30
Ind. Eng. Chem. Process Des. Dev., Vol.
18, No. 1, 1979
G, = noise generator k = integral value of B/T K , = controller gain K , = process gain n, = noise sequence s = Laplace transform variable s, = coefficientsof the polynomial 6(B)(denominator of G,(B)) t = time T = sampling interval X,(B) = transfer function of forward loop controller (Table 111) y = system output ut = controller signal a = relative perturbation in process parameters = relative perturbation in process time delay r ( B ) = polynomial resulting from factorization (Appendix) y, = coefficients of y ( B ) 6(B) = denominator of G,(B) 6, = e-T/rc 0 = process time delay X(B) = numerator polynomial of noise filter (4.1) X, = parameter of X(B) X, = design value of X A, = actual value of X Abase = value of X at the smallest sampling interval u = weight on control effort (5.23) a,2 = variance of a, an2 = variance of n, om2= variance of measurement noise T , = process time constants 4(B) = the autoregressive part in noise model (4.1) 4, = parameters of 4 ( B ) & = design value of 4 & = actual value of 4 #,(El) = defined in (5.11a) w(B) = numerator of G ( B ) w , = coefficients of w ( ~ 7
w =
frequency
L i t e r a t u r e Cited Aiken, P. M., Koppel, L. B., AIChE Annual Meeting, 1974. Astrom, K. J., "Introduction to Stochastic Control Theory", Academic Press, New York, N.Y., 1970. Box, G. E. P., Jenkins, G. M., "The Future of Statistics", p 201, 1968. Box, G. E. P.. Jenkins, G. M., "Time Series Analysis, Forecasting and Control", Holden-Day, San Francisco, Calif., 1970. Cainces, P. E., Proc. I€€, 119 (5), 615 (1972). Dahlin, E. B., Instrum. Control Syst., 61 (6), 77 (1968). Horowitz, I. M., Sidi, M., Int. J . Control, 16 (2), 287 (1972). Hughes, D. J., Electron. Lett., 9, No. 10 (1973). Kalman, R. E., Trans. ASME Basic Eng., 82, 35 (1960). Kalman, R. E., Bertram, J. E., AI€€, 602 (Jan 1959). Kestenbaum, A., Shinnar, R., Thau, F. F., Id.Eng. Chem. ProcessDes. Dev., 15, 2 (1976). Koppel, L. B., "Introduction to Control Theory". Prentice-Hall, Englewood Cliffs, N.J., 1968. Lopez, A. M., Murrill, P. W., Smith, C. L., Instrum. Control Syst., 42 (2), 89 (1969). MacGregor, J. F., Can. J . Chem. Eng., 51, 468 (1973). MacGregor, J. F., Technical Reports No. 306, 307, University of Wisconsin, Department of Statistics, July 1972. Moore, C. F., Smith, C. L., Murrill, P. W., Instrum. Pract., 23 (I), 45 (1969). Newton, G.C., Gould, L. A., Kaiser, J., "Analytical Design of Linear Feedback Controls", Wiley, New York, N.Y., 1957. Paimor, 2. J., Ph.D. Dissertation, City University of New York, 1977. Phiilipson, P. H., Int. J . Control, 21 (5), 785 (1975). Ragazzini, J. R., Franklin, G. F., "SampledData Control Systems", McGraw-Hill, New York, N.Y., 1958. Rosenbrock, H. H., "Computer-Aided Control System Design", Academic Press, New York. N.Y., 1974. Rosenbrock, H. H., IFAC Symposium, 1975. Rovira, A. A,, Murrill, P. W., Smith, C. L., Instrum. ControlSyst., 42 (12), 67 11969).
Sm'ith,-O:.J. M., ISA J . , 6 (2), 28 (1959). Weekman, V. W., Jr., Adv. Chem. Ser., No. 148, 98 (1975). Wiener. N.. "The Extraoohtion. Interoretationand Smmthina of Stationarv Time Seriks", Wiley, New York, N.Y.,' 1949. Wilson, G. T., Technical Report No. 20, University of Lancaster, Department of System Engineering, July 1969.
Received for review May 23, 1977 Accepted June 21, 1978
Heat Transfer in Gas-Solid Packed Bed Systems. 1. A Critical Review Arcot R. Balakrishnan and David C. 1.Pei' Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1
The literature on heat transfer in packed beds subject to flowing gases has been critically reviewed with the emphasis on experimental techniques. It is observed that the two major modes of heat transfer, namely, conduction between the particles in the bed and convection between the flowing gas and the particles, interact with each other. This is believed to be the major reason for the difficulty in obtaining a single generalized experimental correlation or theoreticaVsemiempirica1models to evaluate the total heat transfer rates in packed bed gas-solid systems.
