Thermodynamic analysis of convective heat transfer in an annular packed bed

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1 University of Nebraska - Lincoln From the SelectedWorks of YASAR DEMIREL 000 Thermodynamic analysis of convective heat transfer in an annular packed bed YASAR DEMIREL R Kahraman Available at:

2 International Journal of Heat and Fluid Flow 1 (000) 44±448 Thermodynamic analysis of convective heat transfer in an annular packed bed Y. Demirel *, R. Kahraman Department of Chemical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 3161, Saudi Arabia Received 6 September 1998; accepted 17 March 000 Abstract A combination of the rst and second law of thermodynamics has been utilized in analyzing the convective heat transfer in an annular packed bed. The bed was heated asymmetrically by constant heat uxes. Introduction of the packing enhances wall to uid heat transfer considerably, hence reduces the entropy generation due to heat transfer across a nite temperature di erence. However, the entropy generation due to uid- ow friction increases. The net entropy generations resulting from the above e ects provide a new criterion in analysing the system. Using the modi ed Ergun equation for pressure drop estimation and a heat transfer coe cient correlation for an annular packed bed, an expression for the volumetric entropy generation rate has been derived and displayed graphically. In the packed annulus, the fully developed temperature pro le and the plug ow conditions have been assumed and veri ed with experimental data. The volumetric entropy generation map shows the regions with excessive entropy generation due to operating conditions or design parameters for a required task, and leads to a better understanding of the behavior of the system. Ó 000 Elsevier Science Inc. All rights reserved. Keywords: Thermodynamic analysis; Entropy generation; Annular packed bed; Asymmetric heating; Bejan number; Irreversibility distribution ratio 1. Introduction Much research has been concerned with describing heat transport in packed beds, especially with low bed-to-particle diameter, as the proper design of any xed bed reactor requires the wall heat-transfer coe cient h and e ective thermal conductivity k e. The reported data on the packed bed heat transfer are mainly based on measurements taken for nonreacting beds lled with spheres or cylinders (Dixon, 1988; Borkink and Westerterp, 199; Freiwald and Paterson, 199). Summers et al. (1989) reported new empirical correlations for the heat transfer parameters within an annular packed bed, which simulates the geometry of a steam reformer. The reformer catalyst bed is an annular packed bed with asymmetrically heated walls and small tube diameter to particle diameter ratio D H /D p. A vast amount of work and many correlations for the wall heat-transfer coe cients in a packed tube have been reported in the literature. However the thermodynamic analysis of such systems is mainly neglected. Introduction of packing enhances wall to uid heat transfer considerably (Colburn, 1931; Demirel, 1989; Hwang et al., 199; Demirel et al., 1999a), hence reducing the entropy generation due to heat transfer but increasing it due to uid- ow friction. The net entropy generations resulting from the above e ects provide a new * Corresponding author. address: ydemirel@kfupm.edu.sa (Y. Demirel). criterion in analysing the system (Bejan, 1996). The thermodynamic analysis may help to nd the optimum operating conditions for an existing design. It may also be helpful in a new design with a required task that generates less entropy and less lost work. Recently a packed duct with uniform heating (Demirel, 1995) and a rectangular packed duct with asymmetric heating (Demirel and Al-Ali, 1997) have been analysed thermodynamically. The objective of this study is to provide a thermodynamic analysis for the annular packed bed with asymmetric heating. The experimental data in the range 00 < Re < 800 and D H =D p ˆ 6 for the packed annulus described by Summers et al. (1989) have been employed in the analysis. Assuming a plug ow and fully developed temperature eld, the temperature pro le in the annular packed bed has been derived analytically and validated by the experimental data. With this temperature pro le and the modi ed Ergun equation (Ergun, 195), an expression for the volumetric entropy generation rate and the Bejan number have been derived and displayed graphically for the annular packed bed. The entropy generation map shows the regions where the excessive entropy is generated for a required task.. Entropy generation The Gouy±Stodola theorem states that the lost available work is directly proportional to the entropy generation in a X/00/$ - see front matter Ó 000 Elsevier Science Inc. All rights reserved. PII: S X(00)0003-1

