Modeling of Subcooled Boiling in a Nuclear Reactor Core
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
Troy, New York
December, 2012
i
© Copyright 2013
By
Matthew P. Wilcox
All Rights Reserved
ii
TABLE OF CONTENTS
TABLE OF CONTENTS ................................................................................................. iii
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ABSTRACT ................................................................................................................... viii
1. INTRODUCTION ....................................................................................................... 9
2. MATHEMATICAL FORMULATION ..................................................................... 12
3. THEORY ................................................................................................................... 14
3.1
NATURAL CONVECTION ............................................................................ 14
3.2
TURBULENCE ............................................................................................... 16
3.2.1
Calculating Turbulence Parameters ..................................................... 19
3.3
TWO-PHASE FLOW ...................................................................................... 21
3.4
BOILING HEAT TRANSFER ........................................................................ 24
3.5
POPULATION BALANCE EQUATION ....................................................... 26
3.6
3.5.1
Background .......................................................................................... 26
3.5.2
Equation Formulation .......................................................................... 27
NUMERICAL METHODS.............................................................................. 29
4. NATURAL CONVECTION ..................................................................................... 31
4.1
HORIZONTAL CYLINDER ........................................................................... 31
4.2
VERTICAL PLATE ........................................................................................ 37
5. TURBULENCE ......................................................................................................... 42
5.1
TURBULENT FLOW ...................................................................................... 42
5.2
TURBULENT FLOW WITH HEAT TRANSFER ......................................... 45
6. TWO-PHASE FLOW ................................................................................................ 48
6.1
GAS MIXING TANK ...................................................................................... 48
6.2
BUBBLE COLUMN ........................................................................................ 51
6.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL ............... 56
iii
7. BOILING ................................................................................................................... 60
7.1
POOL BOILING .............................................................................................. 60
7.2
SUBCOOLED BOILING ................................................................................ 63
7.3
SUBCOOLED BOILING WITH POPULATION BALANCE MODEL ........ 72
8. REFERENCES .......................................................................................................... 73
APPENDIX A: ADDITIONAL INFORMATION .......................................................... 75
A.1 LAMINAR FLOW ........................................................................................... 75
8.1.1
Volume of Fluid Model ........................................................................ 78
8.1.2
Mixture Model ..................................................................................... 78
8.1.3
Eulerian Model ..................................................................................... 79
8.1.4
Equation Formulation .......................................................................... 79
8.1.5
Solution Method ................................................................................... 82
iv
LIST OF TABLES
Table 4.1-1: Mesh Validation for Horizontal Cylinder ................................................... 36
Table 4.2-1: Mesh Validation for Vertical Plate ............................................................. 41
Table 5.2-1: Mesh Validation for Turbulence With Heat Transfer ................................. 47
Table 6.1-1: Mesh Validation for Gas Mixing Tank ....................................................... 50
Table 6.2-1: Mesh Validation for Bubble Column .......................................................... 55
Table 6.3-1: Bubble Size Distribution – Surface Tension of 0.072 N/m ........................ 59
Table 6.3-2: Bubble Size Distribution – Surface Tension of 0.0072 N/m ...................... 59
Table 7.1-1: Mesh Validation for Pool Boiling ............................................................... 62
Table 7.2-1: Liquid Properties ......................................................................................... 64
Table 7.2-2: Subcooled Boiling Case Matrix .................................................................. 65
Table 7.2-3: Axial Height Liquid Volume Fraction ........................................................ 66
Table 7.2-4: Subcooled Boiling Case Results ................................................................. 71
Table 7.2-5: Mesh Validation for Subcooled Boiling ..................................................... 71
v
LIST OF FIGURES
Figure 3.2-1: Example of Turbulent Flow ....................................................................... 17
Figure 3.3-1: Flow Regimes ............................................................................................ 21
Figure 3.3-2: Baker Flow Pattern .................................................................................... 22
Figure 3.3-3: Flow Pattern Boundaries for Vertical Upflow of Air and Water ............... 23
Figure 3.4-1: Boiling Heat Transfer Regimes ................................................................. 24
Figure 4.1-1: Horizontal Cylinder Temperature .............................................................. 31
Figure 4.1-2: Horizontal Cylinder Density ...................................................................... 32
Figure 4.1-3: Horizontal Cylinder Velocity Vector ......................................................... 33
Figure 4.1-4: Isotherms Around a Horizontal Cylinder in Free Convection ................... 34
Figure 4.1-5: Temperature at θ = 30° Vs. Radial Distance ............................................. 35
Figure 4.1-6: Temperature at θ = 90° Vs. Radial Distance ............................................. 35
Figure 4.1-7: Temperature at θ = 180° Vs. Radial Distance ........................................... 36
Figure 4.2-1: Vertical Plate Temperature Plot ................................................................. 37
Figure 4.2-2: Vertical Plate Velocity Vector Plot ........................................................... 38
Figure 4.2-3: Isotherms Around a Vertical Plate in Free Convection Flow .................... 39
Figure 4.2-4: Dimensionless Temperature as a Function of Prandtl Number ................. 40
Figure 4.2-5: Dimensionless Temperature as a Function of Prandtl Number (Fluent) ... 40
Figure 5.1-1: Velocity Magnitude Vs. Position ............................................................... 42
Figure 5.1-2: Wall Shear Stress Vs. Axial Distance ........................................................ 43
Figure 5.1-3: Radial Velocity .......................................................................................... 44
Figure 5.1-4: Turbulent Kinetic Energy .......................................................................... 44
Figure 5.1-5: Production of Turbulent Kinetic Energy ................................................... 44
Figure 5.2-1: Temperature ............................................................................................... 45
Figure 5.2-2: Turbulent Kinetic Energy .......................................................................... 45
