Analytical Rate Expressions Accounting for the Elementary Steps in

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Analytical Rate Expressions Accounting for the Elementary Steps in Benzene Hydrogenation on Pt Luis Lozano, Guy B Marin, and Joris W. Thybaut Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 26 Apr 2017 Downloaded from http://pubs.acs.org on April 27, 2017

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Industrial & Engineering Chemistry Research

Analytical Rate Expressions Accounting for the Elementary Steps in Benzene Hydrogenation on Pt Luis Lozano, Guy B. Marin and Joris W. Thybaut* Ghent University, Laboratory for Chemical Technology, Technologiepark 914, Ghent B-9052 Belgium KEYWORDS: Metal catalysis, microkinetic modeling, aromatics hydrogenation, Horiuti-Polanyi

mechanism

ABSTRACT: The complexity of elementary-step reaction networks demands an adequate

methodology to simulate the corresponding kinetics with a comprehensive, yet manageable rate expression. In this framework, a solution strategy has been developed for expressing the aromatics hydrogenation rate according to the Horiuti-Polanyi mechanism with a single analytical equation. Owing to the step-wise atomic hydrogen addition reaction network, a set of algebraic equations, linear in the surface coverage of free sites, is obtained. An approximate 30fold reduction in CPU time could be obtained by using the analytical solution as compared to the numerical one. The corresponding analytical solution produces simulation results that are, within the accuracy requested from the solver, identical to those obtained via the numerical solution. Finally, parameter estimates and statistical tests for the regression of the experimental data sustain the similarity between both simulation approaches. 1 ACS Paragon Plus Environment

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INTRODUCTION Hydrogenation of aromatic compounds is an increasingly important reaction for improving diesel quality and reducing corresponding emissions.1, 2 More particularly, benzene is a compound widely produced in oil refineries, e.g. during the catalytic reforming of nafta. Benzene is frequently used as an industrial solvent; however, it is also known for its high octane number and as an important reactant for considerable (petro)chemical processes.3 Roughly 43 M ton/year of benzene are produced to supply these demands. Nonetheless, the widespread use of benzene leads to approximately 10 kg/ton of diffusive emissions, which raises environmental concerns since aromatic compounds are well-known air pollutants.4 Such concerns have been translated into environmental legislations around the globe to limit aromatic content in liquid fuels. In the last 10 years the maximum allowed concentration of benzene in gasoline has been reduced to j is assumed to be reversible, i.e. hydrogenation/dehydrogenation, and is, hence, calculated according to the general formulation in Eq.13.

z z ri → j = nei → jk (m, n)θiθH Ct − ne j→i k (m, n)−1θ jθ* Ct 2 2

(13)

ANALYTICAL SOLUTION FOR THE OVERALL REACTION RATE A methodology has been devised to calculate the overall benzene hydrogenation rate via an analytical expression while still accounting for the complex SEMK reaction mechanism. No symbolic computation was employed; hence, the arithmetic deduction provides a means to acquire more insight in the relation between the elementary reaction mechanism and its mathematical representation. In this methodology a set of equations is constructed to calculate all surfaces coverage when reactant and product adsorption and desorption are quasi equilibrated and the Pseudo SteadyState Approximation (PSSA) is applied to surface intermediates. Since hydrogen and benzene adsorption are in quasi-equilibrium (Eqs. 9 and 10), two equations are obtained to relate the surface coverage of both species to the corresponding partial pressures, see Eqs. 14 and 15.

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eq

θ H = K H 2 pH 2 θ*

(14)

eq

(15)

θ B = K B pBθ *

The other surface coverages are obtained by applying the PSSA and systematically elaborate the obtained equations. As result of the PSSA for all 13 “j” surface intermediates (Rj=0), Eq. 12 is reduced Eq. 16.

∑r

i→ j

g

=

∑r

(16)

j→k

h

Since θ13 can be assumed to be negligible, the CHA formation rate (rCHA) equals that of surface c-Hexyl (θ12) transformation into surface CHA (θ13) (r12->13) in Eq. 17: ~ z RCHA = rCHA = r12 →13 = ne12→13k (0,2)θ Hθ12 Ct 2

(17)

The number of single events and the rate coefficients are “given” values and θH can be obtained from Eq. 14. As a result, the calculation of the overall reaction rate, is reduced to the determination of θ12 via the equations obtained from the PSSA. Recognizing that in each of the rate expressions θH appears elevated to the first power, the PSSA results in a set of 12 linear algebraic equations in 12 unknowns that is perfectly specified thanks to hydrogen and benzene quasi-equilibrated adsorption. The set of algebraic equations resulting from the PSSA for the surface intermediates can be expressed in terms of the surface coverage of the free sites, preserving the linear character of the set of equations. First, applying the PSSA to surface BH according to Eq. 16 yields Eq. 18. r1→2 = r2→3 + r2→4 + r2→5

(18)

Expanding the reaction rate terms in Eq. 18, results in Eq. 19.



