, entrance effects
dissipate much quicker in turbulent flow than in laminar flow [3] and thus the fluid
velocity reaches a steady state velocity profile quicker. Figure 3.3.1-1 (turbulent flow)
shows that flow reaches a steady profile at about 10 cm from the pipe entrance.
Figure 0-1 (laminar flow) shows that flow reaches a steady profile at about 45 cm from
the pipe entrance. This qualitatively matches experimental data well.
Figure 3.3.1-1: Velocity Magnitude Vs. Position
Figure 3.3.1-2 shows the wall shear stress versus distance from the entrance. The
shear stress is very large at the beginning and decays to the steady state value after about
10 cm (location where steady state profile is reached). The large increase in shear stress
at the beginning of the pipe (~1-2 cm) is caused by the entrance effects. Figure 3.3.1-3
shows that that maximum absolute radial velocity occurs near the pipe entrance. To
conserve momentum, the axial velocity must decrease near the entrance due to the spike
in radial velocity. Figure 3.3.1-4 shows that the greatest reduction in axial velocity
42
occurs near the pipe entrance which is in line with expectations.
Since shear stress is
related to change in 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
43
To further understand the impact of entrance effects, two additional cases were run using
a slightly smaller (Figure 3.3.1-5) and a slightly larger mass flow rate (Figure 3.3.1-6).
(a)
(b)
(c)
Figure 3.3.1-5: Flow Results for Smaller Mass Flow Rate (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 (1.5 kg/s)
(a) radial velocity (b) wall shear stress vs. position (c) daxial-velocity/dx vs. position
44
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 radial
velocity approaches a near zero value (it does not reach zero because turbulent flows
have cross mixing and thus some radial velocity), the change in axial velocity as a
function of position reaches zero and the wall shear stress reaches a steady state. This
distance changes depending on the mass flow rate. A large mass flow rate requires a
greater length to reach steady state.
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 near the wall because the wall generates
turbulent kinetic energy. 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.
45
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 case
information 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 0-2.
46
Table 3.3.2-2: 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.
Figure 3.3.2-1: Temperature
The turbulent kinetic energy shown in Figure 3.3.2-2 is very similar to that shown in
Figure 3.3.1-7 which is expected since the heat addition has a small impact on fluid
velocity.
If the heat transfer rate to the fluid was increased sufficiently such that
buoyancy effects began to influence flow, then the turbulent kinetic energy between the
two scenarios would differ.
Figure 3.3.2-2: Turbulent Kinetic Energy
Figure 3.3.2-3 shows the radial velocity which matches well with Figure 3.3.1-3
because of the same reasons explained in the previous paragraph.
Figure 3.3.2-3: Radial Velocity
47
Comparing the velocity profiles for the two scenarios (Figure 3.3.1-1 and
Figure 3.3.2-4) reveals that the velocity magnitude is slightly larger for the case with
heat transfer. The heat transfer that occurs causes the density of the fluid to decrease and
to maintain a constant mass flow through the pipe, the velocity increases slightly.
Figure 3.3.2-4: Velocity Magnitude Vs. Position
As expected, the wall shear stress shown in Figure 3.3.2-5 matches well with the
wall shear stress shown in Figure 3.3.1-2.
Figure 3.3.2-5: Wall Shear Stress Vs. Axial Position
48
To ensure that the mesh had no impact on the results, a mesh validation was
performed. The results from the mesh validation are shown in Table 3.3.2-1, and prove
that the results given in this section are mesh independent. 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 given in this section are mesh
independent.
Table 3.3.2-1: 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, does not create any localized phase change. Thus, the relationships
developed in Section 3.3.1 are applicable to scenarios with heat transfer as long as the
heat flux is small.
49
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 case information needed to replicate the results shown
in this section.
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
50
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
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
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 shows the gas volume fraction within the mixing tank, Figure 3.4.1-2
shows the liquid vector velocity and Figure 3.4.1-3 shows the gas vector velocity,
respectively, after 5 seconds.
Figure 3.4.1-1: Gas Volume Fraction
51
Figure 3.4.1-2: Liquid Vector Velocity
Figure 3.4.1-3: Gas Vector Velocity
52
The velocities shown in Figure 3.4.1-2 and 3.4.1-3 are similar for the liquid and
gas meaning that the drag forces are strong. The maximum velocity is also greater than
the inlet velocity meaning that buoyancy forces are also playing a large role.
Figure 3.4.1-2 shows that there are a number of small eddies within the tank which is
providing a significant amount of mixing.
