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
© Copyright 2012
By
Matthew P. Wilcox
All Rights Reserved
ii
CONTENTS
Modeling of Subcooled Boiling in a Nuclear Reactor Core ............................................... i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT ..................................................Error! Bookmark not defined.
ABSTRACT .................................................................................................................... vii
1. Introduction.................................................................................................................. 1
2. Background .................................................................................................................. 2
3. Methodology ................................................................................................................ 4
3.1
Turbulence Models............................................................................................. 4
3.2
Two Phase Models (ANSYS Fluent Theory Guide MultiPhase Flow) ............. 5
3.3
3.2.1
Volume of Fluid ..................................................................................... 5
3.2.2
Mixture ................................................................................................... 5
3.2.3
Eulearian ................................................................................................ 6
Solution Models ................................................................................................. 6
4. Equations ....................................................................Error! Bookmark not defined.
5. Numerical Methods ..................................................................................................... 9
6. Natural Convection .................................................................................................... 11
6.1
Introduction ...................................................................................................... 11
6.2
Theory .............................................................................................................. 12
6.3
Examples of Natural Convection ..................................................................... 14
6.3.1
Modeling of a Horizontal Cylinder ...................................................... 14
6.3.2
Modeling of a Vertical Plate ................................................................ 18
7. Laminar Flow with Heat Transfer ............................................................................. 22
8. Turbulence ................................................................................................................. 26
8.1
Calculating Turbulence Parameters ................................................................. 29
9. Turbulence with Heat Transfer .................................................................................. 35
iii
10. Two-Phase Flow ........................................................................................................ 38
10.1 Flow Regimes .................................................................................................. 41
11. Gas Mixing Tank ....................................................................................................... 43
12. Bubble Column .......................................................................................................... 46
13. Population Balance Equation ..................................................................................... 50
13.1 Background ...................................................................................................... 50
13.2 Equation Formulation ...................................................................................... 51
13.2.1 Particle State Vector ............................................................................. 51
13.2.2 Population Balance Equation ............................................................... 52
13.2.3 Particle Growth and Dissolution .......................................................... 52
13.2.4 Particle Birth and Death Due to Breakage and Aggregation ............... 52
13.2.5 Particle Birth by Nucleation ................................................................. 53
13.3 Solution Method ............................................................................................... 54
14. Bubble Column with Population Balance Model ...................................................... 55
15. Pool Boiling ............................................................................................................... 58
16. Subcooled Boiling ..................................................................................................... 61
17. Subcooled Boiling with Population Balance Model .................................................. 62
18. References.................................................................................................................. 63
iv
LIST OF TABLES
Table 11-1: Mesh Independent Quantities ....................................................................... 45
v
LIST OF FIGURES
Figure 12-1: Bubble Column ........................................................................................... 46
Figure 12-2: Instantaneous Gas Volume Fraction ........................................................... 47
Figure 12.3: Instantaneous Liquid Velocity Vectors ....................................................... 48
Figure 12-4: Instantaneous Gas Velocity Vectors ........................................................... 49
Figure 13-1: Particle Size Distribution (ANSYS Fluent PBE Guide Figure 2.1) ........... 54
Figure 14-1: Instantaneous Gas Volume Fraction with PBM ......................................... 56
Figure 14-2: Bubble Column Liquid Vector Velocity with PBM ................................... 57
Figure 15.1: Instantaneous Gas Volume Fraction ........................................................... 59
Figure 15.1: Instantaneous Liquid Velocity Vectors ....................................................... 59
Figure 15.3: Volume Fraction of Vapor on Heated Surface ............................................ 60
vi
ABSTRACT
vii
Nomenclature
viii
1. Introduction
Nuclear reactors have been used for commercial electricity production since
1958. They provide roughly 20% of the electricity in the United States and about 13%
world-wide. There are two distinct types of nuclear reactors, Pressurizer Water Reactors
(PWR) and Boiling Water Reactors (BWR). The more common of the two types is the
PWR which heats water that flows past nuclear fuel rods to produce steam. The benefit
of having a pressurized system is that the water can be heated to very high temperatures
without bulk boiling occurring. This helps to increase efficiency and prevent fuel
failure. Energy is removed from the nuclear fuel rods through an efficient heat transfer
process known as nucleate boiling.
During nucleate boiling, the heated surface
temperature is hotter than the saturation temperature of the fluid causing localized
boiling in the subcooled bulk fluid.
The amount of subcooled boiling is heavily
impacted by fluid inlet temperature, pressure, mass flow and heat flux. Because there
are so many factors that play a role in determining the level of subcooled boiling, the
amount when it occurs in a nuclear reactor is generally unknown.
The power in a reactor core is held constant by keeping the reactivity balance at
zero. Positive reactivity leads to a power increase while negative reactivity leads to a
power decrease. Some of the components that make up the reactivity balance equation
are water temperature, water density, fuel temperature and voiding in the core. The
accuracy at which each parameter can be measured impacts the ability to calculate core
power during a transient. Being able to accurately measure the core power during a
transient removes uncertainty in the safety analyses performed. Currently there are
methods to accurately calculate the reactivity components listed above except for
voiding in the core. If a more accurate method was developed to calculate voiding under
varying conditions, there would be less uncertainty in the power calculation and more
safety analysis margin could be gained thus allowing plants to increase power and
produce more electricity.
The purpose of this thesis is to develop a better understanding of subcooled
boiling and generate a more accurate method to measure voiding at different axial
locations in the core.
2. Background
Electricity is one of the greatest discoveries of the 19th century and its use has
greatly increased the world’s standard of living. It is generated by converting thermal
energy, from a fuel source, into electrical energy. This is done generally through the
Rankine Cycle where fuel is burned to heat water and form steam. The steam is then
used to turn a turbine which spins an electric generator. Electricity production involves
numerous engineering processes but primarily based around heat transfer and fluid flow.
There are many different fuel sources available to electrical power plants, the one in
focus here will be nuclear fuel. Nuclear power plants harness the energy released during
the fission process to heat the surrounding water called the Reactor Coolant System
(RCS) which is then used 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 due to the highly pressurized system; however, a small
amount of localized boiling does occur. This is also known as subcooled boiling. This
thesis will focus on the convective heat transfer that occurs in the 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. Eventually the bubbles detach and enter the bulk fluid.
At this point there is saturated steam in a subcooled liquid. The bubbles have three
options, they can coalesce with other bubbles, they can grow in size or they can shrink in
size.
