Applied Mathematics and Computation 129 (2002) 295–316
www.elsevier.com/locate/amc
Mathematical model for heat transfer
mechanism for particulate system
A.R. Khan *, A. Elkamel
Department of Chemical Engineering, Faculty of Engineering and Petroleum, Kuwait University,
P.O. Box 5969, 13060 Safat, Kuwait
Abstract
Various theoretical models for fluidized bed to surface heat transfer have been
considered to explain the mechanism of heat transport. The particulate fluidized bed
which is the common case for liquid–solid fluidized bed is much simpler and homogeneous and transport operation can be easily modeled. The heat transfer coefficient increases to a maximum and then steadily decreases as the bed void fraction increases
from that of a packed bed to unity. The void fraction emax at which the maximum value
of heat transfer coefficient occurs is a function of the solid–liquid system properties.
An unsteady state thermal conduction model is suggested to describe the heat
transfer process. The model consists of strings of the particles with entrained liquid
moving parallel to surface, during the time interval heat conduction takes place. These
strings are separated by liquid into which the principal mode of transfer is by convection. The model shows a dependence of heat transfer coefficient on void fraction and on
physical properties, which is consistent with the results of experimental work. Ó 2002
Elsevier Science Inc. All rights reserved.
1. Introduction
The high heat transfer coefficient from a hot surface to a fluidized bed facilitates the addition and removal of heat to and from a process efficiently. The
role of various parameters and the mechanism of heat transfer in this field have
been the subject of extensive investigations during the last three decades. Many
*
Corresponding author. Tel.: +965-481-7662; fax: +965-483-9498.
E-mail address: [email protected] (A.R. Khan).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved.
PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 0 3 9 - X
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Nomenclature
CP0
CS
dP
E
f0
g
G
Gmf
h
hb
hcond
hconv
hP
hr
H
Hmf
kg
k0
L
q
R
Ra
RWS
TB
TE
Tf
T P
Tf
TP
u
u0
VP;Axial
Vr
Vi;j
XL
heat capacity for packet of particles ðJ kg1 K1 Þ
heat capacity for solid particles ðJ kg1 K1 Þ
particle diameter (m)
constant in Eq. (1)
fraction of time for which the surface is covered by bubbles
acceleration due to gravity ðm s2 Þ
mass velocity ðkg m2 s1 Þ
mass velocity corresponding to minimum fluidization velocity
ðkg m2 s1 Þ
heat transfer coefficient ðw m2 K1 Þ
heat transfer coefficient for bubbles ðw m2 K1 Þ
heat transfer coefficient for unsteady state thermal conduction
ðw m2 K1 Þ
convective heat transfer coefficient ðw m2 K1 Þ
heat transfer coefficient for packet of particles ðw m2 K1 Þ
equivalent heat transfer coefficient for radiation ðw m2 K1 Þ
Height of the bed (m)
height of the bed corresponding to the minimum fluidization
conditions (m)
thermal conductivity for gas ðw m1 K1 Þ
thermal conductivity for packet of particles ðw m1 K1 Þ
length of the heating element (m)
heat flux ðw m2 Þ
constant in Eq. (1)
resistance offered by the gas–solid packets ðw1 m2 KÞ
resistance offered by gas entrained by the particles close to the
heating surface ðw1 m2 KÞ
average uniform bed temperature (K)
heat transfer element temperature (K)
fluid temperature (K)
particle temperature (K)
dimensionless temperature as defined in Eq. (4c)
dimensionless temperature as defined in Eq. (4c)
fluid average velocity ðm s1 Þ
velocity for packet of particles ðm s1 Þ
Particles’ axial velocity ðm s1 Þ
mean radial velocity ðm s1 Þ
intermediate values of Ti;j for half time step interval (K)
linear dimension of liquid slab (m)
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
XS
297
linear dimension of solid slab (m)
Greek symbols
a
thermal diffusivity ðm2 s1 Þ
Ds
dimensionless time constant as defined in Eq. (4c)
e
void fraction
khX
dimensionless operator in X direction as defined in Eq. (5d)
khY
dimensionless operator in Y direction as defined in Eq. (5d)
lg
gas viscosity ðN s m2 Þ
q
density of fluid ðkg m3 Þ
qS
density of particles ðkg m3 Þ
empirical correlations relating bed to surface heat transfer coefficients for a
range of operating variables have been proposed. They are of restrictive validity because they cannot make adequate allowance for different geometries of
equipment used and varying degree of accuracy of the experimental techniques
used. Furthermore, it is difficult to extrapolate outside the experimental range
of variables studied. Different models have been proposed to explain the different aspects of this complex problem. There are particularly diverse concepts
suggested by different workers regarding the mechanism of heat transfer between a fluidized bed and a heat transfer surface. In this paper an attempt has
been made to explain the mechanism of heat transfer from bed to surface in
liquid fluidized systems. The model results are compared with experimental
data [1] and computed values of the existing models.