To ensure that the mesh had no impact on the results, a mesh validation was
performed. The results from the mesh validation are shown in Table 3.4.1-1, and prove
that the results given in this section are mesh independent. The mesh validation was
performed by comparing the results shown in this section (“Analysis Value” in
Table 3.4.1-1) to a second case with an increased number of nodes and elements (“Mesh
Validation” in Table 3.4.1-1). 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 3.4.1-1, and prove that the results given in this section are mesh
independent.
Table 3.4.1-1: 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
53
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. Per Figure 3.3-1, the flow is bubbly,
meaning the gas is dispersed as bubbles in a continuous volume of liquid. Bubbles form
and travel upwards through the column due to the inlet gas velocity and buoyancy. The
gas introduced through the spargers provides mixing, similar to Section 3.4.1. 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 and decrease in size due to
coalescence and breakup. Coalescence occurs when two or more bubbles collide and the
thin liquid barrier between them ruptures to form a larger bubble. Bubbles breakup
when they collide with turbulent eddies approximately equal to their 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 case information needed to replicate the results
shown in this section.
54
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
55
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 reached the top of the liquid and
caused the surface to change shape. Also, after 5 seconds the liquid level is higher than
that after 1 second by about 5 cm. This 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
56
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 can be seen at both time points.
(a)
(b)
Figure 3.4.2-3: Instantaneous Liquid Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
57
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 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) and wall drag. Figure 3.4.2-4b shows that the greatest gas velocities
occur near the walls. This is in alignment with Figure 3.4.2-2 which showed that the
highest gas volume fractions are near the walls. Higher gas volume fractions lead to
greater buoyancy forces which cause greater gas velocities.
(a)
(b)
Figure 3.4.2-4: Instantaneous Gas Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
58
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
increase than the gas hold shown in Figure 3.4.2-2, which employed used a gas inlet
velocity was 5 cm/s.
Figure 3.4.2-5: Instantaneous Gas Volume Fraction (10 cm/s)
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 results from the mesh
validation are shown in Table 3.4.2-1, and prove that the results given in this section are
mesh independent. The mesh validation was performed by comparing the results shown
in this section (“Analysis Value” in Table 3.4.2-1) to a second case with an increased
59
number of nodes and elements (“Mesh Validation” in Table 3.4.2-1). 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 3.4.2-1, and prove that the
results given in this section are mesh independent.
Table 3.4.2-1: Mesh Validation for Bubble Column
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Max Liquid Volume Fraction
Max Static Pressure (Pa)
Analysis
Value
7006
6750
0.625945
0.994955
0.998733
4929.094
60
Mesh
Validation
8785
8500
0.63157
1.16063
1.00000
4920.58
Difference
25.39 %
25.93 %
0.90 %
16.65 %
0.13 %
-0.17 %
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL
The bubble swarm within the column will not have a uniform size due to growth,
coalescence, and breakup. The bubble column model discussed in Section 3.4.2 was
expanded to include a population balance model. The implementation of a population
balance model allows for the direct calculation of changes in bubble size due to growth,
breakup, and coalescence as they travel up the column.
A population balance model with three discrete bubble sizes was added to the
Section 3.4.2 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
Method
Number of Bins
Bin-0
Bin-1
Bin-2
Bin Distribution
Bin-0
Bin-1
Bin-2
Aggregation Kernel
Model
Surface Tension
Breakage Kernel
Model
Surface Tension
Formulation
Value
Discrete
3
0.0075595 m
0.0047622 m
0.0030000 m
25 %
50 %
25 %
Luo
0.072 N/m
Luo
0.072 N/m
Hagesather
61
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. At both time points,
the gas tends to flow in slugs. After 5 seconds the gas has reached the top of the surface
of the liquid and causes it to change shape. Comparing Figure 3.4.3-1a and 3.4.3-1b
reveals that the liquid level in Figure 3.4.3-1b is higher. This is caused by drag and
displacement of the liquid by the flowing gas. 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 phases at both 1 second and 5 seconds. With the population
balance model implemented, Figure 3.4.3-1, the phase distribution is much more
uniform without any large areas of high gas volume. This is most noticeable at the
bottom of the column.
(a)
(b)
Figure 3.4.3-1: Instantaneous Gas Volume Fraction with PBM
After (a) 1 Second and (b) 5 Seconds
62
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. 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 bubbles at the top of the column than the bottom. The larger bubbles have
more surface area which causes larger drag forces between the liquid and gas. The
greater the drag force the greater the liquid velocity.
