Numerical Calculation
of
Wall-to-Bed Heat-
Transfer Coefficients
in
Gas-Fluidized Beds
J.
A.
M.
Kuipers,
W.
Prins,
and
W.
P.
M.
van
Swaaij
Department of Chemical Engineering, Twente University of Technology,
7500
AE
Enschede, The Netherlands
A
computer model for a hot gas-fluidized bed has been developed. The theoretical
description is based
on
a two-fluid model
(
TFM) approach
in
which both phases
are considered to
be
continuous and fully interpenetrating. Local wall-to-bed heat-
transfer coefficients have been calculated by the simultaneous solution
of
the TFM
conservation of mass, momentum and thermal energy equations. Preliminary cal-
culations suggest that the experimentally observed large wall-to-bed heat-transfer
coefficients, frequently reported in literature, can be computed from the present
hydrodynamic model with no turbulence. This implies that there
is
no
need to explain
these high transfer rates by additional heat transport mechanisms (by turbulence).
The calculations clearly show the enhancement
of
the wall-to-bed heat-transfer
process due to the bubble-induced bed-material refreshment along the heated wall.
By providing detailed information
on
the local behavior
of
the wall-to-bed heat-
transfer coefficients, the model distinguishes itself advantageously .from previous
theoretical models. Due to the vigorous solids circulation in the bubble wake, the
local wall-to-bed heat-transfer coefficient
is
relatively large in the wake
of
the bubbles
rising along a heated wall.
Introduction
Because of their favorable heat-transfer properties fluidized
beds find a widespread application in highly thermal processes.
Many of these applications involve the transfer of heat between
the bed and immersed surfaces but yet it is very difficult to
predict values for the corresponding heat-transfer coefficients
with confidence. Many empirical correlations for bed-wall and
bed-immersed tube heat-transfer coefficients have been pro-
posed in the literature but the use of these expressions is limited
to the experimental conditions on which they are based: pre-
dicted heat-transfer coefficients may differ
by
almost two
or-
ders of magnitude from the actual coefficients in some cases
(Gelperin and Einstein,
1971).
Thus, the designer of fluidized-
bed heat-transfer systems should cautiously use such corre-
lations in estimating heat-transfer coefficients. From a sci-
entific point of view, these empirical correlations are less
attractive because they generally do not contribute to the un-
derstanding of the fundamental transport mechanisms.
Therefore, several investigators have developed mechanistic
models
for the prediction of heat-transfer coefficients, the most
Correspondence
concerning
this
article
should
be
addressed
to
J.
A.
M.
Kuipers
useful type being based upon transient conduction between the
particles and the surface. These type of mechanistic models
can roughly be divided into single particle models and emulsion
phase models.
Single-particle
models
In single-particle models the fluidized bed is considered as
a heterogeneous system consisting of a continuous phase (that
is, the fluidizing medium) and a discrete phase (that is, the
solid particles). The heat-transfer process for single particles
during their residence at the heat exchanging surface is de-
scribed in terms of two separate transient heat conduction
equations.
This model type was developed in its simplest form by Bot-
terill and Williams
(1963).
They considered an isolated particle,
surrounded by a stagnant fluid (that
is,
a gas or a liquid), and
in contact with
a
heat-transfer surface for a certain time, during
which heat was transferred to it. An explicit finite difference
technique was used to solve the transient heat conduction equa-
tions. Botterill and Williams
(1963)
compared their model pre-
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dictions with experimental data obtained from a mechanically
stirred fluidized-bed heat exchanger under closely controlled
particle residence time conditions. Their theoretical predictions
deviated considerably from the experimental data and in order
to remove this shortcoming of the model, they introduce a gas
film
or
gas gap with a thickness of approximately
0.
Idp
between
the particle and the surface. The modified model showed then
good agreement between the predicted and experimental data
for short particle residence times. However, the assumption
of such a gas gap does not correspond to the physical reality
and has been criticized, among others, by Schlunder
(1971).
Besides, the single particle model of Botterill and Williams
(1963)
fails if the heat from the transfer surface penetrates
beyond the first layer of particles which is the case at relatively
long particle residence times.
This failure of the single particle model has been recognized
by Botterill and Butt
(1968)
and Gabor
(1970)
who proposed
respectively a model with two spherical particles normal to the
heat-transfer surface and a model with a string of spherical
particles of infinite length normal to the heat-transfer surface.