2. Heat transfer models for fluidized bed
Both gas, and to a lesser extent liquid fluidized beds have been employed in
chemical engineering practice particularly where the addition or removal of
heat from the bed is required. In the case of gas fluidized beds the more
important aspects have been collected and presented in detail by Zabrodsky
[2]. Most of the workers have examined a limited range of experimental
variables and presented their results in the form of correlations; the power of
any group in the correlations gave some indication of its importance within
the experimental range investigated. It is clear that the scale of the equipment
in which measurements were made has influenced the results. There is no
sufficiently general theory of heat transfer in fluidized beds, although several
different models have been proposed to explain various aspects of this
problem. A brief description of some of the models is given in the following
section.
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2.1. The limiting laminar layer model
Leva and Grummer [3] noted that the core of the bed was isothermal and
offered negligible thermal resistance while the main resistance limiting the rate
of heat transfer between the bed and the heat source lay in a fluid film near the
hot surface. They suggested that particles acted as turbulence promoters, which
eroded the film reducing its resistive effects. Levenspiel and Walton [4] derived
an expression in terms of modified Nusselt and Reynolds numbers for the effective fluid film thickness assuming that the film breaks whenever a particle
touches the transfer surface. They have to modify the coefficients in the model
to account for their own experimental data. Wen and Leva [5] correlated
the published heat transfer results on the basis of a scouring action model in
which particle movement was assumed to be vertical and parallel to the heat
source.
#0:36
pffiffiffi 0:4 "
CS qS dP1:5 g
GdP E
Nu ¼ cons
:
kg
lg R
ð1Þ
In this correlation the fluidization parameters are defined as follows:
1. E is the fluidization efficiency ðG Gmf Þ=Gmf .
2. R is the expansion ration of the bed H =Hmf .
Richardson and Mitson [6] and Richardson and Smith [7] reported that for
liquid fluidized beds the resistance to heat transport lay near the tube wall
within the laminar sub-layer where the effective thickness is reduced by the
presence of particles for two reasons:
1. The particles cause turbulence in the fluid thereby reducing the thickness of
the laminar boundary layer.
2. The particles themselves transport heat as a result of the radial component
of their rapid oscillating motion.
Wasan and Ahluwalia [8] proposed that heat transfer through a fluid film
was promoted by fluid eddies beyond the film boundary. They assumed that
the solid particles were stationary and equally spaced and that heat was
transferred through the film and then spread by fluid convection into the bulk
of the bed. They compared the experimental results of various workers and
found deviations of up to 44%.
These models basically involve a steady-state concept of the heat transfer
but Wasan and Ahluwala [8] included some dynamics transfer features for
transfer through the fluid into the bed.
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299
2.2. Two resistance film model
Wasmund and Smith [9] suggested a modified laminar layer model, in which
they considered particle convective transfer due to radial motion of particles
into the laminar layer. This mechanism contributed 50–60% of the total heat
transferred and the remainder was from fluid convective transfer. Tripathi et al.
[10] used the series model proposed by Ranz [11] for effective transport
properties in packed beds. They compared results obtained by Wasmund and
Smith [12] using radial velocities of the particles and observed deviation of
20%. Brea and Hamilton [13] and Patel and Simpson [14] used a two resistance film model and emphasized that the fluid eddy convection is the main
contributing factor to the heat transfer. Zahavi [15] measured the effective
diffusivity of the fluidized beds and also developed a semi-empirical correlation,
which represented his results with a maximum deviation of 34%.
2.3. Unsteady state heat transfer
Mickley and Trilling [16] suggested that the heat transfer process in a gasfluidized bed was of an unsteady state nature. Later Mickley and Fairbanks [17]
developed a model of heat transfer on the assumption that at any time there is
unsteady state heat transfer within the fluidized bed close to heat source; this
can be broken down into components due to solid/solid, solid/surface, gas/solid,
and gas/surface transfer. Packets of loosely locked particles which are assumed
to have uniform thermal properties constitute the fluidized bed. The mean heat
transfer coefficient can be calculated for packets of particles moving with a
constant speed upassing rapidly along the length of the heat source, then
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
u0
h ¼ pffiffiffi k 0 q0 CP0 :
L
p
Thermal conductivity k 0 , density q0 and heat capacity CP0 for packets can be
estimated by use of the Schumann and Voss [18] correlation. The assumption
that the thermal properties of the bed are uniform in the neighborhood of the
heat source is unrealistic when the source and the bulk of the bed are at
considerably different temperatures. Mickley and Fairbanks [17] calculated the
residence times of packets from resistance fluctuations recorded for a thin
electrically heated platinum strip. The frequency of packets was of the order of
two per seconds and the residence time of 0.4 s. Henwood [19], Catipovic et al.
