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Influence of Reaction Reversibility on the Interpretation of Delplots of the Parallel-Series Reaction Network Nabeel S. Abo-Ghander*,† and Michael T. Klein‡,§ †

Chemical Engineering Department and §Center for Refining & Petrochemicals, The Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia ‡ Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, United States ABSTRACT: The use of the Delplot analysis to interpret the experimental results from the parallel-series reaction network Energy Fuels Downloaded from pubs.acs.org by UNIV OF KANSAS on 01/19/19. For personal use only.

k1

k3

k2

A → B → C; A → D under the influence of reaction reversibility was examined. A reactor model was used to test two experimental feeds, one with zero product species concentrations and the other with non-zero product species concentrations. The equilibrium constants K1, K2, and K3 were varied between 0.2 and 1.5 for each reaction in the network. When CA0 = 2.0 M and CB0 = CC0 = CD0 = 0 M, it was found that reversibility did not change the analysis provided by the first-rank Delplots; i.e., species “B” and “D” are primary, while species “C” is non-primary. When CA0 = 2.0 M and CB0 = CC0 = CD0 = 0.5 M, it was found that the effect of the reversibility altered the information provided by Delplot analysis. For instance, the first-rank Delplot classified species “B” as a primary product when K1 ≥ 0.5 but non-primary when K1 = 0.2. A similar conclusion was also obtained for species D; i.e., species “D” was a primary product when K2 ≥ 0.5 but non-primary when K2 = 0.2. For species “C”, it was found that species “C” is primary at high K3 but non-primary at low K3. In summary, when the feed has non-zero product concentrations and the reaction network contains reversible reactions, the species ranks suggested by standard Delplot analysis can be disguised. This would suggest that extra care should be practiced when Delplot analysis is used for reaction networks in which high reversibility may exist.

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INTRODUCTION The Delplot technique is a method used to classify reaction network products into their product rank, i.e., the number of reaction steps required for formation. It is employed by plotting Yi/xAr, where r = 1, 2, ... on the ordinate versus xA on the abscissa for every product species. When τ → 0, Delplot y intercepts for every species are secured. While first-rank Delplot is used to identify primary products, the higher Delplot ranks are used to classify the non-primary products into secondary, tertiary, etc. Primary products of the parallel-series reaction network exhibit positive y intercepts on the first-rank Delplots, while zero yintercepts are obtained for the non-primary species. Several researchers have used this technique to sort reaction products into primary and non-primary and then reveal the proper reaction network. For instance, Walter et al.1 studied the reaction and kinetics of the thermal and catalytic cracking of 4(naphthylmethyl)bibenzyl. In this work, the construction of the first-rank Delplots for different species has greatly assisted in interpreting the influence of the presence of hydrogen on the reaction network and, consequently, the chemistry of the reaction. Afifi et al.2 investigated the kinetics of the pyrolysis and thermal hydrogenation of guaiacol in tetralin (hydrogen donor) in the presence of the homogeneous catalysts Fe and Ru. In this study, the first- and second-rank Delplots classified catechol and phenol to be primary products, while the non-identified polymeric products were classified as secondary products. The kinetics of thermal cracking and catalytic hydrocracking of asphaltene over NiMo/γ-Al2O3 was studied in a microbatch reactor at 430 °C by Zhao and Yu.3 A three-lump kinetic model involving liquid oil, gas, and coke was proposed. The first-rank © XXXX American Chemical Society

Delplot classified liquid oil and gas as primary products and coke as non-primary. Huelsman and Savage4 studied the kinetics of phenol gasification in supercritical water at the temperature range of 500−700 °C. The first-rank Delplot analysis revealed that dibenzofuran and benzene are primary products. Li et al.5 studied the kinetics of upgrading asphaltene in supercritical water at three temperatures: 400, 425, and 450 °C. Methane, gas, and coke were found to be the major products of the asphaltene upgrading reaction. When the Delplot analysis was used, methane was found to be a primary product, while gas and coke were non-primary. The second-rank Delplots revealed that gas and coke were secondary products. The kinetics of the liquid-phase glycerol hydrogenolysis were studied in the presence of a commercial copper-based catalyst in an isothermal trickle-bed reactor by Rajkhowa et al.6 The first-rank Delplot analysis was used to discriminate among the reaction product, and it was found that acetol was primary, while propylene glycol was non-primary. In terms of the evolution of the technique itself, the discernment of the rank of reactive species by the Delplot method was proposed by Bhore et al.7 and applied for a parallelk1

k3

k2

series reaction network A → B → C; A → D using a feed with zero product concentrations, CB0 = CC0 = CD0 = 0. The analysis of the Delplot technique was then extended to include the network rank, and the results were applied to sequential and parallelReceived: October 20, 2018 Revised: December 18, 2018

