Methane Conversion to Ethylene and Acetylene: Optimal Control with

In this paper, an optimal control strategy is applied to the problem of finding the flux profiles for the conversion of methane to ethylene and acetyl...
0 downloads 0 Views 696KB Size
Download PDF
Ind. Eng. Chem. Res. 1996, 35, 683-696

683

Methane Conversion to Ethylene and Acetylene: Optimal Control with Chlorine, Oxygen, and Heat Flux Atipat Rojnuckarin and Christodoulos A. Floudas Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544

Herschel Rabitz* Department of Chemistry, Princeton University, Princeton, New Jersey 08544

Richard A. Yetter Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544

In this paper, an optimal control strategy is applied to the problem of finding the flux profiles for the conversion of methane to ethylene and acetylene in a plug flow reactor. A chlorinecatalyzed oxidative pyrolysis mechanism is used in the calculations, in which two mechanistic pathways to the C2 products were examined. One involves CH3Cl and/or CH2Cl, and the other involves C2H6 and/or C2H5 as reaction intermediates. Optimal control designs were performed with respect to the final mass fractions of ethylene and acetylene in a plug flow reactor using heat, oxygen, and chlorine fluxes as controls. The simulation results show that for the temperatures (1200 K < T < 1900 K) and pressure (P ) 1 atm) considered, C2H4 is formed initially, which is subsequently converted to C2H2. Because of the abundant supply of H2 formed during the reaction, it is possible to reform C2H4 from C2H2 by controlled extraction of energy. The heat flux plays the most important role in determining the final concentration of the desired C2 products by controlling the temperature and the rate of H-atom radical generation, and thus, the interconversion between C2H2 and C2H4 through the C2H3 radical. The solutions obtained, although not proven to be globally optimal, are of very high quality. More than 40% yield of the desired C2 products, either ethylene or acetylene, can be obtained in all cases. 1. Introduction The chlorine-catalyzed oxidative pyrolysis (CCOP) process has been proposed as a practical method to convert methane to more valuable products such as ethylene and acetylene (Karra and Senkan, 1988). The proposed process involves the formation of C2 species from the intermediate CH3Cl, which in turn is generated from CH4 and Cl2. The byproduct HCl produced can be either converted back to Cl2 via the Deacon reaction and recycled, or it can be used to oxychlorinate methane to form CH3Cl, thus completing the catalytic cycle of chlorine (Karra and Senkan, 1988). Since methane is abundant in natural gas and is often used only as a fuel, the importance of the ability to convert methane to more valuable chemical feedstocks such as ethylene or acetylene is clear. In this paper, we utilize the optimal control strategy to improve the yield of the process by controlling heat and/or mass fluxes into the reactor. Optimization is carried out with a detailed reaction mechanism consisting of 74 species, including up to C6 hydrocarbon species, and 338 reactions. By treating the feedstock as a natural-gas-like mixture (e.g., methane with a trace amount of C2H6), chlorination is performed in situ and optimization can be applied to the overall conversion process from CH4 to the desired C2 products. Furthermore, a nonchlorinated pathway from CH4 to C2H4 and C2H2 products, involving C2H6 and the C2H5 radical, is also examined. After the optimal control designs were achieved, sensitivity analysis was applied to the results to determine the effect of the optimal controls on the mechanistic pathways of the reaction mechanism. * Author to whom correspondence should be addressed.

0888-5885/96/2635-0683$12.00/0

The optimal control approach implemented in the paper (see Appendix) belongs to the class known as gradient methods in function space, and the application of the algorithm to a plug flow reactor is very similar to those described by Rojnuckarin et al. (1993), where the optimal control of various liquid-phase reactions was examined. In section 2, the reaction mechanism is described, and in section 3, the flow reactor model is presented. Then, four optimal control examples are discussed in section 4, and finally, conclusions that consider the potential practical feasibility of the process are presented in section 5. The Appendix presents the details of the optimal control algorithm as well as all other relevant equations. 2. Reaction Mechanism The chemical species included in the reaction mechanism are given in Table 1. Listed in the table are the standard heats of formation and entropies at 298 K, the specific heats as a function of temperature, and a reference for the source of the data or a reference to the development of a previous submechanism, where the original source may be found. The reaction mechanism and forward rate constants are given in Table 2 along with associated references. Reverse rate constants are computed from the principle of detailed balance using the equilibrium and forward rate constants. A thorough understanding of methane pyrolysis, particularly the later stages of reaction that lead to formation of polycyclic aromatic hydrocarbons and eventually solid carbon, is still a subject of current research. The formation of solid carbonaceous products is known to occur during pyrolysis of methane and © 1996 American Chemical Society

