Delplot rank

For reactions that include irreversible steps of higher reaction orders, the same procedure is used. However, a distinction between the Delplot rank and network rank of a participant must now be made. The Delplot rank R is related to the intercepts as for networks with first-order steps only and is obtained by inspection of the plots. The network rank N indicates the provenance N = 1 for primary participants, N = 2 for secondary participants, etc. The Delplot rank of the participant L of a step nKK — L (n th order in K) is related to the Delplot rank of its parent K by... 

For example, if the step K — P in the network 7.1 were replaced by a second-order step 2K — P, the Delplot rank of P would become 1 -I- 2 1 =3, although the network rank of P as a secondary product is 2. 

This procedure introduces uncertainties because the deduction of the network rank from the Delplot rank requires some prior knowledge about the network. For example, suppose the Delplot ranks of K and L are 1, that of M is 2, and that of N is 3, that M is likely to be formed from K, and N from K and L. If so, K and L could react directly to N in a second-order step, and K to M in a separate first-order step (Case I). Instead, K could form M, which then reacts with L in a two-step reaction that is first order in M and zero order in L (Case II, with X + L — N much faster than M — X) ... 

Non-simplicity is caused by intermediates whose concentrations rise above trace level, or by steps in which two or more molecules of intermediates function as reactants. Non-simplicity caused by the first of these possibilities usually becomes apparent immediately, when the known participants in a reaction are sorted into reactants, products, intermediates, and possibly catalysts and silent partners. Where this is not so, say, because the number of participants is very large—not uncommon in hydrocarbon processing and combustion—, Delplot rank ordering can help to distinguish intermediates from end products (see Section 7.1.2). Nonsimplicity caused by reactions of trace intermediates with one another may not be apparent at the outset, only to turn up as the mechanism becomes clearer. If so, the kineticist will have to cross that bridge when he comes to it. 

Reactions with other than unimolecular steps pose still another difficulty. Formula 7.3 yields Delplot ranks if the network is known, but the user wants the reverse, to deduce networks from Delplot ranks. Ambiguities as to provenance may arise. For example, in both the two networks below, the Delplot ranks are 1 for P and Q, so that no distinction is possible without additional information. 

In so simple a case, stoichiometry can settle the issue If P is not an isomer of A, the network on the left is incorrect, and if it is not an isomer of B, the network on the right is incorrect. However, stoichiometry cannot decide between the following two networks, both with Delplot ranks 1 for P and R, and 2 for Q ... 

The rank is a recent concept, developed and formalized at the University of Delaware [2,3], based in part on earlier work by Myers and Watson [4]. The name "Delplot" for the plots to determine it alludes to this origin. The concept and procedure have not yet found their way into standard texts on reaction engineering. 

Primary participants give Delplots with finite intercepts. Participants of higher ranks give Delplots with zero intercepts. 

The corresponding Delplots are shown in the upper diagram of Figure 7.1. The plots for K and Q have finite intercepts while that for P has a zero intercept. This allows K and Q to be identified as primary participants, and P as one of higher rank. 

Figure 7.1. Delplots for K, P, and Q in batch reaction with network 7.1 and rate coefficients fcAK = 1, kKP = 4, kAQ = 2 (arbitrary units). Top first-rank plots bottom second-rank plots. (Adapted from Bhore et al. [3].)...

Example 7.2. Pyrolysis of n-pentadecylbenzene [5]. Upon pyrolysis, n-pentadecyl-benzene decomposes to about sixty different products. The major ones are toluene, styrene, n-tridecane, 1-n-tetradecene, and ethylbenzene. First-rank Delplots of experimental results are shown in Figure 7.2. 

Figure 7.2. First-rank Delplots of major decomposition products in pyrolysis of n-pentadecylbenzene at 400°C (from Savage and Klein [5]).

The two most general features of a reaction are the apparent kinetic orders with respect to the participants (reactants, products, intermediates, catalysts, and silent partners) and the ranks of the intermediates and products. Reaction orders may vary with conversion, so accurate values are not sought. Ranks, established by Delplots, provide an indication of the sequence in which the respective species are formed, and are useful primarily in the study of reactions with many participants and about whose networks little is known to start with. 

This principle is formalized and sharpened in the Delplot method. In first-rank Delplot [3], selectivities Sf = yP //A are plotted versus/A, where yP is the yield of product P and/A is the fractional conversion of the original (limiting) reactant A. 

Figure 7.1. Concentration histories and first-rank Delplots of primary and higher-rank products.

First- and second-rank Delplots are shown in Figure 7.3. In the first-rank plots the extrapolated intercept for butyric acid appears to be close to zero, and that for butanamide close to one (the selectivities must sum up to 1.0). This indicates that butanamide is a primary product and butyric acid is not. 

Figure 7.3. First- and second-rank Delplots of butanamide and butyric acid in hydrolysis of butyronitrile at 330°C (from Iyer and Klein [5]).

To distinguish between higher ranks, the quantity yJ f R is plotted for each product i versus/A successively with/ =1,2, etc. (R-rankDelplots) and extrapolated to zero conversion. Delplots with R = 2 are called second-rank those with R = 3, third-ranked etc. Provided all steps are first or pseudo-first order, ranks of products can be identified as follows ... 

For example, in butyronitrile hydrolysis, the second-rank Delplots for butanamide diverges, and that for butyric acid has a finite intercept (see Figure 7.3). This confirms butanamide as the primary product and butyric acid as the secondary one. 

The primary, but unstable products in autoxidation of paraffins are hydroperoxides, which quickly decay to ketones and more slowly to alcohols and acids. Typical selectivities to hydroperoxides and ketones as functions of fractional conversion are shown in Figure 7.4 (first-rank Delplots). The selectivity to hydroperoxides as sole and unstable primary products is 1.0 at zero conversion and decays steeply. The selectivity to ketones as quickly formed secondary products is zero at zero conversion and goes through an early maximum of about 0.35, other products also being formed. However, if no samples were taken during the first 0.5% conversion, the selectivity to ketones would be judged to extrapolate to about 0.4, as though ketones were primary products ... 

Another indication of provenance can sometimes be gained from a comparison of Delplot slopes. Regardless of rank, a steep downward slope of a plot indicates that the respective product decays rapidly. Such decay must be matched by rapid formation of one or more other products. The latters Delplots of same rank must show comparable steep upward slopes in the same conversion range. Matching steep upward to steep downward slopes can thus help to assign provenance. As an example, see the slopes of the plots for hydroperoxides and ketones in Figure 7.4. 

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