Mathematical model for heat transfer mechanism

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.Box5969,13060Safat,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 homoge-neous and transport operation can be easily modeled.The heat transfer coefficient in-creases to a maximum and then steadily decreases as the bed void fraction increases from that of a packed bed to unity.The void fraction e max 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 convec-tion.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.Ó2002Elsevier Science Inc.All rights reserved.
1.Introduction
The high heat transfer coefficient from a hot surface to a fluidized bed fa-cilitates 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 Applied Mathematics and Computation 129(2002)
295–316
*Corresponding author.Tel.:+965-481-7662;fax:+965-483-9498.
E-mail address:rehman@.kw (A.R.Khan).
0096-3003/02/$-see front matter Ó2002Elsevier Science Inc.All rights reserved.
PII:S 0096-3003(01)00039-X
296 A.R.Khan,A.Elkamel/put.129(2002)295–316
A.R.Khan,A.Elkamel/put.129(2002)295–316297
empirical correlations relating bed to surface heat transfer coefficients for a range of operating variables have been proposed.They are of restrictive va-lidity 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 dif-ferent aspects of this complexproblem.There are particularly diverse concepts suggested by different workers regarding the mechanism of heat transfer be-tween afluidized 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 liquidfluidized systems.The model results are compared with experimental data[1]and computed values of the existing models.
2.Heat transfer models forfluidized bed
Both gas,and to a lesser extent liquidfluidized 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 gasfluidized 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 influidized 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.
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 afluidfilm near the hot surface.They suggested that particles acted as turbulence promoters,which eroded thefilm reducing its resistive effects.Levenspiel and Walton[4]derived an expression in terms of modified Nusselt and Reynolds numbers for the ef-fectivefluidfilm thickness assuming that thefilm 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.
Nu¼cons
C S q
S
d1:5
P
ffiffiffig p
k g
0:4Gd
P
E
l
g
R
"#0:36
:ð1Þ
In this correlation thefluidization parameters are defined as follows:
1.E is thefluidization efficiencyðGÀG mfÞ=G mf.
2.R is the expansion ration of the bed H=H mf.
Richardson and Mitson[6]and Richardson and Smith[7]reported that for liquidfluidized 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 thefluid 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 afluidfilm was promoted byfluid eddies beyond thefilm boundary.They assumed that the solid particles were stationary and equally spaced and that heat was transferred through thefilm and then spread byfluid convection into the bulk of the bed.They compared the experimental results of various workers and found deviations of up toÆ44%.
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 thefluid into the bed.
298 A.R.Khan,A.Elkamel/put.129(2002)295–316
2.2.Two resistancefilm 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 contributed50–60%of the total heat transferred and the remainder was fromfluid 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 Æ20%.Brea and Hamilton[13]and Patel and Simpson[14]used a two resis-tancefilm model and emphasized that thefluid eddy convection is the main contributing factor to the heat transfer.Zahavi[15]measured the effective diffusivity of thefluidized beds and also developed a semi-empirical correlation, which represented his results with a maximum deviation ofÆ34%.
2.3.Unsteady state heat transfer
Mickley and Trilling[16]suggested that the heat transfer process in a gas-fluidized bed was of an unsteady state ter Mickley and Fairbanks[17] developed a model of heat transfer on the assumption that at any time there is unsteady state heat transfer within thefluidized 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 thefluidized bed.The mean heat transfer coefficient can be calculated for packets of particles moving with a constant speed u passing rapidly along the length of the heat source,then
h¼2
ffiffiffip p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k0q0C0
P
u0
L r
:
Thermal conductivity k0,density q0and heat capacity C0
P 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 resistancefluctuations recorded for a thin electrically heated platinum strip.The frequency of packets was of the order of two per seconds and the residence time of0.4s.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 thefluid to the particle with the maximum rate in the vicinity of the contact part.
A.R.Khan,A.Elkamel/put.129(2002)295–316299
300 A.R.Khan,A.Elkamel/put.129(2002)295–316
2.4.Simplified models
The heterogeneity offluidized beds is an important factor enhancing the value of the heat transfer coefficient up to50–100folds for a gas and5–8folds for liquidfluidized 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 thefluidized 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 byfinite 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
purefluid with which a particle was in contact.
