Theses and Dissertations
2007
Heat transfer and mass transfer with heat
generation in drops at high peclet number
Adham SouccarThe University of Toledo
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Recommended Citation
Souccar, Adham, "Heat transfer and mass transfer with heat generation in drops at high peclet number" (2007).Theses and Dissertations.Paper 1326.
Heat Transfer and Mass Transfer with Heat Generation
in Drops at High Peclet Number
Adham Souccar
Submitted as partial fulfillment of the requirements for The Doctor of Philosophy degree in
Engineering The University of Toledo
May 2007
Kronig and Brink published a classic analysis of transport from translating droplets. Their analysis assumed that the bulk of the resistance to transfer was in the droplet phase. It considered the limiting solution as the Peclet number became very large. Their work has been cited in many subsequent studies of droplet transfer. Chapter Three section 3.1 revisits their solution using numerical techniques that were not then available. It was found that only the first mode of their solution is accurate; hence, their solution is accurate at only at large times. In Chapter Three Section 3.2, the transient heat transfer from a droplet with heat generation is investigated. It is assumed that the
Peclet flows and very high Peclet flows.
As expected, it was found that the temperature rise due to the heat generation was less for high Peclet flows. In addition, the temperature profile responds more quickly for high Peclet flows. This analysis is also applicable to mass transfer with a zero-order reaction.
The system of differential equations that describes this behavior was solved using a semi-analytical method and a Matlab software computer program.
To my parents,
my brother
I would like to express my sincere gratitude to my advisor, Dr. Douglas Oliver, for giving me the initial stimulus to work on this project, for his support and guidance through out my graduate studies. His constant encouragements, generosity, constructive comments and inspiring discussion have been of immeasurable value to me. It’s an honor to be one of his PhD students.
I would like to express my deep gratitude to the Mechanical, Industrial and Manufacturing Engineering department for its financial support, and to Dr. A. Afjeh, Dr. M. Hefzy and all the staff for their assistance during my research.
I also want to extend my appreciation to the members of my dissertation committee, Dr. Terry Ng, Dr. Cyril Masiulaniec, Dr. Sorin Cioc and Dr. Ezzatollah Salari.
Many thanks for Prof. Baher Hanna and for Mr. Marc Iskandar for their proofreading my dissertation. I also wish to thank my friend Paul La Fontaine for his initial help.
Lastly, I would like to express my deepest thanks to my beloved parents, my brother and my wife for their love, understanding, and continuous support.
Page
Abstract
iii
Dedication
v
Acknowledgments
vi
Table of contents
vii
List of tables
x
List of figures
xi
Nomenclature xii
CHAPTER
1:
INTRODUCTION
1.1
Motivation
1
1.2
Theoretical
Basis
6
1.3
Droplet
Stream
Lines
7
1.4
Problem
Formulation
10
CHAPTER 2:
LITERATURE REVIEW
2.1 Heat and Mass Transfer from Droplets
13
2.2 Variation of Reynolds numbers
14
2.2.1
Low
Reynolds
numbers
14
2.2.2 Moderate Reynolds numbers
14
2.3.1 External Problem
15
2.3.2
Internal
Problem
16
2.3.3 Conjugate Problem 17
2.4 Heat Generation
19
CHAPTER 3:
MATHEMATICAL ANALYSIS
Introduction
22
Section 3.1 Analysis of Heat Transfer without Heat Generation
3.1.1
Introduction
23
3.1.2 Problem Formulation 24
3.1.3 Solution Procedure 25
Section 3.2 Analysis of Heat Transfer from a Droplet at High
Peclet Numbers with Heat Generation
3.2.1
Introduction
29
3.2.2
Problem
Formulation 29
3.2.3
Solution
Procedure
31
3.2.3.1 Special Case 1: Peclet number = 0
31
3.2.3.1a The steady state part
32
4.1 Heat Transfer from a translating Droplet at High Peclet
Numbers Revisiting the Classic Solution of
Kronig and Brink
48
4.2 Heat Transfer from a droplet at High Peclet Numbers
with Heat Generation
55
CHAPTER 5:
CONCLUSION AND FUTURE WORK
5.1 conclusion
62
5.2 Suggestions for future work 63
5.3 Final Thoughts
64
References
65
Appendix 1
75
Table
No.
Title Page
4-1
Convergence with Respect to
Δξ (without heat generation)49
Figure
No.
Title Page
1-1 Phase identification of drops and bubbles 4
1-2 Spherical coordinate system 8
1-3 motion through liquids Shape regimes for bubbles and drops in unhindered gravitational 9
1-4 Droplet Stream Lines 10
1-5 Stream lines and their orthogonal trajectories in a vertical plane
through the axis of a falling droplet
12
4-1 First two eigen-functions, present vs. Kronig and Brink 50
4-2 Bulk Temperature Θ(τ) 51
4-3
Θ
(
r
,
τ
)
along
θ
=
π
/
2
,
Kronig and Brink vs. present 524-4 Nusselt Number, Kronig and Brink vs. Present 53
4-5
Θ
(
r
,
τ
)
along
θ
=
π
/
2
544-6
Θ
(
r
)
along
θ
=
π
/
2
at several times 574-7 Θas a function of time. 58
4-8 ADI predictions of bulk temperatures. Copied from Fayerweather 59
4-9 Illustration of heat flux directions: High Peclet number vs. low
Peclet numbers 60
a Droplet radius.
An Coefficient, see Eq. (3.1-17).
Bn Coefficient, see Eq. (3.2-80).
cj Molar concentration
cp Specific heat
Djm Mass diffusion coefficient
Eo Eotvos number,
σ ρgL2 Eo= Δ
E& Rate of energy flow
Jj Molar diffusion flux vector
k Thermal conductivity
nmax Truncation limit.
Nu Nusselt number or Sherwood number
Pe Peclet number,
α
Ua Pe=2 .
q& Heat generation
q” Heat flux
Re Reynolds number,
r Radial coordinate made dimensionless with the radius a.
t Time
T Temperature.
U Free Stream lines velocity.
u Radial velocity v tangential velocity Greek Symbols ) (ξ γ Defined by Eq. (3.1-9). T T − μ ρ UL F F viscous inertia = = Re
Θ
Average or bulk temperature )(ξ
β Defined by Eq. (3.1-10)
ξ Dimensionless spatial coordinate, ξ =4r2
(
1−r2)
sin2θ.ψ Dimensionless stream function
μ Dynamic viscosity.
)
(
ξ
nΞ
Eigen-function. ρ Density α Thermal diffusivity φ Azimuth coordinates τ Dimensionless time, 2 a t α τ = .δ Kronecker delta function
θ Tangential coordinate.
Ф Viscous dissipation function
κ Ratio of dynamic viscosities μext/μdrop.
λn Eigen-value
Subscripts
drop Droplet or dispersed phase.
ext Continuous phase.
g With heat generation
init Initial.
max Maximum
CHAPTER ONE
INTRODUCTION
1.1 Motivation
In the eighteenth century and early nineteenth centuries the study of heat transfer was based on the caloric theory which asserts that heat is a mass-less, colorless, odorless, and tasteless fluid substance which can be poured from one body to another. When caloric was added to a body it raises its temperature and vice versa…
The modern physical understanding of the nature of heat developed in the middle of the nineteenth century where heat was defined as the energy associated with the random motion of atoms and molecules.