Introduction Heat transfer in packed bed gas-solid systems is an important operation in the chemical process industries. It is obvious that an extensive knowledge and thorough understanding of the heat transfer phenomena in the bed is essential for the successful design of such systems. The phrase, "packed bed heat transfer," is currently used to describe a variety of phenomena, namely: (1) the convective heat transfer from the walls of the packed bed to the fluid; (2) the convective heat transfer from the particles to the fluid flowing through the bed, sometimes referred to as the fluid-particle mode; (3) the conduction heat 0019-7882/79/1118-0030$01.00/0
transfer from the walls of the bed to the particles constituting the bed; (4) the conduction heat transfer between the individual particles in the bed; this is sometimes referred to as the particle-particle mode; (5) radiant heat transfer; and (6) heat transfer by mixing of the fluid. These modes are illustrated schematically in Figure 1. The fourth mode, namely the conduction between the particles, can be further subdivided into the axial and radial directions. Moreover, at elevated temperatures heat transfer by radiation will also be an important mode. In many industrial applications, it is found that two or more of the modes cited above take place simultaneously. 0 1978 American
Chemical Society
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Moreover, the modes may interact with one another. For example, the conduction between the particles may be affected by the convection between the particles and the fluid. This interaction among the different modes is one of the main reasons for the difficulty in correlating the total heat transfer and analyzing the experimental data in this field. In the next two sections some of the highlights of the experimental and theoretical research studies in the different aspects of heat transfer in gas-solid packed bed systems are presented. Experimental Investigations Table I summarizes the experimental conditions, proposed correlations, range of parameters, etc. of the many investigations which are reviewed here. Two parameters, the effective or apparent conductivity, k,, and the “total” heat tranbfer coefficient, h,, are commonly used to express the heat transfer rates in packed beds. The effective thermal conductivity is an averaged parameter that describes the total thermal performance of the granular medium that constitutes the bed. In other words, the effective conductivity is the conductivity the medium would have were it homogeneous. It should be noted that the effective conductivity is not the same as the conductivity of the material that constitutes the bed. The effective conductivity is dependent both on the thermal properties of the bed and on the flow rate, but in most experimental correlations it is generally expressed as a function of the Reynolds number only. This implies that these correlations are valid only for the particular bed materials which were used in developing them. The other preferred parameter to describe the thermal performance of a packed bed-the total heat transfer coefficient, h,-is also an averaged parameter. This total heat transfer coefficient generally incorporates the conduction mode between the bed particles, the convective mode between the bed particles and the flowing medium, the wall to bed conduction, and the wall to fluid convection. The last two modes can be expected to occur when the walls of the bed are maintained a t isothermal conditions. Therefore, as in the case of the effective conductivity, the total heat transfer coefficient is also dependent both on the thermal properties of the bed and on the flow rate, but most experimental correlations express it (in dimensionless form) as a function of the Reynolds number only. Therefore, the applicability of these correlations is also limited to the particular bed
10
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Figure 2. Typical correlations for total heat transfer in packed beds in the literature.