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z z ne1→2 k (2,2)θ1θ H Ct − ne 2→1k (2,2) −1 θ 2θ* Ct 2 2 z z = ne 2→3k (1,2)θ 2θ H Ct − ne 3→2 k (1,2) −1 θ3θ* Ct 2 2 z z + ne 2→4 k (2,2)θ 2θ H Ct − ne 4→2 k (2,2) −1 θ 4θ* Ct 2 2 z z + ne 2→5 k (2,2)θ 2θ H Ct − ne 4→2 k (2,2) −1 θ5θ* Ct 2 2

(19)

Hydrogen and benzene surface coverage can be determined from Equations 14 and 15 thanks to the quasi equilibrated character of both adsorptions; through the proper substitutions and simplifications a single expression for θ2 is obtained, exhibiting a linear dependence between the surface coverage of BH and the remaining species and the coverage of the free sites, see Eq. 20. ~ ~ ~ ~ −1 −1 −1 ne1→ 2 k (2,2 )K B pB K H 2 pH 2 θ* + ne3→ 2k (1,2 ) θ3 + ne4→ 2 k (2,2 ) θ 4 + ne5→ 2 k (2,2 ) θ5 θ2 = ~ ~ ~ ~ −1 ne2→1k (2,2 ) + ne2→3k (1,2 ) + ne2→ 4 k (2,2 ) + ne2→5k (2,2 ) K H 2 pH 2

[

]

(20)

The coefficients of the surface coverage in Eq. 20 are function of the number of single events, rate coefficients and partial pressures, i.e., of “given” and/or observable quantities. These coefficients are denoted by letters to simplify the mathematical treatment of the different expressions (see Eq. 21). A2,3

~ ne3→ 2 k (1, 2 ) = ~ ~ ~ ~ −1 ne2 →1k (2,2 ) + ne2 → 3k (1,2 ) + ne2 → 4 k (2, 2 ) + ne2 → 5 k (2,2 ) K H 2 pH 2

[

(21)

]

Equation 21 can be therefore expressed in terms of these coefficients (see Eq. 22). θ 2 = A2 , 0θ * + A2 , 3θ 3 + A2 , 4θ 4 + A2 , 5θ 5

(22)

Similar operations can be performed for the surface intermediates further in the hydrogenation sequence, e.g., for 12CHD. Applying the PSSA, see Eqs. 23 and 24, provides a single expression to calculate θ3 in Eq. 25.


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r2 → 3 = r3→ 6 + r3→ 7

(23)

z z ne 2→3k (1,2)θ 2θ H Ct − ne3→2k (1,2) −1 θ3θ* Ct 2 2 z z = ne 3→6k (1,2)θ3θ H Ct − ne 6→3k (1,2) −1 θ 6θ* Ct 2 2 z z + ne 3→7 k (2,2)θ3θ H Ct − ne 7→3k (2,2) −1 θ7θ* Ct 2 2

(24)

~ ~ ~ −1 −1 ne2→3k (1,2 )θ 2 K H 2 pH2 + ne6→3k (1,2 ) θ 6 + ne7→3k (2,2 ) θ 7 θ3 = ~ ~ ~ −1 ne3→2 k (1,2 ) + ne3→6 k (1,2 ) + ne3→7 k (2,2 ) K H 2 pH 2

(25)

[

]

An expression for θ3, independent of θ2, is obtained when Eq. 19 is substituted in Eq. 22 (See Eq. 23). θ3

[n =

e2→3

]

~ ~ ~ −1 −1 k (1,2 )(A2,0θ* + A2,3θ 3 + A2,4θ 4 + A2,5θ 5 ) K H2 pH2 + ne6→3 k (1,2 ) θ 6 + ne7→3 k (2,2 ) θ 7 ~ ~ ~ −1 ne3→2 k (1,2 ) + ne3→6 k (1,2 ) + ne3→7 k (2,2 ) K H2 pH2