The human body rejected heat to its surroundings and the rate of this rejection was related to the human comfort. This rate of heat transfer is controlled by adjusting clothing to the environment. A lot of home appliances are designed by using heat transfer principles, (heating, air conditioning, refrigerating, etc…).
Heat transfer equipment such as boilers, condensers, heaters and heat exchangers are designed on the basis of heat transfer analysis, which in practice can be divided into two major groups:
- Rating problems: which determine the heat transfer rate for an existing system at a specific temperature difference
- Sizing problems: which determine the size of the system in order to transfer
heat at specific rate for a specified temperature difference.
A heat transfer process can be studied either experimentally by taking measurements, or analytically by analysis and calculations. However the first approach deals with actual size systems with fewer experimental errors.
Experimentation is expensive and time consuming. In contrast, the analytical approach has the advantage of being fast and relatively inexpensive. The work done in this study is analytical in nature. Any analytical work uses a set of assumptions to simplify the analysis. The accuracy of the solution depends on the assumptions made in such analysis.
Mass transfer is a similar phenomenon to heat transfer. The extraction of a substance dissolved in fluid droplets, by a second fluid surrounding the droplets, and not miscible with the first one, is a process of considerable technical importance. Bubbles, droplets and particles are essential in many natural processes like boiling, fermentation, air pollution and rainfall, as well as man related activities such as industrial systems, nuclear power plants, etc…
Clouds are assemblages of small water droplets which under certain circumstances coalesce, leading to rainfall. Oceans, seas, and lakes contain air in dissolved form known as bubbles. Particles play a primary role in all sorts of sprays.
In industrial systems, such as chemical reactors, drops and bubbles carry reactants and products. In nuclear power plants, one encounters bubbles in a boiling water reactor
and drops in spray cooling components. In internal combustion and jet engines fuel is
atomized. Engineers and researchers in many branches of engineering and science are
faced with heat and mass transfer problems.
When considering heat or mass transfer near droplets it is often convenient to consider the relative thermal resistance in the droplet as compared with the resistance in the surrounding fluid. If the thermal resistance in the droplet is much higher than that of the surrounding fluid, then it is reasonable to assume that the temperature at the droplet surface is equal to the ambient temperature. Thus, heat transfer is only calculated in the droplet, not in the ambient fluid. This assumption is associated with the so-called
interior problem.
When the thermal resistance is primarily in the surrounding phase, the temperature in the droplet is often assumed to be spatially uniform. This assumption is
associated with the so-called exterior problem. Finally, if the thermal resistance in
the droplet is of the same order of magnitude as that of the ambient fluid, then the heat transfer must be calculated in both the droplet and the ambient fluid. This situation
is the so-called conjugate problem. The work of this study deals with the interior
problem for droplet heat transfer.
A droplet is a mass of liquid in a liquid or gas medium, while a bubble is a mass of gas in an external medium. The fluid particle and the surrounding medium are separated by a well defined interface. In some cases like soap bubble this interface is a thin film. More complex situations, (compound drop) consist of pairs of drops and bubbles as shown in figure (1- 1)
(a) liquid drop (b) gas bubble in a liquid (c) Soap bubble
(d) Compound drop – three interfaces (e) Compound drop – two interfaces
Figure 1-1 Phase identification of drops and bubbles: Ref [73]
Application of transport studies related to droplets includes heat or mass transfer from bubbles rising in a liquid or from drops moving in a second fluid of different properties, combustion processes, and chemical reactions involving fluids.
Drops, bubbles or compound drops are referred to as fluid particles which consist of large numbers of molecules so as to be considered a continuum.
Gas Liquid
More than five decades ago, Kronig and Brink’s [66] work investigated mass transfer of a solute in a droplet. In particular, they obtained a semi-analytic solution
for the limiting case of the interior problem where the Peclet number is very large and the Reynolds number is very low. Their work has been extensively cited in the following decades. One of the problems with the work of Kronig and Brink is that their mathematical model only allowed for parabolic approximations to the concentration contours.
The intent of this work is to investigate the accuracy of the Kronig and Brink solution, especially at small times. In addition, this work will extend their model to include the effects of heat or mass generation.
1.2 Theoretical Basis
The mathematical work of Kronig and Brink was directed at mass transfer. It is equally applicable to heat transfer with the appropriate dimensional conversion. As a
result of the similarity of modeling heat and mass transfer processes, the terms heat or mass transfer are sometimes used interchangeably. Although only heat transfer is modeled in the dissertation, the results are equally applicable to mass transfer See Appendix 1.
Wherever there is temperature gradient serving as a driving force heat flow occurs, this is known as Fourier’s law, and is expressed as
T
k
q
′′
=
−
∇
(1-1)where
q
′′
is the heat flux, k the thermal conductivity and∇
T
is the temperaturegradient.
Mass diffusion occurs where there is a concentration gradient. A linear relation known as Fick’s law can describe this and could be mathematically stated as
J
j=
−
D
jm∇
c
jj
=
1
,
2
,
3
...
N
where
J
j is the molar diffusion flux vector,D
jm is the mass diffusion coefficientand ∇cj is the molar concentration gradient.
Applying Fourier’s law, Eq. (1-1) and the first law of thermodynamics to a controlled volume an energy equation may be developed and expressed as:
⎟
=
∇
⋅
∇
+
+
Φ
⎠
⎞
⎜
⎝
⎛
+
∇
∂
∂
μ
ρ
u
T
k
T
q
t
T
c
p&
(1-2)where
ρ
is density, cp is the specific heat, q& is the volumetric heat production rate,uis the radial velocity, μis the dynamic viscosity and Φ is the viscous dissipation
function.
For many heat transfer applications, the thermal conductivity and density can be assumed to be constant. In addition in the cases of slow flow, the viscous dissipation function is negligible. Under these conditions, the energy equation is stated as:
k
q
T
T
u
t
T
=
∇
+
&
⎟
⎠
⎞
⎜
⎝
⎛
+
∇
∂
∂
21
α
(1-3)where
α
=
k
/
ρ
c
p is the thermal diffusivity. (1-4)In the absence of heat and sink sources the energy equation takes the form of
u
T
T
t
T
21
=
∇
⎟
⎠
⎞
⎜
⎝
⎛
+
∇
∂
∂
α
(1-5)1.3 Droplet Stream Lines
Bubbles and droplets in free rise or fall under the influence of gravity can be classified into three main categories:
¾ Spherical
¾ Ellipsoidal
z r x y θ φ θ φ
The spherical coordinate system is applicable to spherical drops and bubbles. For an illustration of the spherical coordinates system see Fig. (1-2)
Figure 1-2 Spherical coordinate system
where is the azimuth coordinate, r is the radial coordinate, and the tangential coordinate.
Due to the force of surface tension, small bubbles and drops take a spherical shape. This means that the interfacial tension and/or viscous forces are much more important than the inertia forces. Figure (1-3) shows that at low Reynolds Numbers, spherical shapes for fluid particles occur.