3 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44± Notation A, A 0 parameter given by Eqs. (11) and (6), dimensionless Be Bejan number (Eq. (3)) c p speci c heat, J kg 1 K 1 D D H /D p D H hydraulic diameter of the annulus, m D p particle diameter, m h wall to uid heat transfer coe cient, W m K 1 G mass velocity, kg m s 1 J entropy generation number, Sgen 000 eto =Q, dimensionless k f thermal conductivity of uid, W m 1 K 1 k e e ective thermal conductivity of uid, W m 1 K 1 L ow path length, m n q o =q i Nu Nusselt number (Nu ˆ h D p /k f ) Q total heat ux rate, W m r r i =r o R r/r o R' r o = r o r i Re p Reynolds number Re ˆ GD p =l Re Reynolds number Re ˆ GD eq =l S 00 cross-sectional entropy generation, W m K 1 S 000 volumetric rate of entropy generation, W m 3 K 1 St Stanton number St ˆ h=qu b c p T temperature, K u velocity, m s 1 z direction of uid ow Z z= r o r i r direction normal to the ow direction Greeks a e e ective thermal di usivity, m s 1 e void fraction l Newtonian uid viscosity, kg m 1 s 1 q density, kg m 3 / ratio of entropy generation by friction to that of heat transfer (Eq. ()) s dimensionless temperature di erence (Eq. (1)) Subscripts av average e e ective f uid p packing w wall DP nite pressure DT nite temperature nonequilibrium phenomenon of exchange of energy and momentum within the uid and at the solid boundaries. The local rate of entropy generation per unit volume, S 000, of an incompressible Newtonian uid for a two-dimensional, axial and radial, annular ow is represented by (Bejan, 1996): " S 000 ˆ k T ot or ot # l oz T ou : 1 oz Here k and l are the thermal conductivity and dynamic viscosity of the uid, respectively. The terms u and T denote the velocity and temperature of the uid. The rst term on the right side of Eq. (1) shows the entropy generation due to nite temperature di erences in axial z and in radial r directions, while the second term shows the entropy generation by the uid friction. Entropy generation pro les may be constructed using Eq. (1) if the temperature and the velocity elds are known in the heat transfer medium. The control volume in the experiments of Summers et al. (1989) is in the middle part of the packed annulus, 30 cm away from the inlet and exit regions of air ow. Assuming fully developed velocity and temperature pro les for the control volume of annular packed bed the energy equation is: 1 o r or r ot or ˆ u a e dt b dz : Here a e is the e ective thermal di usivity of the bed, and T b the bulk temperature. The axial thermal conduction in the bed has been neglected in Eq. (). The in uence of the axial dispersion depends on Graetz number for the packed bed. Summers et al. (1989) and Tsotsas and Schlunder (1990) justi ed the omission of axial dispersion e ect for RePr L=D p 1; since the value for RePr L=D p ˆ6 for the annular packed considered. A maximum relative error of 1% due to omission of the axial conductance is reported over entire region of 00 < Re < 800 by Summers et al. (1989). We consider the annular packed bed described by Summers et al. (1989). It is assumed that the plug ow conditions u ˆ u av and essentially radially at super cial velocity pro les (Standish, 1984; Vortmeyer and Winter, 1984) prevail through the cross section of the packed ow passage. This is especially a common approach in two dimensional pseudo homogeneous models (Borkink and Westerterp, 199; Demirel and Al-Ali, 1997). The lumped parameter model has often been used to study the performance of a wall-cooled catalytic reactor (Froment, 197). This model assumes a plug ow. Cheng and Hsu (1986) studied three velocity models, e.g., BrinkmanÕs model with variable and constant permeabilities and plug ow in the fully developed, forced convective ow through annular packed-sphere bed. They compared the predicted and experimentally determined Nusselt numbers in an annular packed column in the range 10 < Re < 500; Pr ˆ 0:7; and found that the plug ow assumption was justi able. The uniform heat uxes of q o and q i at each of the two surfaces (Fig. 1) specify the temperature gradients at the surfaces, which provides the necessary boundary conditions with positive heat uxes when the heat ows into the uid: at r ˆ r o k e ot =or ˆq o ˆ constant; 3a at r ˆ r i k e ot =or ˆq i ˆ constant: 3b Eq. () can be directly integrated as the term dt =dz ˆ constant (Kays and Crawford, 1980). The linearity of the energy equation suggests that superposition methods may be employed to build solutions for asymmetric heating by adding the two fundamental solutions: (1) the outer wall heated with the inner insulated, and () the inner wall heated with the outer insulated. The fundamental solutions for the thermal boundary conditions shown in Fig. 1 are: Fig. 1. Superposition method for the annulus.