684

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Table 1. Thermodynamic Data: ∆Hf (298), S (298), and CP (T) for Species Considered in the Chlorine-Catalyzed Oxidative Pyrolysis Mechanism of Methanea

H O OH H2 O2 H2O HO2 H2O2 CO CO2 HCO Cl HCl ClO Cl2 CClO HOCl CH2 1CH 2 CH3 CH4 C2H2 C2H3 C2H4 C2H5 C2H6 CHCl CH2Cl CHCl2 CH3Cl CH2Cl2 C2H3Cl CH2CCl2 CHClCHCl CH2ClCH2 CH3CHCl C2H5Cl CH3CHCl2 C2H4Cl2 CH2ClCHCl2 CHClO CH2O CH3O CH2CO CH2ClO C2H HCCO C2H2Cl C3H7 C4 C4H C4H2 C4H3 C4H4 C4H5 C4H6 C4H7 C4H9 C6H C6H2 C6H5 C6H6 C6H7 c-C6H7 C6H8 c-C6H8 C6H9 c-C6H9 C8H C8H2 N2 a

∆Hf 298

S 298

CP 300

CP 400

CP 500

CP 600

CP 800

CP 1000

CP 1500

ref

52.10 59.56 9.32 0.00 0.00 -57.80 3.00 -32.53 -26.42 -94.06 10.40 28.99 -22.06 24.19 0.00 -4.00 -17.80 92.35 101.44 35.12 -17.90 54.19 67.10 12.54 28.36 -20.24 71.00 29.10 23.50 -19.59 -22.80 8.40 0.62 0.75 20.78 17.51 -26.83 -30.60 -31.01 -35.40 -39.30 -27.70 3.90 -14.60 2.16 132.00 42.58 60.40 22.60 232.02 155.09 111.71 129.89 69.15 86.10 28.29 45.70 12.35 213.17 169.68 79.44 19.81 96.41 56.11 38.00 25.90 50.90 41.40 288.89 226.17 0.00

27.39 38.47 43.88 31.21 49.01 45.10 54.76 55.66 47.21 51.08 53.66 39.45 44.64 54.14 53.29 63.50 56.50 46.32 44.15 46.38 44.48 48.01 56.20 52.39 57.90 54.85 56.17 59.60 67.40 56.01 64.59 63.09 69.25 69.25 68.50 67.31 66.03 72.89 73.78 81.50 61.80 52.26 53.25 57.79 63.27 49.58 60.62 64.46 64.14 54.56 60.90 59.79 69.07 67.35 73.08 70.45 74.31 76.37 74.12 70.94 69.83 64.37 81.81 45.10 79.31 46.90 87.91 76.81 78.41 75.96 45.77

4.97 5.23 7.15 6.90 7.01 8.00 8.35 10.41 6.95 8.91 8.24 5.65 6.27 8.40 8.72 10.80 8.91 8.28 8.28 9.26 8.51 10.60 10.89 10.28 12.26 12.58 8.80 9.32 13.11 9.77 12.26 12.33 15.81 15.81 14.00 14.10 15.06 18.29 18.99 21.01 11.12 8.45 9.01 12.98 11.23 8.88 11.79 11.39 18.10 12.09 14.10 17.74 18.02 17.32 19.38 18.58 20.07 28.08 22.14 24.63 21.01 19.92 27.26 26.34 27.96 22.96 27.43 25.16 27.14 29.51 6.95

4.97 5.14 7.10 6.96 7.22 8.23 8.89 11.44 7.03 9.86 8.78 5.59 6.50 8.47 8.76 11.28 9.56 8.62 8.62 10.05 9.77 11.97 12.47 12.73 14.81 15.77 9.45 10.18 13.90 11.58 14.23 15.31 18.43 18.43 17.38 17.19 18.62 21.87 22.14 24.72 12.46 9.46 10.66 15.17 13.27 9.61 13.51 14.08 22.27 13.71 15.37 20.03 20.82 20.61 23.47 23.16 25.63 31.78 25.22 27.76 27.06 27.09 31.33 31.07 32.59 28.34 32.38 30.59 31.60 34.43 7.01