Figure 5.2-3: Radial Velocity .......................................................................................... 46
Figure 5.2-4: Velocity Magnitude Vs. Position ............................................................... 46
Figure 5.2-5: Wall Shear Stress Vs. Axial Position......................................................... 47
Figure 6.1-1: Gas Volume Fraction ................................................................................. 48
Figure 6.1-2: Liquid Vector Velocity .............................................................................. 49
Figure 6.1-3: Gas Vector Velocity................................................................................... 49
vi
Figure 6.2-1: Bubble Column Reactor............................................................................. 51
Figure 6.2-2: Instantaneous Gas Volume Fraction .......................................................... 52
Figure 6.2-3: Instantaneous Liquid Velocity Vectors...................................................... 53
Figure 6.2-4: Instantaneous Gas Velocity Vectors .......................................................... 54
Figure 6.3-1: Instantaneous Gas Volume Fraction with PBM ........................................ 57
Figure 6.3-2: Bubble Column Liquid Vector Velocity with PBM .................................. 58
Figure 7.1-1: Instantaneous Gas Volume Fraction .......................................................... 61
Figure 7.1-2: Instantaneous Liquid Velocity Vectors...................................................... 61
Figure 7.1-3: Volume Fraction of Vapor on Heated Surface .......................................... 62
Figure 7.2-1: Void Fraction in Various Boiling Regimes ............................................... 63
Figure 7.2-2: Base Case - Temperature (K)..................................................................... 65
Figure 7.2-3: Base Case - Liquid Volume Fraction ......................................................... 66
Figure 7.2-4: Base Case - Mass Transfer Rate (kg/m3-s) ................................................ 66
Figure 7.2-5: Base Case – Liquid Volume Faction Vs. Position ..................................... 67
Figure 7.2-6: Case 1 - Liquid Volume Faction Vs. Position ........................................... 67
Figure 7.2-7: Case 2 - Liquid Volume Faction Vs. Position ........................................... 68
Figure 7.2-8: Case 3 - Liquid Volume Faction Vs. Position ........................................... 68
Figure 7.2-9: Case 4 - Liquid Volume Faction Vs. Position ........................................... 69
Figure 7.2-10: Case 5 - Liquid Volume Faction Vs. Position ......................................... 69
Figure 7.2-11: Case 6 - Liquid Volume Faction Vs. Position ......................................... 70
Figure A.1-1: Laminar Flow Velocity Profile Vs. Positon.............................................. 76
Figure 4.4.3-1: Particle Size Distribution ........................................................................ 82
vii
ABSTRACT
viii
1. INTRODUCTION
Electricity is one of the greatest discoveries of the 19th century and its use has
significantly increased the world’s standard of living. One of the more common ways
electricity is generated is by converting thermal energy, from a fuel source, into
electrical energy. The Rankine Cycle is an energy conversion 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. 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 paper will focus on the convective heat transfer that occurs in a
nuclear reactor core, and more specifically, subcooled boiling.
Subcooled boiling occurs when a fluid comes into 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 and heat flux. 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.
Subcooled boiling is a very complex heat transfer and fluid flow scenario that
can be 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.
9
Krepper et. al. [12] [13] using CFX and Degha et. al. [14] using RELAP have created
Computational Fluid Dynamic (CFD) models to calculate the liquid volume fraction at
various axial heights along a heated surface. A similar investigation will be discussed in
this paper however ANSYS Fluent® will be used to create the CFD model. Fluent is a
widely accepted commercial CFD code that can simulate complex heat transfer and fluid
flow regimes. The objective of this thesis will be to provide insight about the level of
subcooled boiling that occurs at different axial locations and compare the results to those
determined using RELAP and CFX.
Due to its complexity the development of a subcooled boiling model is
performed in stages. With the development of each model, a more complex fluid flow or
heat transfer scenario will be analyzed. The first and most simple model created is
natural convection. The theory of natural convection is described in Section 3.1 and the
analytical modeling results are discussed in Section 4.
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 created is a turbulence model.
The theory of turbulence is described in Section 3.2 and the analytical modeling results
are discussed in Section 5. Section 5 discusses two scenarios, turbulent flow with heat
transfer and turbulent flow without heat transfer. The third model created is two-phase
flow. The theory of two-phase flow is described in Section 3.3 and the analytical
modeling results are discussed in Section 6. 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 is boiling heat transfer. The theory of boiling heat transfer is described
in Section 3.7 and the analytical modeling results are discussed in Section 7. 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.
As an exploratory measure, this paper will also investigate the use of population
balance equations (PBEs) to better determine how the steam bubbles interact after
detaching from the heated surface. This is a relatively new way of subcooled boiling
10
and none of these aforementioned papers implement a population balance model but it is
recommended by Krepper et. al. [13].
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. The PBE model will be used to
determine the number of steam bubbles in the core, reveal how they develop over time
and determine if the bubbles shrink and collapse or coalesce with other bubbles and
grow in size. Due to modeling limitations of Fluent, CFX will be used to perform this
investigation.
11
2. MATHEMATICAL FORMULATION
Continuity equations are used in physics and engineering to describe how various
quantities are conserved. Continuity equations are a local form of conservation laws
which state that mass, energy and momentum as well as other natural quantities must be
conserved.
Therefore, a number of physical phenomena may be described using
continuity equations [5]. In fluid dynamics, two important continuity equations are the
conservation of mass and the conservation of momentum.
Conservation of Mass in Vector Form:
𝜕𝜌
⃑ ∙ 𝜌v
+ (∇
⃑)= 0
𝜕𝑡
Conservation of Mass in Cartesian Form:
𝜕𝜌 𝜕
𝜕
𝜕
(𝜌𝑣𝑥 ) +
(𝜌𝑣𝑌 ) + (𝜌𝑣𝑧 ) = 0
+
𝜕𝑡 𝜕𝑥
𝜕𝑌
𝜕𝑧
Conservation of Momentum in Vector Form:
𝜌
𝐷v
⃑
⃑ 𝜌 + 𝜇∇
⃑ 2v
= −∇
⃑ + 𝜌𝑔
𝐷𝑡
Conservation of Momentum in Cartesian Form:
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑝
𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥
𝜌(
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑥
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜌(
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕𝑣𝑦
𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦
𝜕𝑝
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑝
𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧
𝜌(
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
)=−
+𝜇( 2 +
+
) + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
12
In many instances of fluid dynamics, energy is being 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
The above continuity equations are solved by Fluent to determine pressure,
temperature, mass flux, etc. for various scenarios and boundary conditions.
13
3. 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.