(a)
(b)
Figure 3.4.3-2: Bubble Column Liquid Vector Velocity with PBM
After (a) 1 Second and (b) 5 Seconds
63
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
Figure 3.4.2-4, the shape of the gas as it initially climbs up the bubble column is double
parabolic; 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 to no movement. This is different from Figure 3.4.2-4b where areas of
no movement (in the center of the column) are prevalent.
Figure 3.4.3-3: Bubble Column Liquid Vector Velocity with PBM
After (a) 1 Second and (b) 5 Seconds
64
The population balance model calculates the bubble size distribution at each axial
height using the Luo break up and coalescence model. Table 3.4.3-1 shows the bubble
size population fraction at the inlet and outlet of the bubble column. The table shows
that there is a strong bias for the smaller bubbles to coalesce into larger bubbles. This
means that there is a small amount of turbulence in the column to break up the bubbles
and that there is a strong desire to reduce surface area.
Table 3.4.3-1: 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
To test the impact surface tension has on the Luo model, the surface tension was
reduced by a factor of ten to 0.0072 N/m.
Table 3.4.3-2 shows the bubble size
population fraction at the inlet and outlet of the bubble column with a reduced surface
tension. The smaller surface tension significantly changes the bubble size distribution.
There is less of a bias to form larger bubbles due to a smaller coalescence driving force.
Table 3.4.3-2: 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
Net
(Fraction)
+0.245
-0.165
-0.080
A mesh validation was not performed for this model because it is so similar to the
one developed in Section 3.4.2.
For the bubble column mesh validation see
Table 3.4.2-1.
65
3.5
BOILING FLOWS
3.5.1
POOL BOILING
Pool boiling occurs when a liquid turns 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 rapidly 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 superheated. Pool
boiling flows involve complex fluid motions initiated and maintained by the nucleation,
growth, departure, and collapse of bubbles, and by natural convection. [2]
Table 3.5.1-1 lists the important case information needed to replicate the results
shown in this section.
Table 3.5.1-1: Pool Boiling Input
Input
Value
Geometry
Column Width
Column Height
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Slip
Mass Transfer
Initial Conditions
Bubble Diameter
Initial Fluid Temperature
Heater Temperature (Bottom)
Backflow Temperature (Top)
Backflow Volume Fraction (Top)
66
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) [17]
Density
Specific Heat
Thermal Conductivity
Viscosity
Heat of Vaporization
Material Properties (Vapor) [17]
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: 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. This is the
driving force behind all fluid motion. The second time point was chosen because it
shows how the fluid and vapor interact over the long term.
The evolution of steam generation and upward movement (due to buoyancy) and
liquid refill is shown in Figure 3.5.1 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. The two
areas of significant steam generation are shown in green. Figure 3.5.1-1b shows distinct
regions of fluid and vapor.
67
(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 greatest gas volume fraction (Figure 3.5.1-1). As vapor is
formed on the heated surface, it eventually detaches from the heated surface and enters
the liquid above. Due to buoyancy forces the vapor travels upward through the liquid.
Drag forces between the two phases causes the liquid to also travel upwards but at a
slower rate due to slip. The other large liquid velocity region occurs 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 evaporated liquid. This causes large
velocity gradients and mixing.
68
(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
69
To better visualize the location of the nucleation sites, Figure 3.5.1-4 shows the
volume fraction of vapor on the heated surface after 10 seconds.
Vapor is being
produces significantly at two locations (locations where the vapor volume fraction is at a
maximum), 0.0008 m and 0.0095 m.
There is one location where the vapor volume
fraction is at a local minimum, 0.005 m, where liquid is taking the place of the recently
created vapor.
Figure 3.5.1-4: Volume Fraction of Vapor on Heated Surface
70
To ensure that the mesh has no impact on the results, a mesh validation was
performed. The results from the mesh validation are shown in Table 3.5.1-1, and prove
that the results given in this section are mesh independent. The mesh validation was
performed by comparing the results shown in this section (“Analysis Value” in
Table 3.5.1-1) to a second case with an increased number of nodes and elements (“Mesh
Validation” in Table 3.5.1-1). 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 3.5.1-1, and prove that the results given in this section are mesh
independent.
Table 3.5.1-1: 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
71
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 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. This form of heat
transfer is essential for cooling applications requiring high heat transfer rates, such as
nuclear reactors and fossil boilers. Figure 3.5.2-1 [2] 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 Martinelli-Nelson 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
72
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. [12]
Subcooled boiling involves intense interaction between the liquid and vapor phases.