materials used in developing them. The commonly used dimensionless numbers for expressing ht are the Colburn-J factor and the Nusselt number. They are defined as ht jh, = -(Prf)*i3 CpfGf and
respectively. It is expected that depending on the experimental technique, different modes of heat transfer will contribute to the total heat transfer. For example, in transient experiments performed in beds with adiabatic walls, the modes that contribute will be axial conduction and convection between the bed and the fluid. On the other hand, in beds that have heated walls the significant modes are wall to bed and wall t o fluid heat transfer in addition to the radial conduction and the bed to fluid convection. Therefore, depending on the modes that have contributed, the experimentally obtained correlations relating the total heat transfer rates to the Reynolds number can be expected to show wide variation. Moreover, the thermal properties of the bed have a significant effect on the conduction mode and therefore studies using similar experimental techniques but different bed materials can also be expected to yield different correlations relating the heat transfer rates to the Reynolds numbers. Figure 2 shows the experimental correlations of several authors (all reduced to plots of Nu, vs. Rep for ease in comparison) and the wide variations in the results can be clearly seen. Consequently, in the detailed review of the experimental studies that follows, the various investigations are grouped under the heating technique employed. By this classification a grouping of the studies by the modes or combination of modes that contribute to the total heat transfer is achieved. This provides a convenient basis for the review. 1. Heated Bed Walls Leva (1947) studied the heat transfer from the wall to a gas flowing in a packed bed. The mode that may be
32 Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
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M .-
expected to occur in this configuration is (1)wall to particle conduction, (2) wall to fluid convection, (3) particle to particle conduction radially and finally particle to fluid convection. The study involved the use of different sizes of spheres and found the maximum heat transfer occurred when the particle to the diameter ratio, D,/D, was 0.15. In a subsequent study, Leva and Grummer (1948) extended the work to nonuniform spheres and found the use of a simple arithmetic average of the D, value in their correlation fit the data satisfactorily. Similarly, the use of the nominal diameter for Raschig rings and the diameter of a sphere of equal volume for cylinders was suggested. It may be noted that the authors did not use the wellknown shape factor for the beds of nonuniform shapes. In an extention of this work, Leva et al. (1948) found that if the gas were cooled instead of being heated, the constants in their correlation had to be altered. (See Table I for the correlation.) The differences in the constants for heating and cooling probably arose due to the fact that the thermal gradients within the solid phase of the bed, especially in the radial direction near the wall, were different in the two cases. The electrically heated wall would produce a constant flux and the water-cooled wall in the second case would produce approximately an isothermal surface at the wall. Similar studies were performed by Colburn (1931). He also studied the rate of heat transfer from the tube surface to the air flowing in a packed tube. Steam-jacketed pipes were used and these can be expected to produce a constant temperature wall surface. It is therefore not surprising that the numerical values estimated by Colburn's (1931) correlation are closer to the correlation of Leva et al. (1948) for the cooling case. Bunnell et al. (1949) used hot air a t 400 "C in a bed which was "cooled" by a steam jacket. Using chromelalumel thermocouples, they measured both solid and gas temperatures and found them to be identical at any given radial position. This would normally imply that convective effects are absent as there is no temperature driving force and the only heat transfer mode is conduction through the solid phase in the radial direction. However, at steady state as heat is conducted toward the wall, the packing at the center of the bed would tend to cool unless heat is supplied by the air. Furthermore, the correlation proposed for the effective conductivity shows a strong dependence on Reynolds number proving convective effects are significant. Therefore, it seems reasonable to conclude that the gas temperature measurements were probably in error. Similar studies were performed by Coberly and Marshall (1951), Houghen and Piret (1951), and Plautz and Johnstone (1955). All of these authors assumed the packed bed to be a homogeneous medium and used the heat diffusion equation together with the experimentally obtained values of the temperature profile to calculate he, the effective conductivity of the bed in the radial direction. A significant feature of these studies is that the effective conductivity was obtained experimentally as a function of the Reynolds number. In other words, the convective effect on the conductive mode has been studied. However, the fundamental properties of the bed materials have not been included as a parameter in the correlations and therefore each correlation is restricted to a specific bed material only. 2. Heat Transfer Rates from Drying Experiments Glasser and Thodos (19581, DeAcetis and Thodos (19601, Gupta and Thodos (1962a, b, 1963) and Bradshaw and Myers (1963) used a drying technique to determine heat and mass transfer rates in packed beds. This technique
Ind. Eng. Chern. Process Des. Dev., Vol. 18, No.
