Ind. Eng. Chem. Res. 1990,29,313-316
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The Delplot Technique: A New Method for Reaction Pathway Analysis A method for discernment of the rank (i.e., primary, secondary, etc.) of a reaction product is developed. The first-rank ”delplot” method allows separation of primary from nonprimary products for any reaction order. Higher rnnk products (secondary, tertiary, etc.) can be separated by using (a) higher rank delplots, which are applicable for first-order and many other kinetic expressions, or (b) successive application of the first-rank delplot method. Quantitative kinetic analysis of complex reaction systems involves discrimination between not only rival models for rate expressions but also rival models for the network of reaction pathways. The use of quantitative statistical methods (Hosten and Froment, 1986) provides important information that usually narrows the spectrum of possible models but often fails to provide unequivocal discrimination. The problem is especially severe when integral data are used to discriminate between many candidate rate models within several candidate network models. Clearly a complementary procedure aimed at the independent deduction of a reaction network would be useful. Herein, we present a simple but useful methodology for the discernment of the rank (i.e., primary, secondary, etc.) of a reaction product, which is the first step in the deduction of reaction networks. The literature provides related treatments that are frequently limited by the allowable reaction order or the ease of application (Carberry, 1976; Froment and Bischoff, 1979; Hougen and Watson, 1947). The time-honored “initial rate” method allows separation of primary products but does not provide for the discernment of secondary, tertiary, and higher rank products. Wei and Prater (1962) and Lee (Lee, 1978,1985; Akella and Lee, 1981) have described elegant methods for the synthesis of the structure of a network of first-order reactions. The plot of selectivity (molar yield of P (yp) divided by conversion of reactant “A” (xA)) vs conversion (xA), used very often as a measure of reaction economics (Le., the selectivity vs conversion trade-off), contains very important network information. Originally Myers and Watson (1946), and more recently Klein and Virk (1983), used these plots to discriminate between primary and nonprimary products. Hofmann (1985) and Hosten and Froment (1986) discussed the efficacy of related analyses, e.g., a plot of yield vs conversion, to characterize both activity and selectivity in reacting systems. Herein we offer quantitative proof and extension of the utility of plots of y / x ‘ vs x (selectivity plots for r = 1)in the analysis of reaction networks. The proposed methodology is developed in two parts. The first develops the “first-rank delplot” method, which is valid for any reaction rate law and separates primary from non-primary products. The second part develops the “higher rank delplot” method for the discernment of secondary, tertiary, etc., products. Thus, the overall goal of this work is to extend the quantitative foundations of reaction network analysis.
k2
A-D
The temporal variation of species concentrations for Bo = Co = Do = 0 given in A = A 0e-(ki+kz)r
B=
k1Ao (e-(ki+kz)r- e-ksr) k3 - k1- k2
The first-rank delplot method is a plot of molar yield/conversion (yp/x) vs conversion ( x ) . The information gained from this analysis is in its intercepts as x 0. The method is best explained through an example. Consider the network (the unknown network is, of course, the goal of the analysis; we use eqs 1 and 2 to illustrate the method) composed of
-
(4)
for first-order kinetics allows formal evaluation of the delplot intercepts. For each product P, the plot of YP/XA vs xA is extrapolated to X A 0 or 7 0. This intercept, denoted as lPAfor any product P based on conversion of reactant A, can be evaluated as shown in
- -
P to A , - A
lPA = lim
(7)
by substituting eqs 4-6 for P = B, P = C, and P = D, respectively. Evaluation of eq 7 thus provides basic delplot intercepts:
-
-
Species B and D have positive delplot intercepts as x 0 or 7 0, whereas C has a zero intercept. Specification of the rank of product C is to follow. The first-rank delplot method is independent of the functional form of the kinetics of each step. Expanding the concentrations in a Taylor series allows proof. Consider the intercept for product B, i.e., ‘BA = lime (B/(Ao - A ) ) . Expanding A and B as a Taylor series in 7 gives
..
The First-Rank Delplot: Identification of Products of Primary Rank
(3)
(12)
Since the initial rate of formation of primary product is finite, the delplot gives a finite intercept. For a nonprimary product, the initial rate is zero, and hence, a zero intercept is found. It is important to note that this method does not specify the shape of the curves in the delplot. This can contribute to the limitations of the method. Since kinetic information is always at finite conversion, deduction of delplot intercepts necessarily implies the
0888-5885/90/2629-0313$02.50/0’ 0 1990 American Chemical Society
314
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 0.8 0.6
100
1............................... --.----. D
..
.......... u.0
0.1
I"....---.----.