4.97 5.08 7.07 7.00 7.44 8.44 9.46 12.34 7.14 10.65 9.28 5.53 6.71 8.54 8.80 11.68 10.08 8.99 8.99 10.81 11.10 13.08 13.87 14.91 17.13 18.68 10.13 11.14 14.68 13.20 15.88 17.73 20.56 20.56 20.13 19.79 21.67 24.81 24.74 27.67 13.55 10.49 12.22 16.92 15.01 10.22 14.73 16.35 25.98 15.02 16.56 21.85 23.29 23.62 27.21 27.42 29.94 35.22 27.67 30.26 32.43 33.25 35.45 35.88 37.21 33.94 37.53 36.33 35.15 38.32 7.08

4.97 5.05 7.06 7.02 7.65 8.67 9.99 13.11 7.27 11.31 9.77 5.48 6.91 8.61 8.84 12.02 10.50 9.37 9.37 11.54 12.44 13.97 15.11 16.84 19.24 21.31 10.81 12.14 15.44 14.63 17.26 19.67 22.26 22.26 22.37 21.98 24.28 27.23 26.90 29.99 14.42 11.49 13.69 18.32 16.47 10.73 15.61 18.26 29.23 16.06 17.66 23.24 25.41 26.30 30.54 31.26 33.26 38.41 29.53 32.18 37.05 38.38 39.51 40.58 41.72 39.49 42.66 42.10 37.84 41.23 7.19

4.97 5.02 7.13 7.07 8.07 9.22 10.77 14.29 7.61 12.32 10.74 5.39 7.26 8.72 8.90 12.53 11.13 10.15 10.15 12.90 15.00 15.31 17.15 20.02 22.85 25.80 12.11 14.10 16.83 17.02 19.36 22.47 24.68 24.68 25.81 25.42 28.43 30.87 30.32 33.25 15.70 13.34 16.28 20.31 18.72 11.54 16.96 21.23 34.37 17.52 19.59 25.10 28.56 30.53 35.76 37.44 37.91 44.06 31.81 34.76 43.90 45.87 47.07 49.08 50.03 49.68 52.24 52.81 41.15 44.72 7.50

4.97 5.00 7.33 7.21 8.35 9.87 11.38 15.21 7.95 12.99 11.52 5.31 7.56 8.81 8.95 12.88 11.58 10.88 10.88 14.09 17.20 16.29 18.73 22.45 25.74 29.33 13.22 15.83 17.98 18.87 20.81 24.26 26.19 26.19 28.42 27.99 31.47 33.44 33.06 35.36 16.58 14.86 18.38 21.61 20.28 12.16 18.21 23.38 38.29 18.39 21.15 26.61 30.46 33.20 38.89 41.29 41.11 48.86 33.27 36.81 47.77 51.05 53.42 55.74 57.00 57.76 60.14 61.45 43.23 46.91 7.83

4.97 4.98 7.87 7.73 8.72 11.26 12.48 16.85 8.41 13.93 12.56 5.18 8.13 8.99 9.06 13.40 12.40 12.22 12.22 16.26 20.61 18.31 21.34 26.21 30.54 34.91 14.78 18.31 19.80 21.80 22.90 26.88 28.21 28.21 33.43 32.50 36.27 37.80 38.79 38.91 18.11 16.95 21.56 23.80 22.41 13.32 19.36 26.87 44.31 19.55 23.44 28.96 33.46 37.25 43.16 46.79 47.49 57.66 35.88 39.90 53.26 58.31 62.83 64.18 67.49 68.04 71.43 72.72 46.76 51.00 8.32

Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Weissman and Benson, 1984 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Weissman and Benson, 1984 Weissman and Benson, 1984 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987

Units are cal/mol·K for CP and S and kcal/mol for ∆Hf.