involved soaking the bed material which generally consisted of catalyst carriers. The packing was then dried with air. The several correlations based on this work are listed in Table I. The heat and mass transfer rates obtained from these experiments were expressed in terms of the Colburn J factor: j h for heat transfer and j d for mass transfer. In every case the value of j h was greater than j d . (The two values would have to be equal if the heat transfer consisted of particle-fluid convection alone; this follows directly from the heat transfer-mass transfer analogy.) The reasons for the heat transfer phenomena in the drying experiments consisting of both convection and conduction are as follows. (1) The bed surface temperature which was generally assumed to be constant and uniform a t the wet bulb temperature of the air has been shown by Gupta and Thodos (1964) to be incorrect. (2) To avoid wall effects the bed walls were maintained at the wet-bulb temperature of the air, but the bed itself, being a t a different temperature, leads to even greater temperature gradients. From the above considerations, it may be concluded that the heat transfer in the bed consisted of both conduction through the solid phase and convection between the bed and the drying air. 3. O t h e r Heating Methods Furnas (1930) studied heat transfer in a bed of iron spheres. In this study a step change in inlet temperature in the air was used. The column was insulated ensuring adiabatic conditions. When the bed attained steady state, heated air was introduced in an upward flow through the vertical column. Here the modes of heat transfer that can be expected to contribute to the total heat transfer are convection from the fluid to the spheres and conduction between the particles in the axial direction (walls are insulated to minimize radial conduction). Lindauer (1967) used electrical resistance heating to produce cyclical temperature variations. These temperature variations cause similar variations in the gas but with a change in phase. Therefore, there are always temperature driving forces between the particles and fluids and between the particles. Since the walls were insulated, the contributing heat transfer modes are axial conduction and convection between the fluid and the particles. Electrical induction was used by Baumeister and Bennett (1958). Steady-state measurements were made in a bed with insulated walls and therefore here again axial conduction and fluid to particle convection are the contributing modes of heat transfer. Eichorn and White (1952) used r.f. heating in their transient measurements. While a constant temperature may be achieved in steady-state measurements during the heating cycle resulting in pure convective transfer between the bed and the fluid, during the cooling cycle particle to particle conduction will also be significant. This results in both the conduction mode and convective mode contributing to the total heat transfer. Although the modes contributing to the total heat transfer in the studies of Lindauer (1967), Baumeister and Bennett (1958), and Eichorn and White (1952) are the same, their proposed correlations show wide differences. This is believed to be due to the fact that the bed materials used in these three investigations were each different and their physical and thermal properties can be expected to affect the correlations. The work of Balakrishnan and Pei (1974) and Bhattacharyya and Pei (1975) involved the use of microwaves as the heating medium. When a bed of metallic oxides is subject to microwave radiation, the entire bed attains a uniform temperature instantaneously. This gives rise to
1, 1979
37
a situation where thermal gradients within the bed are eliminated and thereby the particle to particle conduction mode is absent and the fluid to particle convection heat transfer can be determined. Using this technique, Bhattacharyya (1973) proposed the following correlation
z] 0.25
Cihh, = 0.018[
(3)
Subsequently, Balakrishnan (1973) working with nonspherical packings, extended the correlation to (jhlfp= 0.018[Frp,]~0”[&]3 ’i6
(4)
where & is the shape factor of the pellets. Equation 4 may also be written in terms of the Nusselt number as
Nuf, = 0 .O 16 [Arm] [Re,] [ &] 3.76
-
-
(5)
