Ind. Eng. Chem. Process Des. Dev. 1988, 25,229-236
= average of the values of 4'for the experimental set, defined by eq 11
(r$)av
Abbreviation H2 = hydrogen H2S = hydrogen sulfide DBT = dibenzothiophene BPH = biphenyl CHB = cyclohexylbenzene HDS = hydrodesulfurization Registry No. DBT, 132-65-0;Co, 7440-48-4;Mo, 7439-98-7; thiophene, 110-02-1; 2-methylthiophene, 554-14-3; 2-ethylthiophene, 872-55-9;2,5-dimethylthiophene,638-0243, L i t e r a t u r e Cited Angevine, P. J., Becker, M.; Callen, R. B.; Babkowski, M. J.; Granchi, M. P.; Green, L. A.; Heck, R. H.; Simpson, C. A.; Shih, S. S.; Stein, T. R. Report No. EPRI AF-1255, Electric Power Research Institute, Palo Alto, CA, 1979. Altar, A.; Dupuis, F. frepr., Div. Fuel Chem., Am. Chem. SOC. 1979, 24, 1. Betty, J. M. Chem. Eng. f r o g . 1974, 70(5), 78. Broderick, D. H.; Gates, B. C. f r e p r . , Div. Fuel Chem., Am. Chem. SOC. 1980, 25 (I), 53. Broderick, D. H.; Gates, B. C. AIChE J. 1981, 2 7 (4), 663-673. Broderick, D. H.; Schuit. G. C. A,; Gates, 6. C. frepr.-Div. Fuel Chem ., Am. Chem. SOC. 1978, 23, (l), 92. Carr, G. S.. unpublished Master of Science thesis, Thermal and Environmental Engineering Department, Southern Illinois University at Carbondale, Carbondale, IL, 1982. Crynes, B. L. "Chemlcal Reactions As a Means of Separatlon: Sulfur Removal"; Marcel Dekker: New York, 1977; pp 73-148. Espino, R. L.; Sobel, J. E.; Singhal, G. H.; Huff, G. A,, Jr. Prepr., Div. offetr. Chem., Am. Chem. SOC. 1978, 23(1), 46. Frumkin, H. A.; Sulllvan, R. F.; Stangeland, B. E. I n "Upgrading Coal Liquids": Sullivan, R. F., Ed.; American Chemical Society: Washington, DC, 1981; ACS Symp. Ser. No. 156, pp 75-113. Frye, C. G.; Mosby, J. F. Chem. Eng. frog. 1967, 6 3 , 9. Garg, D.: Tarrer, A. R.; &In, J. A,; Clinton, J. H.; Curtis, C. W.; Paranjape, S. M. Fuel R o c . Technol. 1980, 3, 263-284. Gates, B. C.; Katzer, J. R.; Schuit, G. C. A. "Chemlstry of Catalytic Processes"; McGraw-Hill: New York, 1979; pp 390-41 1. Greene, M. I.F u e l f r o c . Technol. 1981, 4, 117-144.
229
Houalla, M.; Nag, N. K.; Sapre. A. V.; Broderick, D. H.; Gates, B. C. AIChfJ. 1978. 24 - (6). ,-,. 1015-1021. Houalla, M.; Nag, N. K.; Sapre, A. V.; Broderick, D. H.; Gates, B. C. Paper presented at the 6th North American Meeting, The Catalysls Soclety, Chicago, IL, March 1979. Houalla, M.; Broderick, D.; deBeer, V. H. J.; Gates, B. C.; Kwart. H. frepr., Div. o f f e t r . Chem., Am. Chem. SOC. 1977, 22(3), 941. Hougen, 0. A.; Watson, K. M. "Chemical Process Principles: Part 111"; Wiley: New York, 1947; pp 906-956. Nag, N. K.; Sapre, A. V.; Broderick, D. H.; Gates B. C. J. Catal. 1979, 57, 509-5 18. Nayak, R. V., unpublished Master of Science thesis, Thermal and Environmental Engineering Department, Southern Illinois University at Carbondale, Carbondab, IL, 1979. Pannetier, G.; Souchay, P. "Chemical Kinetics"; Elsvier: London, 1967; p 317. Qader, S. A.; Wiser, W. H.; Hill, G. R. Ind. Eng. Chem. Process Des. Dev. 1988, 7, 3. Sapre, A. V.; Gates, B. C. Prepr., Div. Fuel Chem ., Am. Chem. SOC. 1980, 25 (I), 66. Satterfield, C. N. "Heterogeneous Catalysis I n Practice"; McGraw-Hill: New York, 1980; pp 259-265. Sinfelt, J. f r o g . Solid-state Chem. 1975, 10 (2). 55. Singhal, G. H.; Espino, R. L. frepr., Div. f e t r . Chem., Am. Chem. SOC. 1978, 23 (l), 36. Singhal, G. H.; Espino, R. L.; Sobel, J. E. Paper presented at the 8th North American Meeting, The Catalysis Society, Chicago, IL, March, 1979. Stein, T. R.; Bendorwitis, J. G.; Cabal, A. V.; Dabkowski, M. J.; Heck, R. H.; Ireland, H. R.; Simpson, C. A. Report No. EPRI AF-444, Electric Power Research Institute, Palo Alto, CA, 1977. Stein, T. R.; Cabal, A. V.; Callen, R. 6.; Dabkowski, M. J.; Heck, R . H.; Simpson, C. A.; Shih, s. s. Report No. EPRI AF-873, Electric Power Research Institute, Paio Alto, CA, 1976. Stiegel, G. J.; Shah, Y. T.; Krishnamarty, S.; Panvelker, S. V. "Reaction Englneering in Direct Coal Liquefaction"; Addison-Welsey Publishing Co.: Boston, 1981; pp 285-381. Thakkar, V. P.; Baldwin, R. M.; Bain, R. L. Fuel R o c . Technol. 1981, 4 , 234-250. Weller. S.; Pelipetz, M. G.; Friedman, S. Ind. Eng. Chem. 1951a, 43 (7), 1572-1 575. Weller, S.;Pellpetz, N. G.; Friedman, S. Ind. f n g . Chem. 195lb, 43 (7), 1575-1579.