chloromethane (Back and Back, 1983; Weissman and Benson, 1984; Benson, 1980; Senkan, 1987). In order to minimize the formation of solid carbon during py-

rolysis, small amounts of molecular oxygen may be added to the initial methane mixture to route the reaction toward formation of oxygenates such as carbon

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 685

oxides and away from the formation of large molecular weight hydrocarbons (Senkan, 1987; Grenada et al., 1987). The addition of chlorine to pyrolyzing methane mixtures produces a new channel (i.e., parallel to the self-reaction of methyl radicals for formation of ethylene and acetylene) through formation of methyl chloride or chloromethyl radicals and their subsequent reaction. The presence of either chlorine or oxygen also lowers the overall activation energy of the initial bond-breaking process, thus allowing the reaction to be initiated at lower temperatures at which solid carbon formation can be reduced. For the present study on the chlorine-catalyzed oxidative pyrolysis of methane, the reaction mechanism was constructed from a moist CO oxidation submechanism (Yetter et al., 1991), a literature review of reactions for the pyrolysis and oxidation of methane (Tsang and Hampson, 1986), a mechanism for methylchloride pyrolysis and oxidation (Roseler et al., 1994; Ho et al., 1992; Weissman and Benson, 1988; Karra and Senkan, 1988), and mechanisms for the pyrolysis of acetylene (Kiefer et al., 1992; Kiefer and Van Drasek, 1990; Benson, 1989; Chanmugathas and Heicklen, 1986; Tanzawa and Gardiner, 1980). Although soot and carbon deposit formation are not explicitly provided for in the mechanism (Frenklach and Wang, 1990; Frenklach et al., 1983, 1984), steps are included for formation of allene and 1,3-butadiene and for cyclization leading to benzene (i.e., indicators or precursors of soot formation in methane pyrolysis). In addition, the formation of polyacetylenes and their radicals up through C8H2 and C8H and of pure carbon species up to C4 are included. In the present model, the formation of any of these species will also be interpreted as being associated with the possibility of forming solid carbon. A schematic diagram illustrating the dominant species and their reaction sequence, obtained from reaction flux and sensitivity analyses of the optimally controlled reactor solutions, is given in Figure 1. Two parallel pathways are shown to lead from CH4 to the desired C2 products: a chlorinated pathway through CH3Cl and a nonchlorinated pathway through C2H6. Of the desired products, ethylene is produced first, which can subsequently be converted to C2H2. Acetylene can then be used to reform C2H4, can be oxidized to CO and CO2, or used to undergo polymerization to form higher order hydrocarbons and soot. Thus, to control the ethylene yield, the reaction could be quenched prior to the formation of acetylene or be allowed to progress to maximum acetylene yield (where the reaction could be quenched if C2H2 is desired). Alternatively, in a second stage of reaction, the acetylene could be used as the feedstock to reform ethylene. 3. Physical Formulations A plug flow reactor (PFR) was chosen to run the pyrolysis reaction. In this calculation, the PFR was taken to be a cylinder with constant cross sectional area A, which permits control by chemical and/or heat flux added through the side wall of the reactor as a function of position l along its length. The reactions, described by the rate of production wi of species i, i ) 1 ... n, proceed along the length of the reactor and terminate at the other end. The control variables are the flux densities of species i, denoted as ji (mass/length-time), and heat flux density (energy/length-time), denoted as q, as a function of position l. The mass fraction of species i in the reactor is denoted as xi, and the mass flow rate is denoted as F(l).

In modeling the PFR, we assume the following: (i) steady one-dimensional plug flow, (ii) instantaneous radial mixing, (iii) no diffusion along the axis of the reactor, (iv) adiabatic reaction conditions, and (v) the heat capacity of each species Cpi is constant and equal to the value at the input temperature To ) T(0), for simplicity. In addition, the temperature of the mass fluxes is also arbitrarily assigned to To. The governing equation of the plug flow reactor is formulated as follows:

F(l) ) F(0) +

n

∫0 ∑ji(l′) dl′ l

i)1

dxi

n 1 ) (wi - xi jk + ji) dl F k)1

1

dT ) dl

∑

n

(1)

n

ji - ∑Hfiwi + q] ∑ i)1 i)1

[C h pf(To - T)