Obviously as Re, 0, Nufp 0; Le., a t no flow there is no forced convection heat transfer. Theoretical a n d Empirical Models In the previous section on experimental studies on heat transfer in packed beds it was seen that most of the studies were directed at obtaining either the effective conductivity or the total heat transfer coefficient as functions of Reynolds number. However, it is obvious that the conduction mode, apart from being dependent on the Reynolds number, is also strongly dependent on the physical, thermal, and transport properties of the bed material. Furthermore, the conduction mode can be subdivided into the radial and axial directions. Radiation will also be a significant contributor to the total heat transfer at elevated temperatures. In view of all this, a more fundamental approach is to analyze the contributions of each mechanism or mode in terms of the properties and parameters which are likely to affect the heat transfer. The implicit assumption, in many of the early models that used this approach, is that the different modes are essentially additive. This type of analysis was first used by Schuman and Voss (1934) and Damkoehler (1937). They summed up the contributions of each mechaism by which heat is transferred radially in the bed. Later, Wilhelm et al. (1948) modified this work for systems where the gas is not flowing. Verschoor and Schuit (1950) divided the effective conductivity of packed beds into two parts, one which is independent of fluid flow and the other which is dependent on the lateral mixing of the fluid in the packed bed. Singer and Wilhelm (1950) suggested separate mechanisms by which heat is conducted radially within the bed, namely: (1)thermal conduction through the fluid phase, (2) thermal solid-fluid -solid convection, (3) thermal solid-solid conduction through contact surfaces, (4) thermal conduction within the solid, and (5) thermal conduction through the fluid film near the contact surface of the packings. Ranz (1952) extended this model by adding contributions due to: (1) radiant heat transfer between packings, (2) radiant heat transfer between neighboring voids, and (3) heat transfer by lateral mixing of the fluid. Agro and Smith (1953) used the model by Ranz (1952), and their experimental results indicate good agreement with their predictions. The work by Ranz (1952) also laid the foundation for the model suggested by Yagi aqd Kunii (1957). They also divided the conductive heat transfer into two parts, namely those dependent and independent of fluid flow. Heat transfer mechanisms independent of fluid
38
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
flow, according to Yagi and Kunii (1957) are: (1) thermal conduction within the solid, ( 2 ) thermal solid-solid conduction through the contact surfaces, (3) radiant heat transfer between packings, and (4) radiant heat transfer between adjacent voids. The mechanisms dependent on fluid flow are: (5) thermal conduction through the fluid film near the contact surface of the two solids, (6) solid-fluid-solid heat transfer by convection, and (7) heat transfer by lateral mixing of fluid. Their generalized correlation, proposed on the basis of contributions dependent and independent of fluid flow is he - k,O _ - - + (ap)PrfRep kf
kf
where, k,O, the effective conductivity of the bed under “no flow” conditions, is given by
On comparison with experimental data, the Yagi and Kunii model (eq 7) is found to underestimate the effective conductivity of the packed bed. They attributed the discrepancies, which were appreciable in some cases, to the unavailability of the physical parameters of the system. However, Bhattacharyya and Pei (1975) suggest that these discrepancies probably resulted from not incorporating the effects of fluid flow on the conductive heat transfer mode. Yagi et al. (1960) were among the first to measure the effective thermal conductivity of packed beds in the axial direction. Their purpose was apparently to test the validity of the Yagi and Kunii (1975) model with reference to the axial conduction. Using a bed with adiabatic walls to eliminate radial gradients and an infrared lamp applied continuously to the fluid flow as the heat source, they were able to correlate their results with eq 6. (cup) took on values between 0.7 and 0.8 in this case. Wakao and Kat0 (1969) and Wakao and Vortmeyer (1971) developed a lattice model to predict the effective thermal conductivity (of beds with stagnant fluids) as compared to the simplified geometric model of Yagi and Kunii (1957). The heat transferred through a unit cell was calculated using a relaxation procedure where the solid and gas were replaced by a network of rods. (A point contact was assumed between the spheres, and the conductance a t the contact point in the grid was evaluated by assuming unidirectional heat transfer.) Numerical solutions were obtained for combined conduction and radiation effects in cubic ( e = 0.476) and orthorhombic ( e = 0.395) lattice models of spheres. The generalized correlation is
%]
1.11
+ 0 . 7 0 i N ~ ~ ~ ~ ~ [(8) for k,/kf = 20-1000, Nu,, = 0-0.3, and Nurp = h,$,/k,. Bhattacharyya and Pel (1975) also used the approach of correlating the contributions of each mode separately. They evaluated the total heat transfer by
where k:/kf is the effective conductivity with motionless fluid evaluated by the Yagi and Kunii (1957) model (eq 7); for Nufc,the effect of fluid flow on conduction, they proposed
Nufc = jhfc = 2.05 Re$rf‘13
X
-
PPCPP
Ptcpf
(10)