The idea of using PBEs for subcooled boiling is a relatively new idea and has
shown potential recently to provide great insights into this regime of boiling heat
transfer. The amount of voiding that occurs in a nuclear reactor core has a direct impact
on fission power due to reactivity feedbacks caused by voiding.
If a better
understanding of the level of voiding due to subcooled boiling was developed, the
accuracy at which fission power is calculated during a transient could be improved and
the amount of uncertainty reduced. Fluid properties such as temperature, pressure, mass
2
flux and heat flux will be varied and their impact on the amount of subcooled boiling at
different axial locations in a nuclear reactor will be determined.
3
3. Methodology
A portion of a fuel bundle will be modeled using ANSYS Fluent. The traditional models
available (energy equation, turbulence, two-phase, etc.) in Fluent will be implemented
along with a Population Balance Equation (PBE) model. Population balance equations
have been introduced in several branches of modern science, mainly in branches with
particulate entities. Population balance equations define how populations of separate
entities develop in specific properties over time. They are nothing more than a balance
on the number of particles in a particular state. The PBE model will be used to
determine the number of steam bubbles in the core, reveal how they develop over time
and decide if the bubbles shrink and collapse or coalesce and grow in size.
Ten models will be created, each more advanced than the previous. The final model will
be three-dimensional, use multiple heated rods, allow for turbulent, two-phase flow and
have the PBE model implemented. For more information about the model progression
and development, see the Model Development section.
After each model is developed, it will be compared to known experimental data
whenever possible in order to validate the information generated by ANSYS Fluent.
After the models have been validated and the final model developed, the initial
conditions, temperature, pressure, mass flux and heat flux, will be altered so that the
voiding at different axial locations can be determined for the various initial conditions.
Once the data has been collected, it will be analyzed to produce either a set of equations
or a set of tables that will allow the user to quickly determine how much voiding occurs
based on the known conditions.
To determine how the bubbles will react within the subcooled fluid, Population
Balance Equations (PBE) will be used. Population Balance Equations allow Fluent to
calculate probabilities instead of keeping track of each individual steam bubble which
would require large amounts of computing power and significant amount of time.
3.1 Turbulence Models
4
3.2 Two Phase Models (ANSYS Fluent Theory Guide MultiPhase
Flow)
A large number of flows encountered in nature and technology are a mixture of
phases. Physical phases of matter are gas, liquid, and solid, but the concept of phase in a
multiphase flow system is applied in a broader sense. In multiphase flow, a phase can be
defined as an identifiable class of material that has a particular inertial response to and
interaction with the flow and the potential field in which it is immersed. For example,
different-sized solid particles of the same material can be treated as different phases
because each collection of particles with the same size will have a similar dynamical
response to the flow field.
3.2.1
Volume of Fluid
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.
3.2.2
Mixture
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.
5
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.
3.2.3
Eulearian
The Eulerian multiphase model in ANSYS 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.
ANSYS FLUENT’s Eulerian multiphase model does not distinguish between fluid-fluid
and fluid-solid (granular) multiphase flows.
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.
3.3 Solution Models
6
4. Mathematical Formulation
A continuity equation in physics is an equation that describes the transport of
a conserved quantity. Since mass, energy, momentum and other natural quantities are
conserved under their respective appropriate conditions; a variety of physical
phenomena may be described using continuity equations. Continuity equations are a
stronger, local form of conservation laws. In fluid dynamics, two important continuity
equations are the conservation of mass and the conservation of momentum. [Transport
Phenomenon]
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
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.
7
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
8
5. 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.” 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 (see
Section 4). “A numerical solution of a differential equation consists of a set of number
from which the distribution of the dependent variable can be constructed.” The goal of
computational fluid dynamics (CFD) is to calculate the temperature, velocity, etc. of a
fluid at a particular location within a control volume and 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 the grid points, the need to find the exact
solution of the differential equation has been replaced. The algebraic equations (also
known as discretization equations) involving the unknown values of the independent
variable at chosen 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.
“A discretization equation is an algebraic relation connecting the values of the
independent variable for a group of grid points. Such an equation is derived from the
differential equation governing the independent variable and thus expresses the same
physical information as the differential equation. The fact that only a few grid points
participate in a given discretization equation is a consequence of the piecewise nature of
the profiles chose.
The value of the independent variable at a grid point thereby
influences the distribution of the independent variable only in its immediate
neighborhood. 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 follows from the consideration that, as the grid points get
9
closer together, the change in phi between neighboring grid points becomes small and
the actual details of the profile assumption become unimportant.” This is where the term
“mesh independent” comes from.
If there are too few grid points, the profile
assumptions impact the solution and the solution is not mesh independent. To ensure
that the results are not dependent on the profile assumptions, the solution should be
checked for mesh independence.
“The most common procedure for deriving the discretization equations is the
through 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 solution of the algebraic equation or in the overall iterative scheme
employed for handling nonlinearity, it is often desirable to speed up or to slow down the
changes, from iteration to iteration, in the values of the dependent variable. This process
is called overrelaxation or underrelaxation, depending on whether the variable changes
are accelerated or slowed down. Underrelaxation is a very useful device for nonlinear
problems. It is often employed to avoid divergence in the iterative solution of strongly
nonlinear equations.”
ANSYS Fluent offers numerous spatial discretization solvers for the various
independent variables such as gradients, pressure, flow, momentum, turbulence, energy,
etc. One of the most common spatial discretization solvers is the upwind scheme which
was first proposed by Courant, Isaacson, and Rees in 1952. “It is part of a class of
numerical discretization methods for solving hyperbolic partial differential equations.
Upwind schemes use an adaptive or solution-sensitive finite difference stencil to
numerically simulate the direction of propagation of information in a flow field. The
upwind schemes attempt to discretize partial differential equations by using differencing
biased in the direction determined by the sign of the characteristic speeds (Wikipedia).”
Other methods are available in Fluent however this is the most commonly used solver.
Other options include QUICK, power law and third-order MUSCL.
10
6. Natural Convection
6.1 Introduction
Convection can be defined as the transport of mass and energy by potential gradients and
by bulk fluid motion. If the fluid motion is induced by some external force, it is
generally referred to as forced convection (Convective mass and heat transfer). Natural
convection is a mechanism, or type of heat transport, 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, which,
in turn, arise from mass concentrations and or temperature gradients in the fluid.
(Convective mass and heat transfer). In natural convection, fluid surrounding a heated
surface absorbs energy, becomes less dense, and rises. Then, the surrounding, cooler
fluid moves 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.