3.1
NATURAL CONVECTION
Convection can be defined as 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 body forces acting on
density gradients. The density gradients can arise from mass concentrations 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 computer chips.
14
Calculating the amount of heat transfer occurring due to natural convection in a system
is determined using 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:
Cp μ
k
The Rayleigh number, Ra, is a dimensionless parameter that represents the ratio
Pr =
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 the critical value for that 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 is occurring 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.
15
3.2
TURBULENCE
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.
The Reynolds number, Re, is a dimensionless number that
represents the ratio of inertial forces to viscous forces; and is defined as:
Re =
ρVA
μ
The Reynolds number is used to determine if a flow laminar or turbulent. For
internal flows, such as within a pipe, laminar flow is characterized by a Reynolds
number less than 2300 whereas turbulent 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]
16
A common example of the transition of laminar flow to turbulent flow is smoke
rising from a cigarette.
Figure 3.2-1: Example of Turbulent Flow
As the smoke leaves the cigarette it travels upward in a laminar fashion as shown by
the single stream of smoke. At a certain distance, the Reynolds number becomes too
large and the flow begins to transition into the turbulent regime. When this happens, the
flow becomes more random and mixes with the air causing the smoke to dissipate.
Modeling turbulent flow requires an exact solution to the Navier-Stokes equations
which can be extremely difficult and time consuming. To reduce the complexity, an
approximation to the Navier-Stokes 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 (time-averaged) and fluctuating components. The RANS equations
can be used with approximations based on knowledge of the turbulent flow to give
approximate time-averaged solutions to the 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.
17
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. They can be written in Cartesian tensor form as:
𝛿𝑝
𝛿
(𝜌𝑢𝑖 ) = 0
+
𝛿𝑡 𝛿𝑥𝑖
𝛿
𝛿
𝛿𝑃
𝛿
𝛿𝑢𝑖 𝛿𝑢𝑗 2 𝛿𝑢𝑙
𝛿
′ ′
̅̅̅̅̅̅
(𝜌𝑢𝑖 ) +
(𝜌𝑢𝑖 𝑢𝑗 ) = −
+
[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
′ ′
̅̅̅̅̅̅
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 incorporated is using 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 a new
model known as 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]
18
3.2.1
CALCULATING TURBULENCE PARAMETERS
The CFD models discussed in this paper use the k-ϵ turbulence model when
applicable. The Fluent turbulence models require certain parameters to be established
prior to initialization to properly set the initial and boundary conditions for the flow
regime.
The following equations were used to determine the boundary and initial
condition inputs for an example situation.
Mass Flow Rate: 0.5 kg/s
Pipe Diameter (D): 0.03 m
Viscosity (μ): 0.001003 kg/m-s
Density (ρ): 998.2 kg/m3
Turbulence Empirical Constant (Cμ) = 0.09 [1]
Hydraulic Diameter (Dh):
𝐷ℎ =
𝐷 2
𝜋 ∗ (2)
4∗𝐴
=
= 𝐷 = 0.03 𝑚
𝑃
4∗𝜋∗𝐷
Flow Area (A):
𝐷 2
0.03 𝑚 2
𝐴 = 𝜋∗( ) =𝜋∗(
) = 0.00070686 𝑚2
2
2
Average Flow Velocity (uavg):
𝑢𝑎𝑣𝑔 =
𝑚̇
=
𝜌∗𝐴
0.5 𝑘𝑔/𝑠
998.2
𝑘𝑔
∗ 0.00070686 𝑚2
𝑚3
= 0.708631
𝑚
𝑠
Reynolds Number (ReDh):
𝑅𝑒𝐷ℎ
𝑚̇𝐷ℎ
=
=
𝜇𝐴
𝑘𝑔
0.5 𝑠 ∗ 0.03 m
= 21157
𝑘𝑔
0.001003 𝑚 − 𝑠 ∗ 0.00070686 𝑚2
Turbulence Length Scale (l):
𝑙 = 0.07 ∗ 𝐷ℎ = 0.07 ∗ 0.03 𝑚 = 0.0021 𝑚
19
Turbulent Intensity (I):
−
1
8
1
𝐼 = 0.16 ∗ 𝑅𝑒𝐷ℎ = 0.16 ∗ 21157−8 = 0.0460721
Turbulent Kinetic Energy (k):
𝑘=
2
3
3
𝑚
𝑚2
2
(𝑢𝑎𝑣𝑔 ∗ 𝐼) = (0.708631 ∗ 0.0460721) = 0.00159885 2
2
2
𝑠
𝑠
Dissipation Rate (ϵ):
ε=
3/4 k
C𝜇
3/2
3/4
𝑙
= 0.09
20
0.001598853/2
0.0021
3.3
TWO-PHASE FLOW
Two-phase flow is simply a flow that contains two different components. These
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 3.4 will focus on two-phase flow involving water and steam.
Depending on the volume fraction of each component in the two-phase flow,
different flow patterns can exist. The flow pattern of a two-phase flow is very important
to understand 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. An image of each of these
flow patterns can be seen in the Figure 3.3-1 [2, Figure 3-2].
Figure 3.3-1: 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
volume fractions. Subcooled boiling can be classified as bubbly flow. Slug flow occurs
21
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. [2]
The flow regime in a given system can be classified by comparing the gas
volume fraction and the flow velocity as shown in Figure 3.3-2 [2, Figure 3-4].
Figure 3.3-2: Baker Flow Pattern
22
The flow regime can also be determined by comparing the gas and liquid fluxes as
shown in Figure 3.3-3 [7].
Figure 3.3-3: Flow Pattern Boundaries for Vertical Upflow of Air and Water
Two-phase flows obey all of the basic laws of fluid mechanics. The equations are
more complicated and more numerous than those for single-phase flow. Additionally,
phenomenon like phase-interface interactions and slip must be considered. Fluent offers
three multiphase flow models, Volume of Fluid, Mixture and Eulerian, each with
varying strengths and computational demand.
