A study of the particle convection contribution to heat transfer in gas fluidized beds

Chemical Enginee+fng S&me, Printed in Great Britain.

OCOS-X09/93 $6.00 + 0.M) Q 1993 Pcrgamon Press Ltd

Vol. 48, No- 16, PP. 2901 2911, 1993.

A STUDY OF THE PARTICLE CONVECTION CONTRIBUTION TO HEAT TRANSFER IN GAS FLUIDIZED BEDS R. S. FIGLIOLA Thermal/Fluid

Sciences Research Laboratory,

and D. E. BEASLEY Clemson University, Clemson, SC 29634-0921,

(First received 5 June 1992; accepted

in revised form 20 December

U.S.A

1992)

Abstract-A model for the particle convection heat transfer component of the overall heat transfer between a submerged surface and a bubbling fluid&d bed is presented. A convective boundary condition combined with surface coverage concepts are introduced to describe the instantaneous wall-to-bed heat transfer. Both thermally lumped and distributed formulations with constant and variable properties for the model are presented and studied. As formulated, the model is not restricted to uniform bed particle size distributions. Experimental measurementsobtained from instantaneouslocal heat flux probes are compared to the model medictions. The thermally distributed,variable nrooerty model formulation movides very good agreement with the experimental dais over a wide range 0-fp&&e sizes and fluidizatibnvelocities, For particle size beds ranging from 256 to 4000~, the model displayed a very accurate trend prediction relative to available data, with magnitude predictions of 2-28% error, dependingupon particle size over a wide range of fluidizationvelocity.-

INTRODUCTTON

The heat transfer to a submerged surface in a bubbling fluidized bed is a result of energy transport to the particles and to the fluidizing gas. Most models of heat transfer in gas fluidized beds separate the heat transfer into the bubble phase and emulsion phase contributions. At high temperatures, radiation may play a significant role in the heat transfer process. The heat transfer to the emulsion phase can further be divided into the particle convection and gas convection contributions. Particle convection is a term typically used to refer to the transport of energy from the heat transfer surface to the particle by conduction through the contact point and through the fluidizing gas in the vicinity of the contact point. The purpose of the present study is to incorporate observations of the behavior of heat transfer during emulsion phase contact in beds composed of a variety of particle sizes into physical models of the heat transfer to a submerged surface. Many models for heat transfer from immersed surfaces within a bubbling fluidized bed have been proposed, these models begin from a variety of physical bases and may contain empirically determined parameters not based directly on the physics of the transport process. These bases include transient conduction to single particles, particle strings, or packets having uniform effective thermal properties. Only those models having specific relevance to this work are discussed. The venerable packet model proposed by Mickley and Fairbanks (1955) assumed that the process could be modeled as a suddenly imposed step change in surface temperature for a semi-infinite continuum which models the emulsion. Upon contact with the heat transfer surface, the interface of the emulsion is assumed to immediately attain the temperature of the

surface. The model implies a steadily decreasing heat transfer rate from the surface to the emulsion phase as the temperature gradient at the interface in the continuum decreases. Because predicted values of the heat transfer coefficient are significantly too large when compared to experimental measurements, especially in beds of small particles or when contact times are short, Baskakov (1964) introduced the notion of a gas film which is conjectured to separate the emulsion phase from the heat transfer surface. This modification to the packet model provides additional resistance to heat transfer at the interface and a finite limiting value for the heat transfer coefficient at small contact times. While others have since improved upon these two ideas to generate models which are capable of predicting heat transfer rates within specific beds [see Xavier and Davidson (1985)], the resulting correlations have not permitted general predictions of heat transfer rates. Often the scale of the laboratory experiments influences strongly the empirical input to these models. Attempts have been made to model the heat transfer between a surface and a single particle, and to extend the resulting predictions of heat transfer rate to the emulsion phase components. The single particle layer approach yields significant errors iti the Reynolds number range where transient conduction effects dominate. Gabor (1970) modeled the heat transfer by examining a string of particles in contact with the surface and having an orthorhombic array. The boundary condition at the surface-emulsion interface imposed a step change in temperature to the surface and also included a gas film at the interface. Decker and Glicksman (1983) proposed a model of heat transfer for large particle fluidization based on the assumption that the thermal time constant of the particles was large compared to their residence time in contact with the surface. They concluded that the