[

]

(26)

Coefficients that only contain kinetic parameters and observable quantities are again defined, e.g., B3,0 in Eq. 27, to simplify the mathematical expression. After Eq. 27 is solved for θ3, a single expression to calculate θ3 is obtained, see Eq. 28. B3, 0

~ ne2→3k (1,2 )A2 ,0 K H 2 pH 2 = ~ ~ ~ ~ −1 ne3→ 2 k (1,2) + K H 2 pH 2 ne3→6 k (1,2 ) + K H 2 pH 2 ne3→ 7 k (2,2 ) − A2,3 K H 2 pH 2 ne2 →3k (1,2 )

θ 3 = B3,0 θ * + B3,4 θ 4 + B3,5 θ 5 + B3,6 θ 6 + B3,7 θ 7

(27)

(28)

The latter equation is independent from surface species to which a lower number of hydrogen atoms have been added. Continuing these operations up to cyclohexyl (species 12 in Figure 2) requires a total of 33 substitutions, and results in a single, yet complex expression to solve the net rate of formation for gas-phase cyclohexane in Eq. 17. The latter equation is linear in the surface coverage of free sites and, hence, allows a straightforward calculation of the cyclohexane 15 ACS Paragon Plus Environment


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formation rate. More elaborate guidelines for the derivation of the analytical rate expression according to the proposed methodology are given in the supporting information. The applicability of the methodology presented in this work is closely related to the generic features as occurring in a Horiuti–Polanyi mechanism, i.e., the surface reactions should constitute a series of, essentially, identical steps occurring on a single model. Species adsorption should be quasi-equilibrated and the Pseudo Steady State Approximation (PSSA) should be applicable to all surface reactions. This allows the formal linearization of the rate equations in the surface coverages of the hydrocarbon species and, hence, the step wise substitution of surface coverages by those of species situated earlier in the reaction network. Finally, product desorption being assumed to be fast and irreversible, ensures that the net rate of formation of the final product is independent of any additional equation related to product re-adsorption. As a result, the extension of the presented methodology to a diversity of hydrogenation reactions such as toluene, xylene, ethylene, etc. is straightforward since the Horiuti–Polanyi mechanism is widely accepted.11-19 Thanks to the symmetry present in benzene, the corresponding hydrogenation reaction network complexity is still relatively limited. The number of surface intermediates and hydrogen addition/abstraction reactions rapidly increases with the presence of substituents. For instance, a similar SEMK model for toluene hydrogenation, comprising 40 surface intermediate species and 100 reversible surface reactions, still exhibits the same characteristics in terms of surface coverage as the benzene hydrogenation network.18 Hence, the presented analytical solution strategy remains valid for such heavier aromatic species. Other reactions that exhibit similar features as the ones described above include the total and partial oxidation of hydrocarbons.41, 42 Also for addressing sub networks of more complex ones, such as the FTS,39, 40 the presented methodology opens up interesting perspectives to derive an 16 ACS Paragon Plus Environment

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analytical expression to simulate the rate of the consecutive insertion of a building block in a growing chain and, consequently, reduce the extent of the reaction network that needs to be solved numerically. ANALYTICAL VERSUS NUMERICAL SIMULATION RESULTS A validation of the developed methodology for the calculation of the overall aromatic hydrogenation rate based on an analytical rate expression is performed via comparison of these simulation results with those previously reported by Bera et al.,17 using a numerical solution strategy as well as with the available experimental data. Deviations between the two solution methodologies, i.e., analytical and numerical are expected to be within tolerance of the numerical routine used. Furthermore, the analytical solution reduces the CPU time from 1.186 s to 0.043 s for the 43 experiments, when it is compared with the numerical routine. The analytical rate expression ensures that a solution is obtained while, numerically, such a solution is not guaranteed when trying to solve the set of equations describing the reaction in steady state. Moreover, even if the CPU time gain from 1.186 to 0.043 s may seem insignificant at first sight, a 30-fold reduction in required CPU time potentially makes the difference in more complex reaction networks or when many simulations need to be performed consecutively, e.g., as in a regression. A parity diagram of experimental benzene conversion against the simulated results with the analytical solution is presented in Figure 3. Simulated values are in good agreement with the experimental data. The observed deviations can entirely be attributed to experimental error.17