Figure 5.1-1: Natural Convection Currents in a Pot
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.
11
6.2 Theory
Like forced convection, natural convection also builds boundary layers on the surfaces
of solid bodies. The equations that govern forced convection are essentially the same as
those for external laminar flow free convection. For natural convection problems, it is
easiest to assume that the fluid is Newtonian, meaning a fluid whose stress vs. strain rate
curve is linear, and the flow field is two dimensional and steady. The continuity
equation for two-dimensional natural convection is as follows:
𝛿(𝜌𝑢) 𝛿(𝜌𝑣)
+
=0
𝛿𝑥
𝛿𝑦
The momentum equation is similar to that shown in Section 4 except it contains the
gravitational body force term (-ρg):
𝜌𝑢
𝛿𝑢
𝛿𝑢
𝑑𝑃
𝛿
𝛿𝑢
+ 𝜌𝑣
=−
− 𝜌𝑔 +
(𝜇 )
𝛿𝑦
𝛿𝑦
𝑑𝑥
𝛿𝑦 𝛿𝑦
The pressure gradient term is due to the hydrostatic pressure field outside the boundary
layer and can be written as:
𝑑𝑃
= −𝜌∞ 𝑔
𝑑𝑥
Therefore the momentum equation becomes:
𝜌𝑢
𝛿𝑢
𝛿𝑢
𝛿
𝛿𝑢
+ 𝜌𝑣
= 𝑔(𝜌∞ − 𝜌) +
(𝜇 )
𝛿𝑦
𝛿𝑦
𝛿𝑦 𝛿𝑦
The energy equation is:
𝜌𝑐𝑢
𝛿𝑇
𝛿𝑇
𝛿
𝛿𝑇
+ 𝜌𝑐𝑣
=
(𝑘 )
𝛿𝑥
𝛿𝑦 𝛿𝑦 𝛿𝑦
A more practical way to measure the amount of natural convection occurring is to use
the Grashof number.
The Grashof number, Gr, is a dimensionless number that
approximates the ratio of the buoyancy to viscous forces acting on a fluid. The Grashof
number can be calculated by:
𝐺𝑟 =
𝑔𝛽(𝑇𝑠 − 𝑇∞ )𝐿3
𝜐2
where β is the thermal expansion coefficient:
1 𝛿𝑃
𝛽=− ( )
𝜌 𝛿𝑇 𝑝
12
The importance of buoyancy forces in a mixed convection flow (i.e., forced and natural
convection are occurring simultaneously) can be measured by the ratio of the Grashof
and Reynolds numbers:
𝐺𝑟
𝑔𝛽Δ𝑇𝐿
=
2
𝑅𝑒
𝜐2
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. In
pure natural convection, the strength of the buoyancy-induced flow is measured by the
Rayleigh number:
𝑅𝑎 = 𝐺𝑟𝑃𝑟
The Prandtl number, Pr, is a dimensionless number that approximates the ratio of
momentum diffusivity (kinematic viscosity) to thermal diffusivity.
𝑃𝑟 =
𝑣 𝐶𝑝 𝜇
=
𝛼
𝑘
the thermal diffusivity (α) is:
𝛼=
𝑘
𝜌𝑐𝑝
Rayleigh numbers less than 108 indicate a buoyancy-induced laminar flow, with
transition to turbulence occurring over the range of 108 < Ra < 1010. (ANSYS Help
Theory Guide)
13
6.3 Examples of Natural Convection
Two of the simplest forms of natural convection are a fluid surrounding a
horizontal cylinder and fluid surrounding a vertical plate. Both will be examined in the
following subsections.
6.3.1
Modeling of a Horizontal Cylinder
In the first example, a uniformly heated cylinder is submerged in an infinite pool. The
cylinder is slightly warmer than the surrounding fluid and therefore energy passes from
the cylinder to the surrounding fluid. As the fluid absorbs the energy, its temperature
begins to increase. The fluid temperature gradient as calculated by Fluent is shown in
Figure 5.3.1-1.
Figure 5.3.1-1: Horizontal Cylinder Temperature Plot
When the temperature increases, the fluid expands and the density decreases. The
hottest fluid is that in direct contact with the cylinder.
As the density decreases,
buoyancy forces take affect and the warmer, less dense fluid begins to rise. The density
changes can be seen in Figure 5.3.1-2. Notice that even some distance away from the
cylinder there is a density change. This is caused by small amounts of conduction within
14
the fluid which causes small density changes in the fluid not even in contact with the
heated cylinder.
Figure 5.3.1-2: Horizontal Cylinder Density Plot
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 since it is denser than the fluid that
surrounds it. This motion begins to forms a large convection cell which eventually
returns the fluid to the bottom where it is reheated by the cylinder. This can be seen in
the velocity vector plot shown in Figure 5.3.1-3
15
Figure 5.3.1-3: Horizontal Cylinder Velocity Vector Plot
Figure 5.3.1-3 shows the magnitude and direction of the fluid in the pool with a
submerged horizontal cylinder.
It can be seen that the as the cylinder heats the
surrounding fluid, it travels around the cylinder, eventually separating and rising. An
interesting point is when the fluid loses heat to its surrounding it begins to fall. Some of
that falling liquid is heated by the warmer fluid rising and it begins to rise without being
heated by the cylinder. This creates a small convection cell 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).
To ensure that the ANSYS Fluent generated figures are correct, the results were
compared to experimental results. The following figure shows isotherms surrounding a
horizontal tube in natural convection flow as revealed by an interference photograph.
16
Figure 5.3.1-4: Horizontal Cylinder Experimental Isotherm Plot
Show Figure 93, page 167 from Introduction to the Transfer of Heat and Mass
The ANSYS Fluent calculated isotherms for a horizontal cylinder is shown in
Figure 5.3.1-5:
Figure 5.3.1-5: Horizontal Cylinder Isotherm Plot
The modeling of a horizontal cylinder submerged in an infinite pool using ANSYS
Fluent 14.0 shows good resemblance to experimental data. Figure 5-5 and 5-6 show
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.
Because the
isotherms calculated by ANSYS Fluent are similar to those found experimentally, it can
confidently be stated that the results for a uniformly heated horizontal cylinder
submerged in a pool are reasonable.
17
6.3.2
Modeling of a Vertical Plate
The second example is similar to the first, except it involves a uniformly heated vertical
flat 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
surrounding fluid. The main difference between the flat plate and the cylinder example
is that in the flat plate example, the fluid has more time in contact with the plate as it
rises and is therefore able to absorb more energy. As the fluid absorbs the energy, its
temperature begins to increase. Because the fluid is heated as it rises along the plate, the
fluid temperature is greater in the flat plate example than in the horizontal cylinder
example.
