On Criteria for Axial Dispersion in Nonisothermal Packed-Bed Catalytic Reactors David E. Mears Union Oil Company of California, Union Research Center, Brea, California 9262 7
Inlet rate criteria are derived for detecting significant effects of axial heat and mass transfer in nonisothermal reactors with uniform wall temperature. The new criteria provide that the reaction rate at reactor inlet will be within 5 YO of the rate which would prevail under plug flow conditions. As in isothermal reactors, increasing the reactor length at constant space velocity is an effective way to reduce dispersion effects.
Axial dispersion of mass or heat can significantly affect the performance of packed-bed catalytic reactors when axial temperature or concentration gradients become large. In certain cases, such as the catalytic muffler, the dispersion effects may be advantageous, but in studies of catalytic kinetics it is necessary to minimize their intrusion. Consequently criteria are needed for detecting or predicting when axial dispersion will significantly distort the reaction rate. Such a criterion is available (Mears, 1971a, b) for the isothermal case. The present paper extends the approach of Young and Finlayson (1973) to derive a rate criterion for the nonisothermal reactor cooled or heated a t the wall. Young and Finlayson focused on the inlet of the reactor, using an approximate model for the problem shown in Figure 1. Radial dispersion effects were neglected on the ground that these effects would be similar in the corresponding plug flow case to which comparison was made. The reaction rate expression was effectively held constant a t the inlet value to decouple the mass and energy equations. Exact solutions were obtained for each differential equation and the corresponding plug-flow solutions subtracted. For minor contribution of axial dispersion of mass and heat, respectively, the deviations a t the inlet were given by
perature response of the reaction, which are not taken into account by these criteria. Thus a criterion is needed which will keep deviations in rate within acceptable bounds.
Inlet Criterion for Significant Deviations in Rate Such a rate criterion will now be derived by the perturbation approach for the case in which To = T,. The dimensionless temperature 8 = ( T - T,)/T, is appropriate. The solutions corresponding to eq 1 and 2 are
X - X, = DaI/Pe,,, 6' - 8, = DadPeh.,
(5)
(6)
The Damkohler numbers, defined in the Nomenclature, express, respectively, the ratio of reaction rate to bulk flow of mass and the ratio of heat generated by reaction to the bulk flow of heat. These groups are defined in terms of particle diameter instead of reactor length to be consistent with the definitions of the Peclet numbers. Dam contains T, in the denominator instead of (To - T,) which P' includes. For simplicity assume an n-order power law rate expression of the Arrhenius type & = A(exp - E / R T ) ( l - X)"
(7)
Linearizing eq 7 with a Taylor series about the plug flow value and truncating after the first terms in X and T yields
X is defined as the dimensionless concentration (CO - C)/C, or conversion, t is the dimensionless temperature ( T - T,)/(To - T,), and the subscript p refers to the cor-
where
responding values which would obtain with plug flow. The groups on the right, which are defined in the Nomenclature, have been modified slightly here to make P and p' dimensionless. The following criteria were given if axial dispersion effects are to be unimportant a t the inlet, Le., if deviations in temperature or concentration from the values that would obtain in plug flow are to be small
+
1 - n ( X - X,) Ar
Substituting the dimensional form of eq 5 and 6 gives
This equation applies only to the relatively small deviations in rate of interest here. If dispersion effects are to affect reaction rate by less than f 5 % , the necessary condition is
(3)
(4) in which the axial Peclet numbers for mass (Pe,,,) and heat (Peh,J are based on particle diameter. Unfortunately, eq 4 does not apply to the most frequently encountered case in which To = T , or to endothermic reactions. Also the relative importance of concentration and temperature deviations depends greatly on the reaction order and tem20
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
Because DaIII is negative for endothermic reactions and positive for exothermic, the effects respectively add or subtract. With endothermic reactions both effects act to diminish the rate. With exothermic reactions the effects oppose; i.e., the heat effect enhances the rate and the mass effect diminishes it. The absolute value sign permits the criterion to apply to cases in which the heat effect is greater than the mass effect (R/R, > 1). For zero-order reactions,
there is no mass transfer effect but the reaction may still be affected by heat transfer. Note that eq 10 is more precise, and hence less conservative, than the criteria of eq 3 and 4, which are expressed as 100, but has values as high as 10 a t lower Reynolds numbers due to the contribution of axial conduction. The Arrhenius number typically has values of 5-40. Thus the left side of eq 19 may be expected to fall in the range 0.1 to 400, indicating that axial dispersion of heat is usually more important than axial dispersion of mass for gases. This is particularly so for cases with high adiabatic rises, high activation energies, and low reaction orders. I t is also instructive to compare the axial heat transfer criterion with the Mears ( 1 9 7 1 ~criterion ) for freedom from significant radial intrareactor heat transfer effects
-
AH(Rdt2
1.6 < 1 8/Bi,
lEIA + r in which Bi,, the Biot number a t the wall, is here defined in terms of the tube diameter. This criterion applies a t the "hot spot" and is conservative when applied a t the inlet. There is no corresponding mass transfer criterion but computer simulations with a two-dimensional model (Carberry and White, 1969; Valstar et al., 1969) suggest that radial mass transfer effects are negligible compared to the radial heat effects. Comparison with eq 10, neglecting the axial mass transfer term, shows that radial heat transfer becomes significant before axial heat transfer if Peh,,(Peh,r)
(2)2 >
+
Nomenclature Ar = Arrhenius number, = E / R T Bi, = Biot number a t reactor wall, = h,dJk,,, C = concentration of limiting reactant, g-mol/cm3 C, = heat capacity of gas a t constant pressure, cal/g "C d , = particle diameter, cm d t = inside diameter of reactor, cm DaI = Damkohler number for bulk mass flow, = (R,d,/aCo DaII = Dsmkohler number of diffusion = (Rrp2/DeCs DaIII = Damkohler number for bulk heat flow, = (- ~ ) ( R d , / G C , T , DaIv = Damkohler number for conduction, = ( - AH)(Rr,*/A T , D e = effective intraparticle diffusivity, cm2/sec D , = axial dispersion coefficient for mass, cm2/sec E = activation energy, cal/g-mol K G = mass velocity based on empty tube cross section, g/cm2 hr h, = heat transfer coefficient a t the reactor wall, cal/cm2 hr fl = heat of reaction, cal/g-mol "C k = reaction rate constant based on unit bulk volume of catalyst, hr-l for first-order reaction k,, = radial effective thermal conductivity of catalyst bed, cal/"C cm hr k,, = axial effective thermal conductivity, cal/"C cm hr L = length of catalyst bed, cm n = order of reaction in power-law rate expression, (R = RO(l - X)" Peh,, = radial Peclet number for intrareactor heat transfer, = GC,d,lk,, Peh,, = axial Peclet number for heat based on catalyst diameter = GC,d,/k,, Pem,, = axial Peclet number for mass based on catalyst diameter, = iid,/D, Pt = Prater number, = (-AH)D,C,/hT, R , = reaction rate a t reactor inlet per unit weight of catalyst, g-mol/hr g of catalyst (R = reaction rate per unit bulk volume of catalyst, = k c " , g-mol/hr cm:' catalyst Re = Reynolds number, = Gd,/fi S.V. = space velocity, = a/L, hr-l t = dimensionless temperature, = ( T - T,)/(Ti - T,) T = absolute temperature, K a = superficial velocity based on empty tube cross section, cm/hr V = volume of catalyst bed, cm3 x = distance in axial direction, cm X = dimensionless concentration of limiting reactant, = (CO - C)/Co z = dimensionless distance in axial direction, = x / L
16 8/Bi,
Greek Symbols
In experimental reactors typical values are 0.5 < Pet,,, < 2, 3 < Peh,, < 12, 5 < d J d , < 20, and 5 < Bi, < 20. Thus the criterion of eq 21 is usually met, and radial heat transport a t the hot spot becomes significant first.
Summary Inlet criteria are derived for axial dispersion of heat and/ or mass in a nonisothermal reactor cooled a t the wall. These criteria provide that the observed rate a t catalyst 22
bed inlet will be within 5% of the rate which would obtain with plug flow. Increasing the catalyst bed length a t constant space velocity is the most effective way to minimize dispersion effects. The new criteria are expressed with precise limits in terms of established dimensionless groups. Finally, the relative importance of the individual intrareactor transport effects in experimental reactors often falls in the order: radial heat > axial heat > axial mass transfer.
Ind. Eng. Chem., Fundam., Vol. 15, No. 1,
1976
p = p&iL/GCo P' = ( - f l ) p ~ R i L / G c , ( T , - T,)
= D,/UL k ez/Gc& R = dimensionless temperature = ( T - T,)/T, h = intraparticle catalyst thermal conductivity, cal/'C cm hr p = density of gas, g/cm3 p~ = bulk density of catalyst, g/cm3 T = space time, = l/S.V., hr o = adiabatic temperature rise group = (-AH)Co/pC,T, y
?' =
Subscripts a = apparent c = center of catalyst particle f = value a t catalyst bed outlet h = heat i = value a t catalyst bed inlet m = mass 0 = value well upstream of reactor inlet p = particle or pressure s = a t catalyst particle surface w = a t reactor wall z = axial
Literature Cited Carberry, J. J., White, D.,lnd. Eng. Chem.. 6 1 (7), 27 (1969). Mears. D.E., Chem. Eng. Sci., 26, 1361 (1971a). Mears, D.E.,h d . Eng. Chem., ProcessDes. Dev., 10, 541 (1971b). Mears, D. E., J. Catal., 20, 127 ( 1 9 7 1 ~ ) . Olson, R.W., Schuler, R . W., Smith, J. M., Chem. Eng. Prog., 46, 614 (1950). Suzuki, M.. Smith, J. M., Chem. Eng. J., 3, 256 (1972). Valstar, J. M., Bik, J. D.,van den Berg, P. J., Trans. Inst. Chem. Eng. (London), 47, CE 136 (1969). Votruba. A,, Hlavacek, V., Marek, M., Chem. Eng. Sci., 27, 1845 (1972). Young, L. C., Finlayson, B. A , , Ind. Eng. Chem., Fundam., 12, 412 (1973).
Receiued for reuieu: J u n e 24, 1974 Accepted September 11, 1975
Continuous Thickening in a Pilot Plant John P. S. Turner and David Glasser' Department of Chemical Engineering, University of the Witwatersrand, Johannesburg, South Africa
The continuous thickening of a uranium plant slurry was studied on a 7-ft diameter pilot plant. Flow patterns and density profiles within the thickener were observed. It was found that there were two distinct stable modes of operation which were named "settler" and "filter" modes. The "settler" mode coincided with the underloaded operation while the "filter" mode corresponded to a fully loaded thickener. The flow patterns, internal circulations, residence-time density functions, overflow turbidities, and visual appearances of these two states were found to be very different. Maximum throughputs were 70-80% compared with those predicted by Coe and Clevenger and flux theory methods. This is attributed mainly to a more rapid settling out of the settling zone of the coarser particles, leaving the remaining pulp with a lower settling flux. Concentration zones in the thickener were not found to agree with the flux theory predictions.
Introduction The existence of concentration zones within a thickener was shown by Coe and Clevenger in 1916. In general, three main zones exist, namely the clear water zone a t the top, the settling zone in the middle, and the compaction zone a t the bottom of the thickener. Coe and Clevenger compared the settling process occurring in both batch settling and continuous thickening and they formulated a method for sizing thickeners from batch settling results. Talmage and Fitch slightly modified the theory and test procedure in 1955, on the basis of the theory of sedimentation put forward by Kynch (1952). It has. however, been shown (Dunstan and Scott, 1969; Scott and Paulsen, 1970; Cross, 1963) that substantial flow patterns can exist in a thickener, particularly the settling zone, and this circulation has definite effects on the process of sedimentation. Nevertheless, the batch settling data of both Coe and Clevenger and Talmage and Fitch have been reliably used for sizing thickeners for many years. Many of the laboratory scale experiments in thickening have used settling tanks with diameters of 30 cm or less. Under these circumstances flow patterns have not been reported as being of much significance. In order to consider the effects not present in laboratory scale equipment, a 7-ft diameter pilot plant was subjected to extensive investigation. The nature of the flocculated pulps has been discussed by many workers (Michaels and Bolger, 1962; Scott, 1968a) and this will not be repeated here.
Experimental Section Material. The slurry was obtained as a waste product from a uranium processing plant. After oxidation and ex-
traction of the uranium by means of a sulfuric acid leach in the presence of manganese dioxide, the filtered pregnant liquor is solvent extracted, and the barren acid solution is treated with lime. The stoichiometrically calculated composition of the plant slime was in weight per cent: Fe(OH)Z, 10.0%; Fe(OH),, 11.9%; Mn(OH):I, 10.8%; CaS04, 58.5%; lime impurities, 8.8%. The pulp used consisted of the plant slime minus most of the coarser particle sized lime impurities. Micrographs revealed that the suspension consisted of fairly discrete particles of iron hydroxide and transparent crystals of calcium sulfate (Turner 1972). Equipment. The continuous thickening experiments were done on a 7-ft diameter pilot plant, with central feed and underflow and peripheral overflow. The thickener was run in closed circuit with the underflow and overflow being combined in the stock tank to make up the feed pulp. The feed pulp was circulated through a steady head tank to maintain a constant feed rate. The level in the stock tank was controlled, if necessary, by make-up water on a ball valve control. The equipment and circuit diagram are shown in Figure 1. A rake mechanism was incorporated with a variable speed drive and consisted of angled rubber pads mounted below two revolving arms. Overflow was effected through 88 0.5-in. diameter holes drilled in the metal perimeter. The overflow could be adjusted in sections by blocking the holes with rubber stoppers. A conventional shallow feed well was used, with a horizontal baffle placed below the feed well to reduce the scouring effect. The feed distributor and baffle were mounted on the central shaft driving the rake mechanism, which revolved. Ind. Eng. Chem., Fundam., Vol. 15, No. 1,
1976
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