Figure 1-3 Shape regimes for bubbles and drops in unhindered gravitational motion
through liquids. Adapted from Fig. [Ref. 12] EOTVOS NUMBER ,Eo
REYNOLDS NUMBE R , Re
μ
ρ
U
L
F
F
viscous inertia=
=
Re
σ
ρ
g
L
2Eo
=
Δ
The steady stream function for the internal flow field for a spherical droplet at low Reynolds numbers is given using dimensionless coordinates by:
(
)
θ μ μ θ ψ 2 2 4 sin2 1 4 ) , (r Ua r r drop ext − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = . (1-6)where U and a are the droplet velocity and radius, and μext and μdrop are the dynamic
viscosities of the continuous and droplet phases. The radius, r, is scaled by the droplet
radius, a. See Figure (1-4) for an illustration of the stream line ψ
Figure 1-4: Droplet Stream Lines
1.4 Problem Formulation
Kronig and Brink assumed that as the Peclet number became larger, the solute concentration contours (or for heat transfer – the temperature contours) became parallel with the stream function contours. See Equation (1-6)
g
Θ
As stated, the temperature contours are assumed to be parallel with the stream function contours: ) , ( ) , , , ( limT r t T t Pe→∞ θ φ = ψ (1-7)
For the interior problem, the bulk of the resistance to heat transfer is assumed
to be in the droplet phase. As such, the temperature at the surface, Ts, is assumed
to be constant.
Following the example of Kronig and Brink, Eq. (1-7) may be restated in terms of the dimensionless spatial variable, ξ, and the dimensionless time,τ:
) , ( ) , , , ( limΘ θ φ =Θ ξ τ ∞ → r t Pe Section 3.1 (1-8a) ) , ( ) , , , ( lim g θ φ g ξ τ Pe→∞Θ r t =Θ Section 3.2 (1-8b) where: 2 a t α τ =
and ξ =4r2
(
1−r2)
sin2θ where 0≤ξ ≤1. (1-9)the radius r is scaled by the droplet radius, a. See Figure (1-5) for an illustration of the position of the function ξ
Θ and are the scaled temperatures in the droplet such that:
s T init T s T T − − =
Θ (without heat generation) Section 3.1 (1-10)
2 ) ( a q k s T T g & − =
Θ (with heat generation) Section 3.2 (1-11)
ξ =00 ξ =0.0067 ξ =0.5 ξ =0.75 ξ=0.93 ξ =1
Figure 1-5: Stream lines and their orthogonal trajectories in a vertical plane through
the axis of a falling droplet
This will involve the solution procedure and results for two different forms of energy equations: Section 3.1 will deal with Eq. (1-5) in the absence of heat generation. Section 3.2 will deal with Eq. (1-3) in the presence of heat generation.
Section 3.1 is a revision of the classic work of Kronig and Brink [66]. In that work, only two modes of eigen-functions were used. In addition, the eigen-functions were approximated with quadratic polynomials. In the present work, up to 30 modes
were used with finite-difference approximations for the eigen-functions. Section 3.2 is a new study in the light that no other known publication has investigated
heat transfer with heat generation inside droplets (interior problem) at large Peclet numbers. ξ =0 ξ =0.005 ξ =0.5 ξ =0-75 ξ =0.9 ξ =1
CHAPTER TWO
LITERATURE REVIEW
2.1 Heat and Mass Transfer from Droplets
The problem of heat transfer from a sphere has been the subject of several investigations, starting with the original work of Fourier [44] in 1822. Fourier’s primary interest was the cooling effect of the planets; his work was applied to rigid spheres.
The subject of heat and mass transfer from droplets has been investigated in many papers. There are two excellent monographs covering the subject:-
¾ Bubbles, Drops, and Particle published in 1978 [12], and
¾ Transport Phenomena with Drops and Bubbles published in 1997 [73]. These monographs present the theory of heat and mass transfer near droplets. Brauer [36] investigated the influence of the distribution coefficient on the physical mass transfer. Two spherical models were considered; a rigid sphere and a fluid sphere with viscosity ratio zero (e.g. a gas bubble).
Heat transfer problems from droplets are often classified as one of the following three cases: external, internal and conjugate problems. These problems may be further categorized by Reynolds number, Peclet number and the rate of heat generation.
2.2 Variation of Reynolds numbers
2.2.1 Low Reynolds number
Carslaw and. Jaeger [42] showed a solution of the problem of transient heat transfer from a sphere at creeping flow (Re=0). An analytical solution of the unsteady heat transfer from a sphere at low Reynolds number under steady velocity conditions was developed by Choudhoury and Drake [62]. Feng and Michaelides [88] have analytically derived an expression for the heat transfer from a small sphere at low Peclet numbers assuming a Stokesian velocity distribution around the sphere. Acrivos and Taylor [1] used an asymptotic method and applied their study to a Stokesian velocity field which implies very small Reynolds numbers; they discussed the effect of small but finite Reynolds number of the rigid sphere when the Peclet Number is large. They also derived a solution for the steady-state heat transfer from a sphere at small but finite Peclet numbers. Abramzon and Elata [4] computed the transient heat transfer coefficient for rigid spheres assuming a Stokesian velocity field implying low Reynolds number outside the sphere. Haywood et al. [67] computed the transport parameters of evaporating droplets.
2.2.2 Moderate Reynolds number
Chiang et al. [11] computed the transient heat transfer from evaporating droplets at various initial temperatures for moderate Reynolds numbers. Abramzon & Elata [4] analyzed the physical heat transfer from a sphere in Stokes flow at different Reynolds numbers.
2.2.3 High Reynolds number
A review on numerical study on the transient heat transfer from a sphere at high Reynolds numbers was made by Feng and al. [26]. Friedlander [69] was the first to determine asymptotically the Sherwood number for a small rigid sphere and for large Peclet numbers. The Sherwood number is roughly equivalent to the Nusselt number, but for mass transfer. Friedlander applied the boundary layer theory assuming a concentration profile with coefficients derived from the boundary conditions.
2.3 Variation of relative resistance
2.3.1 External Problem
Most of the theoretical work is suitable to steady-state solution for the heat
transfer from an isothermal sphere.
The so-called external problem is when the transfer
resistance is assumed negligible inside the sphere as compared to that of continuous phase (ambient). An example of external problem where the resistance is primary in the continuous phase can be found in a rain droplet descending in the atmosphere where the thermal conductivity k= 20/1.
Several authors investigated the external problems where the volumetric heat capacity ratio was infinite. With this assumption the sphere remains at its initial temperature. At Low Peclet numbers it was shown that the Nusselt number approaches 2.00. Brunn [63] analytically developed equations for the steady-state Nusselt numbers at low Reynolds numbers for both fluid and solid spheres.
Abramzon and G.A. Fishbein [8] numerically solved the external problem for the steady-state energy problem of with moderate Peclet numbers up to Pe = 1000. Their results suggest that the boundary layers assumptions are not accurate for Pe < 1000.
2.3.2 Internal problem
When the transfer resistance is assumed negligible in the continuous phase
compared to that inside the sphere, the situation is then called the interior problem.