4 444 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44±448 f 1 ˆ T T o ˆ qo h St ZR0 1 hr0 ro R 1 r lnr ; 4 k e D H f ˆ T T o ˆ qi h St Zr R 0 where R ˆ r ; r o 1 hr R 0 ro R r lnr lnr ; k e D H r ˆ ri r o ; R 0 ˆ r o r o r i ; Z ˆ z r o r i : St is the Stanton number, and h the heat transfer coe cient. The temperature pro le for the annular packed bed can be obtained by adding the fundamental solutions of f 1 and f, and is expressed by: T ˆ T o 1 sa ; 6 where A ˆ St ZR 0 r n n 1 Nuk fro R0 k e D p D H h i R r n r 1 nr lnr r lnr r n and s ˆ Q=h ˆ Tw T b ; n ˆ qo : T o T o q i The hydraulic diameter of the annular bed is D H ˆ r 0 r i ; and D p shows the packing diameter. The Nusselt number Nu and the e ective thermal conductivity k e for the annular packed bed are expressed by (Summers et al., 1989): Nu ˆ hd p k f ˆ 5:9 Re 0:44 ; 7 k e ˆ k f 0:6 0:157 Pr Re : 8 The heat transfer parameters have been derived for an annular packed bed in the range 00 < Re < 800 and D ˆ D H =D p ˆ 6: The reported heat transfer parameters show the wide range of experimental conditions and discrepancies in the correlations (Demirel et al., 1999b). Dixon (1988), Tsotsas and Schlunder (1990) and Freiwald and Paterson (199) provide excellent discussion on the matter. The average uid temperature is obtained from T av ˆ R ro r i rt dr R ro : 9 r i r dr Using the heat transfer parameters of Eqs. (7) and (8), the average air temperature has been obtained from Eq. (9) for Re ˆ 611; D ˆ 6:3; T 0 ˆ 395 K; q o ˆ 0 (insulated outer surface), q i ˆ 6571 W m and St ˆ 0:354 and compared with the experimental data in Fig.. The predictions justify the usage of Eq. (6) as an acceptable temperature pro le in the annular packed bed. The term (dt/dz) may be calculated from the simple energy balance using the mass velocity G: q i 1 n p r o r i Šdz ˆ Gc p p r o r i dt 10 or directly from the di erentiation of Eq. (6) with respect to axial distance z as: 5 Fig.. Comparison of experimental and calculated average air ow temperatures. dt dz ˆ q ir 0 n r ; 11 Pek f D where Pe is the Peclect number. The temperature gradient in the radial direction may be obtained from Eq. (6) and is given by: ot or ˆ qir 0 r o k e D H R n r r R 1 nr : 1 The velocity may be related to the pressure by inviscid- ow behavior dp=q ˆ du b =, and using the Bernoulli equation, the velocity gradient in the ow direction is expressed by (Kays and Crawford, 1980): du dz ˆ 1 dp : 13 G dz The pressure gradient ()dp/dx) can be evaluated from the Ergun equation (Ergun, 195): "! dp dx ˆ 1 C 1 1 e l q e 3 D p C 1 e e 3 D p # G G; 14 where the rst term on the right-hand side represents the viscous ow and the second term the inertial resistance for uid ow. After substituting Eq. (14) into Eq. (13), the expression for du/dz reduces to: du dz ˆ C 1 1 e C 1 e ReŠl e 3 D p q : 15 The constants C 1 and C were given by Foumeny et al. (1993) by taking into account the e ect of con ning walls: D C 1 ˆ 130 and C ˆ 0:335D :8 : 16 The Reynolds number is based on the packing diameter Re ˆ GD p =l and does not include the bed void fraction e because of the uncertainty involved in determining the radial void fraction pro le (Summers et al., 1989). The average void fraction may be related to the packing diameter by (Foumeny et al., 1993): e ˆ 0:383 0:5D 0:93 0:73D 1 1= for D > 1:89: 17

