Non-isothermal effects in a single pellet diffusion reactor

NON-ISOTHERMAL EFFECTS IN A SINGLE PELLET DIFFUSION REACTOR J.J. MEYERS and E. E. WOLF* Department of

Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A. (Accepted

Abshct-A isothermal significant falsification proximation

16 hune 1980)

theoretical analysis of an adiabatic non-isothermal single-pellet diffusion reactor shows that noneffects can alter the value of the centerplane concentration. Deviation from isothermal behavior is under exothermic conditions where multiplicity of solutions can lead to reactor instabilities or of reaction parameters. Conversely, it is shown that the internal isothermal model is a good apto detect non-isothermal pellet behavior from centerplane concentration measurements. lNTROIKMTTlON

is possible to determine the poisoning kinetics. The effect of nonisothermal conditions in the centerplane concentration presented here, considers only the case in which constant activity prevails. it

The single pellet diffusion reactor (SPDR) has proven to be a useful tool to analyze the interaction of diffusion mass

transfer

and

catalytic

activity&31.

Balder

and

have shown that the SPDR can be used advantageously to study the poisoning kinetics occurring during cyclopropane hydrogenolysis. Wolf and Petersen[3,4] have presented further experimental and theoretical studies of catalyst deactivation occurring in a SPDR. In all the previous work, isothermal conditions have been assumed to prevail in the SPDR. In this paper non-isothermal effects in the SPDR are analyzed in terms of the variables that characterize SPDR operation, that is, nonisolhermal effects in the centerplane concentration. Previous work concerning nonisothermal effects in a single catalyst pellet published so far, have related the effectiveness factor with the Thiele modulus[6.7]. Experimental verification of nonisothermal effects in a single catalyst pellet have been presented by Kehoe and Butt[8]. Carberry[9] has analyzed Butt’s results in terms of DamkGhler numbers and found a good correlation with the nonisothermal interphase, isothermal intraphase model. The description of the SPDR, its isothermal theory and operation, have been presented in a recent review paperllO], accordingly, only a brief description will be afforded here. The SPDR consists of a single catalyst pellet which separates the reaction chamber into two sections: a bulk-phase chamber which contains the bulk reactant mixture, and a closed centerplane chamber which is very small in volume compared to the bulk chamber. Interaction between these chambers occurs through diffusional transport of species and a chemical reaction occurring on the catalyst surface. Simultaneous measurement of the reactant’s concentration, in the bulk and centerpiane chamber, allows computation via a diffusion-reaction model, of the reaction rate constants, effective diffusivity, and the effectiveness factor [ IO]. When poisoning occurs, the centerplane concentration changes with time in a manner that is characteristic of the poisoning kinetics. From dimensionless plot of the time dependent activity vs the centerplane concentration, PetersenI

11 and

*Author to whom

Hegedus

and

correspondence

Petersen[2]

THEO~ECAL

CONSIDERATIONS

The SPDR model used, considers the catalyst as a pellet of flat geometry, wherein unidirectional diffusion of mass and heat occurs simultaneously with an irreversible first order reaction which exhibits an Arrhenius temperature dependence. External transport of mass and heat takes place in the bulk chamber, but no transport of mass or heat occurs at the closed centerplane chamber. For a coordinate system located with the origin at the external catalyst face, the dimensionless governing equations are: $$=&exp(-e(i-1))

g=

O
-@#*elp(-‘(i-1))

The corresponding

boundary

O
(1)

(2)

are:

at

-- dt = Bi,, (1 - t) dZ Iz-o

(3)

I

(4)

andatZ=1

!!!I!

dZ Z-I

=0

and

=o g dZ Iz=,

where

+=CIc,;t= Bi,

should be addressed. 363

= k&D_+;

7YT,;Z=zlL; Si,, = hL/kd;

l

= ElRT,

J. J. MEYERS and E. E. WOLF

364

Bi,,, and Bi,, are the mass &nd heat Eliot numbers respectively, 4 is the Thiele modulus and p the dimensionless heat of reaction. Equations (I) and (2) are non-linear, and close analytical solutions are not feasible. Asymptotic solutions can be obtained for cases in which the centerplane concentration is negligible. This condition however, defeats the purpose of the SPDR since its theory and operation rest upon measuring the centerplane concentration. One case that allows a close SOL tion, is the internal isothermal model, wherein the temperature within the pellet is assumed to be constant and equal to T;, the temperature existing at the solid-fluid interface. Under these conditions, the differential heat balance and boundary conditions (eqn 2) are replaced by the integral balance

Fig. I. Centerplane concentration vs Thiele modulus for different values of the dimensionless heat of reaction, Bi, = 50, Bi, = 1.0, E =

L( - AH)k( T,)Cdr = h( T, - T,). The solution is:

of eqn

(I)

for the internal

isothermal

+(Z t_) = cash (QZ) - tanh (a) sinh (QZ) 1I I + @tanh WBi,,,

10.

(5) case

(6)

where @=bexp(-g(i-1). Substituting

$(Z, fi) into eqn (5) gives: Fig. 2. CenterpIane Cptanh @ =+t,1+ @ tanh CbfBi,

concentration vs Thiele modulus for different values of the Arrhenius nun&x, I?&,,= 50, Bi,, = 0. I, p = 0.01.

I).

