ARTICLE pubs.acs.org/EF
Reaction Network Elucidation: Interpreting Delplots for Mixed Generation Products Michael T. Klein,* Zhen Hou, and Craig Bennett Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716, United States
bS Supporting Information ABSTRACT: The upgrading and conversion reactions of coal and biomass can involve complex reaction networks involving products formed in several generations from the reactants. The appearance of these “mixed rank” products in the Delplot method for reaction network analysis is examined. Guidelines are developed for the interpretation of network-oriented experimental results intended to resolve such reaction networks.
’ INTRODUCTION A kinetics model for multi-component reaction systems, such as in coal or biomass liquefaction, contains a list of reactions that constitute the reaction network and a set of rate laws that, with the associated parameter values, allow for quantitative solution. While rate law determination is covered in most kinetics and reaction engineering texts, the issue of network elucidation has received considerably less attention. However, this is an essential modeling step for multi-component systems, because it provides qualitative features of the model that can never be overridden by the quantitative values of the kinetic parameters. Conceptually, a reaction network is a graph, with the reaction component views as vertices and the reaction steps as edges. In principle, it does not have a beginning or an end point. Elucidation of this network, however, is generally accomplished in the context of laboratory experiments with clear and well-defined starting reactants. Consider such an experiment where the starting material is abstracted as A and the products are B, C, D, etc. The experimentalist will, in general, measure the component concentrations as a function of time or space velocity. The essential question in network elucidation, then, is whether a given product arises directly from A or a result of the secondary reaction of another. The purpose of the Delplot analysis1,2 is to sort out the “generations” of reaction products. Primary products are those evolving directly from the reactant. Secondary products are formed from the primary products. Tertiary products are formed from the secondary products, etc. The essence of this method is to perform kinetic experiments over a wide enough range of reactant conversions to allow for confident statistical extrapolation of the selectivities (defined as the yield divided by the conversion = yi/xA) of reaction products to conditions of zero conversion. Species with non-zero intercepts are primary, and species with zero intercepts are secondary or higher rank, with rank being the number of reaction steps removed from the reactant. Successive plots of yi/xAr versus xA expose the rank R of a product when the extrapolation to x = 0 reveals a non-zero intercept. The first-rank Delplot method is actually the classic and wellknown plot of molar selectivity versus conversion. Bhore et al.1,2 r 2011 American Chemical Society
generalized and extended these notions using the network of eqs 1 and 2 for illustration.3 1
3
AfBfC
ð1Þ
AfD
2
ð2Þ
4
ð3Þ
AfC
The first-rank Delplot method is independent of the functional form of the kinetics of each step, as shown in Figures 1 and 2. The more interesting higher rank Delplot method allows for sorting products of rank R > 1. The higher rank Delplot methodology considers integer values of r in ascending order. For r = 2, this consists of plotting (yi/xA2) versus xA. 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. Products of rank R > 2 can be sorted in an analogous manner. This method is completely general for first-order kinetics and can also be used for many other types of kinetics. Thus, the Delplot technique can provide useful elements of the reaction network by identifying the rank of a given product. It does not provide the complete network, however, because products can be of multiple ranks. The formation of COx by primary and secondary routes in selective oxidation is a clear example. Indeed, since the original publication of the Delplot method, the most common question concerns the appearance of Delplots where a product is of multiple ranks. The purpose of the present paper is to clarify that issue, as follows. First, we use the results of the Taylor series expansions summarized in the Supporting Information to expose the mathematical basis of the Delplot technique. We then extend this for Special Issue: 2011 Sino-Australian Symposium on Advanced Coal and Biomass Utilisation Technologies Received: August 1, 2011 Revised: September 6, 2011 Published: September 07, 2011 52
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Figure 3. First-rank Delplot for the network of eqs 13, with k1 = 1 min1, k2 = 1.5 min1, and k3 = 1 min1.
Figure 1. First-rank Delplot for the network of eqs 1 and 2, with k1 = 1 min1, k2 = 1 min1, and k3 = 1 min1.