A

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels series reaction networks.8 The influence of using a feed with non-zero product concentrations, CB0 ≠ CC0 ≠ CD0 ≠ 0, was investigated by Abo-Ghander and Klein9 for the parallel-series network. It was found that the presence of species “B” in the feed can make the discernment of the rank of species by the Delplot technique ambiguous because species “B” may appear primary or secondary depending upon the relative magnitude between the ratios of the reaction rate constants of the parallel reactions of species “A” and the feed ratio of “A” to “B”. The effect of using the temperature instead of isothermal constant time to vary xA was also addressed by Abo-Ghander and Klein10 for the parallelseries network. This analysis showed that, depending upon the difference between the activation energies of the parallel reaction of species “A” giving species “B” and “D”, species “B” can appear to be primary at a low temperature and non-primary at a high temperature. At this point, the influence of the presence of reaction reversibility in the network has not been addressed. This motivated the aim of the present paper to investigate the effect of the reversibility of reactions for the parallel-series network k3

k1

= CC 0, and CD = CD 0. The influence of the non-zero concentration of product species “B”, “C”, and “D” will be explored. The above system of differential equations may be written in matrix form, as follows: ÄÅ ÉÑ ÅÅC A ÑÑ ÅÅ 0 ÑÑ ÅÅ ÑÑ ÅÅC ÑÑ ÅÅ B0 ÑÑ X(τ = 0) = X 0 = ÅÅÅÅ ÑÑÑÑ ÅÅCC0 ÑÑ ÅÅ ÑÑ ÅÅÅ ÑÑÑ ÅÅC D0 ÑÑ ÅÇ ÑÖ

X′(τ ) = AX(τ )

(7)

(8)

The vectors and matrix introduced in eq 7 are defined as follows: ÄÅ ÉÑ ÅÅCA ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ d ÅÅÅÅC B ÑÑÑÑ X′(τ ) = Å Ñ dτ ÅÅÅÅCC ÑÑÑÑ ÅÅÅ ÑÑÑ ÅÅC ÑÑ (9) ÅÇ D ÑÖ

k2

A → B → C; A → D using a feed with zero product concentrations and another feed with non-zero product concentrations.

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REACTION NETWORK The reaction network including the reversible reaction is thus k1

k3

k −1

k −3

A XooY B XooY C

(1)

ÄÅ ÉÑ ÅÅCA ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅC B ÑÑ Å Ñ X(τ ) = ÅÅÅ ÑÑÑ ÅÅCC ÑÑ ÅÅÅ ÑÑÑ ÅÅ ÑÑ ÅÅÇC D ÑÑÖ

k2

A XooY D k −2

(2)

It is a parallel-series reaction network composed of three main reversible reactions. Species “A” in the above reaction network goes through parallel reactions to produce species “B” and “D”, while species “B” goes through a series reaction ending with the formation of species “C”. As in previous publications, eqs 1 and 2 will be used to synthesize data that can, in turn, be used to probe the inference of Delplot analysis.

The system given by eq 7 is recognized to be a homogeneous system whose analytical solution is

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X(τ ) = c1 V1e λ1τ + c 2 V2e λ2τ + c3 V3e λ3τ + c4 V4e λ4τ

MATHEMATICAL MODELING OF THE CHEMICAL REACTOR At the steady-state condition for isothermally operated chemical reactors, the temporal variations of the concentration of species A, B, C, and D are depicted by the solution of the following differential mole balance equations: (3)

k dC B k = k1CA − 1 C B − k 3C B + 3 CC dτ K1 K3

(4)

k dC C = k 3C B − 3 CC dτ K3

(5)

dC D k = k 2CA − 2 C D dτ K2

(6)

(12)