(B)The temperature of the other end the particle was set at the sink temper-
ature,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 resi-dence time of metallic particles were far less than the predicted values.This discrepancy was accounted for by assuming that a gasfilm of thickness equals to about10%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 fluidfilm between the particles and the heat transfer surface.The thickness of thefilm 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 be-tween 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 v r,approached the surface and left it again with the same velocity.The Fourier equations were solved by using the explicitfinite 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 byfluidized particles.The experimental high value was attributable to
A.R.Khan,A.Elkamel/put.129(2002)295–316301 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
h¼h P
ðÀh convÞ1ðÀf0Þþh b f0þh r;ð2Þ
where h P;h conv;h b and h r are the heat transfer coefficients corresponding to packet,convection,bubble and radiation,respectively,and f0is 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:R WS the resistance offered by gas entrained by the particles close to the transfer surface and R a the resistance offered by the gas–solid packets.They have tabulated theirfinal 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 freeflight.
2.The wall to particle heat transfer coefficient is calculated by integrating the
local conduction heatfluxes 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.
302 A.R.Khan,A.Elkamel/put.129(2002)295–316
3.Development of mathematical model
For the purpose of establishing a simplified model of a liquidfluidized bed, the system is assumed to consist of strings of particles with liquidfilling 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.1to separate the strings.
The overall heat transfer coefficient for liquidfluidized bed from immersed surface constitutes solid conductive,liquid conductive and liquid
convective Array Fig.1.Mathematical representation offluidized particle entrained in liquid with initial and boundary conditions.
components based on the void fraction determined by bed expansion charac-teristics and particle axial velocity V P;Axial.The convective component is cal-culated 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 par-ticles with entrained liquid.
3.1.Heat transfer across incompressible boundary layers
The simulated element is considered as aflat plate located on the axis of the column with the large faces parallel to liquidflow.The liquid with an average velocity u and uniform temperature T B passes over the hotflat plate at a constant temperature T E.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 theflat plate.The detailed solution of the energy equation for laminar form assuming linear velocity profile within laminar sublayer is expressed as
y s WðxÞ
l
o T
o x
À
y2d s W
d x
o T
o y
¼a
o2T
o y2
ð3aÞ
by substituting
g¼
y3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðs W=lÞ3
q
9a
R x
x0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðs W=lÞ
p
dx
;
Eq.(3a)can be transformed to ordinary differential equation.
d2T d g2þg
þ
2
3
d T
d g
¼0;ð3bÞ
where
T¼T EÀT
T EÀT B
¼1;
for x¼0and all values of y>0,T¼1,for y¼/and all values of x>0and T¼0,for y¼0and all values of x>0.
The solution is given in the form of incomplete c-function as
TðgÞ¼c1=3;g ðÞCð1=3Þ
:
The calculated values of the heat transfer coefficient are presented together with experimental values[1]in Table2.The experimental data are obtained
A.R.Khan,A.Elkamel/put.129(2002)295–316303
304 A.R.Khan,A.Elkamel/put.129(2002)295–316
from immersed electrically wound heating surface in6-mm glass particleflui-dized bed dimethyl phthalate.The high experimental values are due mainly to the following causes:
1.The plane element was not a trueflat 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 afluidized bed the presence of solid particles decreased the free area available forflow 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 offluidized particles
3.2.1.Particle velocities influidized beds
By means of high speed photography several workers have measured the paths taken by a tracer particle in a transparentfluidized bed.Toomey and Johnstone[31]obtained particle velocities near the wall of dense phase gas-fluidized bed.For the particular case of0.376-mm diameter glass spheresflu-idized in air,they reported particle velocities from60to600mm/s.Kondukov et al.[32]used radioactive tracer particles and radiation detectors to measure the particle trajectories in an airfluidized 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]fluidized1.1-and1.53-mm glass spheres with methyl benzoate in31-and76-mm diameter columns.He reported the radial and axial velocities of the particles for bed voidage ranging from0.67to0.905.He concluded that the velocity of the particles was completely random in a uniformlyfluidized bed.A more extensive study of particle velocities influidized bed was made by Carlos[34]and by Latif[35].Carlosfluidized9mm glass beads with dimethyl phthalate at30°C in a100mm 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 tif[35]extended the work of Carlos to6-mm glass spheres and developed a simple relationship between particle velocity and axial and radial positions.The constants of these equations are listed in Table1.These equations are used to evaluate particle velocities as a function of bed expansion characteristics.
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 reason-able 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
t ¼
L V P ;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 ðT B Þand the temperature of the element ðT E Þare uniform.
2.A string of particles with entrained liquid at a uniform temperature equal to T B arrives quickly in an axial direction at the element.