2902

R. S.

FIGLIOLA

and D. E. B~ASLEY

particles in contact with the heat transfer surface remain at the bulk bed temperature and that the frequency of renewal does not influence the heat transfer. However, Adams (1982), using the previous work of Adams and Welty (1979), examined the heat transfer to individual particles and the gas flow around these particles and concludes that significant cooling of the particle surface did oocur near the contact point, even for large particles. Martin (1981, 1982) treated the problem by assuming that the particle convection could be described using the ideas of accommodation as used in describing heat transfer between a surface and a rarefied gas. The heat transfer boundary condition for a single particle is described using the results of Schlunder (1980, 1981) for the heat transfer to a particle in contact with a surface in the limit of zero contact time. Schlunder’s concept provides a physical basis for the existence of a limiting value of heat transfer rate as contact time approaches zero, an important physical limit for heat transfer in a fluidized bed. Experimental studies undertaken to quantify the gas film layer correction to the unsteady conduction models have been performed using both direct methods in bubbling beds [e.g. Baskakov (1964), Catipovic et al. (1982), Suarez et al. (1983)] and indirect methods in packed beds or beds at incipient fluidixation [e.g. Gloski et al. (1984)]. One result suggested by such studies is that the particles do

make physical contact with the heat transfer surface (Baskakov, 1964; Catipovic et al., 1982; Suarez et al., 1983; Gloski et al., 1984) and that the resulting heat transfer takes place mainly through the fluid to the particle, with the maximum rate occurring in the vicinity of the contact point. The hydrodynamic phenomena of fluidixation near a submerged surface. are complex and difficult to identify. The fluidization characteristics around a submerged cylinder in the bubbling regime for Geldart’s group D particles are shown in Fig. 1. Bubble activity originates at the stagnation point, causing significant stirring of the particles in this region. Flow around the cylinder also creates particle mixing and particle motion along the surface. Visual studies [e.g. Suarez (1983), Kurosaki et al. (1988)] indicate that often the particles are in motion at the surface and there may be significant mixing between surface particles and particles in the bulk region of the bed. At other times the particles contacting the surface are essentially still, and heat transfer occurs by conduction to this “packet” of particles. On the upper portion of the cylinder, the fluidization character changes with particles either sliding or nearly stagnant, depending upon operating conditions, and heat transfer is primarily by conduction through the composite medium. In light of the variations of the hydrodynamic phenomena around a submerged cylinder, it is the purpose of the present paper to examine the heat transfer

STAGNANT

CAP

PARTICLE

SLIDING

FLOW SEPARATION

I

GAS FLOW

Fig. 1. Model of the fluidization cdaracteristks about a submerged cylinder in the bubbling regime.

2903

Heat transfer in gas fluidixed beds

in the region of the submerged surface where bubble action and gas flow create significant particle replacement, or surface renewal. The physical occurrences in the emulsion phase will be examined, and a model which incorporates the primary physics of the heat transfer process proposed. We will show that models of the heat transfer to the emulsion phase based on different physical assumptions can result in similar forms of the equation for the heat transfer coefficient. As such, the fundamental physical assumptions are not unique to a given model result and, as a consequence, it may not be possible to judge the appropriateness of the underlying physics by comparing with limited experimental data. MODELING

A model is proposed here which employs a convective boundary condition at the interface between the heat transfer surface and the emulsion phase. The heat transfer coefficient between the heat transfer surface and the emulsion phase is defined such that 9” = h&CT,

- T(O, t)7

process. These approaches, listed in Table 1, address models to evaluate the properties of the emulsion phase. Each of the three resulting formulations will incorporate various physical assumptions, and predictions of the heat transfer rates will be compared to measured data. Lumped thermal capacitance formulation Consider first a thermally lumped system of depth L from the heat transfer surface. The description of the heat transfer from the surface to the lumped thermal system must be based on a physical assumption. Consider the heat transfer to a spherical particle in contact with a heat transfer surface, which has been analyzed by Schlunder (1981) for zero contact time and expressed as +$)-

a=26

z-1

(Y

.

Since eq. (2) suggests a limiting value of the average heat transfer coefficient over a vanishingly small contact time, the physical description of the heat transfer from the surface to the first layer of particles will bc described by a convective boundary condition applied over the portion of the surface covered by the particle projected area. The fraction of the heat transfer surface covered by the projected area of the particles in contact with the surface is +. This projected area is the same as was used by Schlunder in deriving the heat transfer coefficient in eq. (2). Considering the system

Physics

Assumptions

Finite volume of emulsion phasehaving a uniform temperatureparticipates in

1. Uniform thermophysical properties in emulsion phase. 2. One-dimensional heat transfer normal to surface. 3. Lumped thermal capacitance with uniform temperature.

Distributed temperature constant property formulation

Semi-infinite emulsion phase modeled with spatially varying temperature and a convective boundary condition at the heat transfer surface.

1. Uniform thermophysical properties in emulsion. 2. Temperature varies in one spatial dimension normal to the heat transfer surface. 3. Emulsion phase is semi-infinite in extent.

Variable property distributed temperature formulation

Semi-infinite emulsion phase

1. Void fraction and thermophysical properties vary in the spatial coordinate normal to the heat transfer surface. 2. Temperature varies in one spatial dimension normal to the heat transfer surface. 3. Emulsion phase in semi-infinite in extent.