Figure 5.3.2-1: Vertical Plate Temperature Plot
Since heat is exchanged between the plate and the fluid, a thermal boundary layer is
created. Thermodynamic equilibrium demands that the fluid in contact with plate is
equal to the temperature of the plate. The region in which the temperature changes from
the plate surface temperature to that of the bulk fluid is known as the thermal boundary
layer. Notice how the thermal boundary layer is small at the bottom of the plate and
much larger at the top. The thermal boundary layer expands as the momentum boundary
layer expands which helps pull warm fluid away from the hot plate.
18
For more
information on thermal and momentum boundary layers, see Convective Heat and Mass
Transfer (Reference XX).
Figure 5.3.2-2: Vertical Plate Velocity Vector Plot
Figure 5.3.2-2 shows how the fluid velocity is highly vertical and increases as it travels
up the plate. This is caused by the fluid having lengthy contact time with the heated
surface creating a greater temperature gradient and therefore a large 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 the fluid is in contact with
the heated surface longer. 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.
To ensure that the ANSYS Fluent generated figures are correct, the results were
compared to experimental results. The following figure shows isotherms surrounding a
vertical plate in natural convection flow as revealed by an interference photograph.
19
Figure 5.3.2-3: Vertical Plate Experimental Isotherm Plot
Show Figure 94, page 168 from Introduction to the Transfer of Heat and Mass
The ANSYS Fluent calculated isotherms for a heated vertical plate is shown in the
Figure 5.3.2-4.
Figure 5.3.2-4: Vertical Plate Isotherm Plot
Figure 5.3.2-3 and Figure 5.3.2-4 show 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 heated surface.
Not only do the isotherms match expectations, the momentum boundary layer calculated
by ANSYS Fluent also matches experimental expectations.
Figure 5.3.2-5: Experimental Momentum Boundary Layer
Show Figure 17-2, page 371 from Convective Heat and Mass Transfer
20
Figure 5.3.2-6: Momentum Boundary Layer Plot
The trends shown in Figure 5.3.2-5 and 5.3.2-6 are comparable.
Having similar
isotherms and momentum boundary layers, it can confidently be stated that the results
determined by ANSYS Fluent for a uniformly heated vertical plate submerged in a pool
are reasonable.
21
7. Laminar Flow with Heat Transfer
“Single-phase fluid flow can be characterized into two categories, laminar or
turbulent flow. Laminar flow implies that the fluid moves in sheets, or “laminae,” that
slip relative to each other (Introduction to Thermal and Fluids Engineering). Laminar
flow occurs at very low velocities where there are only small disturbances and little to no
local velocity variations. At low velocities, the fluid tends to flow without lateral mixing,
and adjacent layers slide past one another easily.
There are no cross currents
perpendicular to the direction of flow, nor eddies or swirling of fluid. In laminar flow,
the motion of the fluid particles is very orderly. In fluid dynamics, laminar flow is a
flow regime characterized by high momentum diffusion and low momentum convection
(Wikipedia)”. Turbulent flow, the other category of single-phase flow, is discussed in
Section 7.
The Reynolds number is used to characterize the flow regime. The Reynolds
number, Re, is a dimensionless number that gives a measure of the ratio of inertial forces
(fluid resists change in motion) to viscous forces. This helps to quantify the relative
importance of these two types of forces for given flow conditions. (Wikipedia) The
Reynolds number can be calculated using the following equation:
Re =
ρVA
μ
For internal flow, such as within a pipe, laminar flow is characterized by flow with a
Reynolds number less than 2300 whereas turbulent flow is characterized by a flow with
a Reynolds number greater than 4000. For flow with a Reynolds number between 2300
and 4000, both laminar and turbulent flows are possible. This is called transition flow.
The velocity of laminar flow in a pipe is can be calculated by:
𝑢=
𝑟𝑠2
𝑑𝑃
𝑟2
(− ) (1 − 2 )
4𝜇
𝑑𝑥
𝑟𝑠
However, it is friendlier to express the velocity in terms of the mean velocity, V.
𝑢 = 2𝑉 (1 −
𝑟2
)
𝑟𝑠2
The energy equation for flow through a circular pipe assuming symmetric heat transfer,
fully developed flow and constant fluid properties is:
22
𝛿𝑇
1𝛿
𝛿𝑇
𝛿 2𝑇
𝑢
= 𝛼[
(𝑟 ) + 2 ]
𝛿𝑥
𝑟 𝛿𝑟 𝛿𝑟
𝛿𝑥
Using ANSYS Fluent, a simple laminar flow problem through a cylindrical pipe
with uniform surface termperature was developed.
The Reynolds number for the
problem was 352 which means the flow in the laminar regime. One of the most notable
characteristics of laminar flow is the velocity profile parabolic shape which can be
calculated using the equation for velocity above.
Figure 6-1 shows the velocity
magnitude versus 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
developes, i.e., the entrance effects dissipate, the velocity profile becomes more and
more parabolic until it reaches a steady state 45 cm from the entrance. This is proven by
the fact that the outlet, 50 cm from the entrance, and the velocity profile 45 cm from the
entrance are basically the same.
23
Figure 6-1: Laminar Flow Velocity Profile
Another characteristic of laminar flow is the lack of mixing that occurs within the
fluid as it travels through the pipe. Laminar flow is considered to move in “sheets” and
each fluid molecule or atom tends to stay about the same distance from the centerline as
it travels through the pipe. This is shown by viewing the temperature profile below.
Diffusion and conduction are the primary forms of heat transfer in laminar flow. Notice
how the fluid thermal boundary layer grows slowly as it travels down the pipe towards
the centerline.
Figure 6-2: Laminar Flow Temperature Plot
Figure 6-3 shows the radial flow velocity. The greatest radial velocity occurs at the
entrance and exit of the pipe. However, as expected, because laminar flow moves in
24
“sheets,” the radial velocity for the middle of the pipe is near zero meaning there is little
mixing.
Figure 6-4: Laminar Flow Radial Velocity Plot
Laminar flow also tends to create large momentum boundary layers which cause
frictional force on the wall. The drag force on the wall is shown in Figure 6-5.
Figure 6-5: 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. At the very end, around 49 cm, the wall shear stress begins to
increase due to the pipe exit.