According to Oliver and Chung, [17] there is no equivalent steady-state situation corresponding to a steady-state solution for a constant temperature sphere.
At low Peclet numbers Newman, [2] presented a solution for diffusion of mass into a sphere. From his work it shows that the Nusselt number approaches 6.58. Johns and Beckmann [51] numerically integrated the energy equation for the droplet region for moderate Peclet numbers. They reported oscillations in the Nusselt numbers that were due to the recirculation of the fluid inside the droplet. Dwyer, Kee and Sanders [38] obtained similar results using a promising adaptive grid scheme. For high Peclet numbers, a conventional boundary layer is appropriate only at very small times. Many investigations of the internal problem for high Peclet number included reference to the classic work of Kronig and Brink [66]. This classic work investigated mass transfer of a solute in a droplet. Their work obtained a semi-analytic solution for the limiting case of the interior problem where the Peclet number is very large and the Reynolds number is very low. Kronig and Brink assumed that as the Peclet number becomes large, the solute concentration contours (or for heat transfer – the temperature contours) became parallel with the stream function contours. The mathematical work of Kronig and Brink was
directed to mass transfer. This is equally applicable to heat transfer with the appropriate dimensional conversion.
Due to the recirculation of the fluid inside the droplet, Kronig and Brink [66] solved the energy equation analytically with the assumption that the isotherms are parallel to the stream lines. Their results showed that as time increases the Nusselt numbers approaches 17.9.
The work of Kronig and Brink has been cited in many subsequent works as a limiting bound for heat and mass transfer from droplets at low Reynolds numbers and high Peclet numbers. However, their classic work is a semi-analytic approximation. The trial functions used were only quadratic functions. The Kronig and Brink solution has been shown to be an accurate predictor of heat and mass transfer rates at large times. However, it is not clear if their solution was accurate at small times. Since this classic analysis of Kronig and Brink has been frequently cited, it seems reasonable to revisit their work using more accurate numerical procedures.
2.3.3 Conjugate Problem
When the transfer resistances in both phases are comparable, the conjugate
problem involves calculations of the temperature field in both the continuous and the dispersed phases.
At low Peclet number Cooper [22] developed an analytical solution for the temperature field for various combinations of thermal properties. His work showed that for all conjugate problems, the Nusselt number vanishes for large time intervals with Pe = 0.
Kleinman and Reed [54] analyzed the conjugate transfer from a fluid in the presence of chemical reaction. Latter, Juncu [32] analyzed the conjugate heat and mass transfer between a rigid sphere and an infinite medium in the presence of a chemical reaction, where the chemical reaction takes place inside the particle. Juncu’s work was focused on thermal conductivity, heat capacity, as well as diffusivity.
Analytical results for the conjugate mass transfer problems at high Peclet numbers are available by Levich et al. [78]. The first numerical results for conjugate heat transfer problems were published by Abramzon and Borde [5]. Unsteady conjugate heat transfer from a spherical droplet at low Reynolds numbers was investigated by Oliver and Chung [17]. They solved the energy equation using an implicit finite difference method of
Alternating Directions Implicit (ADI), with a range of 50 ≤ Pe < 1000 for Peclet
numbers. They found that the dimensionless temperature profile asymptotically approaches a steady-state value that is independent of the initial profile in the droplet. For moderate Reynolds numbers Oliver and Chung [18] solved the energy equation using (ADI) finite difference method with fluid motions inside and outside the droplet.
Chao [10] used boundary layer assumptions at high Peclet numbers to estimate the heat transfer rates from spheres. Due to the elliptic nature of the interior region, such boundary layer solution will be accurate only at small times.
For moderate Peclet numbers, Abramzon and Borde [5] used a finite difference
method (ADI) to integrate the energy equation 0 ≤ Pe < 1000. Their work which was
carried at low Reynolds numbers provides a good literature review into the transport process in droplets and solid spheres.
Recently, several investigations have been made to characterize mass transfer near a bubble or droplet with chemical reactions. For example, Klienman and Reed [54] investigated conjugate mass transfer between a droplet and an ambient fluid with a first-order chemical reaction in the ambient flow. Similarly, Juncu [29] investigated conjugate mass transfer to a droplet with a second-order chemical reaction in the droplet.
According to Oliver and Chung [17], for a fluid sphere at moderate Peclet numbers, the Nusselt numbers oscillates with decaying amplitude. The oscillations are due to the circulation of the fluid alternately supplying hot and cold fluid to the fore region of the droplet where most of the heat transfer takes place. As time increases the Nusselt number approaches a steady value. Moderate Reynolds numbers were used in the range of 0 to 50. It was found that by increasing Reynolds number, the predicted rate of heat transfer is significantly increased for fluid spheres as a result of increased fluid motions inside and outside the droplet. It was also found that the increased velocities near the interfacial surface of a drop are a result of an increase in the Reynolds number.
2.4 Heat Generation
Fewer researchers have investigated the effects of distributed heat (or mass)
sources or sinks related to heat and mass transfer near droplet. One aim of the present work is to investigate the heat or mass transfer from a droplet with a uniform heat
source, q&. A heat source may be created by an exothermic chemical or nuclear reaction,
or by an electro-magnetic field. A heat sink may be created by an endothermic reaction. Recently, several investigations have been made to characterize mass transfer near a bubble or droplet with chemical reactions; where there is a chemical reaction a new
species is being created, this will be analogous to heat generation for a mass transfer problem.
The mass and/or heat transfer from a sphere with uniform concentration and/or temperature has been analyzed by Ruckenstein et al. [19]. Ruckenstein analyzed the mass transfer accompanied by a first order irreversible chemical reaction from a single component using Duhamel’s theorem. He derived analytical expressions for the Sherwood number and sphere average concentration. Soung and Sears [83] numerically solved the same problem assuming negligible diffusion in tangential direction and different orders of chemical reactions.
Kleinman and Reed [54] were the first to consider the chemical reaction related to the conjugate problem. They studied conjugate mass transfer between a single droplet (solid, liquid, or gas) and a surrounding ambient flow (liquid or gas) with external chemical reaction. Their work analyzed the surrounding fluid flow with chemical reaction in the continuous phase. The asymptotic regime of mass transfer was investigated for long times. Their investigation was internal for mass transfer inside the particle with Peclet number equals zero and external for mass transfer in the continuous
phase for Pe→∞.
Juncu [26] studied the conjugate heat and mass transfer between a rigid particle and an infinite convective continuous phase in the presence of a non-isothermal chemical reaction. The exothermic and endothermic chemical reactions were analyzed in two hydrodynamic regimes: creeping flow (Re = 0) and moderate Reynolds number. This work assumed a Peclet number of 100 for the continuous phase.
Juncu [29] investigated conjugate mass transfer between a drop and a surrounding fluid flow with second order irreversible chemical reaction inside the drop. The dispersed phase (the drop) reactant was insoluble in the continuous phase. Two sphere models were considered; a rigid sphere and a fluid sphere with internal circulation. For each spherical model two hydrodynamic regimes were used, creeping flow, and moderate Reynolds
numbers. Slow and fast chemical reactions varying from 10-4 to 102 were analyzed at a
moderate value of Pe =100.