[20], Suarez et al. [21] and George and Smalley [22] used a small heat transfer
surface to measure the variations in local heat transfer coefficient within and
adjacent to a rising bubble. They concluded that heat transfer took place
mainly through the fluid to the particle with the maximum rate in the vicinity
of the contact part.
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2.4. Simplified models
The heterogeneity of fluidized beds is an important factor enhancing the
value of the heat transfer coefficient up to 50–100 folds for a gas and 5–8 folds
for liquid fluidized bed. Botterill and Williams [23] have proposed a model for
heat transfer in gas-fluidized systems which are based upon the unsteady state
conduction of heat to spherical particles adjacent to the transfer surface. The
convective transfer through the gas is ignored because the effective diffusivity
of the fluidized bed is much higher than the eddy diffusivity of the gas. The
particle and gas, whose temperatures were initially the same, approached the
surface, the temperature of which remained approximately constant because
of its high heat capacity. The Fourier equations for thermal conduction were
solved by finite difference technique. Apart from the axes of symmetry through
the particles, there were three space limits to the problem.
(A) Close to the heater it was assumed that there was a continuous thin layer of
pure fluid with which a particle was in contact.
(B) The temperature of the other end the particle was set at the sink temperature, taken as the bulk temperature.
(C) Transfer of heat between particles in a direction parallel to the surface
could be neglected because the temperature difference between particles in
adjacent position was very small.
The experimental results for heat transfer coefficients for the shortest residence time of metallic particles were far less than the predicted values. This
discrepancy was accounted for by assuming that a gas film of thickness equals
to about 10% of the particles around the heat source. Botterill [24] tried both
models in which there was triangular and square packing on the particles and a
fluid film between the particles and the heat transfer surface. The thickness
of the film was related to the resistance limiting the heat transfer. Different
workers made observations but no satisfactory conclusion was reached
about the local variation of void fraction in the vicinity of the heat transfer
surface.
Davies [25] considered the unsteady state heat transfer by conduction between the element at one temperature and spherical particles immersed in a
liquid at another temperature. The particles were assumed to enter the thermal
boundary layer with a mean radial velocity vr , approached the surface and left
it again with the same velocity. The Fourier equations were solved by using the
explicit finite difference method and the same boundary condition as reported
by Botterill and Williams [23]. The very low values of particle convective heat
transfer component predicted by the model indicated that only a very small
proportion of the heat transferred from the heated surface to the bed was
carried by fluidized particles. The experimental high value was attributable to
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301
the fact that the effective thickness of the thermal boundary layer had been
substantially reduced; this was mainly due to the following causes:
1. The scouring action of particles.
2. The high interstitial velocity of the liquid.
2.5. Particle replacement model
Gabor [26] has proposed that heat has been absorbed by the particulate bed
based on string of spheres of infinite length normal to heat transfer surface.
Another simplified approach based on series of alternating gas and solid slabs
also provided similar results as the spherical model.
Gelperin and Einstein [27] have developed a more refined model taking into
account other details of the process involved. They considered that heat is
transferred from the heat transfer surface by packets of solid particles by gas
bubbles and by gas passing between the packet and the surface. The total heat
transfer coefficient is expressed as
h ¼ ðhP hconv Þð1 f0 Þ þ hb f0 þ hr ;
ð2Þ
where hP ; hconv ; hb and hr are the heat transfer coefficients corresponding to
packet, convection, bubble and radiation, respectively, and f0 is the fraction of
time for which the surface is covered by bubbles.
They solved the basic equations for their models of bed to surface heat
transfer in terms of two heat resistances: RWS the resistance offered by gas
entrained by the particles close to the transfer surface and Ra the resistance
offered by the gas–solid packets. They have tabulated their final equations
for instantaneous and mean heat transfer coefficient for different boundaries
in their publication [27]. For isothermal conditions of heat source, which
have already been proposed a simplified solution can be used with little
error.
Martin [28] has presented a particle convective energy transfer model for
wall to bed heat transfer from solid surface immersed in gas-fluidized bed. In
his model the following assumptions were applied.
1. The contact time is regarded to be proportional to the time taken to cover
the path with the length of one particle diameter in free flight.
2. The wall to particle heat transfer coefficient is calculated by integrating the
local conduction heat fluxes across the gaseous gap between sphere and plain
surface over the whole projected area of the sphere [29].