Lumped thermal capacitance formulation

(2)

>

Table 1, Model formulation for emulsion phase Model

I]

where

(1)

where JI is the fraction of the heat transfer surface covered by the projected area of the particles. Clearly, a difficulty with this approach is the specification of the heat transfer coefficient, h,,, and its physical interpretation. This model yields a physically realistic behavior at zero time, since the heat transfer coefficient provides a finite rate of heat transfer at zero time, unlike the packet model which predicts infinite heat transfer rates at zero time. Three related ap proaches to develop this model for the emulsion phase will be derived for the description of the heat transfer

+$)ln(I

wP --:4[(1

h”“”

heat transfer with submerged surface through a convection boundary condition at the interface.

having spatiallyvarying thermophysicalproperties and void fraction.

R. S. FIGLIOLA and D. E. BEASLEY

2904

to have a uniform temperature and constant thermophysical properties, the heat flux may be expressed as, neglecting the energy capacity of the gas phase, 4” = ppc,L(l

- 6);

= h,,@(B,).

(3)

This formulation results in an average heat transfer coefficient over a contact time, t,, given by

The convection heat transfer coefficient is assumed to be determined by the wall-to-particle heat transfer coefficient to a single particle as derived by Schlunder (1980). Thus, if the single particle value is multiplied by the fraction of the surface covered by the projected area of the particles, a convective boundary condition again results. The solution of the one-dimensional transient conduction pi-oblem for a semi-infinite medium can be expressed as h = h,,JI exp (a, 5’) erfc (i; ,,&).

Some method is required to specify L/t,, ideally as a function of overall bed parameters. Martin (1981, 1982) introduces a random particle contact at the heat transfer surface, an idea which is fundamentally different from the classical packet theory approach. Martin modeled L/t, as a random velocity, which he expressed as a function of particle size and bed void fraction, and modeled surface coverage as If5= (1 - E). The result is that eq. (4) can be reduced to a form which is identical to that of Martin’s model. It is interesting that the lumped thermal system approach developed herein and the kinetic theory approach proposed by Martin do not result in different forms of predictive equations for the heat transfer coefficient despite their different bases. As such, it is not possible to evaluate the accuracy of the physics underlying the final form of eq. (4). However, the lumped thermal capacitance model reasonably represents any hydrodynamic and thermal behavior which results in a well mixed packet of particles which contact the heat transfer surface, as can be shown by comparison with measured values. Constant property distributed temperatureformulation If the lumped thermal capacitance system approach is replaced with a semi-infinite composite medium having uniform average thermophysical properties which models the emulsion phase, the resulting model becomes equivalent to the packet mddel in terms of its composition and renewal concept, but with a convective boundary condition at the wall-emulsion interface. For an analysis to be valid for mixed and large particle systems, the effects of nonuniform temperature in the emulsion phase must be determined and included. Consider a semi-infinite composite medium having effective thermophysical properties. The governing equation for the temperature distribution as a function of distance from the heat transfer surface, x, is

with boundary conditions

(6)

This model requires information regarding the void fraction and the surface coverage factor, preferably as general expressions of the velocity ratio and the minimum fluidization velocity. The surface coverage factor is based on reasonable physical behavior, and provides a mechanism for modeling the effects of particle size distribution. Optical techniques, such as those proposed by Kurosaki et al. (1988), permit observation of particle contact and motion on a submerged surface within a fluidized bed. The proposed surface coverage factor could be determined from such observations. Models required for the effective thermal conductivity and specific heat of the emulsion phase will be discussed in the following section. Variable property formulation Particle size and shape distributions can affect heat transfer through changes in the local void fraction and surface coverage factor. The presence of a constraining surface in a packed or fluidized bed creates variations in the void fraction so that the local void fraction near a heat transfer surface varies with distance from the surface. The difference between the radius of curvature of the constraining surface and the particle causes the void fraction to be 1.0 at the surface. Bed void fraction displays a damped oscillatory variation and the minimum void fraction is shifted from the constraining surface for a completely fluidized bed (Korolev et al., 1971). As such, the effective thermophysical properties of the emulsion phase are not constant, but rather vary with distance from the heat transfer surface. The governing equations are suitably modified from those stated for the constant property model to account for variable properties. The effective thermophysical properties of the emulsion phase are a function of the void fraction (Jaguaribe and Beasley. 1984). The variations of void fraction at the heat transfer surface affect both the temperature distribution and the heat transfer rate from the surface. The method proposed by Kubie and Broughton (1975) to account for the wall effect on void fraction is used here: for I i 1: E(Z) = 1 - 3(1 - E*)(z - 2z2/3)

(7)

for z > 1: E(Z)

=

Eb

03)

Heat transferin g&sfluid&d beds

2905

where

x 2 = x/d

4; -

The bulk mean void fraction, Ed, varies with the superficial gas velocity, particle diameter and the regime of fluidization. The present model uses the correlation of Geldart (1972) for the minimum fluidization velocity, U m/

=

8

I+l

i-l

I

.

i AX

SG4’

-Q’;+

.

.

i+l _

AX

AX

x 10-4@, - pJp.