In eq 12, λ1, λ2, λ3, and λ4 are the distinct eigenvalues of the coefficient matrix A, V1, V2, V3, and V4 are the corresponding eigenvectors, and c1, c2, c3, and c4 are arbitrary integration constants. The eigenvalues of the coefficient matrix A are obtained from det(A − λ I) = 0

dCA k k = −k1CA + 1 C B − k 2CA + 2 C D dτ K1 K2

(11)

(13)

In which I is the 4 × 4 identity matrix. Equation 13 can be reduced to a fourth-order polynomial known as the characteristic equation whose roots are the intended eigenvalues of the system. The eigenvectors, however, can be secured by applying Gauss−Jordan elimination to the following augmented matrix: (A − λ I|0)

(14)

The integration constants in eq 12 can be secured by applying the initial condition and then solving the following linear system of equations: X(0) = φ(0)C

In eqs 3−6, k1, k2, and k3 are the rate constants for the forward reactions and K1, K2, and K3 are the equilibrium constants. The initial conditions are given at τ = 0, where CA = CA0, CB = CB0, CC

(15)

where φ(0) is the fundamental matrix whose columns are the eigenvectors of the system at the initial condition and C is the 4 B

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX

ÅÄÅ c1 ÑÉÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅÅ C A 0 ÑÑÑ ÅÅ Ñ ÅÅ c ÑÑÑ ÅÅ 2 ÑÑ ÅÅÅ C ÑÑÑ ÅÅ A 0 ÑÑ ÑÑ Ĉ = ÅÅÅÅ Ñ ÅÅ c3 ÑÑÑ ÅÅ Ñ ÅÅ C A ÑÑÑ ÅÅ 0 ÑÑ ÅÅ Ñ ÅÅ c4 ÑÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅÅÇ C A 0 ÑÑÑÖ

Energy & Fuels × 1 column vector of the integration constants containing c1, c2, c3, and c4.

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DEFINITION OF CONVERSION OF A AND SPECIES YIELD The conversion of species “A” is defined as the number of moles of “A” reacted per moles of “A” fed to the reactor. Mathematically, it can be represented as xA =

C A 0 − CA CA0

=1−

CA CA0

(16)

Yi =

CA0

i ∈ {B, C, D} (17)

The expansion of the right-hand side of eq 12 is given as ÄÅ ÉÑ ÄÅ ÉÑ ÄÅ ÉÑ ÄÅ ÉÑ ÅÅÅCA(τ ) ÑÑÑ ÅÅÅ v ÑÑÑ ÅÅÅ v ÑÑÑ ÅÅÅ v ÑÑÑ ÅÅ ÑÑ ÅÅ 11 ÑÑ ÅÅ 21 ÑÑ ÅÅ 31 ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅC B(τ ) ÑÑ ÅÅ v12 ÑÑ ÅÅ v22 ÑÑ ÅÅ v32 ÑÑ ÅÅ ÑÑ Å Ñ Å Ñ Å Ñ λ τ λ τ X(τ ) = ÅÅ ÑÑ = c1ÅÅÅ v ÑÑÑe 1 + c 2ÅÅÅ v ÑÑÑe 2 + c3ÅÅÅ ÑÑÑe λ3τ ÅÅC (τ ) ÑÑ ÅÅ 13 ÑÑ ÅÅ 23 ÑÑ ÅÅ v33 ÑÑ ÅÅ C ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ v ÑÑ ÅÅ v ÑÑ ÅÅ v ÑÑ ÅÅÅC D(τ )ÑÑÑ ÅÅÅ 14 ÑÑÑ ÅÅÅ 24 ÑÑÑ ÅÅÅ 34 ÑÑÑ ÅÇ ÑÖ ÅÇ ÑÖ ÅÇ ÑÖ ÅÇ ÑÖ ÅÄÅ ÑÉÑ ÅÅÅ v41 ÑÑÑ ÅÅ ÑÑ ÅÅ v ÑÑ ÅÅ 42 ÑÑ + c4ÅÅÅ ÑÑÑe λ4τ ÅÅ v43 ÑÑ ÅÅ ÑÑ ÅÅÅ v44 ÑÑÑ ÅÅ ÑÑ (18) ÅÇ ÑÖ