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Figure 3. Experimental and simulated benzene conversion at 5.0 - 10.0 H2/Benzene inlet molar ratio, 423 – 498 K temperature range, 1 – 3 MPa total pressure and 22.5 - 69.2 kgcat s mol-1 space time. The calculated benzene conversion is obtained by computing the outlet molar flow rate of Benzene in Eq. 1. The reaction term is solved by means of the methodology described in the section Analytical Solution for the Overall Reaction Rate, using the set of kinetic parameters estimated by Bera et al.17 Apart from the overall agreement between experimental data and model simulations, a more specific performance evaluation in terms of reproduction of temperature, pressure and hydrogen/benzene inlet molar flow rate effects is studied in Figures 4 and 5. The evolution of the benzene conversion with the total pressure and temperature is simulated by the analytical solution as presented in Figure 4. The maximum in the benzene hydrogenation conversion as a function of the temperature is adequately reproduced and stems from a decrease of the benzene surface concentration with increasing temperature.7


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Figure 4. Benzene conversion as a function of the temperature at H2/Benzene inlet molar ratio of 5 and W/Fbenzene0 of 6.38 kg s mol-1: ● 3 MPa, ▲ 2 MPa, ♦ 1MPa. Full lines are computed by calculating the benzene outlet molar flow rate in Eq.1: __ 3 MPa, _ . 2 MPa, … 1MPa. The reaction term is solved by means of the methodology described in the section Analytical Solution for the Overall Reaction Rate, using the set of kinetic parameters estimated by Bera et al.17 Figure 5 shows the effect of the hydrogen/benzene molar ratio on the overall benzene conversion at 448 K and three total pressures. According to the reaction stoichiometry, three H2 molecules are needed for each benzene molecule to yield gas-phase cyclohexane. However, it is remarkable that only moderate conversions can be achieved even with hydrogen in excess at a molar ratio amounting to 5:1. The potentially competitive character of the H2 and benzene adsorption on the catalyst surface has been extensively discussed in the literature. Our data indicate that increasing benzene inlet partial pressure has a negative effect over the activity at those working conditions.2, 7, 11-13, 17, 18, 20, 30-33



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Figure 5. Benzene conversion as a function of the H2/Benzene molar ratio at 448 K: 3 MPa, ▲ 2 MPa, ♦ 1MPa. Full lines are computed by calculating the Benzene outlet molar flow rate in Eq.1: __ 3 MPa, _ . 2 MPa, … 1MPa. The reaction term is solved by means of the methodology described in the section Analytical Solution for the Overall Reaction Rate, using the set of kinetic parameters estimated by Bera et al.17 It is clear from Figure 5 that the model simulations based on the analytical rate expressions allowed reproducing the experimentally observed trends. A similar conclusion was obtained by Bera et al. when adopting a numerical solution strategy.17 Figure 6 compares the results of the two solution strategies directly with each other, by displaying the differences in the benzene conversion obtained with both solution strategies as a function of the cyclohexane outlet flow rate.


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Figure 6. Percentage difference on benzene conversion between the analytical solution described in section Analytical Solution for the Overall Reaction Rate, and the numerical solution achieved by Bera et al.,17 as a function of the cyclohexane outlet molar flow rate. Benzene conversion is calculated for both cases by computing the outlet molar flow rate of Benzene in Eq. 1, using the set of kinetic parameters estimated by Bera et al.17 The difference between the analytically and numerically calculated benzene conversion is randomly distributed with a maximum absolute value amounting to 8 10-2 %, which is perfectly in line with what could be expected based on the relative tolerance of the numerical solution in the DNSQE solver, which was set to 10-2.26 Kinetic parameter values were estimated making use of the analytical rate expression developed in this work. Table 1 compares the presently obtained estimates with those reported by Bera et al.17 Table 1. Kinetic parameter values (kJ mol-1) for the SEMK model for benzene hydrogenation. Kinetic parameters calculated in this work

Kinetic Parameters calculated by Bera et al.17 21

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Ea(0,2)

57.6 ± 0.3

57.5 ± 0.5

Ea(1,2)

65.3 ± 8.4

65.1 ± 8.8

∆H(0,2)

7.8 ± 0.2

7.9 ± 0.2

∆H(1,2)