-.
Received for review October 3, 1983 Revised manuscript received March 25, 1985 Accepted July 3, 1985
Weighting Factors To Obtain Kinetic Parameters from Integral Reactors with Differential Reactor Methods Sorab R. Vatcha Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
Dady B. Dadyburjor" Depariment of Chemical Engineering, West Virglnia Unlversiw, Morgantown, West Virginia 26506-6 10 1
A new method of analyzing kinetic data is formulated. Kinetic data obtained from integral reactors are analyzed by a differential method, with weighting factors on either the overall reaction rates or on the exit conversions. This method combines the experimental advantages of the integral method with the simplicity and accuracy of the differential method. The weighting factors are evaluated for several common systems of single and multiple reactions. I n general, the weighting factors depend only weakly on the functional form of the kinetic model, and they are independent of the value of the rate constants for single reactions. Hence, this method does not require the kinetlc model to be reliably known a priori. A series reaction example illustrates the power, robustness, and ease of application of the technique.
The evaluation of parameters in a reaction rate model involves either a (pseudo) differential or an integral reactor. The latter requires a more complicated analysis of the
* To whom inquiries should be addressed. 0196-4305/86/1125-0229$01.50/0
experimental data, particularly for non-power law kinetics or multiple reactions, but entails a more tractable experiment. On the other hand, the former has the advantage of an easier analysis, but the disadvantage of an (virtually) impossible experimental constraint. The experimental constraint of the differential reactor refers, of course, to 0 1985 American Chemical Society
230
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
the requirement that the reactant concentration be (virtually) unchanged from inlet to outlet, while the reaction rate through the vessel is monitored by noting the change in concentration and the flow rate. The two conflicting considerations, of accurately measurable but negligibly small changes in concentration, impose compromises in the operation of actual experimental systems, such as the requirement to operate at low conversions, and tend to force errors in the analysis of the data. The experimental systems used-spinning basket reactors, microflow reactors, recycle reactors, and others-are best termed "quasidifferential", since their behavior approximates that of the ideal differential reactor. The general features of these systems, and their merits relative to other systems, have been adequately covered in texts, e.g., Levenspiel (1972) and Froment and Bischoff (1979). Errors associated with the use of data from quasi-differential reactors, no matter how accurate the product and feed stream analyses might be, were discussed by Kraus (1964) and Massaldi and Maymo (1969). Analogous errors arising from temperature gradients in the reactor have been considered by Bercovich and Maymo (1971). These errors cannot always be made negligible by working at low conversions; for instance, Wei (1966) noted that a reactor for cyclohexane dehydrogenation does not behave "differentially" at fractional conversions of the order of 10-3. We show below how the restriction to operate at very low conversions can be relaxed without compromising the accuracy of the results or sacrificing the ease of the differential analysis method. This is accomplished by the application of certain weighting factors to the kinetic data, so that the data can now be obtained at a higher (more convenient) conversion. The basis for judicious selection of the weighting factors for several common systems is discussed. Finally, the advantage of the technique is illustrated for the case of data obtained for a series reaction. The treatment below is given for a steady-state, isothermal, plug flow, heterogeneous catalytic reactor, but an obvious and closely analogous development exists for a steadystate, isothermal, homogeneous reactor or for unsteadystate (batch), isothermal reactors.