Figure 2. Second-rank Delplot for the network of eqs 1 and 2, with k1 = 1 min1, k2 = 1 min1, and k3 = 1 min1.
Figure 4. Second-rank Delplot for the network of eqs 13, with k1 = 1 min1, k2 = 1.5 min1, and k3 = 1 min1.
the network of eqs 13, where product C is both secondary and primary. The Taylor series analysis is then supported by numerical simulations that reveal the shape of the plots for a wide range of rate constants, 0.01 < k4 < 10, which essentially covers the space where C is “secondary only” to where C is strongly primary.
The expansions for the network of eqs 13 are shown in the Supporting Information, where C is formed via both steps 3 and 4. Including step 4 in the formation rate and dividing by the common rate of disappearance of A show the limit to be the sum of two limits. Clearly, the behavior of the lower rank R species controls the plot because its ordinate will diverge earlier in the Delplot than the higher rank species.
’ TAYLOR SERIES ANALYSIS The expansions for the species rates in the network of eqs 1 and 2 shown in the Supporting Information allow for calculation of the various Delplots via the evaluation, using L’Hopital’s rule, of the limit in eq 4 as τ f 0. r
PA ¼ lim
τf0
P=Ao ½1 A=Ao r
’ NUMERICAL SIMULATIONS These notions are illustrated in Figures 3 and 4 as first- and second-rank Delplots for the mixed generation network of eqs 13. Figure 3, a plot of yC/xA versus xA for k4 = 0.01, 0.1, 1.0, and 10, reveals that the primary character of C4 would likely be overlooked at k4 < 0.1. Note also the shape of the curves; i.e., the increase in yC/x versus x at higher xA for k4 = 0.01, 0.1, and even 1.0 reveals the secondary character of C. The foregoing information is mirrored in Figure 4, a plot of yC/xA2 versus xA on a log-lin graph. The divergence is clear for k4 = 10, evident for k4 = 1, and detectable only for xA < 0.5 for k4 = 0.1 and 0.01. While the principal information is in the Delplot intercept as xA f 0, additional network clues can be obtained at higher conversions. This is also shown in Figure 3, a plot of yi/xA versus xA for i = B and D. Clearly, yD/xA is constant at all xA values, whereas the decrease in yB/xA with increases in xA suggests it to be the species that undergoes the secondary reaction to C.
ð4Þ
For the first-rank Delplots, e.g., r = 1, the Taylor series is shown in eq 5, and as developed in the Supporting Information, the limits for B = k1/(k1 + k2) and D = k2/(k1 + k2) are finite, whereas the limit for C = 0. For the second-rank Delplot, the limits for B and D diverge and that for C is finite at k1k3/[2(k1 + k2)2]. 1
PA ¼ lim
τf0
P0 þ ðP0 Þτ ¼ 0 τ þ ::: A0 A0 ðA0 Þτ ¼ 0 τ :::
ð5Þ
Increasing the value of the Delplot exponent r creates divergence for lower rank R products earlier than for higher rank products. 53
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Collectively, the information in Figures 24 would allow for elucidation of the network of eqs 13 for “reasonable” values relative to k1, k2, and k3. For “unreasonable” values of k4, the modeler would make the “reasonable” approximation to ignore the kinetically insignificant step in the network.
’ ASSOCIATED CONTENT
bS
Supporting Information. Taylor series expansions for the network of eqs 1 and 2. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ REFERENCES (1) Bhore, N. A.; Klein, M. T.; Bischoff, K. B. The Delplot Technique: A New Method for Reaction Pathway Analysis. Ind. Eng. Chem. Res. 1990, 29, 313–316. (2) Bhore, N. A.; Klein, M. T.; Bischoff, K. B. Species Rank in Reaction Pathways: Application of Delplot Analysis. Chem. Eng. Sci. 1990, 45 (8), 2109–2116. (3) The unknown network is, of course, the goal of the analysis. We use eqs 13 to illustrate the method. (4) Note that k1 = 1 min1, k2 = 1.5 min1, and k3 = 1 min1.
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