Due to the highly coupled phase interaction, the Eulerian multiphase model is most
appropriate multiphase model.
Additionally, there are three parameters of great
importance when modeling subcooled boiling. The parameters are the active nucleation
site density (Na), departing bubble diameter (dbw) and bubble departure frequency (f)
[19]. 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
73
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 [19]:
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 nucleation site density was created by
Kocamustafaogullari and Ishii. They assumed that the active nucleation site density
correlation developed for pool boiling could be used in forced convective system if the
effective superheat was used rather than the actual wall superheat. This correlation
accounts for both the heated surface conditions and fluid properties. This correlation can
be written as [19]:
−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 herein are applicable at low pressure subcooled flow
boiling. The first was proposed by Tolubinsky and Kostanchuk. It establishes the
bubble departure diameter as a function of the subcooling temperature [19]:
𝑑𝑏𝑤 = 𝑚𝑖𝑛 [0.0006 ∗ exp (−
74
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 [19]:
𝑑𝑏𝑤 = 2.5 ∗ 10−5 (
𝜌𝑙 − 𝜌𝑔
𝜎
) 𝜃√
𝜌𝑔
𝑔 ∗ (𝜌𝑙 − 𝜌𝑔 )
A more comprehensive correlation proposed was by Unal which includes the effect
of subcooling and the convection velocity and heater wall properties [19]:
𝑑𝑏𝑤 =
2.42 ∗ 10−5 ∗ 𝑝0.709 ∗ 𝑎
√𝑏𝛷
where
The most common bubble departure frequency correlation for CFD was developed
by Cole. It is derived from the bubble departure diameter and a balance between
buoyancy and drag forces [19]:
𝑓=√
4𝑔(𝜌𝑙 − 𝜌𝑔 )
3𝜌𝑙 𝑑𝑏𝑤
75
The subcooled flow boiling developed uses the inputs listed in Table 3.5.2-1 to
understand the impact different boiling models and initial conditions have on axial liquid
volume fraction.
Table 3.5.2-1: Subcooled Flow Boiling Input
Input
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Near Wall Treatment
Turbulent Intensity*
Multiphase
Drag
Lift
Heat Transfer
Mass Transfer
Parameters
Interfacial Area
Bubble Diameter
Initial Conditions
Mass Flow Rate
Inlet Fluid Temperature
Wall Heat Flux
Material Properties (Water)
Density
Specific Heat
Thermal Conductivity
Viscosity
Heat of Vaporization
Material Properties (Vapor)
Density
Viscosity
Thermal Conductivity
Value
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
Ranz-Marshall
RPI Boiling
See Table 3.5.2-3
Ia-Symmetric
Sauter-Mean
0.3 kg/s
370 K
90000 W/m2
See Table 3.5.2-2
See Table 3.5.2-2
See Table 3.5.2-2
See Table 3.5.2-2
See Table 3.5.2-2
0.5542 kg/m3
1.34E-05 kg/m-s
0.0261 W/m-K
76
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 0-2.
Table 3.5.2-2: Liquid Properties
368 K
370 K
Density (kg/m3)
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.
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 different combinations of
boiling models were analyzed.
The boiling model combinations are displayed in
Table 3.5.2-3. The liquid volume fraction at different axial heights as well as the
cumulative liquid volume fraction in the pipe will be 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
77
Cole
Cole
Plots of temperature, liquid volume fraction and mass transfer rate for the Case 1 are
shown in Figures 3.5.2-2, 3.5.2-3 and 3.5.2-4, respectively. Although these figures are
specific to Case 1, their trends can be applied to all of the cases analyzed. Figure 3.5.2-3
shows how the liquid temperature increases.
Note that the maximum bulk liquid
temperature is about 373 K which is the fluid saturation temperature.
Figure 3.5.2-2: Case 1 - Temperature (K)
Figure 0-3 shows how the liquid volume fraction decreases as energy is added to the
system and the fluid changes phases.
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction
Figure 3.5.2-4 is of particular interest because it shows both the generation and
destruction of steam bubbles. The light blue and green 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 turn from steam back into liquid. The generation and destruction
of steam bubbles is very characteristic of subcooled flow boiling.
78
Figure 3.5.2-4: Base Case - Mass Transfer Rate (kg/m3-s)
The volume weighted liquid volume fraction for the six cases described in
Table 0-3 are shown in Table 3.5.2-4.
In general, 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.