Nufp,the convective heat transfer, was evaluated by their earlier experimental correlation (eq 5). The model by Bhattacharyya and Pei (1975) uses the Yagi and Kunii (1957) model which is empirical in nature. Furthermore, the effect of fluid flow, Nuf,, was obtained empirically from experimental data in the literature on total heat transfer rates. This was done by evaluating k,O/kf and Nuf,, respectively, and subtracting them from Nu, values available in the literature and subsequently correlating them as shown in eq 10. Therefore, while the convective part of the Bhattacharyya and Pei (1975) model is general and valid for all bed materials, the applicability of the conductive mode, particularly the effect of fluid flow on conduction, Nufc,is limited to bed materials and ranges used in obtaining the Nu, values in the literature. While the models presented up to now treat the problem of heat transfer in packed beds by looking a t the conduction mode theoretically and using experimental studies to evaluate the convective transfer, there are several investigations which treat the bed as a homogenous medium and make no distinction between the contributions of the different modes. Kunii and Suzuki (1967) used the familiar two-dimensional diffusion equation for packed beds in the range of low Peclet number, Le., P e < 10. However, this model over predicted the heat and mass transfer coefficients and this was attributed to channelling or local uneven contacting of fluids with the solids in the real situation. Galloway and Sage (1970), using an instrumented copper sphere in a bed of packed spheres, obtained data on the local heat transfer rates. They discussed their results in terms of boundary layer theory with particular reference to single spheres and cylinders in turbulent flow. Wakao et al. (1973) theoretically showed that the Nusselt number between a spherical particle in a packed bed and stagnant fluid has a limited value of 2 , which is the same as a single sphere in a stagnant fluid environment. They suggest that this should be considered as the lower limiting Nusselt number. Sorensen and Stewart (1974) assumed a constant bed temperature and solved the continuity and conservation equations and obtained numerical solutions for heat and mass transfer rates for creeping flow a t low Peclet numbers. Nelson and Galloway (1975), working on a model for mass transfer, solved the unsteady-state diffusion equation with boundary conditions from the Penetration Theory and obtained an expression for the Sherwood number which is valid for a Reynolds number range of 0.01 to 100. Rowe (1975) modified Nelson and Galloway’s model slightly to make it applicable to liquid fluidized beds. Votruba et al. (1975 a,b) modeled and obtained heat and mass transfer rates experimentally in monolithic honeycomb catalysts whose main application is in pollution control in the automobile industry. Gunn and Khalid (1975) presented an analysis showing the effect of axial and radial thermal dispersion and wall thermal resistance heat transfer to fixed beds of solids. Application of this analysis to experiments yielded axial, thermal, and wall transfer coefficients. The discrepancies on comparison with experimental values of other workers were attributed to the fact that axial dispersion was neglected by some workers and also the thermal characteristics of fixed beds of metallic and nonmetallic particles were neglected. Recently, there has been considerable interest in the use of packed beds of spheres for insulation purposes, espe-
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
cially in cryogenic applications. Here, of course, the convective mode is absent as there is no fluid flow through the bed. Yovanovich (1973) used the concept of thermal constriction resistance within the spheres and conduction resistance of an effective gas gap thickness to predict the apparent conductivity of glass microspheres from atmospheric pressure vacuum. Chan and Tien (1973) have presented an analytical study of the heat transfer through the solid phase of a packed bed of spheres. Both these models use the concept of a finite contact spot between the spheres in the bed. The dimensions of the finite contact spot dimensions may be estimated by the Hertz relation from elasticity theory.
39
h, heat transfer coefficient, J/m2 s K hr,, radiation heat transfer coefficient between solids
hr, = O.l952[
2-e
1001 t , + 273
(5.678) J / m 2 s K
hrf, radiation heat transfer coefficient between gases hrf = hrf =
r
I 1
0.1952
[L [&I[ I+
5 1J 1[ l o o - ] -
-
%?]’(5.678) (5.678) J / m 2 s K
