Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
ReG = gas Reynolds number, pGUGd / p ~ ReL = liquid Reynolds number, p L d d p / b L So = power spectral density function Sc = Schmidt number, pL/(pLDL) Sh = time-averaged Sherwood number, t&d,/ DL Sh,,, = root mean square Sherwood number, ks,d,/DL UG = gas superficial velocity UL = liquid superficial velocity Greek Letters a = thermal diffusivity of the catalytic pellet = bed void fraction pG = viscosity of the gas p L = viscosity of the liquid p~ = density of the gas
= density of the liquid = time constant for heating a single pellet 4 = fractional wetting of pellet
pL 7
L i t e r a t u r e Cited
283
Gianetto, A., Baldi, G., Specchia, V., Quad. rng. Chim. Ita/.,6, 125 (1970). Goto, S., Levec, J., Smith, J. M., Ind. Eng. Chem. Process Des. Dev., 14, 473 (1975). Hirose, T., Mori, Y., Sato, Y., J. Chem. Eng. Jpn., 9, 220 (1976). LemaY, Y., Peneautt, G., Ruether, J. A., Ind. Eng. Chem. Process Des. Dev., 14; 280 (1975). Matsuura, A., Akehata, T., Shirai, T., J. Chem. Eng. Jpn., 9, 294 (1976). Mears, D. E., Adv. Chem. Ser., No. 133, 218 (1974). Mizushina. J.. Adv. Heat Transfer. 7. 87 (1971). Mochizuki; S.’, Matsui, T., Chem. Eng. Sci., 29; 1326 (1974). Onda, K., Mem. Fac. Eng., Nagoya Univ., 24(2), 165 (1972). Puranik, S.S., Vogelpohl, A., Chem. Eng. Sci., 29, 501 (1974). Satterfield, C. N., Van Eek, M. W., Bliss, G. S., AIChE J., 24, 709 (1978). Shah, Y. T., “Gas-Liquld-Solid Reactor Design”, McGraw-Hill New York, N.Y., 1979. Specchia, V., Baldi, G., Gianetto, A,, Proc. 4th Int. Symp. Chem. React. Eng. Heidelberg, Germany, Dechema, 390 (1976). Specchia, V., Baldi, G., Ghetto, A., Ind. Eng. Chem. Process Des. Dev., 17, 362 (1978). Sylvester, N. D., Ptayagulsarn, P., Ind. Eng. Chem. Process D e s . Dev., 14, 421 (1975); correction, ibM., 15, 360 (1976). Weekman. V. W., Jr., Proc. 4th Int. Symp. Chem. React. Eng., HeMelberg, Germany, Dechema, 2, 615 (1976). Weekman, V. W., Jr., Myers, J. E., AIChE J., 11, 13 (1965).
Chou, T. S., Ph.D. Thesis, University of Houston, 1976. Dawson, D. A,, Trass, O., Int. J. Heat Mass Transfer, 15, 1317 (1972). Dharwadkar, A,, Sylvester, N. D., AIChE J., 23, 376 (1977).
Receiued for reuiew October 16, 1978 Accepted April 2, 1979
Effective Thermal Conductivity of Packed Beds Peter W. Dietr Thermal Branch, Mechanical Systems & Technology Laboratory, General Electric Corporate Research and Development, Schenectady, New York 1230 1
A model is developed for the conduction of heat through a packed granular bed of conducting particles in a medium of lower conductivity. Based on this model, an expression is derived for the effective thermal conductivity of the bed in terms of the thermal conductivities of the granular material and the surrounding medium. This result is shown to compare favorably with published data.