5 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44± Substitution of Eqs. (11), (1) and (15) into Eq. (1) yields an expression for the volumetric entropy generation for the packed annulus: S 000 ˆ kf T " q i R 0 r o R n k e D H D pq i R 0 # n r Pek f D H l T r r R 1 nr e l C 1 e Rel e 3 D p q! : 18 Here the rst term on the right-hand side shows the entropy generated due to heat transfer, SDT 000, while the entropy generated due to uid friction, SDP 000, is shown by the second term, hence the entropy generation expression has the following basic form: S 000 ˆ S 000 DT S000 DP : 19 The volumetric entropy generation rate is positive and nite as long as temperature or velocity gradients are present in the medium. The dimensionless entropy generation pro le can be obtained as: J ˆ S 000 k fto Q ; 0 where Q ˆ q i q o : In the entropy generation analysis of convective heat transfer there are two new dimensionless parameters (Bejan, 1996). One of them is the dimensionless temperature di erence: s ˆ Q=h ˆ Tw T b 1 T o T o and the other is the irreversibility distribution ratio: / ˆ S 000 DP =S000 DT : Recently, the alternative irreversibility distribution parameter expressed by: Be ˆ S 000 DT =S000 ˆ 1 / 1 3 was named the Bejan number (Be) (Petrescu, 1994). Be ˆ 1is the limit at which the irreversibility due to heat transfer dominates, Be ˆ 0 is the opposite limit at which the irreversibility due to uid friction is the dominating e ect. In Eq. (18) the local entropy generation has been expressed in terms of s, R, Z and D including the properties of the uid q and c p. The rate of entropy generation over the cross section S 00 may be calculated by integration: S 00 ˆ Z ro r i S 000 r dr: 4 3. Results and discussion The experimental equipment of packed bed simulates the annular bed steam reformer and results in a variable wall temperature as detailed by Summers et al. (1989). The inner tube is 15.4 cm long aluminum with an outside diameter of 3.81 cm. The annular gap between the two tubes is cm wide and is packed with alumna catalyst support rings of cm in height and cm in diameter. The outer tube is insulated. Fig. 3. Temperature pro le (a), nondimensional entropy generation pro le J (b), irreversibility distribution ratio / (c), and the Bejan number Be (d) for Re ˆ 611; D ˆ 6:3; T o ˆ 395 K; q o ˆ 0 (insulated), q i ˆ 6571 W m and St ˆ 0:354 in the packed annulus.

6 446 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44±448 The annulus hydraulic diameter to particle diameter ratio is D ˆ 6:3, which was approximately the same as the proposed ratio for the annular bed reformer. In the analysis D and Q ˆ q i q o were kept unchanged. The adapted ranges from the experimental data of Summers et al. (1989) are 00 < Re < 800; 15:1 < Z < 30; 0:49 < R < 0:99 and 0:08 < s < 0:036: The Prandtl number is This experimental data were used to validate the predicted temperature pro le by Eq. (6), which was shown in Fig.. After that, the temperature gradients (Eqs. (11) and (1) and the velocity gradient (Eq. (13)) are substituted into Eq. (1) to determine the local entropy generation rate. The thermodynamic analysis in the present work consists of evaluating the distribution of the local entropy generation rate in the volume of the packed and empty annulus with various operating conditions and design parameters. Fig. 3 shows the pro les of T, J, / and Be in the axial and radial directions with the outer wall insulated and inner wall heated as was the case in the experiment. The values of J and Be follow the trend of temperature pro le forced by asymmetric heating. The entropy generation is high in the heated wall region. A gradual decrease of J is observed away from the heated wall. The distribution of Be shows that the contribution of heat transfer to the entropy generation decreases in dominance as radial distance increases towards the adiabatic wall. The irreversibility distribution ratio / also shows that only in the vicinity of the adiabatic wall, uid friction contribution to the entropy generation is dominant. Heat transfer to a uid owing in an annulus has technical importance because either or both of the surfaces can be heated independently. Fig. 4 shows the pro les of T, J and Be when n ˆ 0:6: The pro les display the considerable e ect of asymmetric heating from both outer and inner walls. A required task will determine the entropy generation, and the Bejan number pro les which will lead to a