25
8. Turbulence
In fluid
dynamics, turbulence or turbulent
flow is
a
flow
regime
characterized by chaotic and stochastic property changes. It exists everywhere in
nature from the jet stream to the oceanic currents. Giving an exact definition of
turbulence is rather difficult. Instead, it is easier to describe a turbulent flow.
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 and heat and mass
transfer. They all have 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. Finally, turbulent flows are
dissipative. Viscous shear stresses perform deformation work which increases the
internal energy of the fluid at the expense of kinetic energy. 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 decays.
The
Reynolds number decreases and the flow tends to become laminar again (A First
Course in Turbulence).
In flows that are originally laminar, turbulence arises from instabilities at large
Reynolds numbers. As stated previously, laminar flow starts to transition to turbulent at
Reynolds numbers around 2100 (A First Course in Turbulence). Although laminarturbulent transition is not governed by Reynolds number, the same transition
occurs if the size of the object is gradually increased, or the viscosity of the fluid is
decreased, or if the density of the fluid is increased (Wikidpedia). One of the most
common examples of the transition of laminar flow to turbulent flow is smoke rising
from a cigarette.
26
http://askphysics.com/wp-content/uploads/2012/01/cigarette.gif
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.
Because of its irregular nature, modeling turbulence can be difficult. It requires the
solution to the Navier-Stokes equations (see conservation of momentum equation in
Section 4).
The Reynolds-averaged Navier–Stokes equations (or RANS equations) are timeaveraged equations of motion for fluid flow. The idea behind the equations is Reynolds
decomposition, whereby an instantaneous quantity is decomposed into its time-averaged
and fluctuating quantities, an idea first proposed by Osborne Reynolds. The RANS
equations are primarily used to describe turbulent flows. These equations can be used
with approximations based on knowledge of the properties of flow turbulence to give
approximate time-averaged solutions to the Navier–Stokes equations (Wikipedia).
In Reynolds averaging, the solution variables in the instantaneous (exact) NavierStokes equations are decomposed into the mean (ensemble-averaged or time-averaged)
and fluctuating components. For the velocity components:
𝑢𝑖 = 𝑢̅𝑖 + 𝑢𝑖′
where 𝑢̅𝑖 and 𝑢𝑖′ are the mean and fluctuating velocity components (i = 1, 2, 3).
Likewise, for pressure and other scalar quantities:
𝜙 = 𝜙̅ + 𝜙 ′
27
where 𝜙 denotes a scalar such as pressure, energy, or species concentration.
Substituting expressions of this form for the flow variables into the instantaneous
continuity and momentum equations and taking a time (or ensemble) average (and
dropping the overbar on the mean velocity, 𝑢̅) yields the ensemble-averaged 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 Reynolds-averaged Navier-Stokes (RANS)
equations.
They have the same general form as the instantaneous Navier-Stokes
equations, with the velocities and other solution variables now representing ensembleaveraged (or time-averaged) values. Additional terms now appear that represent the
′ ′
̅̅̅̅̅̅
effects of turbulence. These Reynolds stresses,−𝜌𝑢
𝑖 𝑢𝑗 , must be modeled in order to
close the second equation.
One way that the Reynolds stress is modeled is using the k-ϵ turbulence model.
The k-ϵ model was first introduced by Harlow and Nakayama in 1968. (F. H. Harlow
and P. I. Nakayama, Transport of turbulence energy decay rate, Los Alamos Sci. Lab.,
LA-3854, 1968). The k-ϵ model has become the most widely used model for industrial
applications because of its overall accuracy and small computational demand. The k in
the k-ϵ model stands for the turbulent kinetic energy and the ϵ stands for its dissipation
rate.
Turbulent kinetic energy is the mean kinetic energy per unit mass associated
with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterized
by measured root-mean-square (RMS) velocity fluctuations (Wikipedia). Epsilon (ϵ) is
the rate of dissipation of the turbulence energy per unit mass.
Two-equation turbulence models allow the determination of both, a turbulent
length and time scale by solving two separate transport equations. The standard k-ϵ
model in falls within this class of models and has become the workhorse of practical
engineering flow calculations in the time since it was proposed by Launder and
Spalding. Robustness, economy, and reasonable accuracy for a wide range of turbulent
flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-
28
empirical model, and the derivation of the model equations relies on phenomenological
considerations and empiricism.
The standard k-ϵ model is a model based on model transport equations for the
turbulence kinetic energy (k) and its dissipation rate (ϵ). The model transport equation
for k is derived from the exact equation, while the model transport equation for ϵ was
obtained using physical reasoning and bears little resemblance to its mathematically
exact counterpart. 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. The standard kϵ model is therefore valid only for fully turbulent flows.
As the strengths and
weaknesses of the standard k-ϵ model have become known, modifications have been
introduced to improve its performance.
Over the years improvements have been made to the Standard k-ϵ model. The
improvements helped to create a new model known as the Realizable k-ϵ model. The
Realizable k-ϵ model differs from the Standard k-ϵ model in two important ways. First,
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.
8.1 Calculating Turbulence Parameters
When using the Realizable k-ϵ in ANSYS Fluent certain parameters need to be
established to properly set the initial and boundary conditions of the problem. The
following calculations were performed to determine the boundary condition and initial
condition inputs for the turbulence model.
Mass Flow Rate: 0.5 kg/s (randomly chosen flow rate that will give turbulent flow)
Pipe Diameter (D): 0.03 m
Viscosity (μ): 0.001003 kg/m-s
Density (ρ): 998.2 kg/m3
29
Turbulence Empirical Constant (Cμ) = 0.09 [1]
Hydraulic Diameter (Dh):
𝐷 2
4 ∗ 𝐴 𝜋 ∗ (2)
𝐷ℎ =
=
= 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
2
0.001003 𝑚 − 𝑠 ∗ 0.00070686 𝑚
Turbulence Length Scale (l):
𝑙 = 0.07 ∗ 𝐷ℎ = 0.07 ∗ 0.03 𝑚 = 0.0021 𝑚
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
0.001598853/2
0.0021
Using the above calculated turbulent parameters, the following was determined.
Below is a figure of the velocity magnitude versus distance from the 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. The figure shown below is very difference from the similar figure for laminar
30
flow. First the velocity profiles are much flatter. The parabolic shape seen in the laminar
flow plot is not as clear. Second, the flow reaches the steady state velocity profile
quicker for turbulent flow. The flow is stable at about 40 cm where it takes until about
45 cm to become stable for the laminar flow for the same pipe geometry.