The correlation for bulk mean void fraction used by Grewal and Saxena (1980) to estimate the heat transfer between a horizontal tube and a gas fluidized bed is used here as

Fig. 2. Formulation for the governing equations and boundary conditions of the thermally distributed modeI. tion is - [IQ] T:-+I’ + [k, + ICI+, + p,c, Ax*/At]Ti”+ ’

E* =

jgo.4 +[4(d;42p;;p _p)gJ”4’]o~33]. (10)

The thermophysical properties are taken as functions of local void faction. The effective volumetric emulsion heat capacity (c.) is described based on the masses of solid and fluid in a given volume as

ce=

ppcp(l - 4 + PrC/& &,(l - E) + p+

(11)

The general problem of determining the effective thermal conductivity of a porous matrix has been addressed by numerous authors; the present study adopts the model developed by Kunii and Smith (1960). Finally, in order to actually predict the heat transfer rate from a submerged surface, it is necessary to either predict or to measure the emulsion phase contact time, or the bubble fraction. In cases where the contact time is calculated, the present study uses an expression developed by Baskakov et al. (1973):

This expression provides reasonable agreement with horizontal cylinder heat transfer experiments (Catipovic et al., 1982; Anewke, 1982; Suarez, 1983) for Geldart class B and D particles. Method of solution The constant property model is important because an analytical solution exists for the case represented in eq. (6). In this work it was used as a means of verifying the computational solution of the variable property model id>the limit of constant properties. The solution for the distributed temperature, variable property model was formulated numerically based on the methods described by Spalding and Patankar (1970). The solution is accomplished using a fully implicit finite-difference scheme; the formulation results from the control volume shown in Fig. 2. The general equation for the finite-difference formula-

[k,+,]T,“+f,’ = [PiciAx2/At]T:.

(13)

As illustrated in Fig. 2, the Roman letters indicate control volumes and the alpha-numeric indicate nodes. The effective thermal conductivities are calculated at the interface of two adjacent control volumes and the effective densities and heat capacities are calculated at the nodes. The temperature solution described above is employed to evaluate the time-averaged heat transfer coefficient. The result for h,,, is n-1 h,+(At) 0.5 T: + 2 2T:’ + T; - T, j=2 [ ( > h

IYe =

6% - T&c

1

(14)

where T{ is the temperature at ,x = 0 and the jth time step and At is the computational time step. EXPERIMENTS

In order to investigate models and model compon-

ents for fluidized bed heat transfer, an atmospheric fluidized bed facility was developed and experiments performed. The fluidizing vessel consisted of a 200 cm high, 30 by 30cm cross section bed. The steel bed frame used a distributor plate made of 2 cm thick porous polyethylene board which produced a sufficient pressure drop for uniform flow. Nuidizing air was supplied by a compressor and metered with an in-line laminar Bow element. Bed temperatures were measured using thermocouples located symmetrically above and below a single-tube heat-exchanger surface. The nominal steady-state bed temperature during the experiments was 303 K. For these experiments, the mean particle size was varied but the type and total quantity of the fluidized material was kept constant. The bed was charged with 40.8 kg of spheroidal glass beads. This resulted in an average slumped bed height of 31 cm. Mixtures were prepared from two base mixtures, each containing a very narrow particle size distribution of 568 and 256 pm mean particle size, respectively. Three mixtures, the two base mixtures and a SO-50% by weight mixture, were used in separate heat transfer tests. The

R. S. FIGLIOLA

2906

and D. E. BEASLEY

mean (volume to surface area weighted) particle size for each mixture was found using standard ASTM sieve analysis procedures and using screen sizes 20 to 100. Heat transfer measurements were made from a single 5 cm diameter, 25.4 cm long solid cylinder which was situated along a horizontal plane 15.9 cm above the distributor plate. The cylinder was maintained at a constant, isothermal temperature 35 K above that of the surrounding bed by using an internal resistance heater. The cylinder end mounts were designed such that it could be rotated through 360” while the bed was in operation. The overall average, the local azimuthally dependent average and instantaneous local heat transfer rates were measured. The effect of mixed particle sizes based on the overall and azimuthally dependent average heat transfer measurements has been discussed by Figliola et al. (1986). Here we are concerned with the instantaneous measurement information which was obtained from the output of a custom designed platinum film heat flux probe. The heat flux probe was fabricated by vacuum deposition of a platinum film onto a Pyrex substrate and coated with a 9 pm thick layer of aluminum oxide. The design of the instantaneous heat flux sensor has been analyzed and described by Beasley and Figliola (1988) and was constructed to maintain as uniform and constant a surface thermal boundary condition as possible. A metallic sensor has a nearly linear temperature-resistance relation which was exploited here to determine and to control the sensor temperature through its equivalence with resistance. The probe, which was flush mounted onto the cylinder surface, was connected as one arm of an external constant resistance bridge circuit, as shown in Fig. 3. The probe film temperature was set at the cylinder temperature, plus a 0.25 K offset necessary to achieve circuit stability, and maintained. The surrounding cylinder served as a guard heater for the probe. The

tlrtatlic SCIISOP

Fig. 3. Schematic heat flux probe:

diagram

of

the

instantaneous

local

(a) constant resistance control circuit; (h) mounted probe.