One of the most recent papers is the work of Juncu [28]. Juncu investigated conjugate heat/mass transfer from a circular cylinder with an internal heat/mass source in laminar cross flow at low Reynolds numbers. The heat/mass source consisted of a constant temperature/concentration wire imbedded in the cylinder center. Numerical investigations were carried out for such a cylinder with Reynolds numbers of 2 and 20 and a Peclet number of 100.
No work was found that investigated heat or mass transfer from a droplet at high Peclet numbers with heat or mass generation in the droplet.
CHAPTER THREE
MATHEMATICAL ANALYSIS
Introduction
This chapter involves the mathematical analysis of the transport equations given by Eqs. (1-3) and (1-5). Section 3.1 deals with transport without heat generation – Eq. (1-5). Section 3.2 concerns transport with heat generation – Eq. (1-3).
Using the separation of variables method, a revision of the classic solution of Kronig and Brink is prepared and then the results are examined and compared with the classical solution. Heat generation is then added to the droplet at high Peclet numbers.
This chapter is divided into two main sections:-
Section 3.1: Transfer From a Droplet at High Peclet Numbers without heat generation.
Section 3.2: Transfer From a Droplet at High Peclet Numbers with heat generation.
which is further divided into two subsections :-
i. Pure conduction Pe = 0
Section 3.1
Analysis of Heat Transfer without Heat Generation
3.1.1 Introduction
The Kronig and Brink analysis assumed that the bulk of the resistance to transfer was in the droplet phase. It considered the limiting solution as the Peclet number became very large. Their work has been cited in many subsequent studies of droplet transfer. This section revisits their solution using numerical techniques that were not then available. The work of Kronig and Brink has been cited in many subsequent works as a limiting bound for heat and mass transfer from droplets at low Reynolds numbers and high Peclet numbers. However, this classic work was a semi-analytic approximation
using the Ritz method. The trial functions used were only quadratic functions. In addition, solutions for only two modes were sought.
The Kronig and Brink solution was shown to be an accurate predictor of heat and mass transfer rates at large times. Evidently, it is not clear if their solution is accurate at small times. Since their classic work has often been cited, it seems reasonable to revisit their work using more accurate numerical procedures that were not available fifty years ago. The primary purpose is to investigate the minimum time at which the solution of Kronig and Brink is accurate. In this section the same assumptions are made regarding the physical parameters (as applied to heat transfer) and governing equations as were made by Kronig and Brink. However, the solution procedure has been modified to obtain a more accurate solution at small times.
3.1.2 Problem Formulation
The steady stream function for the internal flow field for a spherical droplet at low Reynolds number is given using dimensionless coordinates by:
(
)
θ μ μ θ ψ 2 2 4 sin2 1 4 ) , (r Ua r r drop ext − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = . (3.1-1)where U and a are the droplet velocity and radius, and μext and μdrop are the dynamic
viscosities of the continuous and droplet phases. The radius r is scaled by the droplet
radius, a.
As stated above, the temperature contours are assumed to be parallel with the stream function contours at all times
) , ( ) , , , ( limT r t T t Pe→∞ θ φ = ψ (3.1-2)
Θ is the scaled temperature in the droplet (see Eq.(1-10))
s init s T T T T − − = Θ
Where Tinit is the initial droplet temperature and Ts is the surface temperature. For the
interior problem, the bulk of the resistance to heat transfer is assumed to be in the droplet phase. As such, the temperature at the surface, Ts, is assumed to be constant.
Following the example of Kronig and Brink, Eq. (3.1-2) may be restated in terms of the dimensionless spatial variable, ξ, and the dimensionless time,τ:
) , ( ) , , , ( limΘ θ φ =Θ ξ τ ∞ → r t Pe (3.1-3) where: 2 a t α τ = (3.1-4)
α
is the thermal diffusivityand ξ =4r2
(
1−r2)
sin2θ where 0≤ξ ≤1.3.1.3 Solution Procedure
According to Eq. (8) of Kronig and Brink’s paper (applied to heat transfer), the differential equation for heat transport may be stated as:
τ ∂ ∂ ξ β ξ ∂ ∂ ξ ∂ ∂
γ
ξ
Θ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Θ=
16 ) ()
(
. (3.1-5)The initial and boundary conditions imposed on Eq. (3.1-5) are: 1 ) 0 , ( = = Θ
ξ
τ
( initial condition) (3.1-6) 0 ) , 0 ( = =Θ
ξ
τ
(along the outer stream line), and (3.1-7)) , 1 (
ξ
=τ
Θ is finite at the vortex center. (3.1-8)
The functions γ(ξ) and β(ξ) are given by Eqs. (14) and (15) of Kronig and Brink’s
paper as:
(
)
(
)
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − + = ξ ξ ξ ξ ξ ξ ξ ξ ξ γ 1 1 3 4 1 1 3 4 3 1 2 ) ( E K (3.1-9) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = ξ ξ ξ ξ β 1 1 1 2 ) ( K (3.1-10)where E(x) and K(x) are the complete elliptic integrals.
Equation (3.1-5) is solved using separation-of-variables techniques where
Θ
(
ξ
,
τ
)
isn
λ
where )Ξn(ξ is the eigen-function
The corresponding ordinary differential equation for
n
Ξ
is then:0
)
(
)
(
)
(
⎥+
2Ξ
=
⎦ ⎤ ⎢ ⎣ ⎡γ
ξ
Ξλ
β
ξ
ξ
ξ ξ d n n d d d n . (3.1-12)The boundary conditions imposed on (
ξ
)n Ξ are: 0 ) 0 ( = = Ξ
ξ
n (at the outer stream function of the droplet), and
)
1
(
=
Ξ
ξ
n is finite (at the droplet vortex center).
Both of the functions γ(ξ) and β(ξ) are positive on the interval (0,1). In
addition, lim ( )=0
∞
→
γ
ξ
Pe . Under these circumstances, Eq. (3.1-12) is a Proper
Sturm-Liouville like problem where the corresponding eigen-functions,
n
Ξ , are orthogonal on
the interval (0,1) with respect to the weighting function β(ξ). That is:
[
]
β
ξ
ξ
δ
ξ
ξ
β
ξ ξ ξ n d mn n d m ( ) ( ) 1 0 2 1 0 ) ( ) ( ) (∫
∫
Ξ Ξ = Ξ (3.1-13)where
δ
mn is the Kronecker delta function.The eigen-values, λn, may be evaluated using an energy balance analysis (see
Appendix 3) which requires that:
0 3 8 ) ( ( ) 1 0 2 = Ξ Ξ =
∫
ξ ξ ξβ
ξ
ξ
λ
d d n n n d (3.1-14)Once the eigen-values and corresponding eigen-functions (
ξ
)n
Ξ are ascertained, the
eigen-functions are normalized so that
[
]
∫ Ξ =1 0 2 ) ( ) ( 1 n ξ β ξ dξ (3.1-15)A finite-difference scheme using Matlab has been developed to find the eigen-values and to numerically integrate Eqs. (3.1-13), (3.1-14), etc. See Appendix 2.