Table 3.5.2-4: Boiling Model Case Input
Case Number
1
2
3
4
5
6
Volume-Weighted Liquid Volume Fraction
0.91076716
0.90031346
0.90856631
0.91649488
0.91611270
Figures 3.5.2-4 through 3.5.2-10 show the liquid volume fraction at nine axial
heights for the cases described in Table 3.5.2-3.
Figure 3.5.2-4: Case 1 – Liquid Volume Faction Vs. Position
79
Figure 3.5.2-4: Case 2 – Liquid Volume Faction Vs. Position
Figure 3.5.2-4: Case 3 – Liquid Volume Faction Vs. Position
80
Figure 3.5.2-5: Case 4 – Liquid Volume Faction Vs. Position
Figure 3.5.2-7: Case 5 – Liquid Volume Faction Vs. Position
Figure 3.5.2-7: Case 6 – Liquid Volume Faction Vs. Position
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 model is
determined by the Lemmert and Chwala correlation, the bubble departure diameter is
determined by the Tolubinsky and Kostanchuk correlation and the bubble departure
frequency is determined by the Cole correlation. The liquid properties at three different
inlet temperatures are shown in Table 3.5.2-2 [17]. Seven subcooled flow boiling cases
were analyzed in total. Case 1 is the nominal case to which the other six are compared.
Cases 2 through 6 increase or decrease the inlet temperature, the mass flow or the heat
flux compared to Case 1. The input for the seven cases analyzed is documented in
Table 3.5.2-5.
81
Table 3.5.2-5: Subcooled Boiling Case Matrix
Case Number
Inlet Temperature
(K)
370
370
370
372
368
370
370
1
2
3
4
5
6
7
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 remaining scenarios were analyzed and the liquid volume fraction at nine
axial heights are shown in Table 3.5.2-3. Figure 3.5.2-5 through Figure 3.5.2-11 show
the information contained in Table 3.5.2-3 in graphical form.
Table 3.5.2-6: 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
1.0000
0.9955
0.9834
0.9669
0.9385
0.9146
0.8974
0.8811
0.8589
Case 2
1.0000
0.9941
0.9774
0.9516
0.9206
0.8974
0.8774
0.8542
0.8080
Case 3
1.0000
0.9965
0.9884
0.9780
0.9587
0.9335
0.9154
0.9025
0.8896
Case 4
1.0000
0.9663
0.8769
0.7961
0.7113
0.5771
0.4290
0.3206
0.2517
82
Case 5
1.0000
0.9987
0.9965
0.9930
0.9879
0.9824
0.9713
0.9498
0.9354
Case 6
1.0000
0.9962
0.9868
0.9704
0.9463
0.9215
0.9012
0.8843
0.8669
Case 7
1.0000
0.9944
0.9804
0.9602
0.9298
0.9106
0.8952
0.8766
0.8340
Figure 3.5.2-8: Base Case – Liquid Volume Faction Vs. Position
Figure 3.5.2-9: Case 1 - Liquid Volume Faction Vs. Position
83
Figure 3.5.2-10: Case 2 - Liquid Volume Faction Vs. Position
Figure 3.5.2-11: Case 3 - Liquid Volume Faction Vs. Position
84
Figure 3.5.2-12: Case 4 - Liquid Volume Faction Vs. Position
Figure 3.5.2-13: Case 5 - Liquid Volume Faction Vs. Position
85
Figure 3.5.2-14: Case 6 - Liquid Volume Faction Vs. Position
The liquid volume fraction at various axial heights from the six cases is compared to
the liquid volume fraction of the base case 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
86
The values from Table 3.5.2-3 were plugged into the above three equations and the
change from the base case is shown in Table 3.5.2-4. For example, at an axial height of
10 cm, by increasing the heat flux from 90 kW/m2 to 100 kW/m2 (Base Case to Case 1)
the liquid volume fraction decreased by 0.00060 / kW/m2. Since Cases 1 and 2 alter heat
flux, their change in liquid volume fraction was averaged over the entire control volume.
This shows the relationship that heat flux has on liquid volume fraction. The same
process is followed for inlet temperature (Cases 3 and 4) and mass flow (Cases 5 and 6).