probe-control circuit 90% response time was determined adequate to follow the heat flux variations on the cylinder surface with amplitude fidelity. The instantaneous heat transfer from the probe was determined ,from the equation h(t) =

&ub Tp +&essde h/Rd2 Ii

E(t)2

Rdecade

-

Qk

(15)

where E(t) is the bridge applied voltage and 0, is the conduction loss to the cylinder. The conduction loss was a bias which was determined from the output of thermocouples located along the probe-cylinder interface. The integrated spatial and temporal heat transfer data taken from the instantaneous probe agreed with the overall cylinder heat transfer data within + 7%. More specific calibration, conduction loss correction methods, benchmark tests and operating theory have been given by Suarez (1983) and Beasley and Figliola (1988). The probe output was digitized and stored using a microcomputer based data acquisition system. Additional information regarding overall and azimuthal heat transfer and bed operating conditions were also recorded. The recorded data were reduced to generate plots of heat transfer coefficient vs time. These plots were used to obtain information regarding the emulsion phase and the bubble phase contributions to overall heat transfer, as well as the average emulsion contact times. RESULTS AN0 Experiments

were

DISCUSSION

performed

in

a bubbling

bed

at velocity ratios (u/u,,,,) ranging from 0.86 to 3.6. The upper limit on u/u,,,~was restrained in order to limit bubble size (Bar-Cohen et al., 1981) so as to minimize intrinsic wall effects on the test data. Representative photographs of the instantaneous heat transfer coefficient vs time are shown in Fig. 4 for the three particle sizes tested at two azimuthal positions of 0 (stagnation point) and 90” and at u/u,,,~% 2. Several observations can be made. The heat transfer coefficient increases upon contact with the emulsion phase. In many cases, this emulsion phase component fluctuates due to lateral mixing of particles at the bed temperature, particle motion at the heat transfer surface, and a changing local emulsion void fraction near the surface. Often it was observed to continuously fluctuate about some average maximum value until the emulsion phase was displaced by the void phase. Such a situation is depicted in Fig. 5 and indicated by notation (a). In other cases, the heat transfer coefficient decreased in time, as would result from a transient diffusion of energy into the emulsion phase as depicted in Fig. 5 by notation (b). At the time of displacement of the emulsion phase, the coefficient decreases towards a minimum value. This decrease is rapid, but not abrupt, as the thermal boundary layer regime

Heat transferin gas fluidized

beds

2907

600 500 T.T 4

400

3

300

2 2

200 100 n

(a)

T 00

1.25

1.&o

2.bo

(W

1.00

1 25

t (s)

600,

1.50

1.75

2.00

1.75

2 00

1.75

2.00

t (s)

600

I

i.00

Cd)

1.25

1.50 t (s)

3

7

\53

300 200

E 100

w

0

Loo

1.25

1.50 t (s)

Fig. 4. Instantaneousheat transfercoefficientvs time at U/Q near 2: (a) 256pm, 0”; (b) (c) 340 ,um, 0”; (d) 340 pm, 90”; (e) 568 q,

in the void phase is established and the steady-state minimum value of heat transfer is reached. This process has typically been observed to take on the order of SO-100 ms, which is in agreement with a thermal boundary layer development analysis by Chao and Jeng (1965). Recently, Kurosaki et al. (1988) visually monitored the characteristics of particle motion at a submerged horizontal cylinder in a gas fluidized bed. The local behavior of the particles was either no motion at the surface, sliding along the surface, or particle mixing with the bulk region of the bed. The behavior of the particles at the surface of the cylinder did not relate directly to the fluidization state in the entire bed. The CES48:16-F

256 pm, 90”;

0”; (f) 568 pm, 90”.

particle movement characteristics also varied with angular position on the surface of the cylinder. These observations support the heat transfer characteristics measured in the present study. Clearly, the particle behavior at a heat transfer surface will not be completely specified for all states of fluidization by any single physical mechanism. Experimental data for the emulsion phase contribution to heat transfer were obtained at O-180” at 45” increments. The emulsion and void phase contributions to total heat transfer were quantified using the area integration scheme depicted graphically in Fig. 5. The resulting values were assumed to be symmetric about the cylinder and combined to obtain a mean

R. S. FIGLIOLA and D. E. B~ASLEY

2908

+== e -

MODEL PREDUTIDNS

500-

EXPEF(NENTAL DATA

*

450

-

L-

4Co-

Y

350

.

PmDE 2 250

**W/4

.