0.3
0.2
.(
,
................................. 6
Figure 3. Second-rank delplot for the reaction network A f B -% C; A 5 D with kl/min-' = 1, kl/min-' = 2, and ks/min-' = 4. Table I. Summarv of Information from DelDlot InterceDts order rank r=1 r>l n=l 'PA = 1 "'PA = 0 for m = 1, (r - I);'PA finite "'PA= 0 for m 1, ( r - 1 ) ; 'PA diverges n < la 'PA = 1 n > Io 'PA = 1 "'PA = 0 for m =: I, r
-
a
-
, - i
'3
0.6
X
X Figure 1. Basic delplot for the reaction network A 1,B % C; A 1, D with k,/min-' = 1, k,/min-' = 2, and k3/min-' = 4.
I
0.4 0.5
24
3;
Only the last step has order n # 1.
is of 0/0 form and can be evaluated by expanding the numerator and denominator as a Taylor series:
,5
Y Figure 2. Basic delplot for the reaction network A 1,B L C; A 5 D with k,/min-' = 1,k2/(min-' conc-') = 2, and k,/min-' = 4.
extrapolation of data. It is frequently difficult to find the exact value of the intercept. However, even though the precise numerical value of the intercept provides information about reaction selectivity and stoichiometry, the essential issue in this analysis is whether the intercept is statistically nonzero. Linear regression of low conversion data and subsequent application of the F test for the likelihood of a nonzero intercept (Draper and Smith, 1966) are reasonable quantitative probes of this essential issue. Figure 1 and 2 are first-rank delplots for the reaction network of eqs 1 and 2. Figure 1, for first-order kinetics with kl:k2:k3= 1:2:4, shows clear finite intercepts for products B and D and a zero intercept for C. Furthermore, the intercepts for B and D provide information about the rate constants of the primary steps, e.g., 'BA= 0.33 = 1/(1 + 2). Figure 2, for first-order kinetics save k2 = 2 M-' and A, = 1M-l, shows the first-rank delplot to be useful for non-first-order primary reactions.
Higher Rank Delplots The higher rank delplot method allows sorting products of rank r > 1. This method is completely general for first-order kinetics and can also be used for many other types of kinetics. First-Order Reactions. Recall the network of eqs 1 and 2,where B and D were shown to be primary products and C was shown to be a nonprimary product. The higher rank delplot methodology considers integer values of r in ascending order. For r = 2, this consists of plotting Cy/x2) vs x . Secondary products (r = 2) will have finite intercepts, and higher rank products ( r > 2) will have zero intercepts. The intercept of a primary product on a second-rank delplot will diverge. For the network of eqs 1 and 2, the intercept of the second-rank delplot for product C (13)
c, +
(C'),=OT
+ f/2(C"),=,72 + ...
Since, for the network of eqs 1and 2, (A'),=, = -(kl + k2)A0,(C'),=, = 0, and (C'')r=O= k,klAo, the second-rank delplot intercept of C reduces as 2CA =
k1k3
2!(k, + k2)2
which is finite. Products of rank r > 2 can be sorted in an analogous manner. To illustrate, consider the network of ki
k2
A-B-C-D
ks
with Bo = C, = Do = 0. With these initial conditions, the initial rates are
(C'),=, = kzB,,, = 0 (C"),=o = (C')'r=O= (k,B)',,o
k,B',=o = k,klAo
(17) (18)
and, of course, A',,, = -k,Ao. Substituting eqs 17 and 18 in eq 14 gives
for the second- and third-rank delplots, respectively. Equation 19 shows that the second-rank delplot intercept is finite for a secondary product ( r = 2), whereas eq 20 shows that the third-rank delplot intercept is finite for a product of rank r = 3.
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 315 100
.OO&
..
..
u.0
' 0.1
"
0.2
"
0.3
"
"
0.4 0.5
" 0.6
011
'
uo
X
'
01
'
'
02
'
' 03
'
'
04
'
' 05
'
I 06
x
Figure 4. Second-rank delplot for the reaction network A 1,B 0 C; A . 2 D with k,/min-' = 1, k2/min-' = 2, and k,/(min-' con&) = 4.
Figure 5. Third-rank delplot for the reaction network A I, B 0 C; A % D with k,/min-' = 1, k2/min-' = 2, and k,/(min-' conc-l) = 4.
Figure 3 is a second-rank delplot for products B, C, and D for the reaction network of eqs 1 and 2 and k,:k2:k3= 1:2:4. The second-rank delplot intercepts for B and D diverge, but the intercept for C is finite at 0.22.
m > r diverge. For a product P of rank r > 1formed via a reaction with order n < 1 (all the other reaction steps are first order), the rth-rank delplot diverges. The rth delplot intercept of product of rank T > 1and n > 1is zero.