better understanding of the behavior of the system in the thermodynamic sense that is the equipartition of the entropy. In order to compare the entropy generation distribution in the packed and empty annulus with the same assumptions, the temperature pro le in the empty annulus can be obtained for fully developed ow with parabolic velocity eld using a similar procedure: u ˆ u av M 1 R BlnR with B ˆ r 1 lnr and M ˆ 1 r B 5 and the temperature pro le is given by using the energy equation (Eq. ()): A 0 ˆ T o 1 st 0 ; 6 where A 0 ˆ St ZR 0 1 mr m 1 hr or 0 8k 1 r 1 mr 4R R 4 4BR lnr 4BR 3 4B mr 5 4mr 3 B 1 Blnr B 1 ln R r 4r 4 and m ˆ q i =q o or m ˆ 1=n: 4B 8 8B lnr r lnr Fig. 4. Temperature pro le (a), nondimensional entropy generation pro le J (b), and the Bejan number Be (c) for Re ˆ 611; D ˆ 6:3; T o ˆ 395 K; q o ˆ 460 W m ; q i ˆ 4100 W m ; n ˆ 0:6, and St ˆ 0:354 in the packed annulus. The complete set of functions of the Nusselt numbers for the empty annulus, with fully developed laminar ow, are given by Kays and Crawford (1980). For r ˆ r i =r 0 ˆ 0:487 the Nusselt number is approximately given as: 6:45 Nu ˆ 0:5 1 n0:535 5:040 1 n0:14 : 7 Using the temperature pro le given in Eq. (6), velocity pro le from Eq. (5) and the Nusselt number from Eq. (7) the volumetric entropy generation rate for the empty annulus has been calculated and displayed in Fig. 5 for n ˆ :5: The distributions of the entropy generation and the Bejan number are the result of temperature and ow elds in the empty annulus, and are highly di erent from the distributions in the packed annulus, which are shown in Figs. 3(b) and (d). This distinction indicates the equipartition of entropy distribution in the packed annulus relative to that of the empty annulus. The lost energy/work is minimal in energy and momentum transfer processes when the driving forces of DT and DP, and hence the entropy generation, are distributed uniformly along the space variable of the annulus (Tondeur and Kvaalen, 1987; Demirel and Al-Ali, 1997; Bejan and Tondeur, 1998).

7 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44± Fig. 5. Temperature pro le (a), nondimensional entropy generation pro le J (b), and the Bejan number Be (c) for Re ˆ 3850; T o ˆ 395 K; q o ˆ 4650 W m ; q i ˆ 1860 W m ; n ˆ q o =q o ˆ :5; and St ˆ 0:003 in the empty annulus. Fig. 6. E ect of Re on temperature pro le (a), nondimensional entropy generation pro le J (b), and the Bejan number Be (c) for Z ˆ 0; T o ˆ 395 K; q o ˆ 460 W m ; q i ˆ 4100 W m ; and n ˆ 0:6 in the packed annulus. The e ect of the Reynolds number on the pro les of T, J and Be are shown in Fig. 6 for n ˆ 0; Z ˆ 0: The changes of temperature in the radial direction at the low values of Re a ect the distribution of Be which decreases sharply in the region where the temperature gradient changes its sign due to asymmetric heating. Fig. 7 shows the e ect of dimensionless temperature difference s, given in Eq. (1), on the pro les of Be at R ˆ 0.6. Here the value of T T b remains unchanged while the inlet temperature T o changes. By decreasing T o wall-to- uid heat transfer increases and Be shows slight increase. 4. Conclusions Using the combination of the rst and second law of thermodynamics, together with the temperature and velocity elds, an expression for the volumetric entropy generation in an annular packed bed with asymmetrical heating has been derived and displayed graphically. A modi ed Ergun equation with inviscid ow behavior and fully developed ow conditions are used. The in uences of asymmetric thermal boundary Fig. 7. E ect of s on the Bejan number Be for R ˆ 0:6; Re ˆ 611; T o ˆ 395 K; q o ˆ 0 (insulated) q i ˆ 6571 W m and n ˆ 0 in the packed annulus. conditions, Re and s, on the entropy generation pro les and the Bejan numbers, Be, are evaluated. For a speci ed heat transfer duty in the packed annulus, the local rate of entropy generation is closer to the con guration uniformly distributed (equipartitioned) along the space compared with that of empty annulus. The equipartition of the entropy generation is