(7)

For a given value of 4, eqn (7) can be solved by an iterative procedure to obtain t,, thus fixing a, which is then substituted into eqn (6) to obtain the centerplane concentration -$(l, k). The solutions of the internal isothermal model, eqns (6) and (7), are compared with the numerical solutions of eqns (I) and (2). The latter were obtained using the McGinnis shooting numerical technique [5]. RESULTSAND DlSCUS?5ION

Figures l-3 show the centerplane concentration in a log scale YS Thiele modulus cwves resulting from the numerical solution of eqns (1) and (2), and some solutions obtained from the internal isothermal model. Also shown is the solution of the internalexternal isothermal model, that is 6 = 0, Bi, = TV.Parametric sensitivity of the centerplane concentration curves was studied using realistic discrete values of the physico-chemical parameters I$, E, 8, Bi,,, and Bi,,. Figures l-3 show that, for the value of the parameters investigated, there is a substantial effect of the temperature in the centerplane concentration curves, even for values of p as low as 0.01. As shown in Fig. 1, the centerplane concentration increases for endothermic reactions and decreases for exothermic reactions. The % change is much more dramatic for exothermic reactions

Fig. 3. Centerplane concentration vs Thiele modulus for different values of Si, and Bi,,, 0 = 0.01, c = IO.

than for endothermic reactions. The centerplane concentration decreases rapidly as /3 increases, whereas the same drastic increase does not occur for the same negative value of fl (endothermic). Results presented in subsequent Figs. show exothermic effects only. Figure 2 shows the effect of increasing Arrhenius number 6 in +(l, t) for values of Bib = 0.1, Bi,,, = 50 and ,9 = 0.01. It can be seen that a sharp decrease in centerplane concentration occurs in a narrow range of values of the Thiele modulus. Furthermore, as l increases, multiple solutions

Non-isothermal effects

in a single pellet diffusion reactor

exist, thus for a given value of + a high and low steady state value of the centerplane concentration arise. This characteristic of nonisothermal operation[6] poses experimental difficulties, since changes from one steady state to another occur with narrow changes of .$ which could occur due to changes in fluid temperature. Also shown in Fig. 2, as solid points, are the results obtained from the internal isothermal model which are barely discernible from the nonisothermal intraphase case. These results show that, as already documented by others [S, 91, the main resistance to heat transfer in catalyst pellets occur in the gas film, which provides a simpler computing procedure for SPDR analysis. The effect of Bib in $(l, t) is shown in Fig. 3. As expected from the above discussion, keeping all other parameters the same, decreasing values of Bi, from 1.0 to 0.1, causes a drastic decrease in the value of the centerplane concentration. Figure 3 shows that if Bi,,, increases c(r(l, t) decreases, since the reaction rate increases within the pellet. In practice however, an increase in Bi, is accompanied by an increase in Bi,,. The results presented above constitute a first order approximation to non-isothermal effects in a SPDR, and show that such effects can significantly alter the value of the centerplane concentration, especially under exothermic conditions where multiplicity of solutions can lead to reactor instabilities or falsification of reaction parameters. Conversely, it is shown that the internal isothermal model constitute an alternative to detect nonisothermal pellet behavior based on centerplane concentration measurements. The question of the effect of the steel housing in which the pellet is pressed has been circumvented here since adiabatic conditions have been assumed to exist at the pellet outer radius. Analysis of this and other temperature (radiation) and transient effects require more computational effort than the first order analysis presented here. The internal isothermal model teaches that conductive transport inside the pellet is very effective as opposed to transport of heat in the gas film. Thus, if the steel housing temperature equals pellet temperature, an adiabatic condition will be attained and the analysis presented here applies. In practice, the start up will be carried out by non-adiabatic operation until an intermediate thermal condition is achieved. The validity of the results presented here depends on how close are the pellet and housing equilibrium temperatures, that is, it will depend on the temperature controller used. Adiabatic operation is likely to be attained in endothermic

365

reactions,

but might not be so easily accomplished in systems. in which case, a more elaborate analysis including non-adiabatic effects is required. Experimental design of a SPDR should provide for measurements of the catalyst temperature, pellet housing temperature and bulk temperature to evaluate and control non-isothermal and non-adiabatic contributions and thus validate calculations of rate parameter based on centerplane concentrations. Operation of SPDR should also aim at attaining the highest possible Bib.

exothermic

NOTATION

Bi c D eR E h AH k k2 L. : G

Biot number; L?b= Biot heat, Bi, = Eliot mass reactant concentration effective diff usivity activation energy heat transfer coefficient heat of reaction reaction rate constant mass transfer coefiicient effective thermal conductivily pellet thickness dimensionless temperature temperature distance from outer face of pellet dimensionless distance, z/L

Greek symbols

dimensionless heat of reaction, AH De,, C,JkenT, Arrhenius number, E/RT, Z$ Thiele modulus, _Ld(kJl&) @ non-isothermal Thiete modulus, 4 exp - [($2)((l/t)) dimensionless concentration, C/C, ,5

c

I)]

Subscripts f bulk fluid value i internal isothermal value RFSERENCE!i

[II Balder J.

R. and Petersen E. E., Chem. Engng Sci 1968 23,

1287. 121 Hegedus L. L. and Petersen E. E., Chem. Engng Sci. 1973 28 79. [3j Wolf E. E. and Petersen E. E., Chem. Engng Sci. 1977 32, 493. [41 Wolf E. E. and Petersen E. E.. L Cutal. 3977 46 190. [5] McGinnis P. H. Jr., Chem. Engng Prog. Symp. Ser. 1965 61(55) 2. [61 Aris R., The Mathematical Theory oj Difusion and Reaction in Permeable Catalysts. Clarenden Press, Oxford 1975. [71 Sattefield Ch. and Sherwood T. K., The Role of Difuusion in Catalysb Addison-Wesley, Reading, Mass. 1%3. 181Kehoe 1. P. G. and Butt 1. B., A.LCh.E.J. 1972 18 347.

I91 Carberrv J. J.. Ind. En.eng Chem. Fund 197514 129. [iOj Hegedus L. i. and P%rsen E. E., Cat. Reo. Sci. 1974 9 245.

Engng