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given by

ÄÅ v ÉÑ ÅÅ 12 ÑÑ ÅÅ ÑÑ c1 Y ÅÅ v ÑÑ λ1τ Å 13 ÑÑe = λ1τ λ2τ λ3τ λ4 τ Å xA C A 0 − c1v21e − c 2v21e − c3v31e − c4v41e ÅÅÅÅ v ÑÑÑÑ ÅÇ 14 ÑÖ ÄÅ ÉÑ ÅÅ ÑÑ ÅÅ v22 ÑÑ ÅÅ ÑÑ c2 ÅÅÅ v23 ÑÑÑe λ2τ + Å Ñ C A 0 − c1v21e λ1τ − c 2v21e λ2τ − c3v31e λ3τ − c4v41e λ4τ ÅÅÅÅ v ÑÑÑÑ ÅÅ 24 ÑÑ ÅÇÅ ÑÖÑ ÅÄÅ ÑÉÑ ÅÅ v32 ÑÑ ÅÅ ÑÑ ÅÅ ÑÑ λ τ c3 ÅÅ v33 ÑÑe 3 + Ñ λ1τ λ2τ λ3τ λ4 τ Å C A 0 − c1v21e − c 2v21e − c3v31e − c4v41e ÅÅÅÅ v ÑÑÑÑ ÅÅ 34 ÑÑ ÅÅÇ ÑÑÖ ÄÅ ÉÑ ÅÅ ÑÑ ÅÅ v42 ÑÑ ÅÅ ÑÑ c4 ÅÅ v ÑÑ λ4τ Å 43 ÑÑe + λ1τ λ2τ λ3τ λ4 τ Å C A 0 − c1v21e − c 2v21e − c3v31e − c4v41e ÅÅÅÅ v ÑÑÑÑ ÅÅ 44 ÑÑ ÅÅÇ ÑÑÖ É ÄÅ ÅÅC B ÑÑÑ ÅÅ 0 ÑÑ Ñ ÅÅ 1 ÅÅC ÑÑÑ ÅÅ C0 ÑÑ − ÑÑÑ C A 0 − c1v21e λ1τ − c 2v21e λ2τ − c3v31e λ3τ − c4v41e λ4τ ÅÅÅÅ Ñ ÅÅÅC D0 ÑÑÑ (25) ÑÖ ÅÇ

CA0 (19)

(20)

Equation 20 can be reduced to the following form: Y(τ ) = ρ(τ )Ĉ − Ĉ 0

where ρ(τ), Ĉ , and Ĉ 0 are given by ÅÄÅ ÑÉ ÅÅ v12e λ1τ v22e λ2τ v32e λ3τ v42e λ4τ ÑÑÑ ÅÅ ÑÑ ÅÅ ÑÑ Å λ τ λ τ λ τ λ τ 1 2 3 4 ρ(τ ) = ÅÅÅ v13e v23e v33e v43e ÑÑÑÑ ÅÅ ÑÑ ÅÅ Ñ ÅÅ v e λ1τ v e λ2τ v e λ3τ v e λ4τ ÑÑÑ ÅÇ 14 ÑÖ 24 34 44

FIRST-RANK DELPLOT

plotting Yi/xA versus x. The abscissa is xA and the ordinate is

C A 0 − c1v21e λ1τ − c 2v21e λ2τ − c3v31e λ3τ − c4v41e λ4τ

The yields of species “B”, “C”, and “D” are given by ÄÅ ÉÑ ÄÅ ÉÑ ÄÅ ÉÑ ÅÅÅ v12 ÑÑÑ ÅÅÅ v22 ÑÑÑ ÅÅÅ v32 ÑÑÑ Å Ñ Å Ñ Å Ñ c3 ÅÅÅÅ ÑÑÑÑ λ3τ c1 ÅÅÅÅ v ÑÑÑÑ λ1τ c 2 ÅÅÅÅ v ÑÑÑÑ λ2τ Y= ÅÅ 13 ÑÑe + ÅÅ 23 ÑÑe + Å v33 Ñe C A 0 ÅÅÅ ÑÑÑ C A 0 ÅÅÅ ÑÑÑ C A 0 ÅÅÅÅ ÑÑÑÑ v v ÅÅ 14 ÑÑ ÅÅ 24 ÑÑ ÅÅ v34 ÑÑ ÅÅÇ ÑÑÖ ÅÅÇ ÑÑÖ ÅÅÇ ÑÑÖ ÅÄÅ ÑÉÑ ÅÄÅC ÑÉÑ ÅÅ v42 ÑÑ ÅÅ B0 ÑÑ Å Ñ ÅÅ ÑÑ ÅÅ ÑÑ c4 ÅÅÅÅ ÑÑÑÑ λ4τ 1 ÅÅCC ÑÑ i ∈ {B, C, D} + ÅÅ v43 ÑÑe − C A 0 ÅÅÅ ÑÑÑ C A 0 ÅÅÅÅ 0 ÑÑÑÑ v ÅÅÅ 44 ÑÑÑ ÅÅÅC D0 ÑÑÑ ÅÇ ÑÖ ÅÇ ÑÖ