; DlsTRlBlJTED

: .

rm-

0.5

::

Time (WC) Fig. 5. Reproduction of a typical instantaneous heat flux signs1 with the interpretation scheme used: (a) signal suggests particle motion at surface; (b) signal suggests unsteady conduction to emulsion phase.

value for emulsion

phase

heat transfer

coefficient,

1.0

1.5

2.0

30

2.5

35

4.0

“‘“In1

Fig. 6. Experimentally determined and predicted values for particle convection and gas conduction components of surface to bed heat transfer for 256 pm particles.

h,,

by d,*568 -

,m MOOEL PREMCTIDNS

.

EXPERMBTAL

DATA

It was not possible to separate between the contributions of the particle convection and the gas convection to measured emulsion phase heat transfer with the probe used. Instead, the particle convection component was calculated from the emulsion phase value by subtracting the predicted value of gas convection component using the method of Baskakov (1973):

1.50 100

-

50

h

e=

=

k I)@)9 .

0

_.&-0.’ pp33.

4

(16)

were used for direct comparison with theparticle convection model predictions. Overall, particle convection contributed over 90% of the measured emulsion phase heat transfer with 256 and 340,um particles, and 80% for the 568 p particles. The lumped thermal capacitance model was used to predict particle convection for each of the three particle sizes experimentally tested. A coverage factor of @ = (1 - a*) was assumed. Equation (12) was used to estimate the contact times and eq. (10) for the void fraction for this model. The thermally distributed model was tested for both the constant and variable property cases. The value for the contact time was evaluated from eq. (12) and the void fraction from eq. (10). Only values for the variable property case will be reported. It was confirmed that the variable property predictions were in complete agreement with the constant property approach when properties were held constant. We report on the predicted results based on two representative models for surface coverage: a constant surface coverage value of I,+= x/4 and a variable surface coverage of $ = 1 - Q,. The former value represents The values

obtained

by these methods

a limiting condition and refers to a cubic packing of uniform sized spheres. The latter value models surface

coverage as a function of bed fluidizing conditions. The results of the thermally lumped model are compared to the experimental data with uniform par-

0 1.0

s 1.5

2.0

2.5

3.0

35

I 4.0

“/“mar

Fig. 7, Experimentally determined and predicted values for particle convection and gas conduction components of surface to bed heat transfer for 568 nm particles.

title sizes in Figs 6 and 7. This formulation tends to overpredict the particle convection heat transfer within a total rms error of 29% for the 256 pm and 9% for the 568 pm particles. However, in both cases the error is essentially all bias, with the largest errors occurring at U/Q at just above 1. Nonetheless, the predictions and experimental data are in reasonable agreement in both magnitude and trend as fluidization velocity varies over the range studied. The thermally lumped model describes a physics associated with limited particle mixing at the heat transfer surface, over a depth related to the parameter L. This model describes clearly and correctly some of the physical aspects of the transport near the surface. The results of the variable property thermally distributed model are compared with the experimental data for 256 pm particles in Fig. 6. For ti = x/4, agreement is very good in both magnitude and trend to within a 11% rms error, the largest contribution to this error occuring below p/p,,,/= 2. In Fig. 7, the model results compare even better with the 568 pm particle data to within a 2% rrns error. For $ = 1 - s,,,the errors for the two datasets increase to 18 and 12%, respectively. It is seen that the major

Heat transferin gas fluid&d beds

2909 dp=2

mm

-MooEL0

01 1.0

m

I.5

’

2.0

-

’

25

’

a

30

’

’

3.5

’

’

40

face to bed heat transferfor a mixtureof 34Opm mean size particles. component for error is a bias with exceptional agreement in trend. This formulation of the model is an improvement on the thermally lumped formulation for beds containing a uniform particle size. However, the thermally lumped model does not show good agreement with the 340 pm binary mixture data in Fig. 8. In general, for 340 pm, the measured particle convection contribution is lower than that predicted by the model within an rms error of 39%. The thermally distributed formulation compares with this experimental data to within a 22% error for + = x/4 and a 35% error for $ = 1 - Ed_As an explanation, consider Fig. 4. It is apparent that the binary sized bed fluidization characteristics are somewhat different from the uniform beds. In particular, a relative reduction in void phase frequency is noted and this is not accounted for in the calculation of the contact time, void fraction, or surface coverage values within either modeling formulation. It has been suggested in earlier works (Golan et al., 1969; Figliola et al., 1986) that particle mixtures suppress bed expansion so as to increase the overall heat transfer and affect the particle convection component, a notion which is supported by a comparison of the test results of these three bed compositions. The overall heat transfer coefficient within the binary bed was of the order of a bed composed of all 256 pm particles. For binary beds with a size ratio of OS (as is the 340 pm sized blend), these previous studies indicate that the augmentation to the overall heat transfer should be most pronounced with 50-50% mixture blends. As such, the 340 pm binary mixture data should represent a worst-case comparison between the data and the model. In an effort to evaluate the model over a larger range of particle sizes, the two-dimensional fluid&d bed experimental data of Catipovic et al. (1982) were used. Catipovic developed a film Sensor for use in his two-dimensional bed of large dolomite particles. Although the film Sensor and its operation differed considerably from the one used in this study, it did provide an indication of the temporal variation in heat

et

(II.. 1962

Fit, , , , , . , 1.0

1.5

U/U&

Fig. 8. Experimentally determined and predicted values for particle convection and gas conductioncomponentsof sur-

cATPovK

2.0

2.5

3.0

3.5

4.0

uwnf

Fig. 9. Thermally distributed model prediction compared with the 2 pm particle data from Reference 15.