FC hp

Here, C h pf denotes the mean specific heat of influx species (mass units), C h p denotes the mean specific heat of the mixture (mass units), and Hfi denotes the enthalpy of the ith species (mass units). The objective function J(xi, T, ji, q) for the optimal control problem can be formulated in many different ways, but it is desirable that J be a smooth and convex function that is bounded from below. In this presentation, J was simply chosen as

J)

∑i Ci(xi(L) - xfi)2

(2)

to force the mass fraction of the controlled species i at the end of the reactor xi(L) to the desired value xfi given the weighting factor Ci. The importance of the weighting factor will be discussed in the following section. In the examples considered, either acetylene or ethylene as well as C6H6 are chosen to be the controlled species at the output position L. Satisfactory results were obtained without specific costs put on the total amount of control fluxes. A full summary of the optimal control algorithm is given in the Appendix. The CHEMKIN II chemical kinetics package was used to calculate the rates of reaction and thermodynamic properties (Kee et al., 1989) from the data given in Tables 1 and 2. The cgs system of units is used except where stated otherwise. 4. Examples The design algorithm was applied to numerous optimal control cases. In each example, a few runs were made with increasingly demanding objectives. The optimal flux profiles of the previous runs were found to be good initial points for subsequent runs. Four examples will be presented in this section. The first example deals with the optimal control flux profiles of Cl2 and heat that aim to maximize the mass fraction of C2H4 in the output as well as minimizing those species which could lead to soot formation or those species which when present are a good indication of the presence of soot (specifically, benzene, butadiene, and vinyl acetylene were chosen here). The second example considers the optimal control fluxes of O2 and heat that also aim to maximize the mass fraction of C2H4 in the output, while minimizing C6H6, C4H6, and C4H4 formation. The third example deals with the optimal control

686

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Table 2. Reaction Mechanism for Chlorine-Catalyzed Pyrolysis of Methanea (k ) ATb exp(-E/RT)) no.

reactions

A

b

E

ref

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77

H + O2 h O + OH O + H2 h H + OH H2 + OH h H2O + H OH + OH h O + H2O H2 + M h H + H + M O + O + M h O2 + M O + H + M h OH + M OH + H + M h H2O + M H + O2 + M h HO2 + M HO2 + H h H2 + O2 HO2 + H h OH + OH HO2 + O h O2 + OH HO2 + OH h H2O + O2 H2O2 + M h OH + OH + M HO2 + HO2 h H2O2 + O2 H2O2 + H h H2O + OH H2O2 + H h HO2 + H2 H2O2 + O h HO2 + OH H2O2 + OH h H2O + HO2 CO + O2 h CO2 + O CO + O + M h CO2 + M CO + OH h CO2 + H CO + HO2 h CO2 + OH HCO + M h H + CO + M HCO + O2 h CO + HO2 HCO + H h CO + H2 HCO + O h CO + OH HCO + OH h CO + H2O H + Cl + M h HCl + M Cl + HO2 h HCl + O2 HCl + H h H2 + Cl HCl + O h OH + Cl HCl + OH h H2O + Cl H2O2 + Cl h HO2 + HCl HCO + Cl h CO + HCl Cl + Cl + M h Cl2 + M Cl2 + H h HCl + Cl ClO + O h Cl + O2 HO2 + Cl h ClO + OH ClO + CO h CO2 + Cl Cl2 + O h ClO + Cl HOCl h OH + Cl HOCl h H + ClO ClO + H2 h HOCl + H HOCl + H h HCl + OH HOCl + O h OH + ClO HOCl + OH h H2O + ClO HCO + ClO h HOCl + CO HOCl + Cl h ClO + HCl HOCl + Cl h Cl2 + OH CClO + M h CO + Cl + M CClO + O2 h CO2 + ClO CClO + H h CO + HCl CClO + O h CO + ClO CClO + O h CO2 + Cl CClO + OH h CO + HOCl CClO + Cl h CO + Cl2 CH2O + M h HCO + H + M CH2O + O2 h HCO + HO2 CH2O + H h HCO + H2 CH2O + O h HCO + OH CH2O + OH h HCO + H2O CH2O + HO2 h HCO + H2O2 CH2O + Cl h HCO + HCl CH2O + ClO h HCO + HOCl CHClO h HCO + Cl CHClO + H h HCO + HCl CHClO + H h CH2O + Cl CHClO + O h CClO + OH CHClO + OH h CClO + H2O CHClO + Cl h CClO + HCl CH3O + M h CH2O + H + M CH3O + O2 h CH2O + HO2 CH3O + H h CH2O + H2 CH3O + ClO h CH2O + HOCl CH2ClO h CH2O + Cl CH2ClO h CHClO + H