Table 3.5.2-4: Subcooled Boiling Case Results
Height
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
Average
Case 1
Case 2
0.00000
0.00000
-0.00014
-0.00010
-0.00060
-0.00050
-0.00153
-0.00111
-0.00179
-0.00202
-0.00172
-0.00189
-0.00200
-0.00180
-0.00269
-0.00214
-0.00509
-0.00307
-0.00157
Case 3
Case 4
0.00000
0.00000
-0.01460
-0.00160
-0.05325
-0.00655
-0.08540
-0.01305
-0.11360
-0.02470
-0.16875
-0.03390
-0.23420
-0.03695
-0.28025
-0.03435
-0.30360
-0.03825
-0.08017
Case 5
Case 6
0.00000
0.00000
0.02333
0.03667
0.11333
0.10000
0.11667
0.22333
0.26000
0.29000
0.23000
0.13333
0.12667
0.07333
0.10667
0.15000
0.26667
0.83000
0.17111
To ensure that the mesh has no impact on the results, a mesh validation was
performed for the base case.
The results from the mesh validation are shown in
Table 3.5.2-5, and prove that the results given in this section are mesh independent.
Table 3.5.2-5: 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 Mesh Validation
9568
11968
8955
11205
0.81464
0.81724
1.00499
1.00572
0.50594
0.49853
21.4428
21.0718
87
Difference (%)
25.08%
25.13%
0.32%
0.07%
-1.46%
-1.73%
4. DISUSSION AND CONCLUSIONS
88
5. REFERENCES
1.
ANSYS, Inc., “ANSYS Fluent 14.0 Theory Guide,” 2012.
2.
Tong, L. S. “Boiling heat Transfer and Two-Phase Flow,” Wiley & Sons Inc., 2nd
Edition, 1965.
3.
Kays, William, Crawford, Michael, Bernhard, Weigand, “Convective Heat and
Mass Transfer,” McGraw-Hill, 4th Edition, 2005.
4.
Incropera, DeWitt, Bergman, Levine, “Introduction to Heat Transfer,” Wiley &
Sons Inc., 5th Edition, 2007
5.
Bird, R. B., Steward, W. E., Lightfoot, E. N., “Transport Phenomenon,” Wiley &
Sons Inc., 2nd Edition, 2007.
6.
Patankar, S. V., “Numerical Heat Transfer and Fluid Flow,” Hemisphere
Publishing Co., 1st Edition, 1980.
7.
Wallis, Graham B, “One-dimensional Two-phase Flow,” McGraw-Hill, 1st
Edition, 1969.
8.
Hinze, J. O., “Turbulence,” McGraw-Hill, 1st Edition, 1959.
9.
Eckert, E. R. G., “Introduction to the Transfer of Heat and Mass,” 1st Edition,
1950.
10.
Ingham, D. B., “Free-Convection Boundary Layer on an Isothermal Horizontal
Cylinder,” Journal of Applied Mathematics and Physics, Vol. 29, 1978.
11.
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.
12.
Krepper, E.; Rzehak, R., “CFD for Subcooled Flow Boiling: Simulation of
DEBORA Experiments,” Elsevier B.V., 2011.
13.
Krepper, E.; Koncar, B.; Egorov, Y., “CFD Modeling of Subcooled Boiling –
Concept,
Validation
and
Application
to
Fuel
Assembly
Design,”
Elsevier B.V., 2006.
14.
Degha, A. L.; Chaker, A., “Numerical Study of Subcooled Boiling In Vertical
Tubes Using Relap5/Mod3.2,” Journal of Electronic Devices, Vol. 7, 2010,
p. 240-245.
89
15.
Tennekes, H; Lumley, J. L., “A First Course in Turbulence,” The MIT Press,
1972.
16.
Faghri, A.; Zhang, y.; Howell, J., “Advanced Heat and Mass Transfer,” Global
Digital Press, 2010.
17.
NIST/ASME Steam Properties, Database 10, Version 2.11, 1996.
18.
F. H. Harlow; P. I. Nakayama, “Transport of Turbulence Energy Decay Rate,”
Los Alamos Sci. Lab., LA-3854, 1968.
90
91
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. [1]
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. [1]
92
A.3
EULERIAN MODEL
The Eulerian multiphase model in Fluent allows for the modeling of multiple
separate, yet interacting phases. The phases can be liquids, gases, or solids in nearly any
combination. An Eulerian treatment is used for each phase, in contrast to the EulerianLagrangian treatment that is used for the discrete phase model.
With the Eulerian multiphase model, the number of secondary phases is limited
only by memory requirements and convergence behavior. Any number of secondary
phases can be modeled, provided that sufficient memory is available. For complex
multiphase flows, however, you may find that your solution is limited by convergence
behavior. See Eulerian Model in the User's Guide for multiphase modeling strategies.
The Fluent Eulerian multiphase model does not distinguish between fluid-fluid and
fluid-solid (granular) multiphase flows.
93