The coefficients, An may be obtained using the orthogonality of the
eigen-functions,Ξn, coupled with the initial condition, Eq. (3.1-6):
)
(
1 1ξ
n n A nΞ
∑
∞ = = (3.1-16)The inner products of both sides of Eq. (3.1-16) are taken with (
ξ
)m
Ξ . Since the eigen-
functions, Ξn , are orthonormal with respect to the weighting function β(ξ), then ∫ Ξ
=
1 0 ) ( ) (ξ
β
ξ
dξ
n nA
. (3.1-17)The bulk or average temperature
Θ
(
τ
)
may be calculated by taking the weightedaverage of the temperature, with:
τ λ ξ ξ β ξ
τ
1 2 0 16 1 ) ( ) ()
(
n n ne
A
A n n d − ∞ = ⎥⎦ ⎤ ⎢⎣ ⎡ ∫ Ξ∑
=
Θ
4
4 8
4
4 7
6
or τ λτ
2 16 2 1 8 3 ) ( n ne A n − ∞ = ∑ = Θ (3.1-18)Solving Eq. (3.1-12) involves finding the eigen-functions,Ξn(ξ), the associated
eigen-values, λn, and the corresponding coefficients, An. Second-order finite difference
techniques were used to solve Eq. (3.1-12). The values of λn were iteratively adjusted
Ξn(ξ), and associated eigen-values, λn, were obtained, the eigen-functions were then
normalized.
As a practical matter, Eq. (3.1-11) must be truncated with:
τ λ ξ τ ξ 2 max 16 ) ( 1 ) , (
e
n n n n A n − Ξ∑
= = Θ (3.1-19)where nmax is the truncation limit, for small times nmax has to be very large for accuracy
we used 30 modes which was
Finally, the coefficients An were obtained using Eq. (3.1-17). All integrations
were performed using the trapezoidal rule. The grid for integration was equally spaced in
ξ and was identical with the finite-difference grid spacing.
The bulk or average temperature of the droplet may be determined using a truncated form of Eq. (3.1-18): τ λ τ 2 16 2 1 max 8 3 ) ( n ne A n n − =
∑
= Θ (3.1-20)Another measure of the mass transfer is the Nusselt number (or Sherwood number for mass transfer). Nusselt number may shown as:
Θ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Θ
−
=
3 2 τ ∂ ∂Nu
. (3.1-21)Substitution of Eq. (3.1-20) into Eq. (3.1-21) yields:
τ λ τ λ
λ
2 2 16 2 1 16 2 2 3 max 132
n n e A e A n n n n n nNu
− ∞ = − ∑∑
=
= (3.1-22)Section 3.2
Analysis of Heat Transfer from a Droplet at High
Peclet Numbers with Heat Generation
3.2.1 Introduction
This section investigates the heat or mass transfer from a droplet with a uniform
heat source,q&, Eq. (1-3). A heat source may be created by an exothermic chemical,
nuclear reaction, or by an electro-magnetic field.
3.2.2 Problem Formulation
For simplicity, further discussion of this problem will emphasize the heat transfer problem. Consider a droplet in an ambient fluid. The droplet is experiencing a uniformly
distributed rate of heat generation;q&. the ambient fluid has no corresponding heat
generation. This investigation is limited to the so-called interior problem. For interior problems, all resistance to heat transfer is assumed to reside in the droplet. As such, the temperature at the surface of the droplet, Ts, is assumed to be the free-stream temperature
ext s T
T = . (3.2-1)
With this assumption, only the heat transfer in the droplet phase needs to be considered. Assuming symmetry about the azimuth, the temperature profile in the droplet will be a function of two spatial coordinates, r and θ, as well as time, t, or T(r,θ, t).
τ ∂ ∂ θ ∂ ∂ ∂ ∂ θ ∂ ∂ θ θ ∂ ∂ θ ∂ ∂ ∂ ∂ T T r v r T u Pe k a q T r r T r r r ⎟⎟⎠+ ⎞ ⎜⎜ ⎝ ⎛ + = + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 sin sin 1 1 2 2 2 2 & (3.2-2)
where r is the radial coordinate scaled by the droplet radius a, u and v are the radial and
tangential velocities, scaled by the droplet velocity U. Pe is the Peclet number and τ is
the dimensionless time with
α Ua
Pe= 2 and 2
a tα
τ = , where α is the thermal diffusivity. Finally, the temperature may be made dimensionless with:
[
]
2 ) , , ( ) , , ( a q T r T k r s g & − = Θ θ τ θ τ . (3.2-3)With this substitution, Eq. (3.2-2) becomes:
τ ∂ ∂ θ ∂ ∂ ∂ ∂ θ ∂ ∂ θ θ ∂ ∂ θ ∂ ∂ ∂ ∂ g g g g g r v r u Pe r r r r r Θ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Θ + Θ = + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Θ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Θ 2 1 sin sin 1 1 2 2 2 (3.2-4)
The initial and boundary conditions imposed on Eq. (3.2-4) are: 0 ) 0 , , ( = =
Θg r θ τ (Initial condition), (3.2-4a)
0 ) , , 1 ( = =
Θg r θ τ (Outer edge of the droplet), and
0 0 = Θ = Θ = = θ π θ ∂θ ∂ θ ∂ ∂ g g (Azimuthally symmetric).
For a spherical droplet at low Reynolds number and without surface agents, the interior steady stream function is given using dimensionless coordinates by Ref [12] as:
(
)
) 1 ( 4 sin ) , ( ) , ( 2 2 4 2 κ θ θ ψ θ ψ + − = = r r Ua r r . (3.2-5)Where κ is the ratio of dynamic viscosities, μext/μdrop. The scaled velocities are then:
(
)
) 1 ( 2 cos 1 sin 1 ) , ( 2 2 κ θ θ ∂ ψ ∂ θ θ + − = = r r r u , and (3.2-6)(
)
) 1 ( 2 sin 2 1 sin 1 ) , ( 2 κ θ ∂ ψ ∂ θ θ + + − = − = r r r r v (3.2-7)3.2.3 Solution Procedure
The solution to Eq. (3.2-4) has been obtained for two limiting cases: first, an
analytic solution was obtained for pure diffusion with Pe = 0; second, a semi-analytic
solution was obtained for limiting case of very high Peclet numbers.