General Formulation Consider a general rate expression for a single reaction, R A ( K , XA),where K represents the matrix of rate parameters and XAis the fractional conversion of reactant A, relative to its concentration at the reactor inlet. The elements of K , which are to be determined by experiment, depend on temperature and basis (inlet) concentration but should be independent of XAif the appropriate functional form of R A is chosen. Typically, the reaction rate monotonically decreases with increasing conversion, as illustrated in Figure 1. For such a reaction occurring in a steady-state, isothermal, plug flow, heterogeneous catalytic reactor, the number of moles of A lost per unit time and unit weight of catalyst in the reactor, i.e., the overall rate, is given by FA = F A X A , o u t / W (la) Further, a macroscopic mass balance around the reactor yields
If the functional form of R A is known, eq I b can be used to obtain estimates for the elements of K by fitting the experimental data ( W / F A vs. x A , o u t ) . However, depending upon the functional form of R A , the expression resulting
Figure 1. Geometric representation of the general rate expression RA(K,XA). The ordinate of the f i e d circle represents the (measured) fractional conversion of the exit stream from an integral reactor, and FA represents the (measured) overall rate in the reactor. The two shaded portions of the figKre have equal areas. Note that w, is the weighting factor such that XA ( ~ ( 1 -uX)XAm+ w,XA,~,J corresponds to ik Similarly, w, is the weighting factor relating R,, R,,,, and FA.
from the analytic integration of eq l b may cause problems in fitting values to the elements of K. If the functional form of R A is not known with complete confidence, and various alternative forms are to be fitted to the experimental data, then the resulting integrated expressions may cause problems in discriminating between the various alternative functional forms of R A . Instead, note that eq la,b can be combined to yield
The graphical equivalent of eq 2 following Levenspiel (1972) is shown in Figure 1. The area under the rectangle bounded by ( l / F A ) and X A , o u t is equal to the area under the curve from X A = 0 to X A = XA,out. Also note that the overall rate FA lies between the point values of the rates at the inlet and outlet, Rinand Rout,respectively, in Figure 1, and the three reaction rates can be related via a weighting factor w, such that FA = (1- w r ) R i n
+ WPout
(34
If w, is known and the overall reaction rate, F ~ is, measured, then, for a given rate expression, Rh can be calculated and the value of Rout can be obtained from eq 3a. We emphasize that Rout is the point value of the rate corresponding to the measured outlet conversion X A , o u t , Le., Rout
= RA(K,
XA,out)
(3b)
If the value of Rout is used, rather than the measured overall (integral) rate FA, then Routvs. X A , p u t data can be fitted to obtain K without the necessity of integrating the rate expression of eq l b (as in the integral method) or the necessity of using low conversions so FA and Routare approximately equal (as in the differential method). Alternatively, again from Figure 1, a fractional conversion XAcan be defined such that the point value of the reaction rate correspondingto XAis identically equal with the measured overall reaction rate, i.e., FA = R A ( K , X A ) (44 and , X A , o u t via Then a weighting factor wx relates XA,X A , ~
2.4 = (1 - W x ) X A , i n + W x X A , o u t
(4b)
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 231 Table I. Weighting Factors for Single Reactions" reaction rate expression RA(K,XA) weighting on rate w,
weighting on conversion w,d b
h
nXA,in1 0 for all cases. c 10 except for the first-order case. *For a zeroth-order reaction, weighting parameters cannot be uniquely defined. However, the assumptions of a differential reactor are explicitly valid for this special case. For M # 1. If M = 1, use the relations for knCAnwith n = 2. dOnly the value of w, in [0,1] should be used. I .oo
0.60
0.40 0.00
0.25
0.50
0.75
1.00
*A,ouI-
0.40 0.00
0.25
0.50
0.75
1.00
'A,QUt
Figure 2. Weighting factors for single-reaction, first-order systems. Curve a represents w, for constant-density systems (c = 0), w, for constant- and variable-density systems, and w, for Langmuir-Hinshelwood systems. Curves b and c represent w, for z = 1 and 4, respectively.
Figure 3. Weighting factors on rate, w,, for higher order, singlereaction systems. Curves a, b, and c correspond to reaction rate expressions that are first-order in reactants A and D with M = 1,2, and 3, respectively. Curve d corresponds to a third-order system (n = 3) with M = 3.
In the present work, XA,inis identically zero, but the extension to non-zero values is trivial. Analogous to the previous case, a knowledge of w, and measured values of X+,outand i;A allow the parameters K to be obtained, again using the experimental convenience of the integral reactor and the analytical ease of the differential model. We term w, and w, the weighting factors on the rate and conversion, respectively. Values of w, and w, can be expected to vary with the functional form of RA, the measured values of Xhout,and the relation between the values of the elements of K. This last dependence would be cause for concern-since our objective is the evaluation of these elements, we cannot expect to know a priori the relation between them. Fortunately, however, the dependence of at least one of the weighting factors on K is relatively weak. In general, no more than a few iterations (or one) are required. The above analysis implicitly assumes a monotonic decreasing reaction rate with increasing conversion of the reactant, as illustrated in Figure 1. Although most reactions follow such rate laws, there are some exceptions. Notable among these is the so-called autocatalytic reaction, where the product itself catalyzes the reaction. The rate of such a reaction passes through a maximum with increasing conversion (for details, see the texts cited earlier).