U/U&

Fig. 10. Thermally distributedmodel prediction compared with the 4 w

particle data from Reference 1.5.

transfer at the surface of a submerged cylindrical surface. Information regarding the particle convection component of the total heat transfer was extracted in the manner discussed previously. Particle convection was found to account for about 60 and 40% of the total heat transfer measured in the 2000 and 4000 Frn uniform particle size beds. Comparisons between the variable property thermally distributed model and the data of Catipovic et al. (1982) are given in Figs 9 and 10. The model tends to overpredict the measured data with about a 25% rms bias error for either particle size using I& = n/4 and 28% bias error with $ = 1 - Q,. In this comparison, it is important to note that the effect of the narrow two-dimensional bed on particle dynamics, particularly on the suppression of lateral mixing, cannot be evaluated quantitatively. Mixing would be. encouraged by a wider bed, thereby increasing the particle convection heat transfer as suggested by the model. However, the agreement with the data trend remains excellent as it was shown to be for all the uniform particle size beds, a range covering more than an order of magnitude in particle sizes and a wide range of fluidization velocities.

R. S. FIGLIOLA

2910 CONCLUSIONS

A model for the particle convective heat transfer component to overall heat transfer has been proposed. In particular, a convective boundary condition at the particle-heat transfer surface interface is applied and studied. A convective boundary condition combined with surface coverage concepts was introduced to describe the instantaneous wall-to-bed heat exchange. Both thermally lumped and distributed approaches were applied to the model formulation using constant and variable properties. An instantaneous heat transfer probe was used to

measure experimentally the emulsion phase heat transfer between a fluidized bed and a submerged surface. Local, azimuthally dependent information was obtained and analyzed for comparison with the model. In general, the thermally distributed, variable property model formulation provided the best agreement both with the measured data and with data available in the literature. For particle beds of between 256 and 4000 e, the model gave an extremely accurate trend prediction and an acceptable magnitude prediction of between 2 and 28% error, depending upon particle size, over a wide range of fluidization velocities. Comparisons of the physics underlying several models for particle convection heat transfer demonstrate the ambiguity which results when the predictions are averaged over the emulsion phase contact time, and the results compared to limited experimental data. The instantaneous heat transfer data suggest that the particle behavior at a heat transfer surfq~ cannot be completely specified by any single physical action.

NOTATION

9 G h h a*s h VP h-” WlJ k k, L d’ t tc

D.

and

effective volumetric specific heat fluid specific heat particle specific heat parameter in equation for L/t, average particle diameter cylinder diameter or characteristic length for the heat transfer surface acceleration due to gravity mass velocity of the gas heat transfer coefficient average heat transfer coefficient over the contact time t, heat transfer coefficient between the waH and a particle wall-to-particle heat transfer coefficient in the limit of zero contact time gas phase thermal conductivity effective thermal conductivity of the emulsion phase depth for the lumped thermal capacitance system heat flux to the emulsion phase time contact time

E. BEASLEY

At

TOGt) TW Tb urn/

u

x z

Greek a,

Y &

bf

e 0, ;L p * c Pe Pf PP

u 4

time step temperature heat transfer surface temperature bed bulk mean temperature minimum fluidization velocity fluid velocity distance from heat transfer surface nondimensional distance from the heat transfer surface letters effective

thermal

diffusivity

{ = k,/[p,c,(l

- 4) gas phase accommodation

coefficient void fraction void fraction at minimum Ruidization (T- Tt.) (Tw - TB) gas phase mean free path gas phase dynamic viscosity fraction of heat transfer surface covered by projection of particles parameter h,$/kc effective density gas phase density