3. The average kinetic energy for the random motion of particles comes from a
corresponding potential energy.
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3. Development of mathematical model
For the purpose of establishing a simplified model of a liquid fluidized bed,
the system is assumed to consist of strings of particles with liquid filling the
intervening spaces. It is proposed that unsteady state thermal conduction takes
place into both the liquid and the solid particles in the string as reported by
Gabor [26]. Liquid layers into which the principal mode of heat transfer is
forced convection as shown in Fig. 1 to separate the strings.
The overall heat transfer coefficient for liquid fluidized bed from immersed
surface constitutes solid conductive, liquid conductive and liquid convective
Fig. 1. Mathematical representation of fluidized particle entrained in liquid with initial and
boundary conditions.
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
303
components based on the void fraction determined by bed expansion characteristics and particle axial velocity VP;Axial . The convective component is calculated for liquid moving with interstitial velocity parallel to the heating
surface. The conductive components for solid and liquid are evaluated based
on contact time using unsteady state conduction equations for string of particles with entrained liquid.
3.1. Heat transfer across incompressible boundary layers
The simulated element is considered as a flat plate located on the axis of the
column with the large faces parallel to liquid flow. The liquid with an average
velocity u and uniform temperature TB passes over the hot flat plate at a
constant temperature TE . At high Prandtl numbers the thermal boundary layer
is always confined entirely within the laminar sublayer. This limiting case of
forced convection across a turbulent boundary layer can be solved analytically.
Kestin and Persen [30] based their analysis on the laminar form of the energy
equation and confined their attention to the laminar sublayer only. The other
assumption made is that the velocity varies linearly with distance perpendicular
to the flat plate. The detailed solution of the energy equation for laminar form
assuming linear velocity profile within laminar sublayer is expressed as
y
sW ðxÞ oT y 2 dsW oT
o2 T
¼a 2
l ox
oy
dx oy
ð3aÞ
by substituting
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y 3 ðsW =lÞ3
g ¼ R x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
9a x0 ðsW =lÞ dx
Eq. (3a) can be transformed to ordinary differential equation.
d2 T
2 dT
¼ 0;
þ
g
þ
dg2
3 dg
ð3bÞ
where
T ¼
TE T
¼ 1;
TE TB
for x ¼ 0 and all values of y > 0, T ¼ 1, for y ¼/ and all values of x > 0 and
T ¼ 0, for y ¼ 0 and all values of x > 0.
The solution is given in the form of incomplete c-function as
T ðgÞ ¼
cð1=3; gÞ
:
Cð1=3Þ
The calculated values of the heat transfer coefficient are presented together
with experimental values [1] in Table 2. The experimental data are obtained
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
from immersed electrically wound heating surface in 6-mm glass particle fluidized bed dimethyl phthalate. The high experimental values are due mainly to
the following causes:
1. The plane element was not a true flat plate.
2. The edge effects of the element caused a high value.
3. The temperature of the element might not be uniform.
In the case of a fluidized bed the presence of solid particles decreased the free
area available for flow and caused an increase in the liquid velocity near the
heat source. The contribution of heat transfer due to liquid alone was assessed
on the basis of the interstitial velocity which was the factor determining the
thickness of the thermal boundary layer.
3.2. The contribution of fluidized particles
3.2.1. Particle velocities in fluidized beds
By means of high speed photography several workers have measured the
paths taken by a tracer particle in a transparent fluidized bed. Toomey and
Johnstone [31] obtained particle velocities near the wall of dense phase gasfluidized bed. For the particular case of 0.376-mm diameter glass spheres fluidized in air, they reported particle velocities from 60 to 600 mm/s. Kondukov
et al. [32] used radioactive tracer particles and radiation detectors to measure
the particle trajectories in an air fluidized bed. Their results were similar to
those of Toomey and Johnstone [31] and gave additional information on
particle behavior in the interior of the bed.
Handley [33] fluidized 1.1- and 1.53-mm glass spheres with methyl benzoate
in 31- and 76-mm diameter columns. He reported the radial and axial velocities
of the particles for bed voidage ranging from 0.67 to 0.905. He concluded that
the velocity of the particles was completely random in a uniformly fluidized
bed. A more extensive study of particle velocities in fluidized bed was made by
Carlos [34] and by Latif [35]. Carlos fluidized 9 mm glass beads with dimethyl
phthalate at 30 °C in a 100 mm diameter column. He reported the particle
velocities (radial, angular, axial, horizontal and total) over the voidage range
0.53–0.7. A set of differential equations governing the mixing process was
numerically solved by computer taking into account the effect of radial and
axial dispersion. Latif [35] extended the work of Carlos to 6-mm glass spheres
and developed a simple relationship between particle velocity and axial and
radial positions. The constants of these equations are listed in Table 1. These
equations are used to evaluate particle velocities as a function of bed expansion
characteristics.