From the correspondingcurve of 1/R, vs. XAfor this case, two values of w, (and XA)can be seen to exist for a single value of FA and XA,out. Only a single value of w, will be obtained, but it need not lie between 0 and 1. While the weighting factor technique can still be used in such a case, the application is more involved and will be described in a later paper. The present work deals only with monotonic decreasing reaction rate laws, as in Figure 1. Below, we present the values of the weighting factors w, and w, as functions of the exit fractional conversion (and various values of K , where necessary) for different functional forms of the rate expression. The results are noted first for the case of single reactions; then we show the extension to multiple reaction sets.
Results and Discussion Single Reactions. Functions for w, and w, corresponding to some common rate expressions are collected in Table I. The functions for w, are obtained by equating the right-hand sides of eq 2 and the definition of w, given in eq 3a and substituting the appropriate functional form of the rate-conversion dependence for Rh and Routin turn. The functions for w, are derived by equating eq 2 and 4a, substituting the appropriate dependence for RA in the integral, and using the definition of w, given in eq 4b. For
232 Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
"0°------1
0 40 000
025
050 X*,O"t
-
075
IO0
Figure 4. Weighting factors on conversion, w,, for higher-order, single-reaction systems. Curves a-d correspond to those in Figure 3. Note how the lines are much closer together in this figure than in Figure 3.
ease of subsequent application, the weighting factors for the first-order systems are plotted as a function of X A , o u t in Figure 2. For higher order systems, w, and w, are plotted in Figures 3 and 4, respectively. The weighting factors are to be used as follows. A series of runs (with finite values of exit conversions) results in a set of ( w / F A ) vs. X A , data. ~ ~ ~For a general nth-order reaction, if the weighting factors on the rate are to be used, eq l a and 3a yield
-- 1
-
(W/FA) kn(CA,inn/XA,out)(l
- w r + w,(l
- XA,out)n/(l
+ tXA,out)nj (54
For every value of XA,out, a value for w, can be read from Figure 2 or 3. Then a straight line plot can be drawn to yield k, as the slope. Alternatively, if only a single run can be made, eq 5a can be rearranged to yield an estimate of kn.
If the weighting factors on conversion are to be used, then from eq l a and 4
-- 1
-
(W/FA) kn(CA,inn/XA,out)(l
- wxXA,out)n/(l
+ EWJA,ouJn
(5b)
Again a value of w, can be read from Figure 2 or 4,and the value of k, can be obtained as described above. For first-order systems at constant density (i.e., t = 0), w, and w, are identical and correspond to taking the logarithmic mean of the initial and final values (see Table I). This feature is of course common to all such systems obeying a linear rate law. Furthermore, first-order reto be versible systems behave analogously, with XA,out replaced by the final fractional approach to equilibrium, qout
= XA,out/Xeqbm*
Lifting the restriction of the constant density has no effect on wx, but does appreciably affect w, (see Figure 2). Similarly, for systems described by the Langmuir-Hin-
shelwood mechanism with a first-order driving force term, the same w, is applicable, but w, now depends on the adsorption equilibrium group (KACA,in). For second-order systems, the weighting factors correspond to taking the geometric mean of the terminal concentrations or rates. The curves for w, (Figure 4)at various feed compositions M have a much narrower spread than those for w, (Figure 3). The behavior of a second-order bimolecular system tends to first order (in the limiting reactant, A) as M grows large, and this trend is reflected in the weighting factors as well. For instance, for X A , o u t up to 0.5, the curves of w, and w, for M = 3 approach those for the first-order case to within 2%. In all the cases considered here, the weighting factors tend to unity in the limit of complete conversion, generally following a very steep rise near X A , o u t = 1. The limiting case arises because complete conversion is mathematically possible only for an infinite space time. For an infinitely large reactor (or a batch reactor to infinite time), the weighted mean would lie a t the "final" value. Operation of an integral reactor to generate kinetic data at conversions approaching completion, in order to take advantage of this unity weighting factor, is not recommended. Due to the steep rise in the curves in this region, conversions near unity may have weighting factor values that are quite far from unity and very sensitive to the actual value of the conversion. For the first-order case, for example, Figure 2a indicates a value of w, = 0.8 at X A , o u t 0.99, even though W, 1 as X A , o u t --* 1. It is also worth noting that the other limiting case, of zero conversion, yields values of 0.5 for all weighting factors. This is of course the value implicitly used in the "differential" analysis of rate data. Multiple ,Reactions. The evaluation of all the rate constants in a system of two or more independent reactions