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Mathematical model for heat transfer mechanism for particulate system
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)
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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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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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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.
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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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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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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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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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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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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
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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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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
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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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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.
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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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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
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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. References [1] J.F. Richardson, M.N. Romani, K.J. Shakari, Heat transfer from immersed surfaces in liquid fluidized beds, Chem. Eng. Sci. 31 (1976) 619–624. [2] S.S. Zabrodsky, Hydrodynamics and Heat Transfer in Fluidized Beds, MIT Press, Cambridge, MA, 1966. [3] M. Leva, M. Grummer, M. Weintraub, M. Pollchik, Introduction to fluidization, Chem. Eng. Prog. 44 (1948) 511–520. [4] O. Levenspiel, J.S. Walton, Bed-wall heat transfer in fluidized systems, Chem. Eng. Symp. Series 50 (9) (1954) 1–13. [5] C.Y. Wen, M. Leva, A generalized dense-phase correlation, AIChE J. 2 (1956) 482–488. [6] J.F. Richardson, A.E. Mitson, Sedimentation and fluidization, Part III Heat transfer from a tube wall to a liquid-fluidized systems, Trans. Inst. Chem. Eng. 36 (1958) 270–282. [7] J.F. Richardson, J.W. Smith, Heat transfer to liquid fluidized systems and to suspensions of course particles in vertical transport, Tran. Inst. Chem. Eng 40 (1962) 13–22. [8] D.T. Wasan, M.S. Ahluwalia, Consecutive film and surface renewal mechanism for heat and mass transfer from a wall, Chem. Eng. Sci. 24 (1969) 1535–1542. [9] L. Wasmund, J.W. Smith, The mechanism of wall to fluid heat transfer in particulately fluidized beds, Can. J. Chem. Eng. 43 (1965) 246–251. [10] G. Tripathi, G.N. Pandy, R.L. Varma, Mechanism of heat transfer in liquid fluidized bed, Indian J. Technol. 9 (1971) 277–280. [11] W.E. Ranz, Friction and transfer coefficients for single particle and packed beds, Chem. Eng. Prog. 48 (1952) 247–253. [12] L. Wasmund, J.W. Smith, Wall to fluid heat transfer in liquid fluidized beds, Can. J. Chem. Eng. 45 (1967) 156–165. [13] W. Hamilton, F.M. Brea, Heat transfer in liquid fluidized beds with a concentric heater, Trans. Inst. Chem. Eng. 49 (1971) 196–203. [14] R.D. Patel, J.M. Simpson, Heat transfer in aggregative and particulate liquid fluidized beds, Chem. Eng. Sci. 32 (1977) 67–74. [15] E. Zahavi, Heat transfer in liquid fluidized beds, Int. J. Heat and Mass Transfer 14 (1971) 835–857.
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[16] H.S. Mickley, C.A. Trilling, Heat transfer characteristics of fluidized beds, Ind. Eng. Chem. 41 (1949) 1135–1147. [17] H.S. Mickley, D.F. Fairbanks, Mechanism of heat transfer to fluidized beds, AIChE J. 1 (1955) 374–384. [18] E.W. Schumann, V. Voss, Heat flow through granulated material, Fuel (London) 13 (1934) 249–256. [19] G.A. Henwood, Heat Transfer in a Gas Fluidized Bed, UK Atomic Energy Auth. Res. Group Report No. AERER 6027 (1969) 1–9. [20] N.M. Catipovic, G.N. Jovanovic, T.J. Fitzgerald, O. Levenspiel, A model for heat transfer to horizontal tubes immersed in fluidized bed of large particles, in: J.R. Grace, J.M. Matsen (Eds.), Fluidization, Plenum, New York, 1980. [21] E.G. Suarez, R.S. Figliola, D.R. Pitts, Instantaneous azimuthal heat transfer from a horizontal tube to a mixed particle size fluidized bed, ASME-AIChE, National Heat Transfer Conference, ASME Paper 83–93, 1993. [22] A.H. George, J.L. Smalley, Instrumented cylinder for the measurement of instantaneous local heat flux in high temperature fluidizes beds, Int. J. Heat and Mass Transfer 34 (1991) 3025– 3036. [23] J.S.M. Botterill, J.R. Williams, The mechanism of heat transfer to gas fluidized beds, Trans. Inst. Chem. Eng. 41 (1963) 217–230. [24] J.S.M. Botterill, Fluid-Bed Heat Transfer, Academic Press, New York, 1975. [25] R. Davies, Local heat transfer in liquid–solid fluidized beds, Ph.D. Thesis, University of Wales, UK, 1975. [26] J.D. Gabor, Wall to bed heat transfer in fluidized and packed beds, Chem. Eng. Prog. Symp. 68 (1970) 76–86. [27] N.I. Gelperin, V.G. Einstein, in: J.F. Davidson, D. Harrison (Eds.), Fluidization, Academic Press, London, 1971. [28] H. Martin, Fluid-bed heat exchangers A new model for particle convective energy transfer, Chem. Eng. Commun. 