The Volume of Fluid model solves a single set of momentum equations for two
or more fluids and tracks the volume fraction of each fluid throughout the domain. The
Mixture model solves for the momentum equation of the mixture and prescribes relative
velocities to describe the dispersed phases. The Eulerian model solves momentum and
continuity equations for each of the phases, and the equations are coupled through
pressure and exchange coefficients. This paper implements the Eulerian model because
of the complex nature of the problems being investigated.
23
3.4
BOILING HEAT TRANSFER
Boiling heat transfer is defined as a mode of heat transfer that occurs with a
change in phase from liquid 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.
Boiling heat transfer can be broken down into four regimes which are shown in
Figure 3.4-1 [16].
Figure 3.4-1: 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 but some vapor is generated. Instead, heat is
transferred from the surface to the bulk fluid by natural convection. The heat transfer
5/4
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
24
in the surface. When the liquid near the wall superheats, it evaporates forming bubbles
at the nucleation sites. When the liquid evaporates, a significant amount of energy is
removed from the heated surface due to the latent heat of the vaporization.
The
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]
25
3.5
POPULATION BALANCE EQUATION
3.5.1
BACKGROUND
Several industrial fluid flow applications 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
population, 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 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 is caused by the breakage of a single large bubble into
multiple smaller bubbles due to liquid turbulence eddies. Particle death 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
26
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. The
population of active sites is found to be:
̅ = 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]
3.5.2
EQUATION FORMULATION
The goal of this section is to present an overview of the theory and governing
equations for the methods used to calculate particle growth and nucleation.
3.5.2.1 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
27
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.
3.5.2.2 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.
28
3.6
NUMERICAL METHODS
“The numerical solution of heat transfer, fluid flow, and other related processes can
begin when the laws governing these processes have been expressed in mathematical
form, generally in terms of differential equations. The individual differential equations
that are encountered express a certain conservation principle. Each equation employs a
certain physical quantity as its dependent variable and implies that there must be a
balance among the various factors that influence it. [6]” 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.
The goal of CFD is to calculate the temperature, velocity, pressure, etc. of a fluid at
a particular location within a control volume. Thus the independent variable in the
differential equations is a physical location.
Due to computational limitations, the
number of locations (also known as grid points or nodes) is finite. By only focusing on
the solution of the differential equations at various 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
independent variable for a group of grid points within a control volume. This type of
equation is derived from a differential equation governing the independent variable and
thus expresses the same physical information as the differential equation. The piecewise
nature of the profiles chose is created by the finite number of grid points that participate
in a given discretization equation. The value of the independent variable at a grid point
thereby influences the value of the independent 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
29
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. A common spatial
discretization solver employed by Fluent is the upwind scheme which was first proposed
by Courant, Isaacson, and Rees in 1952. Other options include QUICK, power law and
third-order MUSCL.
30
4. NATURAL CONVECTION
Two examples of natural convection that are examined in the following
subsections is a heated horizontal cylinder and a heated vertical plate submerged in an
infinite pool.
These examples were chosen because of their simplicity and are
commonly found in nature.
4.1
HORIZONTAL CYLINDER
In this scenario, a cylinder with a constant surface temperature is submerged in
an infinite pool. The cylinder is slightly warmer than the surrounding fluid and therefore
energy passes from the cylinder to the fluid causing its temperature to increase. The
height of the control volume is equal to five times the diameter while the width is equal
to four times the diameter. The cylinder is at 310 K while the fluid (water) has an initial
temperature of 300 K. The Boussinesq approximation is used to calculate buoyancy and
a laminar model is used to determine flow.
The fluid temperature gradient after
20 seconds is shown in Figure 4.1-1.
Figure 4.1-1: 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
31
rises. The density change is shown in Figure 4.1-2. 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.
Figure 4.1-2: Horizontal Cylinder Density
32
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 shown in the
velocity vector plot, Figure 4.1-3.
Figure 4.1-3: Horizontal Cylinder Velocity Vector
33
To ensure that the model is giving realistic results, the solution was compared to
experimental data.
Figure 4.1-4 shows isotherms surrounding a horizontal tube in
natural convection flow as revealed by an interference photograph.
(a)
(b)
Figure 4.1-4: Isotherms Around a Horizontal Cylinder in Free Convection
(a) is from [9] and (b) is from Fluent
The model of a horizontal cylinder submerged in an infinite pool is in qualitative
agreement to experimental data. Figure 4.1-4 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 prove the accuracy of the model. Figure 4.1-5, Figure 4.1-6 and Figure 4.1-7
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.
34
(a)
(b)
Figure 4.1-5: Temperature at θ = 30° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
(a)
(b)
Figure 4.1-6: Temperature at θ = 90° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
35
(a)
(b)
Figure 4.1-7: Temperature at θ = 180° Vs. Radial Distance
(a) is from [10] and (b) is from Fluent
The heated horizontal cylinder model developed in Fluent shows very similar
temperature results 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
has no impact on the results, a mesh validation was performed. The results from the
mesh validation are shown in Table 4.1-1, and prove that the results given in this section
are mesh independent.
Table 4.1-1: Mesh Validation for Horizontal Cylinder
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (°F)
Min Density (kg/m3)
Analysis Value
19716
38688
0.01627
309.9239
993.1765
Mesh Validation
23636
46400
0.01621
309.9531
993.1625
Difference (%)
19.882
19.934
-0.369
0.009
-0.001
Because the model for a uniformly heated horizontal cylinder submerged in a pool as
calculated by Fluent are similar to those calculated experimentally and are mesh
independent, it can confidently be stated that the results for a are accurate.
36
4.2
VERTICAL PLATE
The second scenario of natural convection involves a constant surface
temperature heated vertical plate submerged in an infinite pool. 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. The control volume is 20 cm by
12 cm. The plate is 18 cm long and has a thickness of 1 cm. It is located 2 cm above the
bottom of the control volume. The plate has a constant surface temperature of 310 K
and the fluid (water) has an initial temperature of 300 K. The Boussinesq approximation
is used to calculate buoyancy and a laminar model is used to determine flow. The fluid
temperature gradient after 20 seconds is shown in Figure 4.2-1.
Figure 4.2-1: 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 equal 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
37
plate.
For more information on thermal and momentum boundary layers, see
Reference 3.