requires measurements on several components. The issues of which of the independent components to monitor, should a choice be available, and of the advantages of monitoring more than the independent number of components, have not been completely resolved, but Ravimohan et al. (1970) and others have formulated guidelines for certain simple systems. In the present work, we restrict our attention separately to two single sets: a set of two parallel reactions and a set of two reactions in series. Extensions of these situations are straightforward. Further, we assume that we have available, as functions of W/FA, conversion data for the reactant (again on an inlet concentration basis, Le., XA,h= 0) and formation data for one of the products. To estimate the two rate constants then requires two weighting factors. In principle, both could be on rate, or on conversion, or one could be on each of the two parameters. Parallel Reactions. Extension from the single reaction solution is the most straightforward for the case of the parallel reaction set
-
/ A
C
particularly if the two reactions are of the same order, n. Then the two rate constants kB,, and kc,, can be added to give the overall nth-order rate constant kA,, for the loss of A. This parameter is estimated from the known value of X A , o u t and by using eq 5, with w, or w, obtained from Figures 2-4 or Table I. Further, if C is the product whose exit concentration is known, then kc,, can be estimated by
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 233
Table 11. ExDressions for Parallel Reactions, First and Second Order
Table 111. Expressions for Series Reactions, First Order Darameter weighting on rate weighting factor, w (kA = kc) [1 - (1 + t ) exp(-t)]/[t2 exp(-t)]
weighting on conversion
( l / t ) l ( ( t - 1 + exp(-t))/[l - (1 + t ) exp(-t)l) - 11 weighting factor, w ( k A # k c ) [P - 1 + exp(-pt) - P e x ~ ( - t ) l / [ ~ t ( e x ~ (-- texp(-pt))l ) ( l / t N ( p- l ) ( t - 1 + exp(-t))/[p - 1 + exp(-pt) - P exp(-t)l) - (l/p)I design equation: Y c ,[ ( ~ W/FA)~CA= , ~ ~WxXA,out I - wyY~,iut ~ , ( X ~ , ou tYc,nut) exit conversion, XA,out 1 - exp(-t) fractional formation, Yc,out ( k A = kc) 1 - (1 + t ) exp(-t) fractional formation, YC.,, (kA # kc) (p - 1 + exp(-pt) - p exp(-t))/(p - 1)
-
noting that eq 5 can be applied to the single reaction A C. This is done by substituting the exit fractional conversion to C, YC,out,for XA,out in eq 5, and by obtaining w,or w,by using Yc,out,instead of XA,out,in Figures 2-4 or Table I. If the two parallel reactions are of different orders, the situation is more complicated. In this work, we treat only one such case, where the reactions are of first and second order. Then the overall rate is given by RA = k,cA + k,CA2 (7)
For this case, the weighting factors and the analogues of eq 5 for the analysis are shown in Table 11. The functions for w,and w, depend not only on XA,out but also on the nondimensionalized ratio of the rate constants, q. Figure 5 depicts curves for w,and w, for a few values of q. The curves are all in the region between those for pure firstorder reactions and pure second-orderreactions in Figures 2-4, corresponding to the limiting cases of q 0 and q 0 3 , respectively. Hence, the weighting factors are only weakly dependent on q. This is particularly true for the case of w,,for which the first- and second-order curves are close together. Further, for values of q greater than 10, values of both w,and w,approach their second-order value to within 1.5%. Consequently, in a large range of values of q, one of the two limiting cases (first or second order) can be used to obtain values of the weighting factors. Series Reactions. For the series reaction set
- -
kA
kc
A-B-C we consider that the exit concentrations of reactant A and final product C are measured. A detailed treatment is presented only for the case of two first-order reactions. Either of the two types of weighting factors can be chosen for determining the first rate constant kA. However, since the exit concentration of the product C depends on both kAand kc, the value of whichever type of weighting factor is used for kc depends upon the type of weighting factor chosen for kA. The weighting factors for C used here have been determined as follows. For the weighting factor on conversion (to product C), wy,the fractional conversion of A appearing in the rate expression is the exit conversion multiplied by its own weighting factor on conversion, w,, obtained from Figure la. For the weighting factor on the rate (of formation of C), w,,the fractional conversions appearing in the rate expression are the actual (unweighted) exit values, Le., XA,out and Y C , ~ , Other ,~ choices are possible, of course, but based on our experience with single reaction systems, we expect that wy and w,determined in this way would have the weakest dependences
1
0 40 000
1
025
050
I
075
IO0
XA,O"+-
Figure 5. Weighting factors for parallel reactions, one each of first and second order, as functions with exit conversion. Unbroken curves represent the weighting on conversion, w,; dashed lines represent the weighting on rate, w,. Numbers indicate values of the parameter q = k2CA,,,/k,.