Below is the wall shear stress versus distance from the entrance. Notice how the
shear stress is very large in the beginning and quickly reduces by about 3 cm. The large
shear stress at the beginning is due to entrance effects.
31
Also note that the distance needed to dissipate increased shear stress due to entrance
effects is much shorter than for laminar flow. It took the laminar flow about 5 cm to
dissipate the impact of the entrance on the shear stress.
Once the entrance effects
dissipate the wall shear stress slowly decreases as the flow becomes more and more
stable. At about 22 cm the shear stress reduces again. This can be explained by
boundary layer separation. The structure and location of boundary layer separation
often changes, sometimes resulting in a reduction of overall drag (Wikipeida). At
the very end of the pipe, around 49 cm, the wall shear stress begins to increase. This is
caused by the pipe exit.
Another very different feature of turbulent flow compared with laminar flow is the
mixing that occurs radially. The figure below shows the radial flow velocity. The
greatest radial velocity occurs at the entrance and exit of the pipe. This is also true of
laminar flow.
However, the radial velocity is non-zero in the middle of the pipe
meaning that mixing is occurring.
32
The radial velocity figure for turbulent flow is very different than that for the
laminar flow. That is because laminar flow is characterized by moving in “sheets” while
turbulent flow is characterized by random motion.
The next two figures show the turbulent kinetic energy and the production of
turbulent kinetic energy.
Notice how all of the turbulent kinetic energy is near the wall. This is because the
wall helps generate turbulent kinetic energy. The shape of the production of turbulent
kinetic energy versus distance from the entrance has the same trend as that of the wall
shear stress. This makes sense because shear stress, caused by the wall, produces
turbulent kinetic energy.
33
The entrance effects also help to produce turbulent kinetic energy which is why the
most amount of turbulent kinetic energy is produced at the wall near the entrance.
34
9. Turbulence with Heat Transfer
The problem of turbulent flow can be further complicated by adding energy transfer.
The flow components with heat transfer are basically the same as that for solely
turbulent flow however temperature changes and therefore density changes do have a
slight impact. The radial energy equation given in Section 4 can be applied for turbulent
flow as well as laminar flow. The ANSYS Fluent problem discussed in Section 7 was
expanded to include a constant surface heat flux. The figure below shows the fluid
temperature change caused by the constant surface heat flux.
The turbulent kinetic energy is the same between the two cases which is expected
since turbulent kinetic energy is momentum based not thermal based. However, if the
heat transfer was large enough that buoyance effects began to influence the flow, then
there would be an increase in turbulent kinetic energy in the example with heat transfer.
The radial velocity is the same between the two cases, which is expected since the
radial velocity is caused by turbulent flow.
35
The velocity profiles for each distance from the entrance except that of the entrance
is shifted up a small amount. The heat transfer that occurs causes the density of the fluid
to decrease. To keep a constant mass flow through the pipe, the velocity increases
slightly.
The shear stress is the same which is expected since it is based solely on the flow
and momentum boundary layer.
36
37
10.Two-Phase Flow
A large number of flows encountered in nature and technology are a mixture of
phases. Physical phases of matter are gas, liquid, and solid, but the concept of phase in a
multiphase flow system is applied in a broader sense. In multiphase flow, a phase can be
defined as an identifiable class of material that has a particular inertial response to and
interaction with the flow and the potential field in which it is immersed. For example,
different-sized solid particles of the same material can be treated as different phases
because each collection of particles with the same size will have a similar dynamical
response to the flow field. Multiphase flow regimes can be grouped into four categories:
gas-liquid or liquid-liquid flows; gas-solid flows; liquid-solid flows; and three-phase
flows. The area that will be focused on is gas-liquid flow and more specifically gasliquid flow caused by boiling heat transfer. (Theory Guide).
Boiling heat transfer is defined as a mode of heat transfer that occurs with a
change in phase from liquid to gas. 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. Since the heat transfer rate in
boiling is generally very high, it has been used to cool devices requiring high heat
transfer rates such as nuclear reactors.
There are four regimes of boiling shown in the figure below.
http://www.thermalfluidscentral.org/e-resources/download.php?id=1
The first regime of boiling is natural convection boiling, up to point A. During
this regime, no bubbles form. Instead, heat is transferred from the surface to the bulk
38
fluid
by
natural
convection
(http://www.thermalfluidscentral.org/e5/4
resources/download.php?id=1). The heat transfer rate is proportional to 𝛥𝑇𝑠𝑎𝑡 (Tong).
The second regime of boiling, from point A to point C, is called nucleate boiling.
During this stage vapor bubbles are generated at certain preferred locations on the heated
surface called nucleation sites. Nucleation sites are often microscopic cavities or cracks
in the surface (http://www.thermalfluidscentral.org/e-resources/download.php?id=1).
When the liquid near the wall superheats it evaporates, forming bubbles at the various
nucleation sites. The bubbles transport the latent heat of the vaporization and also
increase the convective heat transfer by agitating the liquid water near the heated
surface. There are two subregimes of nucleate boiling. The first is local boiling which
is boiling in a subcooled liquid, where the bubbles form at the heating surface tend to
condense locally. Subcooled boiling is discussed further in Section 11. Bulk boiling is
nucleate boiling in a saturate liquid; in this case, the bubbles do not collapse (Tong).
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.
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 over the heated surface and thereby dramatically increases the
surface temperature. This is called the boiling crisis or departure from nucleate boiling.
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 (Tong).
If the temperature difference between the surface and the fluid continues to increase,
stable film boiling is achieved. At this point, there is a continuous vapor blanket that
39
covers 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
(http://www.thermalfluidscentral.org/e-resources/download.php?id=1) (Advanced Heat
and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell).
To know the flow pattern of a two-phase flow is as important as to know whether
the flow is laminar or turbulent in a single-phase flow. The number of characteristic
flow patterns and the name used for them vary somewhat from investigator to
investigator. Some of them include, in order of increasing quality 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 types can be seen in the figure
below.
Add Tong, Figure 3-2, page 50
The flow patterns stated above can be further classified into three categories, bubbly
flow, slug flow and annular flow. Bubbly flow is when the liquid phase in continuous
and the vapor phase is discontinuous. The vapor is distributed in the liquid in the form
of bubbles. The flow pattern occurs at low void fractions. This is the flow pattern of
subcooled boiling. Slug flow is when there are relatively large liquid slugs in the flow as
a result of the beginning of agglomeration of vapor bubbles. This flow occurs at
moderate void fractions and relatively low flow velocities. Annular flow is when the
liquid phase is continuous in an annulus along the wall and the vapor phase is continuous
in the core. This flow pattern occurs at high void fractions and high flow velocities.