Introduction The determination of average properties of granular materials is a commonly encountered engineering problem. As a result, beginning with Maxwell (1891), numerous authors have investigated simple Fourier conduction through mixtures, suspensions, and packed beds. However, in spite of this long history, the complicated geometries have precluded a detailed analysis of conduction through a packed bed. In particular, existing equations for the effective thermal conductivity of a packed bed were developed on an “averaged” basis and, thus, do not necessarily incorporate the effects of the local concentration of heat flux near particle-particle contacts. For excellent reviews of the literature, the reader is referred to the articles by Crane et al. (1978) and by van Beek (1967). The present work considers a special case of the packed bed-a hexagonal array of touching spheres. An approximate, analytic solution to the steady-state energy equation is obtained for this idealized geometry. However, the resulting expression for the effective bed conductivity is shown to be a weak function of bed geometry and can, therefore, be applied to a variety of packings. Analysis The granular bed is modeled as being composed of identical spheres (radius R , thermal conductivity kp) arranged in a uniform lattice (see Figure 1). The lattice is
formed by stacking hexagonally packed planes. Thus, each sphere is in contact with six neighbors in the X-Y plane and with the particle directly above and below in the 2 direction. The heat flux is assumed to be in the negative 2 direction. Because of the symmetric arrangement of the particles, all planes described by 2 = NR, where N is an integer, are isotherms. Thus, the problem can be reduced to that shown in Figure 2. Because the particle conductivity is assumed to be much greater than that of the fluid, the isotherms within the spheres intersect the surface of the sphere at almost right angles. Consequently, when calculating heat flow within a sphere, the isotherms can be closely approximated by surfaces that intersect the surface a t right angles. Since the heat flux through the fluid can be calculated in terms of the particle surface temperature, an energy balance can be applied to the differential volume between adjacent isotherms to yield an ordinary differential equation for the surface temperature. This technique is analogous to that employed to calculate the temperature distribution along a thin fin. In the limit of k , >> k f , the isotherms within the sphere can be approximated by spherical caps which intersect the surface of the particle at right angles (see Figure 3). Although this is not an exact solution to the problem, it has two important features: (1)the plane 0 = 7r/2 is an isotherm, and (2) the heat flux is radially inward near the contact point. In this limit ( k , >> k f ) ,most of the tem-
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Ind. Eng. Chern. Fundarn., Vol. 18, No. 3, 1979 X
Conservation of energy requires that
-1 -dQ- - 27rrqf = 2irR sin Oqf R d0
(4)
where qf is the heat flux from the particle into the surrounding medium. Near the contact the temperature gradient along the surface of the particle is much less than the gradient across the gap. Thus, the heat flux between the spheres can be modeled as that between diverging (angle e), isothermal (temperature T,(O)) plates (see Figure 4). T, - k f sin OT, (5) q f = k f ( r t r'?o RO(I - cos 0)
+
Equations 3, 4, and 5 can be combined to yield one equation in T,, namely 1
1
1
d0
HEAT FLUX
Figure 1. Two views of the idealized geometry for a packed granular bed.
Figure 2. Definition of local coordinate system. Note (To/R)is the average temperature gradient across the packed bed.
[
3 1= (1 +
sin20 1 + sin 0 d0
>> 1.
perature drop occurs in the vicinity of the contact. In this region, the isotherms are nearly concentric and thus are also surfaces of constant heat flux. The present analysis will assume that all of the isothermal surfaces are surfaces of constant heat flux. The value of the heat flux parallel to the surface of the particle (in the negative 0 direction) is given by Fourier's law of conduction
where T, is the surface temperature of the sphere. Since the heat flux is assumed to be constant over the spherical cap, the total heat flux through the cap is given by Q(0) =
12rl" v=o J/=o
qp(B) (R'12sin 0' do' dJ/'
= 2a(R'12(1- COS 6')q,(B)
(2)
Simple geometry gives dTB Q(B) = 27rR tan2 B ( 1 - sin B)k de
(3)
6)
(2>. kf
(6)
subject to the boundary conditions T,(O = 0) = 0
The only dimensionless group that appears in eq 6-8 is the conductivity ratio ( k , / k f ) ;there is no dependence on the particle radius (R). Since the principal temperature drop occurs in the vicinity of the contact (at 0 = 0), eq 6 can be expanded about this point as 02
Figure 3. Approximate isotherms within sphere for k , / k f
COS
d2T, do2
kfl k,O
dT, d0
- + 20 - - 2--T,
=0
(9)
Although this linearization is only accurate for 0 5 ir/16, the solution to this equation will be valid for large values of the conductivity ratio ( k , / k f >> l ) , since, in this limit, most of the temperature drop will occur in the specified region. Thus, the temperature drop between 0 = 7r/2 and 0 N 7r/16 is negligible and eq 11 can be replaced by (10) T,(O = 00) = To where 0, is an angle near ir/16. (Note: the solution should be a weak function of the angle O?.) Solutions to eq 9 are modified Bessel functions. Matching boundary conditions (eq 7 and 10) gives
The effective thermal conductivity of the bed can be computed by comparing the actual heat flux through a particle (eq 4 and 11)to that which would flow if the region were occupied by a uniform material with thermal conductivity keff
I;:)"( I';')"(
KO[ Oak,
Kl[
(12)
Bok,
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 285
agreement is obtained over the entire range 1 Ikp/kf I lo4,even though the expression for the effective thermal conductivity was developed for large-conductivity ratios. Crane and Vachon (1977) have compared these data with the models of Rayleigh (1892, corrected by Runge, 19251, Meredith and Tobias (1960), and Bruggeman (1935). Average errors of 26 to 33% are reported.