Figure 4.2-2: Vertical Plate Velocity Vector Plot
Figure 4.2-2 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 the vertical flat plate to the horizontal cylinder, it is expected that the
vertical plate would have a greater maximum fluid velocity because of the larger surface
area of the heated region.
The maximum fluid velocity for the vertical plate is
0.0149 m/s while the maximum fluid velocity for the horizontal cylinder is 0.0177 m/s.
This is quite counterintuitive. The reason why the horizontal cylinder actually has a
larger maximum velocity is because the buoyancy driving head is allowed to work freely
without any drag from the plate. Although the plate is continuing to heat the fluid as it
travels up the plate, the velocity is limited due to friction. For this reason, the plate
actually has a smaller maximum velocity.
38
To ensure that the model is giving realistic results, the solution was compared to
experimental data. Figure 4.2-3 shows isotherms surrounding a vertical plate in natural
convection flow as revealed by an interference photograph.
(a)
(b)
Figure 4.2-3: Isotherms 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 prove the accuracy of the model. Figure 4.2-4 and Figure 4.1-5 show a
comparison of dimensionless temperature versus dimensionless distance for various
Prandtl numbers. Figure 4.2-4a shows theoretical values and Figure 4.2-4b 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.
39
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.
(a)
(b)
Figure 4.2-4: Dimensionless Temperature as a Function of Prandtl Number
(a) Theoretical Values and (b) Experimental Values [11]
Figure 4.2-5: Dimensionless Temperature as a Function of Prandtl Number
(Fluent)
40
The heated vertical plate model developed in Fluent shows very similar temperature
results to 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
has no impact on the results, a mesh validation was performed. The results from the
mesh validation are shown in Table 4.2-1, and prove that the results given in this section
are mesh independent.
Table 4.2-1: Mesh Validation for Vertical Plate
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (°F)
Min Density (kg/m3)
Analysis Value
12310
23572
0.01376
309.8089
993.2319
41
Mesh Validation
18081
35168
0.01380
309.7991
993.2365
Difference (%)
46.881
49.194
0.276
-0.003
0.001
5. TURBULENCE
5.1
TURBULENT FLOW
Due to its random and chaotic nature, turbulent flows are generally homogenized
which increases the heat removal ability of a system. A simple axisymmetric flow
model was developed in Fluent to gain a better understanding of turbulence in a pipe.
The pipe has a diameter of 3 cm and is 50 cm long. The realizable k-ϵ turbulence model
with enhanced wall treatment is implemented. The working fluid is water and there is
no heat transfer. The boundary conditions and inputs to the turbulence model are
calculated in Section 3.2.1. Figure 5.1-1 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 copared the velocity profile of
laminar flow (shown in Appendix A). 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, the fluid velocity reaches the steady state velocity profile
quicker for turbulent flow. Figure 5.1-1 shows that flow is stable at about 40 cm from
the pipe entrance for turbulent flow. Figure A.1-1 shows that flow is stable at about
45 cm from the pipe entrance for laminar flow. This qualitatively matches experimental
data well.
Figure 5.1-1: Velocity Magnitude Vs. Position
42
Figure 5.1-2 shows the wall shear stress versus distance from the entrance.
The
shear stress is very large at the beginning and quickly decays after about 3 cm. The
large shear stress at the beginning is due to entrance effects.
Figure 5.1-2: Wall Shear Stress Vs. Axial Distance
As the flow becomes more and more stable, the entrance effects dissipate and wall
shear stress slowly decreases. At about 22 cm the shear stress reduces again. This is due
to boundary layer separation which results in a reduction of overall drag. At the very
end of the pipe, around 49 cm, the wall shear stress begins to increase. This is caused by
the pipe exit boundary condition.
Turbulence differs greatly from laminar flow when comparing radial velocity.
Laminar flow usually has very little radial velocity whereas turbulent flow can have a
significant amount. Figure 5.1-3 shows the radial flow velocity of turbulent flow within
a pipe. The greatest radial velocity occurs at the entrance and exit of the pipe due to the
boundary conditions applied. However, the radial velocity is non-zero in the middle of
the pipe meaning that mixing is occurring.
43
Figure 5.1-3: Radial Velocity
Figure 5.1-4 and Figure 5.1-5 show the turbulent kinetic energy and the production
of turbulent kinetic energy as a function of distance.
Figure 5.1-4: Turbulent Kinetic Energy
Figure 5.1-5: Production of Turbulent Kinetic Energy
All of the turbulent kinetic energy is near the wall because the wall helps generate
turbulent kinetic energy. The trend of Figure 5.1-5 is similar to that of Figure 5.1-2
which is appropriate because shear stress, created by the wall, produces turbulent kinetic
energy.
44
5.2
TURBULENT FLOW WITH HEAT TRANSFER
Because it is very rare to have a system with zero heat transfer, the turbulent flow
model described in Section 5.1 has been altered to include heat transfer from the pipe
wall to the fluid. The pipe wall has a constant heat flux of about 450kW and the initial
fluid temperature is 300K. Heat transfer causes the fluid temperature to increase which
and therefore density to decrease which has a minor impact on the fluid flow described
in Section 5.1. Figure 5.2-1 shows the fluid temperature change caused by the constant
surface heat flux.
Figure 5.2-1: Temperature
The turbulent kinetic energy shown in Figure 5.2-2 is very similar to that shown in
Figure 5.1-4 which is expected since turbulent kinetic energy is momentum based not
thermal based. If the heat transfer rate to the fluid was large enough such that buoyance
effects began to influence flow, then the turbulent kinetic energy would be different for
the two scenarios.
Figure 5.2-2: Turbulent Kinetic Energy
45
Figure 5.2-3 shows the radial velocity which matches well with Figure 5.1-3
because of the same reasons explained in the previous paragraph.
Figure 5.2-3: Radial Velocity
Comparing the velocity profiles for the two scenarios (Figure 5.1-1 and
Figure 5.2-4) 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 5.2-4: Velocity Magnitude Vs. Position
46
As expected, the wall shear stress shown in Figure 5.2-5 matches well with the wall
shear stress shown in Figure 5.1-2.
Figure 5.2-5: Wall Shear Stress Vs. Axial Position
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 5.2-1, and prove
that the results given in this section are mesh independent.