13 (1985) 1–16. [29] E.U. Schlunder, Heat transfer between packed agitated and fluidized beds and submerged surfaces, Chem. Eng. Commun. 9 (1981) 273–302. [30] J. Kestin, L.N. Persen, The transfer of heat across a turbulent boundary layer at very high Prandtl number, Int. J. Heat and Mass Transfer 5 (1962) 355–377. [31] R.D. Toomey, H.F. Johnstone, Heat transfer between beds of fluidized solids and the walls of the container, Chem. Eng. Progress Symp. Series 49 (5) (1953) 51–63. [32] N.B. Kondukov, A.N. Nornilaru, I.M. Shachko, A.A. Akhromenkor, A.S. Kruglov, An investigation of parameters of moving particles in a fluidized bed by a radioisotope method, Int. Chem. Eng. 4 (1964) 43–47. [33] D. Handley, Studies of the Motion of Particles in Liquid Fluidized Beds, Ph.D. Thesis, University of Leeds, UK, 1957. [34] C.R. Carlos, Solid Mixing in Fluidized Beds, Ph.D. Thesis, University of Wales, UK, 1967. [35] B.A.J. Latif, Variation of Particle Velocities and Concentrations in Liquid-solid Fluidized Beds, Ph.D. Thesis, University of Wales, UK, 1971. [36] R.S. Figliola, D.E. Beasley, A study of the particle convective contribution to heat transfer in gas fluidized beds, Chem. Eng. Sci. 48 (1993) 2901–2911. [37] S. Carnhan, H.A. Luther, H.J.O. Wilkes, Applied Numerical Methods, Wiley, New York, 1969. [38] A.P. Baskakov, The mechanism of heat transfer between fluidized bed and a surface, Int. Chem. Eng. 4 (1964) 320–324. [39] A.R. Khan, A.K. Juma, J.F. Richardson, Heat transfer from a plane surface to liquids and liquid-solid fluidized beds, Chem. Eng. Sci. 38 (1983) 2053–2066.
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)
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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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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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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.
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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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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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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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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
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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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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
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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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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.
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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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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
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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. References [1] J.F. Richardson, M.N. Romani, K.J. Shakari, Heat transfer from immersed surfaces in liquid fluidized beds, Chem. Eng. Sci. 31 (1976) 619–624. [2] S.S. Zabrodsky, Hydrodynamics and Heat Transfer in Fluidized Beds, MIT Press, Cambridge, MA, 1966. [3] M. Leva, M. Grummer, M. Weintraub, M. Pollchik, Introduction to fluidization, Chem. Eng. Prog. 44 (1948) 511–520. [4] O. Levenspiel, J.S. Walton, Bed-wall heat transfer in fluidized systems, Chem. Eng. Symp. Series 50 (9) (1954) 1–13. [5] C.Y. Wen, M. Leva, A generalized dense-phase correlation, AIChE J. 2 (1956) 482–488. [6] J.F. Richardson, A.E. Mitson, Sedimentation and fluidization, Part III Heat transfer from a tube wall to a liquid-fluidized systems, Trans. Inst. Chem. Eng. 36 (1958) 270–282. [7] J.F. Richardson, J.W. Smith, Heat transfer to liquid fluidized systems and to suspensions of course particles in vertical transport, Tran. Inst. Chem. Eng 40 (1962) 13–22. [8] D.T. Wasan, M.S. Ahluwalia, Consecutive film and surface renewal mechanism for heat and mass transfer from a wall, Chem. Eng. Sci. 24 (1969) 1535–1542. [9] L. Wasmund, J.W. Smith, The mechanism of wall to fluid heat transfer in particulately fluidized beds, Can. J. Chem. Eng. 43 (1965) 246–251. [10] G. Tripathi, G.N. Pandy, R.L. Varma, Mechanism of heat transfer in liquid fluidized bed, Indian J. Technol. 9 (1971) 277–280. [11] W.E. Ranz, Friction and transfer coefficients for single particle and packed beds, Chem. Eng. Prog. 48 (1952) 247–253. [12] L. Wasmund, J.W. Smith, Wall to fluid heat transfer in liquid fluidized beds, Can. J. Chem. Eng. 45 (1967) 156–165. [13] W. Hamilton, F.M. Brea, Heat transfer in liquid fluidized beds with a concentric heater, Trans. Inst. Chem. Eng. 49 (1971) 196–203. [14] R.D. Patel, J.M. Simpson, Heat transfer in aggregative and particulate liquid fluidized beds, Chem. Eng. Sci. 32 (1977) 67–74. [15] E. Zahavi, Heat transfer in liquid fluidized beds, Int. J. Heat and Mass Transfer 14 (1971) 835–857.