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305
Table 1
Coefficients for calculation of axial velocity component [35]
VP;Axial ¼ A r þ B where r is normalized radial coordinate of particle
2
A ¼ a0 þ a1 z þ a2 z2 þ a3 ea4 z
B ¼ b0 þ b1 z þ b2 z2 þ b3 eb4 z
2
e
a0
a1
a2
a3
a4
b0
b1
b2
0.55
0.65
0.75
0.85
0.95
)0.76
)2.74
)1.42
0.55
)4.27
7.25
9.66
4.78
)3.3
)17.98
)8.74
)6.92
)2.84
2.58
22.27
)87.4
)74.22
)128.2
)162.3
)222.0
)20.2
)10.09
)15.11
)20.0
)27.5
0.42
1.17
0.96
)0.18
2.22
)3.08
)1.55
)3.17
3.12
12.04
3.83 31.8
0.38 46.5
1.94 74.56
)2.92 86.7
)15.09 119.7
b3
b4
)21.9
)17.03
)18.34
)22.7
)29.0
3.2.2. Residence time of particle in the vicinity of hot surface
Davies [25] assumed that the high heat transfer coefficient for the fluidized
bed was attributable to effects arising from the radial velocity of the particles as
reported by Figliola and Beasley [36]. His model predicted very small values of
heat transfer coefficient because the thermal boundary layer thickness was
small compared with the diameter of the particle and the residence time in the
thin boundary layer was short.
However from the average calculated values of radial and axial particle
velocity components at the center of the fluidized bed, it was obvious that the
axial component of velocity was dominant. On this basis it appeared reasonable to assume, as a first approximation, that particles and fluid both at
the bulk temperature of the bed approached the heat transfer surface at a
velocity approximately equal to the average axial component of a particle
velocity. The string of particles separated by intervening liquid thus traveled
parallel to the hot surface and unsteady state heat conduction took place
through the solid and liquid in parallel. The residence time for a particle could
be given as
t¼
L
VP;Axial
:
3.2.3. Unsteady state thermal conduction for liquid and solid
In the present model in which it is assumed that heat transfer is because
of unsteady state thermal conduction in both liquid and solid, the following
assumptions are made:
1. The temperatures of the bulk of the fluidized bed ðTB Þ and the temperature
of the element ðTE Þ are uniform.
2. A string of particles with entrained liquid at a uniform temperature equal to
TB arrives quickly in an axial direction at the element.
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
3. The particles and entrained liquid absorb heat by unsteady state conduction
as they travel along the surface. Immediately the particles leave the vicinity
of the element they exchange heat with the surrounding liquid.
The model explains the way in which the fluidized particles contribute towards the transfer of heat between the hot surface and the bed. At any time the
heat contents of both the liquid and the particles may be obtained from a
knowledge of the temperature distribution within the particle and liquid. The
temperature distribution in the solid and liquid in contact with the surface may
be obtained by the heat conduction equation for both the liquid and solid over
the residence time for which they are present at the hot surface. The heat
conduction equation in spherical coordinates necessitates the use of three space
dimensions; this may cause complications in solving the equation with its appropriate boundary conditions.
The situation may be simplified by defining the system in terms of Cartesian
coordinates and assuming that the particles may be replaced by cubes, the
length of the side of each of which is equal to the diameter of the particle. Each
cube moves with one face in contact with the surface and liquid occupies the
intervening spaces. Symmetry is assumed along the plane perpendicular to the
surface. The length of the liquid slug between the particles and the thickness of
the liquid layer separating the strings will be calculated as follows:
In Fig. 1 a cube of dimension ðXS þ XL Þ is considered and the void fraction
in the vicinity of the surface assumed to be the same as in the bulk where XS
and XL are dimensions of particle and liquid slug, respectively. The volume of
the particle is XS3 and of the liquid slug XS2 XL and void fraction is expressed as
e¼
ðXS þ XL Þ3 XS3
ðXS þ XL Þ
On rearranging
e¼1
3
XS
XS þ XL
:
3
or
"
XL ¼ XS
1
1e
#
1=3
1 :
The Fourier equations for unsteady state thermal conduction within the two
homogeneous phases of the system are
For solid phase
o2 TP o2 TP
1 oTP
:
þ 2 ¼
aP ot
ox2
oy
ð4aÞ
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307
For liquid phase
o2 Tf o2 Tf
1 oTf
:
þ 2 ¼
af ot
ox2
oy
ð4bÞ
These equations are put in dimensionless form by defining
TP ¼
TE TP
;
TE TB
and s ¼ t aP ;
Tf ¼
TE Tf
TE TB
s ¼ t af
o2 T P o2 T P oT P
;
þ
¼
ox2
oy 2
os
ð4cÞ
o2 T f o2 T f oT f
:
þ 2 ¼
ox2
oy
os
ð4dÞ
Initial and boundary conditions, at t ¼ 0, the particle and liquid slug both are
divided to give a mesh n n and all points within the particle and liquid slug
are at the bulk temperature
T P ¼ T f ¼ 1;
TP ¼ Tf ¼ TB :
ð5aÞ
The temperature at the face of the liquid slug at x ¼ 0 and at all distances in
the y-direction perpendicular to the surface is considered to be at the bulk
temperature. The temperature of the similar face of the solid particle is taken as
the computed values of the liquid temperature at the end of the first time step at
x ¼ XL
at y ¼ 0
and
x ¼ 0;
T f ¼ 1;
Tf ¼ TB ;
TP ¼ Tf
and
TP ¼ Tf
ðAfter first time step at x ¼ XL Þ:
ð5bÞ
At t P 0, the faces of both particle and liquid which are in contact with the
surface are at all times at the surface temperature.