(24)

The first-rank Delplot is constructed for different species by

which can be used to evaluate the concentration variations of all species as functions of time. The conversion of species “A” is given by xA =

(23)

ÄÅ É ÅÅ C B ÑÑÑ ÅÅ 0 ÑÑ ÅÅ Ñ ÅÅ C A ÑÑÑ ÅÅÅ 0 ÑÑÑ ÅÅ Ñ ÅÅ CC ÑÑÑ ÅÅ 0 ÑÑ ̂ C0 = ÅÅ Ñ ÅÅ C A ÑÑÑ ÅÅ 0 ÑÑ ÅÅ Ñ ÅÅ C ÑÑÑ ÅÅ D0 ÑÑ ÅÅ Ñ ÅÅ C ÑÑÑ ÅÅÇ A 0 ÑÑÖ

The yields of species “B”, “C”, and “D” are defined as Ci − Ci0

Article

The y-intercept can be evaluated as the limit of eq 25 as τ goes to (21)

zero. Yintercept = lim

τ→ 0

Y xA

(26)

However, the application of the limits will give 0/0, which (22)

requires the application of L’Hôpital’s rule, which results in C

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX

ÄÅ v ÉÑ ÅÅ 12 ÑÑ ÅÅ ÑÑ c1λ1 ÅÅ v ÑÑ Yintercept = − Å 13 Ñ c1v21λ1 + c 2v21λ 2 + c3v31λ3 + c4v41λ4 ÅÅÅÅ ÑÑÑÑ ÅÅÇ v14 ÑÑÖ ÄÅ v ÉÑ ÅÅ 22 ÑÑ ÅÅ ÑÑ c 2λ 2 ÅÅ v ÑÑ − Å 23 Ñ c1v21λ1 + c 2v21λ 2 + c3v31λ3 + c4v41λ4 ÅÅÅÅ ÑÑÑÑ ÅÅÇ v24 ÑÑÖ ÄÅ v ÉÑ ÅÅ 32 ÑÑ ÅÅ ÑÑ c3λ3 ÅÅ v ÑÑ − Å 33 Ñ c1v21λ1 + c 2v21λ 2 + c3v31λ3 + c4v41λ4 ÅÅÅÅ ÑÑÑÑ ÅÅÇ v34 ÑÑÖ ÄÅ v ÉÑ ÅÅ 42 ÑÑ ÅÅ ÑÑ c4λ4 ÅÅ v ÑÑ − Å 43 Ñ c1v21λ1 + c 2v21λ 2 + c3v31λ3 + c4v41λ4 ÅÅÅÅ ÑÑÑÑ ÅÅÇ v44 ÑÑÖ

Article

Energy & Fuels

(27)

Equation 27 can be written in a compact form as Yintercept = −ρ(0)P

where the P is defined as ÄÅ ÅÅ c1λ1 ÅÅ ÅÅ ÅÅ c1v21λ1 + c 2v21λ 2 + c3v31λ3 ÅÅ ÅÅ ÅÅ c 2λ 2 ÅÅ ÅÅ c v λ + c v λ + c v λ ÅÅ 1 21 1 2 21 2 3 31 3 P = ÅÅÅÅ c3λ3 ÅÅÅ ÅÅ ÅÅ c1v21λ1 + c 2v21λ 2 + c3v31λ3 ÅÅ ÅÅ ÅÅ c4λ4 ÅÅ ÅÅ ÅÅ c1v21λ1 + c 2v21λ 2 + c3v31λ3 ÅÇ

(28)