1.92 × 1014 5.08 × 104 2.16 × 108 1.23 × 104 4.57 × 1019 6.17 × 1015 4.72 × 1018 2.23 × 1022 6.70 × 1019 6.63 × 1013 1.69 × 1014 1.81 × 1013 4.05 × 1016 1.20 × 1017 3.02 × 1012 1.00 × 1013 4.82 × 1013 9.55 × 106 7.00 × 1012 2.53 × 1012 2.51 × 1013 1.50 × 107 6.02 × 1013 1.86 × 1017 7.58 × 1012 7.23 × 1013 3.02 × 1013 3.02 × 1013 7.20 × 1021 1.08 × 1013 1.79 × 1012 3.37 × 103 2.71 × 107 6.62 × 1012 1.00 × 1014 4.68 × 1014 8.59 × 1013 5.70 × 1013 2.42 × 1013 6.03 × 1011 2.52 × 1012 1.76 × 1020 8.12 × 1014 6.03 × 1011 9.55 × 1013 6.03 × 1012 1.81 × 1012 3.16 × 1013 7.62 × 1012 1.81 × 1012 1.30 × 1014 7.94 × 1010 1.00 × 1014 1.00 × 1014 1.00 × 1013 3.30 × 1012 4.00 × 1014 5.00 × 1016 2.05 × 1013 2.50 × 1013 3.50 × 1013 3.00 × 1013 1.00 × 1012 5.00 × 1013 5.50 × 1013 8.86 × 1029 8.33 × 1013 6.99 × 1014 8.80 × 1012 7.50 × 1012 1.25 × 1013 1.00 × 1014 1.00 × 1013 2.00 × 1013 2.41 × 1013 4.53 × 1031 1.83 × 1027

0.00 2.67 1.51 2.62 -1.40 -0.50 -1.00 -2.00 -1.42 0.00 0.00 0.00 -1.00 0.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 1.30 0.00 -1.00 0.00 0.00 0.00 0.00 -2.00 0.00 0.30 2.87 1.65 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -3.01 -2.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.81 -5.15 0.00 -0.58 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -6.41 -5.13

16.40 6.29 3.43 -1.88 104.00 0.00 0.00 0.00 0.00 2.13 0.87 -0.40 0.00 45.50 1.39 3.59 7.95 3.97 1.43 47.70 -4.54 -0.77 23.00 17.00 0.41 0.00 0.00 0.00 0.00 0.10 3.80 3.51 -0.22 1.95 0.00 -1.80 1.17 0.36 2.30 74.0 2.72 56.72 93.69 14.10 7.62 4.37 0.99 0.00 0.18 0.26 8.00 3.30 0.00 0.00 0.00 0.00 0.80 76.23 38.95 3.99 3.51 1.19 8.00 0.50 5.86 92.92 7.40 6.36 3.50 1.20 0.50 25.10 7.17 0.00 0.00 22.56 21.17

Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Yetter et al., 1991 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Tsang and Hampson, 1986 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 687 Table 2. (Continued) (k ) ATb exp(-E/RT)) no.