At y ¼ 0;
T P ¼ T f ¼ 0;
TP ¼ Tf ¼ TE :
ð5cÞ
Eqs. (4c) and (4d) are solved simultaneously using finite difference techniques
for a fixed and for a variable boundary. For t > 0 the fixed boundary is expressed by Eq. (5c) and the variable boundary is at
y¼0
at x ¼ 0
for liquid
Tf ¼ TP
and for particle
and
Tf ¼ TP
ðAfter previous time step at x ¼ XS Þ
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
TP ¼ Tf
and
TP ¼ Tf
ðAfter current time step at x ¼ XL Þ:
ð5dÞ
The explicit method used by Botterill and Williams [23] and by Davies [25]
restricts the time and space increments to ensure stability according to the
equation
1
i:
Ds 6 h
2
2
ðDX Þ þ ðDY Þ
Their difference equation is
Tði;jÞkþ1 Tði;jÞk
Tði1;jÞ 2Tði;jÞ þ Tðiþ1;jÞ
¼
Ds
DX 2
k
Tði;j1Þ 2Tði;jÞ þ Tði;jþ1Þ
þ
:
DY 2
k
ð6aÞ
The above-mentioned method is very sensitive to the value of the operator
kh which is given as
khX ¼ ah
Dt
DX 2
and
khY ¼ ah
Dt
:
DY 2
For the sake of simplicity equal increments are taken in the X - and Y directions, i.e. DX ¼ DY . An implicit method can make the equations independent of the operator value as well as of space and time increments. The
difference equation is then
Tði;jÞkþ1 Tði;jÞk
Tði1;jÞ 2Tði;jÞ þ Tðiþ1;jÞ
¼
Ds
DX 2
Tði;j1Þ 2Tði;jÞ þ Tði;jþ1Þ
þ
:
ð6bÞ
DY 2
kþ1
On rearranging one gets a pentadiagonal matrix
1
1
¼ Tði;jÞk ;
Tði1;jÞ Tðiþ1;jÞ þ
þ 4 Tði;jÞ Tði;j1Þ Tði;jþ1Þ
kh
k
h
kþ1
ð6cÞ
where
ah Dt
;
DX 2
which can be solved by either the Gaussian elimination method or the GaussSeidel iterative method to give five unknowns.
The implicit alternating direction method discussed by Carnhan et al. [37],
which avoids all the disadvantages discussed above, is thought to be most
suited for this type of problem. The two difference equations are used in turn
over successive time steps, each of duration Ds=2. The first Eq. (7a) is implicit
kh ¼
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
309
only in the X -direction and the second (7b) is implicit in the Y -direction. Thus
if Vði;jÞ is an intermediate value at the end of the first time step then
Vði;jÞ Tði;jÞK Vði1;jÞ 2Vði;jÞ þ Vðiþ1;jÞ
Tði;j1Þ 2Tði;jÞ þ Tði;jþ1Þ
¼
þ
;
Ds=2
DX 2
DY 2
k
ð7aÞ
Tði;jÞKþ1 Vði;jÞ Vði1;jÞ Vði;jÞ þ Vðiþ1;jÞ
Tði;j1Þ 2Tði;jÞ þ Tði;j1Þ
¼
þ
2
Ds=2
DX
DY 2
:
kþ1
ð7bÞ
On rearranging, these equations become
1
Vði1;jÞ þ 2
þ 1 Vði;jÞ Vðiþ1;jÞ
khX
kþ12
khY
1
¼
Tði;j1Þ þ 2
;
1 Tði;jÞ þ Tði;jþ1Þ
khY
khX
k
1
Tði;j1Þ þ 2
þ 1 Tði;jÞ Tði;jþ1Þ
khY
kþ1
khX
1
¼
Vði1;jÞ þ 2
:
1 Vði;jÞ þ Vðiþ1;jÞ
khX
khY
kþ1
ð7cÞ
ð7dÞ
2
These equations give a tridiagonal matrix, which is readily solved by the
Gaussian elimination method. After a residence time t the temperature domain
for solid and for liquid is fully defined. The temperatures at the mesh points for
both solid and liquid were obtained by taking several different step
lengths (20, 40, 80, 100, 200 mesh points). For mesh sizes of 100 100 points
or more, the variation in the calculated values of temperatures at different mesh
points was very small confirming the consistency and stability of the procedure.