It is now apparent (Hallmann et al., 1973) that the use of a model which does not take into account the finite contact spot between individual spheres (as opposed to point contact in earlier models) would yield less realistic results. Summary The following summarizes the conclusions that may be drawn from this review. (1)The majority of the experimental studies were directed toward correlating the total heat transfer rates (generally in dimensionless form) with Reynolds number. (2) The total heat transfer consisted, in most cases, of both the conduction mode and the convective mode. Since the conduction mode depends on the physical and transport properties of the bed materials, these correlations although often reliable are applicable to the particular bed materials for which they were developed only. One exception is the correlation for the convective heat transfer alone proposed by Pei and coworkers (eq 5). (3) A number of analytical and empirical models to predict the conductive mode have also been discussed. However, many of these models, particularly the earlier ones, are highly empirical in nature and need considerable refinement before they can be used with confidence for design purposes. The recent models by Chan and Tien (1973) and Yovanovich (1973) take a more fundamental approach and can be used to estimate the conduction heat transfer in packed beds which are not subject to convective effects. The models of Chan and Tien (1973) and Yovanovich (1973) use t.he concept of a finite contact spot between the individual particles in the bed, the dimensions of the contact spot being evaluated from the Hertzian theory of elasticity. Nomenclature a, radius of bed particle, m Arm, Archimedes number, D : g p f ( p p mensionless c , specific heat, J/kg K d,, particle diameter, m D,, tube diameter, m e , emissivity, dimensionless E , Young’s modulus, N/m2 F, force acting on contact spot, N Frpm,modified Froude number
g, acceleration due to gravity, m/sz
G f , fluid mass velocity, kg/m2 s
h, thermal conductivity, J/m s K Nu, Nusselt number, dimensionless Pe, Peclet number, dimensionless Prf, Prandtl number, dimensionless r,, contact radius, m Re,, particle Reynolds number, dimensionless t , temperature, “C u f , fluid velocity, m/s
Greek Letters a , mass velocity of fluid in direction of heat transfer (Yagi
and Kunii (1957) model), kg/m2 s
p, ratio of average length between centers of two neighboring solids to mean diameter of packing (Yagi and Kunii model), dimensionless 7, ratio of effective length of solid relating to conduction to mean packing diameter (Yagi and Kunii model), dimensionless t, bed porosity, dimensionless p f , fluid viscosity, N s/m2 v , Poisson’s ratio, dimensionless p , density, kg/m3 6, ratio of effective thickness of fluid film adhering to solid to mean packing diameter (Yagi and Kunii model), dimensionless q5s, shape factor, dimensionless Subscripts e, effective f, fluid fc, effects of fluid on conduction fp, fluid particle p, bed particle rf, radiation between fluid spaces rp, radiation between bed particles t, total 0, at “no-flow’’condition Literature Cited Agro, W. B., Smith, J. M., Chem. Eng. Prog., 49, 443 (1953). Balakrishnan, A. R., M.A.Sc. Thesis, University of Waterloo, Waterloo, Ont., Canada, 1973. Balakrishnan, A. R., Pei, D. C. T., Ind. Eng. Chem., Process Des. Dev., 13, 441 (1974). Baumeister, E. E., Bennett, C. O., AIChE J . , 4, 69 (1958). Bhattacharyya, D., Ph.D Thesis, University of Waterloo, Waterloo, Ont., Canada, 1973. Bhattacharyya, D., Pei, D. C. T., Chem. Eng. Sci., 30,293 (1975). Bradshaw, R. C., Myers, J. E., AIChE J., 9 , 590 (1963). Bunnell, D. G., Irvin, H. B., Olson, R. W., Smith. J. M., Ind. Eng. Chem., 41, 1977 (1949). Chan, C. K., Tien, C. L . , J . Heat Transfer, 95, 302 (1973). Coberly, C. V., Marshall, W. R., Chem. Eng. Prog., 47, 141 (1951). Colburn, A. P., Ind. Eng. Chem., 23, 910 (1931). Damkohler, G., Chem. Ing., 3, 441 (1937). DeAcetis, J.. Thodos, G., I n d . Eng. Chem.. 52, 1003 (1960). Eichorn, J., White, R. R., Chem. Eng. Prog. Symp. Ser., 4, 11 (1952). Furnas, C.C., Ind. Eng. Chem.. 22, 26 (1930). Galloway, T. R., Sage, B. H., Chem. Eng. Sci., 25,495 (1970). Gamson, B. W., Thodos, G., Hougen, 0. A., Trans. AIChE, 39, 1 (1943). Glasser, H., Thodos, G., AIChEJ., 4, 65 (1958). Gunn, D. J., Khalid, M., Chem. Eng. Sci., 30, 261 (1975). Gupta, A. S., Thodos, G., Chem. Eng. Prog., 58 (7),58 (1962a). Gupta, A. S.,Thodos, G., AIChE J . , 8, 608 (1962b). Gupta, A. S., Thodos, G., AIChE J . , 9 , 751 (1963).
40
Ind. Eng. Chem. Process Des. Dev., Vol.
18,
No.
1, 1979
Gupta, A. S., Thodos, G., Ind. Eng. Chem. Fundam., 3, 218 (1964). Hallmann, V. W., Paas. G., Wolpert, H. D., Vai. Tech., 6, 169 (1973). Houghen. J. O., Piret, E. L., Chem. Eng. Prog., 47, 295 (1951). Kunii, D., Sujuki, M., Int. J . Heat Mass Transfer, IO, 884 (1967). Leva, M., Ind. Eng. Chem.. 39,857 (1947). Leva, M., Grummer, M., Ind. f n g . Chem., 40, 415 (1948). Leva, M., Weintraub, M., Grummer, M..Clark, E. L., Ind. Eng. Chem.. 40, 747 (1948). Lindauer, G. C., AIChE J., 13, 1181 (1967). Nelson, P. A., Galloway, T. R.. Chem. f n g . Sci., 30, 1 (1975). Plautz, D. A,, Johnstone, H. F., AIChE J., 1, 193 (1955). Ranz, W. E., Chem. Eng. Prog., 48, 247 (1952). Rowe, P. N., Chem. f n g . Sci., 30, 7 (1975). Schumann, T. E. W., Voss, V., J . Fuel, 13, 249 (1934). Singer. E., Wilhelm, R. H., Cbem. Eng. Prog., 46, 343 (1950). Sorenson, J. P., Stewart, W. E., Chem. Eng. Sci., 29, 827 (1974). Verschoor, H., Schuit, G. C. A., Appl. Sci. Res., 42, (Part A2, No. 2), 97 (1950).