In both studies a finite difference technique was used to solve
the transient heat conduction equations on the multiple particle
domain. Satisfactory agreement between the theoretical pre-
dictions and the experimental data was obtained in both studies
under closely controlled particle residence time conditions.
Gabor
(1970)
also proposed, as an alternative to his “string
of spheres model,” a computationally less complicated model
based
on
transient heat conduction through a series
of
alter-
nating gas and solid slabs.
Emulsion-phase
models
The emulsion-phase or packet-renewal models are in some
respects similar to the continuum approach adopted in the
present study and use the analogy between a fluidized bed and
a liquid. Here the emulsion phase is considered to be the con-
tinuous phase and the bubbles to be the discrete phase. Mickley
and Fairbanks
(1955)
proposed such a packet renewal model
postulating that heat
is
transferred by “packets” of emulsion
phase which are periodically replaced from the heat-transfer
surface by bubbles. According to the packet renewal model
the local instantaneous heat-transfer coefficient
hi
is given by:
where,
tp
is the time for which the packet was in contact with
the heat-transfer surface. In terms of the packet renewal models,
the thermophysical properties within the packet are considered
to be uniform and are usually evaluated at bed conditions
corresponding to incipient fluidization. However, due to the
bed voidage variation near the constraining heat-transfer sur-
face, the thermophysical properties of the packet close to the
surface will differ from those in the bulk of the emulsion phase.
In
fluidized beds the principal voidage variation occurs within
one particle diameter from the surface (Korolev et al.,
1971)
which implies that the thickness of the surface layer of altered
thermophysical properties is approximately one particle di-
ameter. Most probably, the effect of this surface layer
or
packet
heterogeneity is negligible for packets with a heat penetration
depth
6>>
dp,
thus for packets with relatively large residence
times. However, at relatively short packet residence times the
effect of this surface layer becomes increasingly important and
the simple emulsion phase model fails. Therefore, this model
type has been refined by several workers (Baskakov,
1969;
Kubie and Broughton,
1975)
to extend its validity to relatively
short packet residence times.
Baskakov
(1969)
introduced the concept of a time-inde-
pendent contact resistance to account for the spatial voidage
variation near the surface. Kubie and Broughton
(1975)
mod-
ified the simple packet renewal theory to allow for property
variations near the heat-transfer surface. They used simple
geometrical considerations to model the spatial voidage vari-
ation normal to the heat-transfer surface and subsequently used
the resulting expression for the voidage profile to describe the
corresponding property variations. Despite their simplicity, the
use of these (refined) emulsion phase models has limited ap-
plicability, because they all require information regarding the
actual packet residence times that exist in fluidized beds. Un-
fortunately, such hydrodynamic parameters cannot easily be
obtained by experimental methods. Ozkaynak and Chen
(1980)
reported measurements of the particle residence time distri-
bution on
a
vertical tube immersed in a fluidized bed, obtained
by a fast response capacitance probe mounted in the surface
of the tube. They obtained good agreement between experi-
mentally determined heat-transfer coefficients and the predic-
tions according to the packet renewal model provided that the
measured root-square-mean packet residence times were used
in the model. From a scientific point of view, their approach
is less satisfactory because it requires the use
of
a
II
posteriori
empirical constant.
The two-fluid model (TFM) approach, adopted in the pres-
ent study, has the advantage over previous models reported in
literature that it does not require the input of empirical pa-
rameters such as the average particle
or
packet residence time;
the bed hydrodynamics evolves naturally from the solution of
the TFM conservation of mass and momentum equations.
However, to account for microscale (that is,
on
the scale of a
representative unit cell of particles and interstitial fluidum)
momentum and heat transfer between the phases, the present
mesoscale model incorporates two empirical expressions for
respectively the interphase momentum transfer coefficient and
the interphase heat-transfer coefficient.
For
the further de-
velopment
of
the mesoscale model, mechanistic models should
be developed which describe momentum and heat transfer on
microscale.
Previous work has shown that the hydrodynamic model can
predict bubble sizes (Kuipers et al.,
1991)
and void distributions
(Kuipers et al.,
1992b)
in a cold-flow two-dimensional gas-
fluidized bed satisfactorily without the use of any fitted pa-
rameters. These results provide an indirect experimental val-
idation of the empirical expression for the interphase
momentum transfer coefficient. In the present study, the wider
applicability of this model to predict wall-to-bed heat-transfer
coefficients in gas-fluidized beds will be explored. It must be
emphasized, that the present model should be considered as a
learning model and not as the most efficient way to predict,
for example, average wall-to-bed heat-transfer coefficients.