3.2.3.1 Special Case 1: Pe = 0.
As the Peclet number approaches zero, the problem loses its dependence on the tangential
coordinate. Thus, the dimensionless temperature is a function of r and τ only. In this
case, Eq. (3.2-4) becomes: τ ∂ ∂ ∂ ∂ ∂ ∂ g g r r r r Θ = + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Θ 1 1 2 2 (3.2-8) or 1 2Θ = Θ − ∇ τ d d g g (3.2-9)
The initial and boundary conditions imposed on Eq. (3.2-9) are: 0 ) 0 , ( = = Θg rτ ( Initial condition) 0 ) , 1 ( = = Θg r τ (Boundary conditions) 0 ) , 0 ( = = ⇒ Θ = Θ g g dr d finite r τ
) , ( ) ( ) , (rτ K r M rτ g = + Θ (3.2-10)
Substitute Eq. (3.2-9) into Eq. (3.2-10) we obtain
∇2K =−1 (Steady-state part) (3.2-11) and τ ∂ ∂ = ∇2M M (Transient part) (3.2-12)
3.2.3.1a The steady-state part
1
2 =−
∇ K Boundary conditions: K(0)= finite
K(1)=0 1 1 2 2 ⎟⎠=− ⎞ ⎜ ⎝ ⎛ dr dK r dr d r 1 3 2 3 C r dr dK r = − +
∫
∫
= − + dr r C r dK 21 3[ ]
1 0 6 ) ( 1 2 1 2 = ⇒ + − − = r C BC from C r C r r K 6 1 ] 2 [ 6 ) (r =C2 − r2 ⇒ from BC C2 = K 6 1 ) ( 2 r r K = − (3.2-13)3.2.3.1b The transient part τ ∂ ∂ = ∇2M M , Initial condition: M(r,τ =0)=−K, Boundary conditions: M(r =1,τ)=0, and
M(r=0,τ) is finite.
Applying the one-dimensional laplacian in spherical coordinates results in:
τ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ M dr dM r dr d r 2 2 1 , or τ ∂ ∂ = + ∂ ∂ M dr dM r r M 2 2 2 . (3.2-14) To solve for M(r,τ), let
) , ( ) , (r τ r M r τ U = (3.2-15)
The initial condition and boundary conditions for U are:
Initial conditions: 6 ) 1 ( ) 0 , ( 2 − = = r r r U τ
Boundary conditions: U(r =1,τ)=0and 0 ) , 0 (r = τ = U .
The following equations convert Eq. (3.2-14) into an equivalent equation for U(r,τ):
M r M r r U + ∂ ∂ = ∂ ∂ (3.2-16) r U r M r r U + ∂ ∂ = ∂ ∂ then r r U r U r M 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∂ ∂ = ∂ ∂ (3.2-17) From Eq. (3.2-16) r M r M r M r r U ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2
r M r M r r U ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2 2 (3.2-18) From Eq. (3.2-18) r r M r U r M 1 2 2 2 2 2 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ − ∂ ∂ = ∂ ∂ (3.2-19) Substitution of Eq. (3.2-17) into Eq. (3.2-19) results in:
r r r U r U r U r M 1 1 2 2 2 2 2 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∂ ∂ − ∂ ∂ = ∂ ∂ (3.2-20) τ τ ∂ ∂ = ∂ ∂ M r U r U M 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ∂ ∂ τ τ (3.2-21)
Finally, with substitution of Eqs. (3.2-17), (3.2-20) and (3.2-21) into Eq. (3.2-14) we obtain: τ ∂ ∂ = ∂ ∂ U r U 2 2 (3.2-22)
Equation (3.2-22) may be solved using separation of variables with:
) ( ) ( ) , (r τ R r V τ U = (3.2-23)
The boundary conditions on R(r) are: 0 ) 1 (r = = R , and 0 ) 0 (r = = R .
Substitution of Eq. (3.2-23) into the components of Eq. (3.2-22) results in the following: 2 2 2 2 dr R d V r U = ∂ ∂ , and (3.2-24)
τ τ d dV R U = ∂ ∂ (3.2-25)
Substitute Eqs. (3.2-24) and (3.2-25) into Eq. (3.2-22)
τ d dV R dr R d V 22 = ' " RV VR = 2 " λ − = = V V R R &
There exists an infinite number of solutionsλ2 to the above differential equation.
Hence, following standard separation-of-variables techniques, let: ) sin( ) ( cos ) (r A r B r Rn = n λn + n λn (3.2-26)
Using the boundary condition for R at r = 0 and r = 1 with Eq. (3.2-26) yields: ) ( sin ) (r B r Rn = n λn .
The corresponding equation for Vn(τ) is then
2 n n n V V λ − = & τ λ 2 n
e
V
=
− , where λn =nπ . (3.2-27)Equations (3.2-26) and (3.2-27) may be substituted into Eq. (3.2-23):
∑
∞ = − = 1 2 ) ( sin ) , ( n n n r e n B r U τ λ λ τ (3.2-28)Using the initial condition:
( , 0) ( 1) 2 − = = r r r U τ
∑
∞ = − − = 1 3 2 ) ( sin ) 1 ( 2 ) , ( n n n n n e r r U λ λ τ λ τ .Hence, the value for M(r,τ) is:
∑
∞ = − − = = 1 3 2 ) ( sin ) 1 ( 2 ) , ( n n n n e r r r U r M λ λτ λ τ (3.2-29)Since
Θ
g(
r
,
τ
)
=
K
(
r
)
+
M
(
r
,
τ
)
, then the temperature for the special case 1when Pe = 0, can be stated as:
∑
∞ = − − + − = Θ 1 3 2 2 ) ( sin ) 1 ( 2 6 1)
,
(
n n n n g r e r r λ λτ λτ
ξ
(3.2-30)A parameter of interest is the bulk or average temperature of the droplet, Θg,
∫
∫
Θ = Θ 1 0 1 0)
,
(
vol d vol d g gτ
ξ
(3.2-31) 3 3 4 r vol= π (3.2-32) dvol =4πr2dr (3.2-33) π π π 3 4 3 4 4 1 0 3 2 1 0 1 0 = = =∫
∫
dvol r dr r (3.2-34)Thus, the bulk temperature is:
π π π λ λ τ λ
τ
3 4 ) 4 ( 6 1 ) 4 ( ) ( sin ) 1 ( 2 2 2 1 0 1 2 3 1 0 2)
(
dr r r dr r e r r n n n n g − + − = Θ∫
∑
∫
∞ = −Integration yields the following: τ λ λ 2 1 4 6 15 1 n e n n g − ∞ =
∑
− = Θ whereλ
n=
n
π
(3.2-35)Another parameter of interest is the maximum temperature in the droplet,
max
g
Θ . For Pe = 0, the maximum temperature will occur at the center of the droplet.
To find this value, the value of Eq. (3.2-30) must be evaluated at r = 0.
n n n n r
r
r
r
r
r
r
λ
λ
λ
λ
=
−
+
=
→3
!
...
)
(
sin
lim
3 0 Hence, λ τ πτ
ξ
2 1 2 2 ) 1 ( 2 6 1 ) , ( max n n e n n g − ∞ =∑
− + = Θ (Pe = 0) (3.2-36) 3.2.3.2Special Case 2: Pe →∞As the Peclet number becomes very large, the temperature contours are assumed to be parallel to the stream function contours [66]. With this assumption, Eq. (3.2-4) may be solved using the procedure described in Chapter 3 Section 3.1. Specifically, the
dimensionless temperature is assumed to be a function of the stream function, ψ, and τ :
) , ( ) , , ( lim g θ τ g ψ τ pe→∞Θ r =Θ (3.2-37)
Following the method proposed by Kronig and Brink, [66], Eq. (3.2-37) may be restated in terms of the dimensionless spatial variable, ξ, and the dimensionless time,τ:
) , ( ) , , ( lim g θ τ g ξ τ pe→∞Θ r =Θ (3.2-38)
Applying the First law of thermodynamics to a controlled volume:
st gen out
in E E E
E& − & + & = & . (3.2-40) where E&inis the rate of energy flow into the controlled volume, E&out is the rate of energy flow out of the controlled volume, E&gen is the rate of heat generation in the controlled volume, and E&st is the time-rate-of-change of the energy stored in the controlled volume.