1.1 ± 0.5

1.2 ± 0.5

∆HB

56.0 ± 0.5

56.0 ± 0.5

∆HH2

-59.4 ± 0.5

-59.4 ± 0.5

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Values obtained by regression are given with the corresponding 95% confidence interval, using the net rate of cyclohexane in Eq. 17 and the reactor model in Eq. 1. The reaction term is solved by means of the methodology described in the section Analytical Solution for the Overall Reaction Rate. Only small differences are encountered in the point estimates when using the analytical expression versus when relying on the numerical simulations. Moreover, the differences between the parameter estimates obtained according to both methodologies are statistically insignificant, and the already narrow confidence intervals are slightly reduced for the estimates obtained with the analytical rate expression. With the analytical rate expression a slightly higher F value for the global significance of the regression is obtained as compared to with the numerical solution, i.e., 4163 in the present work as compared to 4150 reported by Bera et al.17 The above statistical analysis, hence, indicates that the method to derive an analytical rate expression for aromatic hydrogenation leads to statistically identical parameter estimates. The differences, if any, are associated to the tolerance employed in the numerical solution and result in the latter being slightly less significant compared to the regression performed with the analytical expression. CONCLUSIONS Microkinetic models for aromatics hydrogenation are characterized by complex reaction networks involving a large number of surface intermediates. While is tempting to recur to


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numerical routines to solve the corresponding set of equations describing the kinetics in such a network, the particular nature of hydrogenation reactions brings an analytical solution within reach even when it doesn’t seem to be feasible at first sight due to highly tangled reaction network. According to the Horiuti – Polanyi mechanism as implemented for aromatics hydrogenation, reactant adsorption is quasi-equilibrated, hydrogen atom additions to unsaturated carbon atoms occurs step-wise, and desorption of the saturated product is fast and irreversible. Following these three assumptions, it was possible to solve a linear set of equations in the surface coverage of intermediate species and free sites allowing to calculate the cyclohexane formation rate, not having to rely on symbolic computation. The strategy proposed in our work yields statistically identical results as the numerical solution of the original set of equations and simultaneously reduces the required CPU time by a factor of 30, providing the guarantee that a solution is found in steady state simulations. The methodology has a generic character and can be directly extended towards, more complex aromatic components such as toluene and (ortho-,meta-,para-)xylene as well as to alternative reactions such as hydrocarbon oxidation or Fischer Tropsch synthesis. ASSOCIATED CONTENT

Supporting Information Further details on the methodology to derive analytical rate expressions for Benzene Hydrogenation on Pt discussed in this work. AUTHOR INFORMATION

Corresponding Author 23 ACS Paragon Plus Environment


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*Email: [email protected]. NOMENCLATURE *

free site

1235THB

tetrahydrobenzene

1245THB

tetrahydrobenzene

123THB

trihydobenzene

124THB

trihydobenzene

135THB

trihydobenzene

12CHD

cyclohexadiene

14CHD

cyclohexadiene

13DHB

dihydrobenzene

A

pre-exponential factor (s-1, Pa-1 s-1)

B

benzene

BH

hydrobenzene

Ct

Total concentration of active sites (mol/kgcat)

c-Hexyl

cyclohexyl

CHA

cyclohexane

CHE

cyclohexene

DFT

Density Functional Theory

F

molar flow rate (mol s-1)

Fvalue

F value for the significance of the regression

b

model vector

h

Planck constant (6.63 x 10-34 J s)

H

enthalpy (J mol-1)

k

rate coefficient (s-1, Pa-1 s-1)

kb

Boltzmann constant (1.38 x 10-23 J K-1) 24 ACS Paragon Plus Environment

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m

number of unsaturated neighbor carbon atoms

n

saturation degree of the reactant carbon

ne

number of single events

nob

number of observations

npar

number of parameters

nresp

number of responses

p

pressure (bar)

S

entropy (J mol-1 K-1)

SSQ

residual sum of squares between the experimental and model calculated outlet molar flow rates (mol2 s-2)

T

temperature (K)

r

reaction rate (mol kgcat-1 s-1)

R

universal gas constant (8.314 J mol-1 K-1)

Rj

net production rate of component j (mol kgcat-1 s-1)

Wcat

mass catalyst (kg)

Greek symbols β

real parameter vector

σ

symmetry numbers

θ

surface coverage

∆

difference

Subscripts ext

external symmetry

int

internal symmetry

gl

global symmetry

rot

rotational

t

total

Superscripts 25 ACS Paragon Plus Environment


n

number of chiral centers

r

reactant

≠

transition state

~

single event

˄

model calculated value

o

standard state

0

inlet

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Analytical Rate Expressions Accounting for the Elementary Steps in

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