Table 1
Coefficients for calculation of axial velocity component [35]
V P ;Axial ¼A Ár þB where r is normalized radial coordinate of particle A ¼a 0þa 1z þa 2z 2þa 3e a 4z
2
B ¼b 0þb 1z þb 2z 2þb 3e b 4z
2
e a 0a 1a 2a 3a 4b 0b 1b 2
b 3b 40.55)0.767.25)8.74)87.4)20.20.42)3.08 3.8331.8)21.90.65)2.749.66)6.92)74.22)10.09 1.17)1.550.3846.5)17.030.75)1.42 4.78)2.84)128.2)15.110.96)3.17 1.9474.56)18.340.850.55)3.3 2.58)162.3)20.0)0.18 3.12)2.9286.7)22.70.95
)4.27
)17.98
22.27
)222.0
)27.5
2.22
12.04
)15.09119.7
)29.0
A.R.Khan,A.Elkamel /put.129(2002)295–316
305
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 thefluidized particles contribute to-wards 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 ap-propriate 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.1a cube of dimensionðX SþX LÞis considered and the void fraction in the vicinity of the surface assumed to be the same as in the bulk where X S and X L are dimensions of particle and liquid slug,respectively.The volume of
the particle is X3
S and of the liquid slug X2
S
X L and void fraction is expressed as
e¼ðX SþX LÞ3ÀX3
S ðX SþX LÞ
:
On rearranging
e¼1À
X S
X SþX L 3
or
X L¼X S
1
1Àe
1=3
"
À1
#
:
The Fourier equations for unsteady state thermal conduction within the two homogeneous phases of the system are
For solid phase
o2T P o x2þ
o2T P
o y2
¼
1
a P
o T P
o t
:ð4aÞ
306 A.R.Khan,A.Elkamel/put.129(2002)295–316
For liquid phase
o2T f o x2þ
o2T f
o y2
¼
1
a f
o T f
o t
:ð4bÞ
These equations are put in dimensionless form by defining
T P¼T EÀT P
T EÀT B
;T f¼
T EÀT f
T EÀT B
and s¼tÁa P;s¼tÁa f
o2T P o x2þ
o2T P
o y2
¼
o T P
o s
;ð4cÞ
o2T f o x2þ
o2T f
o y2
¼
o T f
o s
:ð4dÞ
Initial and boundary conditions,at t¼0,the particle and liquid slug both are divided to give a mesh nÂn and all points within the particle and liquid slug are at the bulk temperature
T P¼T f¼1;T P¼T f¼T B:ð5aÞThe temperature at the face of the liquid slug at x¼0and 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 thefirst time step at x¼X L
at y¼0and x¼0;
T f¼1;T f¼T B;
T P¼T f and T P¼T fðAfter first time step at x¼X LÞ:ð5bÞAt t P0,the faces of both particle and liquid which are in contact with the surface are at all times at the surface temperature.
At y¼0;T P¼T f¼0;T P¼T f¼T E:ð5cÞEqs.(4c)and(4d)are solved simultaneously usingfinite difference techniques for afixed and for a variable boundary.For t>0thefixed boundary is ex-pressed by Eq.(5c)and the variable boundary is at
y¼0at x¼0
for liquid
T f¼T P and T f¼T PðAfter previous time step at x¼X SÞ
and for particle
A.R.Khan,A.Elkamel/put.129(2002)295–316307
T P¼T f and T P¼T fðAfter current time step at x¼X LÞ:ð5dÞThe 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
D s6
1
ðD XÞÀ2þðD YÞÀ2 h i:
Their difference equation is
Tði;jÞkþ1ÀTði;jÞk
D s ¼
TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞ
D X2
k
þ
Tði;jÀ1ÞÀ2Tði;jÞþTði;jþ1Þ
D Y2
k
:ð6aÞ
The above-mentioned method is very sensitive to the value of the operator k h which is given as
k hX¼a h
D t
D X2
and k hY¼a h
D t
D Y2
:
For the sake of simplicity equal increments are taken in the X-and Y-directions,i.e.D X¼D Y.An implicit method can make the equations inde-pendent of the operator value as well as of space and time increments.The difference equation is then
Tði;jÞkþ1ÀTði;jÞk
D s ¼
TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞ
D X2
þ
Tði;jÀ1ÞÀ2Tði;jÞþTði;jþ1Þ
D Y2
kþ1
:ð6bÞ
On rearranging one gets a pentadiagonal matrix
ÀTðiÀ1;jÞÀTðiþ1;jÞþ
1
k h
þ4
Tði;jÞÀTði;jÀ1ÞÀTði;jþ1Þ
kþ1
¼
1
k h
Tði;jÞk;
ð6cÞ
where
k h¼a h D t D X2
;
which can be solved by either the Gaussian elimination method or the Gauss-Seidel iterative method to givefive 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 D s=2.Thefirst Eq.(7a)is implicit 308 A.R.Khan,A.Elkamel/put.129(2002)295–316。