(TONG)
The following figure depicts the three flow patterns described above as a fluid is heated
while travelling in a pipe.
Add Tong, Figure 3-6, page 56
40
10.1 Flow Regimes
The first step in solving any multiphase problem is to determine which of the regimes
described in Multiphase Flow Regimes in the Theory Guide best represents your flow.
Model Comparisons in the Theory Guide provides some broad guidelines for
determining appropriate models for each regime, and Detailed Guidelines provides
details and how to determine the degree of interphase coupling for flows involving
bubbles, droplets, or particles, and the appropriate model for different amounts of
coupling.
41
The gas volume fraction and the ratio of liquid velocity to gas velocity determine the
flow regime that describes a given flow.
Show Figure 1.2 from One-Dimensional Two Phase Flow (p.9)
Show Figure 3-4 from TONG (p. 54)
The following regimes are gas-liquid:




Bubbly flow: This is the flow of discrete gaseous or fluid bubbles in a continuous
fluid.
Droplet flow: This is the flow of discrete fluid droplets in a continuous gas.
Slug flow: This is the flow of large bubbles in a continuous fluid.
Stratified/free-surface flow: This is the flow of immiscible fluids separated by a
clearly-defined interface.
42
11.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 and nonmetallic inclusions. The advancements made in mixing
have increased the amount on control available over the steelmaking process which has
led to improved the quality of steel produced. Gas injection uses a porous plug that is
located at the bottom of a tank that hold molten metal. The porous plug allows the
addition of a gas into the mixing tank at a given velocity and an approximate bubble
diameter. The gas travels through the liquid due to buoyancy forces and causes mixing
of the liquid due to drag forces. In this model, the Schiller-Nauman drag model is used
drag model will be used which is generally acceptable for all multiphase calculations.
To better understand how gas injection impacts fluid movement and therefore
mixing, CFD modeling is used. This model implements the Eulerian multiphase model
with the standard k-ϵ turbulence model, a bubble diameter of 0.00005 m and an inlet
velocity of 0.05 m/s.
43
44
Mesh Independence:
Table 11-1: Mesh Independent Quantities
Nodes
2701
Elements
2592
Max Liquid Velocity
0.37243
Max Gas Velocity
0.37249
Max Liquid Dynamic Pressure
72.6646
Max Gas Dynamic Pressure
0.08920
Max Liquid Volume Fraction
0.94933
Min Gas Volume Fraction
0.05067
45
12.Bubble Column
A bubble column reactor is an apparatus used for gas-liquid reactions. A bubble
column is a vertical column of liquid with gas introduced continuously at the bottom
through a sparger. Bubbles form and travel upwards through the column due to the inlet
gas velocity and buoyancy. The mixing is done by the gas sparging and it requires less
energy than mechanical stirring. Bubble column reactors are characterized by a high
liquid content and a moderate phase boundary surface. The bubble column is particularly
useful in reactions where the gas-liquid reaction is slow in relation to the absorption
rate.
Bubble column reactors are often used in industry to develop and produce
chemicals and fuels for use in chemical, biotechnology, and pharmaceutical processes.
The reactors involve gas-liquid flows where the gas is dispersed as bubbles in a
continuous volume of liquid. When reactants are introduced into the flow, the interaction
of gas and liquid can cause chemical reactions.
Figure 12-1: Bubble Column
http://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bubble_column.svg/200px
-Bubble_column.svg.png
In all gas-liquid flows, the bubbles can increase and decrease in size due to
coalescence and breakup. Coalescence is two or more bubbles colliding, whereby the
thin liquid barrier between ruptures to form a larger bubble. Breakup of bubbles is
46
caused by collisions with turbulent eddies, approximately equal in size to the bubbles
(BC Master Thesis). The ability to calculate the change in bubble size due to turbulent
eddies is discussed in Section 14.
The results shown in this section implement the Eulerian multiphase model with the
standard k-ϵ turbulence model, a bubble diameter of 0.0048 m and an inlet velocity of
0.05 m/s.
Figure 12-2 shows a comparison between gas volume fraction at 1 second and 5
seconds after gas has started flowing through the bubble column. Note that in both time
points the gas tends to flow in slugs. After 5 seconds the gas has reached the top of the
liquid and interaction between the liquid-gas interface has occurred causing it to change
shape. It can also be seen that the liquid level after 5 seconds 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.
Figure 12-2: Instantaneous Gas Volume Fraction
(left image: 1 second; right image: 5 seconds)
47
Figure 12.3 shows a comparison between liquid velocity vectors fraction at 1 second
and 5 seconds after gas has started flowing through the bubble column. Distinct paths of
liquid movement can be seen in both 1 second and 5 seconds. Most of the liquid seems
to be pushed along the wall and the center of the column.
Figure 12.3: Instantaneous Liquid Velocity Vectors
(left image: 1 second; right image: 5 seconds)
48
Figure 12-4 shows a comparison between gas velocity vectors fraction at 1 second
and 5 seconds after gas has started flowing through the bubble column. It can be easily
observed from Figure 12-4 where the gas particles are after 1 second in the bubble
column. It is also interesting that the original gas-liquid interface is not flat but two
parabolas. This is most likely due to bubble coalescence and wall drag. The gas
velocity after 5 seconds shows that the greatest gas velocity occurs near the walls which
is surprising due to friction effects caused by the walls. The greatest velocity is shown at
the liquid-gas interface.
Figure 12-4: Instantaneous Gas Velocity Vectors
(left image: 1 second; right image: 5 seconds)
49
13.Population Balance Equation
13.1 Background
“Several industrial fluid flow applications involve a secondary phase with a size
distribution. The size distribution of particles, including solid particles, bubbles, or
droplets, can evolve in conjunction with transport and chemical reaction in a multiphase
system. The evolutionary processes can be a combination of different phenomena like
nucleation, growth, dispersion, dissolution, aggregation, and breakage producing the
dispersion. 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. Cases in which a population balance could apply include crystallization,
precipitative reactions from a gas or liquid phase, bubble columns, gas sparging, sprays,
fluidized bed polymerization, granulation, liquid-liquid emulsion and separation, and
aerosol flows.