The foregoing results can be generalized for an infinite series reactions
A
- -... ki
B
ka
k,
X
where the rth-rank delplot intercept for product X is
Non-First-Order Kinetics. The higher rank delplot method provides an intercept that is linked to the reaction order for non-first-order reaction kinetics. The reaction scheme of eq 1where the rate of step 2 is second order in B (n = 2) allows illustration. In this case, the second-rank delplot intercept (2CA)diverges. However, the third-rank delplot intercept is finite (3cA). More generally, consider nth-order kinetics for step 2 in eq 1, which indicates (C') = k i B "
- k3C
(22)
The second-rank delplot intercept for C can be evaluated as
Since Bo = 0, the second-rank delplot intercept diverges for n < 1 and is zero for n > 1. The network of eqs 1 and 2 and values of kl:k2:k3= 1 s-':2 s-':4 s-l M-l and A. = 1M-I allow a final illustrative example. The second-rank delplot intercept for C, shown in Figure 4, is zero since the second reaction is second order; the third-rank delplot intercept for C, shown in Figure 5, is finite.
Characteristics of Delplot Intercepts The information derived from delplot analyses is summarized in Table I. For the sake of simplicity, parallel reaction steps are not included, and the stoichiometric coefficients are assumed to be unity. The second column shows that the first-rank delplot intercept for a product of rank one is unity. This intercept is independent of the order of the reaction. In contrast, the third column shows the dependence of the higher rank delplot intercepts on the reaction order. For n = 1 and r > 1, the rth-rank delplot intercepts of products with rank r is finite. Furthermore, for n = l, all the mth delplot intercepts for m < r are zero, whereas the mth-rank delplot intercepts for
Conclusions A new method for discernment of the rank of a reaction product has been developed. The advantages of this method are its simplicity and ease of application: higher rank delplots require no additional experimental data beyond those obtained for the fiist-rank plot. In many cases, this will allow the sequential discrimination of product ranks. This clearly is an important step in the formulation of reaction networks. In conclusion, this work provides proof and extension of the Myers and Watson (1946) method for the delineation of product ranks. The higher rank delplot is the essential new advance. Further extension of these ideas to address noninteger-order kinetics, the coupling between reaction order and species rank, and intrarank reaction steps is discussed elsewhere (Bhore, 1989). Acknowledgment We thank Dr. W. H. Manogue, Dr. G. Alex Mills, and Abhash Nigam for their suggestions.
Nomenclature A, ..., 2 = species or concentration of species ki = rate constant of the ith step n = order of reaction r = rank of species or rank of delplot xA = conversion of reactant A yp = molar yield of species P 'PA= intercept of rth-rank delplot of P based on conversion of A
Literature Cited Akella, I. M.; Lee, H. H.Generalized Methods of Synthesis of Kinetic Structures of Reaction Mixtures. Chem. Eng. J. 1981, 22, 25. Bhore, N. A. Modifiers in Supported Rhodium Catalysts for Carbon Monoxide Hydrogenation: Structure-Activity Relationships. Ph.D. Thesis, University of Delaware, Newark, 1989. Carberry, J. J. Chemical and Catalytic Reaction Engineering; McGraw-Hill: New York, 1976. Draper, N. R.;Smith, H. Applied Regression Analysis; Wiley: New York, 1966. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. Hofmann, H. How to Characterize Activity and Selectivity of Heterogeneous Catalysts. Appl. Catal. 1985,15,79. Hosten, L. H.; Froment, G. F. Kinetic Modelling of Complex Reactions. In Recent Advances i n the Engineering Analysis of Chemically Reacting Systems; Doraiswamy, L. K., Ed.; Wiley Eastern: New Delhi, India, 1986.
316 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 Hougen, 0. A,; Watson, K. M. Chemical Process Principles Part Three: Kinetics and Catalysis; Wiley: New York, 1947; Figure
'Current addreas: Mobil Research and Development Corporation, Paulsboro, NJ 08066.
172.
Klein, M. T.; Virk, P. S. Model Pathways in Lignin Thermolysis 1. Phenethyl Phenyl Ether. Ind. Eng. Chem. Fundam. 1983,22,35. Lee, H. H.Synthesis of Kinetic Structure of Reaction Mixtures of Irreversible First-Order Reaction. AZChE J. 1978, 24, 116. Lee, H. H.Heterogeneous Design; Butterworth Boston, 1985. Lee, H. H. Heterogeneous Reactor Design; Butterworth: Boston, 1985. Myers, P. S.;Watson, K. M. Principles of Reactor Design. Natl. Pet. 1946,38, 388. News Tech. SOC. Wei, J.; Prater, C. D. The Structure and Analysis of Complex Reaction Systems. Advances in Catalysis; Academic Press: New York, 1962.
Nazeer A. Bhore; Michael T. Klein* Kenneth B. Bischoff Center for Catalytic Science and Technology Department of Chemical Engineering University of Delaware Newark, Delaware 19716 Received for review April 17, 1989 Revised manuscript received October 18, 1989 Accepted November 14, 1989