on system parameters. Expressions for w,and w y ,and relations for their use analogous to eq 5, are shown in Table 111. Plots of w,and wyare given in Figure 6 as functions of YC,~,,~ up to the inflexion point. First note that the functional relationships between w,,wy,and YC,,,, are unchanged on interchanging kA and kc. This stems from the fact that Yc,outitself exhibits this property. Further, note that the curves for w yshow a much narrower spread than those for w,over a 64-fold range of values of kc/kA ( = p ) . In other words, the weighting factors for conversion are much less sensitive to the values of the actual kinetic parameters than are the weighting factors for rate. This is not surprising, inasmuch as in the former case the observations on A are already independently optimally weighted and have been "taken care of", leaving w y to optimize the weighting on the terminal conversion of C alone. Finally, in the limit as Y,-,,ut tends to zero, w,tends to 0.5, just as for single reactions. Values of wy are much lower than 0.5; in fact, Table I11 shows that w y tends to --oo as Yc,outtends to 0. However, for most measurable
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
234
Table IV. Illustration Using Series Reaction of Prasad a n d Doraiswamy (1974) mole fractions
WIFA, g-hlgmol 19.5 34.0 70.0 88.5 117.5 240.6 std deviation of value av value value from original ref u k AI
=-
XA,out
yC,out
0.165 0.270 0.475 0.568 0.660 0.899
0.002 0.000 0.015 0.028 0.048 0.144
103kAC,4,,,; units mol/g/h. *kC'
first reaction _____ w,,w, kAfa
wy kc' (k,/k, = 1)
W , kc'* (kA/kc = 8)
0.52
9.26
0.34
1.21
0.34
1.21
0.51
1.23
0.51
C
C
C
c
c
C
C
C
C
0.34 0.35 0.35 0.37
0.85 1.03
0.35 0.36 0.37 0.40
0.54 0.56 0.58 0.68
9.12 9.41 9.10 9.61 fO.19 9.30 9.32
1.12
1.07 rt0.12 1.06 1.05
0.85 1.03 1.12 1.08 f0.12 1.06 1.05
0.53 0.54 0.56 0.63
0.88 1.08 1.19 1.26 f0.14 1.13 1.05
0.55 0.58 0.60 0.72
1.23 c
0.85 1.01 1.11 1.10 zk0.13 1.06 1.05
lO3kCCA,,,,; units mol/g/h. 'This datum discarded due to error in Yc,,,,
an average value of k c is obtained by plotting XA,outand YB,out as functions of ("/FA) and noting that
0 75
kA/kC 0 65
0 55
t %,Wy
0 45
0 35
0 25 01
02
0.3
YC,O"t-f
Figure 6. Weighting factors for the second of two series reactions, both of first order, as functions of the exit formation of the final product. Unbroken curves represent the weighting on formation, wr; dashed lines represent the weighting on rate, w,. Numbers indicate values of the parameter p k,-/kA.
values of Yc,out,the values for wy are greater than zero, as can be seen from Figure 6.
Illustrative Example Here we consider the data of Prasad and Doraiswamy (1974) obtained for the series reaction tetrachloroethane (A)
second reaction wy kc' wy kc' (kA/k, = 8 ) (kA/kc = 1)
kA
kC
pentachloroethane (B) hexachloroethane (C)
The chlorinations were run at 200 "C over an activated silica gel. An excess of chlorine was used so reaction rates could be assumed to be first order in the substituted ethanes. Data were obtained on product distribution (mole fractions of A, B, and C on a chlorine-free basis) as a function of w/FA and the time on stream. The latter variable will be ignored since the present work is not concerned with catalyst deactivation. The original treatment calls for a straightforward integral analysis of the data for the disappearance of A, yielding more-or-less constant values of kA with changing (w/FA)values. Then,
=
y B , ~ ~ t / Y A , ~at ~ t dYB,out/d(W/FA)
= 0 (9)
Le., at the maximum value of the intermediate product. The present treatment uses w, and w, independently to estimate kA, with wy and w, independently (each in conjunction with w,) used to estimate kc. The results are shown in Table IV for values of ( W/FA) up to the inflection point for C . There is an obvious error, typographical or experimental, in the value of Yc,outat W/FA = 34.0; hence, that datum is not used in what follows. First we note that for this constant-density, first-order system, both weighting factors for k A are identical (see Table I), as are the equations that allow kA to be calculated for each factor in turn (eq 5). Hence it is not surprising that identical values of kACA,inare obtained for both of the weightings on overall rate and that on exit conversion. The value of the present estimate of kACA,inagrees well with that obtained by Prasad and Doraiswamy (1974). Indeed, the differences can only be attributed to round-off errors, since the original method and the present one are mathematically identical for this simple first-order case, although the two approaches are different. To estimate kc, the weighting is first performed on the exit fractional formation Yc,out;i.e., w,,is used, together with w,. From Figure 6, an estimate of p ( = k c / k A ) is needed. For this example, we can obtain a good estimate of this value from Prasad and Doraiswamy (1974). However, to demonstrate the power of the present technique, we use a base case value of p = 1 as a first iteration and show that no more than a single additional iteration is necessary, and that too is for confirmation purposes only. Based on the values of k,$A,in obtained in Table IV by using p = 1 to obtain wy,we can obtain a better estimate of p , and we then use this value (p = 8) to obtain a second set of values of w,,and kCCA,in.The two sets of values of kcCk, in Table IV are identical to three significant figures, and the average value of kCCA,in from this approach differs only in the third significant figure from the single value estimated by Prasad and Doraiswamy (1974). When the weighting is carried out on the rate of formation of C using w,, the results are not qualitatively different. These are also shown in Table IV. The results of the fist iteration, using p = 1to estimate w,,are slightly different from those evaluated by using w, and the improved estimate, p = 8. The standard deviation for the second iteration using w, is smaller than that for the first iteration with w,.The average of the second iteration with w, is consistent with the average values corresponding to w y and either p = 1 or p = 8.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 235
In all four evaluations of kc, it can be seen that the point values do not differ from the average value in a systematic manner. A trend in the variation of kC values with WIFA could be interpreted as suggesting a reaction order other than that which is assumed. Consequently, the assumption of first-order reaction rates is probably valid. A comparison of the present treatment and that of Prasad and Doraiswamy (1974) would seem to indicate no particular advantage of using the weighting functions of kA, except perhaps the greater ease in reading a value from a graph. This lack of advantage is to be expected for the single, first-order reaction. (The present treatment would be expected to be advantageous for the more complicated higher order, but still single, reactions.) For the second reaction in the series set of multiple reactions, the present treatment sidesteps the inconvenience of having to draw graphs for each product, especially to determine the maximum point of the intermediate produce so as to use eq 9. Further, the present method provides estimates of kc for each (W/FA) used. This allows for a check of the first-order assumption for kc,which is not found with the other method.