Table 5.2-1: Mesh Validation for Turbulence With Heat Transfer
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (°F)
Min Density (kg/m3)
Max Dynamic Pressure (Pa)
Analysis Value
41013
39040
1.66957
328.293
984.359
1387.069
47
Mesh Validation
44499
42360
1.67111
329.0477
983.997
1389.619
Difference (%)
8.50%
8.50%
0.09%
0.23%
-0.04%
0.18%
6. TWO-PHASE FLOW
6.1
GAS MIXING TANK
In steelmaking processes, gas injection techniques have been extensively utilized to
enhance chemical reaction rates, homogenize temperature and chemical compositions,
and remove impurities. 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.
The mixing tank geometry in this analysis is planar, has a height of 60 cm, a width
of 30 cm and a porous plug width of 2 cm. Water is the primary phase and air is the
secondary phase. The Eulerian multiphase model, the standard k-ϵ turbulence model and
the Schiller-Nauman drag model are implemented with a bubble diameter of 0.005 cm
and a gas inlet velocity of 5.0 cm/s. The drag model is Schiller-Nauman which is
generally acceptable for all multiphase calculations. Figure 6.1-1 shows the gas volume
fraction within the mixing tank, Figure 6.1-2 shows the liquid vector velocity and
Figure 6.1-3 shows the gas vector velocity, respectively, after 10 seconds.
Figure 6.1-1: Gas Volume Fraction
48
Figure 6.1-2: Liquid Vector Velocity
Figure 6.1-3: Gas Vector Velocity
49
The velocities shown in Figure 6.1-2 and 6.1-3 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 6.1-2
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 has no impact on the results, a mesh validation was
performed. The results from the mesh validation are shown in Table 6.1-1, and prove
that the results given in this section are mesh independent.
Table 6.1-1: Mesh Validation for Gas Mixing Tank
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Max Liquid Dynamic Pressure (psia)
Max Gas Dynamic Pressure (psia)
Max Liquid Volume Fraction
Analysis
Value
24747
24416
0.47065
0.47074
111.744
0.13718
1.0000
50
Mesh
Validation
30625
30256
0.46361
0.46370
107.876
0.13602
1.0000
Difference
(%)
23.75%
23.92%
-1.50%
-1.50%
-3.46%
-0.85%
0.00%
6.2
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 6.1.
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 6.2-1
shows a rudimentary image of a bubble column reactor.
Figure 6.2-1: 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 ability to
calculate the change in bubble size due to turbulent eddies is discussed in Section 6.3.
The bubble column geometry in this analysis is planar, has a height of 75 cm and a
width of 10 cm. Water is the primary phase and air is the secondary phase. The
Eulerian multiphase model, the standard k-ϵ turbulence model and the Schiller-Nauman
51
drag model are implemented with a bubble diameter of 0.48 cm and an inlet velocity of
5.0 cm/s.
Figure 6.2-2 shows a comparison between gas volume fraction 1 second and
5 seconds after gas has begun flowing through the bubble column. At both time points
the gas tends to flow in slugs. After 5 seconds the gas has reached the top of the liquid
and caused it to change shape. Also, after 5 seconds the liquid level is higher than that
after 1 seconds. This shows that the gas flowing through the bubble column, through
drag forces and displacement pushes the liquid level higher.
(a)
(b)
Figure 6.2-2: Instantaneous Gas Volume Fraction
After (a) 1 Second and (b) 5 Seconds
52
Figure 6.2-3 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. Most of the liquid is pushed along the
wall and the center of the column.
(a)
(b)
Figure 6.2-3: Instantaneous Liquid Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
53
Figure 6.3-4 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 6.3-4a 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 most likely caused by bubble coalescence and wall drag. Figure 6.3-4b shows
that the greatest gas velocities occur near the walls. This is in line with Figure 6.3-2
which showed that the highest gas volume fractions are near the walls. Higher gas
volume fractions lead to greater buoyancy forces which means greater gas velocities.
(a)
(b)
Figure 6.2-4: Instantaneous Gas Velocity Vectors
After (a) 1 Second and (b) 5 Seconds
54
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 6.2-1, and prove
that the results given in this section are mesh independent.
Table 6.2-1: 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
0.9949547
0.9987329
4929.094
55
Mesh Validation
8500
8217
0.604716
1.021839
0.9999541
4930.642
Difference (%)
21.32%
21.73%
-3.39%
2.70%
0.12%
0.03%
6.3
BUBBLE COLUMN WITH POPULATION BALANCE
MODEL
The bubble population within the column will not have a uniform size due to
growth, coalescence, and breakup. The bubble column model discussed in Section 6.2
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 (0.30 cm, 0.48 cm
and 0.76 cm) was added to the Section 6.2 model. At the gas inlet, the bubble diameter
distribution is 25% 0.30 cm, 50% 0.48 cm and 25% 0.76 cm. The coalescence and break
up of bubbles is determined using the Luo model with a surface tension of 0.072 N/m.
Figure 6.3-1 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 6.3-1a and 6.3-1b reveals
that the liquid level in Figure 6.3-1b is higher. This is caused by drag and displacement
of the liquid by the flowing gas. When comparing Figure 6.3-1 to Figure 6.2-2, 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 6.3-1, 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.
Another major difference is shape of the gas as it initially climbs up the bubble column.
The double parabolic shape is much more sever in Figure 6.3-3a compared with
Figure 6.2-4a. This is attributed to bubble coalescence determined by the population
balance model.
56
(a)
(b)
Figure 6.3-1: Instantaneous Gas Volume Fraction with PBM
After (a) 1 Second and (b) 5 Seconds
57
Figure 6.3-2 shows a comparison between liquid velocity vectors at 1 second and
5 seconds after gas has begun flowing through the bubble column.
Similar to
Figure 6.2-3, there are distinct paths of liquid movement that are visible at both 1 second
and 5 seconds. Figure 6.3-2b shows a uniform liquid velocity distribution throughout
the bubble column where there are no sections of little to no movement. This is different
from Figure 6.2-3b where areas of no movement are prevalent.