Figure 4. Geometry for evaluating the heat flux normal to the surface of the particle.
which is, indeed, a weak function of Bo for large values of the conductivity ratio. Evaluating the constants in eq 1 2 for Bo = a116 gives
The specific choice of Bo = a116 is dictated by the requirement that keff = kf when kp/kf = 1. As discussed above, this choice has little effect on the computed values of keflfor kp/kf 2 100 (where the analysis presented here is valid). In fact, slightly better agreement with the data is obtained in the range kp/kf 2 20 with Bo = a/8. However, these larger values of Bo are not consistent with the linearization that results in eq 12. Equation 13 is plotted in Figure 5. The data are those tabulated by Crane and Vachon (1977) for void fractions between 0.49 and 0.38 and for particles with conductivities between 0.2 and 230 Btu/h ft2 O F in fluids with conductivities between 0.01 and 0.4 Btu/h ft2 O F . Good
‘I
10
Discussion A model of Fourier heat conduction through a packed bed of granular material was developed to predict the effective thermal conductivity of the bed. Implicit in this model are certain assumptions; namely (i) the particleparticle contact is a point contact; (ii) heat is transferred in a steady-state conduction-no convection or radiation effects were included; (iii) the particles are uniformly sized spheres packed in a hexagonal array. The theory agrees well with reported data, including those for beds of irregular, polydisperse, randomly packed particles. The applicability of the theory to such systems can be attributed to several factors. (1)The effective conductivity is an average. Thus, deviations can be expected to “average-out”. (2) The result is independent of radius. Thus, conduction through polydisperse media may also be explained. (3) The assumed packing geometry is not critical. In fact, for the high conductivity ratios assumed here, the interparticle resistance is completely determined by the contact geometry and these results can be directly applied to other packings. For body-centered cubic packing arrays (or close-packed hexagonal), this modification of the result only increases the effective conductivity by 10%. (4) Irregular particles are effectively modeled as spheres because the geometry of the region near the contact is, on the average, approximately spherical. In summary, an analytic model has been developed for the conduction of heat through a packed granular bed. Although the model was developed for large values of the conductivity ratio (kp/kf > loo), the resultant expression was shown to agree favorably with published data over the entire range 1 Ikp/kf S 3 X lo3.
100 Thermal Conductivity d Particle Thermal Conductivity of Medium
1000
10000
Figure 5. Normalized effective conductivity as a function of the conductivity ratio: comparison between theory and experimental data (Crane and Vachon, 1977).
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Literature Cited Bruggeman, D. A. G., Ann. Phys., 24, 636 (1935). Crane, R. A., Vachon, R. I., Int. J . Heat Mass Transfer, 20, 711 (1977). Crane, R. A., Vachon, R. I., Khader, M. S.,“Proceedings of the Seventh Symposium on Thermophysical Properties”, 1978. Maxwell, J. C., “A Treake on EkOkity and Magnetism”,Vd. 1, p 452, Chrendon, Oxford, 1891.
Meredith, R. E., Tobias, C. W., J . Appl. Phys., 31, 1270 (1960). Rayleigh, Lord, Phil. Mag. J . Sci., 34, 481 (1892). Runge, I . , Z . Tech. Phys., 6, 61 (1925). vanBeek, L, K, H,,Prw,Die,ectr,, , 69 (1967).