Theoretical
Model
The present heat-transfer model is based on
a
TFM approach
in which both phases are considered to be continuous and fully
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Table
1.
Two-Fluid Model Conservation
of
Mass, Momentum
and Thermal Energy Equations
in
Vector Form
Continuity Equations
Fluid
phase:
w+
(V
.+)
=o
at
(Tl-I)
Solid
phase:
a[(1
-
E)pSI
+
[
v
.
(1
-
E)P,ij]
=
0
(Tl-2)
at
gomenlum
Equations
Fluid
phase:
Solid
phase:
-
a[(l-C)~~U~+[V.(l-€)p,uT;]=
-(l-€)vp
at
+
(U
-5)
-
G
(6)
VC
+
(I
-
~)p,g
(Tl-4)
Thermal
Energy
Equations
Fluid
phase:
specification of the constitutive equations which define the
remaining variables in terms of the basic variables. Here, the
porosity
E,
the pressure
p,
the fluid phase temperature
Tf,
the
solid phase temperature
T,,
the fluid phase velocity vector
Ti
and the solid phase velocity vector
5
constitute the basic vari-
ables. The additional constitutive equations provide the nec-
essary empirical information. The major empirical input in the
hydrodynamic model is the interphase momentum transfer
coefficient
(3
which was obtained from well-established liter-
ature correlations (Kuipers et al., 1992a). Only the constitutive
equations associated with the thermal energy equations will be
considered here.
Internal energies
If
and
I,
respective temperatures by the caloric equations of state:
The internal energies
of
both phases are related to their
and
+
(V .EKJV
T’)
-a(
TJ-
T,)
(TI-5)
Solid
phase:
where
C,,f
and
C,,s
represent respectively the fluid phase and
solid phase heat capacity. In the present study, both are as-
sumed to be independent of the temperature.
Interphase heat-trans fer coefficient
a
1
a(i
-
a[(l-+JJ+[V.(l
-E)pSJij]=
-p
-
+
[V
.(1
-€)El
at
i
at
+
[
V
-
€)KsV
Ts]
+
a(
TI-
T,)
(Tl-6)
The interphase heat-transfer term in equations (T
1-5)
and
(Tl-6)
is written in standard form as the product of a transfer
interpenetrating’
and
shows
the
TFM
mass,
momentum
energy
conservations
equations for
both
phases
coefficient
01
and a driving force
(T,- T,).
Here,
a
represents
the volumetric interphase heat-transfer coefficient which equals
in vector form.
As
evident from these equations, the present
our
point of view (which is supported by experimental evidence)
the macroscopic (that is, on the scale of the size of bubbles)
the product
of
the specific
interfacial
exchange
area
and
the
spherical
particles
a
can be
obtained
from:
hydrodynamic
does
not
contain
terms*
In
fluid-particle heat-transfer coefficient ffp.
For
mono-+ized
turbulent transport in dense gas-solid systems such as gas-
fluidized beds does not constitute
a
dominant transport mech-
anism. This has been demonstrated in a qualitative sense (Rowe,
1971) by injecting a tracer gas
(NOz)
in a bubbling two-di-
mensional gas-fluidized bed. In these experiments no appre-
ciable lateral mixing of the injected NO2 could be observed
which implies the absence of turbulence in the gas phase and
therefore the absence of turbulence in the solid phase since the
motion of solid particles in gas-fluidized systems is driven by
the gas motion and not vice versa. However, for very dilute
systems turbulent transport becomes more important due to
the low concentration of particles which dampen the gas phase
turbulence.
In addition it should be noticed that the viscous stresses in
both momentum equations have been omitted. This approx-
imation is valid due to the fact that the interphase momentum
transfer coefficient
/3
is the dominant term in the momentum
equations. Our computational experience has revealed that the
sensitivity of the models predictions (that is, calculated bubble
parameters such as shape and size) with respect to the value
of
the solid phase viscosity is low.