The controlled volume is assumed to be a differential volume between two streamline. Since the temperature profile is assumed to be parallel to the stream lines, the heat flux will be due to conduction only.
With this assumption, the energy entering the controlled volume may be found using Fourier’s law of heat conduction:
ξ ξ ε ξ θ π ξ ξ dS a r dS d d dT k
E
&
in =− 2 sin (3.2-41)Similarly, the energy exit the controlled volume is given as:
ξ ξ ξ ξ ξ ξ θ π ξ ξ dS a r dS d d dT k
E
out 2 +Δ sin Δ + − =&
(3.2-42)Combining these yields:
ξ ξ ξ ξ ξ ξ ξ ξ π θ ξ ξ ξ dS a dS d d dT r d dT r k
E
E
in out ( ) ( ) ⎥⎥ 2 sin ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = − Δ + Δ +&
&
(3.2-43)Δ
=
θ
ξ
ξsin
8
r
a
d
dS
(3.2-44) Δ − =θ
ζ
ζ 3 3 2 2 cos 4 ) 1 2 ( r d r a dS (3.2-45)ϕ
θ
ϕar
d
dS
=
sin
(3.2-46)Where Δ=(1−r2)2cos2θ+(2r2 −1)2sin2θ (3.2-47)
Thus, the net heat transfer due to conduction into the controlled volume is:
ξ ξ ξ ξ ξ ξ ξ ξ ξ θ πa θdS a r d dT r d dT r k
E
E
in out 2 sin sin 8 ) ( ) ( Δ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = − Δ + Δ +&
&
(3.2-48) θ ζ θ π ξ ε ξ ξ ξ ξ ξ ξ 3 3 2 2 2 cos 4 ) 1 2 ( sin 16 ) ( ) ( r d r a r d dT r d dT r kE
E
in out − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = − Δ + Δ +&
&
(3.2-49)Again, following ref. [66] this may be written as:
k a T
E
E
in out π ξ ξ γ ξ ( ) ⎟⎟⎠ 4 ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = −&
&
(3.2-50)Where from Eq. (3-9)
(
)
(
)
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − + = ξ ξ ξ ξ ξ ξ ξ ξ ξ γ 1 1 3 4 1 1 3 4 3 1 2 ) ( E KThe term on the right hand side of Eq. (3.2-40) may be evaluated as:
dV t T c dt dE p st ∂ ∂ = ρ (3.2-51)
) ( 4 3 ξ β π a q
E
&
q = & ϕ ζ ξρ
S
S
S
t
T
c
dt
dE
p st∂
∂
∂
∂
∂
=
(3.2-52)Substitution of Eqs. (3.2-44),(3.2-45)and(3.2-46) into Eq. (3.2-52) results in:
∫
∂ ⋅ ∂ ∂ ⋅ ⋅ =ρ β ε ϕ 8 ) ( 3 a t T c dt dE p st (3.2-53) Where,∫
∂ϕ=2π. From Eq. (3-10): ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = ξ ξ ξ ξ β 1 1 1 2 ) ( K . Equation (3.2-40) is then, π ξ β ρ 4 ) ( 3 a t T c dt dE p st ⋅ ∂ ∂ ⋅ ⋅ = (3.2-54)The heat generation term may be calculated as: V
Δ
=q
E
&
q &or, (3.2-55)
Substitute Eqs. (3.2-50),(3.2-54)and (3.2-55) into Eq. (3.2-40) yields:
E
E
E
q st out dt dE in&
&
&
− = − (3.2-56)⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
∂
Θ
∂
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
Θ
∂
∂
∂
1
16
)
(
)
(
τ
ξ
β
ξ
ξ
γ
ξ
g g ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∂ ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ q t T c a k a T p &ρ
π
ξ
β
π
ξ
ξ
γ
ξ
( ) 4 ( ) 4 3Dividing by 4πak yields the partial differential equation:
⎟
⎠
⎞
⎜
⎝
⎛
−
∂
∂
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
q
t
T
c
k
a
T
p&
ρ
ξ
β
ξ
ξ
γ
ξ
(
)
(
)
16
2 (3.2-57)The above equation in dimensionless form is:
(3.2-58)
The functions γ(ξ) and β(ξ) as defined in Section 3.1 as Eqs. (3-9) and (3-10)
(
)
(
)
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − + = ξ ξ ξ ξ ξ ξ ξ ξ ξ γ 1 1 3 4 1 1 3 4 3 1 2 ) ( E K , (3.2-59) And ⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = ξ ξ ξ ξ β 1 1 1 2 ) ( K .Where E(x) and K(x) are the complete elliptic integrals, [66]. The boundary conditions imposed on Eq. (3.2-59) are:
0 ) , 0
( = =
Θg
ξ
τ
(along the exterior of the droplet), and (3.2-60)) , 1 (
ξ
=τ
Θg is finite, (at the vortex center). (3.2-61)
Unlike the previous analysis of Section 3.1, the initial condition imposed on Θg is:
0 ) 0 , ( = = Θg
ξ
τ
. (3.2-62)Equation (3.2-59) is solved using separation-of-variables techniques where F is the
steady-state solution, and G transient solution .
)
,
(
)
(
)
,
(
ξ
τ
=
F
ξ
+
G
ξ
τ
Θ
(3.2-63)The initial and boundary conditions imposed on Eq. (3.2-63) are: 0 ) 0 , ( = = Θ
ξ
τ
,0
)
,
0
(
=
=
Θ
ξ
τ
(along the outer stream line), and)
,
1
(
ξ
=
τ
Θ
is finite at the vortex center.Substitution of Eq. (3.2-63) into (3.2-58) yields the following:
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +
=
+
−
1
)
(
)
(
16 ) ( ) (τ
∂
∂
ξ
γ
β ξ ξ ∂ ∂ ξ ∂ ∂ F GF
G
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +=
(
∂
+
)
−
1
)(
(
16 ) ( )∂τ
∂
τ
∂
ξ
γ
β ξ ξ ∂ ∂ ξ ∂ ∂ ξ ∂ ∂ F GF
G
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛+
−
=
∂τ
∂
ξ
∂
∂
ξ
γ
∂ξ
∂
ξ
γ
β ξ β ξ ∂ξ ∂ ∂ξ ∂F
G
G
16 ) ( 16 ) ()
(
)
(
(3.2-64)The differential equation for F(ξ) is:
16 ) ( ) ( β ξ ∂ξ ∂
ε
∂
∂
ξ
γ
⎟⎟=− ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ F (3.2-65) And for G(ξ,τ): ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =∂τ
∂
ξ
∂
∂
ξ
γ
β ξ ∂ξ ∂ G G 16 ) ( ) ( . (3.2-66)Equation (3.2-65) may be solved using finite-difference methods with the following boundary conditions: