Ind. Eng. Chem. Res. 1987,26, 2306-2312
2306
Petersen, R. C.; Mataon, D. W.; Smith, R. D. J. Am. Chem. SOC.1986, 108, 2100. Randall, L. G. Chemical Engineering at Supercritical Fluid Conditions; Paulaitis, M. E., Penninger, M. E., Gray, R. D., Davidson, P. Eds.; Ann Arbor Science: Ann Arbor, MI, 1983; pp 477-498. Shapiro, A. H. Compressible Fluid Flow; Ronald: New York, 1953; Vol. I., pp 159-186. Smith, R. D. U. S. Patent 4582731, 1986. Smith, R. D.; Fulton, J. L.; Petersen, R. C.; Kopriva, A. J.; Wright, B. W. Anal. Chem. 1986,58, 2057. Smith, R. D.; Udseth, H. R. Anal. Chem. 1983,55, 2266.
Springer, G . S. Adv. Heat Transfer 1978, 14, 281. Walther, J. V.; Helgeson, H. C . Am. J. Sci. 1977, 277, 1315. Wegener, P. P. In Nonequilibrium Flows; Wegener, P. P., Ed.; Marcel Dekker: New York, 1969; pp 163-243. Wilson, C. T. R. Phil. Trans. R. SOC.(London) 1900, A193, 289. Yajima, S.; Hasegawa, Y.; Hayashi, J.; Iimura, M. J . Mat. Sei. 1978, 13, 2569.
Received for review February 18, 1986 Revised manuscript received June 30, 1987 Accepted July 27, 1987
Model Reduction for Two-Dimensional Catalyst Pellets with Complex Kinetics Anthony G. Dixon* and David L. Cressw9llt Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, and New Science Group, ICI, Runcorn, Cheshire, U.K.
A systematic approach to model simplification is presented for diffusion and reaction in finite hollow and solid cylinders, using a consecutive-parallel kinetic scheme as a test case. Over a wide range of conditions typical of industrial practice, the one-dimensional infinite cylinder model shape with d, = 4(Vp/Sx)gives very close approximations to effectiveness factors and pellet selectivities calculated from model simulations in the full two-dimensional geometry. A linear model in the significant process variables can be developed by response surface analysis of the simulations, giving algebraic equations for effectiveness factors and selectivities. The interaction between diffusion and reaction in porous catalyst pellets has probably received more attention than any other aspect of chemical reaction engineering (see, for example, the monographs by Satterfield (1970),h i s (1975), and Jackson (1977)). One reason for this is that to simulate fixed-bed reactor behavior, the actual reaction rates inside the pellet must be calculated, given the bulk gas conditions, at a large number of positions along the tube length. For spherical pellets of one-dimensional geometry, a boundary-value ordinary differential equation must be solved at each position. For cylinders and rings, which are twodimensional geometries, an elliptic partial differential equation must be solved repetitively instead. There is, therefore, a strong incentive to develop simpler methods of including the effects of the catalyst pellets in a reactor model. One convenient approach is the use of the effectiveness factor 7 which represents the ratio of the average pellet reaction rate to that at bulk or surface conditions. For an isothermal first-order reaction in a sphere, the classical results are (Froment and Bischoff, 1979) 1 (34 coth (34) - 1)
v=-
- -
4
-
34
(1)
-
where 4 = (ds/6)(kv/D)1/2.This has the asymptotic results 7 1as 4 0 and 7 1 / 4 as 4 The three, ideal, one-dimensional geometries of slab, infinite cylinder, and sphere give 7-4 curves that agree asymptotically if 4 is generalized to q5 = (V /S,)(kv/D)1/2(Wheeler, 1951; Aris, 1957). More recently hiller and Lee (1983) have presented a shape normalization that unifies the 7-4 curves for the entire range of 4, based on an infinite slab model geometry. Two-dimensional geometries have been less frequently studied, and the usual practice is to treat them by ideal *Worcester Polytechnic Institute. +
ICI.
03.
one-dimensional shapes using Vp/Sxas the characteristic length. Extensions to any form of rate expression may be made by using the generalized modulus (Bischoff, 1965)
for a single reaction in a one-dimensional model shape. In practice, for commercial catalysts, the effectiveness factor may depend in a complicated way on many process parameters, under conditions which make it hard to apply the above theoretical developments directly. In the case of multiple reactions, the effectiveness factors vi for each reaction will in general depend on all reactions. In addition, there may be cases for which the effectiveness factor concept is not meaningful (see later). The effects of product inhibition and other forms of competition for adsorption sites lead to complex rate expressions, again making it difficult to use analytical theories and serving to obscure the relation between pellet performance and process variables. In nonisothermal cases, the temperature drop between pellet surface and bulk gas must also be taken into account. Finally, it may be important to investigate changes in pellet geometry, especially for solid and hollow cylinders. Hollow cylinders, or raschig rings, are becoming of more interest as catalyst supports as energy costs increase, due to the smaller pressure drop than with solid pellets. Their main disadvantage is the lowered crush strength, which limits the diameter of the hole. An improvement in diffusional characteristics may also be expected, due to the extra external surface area provided. An extensive analysis of diffusion and reaction in isothermal solid and hollow cylinders was carried out by Gunn (1967), for first-order reversible and irreversible reactions. The effectiveness factor was shown to depend strongly on geometrical dimensions, but Aris (1975) indicates that a suitable definition of 4 brings Gunn's calculations in line 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2307 with one-dimensionalmodel results. Srarensen et al. (1973) used orthogonal collocation to investigate the first-order reaction in a nonisothermal solid cylinder and found that the Wheeler-Aris shape correction with the sphere as the model shape gave the best results for q, away from ignition or extinction points. More recently, Nigam et al. (1985) have looked a t deactivation in isothermal finite hollow cylinders using Gunn's results, while De Deken et al. (1983) used the generalized modulus of eq 2 in their analysis of steam reforming of natural gas with approximately 1/2-in. catalyst rings. Van Parijs and Froment (1984) solved the full two-dimensional pellet equations for hydrodesulfurization of naphtha in a finite solid cylinder and compared their computed effectiveness factors with simpler methods. They obtained reasonable agreement with results from a one-dimensional sphere model with equivalent diameter and also with results from a generalized modulus approach. For the series reactions considered, some fairly restrictive assumptions had to be made to apply the generalized modulus method, which essentially reduced the scheme to a single reaction. The present work reports a systematic simplification of a finite hollow cylinder pellet model for complex kinetics under realistic industrial conditions. In particular, two questions are addressed: (i) what errors are introduced in approximating a finite hollow cylinder by a one-dimensional idealization, such as a sphere, infinite cylinder, or infinite slab, and (ii) can a simple, yet accurate, algebraic equation be found for predicting pellet effectiveness and selectivity? Chemical Reactions a n d Reaction Kinetics The following consecutive-parallel reaction scheme describes several processes of industrial significance, including the proprietary one for which the present study was undertaken:
+ vlB -% C A +Y ~ B v~D +v~E C + - v1)B vBD + v ~ E A
(3)
72
-
+
(4)
rS
(5) Species A represents a hydrocarbon, which reacts with an oxidant B to form the partial oxidation desired product C. It may also undergo complete oxidation to unwanted combustion products D and E. Furthermore, the desired product C may also undergo complete oxidation to D and E. In addition to these five components,an inert I is added to the reactor feed in excess to modify the A-B explosion limits. A trace amount of a modifier M is also added, which acts as a selective poison. The true reaction kinetics for this process are undoubtedly very complex, but for an industrial "fuel-rich" feed gas over a restricted range of conditions, it is possible to develop tractable expressions. As a fist step, it is common to neglect the further oxidation of product C. An investigation of reaction kinetics over a commercial catalyst was carried out in an isothermal integral tubular reactor packed with chips of impregnated catalyst. Total pressure was held constant and pressure drop over the reactor was negligible. The mole fraction of hydrocarbon A was also kept constant, at about the level of commercial practice; however, the oxidant mole fraction, modifier level, reactor temperature, space velocity, and initial mole fraction of undesired product D were all varied over suitable ranges during the course of the experiments. The last variable was included, as there was evidence that reaction products significantly inhibited the reaction rates. ( ~ 2
P,=Pio
T=To
2L-
Figure 1. Schematic of finite hollow cylinder catalyst pellet.
The following expressions were found to'represent the best balance between physical representation and practical use of the data:
Here Tkis a reference temperature in K, and M is the modifier level in ppm. The experiments covered 6 months of continuous operation, during which time the catalyst suffered some deactivation, so the data were analyzed in blocks of 20-30 days of operating time, during which the catalyst activity stayed reasonably constant. Thus, several sets of the constants kl-k, were obtained, corresponding to different catalyst ages. These kinetics are appropriate for conditions of excess hydrocarbon. They show that the reactions are productinhibited and that the selectivity is affected by the products, as well as the activity. The selectivity is, however, only weakly temperature dependent. Catalyst Pellet Models The catalyst pellet is assumed to reside at some arbitrary point in a fixed bed and to be bathed in a flowing gaseous stream, the composition and temperature of which, whilst changing significantly over the reactor as a whole, can be considered constant over the length scale of a pellet. The dimensions of the hollow cylinder pellet modeled here are shown in Figure 1; the finite solid cylinder corresponds to the special case of a = 0. The finite hollow cylinder or finite solid cylinder are examples of two-dimensional geometries, under angular symmetry. A first step in reducing computational complexity would be to see whether the effectiveness factor and selectivity could instead be calculated from an ideal one-dimensionalpellet model, using the appropriate shape normalization. As model shapes, consider the sphere, infinite cylinder, and infiiite flat plate with diameters d, and d, and thickness d,, respectively. The model equations and their underlying assumptions are presented in the Appendix, for both the actual (twodimensional) and idealized (one-dimensional)geometries. The former case leads to a set of coupled elliptic partial differential equations, while the latter leads to coupled boundary-value ordinary differential equations. The numerical solution of each is described in the final part of the Appendix. Two-Dimensional Model Simulations In order to investigate which process variables have significant influence on q1 and sp,the factors shown in Table I were studied computationally at two levels in a 26-2 fractional factorial design (Box et al., 1978). Earlier screenings had determined that pressure had no significant effect over commercial ranges, and the pellet geometry was fixed at 8- X 8-mm outside dimensions. The composition
2308 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 Table I. Factorial Design Factors variable 1. 2. 3. 4. 5. 6.
temp of bulkgas, T, hole diameter in ring modifier level YB in bulk gas yc in bulk gas yD in bulk gas
T,- 20 0 2 0.04 0 0.04
+ T,+ 20 3 8 0.08 0.03 0.10
units
K mm PPm
Table 11. Factorial Design Levels (2&*Fractional Factorial with 5 = 123 and 6 = 234) variables run 1 2 3 4 5 6 1 2 + + 3 + + + 4 + + + 5 + + 6 + + + 7 + 8 + + + 9 + + 10 + + 11 + + 12 +++ 13 + + + 14 +15 + + 16 + + + + + +
+
+
+
+
+
+ +
+
+
+
and temperature levels were chosen to cover the ranges typically seen by a pellet in an industrial reactor, and the hole diameters represent the likely limiting values and allow a comparison between solid and hollow cylinders. The hydrocarbon mole fraction yAwas held constant at 0.3, and the remaining mole fractions of product E and inert I were adjusted to enable the variations in Table I to occur. The fractional factorial design involved a total of 16 runs. The settings of the variables for these runs are given in Table 11, in terms of the high (+) or low (-) values. Variables 1-4 follow a cyclic permutation, while variables 5 and 6 were obtained from the generating formulas 5 = 123 and 6 = 234. The use of this 26-2design means that two-variable interactions were confounded, i.e., could not be uniquely resolved; however, the primary effects of the six variables were only confounded with the three-variable interactions, which were assumed negligible. In all, four sets of the kinetic parameters k,-K7 were available, corresponding to different catalyst ages. The analysis was repeated for each age, to investigate any changes with time. The values of the responses q1 and sp are given in Tables I11 and IV for the first catalyst age, where they are regarded as the "exact" values for comparison with the simplified models to be discussed shortly. The effects found to be significant for each catalyst age are shown in Table V. Significance was determined based on standard tests involving comparison to a normal distribution (Box et al., 1978) with variance determined by the smallest (in absolute magnitude) effects. All main effects, with the exception of the limiting reactant B, were important for the effectiveness factor. The pellet selectivity was influenced by fewer factors, mainly the presence of modifier M and desired product C. Effectiveness Factor. As reaction rates increase, diffusion becomes more limiting, implying a decrease in 7: this rule governs the influence of effects. The strongest effects are of temperature and partial oxidation product C. Temperature increases force higher reaction rates and decrease effectiveness, while higher C levels retard reaction
Table 111. Effectiveness Factor ql for First Catalyst Age sphere infinite d, E: cylinder plate linear run exacts, 6V,/S, d, = dJ1.5 d, = d.13 model 0.685 0.591 1" 0.546 0.470 0.551 0.709 0.494 2" 0.526 0.464 0.555 0.993 1.049 0.982 3b 0.986 0.971 0.716 0.552 4b 0.586 0.479 0.568 0.991 1.042 0.966 0.978 5" 0.974 0.601 0.744 0.545 6" 0.571 0.512 0.833 0.884 0.940 7b 0.901 0.837 0.881 0.736 0.698 0.774 8b 0.786 0.899 0.724 0.815 9" 0.800 0.752 0.824 0.627 0.686 10" 0.650 0.598 0.971 0.916 0.902 0.935 llb 0.947 0.419 0.238 R 12b 0.327 0.182 0968 0.909 13O 0.920 0.897 0.931 0.453 0.412 0.315 14" 0.353 0.248 0.940 0.973 0.966 15b 0.952 0.911 0.869 0.837 0.926 16b 0.851 0.770
" Cylinder, 8 X 8 mm.
Ring, 8
X
8
X
3 mm; R = runaway.
Table IV. Pellet Selectivities sp for First Catalyst sphere infinite d, = cylinder plate run exact sp 6 V p / S , d, = d,/1.5 dp = d,/3 la 0.806 0.802 0.805 0.809 2' 0.770 0.767 0.768 0.770 3b 0.783 0.782 0.782 0.782 4b 0.799 0.796 0.798 0.802 5" 0.799 0.798 0.798 0.798 6" 0.815 0.812 0.814 0.817 7b 0.833 0.831 0.833 0.834 8b 0.796 0.793 0.794 0.796 9" 0.802 0.801 0.803 0.805 10" 0.775 0.772 0.774 0.776 llb 0.770 0.769 0.770 0.770 12b 0.800 0.785 0.787 R 13" 0.790 0.789 0.790 0.790 14' 0.815 0.802 0.805 0.810 15b 0.819 0.818 0.818 0.819 16b 0.791 0.788 0.789 0.791 'Cylinder, 8
X
8 mm. bRing,8
X
8
Table V. Significant Effects catalvst aee nl 1 1,5,3,2,6 2 1,5,3,2,6 3 5,1,3,2,12+35 4 1,5,3,2,6,15+23+46
X
Age linear model 0.802 0.775 0.775 0.802 0.794 0.821 0.821 0.794 0.802 0.775 0.775 0.802 0.794 0.821 0.821 0.794
3 mm; R = runaway.
S.
593 3,6,5,4,12+35 3,6,4,14+56,12+35 3,5
rates and increase effectiveness. Similarly increases in modifier level and complete oxidation product D, present in recycle, will retard reaction rates, and increase 7, although not as strongly, especially in the case of D. The increase in available surface area when a hole is present in the pellet is seen to improve effectiveness, suggesting that hollow cylinders should be preferred to solid ones for diffusional reasons as well as pressure drop considerations. It should be remembered, however, that it has been assumed that the hole surface is as effective for particlefluid transport as the "external" surface, which may not be the case. Pellet Selectivity. The modifier M is the single factor causing an appreciable increase in selectivity, while the presence of C strongly decreases selectivity. Some small decrease in selectivity is also seen as D and B levels rise. Temperature and pellet geometry have little or no effect on selectivity. Interactions. Some small interaction effects appear, which are confounded with one another. The most likely
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2309 interaction is between the factors decreasing reaction rate (C, M, D); however, this is not strong enough to merit much attention or to cast doubt upon the main effects. Catalyst Aging. Selectivity decreases slowly with catalyst age, while q1 decreases at first and then appears to increase again. The only discernible difference between ages is the appearance of D and B as factors in pellet selectivity, in periods 2 and 3, and the decreased importance of C. This soon disappears, however, and the fourth period shows the same characteristics as the first. No phenomenological explanation for this is apparent. Evaluation of Simplified Pellet Models The results of the one-dimensional model simulations for the first catalyst age are presented for comparison with the exact or two-dimensional results in Tables I11 and IV. These comparisons show that the finite hollow cylinder and finite solid cylinder can be very accurately modeled (