(a)
(b)
Figure 6.3-2: Bubble Column Liquid Vector Velocity with PBM
After (a) 1 Second and (b) 5 Seconds
58
The population balance model calculates the bubble size distribution at each axial
height using the Luo break up and coalescence model. Table 6.3-1 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 6.3-1: 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 6.3-2 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 6.3-2: 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 6.2. For the bubble column mesh validation see Table 6.2-1.
59
7. BOILING
7.1
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 is a complex fluid motion that is initiated and maintained by the nucleation,
growth, departure, and collapse of bubbles; and by natural convection. [2]
The pool boiling model developed is planar, has a height of 15 cm and a width of
3 cm. Water is the primary phase and steam is the secondary phase. The Mixture
multiphase and laminar flow models are implemented. Laminar flow is chosen since
most of the fluid is stanant and there is only an outlet boundary condition (no inlet). The
fluid starts 1 degree subcooled and is heated by a constant temperature surface (at the
bottom) that is 10 degrees above the saturation temperature.
Figure 7.1-1 shows the instantaneous gas volume fraction after 1 second and 2
seconds of heating. After 1 second, the entire bottom of the control volume is heated
and some steam begins to form. Figure 7.1-2 shows that there is no movement of the
liquid and therefore all heat transfer is occurring via conduction. However, just one
second later, enough energy has been absorbed by the fluid that buoyancy effects have
taken affect and the fluid begins to move. The steam created on the bottom surface
moves upward and cooler liquid takes its place creating eddies which can also be seen in
Figure 7.1-2. Figure 7.1-1 shows four distinct nucleation sites on the heated surface
where steam is being formed. At these sites, bubbles nucleate, grow and detach from the
heated surface.
60
(a)
(b)
Figure 7.1-1: Instantaneous Gas Volume Fraction
After (a) 1 Second and (b) 2 Seconds
(a)
(b)
Figure 7.1-2: Instantaneous Liquid Velocity Vectors
After (a) 1 Second and (b) 2 Seconds
61
To graphically see the location of the nucleation sites, Figure 7.1-3 was created
which shows the volume fraction of vapor on the heated surface after 10 seconds. Vapor
is being produces significantly at four distinct location (locations where the vapor
volume fraction spikes), 0.000 m, 0.013 m 0.025 m and 0.030 m.
There are three
locations where the vapor volume fraction is at a local minimum, 0.002 m, 0.016 m and
0.028 m where liquid is taking the place of the recently created vapor.
Figure 7.1-3: Volume Fraction of Vapor on Heated Surface
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 7.1-1, and prove
that the results given in this section are mesh independent.
Table 7.1-1: Mesh Validation for Pool Boiling
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Min Liquid Volume Fraction
Max Dynamic Pressure (Pa)
Max Phase Transfer (kg/m3-s)
Analysis
Value
31995
31520
0.14404
0.16059
0.84741
10.2384
1.76579
62
Mesh
Validation
38715
38192
0.145342
0.159012
0.838169
9.690999
1.84838
Difference
(%)
21.00%
21.17%
0.91%
-0.98%
-1.09%
-5.35%
4.68%
7.2
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 7.2-1 [2] shows the various boiling regimes as
a function of void fraction.
Figure 7.2-1: Void Fraction in Various Boiling Regimes
If the heat flux of a heated wall 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 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
63
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 portino 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]
The subcooled flow boiling model developed is an axisymmetric pipe with a
diameter of 3 cm and a length of 50 cm. Water is the primary phase and steam is the
secondary phase. The Eulerian multiphase model and the realizable k-ϵ turbulence
model with enhanced wall treatment are used. The phase interactions are schillernauman for drag, boiling-moranga for lift, ranz-marshall for heat transfer coefficient and
phase change is determined using the Fluent boiling model. The liquid properties at
three different inlet temperatures are shown in Table 7.2-1 [17].
Table 7.2-1: Liquid Properties
368 K
370 K
Density (kg/m )
961.99
960.59
Specific Heat (J/kg-K)
4210.0
4212.1
Viscosity (kg/m-s)
0.0002978
0.0002914
Conductivity (W/m-K)
0.6773
0.6780
Heat of Vaporization (J/kgmol)
N/A
N/A
Surface Tension (N/m)
N/A
N/A
* Saturation temperature at atmospheric pressure.
3
64
373.15 K*
958.46
4215.5
0.0002822
0.6790
40622346
0.0589
Seven subcooled flow boiling cases were analyzed and compared to gather
insight into how heat flux, inlet temperature and mass flow impact liquid volume
fraction. The base case uses nominal values for inlet temperature, mass flow and heat
flux. The following six cases increase or decrease inlet temperature, mass flow or heat
flux. The input for the seven cases analyzed is documented in Table 7.2-2.
Table 7.2-2: Subcooled Boiling Case Matrix
Base
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Inlet Temperature
(K)
370
370
370
372
368
370
370
Mass Flow
(kg/s)
0.30
0.30
0.30
0.30
0.30
0.33
0.27
Heat Flux
(kW/m2)
90
100
80
90
90
90
90
Plots of temperature, liquid volume fraction and mass transfer rate for the base case
are shown in Figures 7.2-2, 7.2-3 and 7.2-4, respectively. Although these figures are
specific to the base case, their trends can be applied to all of the cases. Figure 7.2-4 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.
Figure 7.2-2: Base Case - Temperature (K)
65
Figure 7.2-3: Base Case - Liquid Volume Fraction
Figure 7.2-4: Base Case - Mass Transfer Rate (kg/m3-s)
The remaining scenarios were analyzed and the liquid volume fraction at nine axial
heights are shown in Table 7.2-3.
Figure 7.2-5 through Figure 7.2-11 show the
information contained in Table 7.2-3 in graphical form.