+ + + +

ÉÑ ÑÑ ÑÑ Ñ c4v41λ4 ÑÑÑ ÑÑ ÑÑ ÑÑ ÑÑ c4v41λ4 ÑÑÑÑ ÑÑ ÑÑ ÑÑ ÑÑ c4v41λ4 ÑÑÑÑ ÑÑ ÑÑ ÑÑ ÑÑ c4v41λ4 ÑÑÑÑÖ

Figure 1. Comparison of the numerical solution of the mole balance equations and the analytical expressions of species concentration for k1 = k2 = k3 = 4.0 min−1, K1 = K2 = K3 = 10, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0.5 M.

plotting Yi/xA, i ∈ {A, B, C} versus xA. Results are shown in Figures 2, 3, and 4 for three different cases: irreversible case, case of K1 = 0.1, K2 = 0.1, or K3 = 0.1, and reversible case when K1 = K2 = K3 = 0.1. The y-intercepts are evaluated and plotted on the same figures using eq 27. It was found that, when only species “A” is present in the feed, the existence of reaction reversibility did not affect the information obtained from the Delplot analysis. Consequently, the parallel-series reaction network will

(29)

It is clear from eqs 28 and 29 that the y intercepts of the first-rank Delplots of species “B”, “C”, and “D” are combinations of the initial condition, the eigenvalues, and the corresponding eigenvectors of the system. It worth mentioning that the eigenvalues and the corresponding eigenvectors have been evaluated using MATLAB functions as the eigenvalues representing the roots of a fourth-order polynomial.

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RESULTS AND DISCUSSION The influence of the variation of reversibility on the information provided by Delplot analysis was examined by solving this model for two feed conditions. The first one has zero concentration of product species B, C, and D, while the second has a non-zero concentration of product species. Figure 1 illustrates the kinetics for k1 = k2 = k3 = 4.0 min−1, K1 = K2 = K3 = 10, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0.5 M by comparing the numerical predictions to that obtained by eq 12. Both predictions show that the concentration of species “A” drops with time because of the parallel reactions forming both species “B” and “D”, while the concentrations of species “D” and “C” increase with time. At the beginning of the reaction, the rate of “B” formation is significant, making the concentration of species “B” increase until a maximum is observed. Then, the concentration of species “B” falls because the rate of consumption of species “B” to form species “C” overcomes the rate of production of species “B”. For the feed with zero product initial concentrations, the firstrank Delplots of species “B”, “C”, and “D” are constructed by

Figure 2. Influence of the reversibility on the first-rank Delplot of species B for k1 = k2 = k3 = 4.0 min−1, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0 M. D

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels

Figure 5. Influence of the reversibility of A → B on the first-rank Delplot of species B for k1 = k2 = k3 = 4.0 min−1, K2 = K3 = 4 × 104, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0.5 M.

Figure 3. Influence of the reversibility on the first-rank Delplot of species C for k1 = k2 = k3 = 4.0 min−1, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0 M.

M, and CB0 = CC0 = CD0 = 0.5 M for different values of the equilibrium constant K1. At high values of K1, the first-rank Delplot identifies species “B” as a primary product. However, as the reversibility increases toward K1 = 0.2, a non-positive y intercept of the first-rank Delplot is eventually secured, which would suggest that species “B” is non-primary. This indicates that the reversibility of the formation reaction of species “B” can make the analysis of the first-rank Delplot ambiguous because it suggests a different reaction scheme than that used here to create the synthetic data. The first-rank Delplot for species “D” is shown for CB0 = CC0 = CD0 = 0.5 M in Figure 6. The influence of the reversibility of the k2

Figure 4. Influence of the reversibility on the first-rank Delplot of species D for k1 = k2 = k3 = 4.0 min−1, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0 M.

reaction A → D was addressed by varying the equilibrium constant K2 between 1.5 and 0.2. It was found that species “D” appears to be a primary product when the value of K2 is greater than or equal to 0.5. However, species “D” was suggested to be non-primary when the influence of the reversibility became more pronounced, as shown for the case of K2 = 0.2. The first-rank Delplot for species “C” is shown in Figure 7, where YC/xA is plotted versus xA for a feed with non-zero product concentrations; i.e., CB0 = CC0 = CD0 = 0.5 M. In this figure, the effect of the equilibrium constant K3 of the reaction

still be suggested because species “B” and “D” are still primary, i.e., giving non-zero y intercepts, while “C” is non-primary, i.e., giving a zero y-intercept. Hence, the reversibility when the feed has zero product concentrations will not affect the classification of the product species but will influence only the molar qualities of the species. Figure 5 analyzes the situation for non-zero initial concentration of the products. The first-rank Delplot of species B is shown at k1 = k2 = k3 = 4.0 min−1, K2 = K3 = 4 × 104, CA0 = 2.0