Conclusions Tables 1-111 are intended to be illustrative rather than exhaustive, although the entries do represent the large majority of real chemical systems. In any case, the theory and concepts used are quite general and can be readily adapted or extended to other more complex systems. For a reacting system for which the stoichiometry and the functional form of the kinetic model are known with confidence, the technique and the results presented here can be applied directly to the kinetic data to yield accurate values of the rate constants, free from errors associated with the differential reactor approximation, when operating even a t a moderate extent of reaction. Even when the exact rate function is not known, it can be seen from Figures 2-6 that a reasonable first estimate of a t least one of the weighting factors (at some known extent of reaction) is still possible. This arises from the weak dependence of a t least one of the weighting factors on the rate function. This first estimate can then be used to find the appropriate dimensionless group of rate constants, from which a more refined estimate of the weighting factors can be made. This feature considerably enhances the utility of the techniques presented here and leads to results much more accurate than by the heretofore common practice of arbitrarily assigning values of the rate to the inlet concentrations, or to the arithmetic mean of the inlet and outlet concentrations. Further, these techniques are useful for analyzing existing data as well as in the planning of new kinetic experiments. The decision as to whether to weight the exit conversion/formation or the overall rate must be based on the particular reaction system under consideration. For the example considered in Table IV, in particular, weighting the exit formation led to an accurate estimate of the rate constant for the second reaction in a single iteration, even though the initial estimate of the rate constant was off by almost an order of magnitude, by design. In this case, weighting the overall rate was seen to require an additional iteration. In fact, in all the cases considered in the present work, weighting the exit conversion/formation appears to be at least as good an approach, or a better approach, than weighting the overall rates. A possible counterexample may be the case of the autocatalytic reaction, only briefly discussed above. Here, weighting the overall rate may be the preferable route, since there is only one value of w,, while two values exist for w,. To generalize, the preferable approach is the one in which the corresponding weighting
factor is less sensitive to the unknown kinetic parameters and their dimensionless combinations. Acknowledgment Acknowledgment is made to NSF EPSCoR, Institutional Matching Project, RII-8011453, for partial support of this work.
Nomenclature reactant products in parallel or series reactions, eq 6 or 8 concentrations of A, B, C, and D reactant molar flow rate of A into reactor adsorption equilibrium constant in LangmuirHinshelwood rate expression, Table I first-order rate constants in series reaction set, eq 8 nth-order rate constants in parallel reaction set, eq 6, kAn = kBn -k kcn nth-order rate constant for single reaction set, Table I first- and second-order rate constants for parallel reaction set, eq 7 ratio of inlet concentration and stoichiometric coefficient of reactants, (vACD,in)/(vDCA,in) number of moles of A and C order of reaction +lka =kicA:ii/kl rate expression for the loss of reactant, as a function of K,-XA,and Yc point values of RA at reactor inlet and outlet overall (measured) rate of disappearance of A overall (measured) rate of formation of C in series reaction -
[1 - XA,outl
volume of given reaction mixture before reaction starts and after complete conversion, used in the definition of weight of catalyst in reactor weighting factor on FA, eq 3a weighting factor on ?c in series reaction, analogue of w, weighting factor on XA,out, eq 4b weighting factor on YC,,,, in series reaction, analogue of w, fractional conversion of A, with the inlet as the basis, =(NA,in - N A ) / N A , i n , =(CA,in - C A ) / C A , i n if t=O value of XA at which the point reaction rate R A equals the observed rate in the reactor, Figure 1
Greek Letters fractionalvolume expansion factor, =( V , - Vo)/Vo; t 0 for a constant density reaction K matrix of reaction rate parameters, for a single nth order reaction set, there is only one element, k, V stoichiometric coefficient Subscripts in condition at reactor inlet out condition at exit of reactor Literature Cited t
Bercovich, S. E.; Maymo, J. A. J . Cefel. 1971, 22, 64. Froment, G. F.; Blschoff, K. B. "Chemical Reactor Analysis and Design"; Wlley: New York, 1979. Kraus, M. Collect. Czech. Chem. Commun. ISS4, 2 9 , 2710.