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[16] H.S. Mickley, C.A. Trilling, Heat transfer characteristics of fluidized beds, Ind. Eng. Chem. 41 (1949) 1135–1147. [17] H.S. Mickley, D.F. Fairbanks, Mechanism of heat transfer to fluidized beds, AIChE J. 1 (1955) 374–384. [18] E.W. Schumann, V. Voss, Heat flow through granulated material, Fuel (London) 13 (1934) 249–256. [19] G.A. Henwood, Heat Transfer in a Gas Fluidized Bed, UK Atomic Energy Auth. Res. Group Report No. AERER 6027 (1969) 1–9. [20] N.M. Catipovic, G.N. Jovanovic, T.J. Fitzgerald, O. Levenspiel, A model for heat transfer to horizontal tubes immersed in fluidized bed of large particles, in: J.R. Grace, J.M. Matsen (Eds.), Fluidization, Plenum, New York, 1980. [21] E.G. Suarez, R.S. Figliola, D.R. Pitts, Instantaneous azimuthal heat transfer from a horizontal tube to a mixed particle size fluidized bed, ASME-AIChE, National Heat Transfer Conference, ASME Paper 83–93, 1993. [22] A.H. George, J.L. Smalley, Instrumented cylinder for the measurement of instantaneous local heat flux in high temperature fluidizes beds, Int. J. Heat and Mass Transfer 34 (1991) 3025– 3036. [23] J.S.M. Botterill, J.R. Williams, The mechanism of heat transfer to gas fluidized beds, Trans. Inst. Chem. Eng. 41 (1963) 217–230. [24] J.S.M. Botterill, Fluid-Bed Heat Transfer, Academic Press, New York, 1975. [25] R. Davies, Local heat transfer in liquid–solid fluidized beds, Ph.D. Thesis, University of Wales, UK, 1975. [26] J.D. Gabor, Wall to bed heat transfer in fluidized and packed beds, Chem. Eng. Prog. Symp. 68 (1970) 76–86. [27] N.I. Gelperin, V.G. Einstein, in: J.F. Davidson, D. Harrison (Eds.), Fluidization, Academic Press, London, 1971. [28] H. Martin, Fluid-bed heat exchangers A new model for particle convective energy transfer, Chem. Eng. Commun. 13 (1985) 1–16. [29] E.U. Schlunder, Heat transfer between packed agitated and fluidized beds and submerged surfaces, Chem. Eng. Commun. 9 (1981) 273–302. [30] J. Kestin, L.N. Persen, The transfer of heat across a turbulent boundary layer at very high Prandtl number, Int. J. Heat and Mass Transfer 5 (1962) 355–377. [31] R.D. Toomey, H.F. Johnstone, Heat transfer between beds of fluidized solids and the walls of the container, Chem. Eng. Progress Symp. Series 49 (5) (1953) 51–63. [32] N.B. Kondukov, A.N. Nornilaru, I.M. Shachko, A.A. Akhromenkor, A.S. Kruglov, An investigation of parameters of moving particles in a fluidized bed by a radioisotope method, Int. Chem. Eng. 4 (1964) 43–47. [33] D. Handley, Studies of the Motion of Particles in Liquid Fluidized Beds, Ph.D. Thesis, University of Leeds, UK, 1957. [34] C.R. Carlos, Solid Mixing in Fluidized Beds, Ph.D. Thesis, University of Wales, UK, 1967. [35] B.A.J. Latif, Variation of Particle Velocities and Concentrations in Liquid-solid Fluidized Beds, Ph.D. Thesis, University of Wales, UK, 1971. [36] R.S. Figliola, D.E. Beasley, A study of the particle convective contribution to heat transfer in gas fluidized beds, Chem. Eng. Sci. 48 (1993) 2901–2911. [37] S. Carnhan, H.A. Luther, H.J.O. Wilkes, Applied Numerical Methods, Wiley, New York, 1969. [38] A.P. Baskakov, The mechanism of heat transfer between fluidized bed and a surface, Int. Chem. Eng. 4 (1964) 320–324. [39] A.R. Khan, A.K. Juma, J.F. Richardson, Heat transfer from a plane surface to liquids and liquid-solid fluidized beds, Chem. Eng. Sci. 38 (1983) 2053–2066.