As
discussed in Kuipers et al. (1992a), the solution of the
balance equations, listed in Table 1, involves the specification
of
the so-called “primary” or basic variables and subsequent
6(1-
E)
ffP
ff=-
dP
(3)
where
01,
has to be estimated from empirical correlations. There
are
a
large number of empirical correlations available
for
the
estimation of both packed bed and fluidized bed fluid-particle
heat-transfer coefficients. In the present study, the correlation
proposed by Gunn (1978) was selected to obtain an expression
for the fluid-particle heat-transfer coefficient
a,.
This corre-
lation relates the Nusselt-number
Nu, to the Reynolds number
Re, and the Prandtl number
Pr
for heat transfer to fixed and
fluidized beds of particles within the porosity range of 0.35-
1.00. Experimental data are correlated up to
Re,=
lo5.
ad
KfJJ
Nu,=>=(7-
10e+5c2)[1
+0.7(Re,)0~2(Pr)”31
+
(1.33
-
2.406
+
1 .20~’) (Re,)’.’ (Pr)”?
(4a)
where
EP/lF
-
6
I
dp
Re,
=
-
Pf
(4b)
1081
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and
As
expected, the Nusselt number
Nu,,
given by Eq. 4a ap-
proaches 2 for an isolated particle in an infinite stagnant
fluidum:
Several other, less evident, asymptotic conditions have also
been incorporated in Eq. 4a. For a detailed discussion refer
to the paper by Gunn (1978).
The use of Eq. 4a inevitably introduces some inaccuracies
in our model. However, due to the large volumetric interphase
heat-transfer coefficient the difference in temperature between
the phases will remain small which implies that the error due
to the introduction of this empiricism can probably be ne-
glected. Trial simulations predicted, as expected, rapid quench-
ing of hot fluidizing air flowing into a cold fluidized bed of
glass beads.
Thermal conductivities
KJ
and
us
The proper specification of the constitutive equations for
the thermal conductivities Kfand
K,
poses a major difficulty in
the theoretical formulation.
In
terms of the
TFM
approach
K/
and
K,
must both be interpreted as effective transport properties
which means that the corresponding microscopic transport
coefficients
K~,,
and
K,,~
cannot be used. Both
~f
and
K>
are
expected
to
depend on
E,
Kf,o,
K~,~
and the particle geometry,
where the functional dependency must be specified partly by
recourse to experimental data.
To derive the corresponding expressions it is necessary to
consider the heat conduction terms in the thermal energy equa-
tions (TI-5) and (Tl-6) in more detail. These equations show
that Fourier’s law of heat conduction has been used to represent
the conductive heat transport in both phases. Accordingly the
expressions
for
the conductive heat fluxes
3f
and
s,,
in the
fluid and the solid phase, respectively, should be formulated
as
:
and
In
terms of the theoretical model, the total conductive heat
flux
3
can be written as follows:
which reduces to:
-
@=
-
(tKf+(l-€)KS)
VT= -KmVT (64
in case of thermodynamic equilibrium
(
Tf=
T,
=
T) between
the phases. Thus, according to Eq. 6d the “mixture conduc-
tivity”
K,
is defined as:
This mixture conductivity
K,
corresponds to the familiar “ef-
fective bed conductivity”
Kb
which can be determined exper-
imentally.
In
the most general case, as discussed by Gelperin
and Einstein (1971), this mixture
or
bed conductivity
(K@~
or
Kb) includes conductive, convective and radiative components.
However, due to its insignificance at the conditions
of
the
present study, radiative heat transfer will not be considered
further. Because, the convective transport components are ac-
counted for separately in the thermal energy equations
(TI
-5)
and (Tl-6), they neither need further consideration. The re-
maining conductive transport mechanism is a very complex
phenomenon involving contact conductance among the par-
ticles and conduction through
a
fluid layer surrounding each
particle.
In
the present study we use the model of Zehner and Schlun-
der (1970) to obtain
an
approximate expression for the effective
thermal bed conductivity
Kb.