236
Ind. Eng. Chem. Process Des. Dev. 1986, 2 5 , 236-241
Levenspiel, 0. "Chemical Reaction Engineering", 2nd ed.: Wiiey: New York, 1972. Mas&& H. A.; Maymo, J. A. J . c8&/.igse, 74, 61. Prasad, K. E. S.; Doraiswamy, L. K. J . Catal. 1974, 32, 384. Ravimohan, A. L.; Chen, W. H.;Seinfeld, J. H. Can. J . Chem. Eng. 1970, 48, 420.
Wei, J. Ind. Eng. Chem. 1966, 58, 38.
Received for review October 22, 1984 Revised manuscript received June 4, 1985 Accepted July 3, 1985
Deactivation Kinetics of Platinum-R henium Re-forming Catalyst Accompanying the Dehydrogenation of Methylcyclohexane Ajlt K. Pal, Madhumlta Bhowmlck, and Rameshwar D. Srlvastava' Department of Chemical Engineering, Indian Institute of Technology, Kanpur -2080 76, India
The kinetics and mechanism of deactivation by coking of Pt-Re-AI,O, catalyst for the dehydrogenation of methylcyclohexane have been examined along with the kinetics of the main reaction. A statistically best rate expression for the main reaction, developed on the basis of the single-site adsorption of methylcyclohexane,was determined from the experimental data. Deactivation occurred in parallel with the main reaction where methylcyclohexane was adsorbed in different ways in the main and the deactivation reactions. The deactivation kinetic equation was governed by the reaction of two adjacent adsorbed methylcyclohexane molecules, resulting in the coke precursor.
Catalysts for re-forming reactions are small crystallites of Pt or Pt-Re supported on alumina. The addition of Re greatly improves the catalyst's resistance to poisoning by coke. The deposition of carbon may be due to side reactions or due to the decomposition of the organic reactant. As a consequence, the kinetic study of the main reaction in itself becomes seriously complicated. Although investigationsto determine the role of rhenium in the platinum-re-forming catalyst have been made for some time, deactivation-kinetic analysis has been slow to develop. It is only recently that some rate data have been obtained (Jossens and Petersen, 1982; Pacheco and Petersen, 1984a, 1984b). A recent bibliographic review on how Re addition brings about the promotional effects can be found in the references (Jothimurugasan et al., 1985). Jossens and Petersen (1982) and Pacheco and Petersen (1984a, 1984b) studied the deactivation kinetics of the Pt-Re re-forming catalyst accompanying the dehydrogenation of methylcyclohexane (MCH) in a single pellet diffusion reactor. Their analysis with the use of a single pellet diffusion reactor required an accurate measurement of the "center plane concentration". Furthermore, in their models, the fouling analyses are restricted to simple power law model relations. Recently, Corella and Asua proposed (1981) and generalized (1982) theoretically the kinetics of deactivation by coking that relates activity directly to the deactivation reaction and obtained the deactivation equation of the form
where $(pi,") is the deactivation function and d = (m + h - l)/m, m and h are the number of active sites involved in the controlling step of the main reaction and deactivation reaction, respectively. This method has been suc-
* Present address: Department of Chemical Engineering, University of Delaware, Newark, D E 19716. 0196-4305/86/1125-0236$01.50/0
cessfully applied in the studies of catalyst deactivation by coke formation in furfural decarbonylation on Pd-Al2O3 (Srivastava and Guha, 1985) as well as in the dehydration of isoamyl alcohol over silica-alumina catalyst (Corella and Asua, 1981). The above proposal is a more fundamental approach to the analysis of catalyst deactivation. As an example of the application of this approach and in keeping with the goal of this study, .it is appropriate to consider the case of MCH dehydrogenation over Pt-Re-A1203 catalyst. The experimental results are analyzed on the basis of LangmuirHinshelwood kinetics for the main reactiion as well as for the deactivation reaction with statistical data interpretation. Experimental Section Catalyst Preparation. Platinum-rhenium catalyst on
alumina support was prepared by the impregnation (incipient wetness) technique. The chemicals used were chloroplatinic acid (Johnson Mathey, London), rhenium heptaoxide (Riedel-De Haen, West Germany), and 7-alumina with a BET surface area of about 220 m2/g. After impregnation, the catalyst was dried in air a t 373 K and calcined in an air stream for 5 h a t 723 K. The catalyst composition was alumina with 0.3 wt ?& Pt and 0.3 wt 7'0 Re (Pt-Re-Al,03; metal area = 0.35 m2/g). Individual metal catalysts with 0.3 wt % Pt and 0.3 w t % Re on y-alumina were also prepared under the same conditions except for Re-A1203 where calcination was not performed. The catalysts have been recently characterized by the use of proton-induced X-ray emission and Rutherford backscattering spectrometry (together with electron microscopy and chemisorption studies) for the dehydrogention of methylcyclohexane (Jothimurugesan et al., 1985). Kinetic and Deactivation Runs. Catalyst was tested primarily for the initial activity and resistance to deactivation by using a micropulse reactor-chromatograph assembly, Hewlett-Packard Model 5880. With this apparatus, either single-pulse or a train of pulses of known volume of MCH and frequency can be automatically 0 1985 American Chemical Society