B → C is examined by changing its magnitude between 1.5 and 0.2. The first-rank Delplot of species “C” suggested that species “C” was primary then changed to non-primary as the influence of the reversibility increases. This happens because of the presence of species B in the feed, which is the only species that can give species “C”. This analysis thus illustrates that extra care should be practiced when the feed has non-zero product concentrations and the effect of reaction reversibility is important. These factors can affect the information drawn from Delplot analysis and could lead to ambiguity in proposing reaction schemes.

k3

E

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels

pronounced, the analysis of the Delplots can be ambiguous and the conclusions should be handled with extra care.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Nabeel S. Abo-Ghander: 0000-0002-8810-997X Michael T. Klein: 0000-0001-5444-1512 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Michael T. Klein acknowledges collaborations with and support of colleagues via the Saudi Aramco Chair Program at King Fahd University of Petroleum & Minerals (KFUPM) and Saudi Aramco. Nabeel S. Abo-Ghander thanks KFUPM for its support.

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REFERENCES

(1) Walter, T. D.; Casey, S. M.; Klein, M. T.; Foley, H. C. Reaction of 4-(naphthylmethyl)bibenzyl, thermally and in the presence of Fe(CO)3(PPh3)2. Catal. Today 1994, 19 (3), 367−379. (2) Afifi, A. I.; Chornet, E.; Thring, R. W.; Overend, R. P. The aryl ether bond reactions with H-donor solvents: Guaiacol and tetralin in the presence of catalysts. Fuel 1996, 75 (4), 509−516. (3) Zhao, Y.; Yu, Y. Kinetics of asphaltene thermal cracking and catalytic hydrocracking. Fuel Process. Technol. 2011, 92 (5), 977−982. (4) Huelsman, C. M.; Savage, P. E. Reaction pathways and kinetic modeling for phenol gasification in supercritical water. J. Supercrit. Fluids 2013, 81, 200−209. (5) Li, N.; Yan, B.; Xiao, X.-M. Kinetic and reaction pathway of upgrading asphaltene in supercritical water. Chem. Eng. Sci. 2015, 134 (29), 230−237. (6) Rajkhowa, T.; Marin, G. B.; Thybaut, J. W. A comprehensive kinetic model for Cu catalyzed liquid phase glycerol hydrogenolysis. Appl. Catal., B 2017, 205, 469−480. (7) 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. (8) 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 (2), 313−316. (9) Abo-Ghander, N. S.; Klein, M. T. Extension of the Delplot analysis for experiments with non-zero initial product concentrations. Energy Fuels 2018, 32 (5), 6234−6238. (10) Abo-Ghander, N. S.; Klein, M. T. Influence of reaction temperature on the interpretation of Delplots for the parallel-series reaction network. Energy Fuels 2018, 32 (10), 10904−10909.

Figure 6. Influence of the reversibility of A → D on the first-rank Delplot of species D for k1 = k2 = k3 = 4.0 min−1, K1 = K3 = 4 × 104, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0.5 M.

Figure 7. Influence of the reversibility of B → C on the first-rank Delplot of species C for k1 = k2 = k3 = 4.0 min−1, K1 = K2 = 4 × 104, CA0 = 2.0 M, and CB0 = CC0 = CD0 = 0.5 M.

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CONCLUSION The influence of reaction reversibility of the parallel-series k1

k3

k2

reaction network A → B → C; A → D was examined using two feeds, one with zero product species concentrations and the other with non-zero product species concentrations. It is found that the influence of the reversibility is insignificant and the analysis of Delplot is unambiguous when the feed has zero concentrations of product species. However, when the feed has non-zero product concentrations and the effect of reversibility is F

DOI: 10.1021/acs.energyfuels.8b03649 Energy Fuels XXXX, XXX, XXX−XXX