particle phase density particle accommodation particle sphericity

coefficient

REFERENCES

Adams, R. L., 1982, Coupled gas convection and unsteady conduction effects in fluid bed heat transfer based on a single particle model. Int. J. Heat Mass Transfer 25, 1819. Adams, R. L. and W&y, J. R., 1979, A gas convection model of heat transfer in large particle fluid&d beds. A.1.Ch.E. J. 25, 395. Anewke, C. I., 1982, Heat transfer in horizontal tubes immersed in fluidizd beds. Ph.D. Dissertation, West Viiginia University, Morgantown. Bar-Cohen, A., Glicksman, L. R. and Hughes, R. W., 1981, Semi-emnirical arediction of bubble diameter in eas fluidized’beds. Iit. J. Multi&se Flow 7, 101. Baskakov, A. P., 1964, The mechanism of heat transfer between a fluidized bed and a surface. fnt. them.Engng 4, 320. Baskakov, A. P., Berg, B. V., Vitt, 0. K. et al., 1973, Heat transfer to objects immersed in fluidlzed beds. Powder Technol. 8, 273. Beasley, D. and Figliola, R. S.,1988,A generalizedanalysisof a local instantaneousheat flux probe. J. Phys. E 21, 316. Benenati, R. F. and Brosilow, C. B., 1962, Void fraction distribution in beds of spheres. A.1.Ch.E. J. 8, 359. Catipovic, N. M., Fitzgerald,T. J.,George, A. H. and Welty, J. R., 1982, Experimental validation of the Adam+Welty model for heat transfer in large particle fluid&d beds. A.1.Ch.E. J. Za 6. Chao, B. T. and Jeng, D. R., 1965, Unsteady stagnation point heat transfer. ASME J. Heat Transfb 2. 221. Chung, B. T. F., Fan, L. T. and Hwang, C. L., 1972, A model of heat transfer in fluidized beds. J. Heat Transfer 94,105. Decker. N. and Glicksman. L. R. 1983. Heat transfer in lartze particle fluidized beds. IAt. J. Heat Mass Transfer 26,130?. Figliola, R. S., Suarez, E. G. and Pitts, D. R., 1986, Mixed particle size distribution effects on heat transfer in a fluidized bed. J. Heat Transfer 108, 913.

Heat transfer in gas fluid&d beds Gabor, J. D., 1970, Wall-to-bed heat transfer in fluid&d and packed beds. Chem. Engng Prog. Symp. Ser. 66, 76. Geldart, D., 1972, The effect of particle size and size distribution on the behavior of gas-fluidized beds. Powder Technol. 6, 201. Gloski, D., Glicksman, L. and Decker, N., 1984, Thermal resistance at a surface in contact with fluidized bed particles. Int. .I. Heat Mass Transfer 27, 559. Golan, L. P., Cherrington, D. C., Deiver, R., Scarborough, C. E. and Weiner, S. C., 1969, Particle size effects in fluidized bed combustion. Chem. Engng Prog. 75, 63. Grewal, N. S. and Saxena, S., 1980, Heat transfer between a horuontal tube and a gas-fluidized bed. Int. .I. Heat Mass Transfer 23, 1505. Higbie, R., 1935, The rate of absorption of a pure gas into a still liquid during short period of exposure. A.1.Ch.E. Trans. 31, 365. Jaguaribe, E. F. and Beasley, D. E., 1984, Modeling of the effective thermal conductivity and diffusivity of a packed bed with stagnant fluid. Int. J. Heat Mass Transjer 27,399. Korolev, V. N., Syromyatnickov, N. Y. and Tolmachev, E. M., 1971, Structure of a fixed and fluidized bed of granular material near an immersed surface (wall). J. Engng Phys. 21, 1475. Kubie, J. and Broughton, J., 1975, A model of heat transfer in fluidized beds. Int. .I. Heat Mass Transfer 18, 289. Kunii, D. and Smith, J. M., 1960, Heat transfer characteristics of porous rocks. A.I.CA.E. J. 6, 71. Kurosaki, Y., Ishiguro, H. and Takahashi, K., 1988, Fundamental study of fluidization and heat transfer characteristics around a horizontal heated circular cylinder im-

2911

mersed in a fluidized bed. Znt. 3. Heat Mass Transfer 31, 349. Martin, H., 1981, Fluid-bed heat exchangers-a new model for particle convective energy transfer. Chem. Engng Commun. 13, 1. Martin, H., 1982, Heat and mass transfer in fluidized beds. Znt. them. Engng 22, 30. Mickley, H. S. and Fairbanks, D. F., 1955, Mechanics of heat transfer to fluid&d beds. A.I.Ch.E. J. 1, 374. Ridgeway, K. and Tarbuck, K. J., 1968, Particulate mixture bulk densities. Chem. Process Engng 103, 49. Schlunder, E. U., 1980, Heat transfer to moving spherical packings at short contact times. Int. them. Engng 20,550. Schlunder, E. U., 1981, Heat transfer between packed, agitated, and fluidized beds and submerged surfaces. Chem. Engng Commun. 9, 273. Spaulding, D. B. and Patankar, S. V., 1970, Heat and Mass Zkansfer in Boundary Layers. Intertext Books, London. Suarez, E. G., 1983, Instantaneous, azimuthal and average heat transfer between a horizontal cylinder and a gas fluidiid bed. Ph.D. Dissertation. Clemson University. Clemson, SC. Suarez, E. G., Figliola, R. S. and Pitts, D. R., 1983, Instantaneous azimuthal heat transfer from a horizontal tube. to a mixed particle size fluidii bed. ASME4IChE National Heat Transfer Conference, ASME Paper No. 83-HT-93. Xavier, A. M. and Davidson, J. F., 1985, Heat transfer in fluid&d beds, in Fluidization (Edited by J. F. Davidson, R. Clift and D. Harrison), 2nd Edition. Academic Press, New York.