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.” (ANSYS Fluent PBE Guide,
Section 1.0)
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,
coalescence of multiple bubbles into one larger bubble or shrinkage due to transfer of
energy from the bubble to the surrounding fluid. The population balance models allows
for a more accurate calculation of void fraction during subcooled nucleate boiling.
In this analysis the Discrete method is used. “In the discrete method, the particle
population is discretized into a finite number of size intervals. This approach has the
advantage of computing the particle size distribution (PSD) directly. This approach is
also particularly useful when the range of particle sizes is known a priori and does not
50
span more than two or three orders of magnitude. In this case, the population can be
discretized with a relatively small number of size intervals and the size distribution that
is coupled with fluid dynamics can be computed. The disadvantage of the discrete
method is that it is computationally expensive if a large number of intervals is needed.”
(ANSYS Fluent PBE Guide Section 1.1)
13.2 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.
13.2.1 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
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.
51
13.2.2 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.
13.2.3 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.
13.2.4 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
52
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.
13.2.5 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
̅ = 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.
(TONG)
53
13.3 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.
(ANSYS Fluent PBE Guide Section 2.3.1)
Figure 13-1: Particle Size Distribution (ANSYS Fluent PBE Guide Figure 2.1)
54
14.Bubble Column with Population Balance Model
The work discussed in Section 12 was extended to include a population balance
model. The entire bubble population may not be a single size formed from the inlet at
the sparger due to growth, coalescence, and breakup.
The implementation of a
population balance model allows for the direct calculation of the growth, breakup, and
coalescence of bubbles as they travel up the bubble column. The population balance
model also does not necessitate estimating the proper characteristic bubble size but
rather permits a specified bubble size distribution.
The same model used in Section 12 was utilized with the addition of a population
balance model with 3 discrete bubble sizes (0.0030 m, 0.0048 m and 0.0076 m). The
inlet uses a gas bubble diameter distribution of 25% 0.0030 m, 50% 0.0048 m and 25%
0.0076 m. The coalescence and breakage of bubbles is determined using the luo model
with a surface tension of 0.072 N/m
Figure 14.1 shows a comparison between gas volume fraction at 1 second and 5
seconds after gas has started flowing through the bubble column. Note that at both time
points the gas tends to flow in slugs. After 5 seconds the gas has reached the top of the
liquid and interaction between the liquid-gas interface has occurred causing it to change
shape. It can also be seen that the liquid level after 5 seconds 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.
When comparing this to
Figure 12.2, 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 PBM implemented
the phase distribution seems to be much more uniform without any large areas with high
gas volume. This is especially noticeable at the bottom. Another difference is the
liquid-gas interface. At 5 seconds, the liquid-gas interface appears to have increased in
elevation only slightly due to displacement and drag forces compared with the bubble
column results without PBM. Another major difference is that after 1 second the gas
seems to have traveled farther into the bubble column compared with Figure 12.2.
55
Figure 14-1: Instantaneous Gas Volume Fraction with PBM
(left image: 1 second; right image: 5 seconds)
Figure 14-2 shows a comparison between liquid velocity vectors at 1 second and 5
seconds after gas has started flowing through the bubble column. Similar to Figure 12-3,
there are distinct paths of liquid movement can be seen in both 1 second and 5 seconds.
Figure 14-2a shows that the liquid velocity in the upper portion of the column is greater
than zero which matches the gas volume fraction results in Figure 14-1. Figure 14-2
shows a more uniform mixing of liquid velocity throughout the bubble column, no
sections of little to no movement. The maximum velocity in Figure 14-2 is also less than
that of Figure 12-3.
56
Figure 14-2: Bubble Column Liquid Vector Velocity with PBM
(left image: 1 second; right image: 5 seconds)
Table 14-1 shows the bubble population at different heights in the bubble column.
Table 14-1: PBM Bubble Size
Bin-0 (0.0076 m)
Bin-1 (0.0048 m)
Bin-2 (0.0030 m)
Inlet
(Fraction)
Outlet
(Fraction)
Net
(Fraction)
0.250
0.500
0.250
0.865
0.117
0.018
+0.557
-0.308
-0.134
Weighted
Average
(Fraction)
0.776
0.174
0.051
This shows that as the bubbles travel up the column, the smaller bubbles primarily
coalesced into larger bubbles. This means that the turbulent forces were weak and did
not break apart the bubbles.
57
15.Pool Boiling
“Pool boiling is the type of boiling that occurs when a heater is submerged in a pool of
initially stagnant liquid. When the surface temperature of the heater sufficiently exceeds
the saturation temperature of the liquid, vapor bubbles nucleate on the heater surface.
The bubbles grow rapidly in the superheated liquid layer next to the surface until they
depart and move out into the bulk liquid. While rising is the result of buoyancy, the
bubbles either collapse or continue their grown, depending upon whether the liquid is
locally subcooled or superheated. Thus, in pool boiling, a complex fluid motion around
the hater is initiated and maintained by the nucleation growth, departure, and collapse of
bubbles; and by natural convection” (TONG p. 5).
This model implements the mixture multiphase model with laminar flow. Laminar
flow was chosen sine most of the fluid is not moving and there is only an outlet
boundary condition (no inlet). The fluid starts 1 degree subcooled and is heated by a
wall (at the bottom) that is constant temperature of 10 degrees above saturation.
Figure 15.1 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 15.2 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 take affect and
the fluid begins to move. The steam created on the bottom surface moves upward and
cooler liquid moves down to take its place creating eddies which can also be seen in
Figure 15.2. Figure 15.1 shows four nucleation sites on the heated wall where steam is
being formed. From these sites a phase changes occurs where bubbles nucleate, grow
and detach from the heated surface.
58
Figure 15.1: Instantaneous Gas Volume Fraction
(left image: 1 second; right image: 2 seconds)
Figure 15.1: Instantaneous Liquid Velocity Vectors
59
(left image: 1 second; right image: 2 seconds)
To better understand the vapor production on the heated surface and graphically see
the location of nucleation sites, Figure 15.3 was created. Figure 15.3 shows the volume
fraction of vapor on the heated surface at 10 seconds. Most of the vapor production
occurs on the left half of the heated surface, as can be seen by the spikes in vapor
volume fraction from 0 m to 0.15 m.
Figure 15.3: Volume Fraction of Vapor on Heated Surface
60
16.Subcooled Boiling
Show Figure 3-8 from TONG (p. 60)
61
17.Subcooled Boiling with Population Balance Model
62
18.References
[1]
ANSYS, Inc., 2012, ANSYS FLUENT 13.0 Theory Guide.
[2]
63