Votruba. J., Sinkule, J., Hlavacek, V.. Skrivarek, J., Chem. Eng. Sci., 30, 117 (1975a). Votruba, J.. Mikus, O., Nguen, K., Hhvacek, V., Skrivanek. J., Chem. f n g . Sci., 30, 201 (1975b). Wakao, N., Kato, K., J . Chem. Eng. Jpn., 2, 24 (1969). Wakao, N., Vortmeyer, D., Chem. Eng. Sci., 26, 1753 (1971). Wakao, N., Takano. Y.. Pei, D. C. T., J . Chem. Eng. Jpn., 6, 269 (1973). Wilhelm, R. H., Johnson, W. C., Wynkoop, R., Collin, D. W., Chem. Eng. Prog., 44, 105 (1948). Yagi, S.,Kunii, D., AIChEJ., 3, 373 (1957). Yagi, S.,Kunii, D., Wakao, N., AICh€ J., 6, 543 (1960). Yovanovich, M. M., ASME Paper 73-HT-43, ASME-AIChE Heat Transfer Conference, Atlanta, Ga., 1973.
Received for review February 3, 1977 Accepted July 6, 1978
Heat Transfer in Gas-Solid Packed Bed Systems. 2. The Conduction Mode Arcot R. Balakrishnan and David C. T. Pei' Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G I
The conduction mode of heat transfer in packed beds subject to flowing gases has been studied analytically. The analysis is based on the more realistic assumption of a finite contact spot between the spheres in the bed, the dimensions of which may be obtained from the Hertzian elasticity theory, rather than the usual assumption of a point contact. Moreover, the convective effects of the flowing gas were also incorporated in the analysis as boundary conditions. The effect of parameters such as contact spot dimensions, packing geometry, Biot modulus (convective effects), and radiation on the conduction mode have also been examined.
Introduction In a previous paper (Balakrishnan and Pei, 1978), a review of the several analytical and empirical models to predict the conduction mode of heat transfer in packed beds was presented. It was pointed out that many of the models, particularly the earlier ones, are highly empirical in nature and need considerable refinement before they can be used with confidence for design purposes. The recent studies by Chan and Tien (1973) and Yovanovich (1973) have taken a more realistic approach of assuming finite contact spots between the particles constituting the bed. This was used in estimating the conduction heat transfer in super insulation problems. However, they are applicable only for the case where there is no fluid flowing through the bed. On the other hand, it is believed (Bhattacharyya and Pei, 1975) that the flowing fluid does have an effect on the conduction mode. This paper presents an analysis which may he used to estimate the conduction heat transfer in packed beds subject to convective effects. As in the models of Chan and Tien (1973) and Yovanovich (1973), the present model makes use of the concept of a finite contact spot between the individual spheres in the bed, the dimensions of the contact spot being evaluated from the Hertzian theory of elasticity. The conductive effects, which are used as boundary conditions in the analysis, are estimated from an earlier experimental correlation of Pei and co-workers (Bhattacharyya and Pei, 1975; Balakrishnan and Pei, 1974). Analysis Consider a single sphere of radius r = a , between two planes A and B where A is a t a higher temperature than B. Gas, a t a bulk temperature of tf, flows past the sphere 0019-7882/79/1118-0040$01.00/0
and heat is transferred by conduction through the sphere between the two planes and by convection between the sphere and the gas. This configuration is shown in Figure 1.
The flux from plane A to the sphere is qo and a flux of q1 leaves the sphere into plane B. The convective flux from the sphere to the air stream is equal to hfp(t- tf), where
hfpis the convective heat transfer coefficient. There is a finite contact spot between the spheres and each of the other two spheres at planes A and B, respectively, and the contact radius is given by the Hertz relation
For the situation described above, the temperature field in the sphere is described by the Laplace equation
with the following boundary conditions
= -hfpT
0, 5 0 5 R - Bo, r = a 0, 5 0 IR , r = a
(3) = -ql where T = t - tf and 6'" = sin-' ( r J a ) . The general solution is A -
(4) where x = cos 0. @ 1978 American
Chemical Society