This conductivity model was orig-
inally developed for the estimation of the effective radial ther-
mal conductivity in packed beds. However, it can also be
applied to estimate the effective dense phase thermal conduc-
tivity in fluidized beds (Biyikli et al., 1989). According to the
Zehner and Schlunder (1970) model the radial bed conductivity
Kb
consists
of
a contribution
Kb,f
due only to the fluid phase
and a contribution Kb., due to
a
combination of the fluid phase
and the solid phase:
Kb
=
Kb,
f
4-
Kb,s
where
and
1019
B=
1.25(?)
for spherical particles
A=%
Kf*o
w
=
7.26
x
Figure 1 shows the effective bed conductivity
Kb
as a function
of
the bed porosity
c
according to the Zehner and Schlunder
model for spherical glass beads in air. It is interesting to note
that in the operating region of fluidized beds
(t
>
0.40),
Kb
is
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AIChE
Journal
W/(mK) 1'20
I
6Y
Ax
*
t
'4,
1
.oo
0.80
0.60
0.40
0.20
0.00
0.20
0.40
0.60
0.80
1.00
Ed
Figure
1.
Effective bed conductivity
Kb
as a function
of
the bed porosity
c
according to the Zehner and
Schlunder
(1970)
model.
Fluid-particle
system: air
[K,,.
=
0.0257
W/(m
.K)];
glass
beads
[K~,~=
1.00
W/(m.K)].
significantly smaller than
K,,~.
The desired constitutive equa-
tions for
~f
and
K,
can finally be obtained by comparing Eq.
6e with Eq. 7a:
KbJ
Kf=-
E
Syamlal and Gidaspow (1985) have also developed a nu-
merical model for a hot gas-fluidized bed based on a TFM
approach. They too used the Zehner and Schlunder model to
obtain the constitutive equations for
Kf
and
K~.
However, their
constitutive equations show some remarkable differences with
the ones derived in the present study. They assumed that the
fluid phase thermal conductivity
~f
equals the corresponding
microscopic thermal conductivity
Kr0
and that the solid phase
thermal conductivity
K,
equals the mixture or effective thermal
bed conductivity
Kb,
thus:
and
In the gas-particle system considered by Syamlal and Gi-
daspow (1985), the bed conductivity
Kb
is dominated by the
contribution of the solid phase
Kb,s
which implies that their
expression for the solid phase thermal conductivity
K~
differs
approximately a factor 1/(
1
-
E)
from the present one.
Numerical Simulation
The TFM conservation equations have been solved numer-
ically by a finite difference technique, described in more detail
by Kuipers et al. (1990). This technique has been embodied in
a
computer model which calculates the "primary"
or
basic
variables in two-dimensional Cartesian
or axisymmetrical cy-
Y
X
~x.=(L{)i*6x;
i=
~...n-l
12
I
hydrodynamic
cell
n-1
Axn
=
Ax
Figure
2.
Subdivision of hydrodynamic computational
cells into subcells near a heated wall.
n
=
number
of
subcells.
lindrical coordinates. Previous work (Kuipers et al., 1991,
1992b) was concerned with the theoretical prediction of the
cold-flow hydrodynamics in a two-dimensional gas-fluidized
bed. In the present study the numerical model has been ex-
tended to enable the additional calculation
of
wall-to-bed heat-
transfer coefficients in fluidized beds.
Preliminary calculations showed the existence of very sharp
temperature gradients near the heated wall which necessitated
the use of a grid refinement technique
in
order to accurately
represent the temperature profiles in the thin thermal boundary
layer. This technique was not incorporated in the basic nu-
merical model and will be briefly discussed below.
Grid refinement technique
While applying
a
finite difference technique for the solution
of the complex partial differential equations, the considered
region of the two-phase flow was divided into a number
of
equally sized cells (as shown in Figure 2). The original cell
structure
of
the hydrodynamic model is now refined in one
dimension, viz normal to the heated wall, by a subdivision as
sketched in Figure 2.
As
Figure 2 shows, the subcell dimension
normal to the heated wall
Ax,
decreases with decreasing dis-
tance from the wall: the finest subcells are situated immediately
near the heated wall where the maximum temperature gradients
are to be expected.
Syamlal and Gidaspow (1985) have used uniform subcell
dimensions in their numerical model. This approach is less
efficient from a computational point
of
view, because the total
number
of
subcells required
to
achieve the same degree
of
grid
refinement near the heated wall is considerably higher. The
required number
of
subcells
n
is problem dependent and should
be found by performing a convergence study with respect to
the numerically calculated heat-transfer coefficients. Com-
putational experience has shown that for the conditions
of
the
present study, division into 7 subcells is sufficient to establish
a grid-independent solution.
Fluid-particle system
For the theoretical calculations reported in this article, the
fluid-particle system consisted
of
air as the fluidizing medium
and mono-sized spherical glass beads as the fluidized solid
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