8 448 Y. Demirel, R. Kahraman / Int. J. Heat and Fluid Flow 1 (000) 44±448 equivalent to the uniform distribution of the driving forces and of heat and momentum uxes, hence leads to less dissipation and lost energy as Tondeur and Kvaalen (1987) suggested. Therefore such a con guration is recommended in the thermodynamic analysis. The uniform distribution of entropy generation and the Bejan number lead to a better match between the operating conditions and design parameters for an annular packed reactor, hence produces thermodynamically optimum design with minimum lost work/energy. Acknowledgements Authors are grateful to King Fahd University of Petroleum & Minerals for the support provided. References Bejan, A. (Ed.), Entropy Generation Minimization. CRS Press, Boca Raton, pp. 75. Borkink, J.G.H., Westerterp, K.R., 199. In uence of tube and particle diameter on heat transport in packed beds. AIChE J. 38, 703±715. Cheng, P., Hsu, C.T., Fully-developed, forced convective ow through an annular packed-sphere bed with wall e ects. Int. J. Heat Mass Transfer 9, 1843±1853. Colburn, A.P., Heat transfer and pressure drop in empty, ba ed, and packed tubes. Ind. Eng. Chem. 3, 910±915. Demirel, Y., Experimental investigation of heat transfer in a packed duct with unequal wall temperatures. Exp. Thermal Fluid Sci., 45±430. Demirel, Y., Thermodynamic optimization of convective heat transfer in a packed duct. Energy 0, 959±967. Demirel, Y., Al-Ali, H.H., Thermodynamic analysis of convective heat transfer in a packed duct with asymmetrical wall temperatures. Int. J. Heat Mass Transfer 40, 1145±1153. Demirel, Y., Abu-Al-Saud, B.A., Al-Ali, H.H., Makkawi, Y., 1999a. Packing size and shape e ects on forced convection in large rectangular packed ducts with asymmetric heating. Int. J. Heat Mass Transfer 4, 367±377. Demirel, Y., Sharma, R.N., Al-Ali, H.H., 1999b. On the e ective heat transfer parameters in a packed bed. Int. J. Heat Mass Transfer 43, 37±33. Dixon, A.G., Wall and particle-shape e ects on heat transfer in packed beds. Chem. Eng. Comm. 71, 17±37. Ergun, S., 195. Fluid ow through packed columns. Chem. Eng. Prog. 48, 89±94. Foumeny, E.A., Benyahia, F., Castro, A.A., Moallemi, H.A., Roshani, S., Correlations of pressure drop in packed beds taking into account the e ect of con ning wall. Int. J. Heat Mass Transfer 36, 536±540. Freiwald, M.G., Paterson, W.R., 199. Accuracy of model predictions and reliability of experimental data for heat transfer in packed beds. Chem. Eng. Sci. 47, 1545±1560. Froment, G.F., 197. Analysis and design of xed bed catalytic reactors. Chem. Reaction Eng. 109, 1±34. Hwang, T.H., Cai, Y., Cheng, P., 199. An experimental study of forced convection in a packed channel with asymmetric heating. Int. J. Heat Mass Transfer 35, 309±3030. Kays, W.M., Crawford, M.E., Convective Heat and Mass Transfer, second ed. McGraw-Hill, New York, p. 98. Petrescu, X.S., Comments on the optimal spacing of parallel plates cooled by forced convection. Int. J. Heat Mass Transfer 37, 183. Standish, Y., Comments on the velocity pro les in packed beds. Chem. Eng. Sci. 39, Summers, W.A., Shah, Y.T., Klinzing, G.E., Heat transfer parameters for an annular packed bed. Ind. Eng. Chem. Res. 8, 611±618. Tondeur, D., Kvaalen, E., Equipartition of entropy production. An optimality criterion for transfer and separation processes. Ind. Eng. Chem. Res. 6, 50±56. Tsotsas, E., Schlunder, E.-U., Heat transfer in packed beds with uid ow: remarks on the meaning and calculation of a heat transfer coe cient at the wall. Chem. Eng. Sci. 45, 819±837. Vortmeyer, D., Winter, R.P., On the validity limits of packed bed reactor continuum model with respect to tube to particle diameter ratio. Chem. Eng. Sci. 39, 1430.