4. Calculation of heat transfer coefficient
Each string of particles and entrained liquid is separated from neighboring
string by liquid layer of thickness XL while moving upward parallel to the heat
transfer as shown in Fig. 1. In this layer of thickness XL , it is assumed that
the convection is the principal mode, contributing to the heat transfer which
is calculated on the basis of interstitial velocity as given in Section 3.1. The
overall heat transfer coefficient for fluidized bed is dependent on convective,
liquid conductive and solid conductive components; each contributing equally
to the fraction of the contact of the heat transfer surface area similar to the
calculation of Baskakov [38] for the effective thermal conductivity for gasfluidized bed as a function of thermal conductivities of solid and liquid and the
area fraction occupied by each phase.
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
The conductive components for particle and entrained liquid are evaluated
from the numerical solution of partial differential equations which yields the
temperature profile within the solid particle and entrained liquid at each time
step for conduction. From the temperature gradient oT =oy, the heat transfer
coefficient h can be evaluated at any position at the surface.
oT At y ¼ 0; the total heat transfer is: q ¼ kA hðt;xÞ ADT ;
ð8Þ
oy y¼0
where hðt;xÞ is the point value of heat transfer at time t and position x and DT is
the overall temperature difference between the surface and the bulk of the bed.
For each time step, instantaneous values of heat transfer coefficient h were
calculated for solid particle as well as for entrained liquid. The average value as
they passed over the surface was then used for evaluating the mean heat
transfer coefficient by heat conduction, hcond .
hcond ¼
XS2
XL XS
hcond;S þ
hcond;L :
ðXL þ XS ÞXS
ðXL þ XS ÞXS
ð9Þ
The overall heat transfer coefficient for the fluidized bed is given as
hADT ¼ hcond Acond DT þ hconv Aconv DT ;
h¼
XS ð XL þ XS Þ
ð XL þ XS Þ
2
hcond þ
XL ðXL þ XS Þ
ð XL þ XS Þ
2
hconv :
ð10Þ
For low void fraction values the residence time was quite large and the system
approached the steady-state; hence in calculating conductive component values
were taken as corresponding to the last time step. For higher void fraction
values the particle movement was quite rapid and the corresponding residence
time was short [35]. The arithmetic mean of the conductive component values
was taken. The evaluated and experimental values for 6-mm glass spheres fluidized by dimethyl phthalate are listed in Table 2 and are shown in Fig. 2.
5. Comparison and discussion
Most of the models previously suggested for heat transfer in liquid fluidized
beds involve the use of an effective thermal diffusivity of the bed. These models
showed a considerable deviation from the experimental values obtained, as
indicated in Table 2. Wasan and Ahluwalia [8] compared the experimental data
of various workers with their suggested model and they concluded that the
proposed expression fitted remarkably well for data obtained for both gas and
liquid fluidized beds. Patel and Simpson [14] compared the experimental data
of Wasmund and Smith [9,12] with the above model and reported that, contrary to the statement of Wasan and Ahluwalia [8], the model is inaccurate for
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
311
Table 2
Comparison of experimental and theoretical results for 6 mm glass spheres fluidized by dimethyl
phthalate [1]
Voidage
0.405
0.43
0.45
0.49
0.54
0.55
0.58
0.605
0.63
0.65
0.745
0.83
0.88
0.93
h ðW=m2 CÞ
Experimental results
Theoretical results
Liquid
alone
Fluidized
bed
Liquid
convective
Liquid
conductive
Solid
conductive
223.2
239.5
249.4
271.6
302.1
309.0
325.2
340.5
356.9
369.9
424.0
476.5
509.2
539.5
860.1
902.5
952.0
969.8
1056.9
1031.9
1002.2
1042.7
1047.2
1039.9
965.8
878.4
806.6
713.4
140.7
149.6
154.2
165.2
180.8
184.5
192
199.5
208.1
214.7
239.7
264.4
280.0
293.7
491.7
493.7
493.8
497.2
504.3
504.9
510.8
513.7
518.3
522.7
543.7
566.2
579.1
579.2
1062
1123
1172
1269
1390
1415
1488
1549
1609
1658
1889
2095
2217
2339
839.4
867.5
887.4
923.0
958.7
964.2
978.7
987.2
993.1
995.6
973.5
901.0
826.1
708.1
Fig. 2. Comparison of different heat transfer coefficients for 6-mm glass spheres fluidized in
dimethyl phthalate.