Table 7.2-3: Axial Height Liquid Volume Fraction
Height
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
Base
1.0000
0.9955
0.9834
0.9669
0.9385
0.9146
0.8974
0.8811
0.8589
Case 1
1.0000
0.9941
0.9774
0.9516
0.9206
0.8974
0.8774
0.8542
0.8080
Case 2
1.0000
0.9965
0.9884
0.9780
0.9587
0.9335
0.9154
0.9025
0.8896
Case 3
1.0000
0.9663
0.8769
0.7961
0.7113
0.5771
0.4290
0.3206
0.2517
66
Case 4
1.0000
0.9987
0.9965
0.9930
0.9879
0.9824
0.9713
0.9498
0.9354
Case 5
1.0000
0.9962
0.9868
0.9704
0.9463
0.9215
0.9012
0.8843
0.8669
Case 6
1.0000
0.9944
0.9804
0.9602
0.9298
0.9106
0.8952
0.8766
0.8340
Figure 7.2-5: Base Case – Liquid Volume Faction Vs. Position
Figure 7.2-6: Case 1 - Liquid Volume Faction Vs. Position
67
Figure 7.2-7: Case 2 - Liquid Volume Faction Vs. Position
Figure 7.2-8: Case 3 - Liquid Volume Faction Vs. Position
68
Figure 7.2-9: Case 4 - Liquid Volume Faction Vs. Position
Figure 7.2-10: Case 5 - Liquid Volume Faction Vs. Position
69
Figure 7.2-11: 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
The values from Table 7.2-3 were plugged into the above three equations and the
change from the base case is shown in Table 7.2-4. For example, at an axial height of
70
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 7.2-4: 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 7.2-5, and prove that the results given in this section are mesh independent.
Table 7.2-5: 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
71
Difference (%)
25.08%
25.13%
0.32%
0.07%
-1.46%
-1.73%
7.3
SUBCOOLED BOILING WITH POPULATION BALANCE
MODEL
72
8. 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.
73
[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.
74
APPENDIX A: ADDITIONAL INFORMATION
A.1
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.
Using Fluent, a simple axisymmetric flow model was developed to gain a better
understanding of laminar flow in a pipe. The pipe analyzed has a diameter of 3 cm and a
length of 50 cm.
The laminar flow model is implemented with uniform surface
termperature of 305K and an inlet temperature of 300K. The Reynolds number for the
scenario was selected as 352 which is well within the laminar regime.
One of the most notable characteristics of laminar flow is the parabolic shape of
its velocity profile. Figure A.1-1 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.
75
Figure A.1-1: 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 A.1-2 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 A.1-2.
Figure A.1-2: Laminar Flow Temperature
76
Figure A.1-3 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 and exit of the pipe due to pipe boundary
conditions but this has little impact on system as a whole.
Figure A.1-4: Laminar Flow Radial Velocity
Laminar flow also tends to create momentum boundary layers which cause
frictional force on the wall. Figure A.1-5 shows the computed drag force on the wall.
Figure A.1-5: 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. Near the outlet of the pipe, around 49 cm, the wall shear
stress begins to increase due to pipe exit effects.
77
APPENDIX B: ADDITIONAL INFORMATION
8.1.1
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]
8.1.2
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]
78
8.1.3
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.
8.1.4
EQUATION FORMULATION
The goal of this section is to present an overview of the theory and governing
equations for the methods used to calculate particle growth and nucleation.
8.1.4.1 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
𝑉⃑𝑥 𝑉𝜑
The local average number density in physical space (that is, the total number of
particles per unit volume is given by
79
𝑁(𝑥, 𝑡) = ∫ 𝑛𝑑𝑉𝜑
𝛺𝜑
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.
8.1.4.2 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.
8.1.4.3 PARTICLE GROWTH AND DISSOLUTION
In the population balance equation given in Section 12.2, ∇𝑉 ∙ [𝐺𝑉 𝑛(𝑉, 𝑡)] is the
particle growth term. The growth rate is based on particle volume, 𝐺𝑉 , 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. Because of this, the growth rate is set
to a negative value.
80
8.1.4.4 PARTICLE BIRTH AND DEATH DUE TO BREAKAGE AND
AGGREGATION
The birth and death of particles occur due to breakage and aggregation processes. In
the case of subcooled nucleate boiling, turbulence plays an important role in the birth
and death of steam bubbles. During mixing processes, mechanical energy is supplied to
the fluid. This energy creates turbulence within the fluid. The turbulence creates eddies,
which in turn help dissipate the energy. The energy is transferred from the largest eddies
to the smallest eddies in which it is dissipated through viscous interactions. Particle
birth is caused by the breakage of a single large bubble into multiple smaller bubbles due
to liquid turbulence eddies. Particle death is due to the coalescence of multiple small
bubbles into one larger bubble. The Luo model is used in this analysis because it
encompasses both the breakage frequency and the PDF of breaking particles and only
requires the specification of surface tension.
8.1.4.5 PARTICLE BIRTH BY NUCLEATION
Depending on the application, spontaneous nucleation of particles can occur due to
the transfer of molecules from the primary phase. In boiling applications, the creation of
the first vapor bubbles is a nucleation process referred to as nucleate boiling.
There are two types of nucleation sites. The first is formed in a pure liquid and can
either be a high energy molecular group or a cavity resulting from a local pressure
reduction such as in accelerated flow (cavitation). The other type forms on a foreign
object such as a cavity on a wall or a suspended foreign material. In subcooled nucleate
boiling, the nucleation sites are created at the cavities of the heated surface. 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
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
nucleation sites. The population of active sites was found to be
81
̅ = 𝑁0 exp (−
𝑁
𝐾
3
𝑇𝑤𝑎𝑙𝑙
)
Where N0 and K represent the liquid an surface conditions. There is no possible
way to predict N0 and K for a particular boiling system. However, it can be seen that the
population of active sites is a strong function of wall temperature and therefore heat flux.
[2]
8.1.5
SOLUTION METHOD
The discrete method (also known as the classes or sectional method) was developed
by Hounslow [10] (p. 65), Litster [16] (p. 65), and Ramkrishna [25] (p. 66). It is based
on representing the continuous particle size distribution (PSD) in terms of a set of
discrete size classes or bins, as illustrated in Figure 12.3-1. The advantages of this
method are its robust numerics and that it gives the PSD directly. The disadvantages are
that the bins must be defined a priori and that a large number of classes may be required.
[1] (ANSYS Fluent PBE Guide Figure 2.1)
Figure 4.4.3-1: Particle Size Distribution
82