Received for review December 8, 1978 Accepted April 26, 1979
Parameter Sensitivity and Kinetics-Free Modeling of Moving Bed Coal Gasifiers Morton M. Denn” and Wen-Ching Yu Department of Chemical Engineering, University of Delaware, Newark, Delaware 1971 1
James Wei Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
A detailed model of a moving bed coal gasifier can provide temperature and composition profiles throughout the reactor, as well as effluent gas composition and temperature. The former are sensitive to certain model inputs, but the latter are insensitive to large variations in the same parameters. Thus, complete model validation requires temperature and composition profiles as well as effluent measurements. The insensitivity of overall model behavior to certain model parameters suggests that a “kinetics-free’’ model would be useful in estimating reactor performance in regions of high carbon conversion. Agreement between the kinetics-free calculation and the detailed model is very good for coals with negligible reactivity to hydrogen.
Introduction
A mathematical model of moving bed coal gasification reactors has been described recently by Yoon et al. (1978). The model, commonly referred to as the University of Delaware (UD) model, employs mass and energy balances, together with information about rates of chemical reactions and physical transport processes. Other models have been published by Amundson and Arri (1978) and Biba et al. (1978); these differ from the UD model in the details of the physical and chemical processes. All models predict temperature and composition profiles in the reactor, as well as effluent compositions and temperatures. The only reliable data available for model validation on pilot and commercial scale gasifiers are effluent gas compositions and temperatures. The UD model predicts effluent properties accurately for both dry ash and slagging moving bed gasifiers (Yoon et al., 1978),and it also predicts that the maximum temperatures are in the feasible operating range; i.e., maxima are below the ash softening temperature for dry ash operation and above the ash melting temperature for slagging operation. Model inputs are taken from independent experiments, but there are uncertainties regarding some parameters. We examine here the sensitivity of the model predictions to these parameters. We find that the detailed temperature and composition profiles in the reactor are sensitive to the parameter values, but the effluent is quite insensitive. These results lead to a “kinetics-free’’ model of gasifier output that can be applied in a region of efficient operation, and they illustrate the need for a more extensive experimental program for complete model validation. Initial Oxidation Product CO/C02 Distribution
The initial product distribution of carbon oxidation under the complex conditions in a gasifier is not known. 0019-7874/79/1018-0286$01 .OO/O
Table I. Optimum Feed Conditions Computed by Yoon et al. (1979a) for Lurgi Gasification at 25 atm Pressure with a 700 “ F Blast (Steam + Oxidant) Temperature coal
oxidant
Illinois Illinois Wyoming Wyoming
oxygen air oxygen air
C/O, steam/O, molar molar feed ratio feed ratio 2.98 2.8
3.7 3.28
9.6 6.7 6.8 4.8
This CO/C02 distribution is a model input that cannot be estimated from independent experiments at the present time. Using the initial fractional conversion of C to CO as a model input, the computed maximum temperature is shown in Figure 1 for oxygen-blown gasification of 11linois No. 6 coal in a pressurized Lurgi gasifier with 20-mm diameter coal particles. The final effluent CO/C02 ratio depends also on subsequent attack of C by C 0 2 and HzO, and by the water-gas shift reaction. Feed conditions are given in Table I; these correspond to the optimum conditions computed by Yoon et al. (1979a) for the Illinois coal a t the specified blast temperature. This variation of fractional conversion to CO caused a variation of more than 300 O F in the computed maximum temperature, but no variation in the properties of the gas leaving the reaction zone, which are listed in the first column in Table 11. Effluent calculations for initial CO-C02 distributions of 100% C M % C 0 2 ,50% C0-50% C 0 2 ,and 0% CO-lOO% C 0 2 agree to three significant figures; therefore only one set of figures is listed. Similar results are obtained at the optimal feed conditions listed in columns 3, 5, and 7 of Table I for the air-blown Lurgi gasification of the Illinois coal, and oxygen- and air-blown Lurgi gasification of a high activity Wyoming coal. 0 1979 American Chemical Society