predicting wall to bed heat transfer in liquid fluidized beds. Davies [25] gave
a model for the calculation of the heat transfer attributable to the solid
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
convective component and concluded that it gave very low values compared
with the experimental values of heat transfer coefficient. Other mechanisms
were therefore of dominant importance.
The experimentally observed behavior of heat transfer coefficient as a
function of void fraction cannot be predicted by any of the models suggested so
far and it is fair to say that the suggested models are quite inaccurate in explaining the mechanism of heat transfer in liquid fluidized systems. From the
present unsteady state thermal conduction model the temperatures of the grid
points at different time intervals for both liquid and solid are known and so the
instantaneous values of heat transfer coefficient can be readily estimated. These
calculated values showed a similar dependence on voidage to that reported in
the experimental results of all the previous investigators as well as Khan et al.
[39] results. The magnitude of the heat transfer coefficient calculated from the
model is different for the following reasons:
Fig. 3. Temperature profiles on XY plane of liquid slug for voidage 0.83.
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
313
1. The element is considered as a plane surface at a uniform constant temperature.
2. The voidage is assumed to be uniform throughout the bed.
3. The reported particle velocities were measured in the absence of the element
and the effect of only the axial component is taken into account.
Despite all these limitations the results calculated from the model give a fair
indication of the trend of the measured values of heat transfer coefficient in
fluidized beds. The temperature profiles after a residence time t for unsteady
state thermal conduction for liquid and solid are shown in Figs. 2 and 3. The
temperature profile for liquid alone for forced convection is shown in Figs. 4
and 5. Figs. 2 and 3 show the temperature distributions in a liquid slug and in a
particle, respectively. Due to symmetry in the z-direction only the XY -plane has
been considered where X is the distance in the direction of flow and Y is the
perpendicular distance from the heat source. In these figures the planes parallel
to the XY -plane are isothermal planes and the lines show the temperature at the
face XY of the cube; this gives the temperature distribution in the whole cube.
6. Conclusion
The various models previously suggested show a considerable deviation
from the experimental values and fail to explain the mechanism involved in
heat transfer from a fluidized bed. An unsteady state thermal conduction
model is proposed. The values of heat transfer coefficient calculated using the
Fig. 4. Temperature profile for forced convection for dimethyl phthalate.
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A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
Fig. 5. Temperature profiles on XY plane of solid particle for voidage 0.83.
model show the same dependence on voidage as is found in the experimental
work. The heat transfer coefficient increases to a maximum and then steadily
decreases as the bed voidage increases from that of a packed bed to unity. The
existence of the maximum is due to the fact that two factors have opposing
effects on the heat transfer in the bed expands:
1. There is a decrease in solid concentration.
2. There is an increase in particle and liquid velocities.
The second factor is controlling for voidage upto the value emax (voidage at
which maximum heat transfer occurs) corresponding to the maximum heat
transfer coefficient. With further expansion the first factor becomes limiting
and heat transfer decreases. In the model the principal mechanisms of heat
transfer which apply in the case of liquid–solid systems are assumed to be
A.R. Khan, A. Elkamel / Appl. Math. Comput. 129 (2002) 295–316
315
1. Unsteady state thermal conduction in which the particle and entered liquid in
contact with the heat transfer surface absorb heat during a residence time t.
2. Forced convection in the fluid: the presence of the fluidized particle causes
an increase in the free area available for flow and gives rise to a modified
flow pattern and an increased interstitial liquid velocity.
The model also predicts an increase in heat transfer coefficient with the
increase in particle diameter for constant bed voidage because the flow rate is
high with large particles and the liquid convective component is increased.
Furthermore, as the velocity is increased, the particle and liquid velocities
within the uniform sized particle bed are also increased, and the residence time
and thermal boundary layer thickness are both reduced. At the same time the
concentration of the particle decreased and the transfer due to thermal conduction through the particles is reduced. At a critical voidage the combined
effects lead to a maximum in the heat transfer coefficient.
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