reactions

A

b

E

ref

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154

CH2 + O2 h CH2O + O CH2 + O h CO + H + H CH2 + OH h CH2O + H 1CH + M h CH + M 2 2 1CH + O h CO + H O 2 2 2 1CH + O h CO + H + H 2 1CH + O h CO + H 2 2 1CH + OH h CH O + H 2 2 CHCl + O2 h CHClO + O CH3 + O2 h CH2O + OH CH3 + O2 h CH3O + O CH2 + H2 h CH3 + H 1CH + H h CH + H 2 2 3 CH3 + O h CH2O + H CH3 + OH h CH3O + H CH3 + HO2 h CH3O + OH CH3 + ClO h CH2O + HCl CH3 + ClO h CH3O + Cl CH2Cl + O2 h CH2O + ClO CH2Cl + H2 h CH3Cl + H CH2Cl + H h CH3Cl CH2Cl + H h CH3 + Cl CH2Cl + O h CH2ClO CH2Cl + O h CH2O + Cl CH2Cl + OH h CH2O + HCl CH2Cl + OH h CH3O + Cl CH2Cl + HO2 h CH2ClO + OH CH2Cl + ClO h CHClO + HCl CH2Cl + ClO h CH2ClO + Cl CHCl2 + H h CH2Cl + Cl CH4 h CH3 + H 1CH + H h CH 2 2 4 CH4 + O2 h CH3 + HO2 CH4 + H h CH3 + H2 CH4 + O h CH3 + OH CH4 + OH h CH3 + H2O CH4 + HO2 h CH3 + H2O2 CH3 + HCO h CH4 + CO CH4 + HCO h CH3 + CH2O CH4 + Cl h CH3 + HCl CH4 + ClO h CH3 + HOCl CH4 + 1CH2 h CH3 + CH3 CH3Cl h CH3 + Cl CH3Cl h 1CH2 + HCl CH3Cl + O2 h CH2Cl + HO2 CH3Cl + H h CH3 + HCl CH3Cl + O h CH2Cl + OH CH3Cl + OH h CH2Cl + H2O CH3Cl + HO2 h CH2Cl + H2O2 CH2Cl + CH2O h CH3Cl + HCO CH3Cl + Cl h CH2Cl + HCl CH3Cl + ClO h CH2Cl + HOCl CH3Cl + CH3 h CH2Cl + CH4 CH2Cl2 h CHCl + HCl CH2Cl2 h CH2Cl + Cl CHCl2 + H h CH2Cl2 CH2Cl2 + H h CH2Cl + HCl CHCl2 + H2 h CH2Cl2 + H CH2Cl2 + OH h CHCl2 + H2O CH2Cl2 + HO2 h CHCl2 + H2O2 CH2Cl2 + Cl h CHCl2 + HCl CH2Cl2 + CH3 h CHCl2 + CH4 CH2Cl2 + CH3 h CH3Cl + CH2Cl C2 + H + M h C2H + M C2H + O2 h CO + HCO C2H + O2 h HCCO + O C2 + H2 h C2H + H C2H + OH h HCCO + H HCCO + H h 1CH2 + CO HCCO + O h CO + CO + H CH2CO + H h CH3 + CO CH2CO + OH h HCO + CH2O C2H2 + M h C2H + H + M C2H2 + H h C2H + H2 C2H2 + O h CH2 + CO C2H + OH h C2H2 + O C2H2 + OH h CH2CO + H

1.00 × 1014 5.00 × 1013 2.50 × 1013 1.00 × 1013 2.41 × 1011 1.50 × 1013 1.50 × 1013 3.00 × 1013 1.00 × 1014 3.59 × 109 2.88 × 1015 3.01 × 109 1.27 × 1014 7.00 × 1013 3.87 × 1012 2.00 × 1013 3.47 × 1018 3.33 × 1011 1.91 × 1014 1.79 × 1012 3.15 × 1027 6.95 × 1015 1.29 × 1015 5.59 × 1013 1.24 × 1022 2.00 × 1012 1.00 × 1013 4.13 × 1019 4.15 × 1012 1.25 × 1014 1.03 × 1033 3.82 × 1025 4.04 × 1013 1.55 × 1014 1.20 × 107 1.60 × 106 2.00 × 1013 1.20 × 1014 7.30 × 103 1.10 × 107 6.03 × 1011 3.45 × 1022 9.98 × 1022 1.10 × 1028 2.02 × 1013 9.55 × 1013 1.54 × 1013 9.30 × 107 1.00 × 1013 2.00 × 1011 2.30 × 107 3.03 × 1011 3.30 × 1011 8.73 × 1037 7.40 × 1040 4.81 × 1026 7.00 × 1013 4.63 × 1012 2.83 × 1012 6.67 × 1012 2.79 × 1013 6.76 × 1010 1.40 × 1011 1.00 × 1023 2.41 × 1012 6.02 × 1011 3.02 × 1013 2.00 × 1013 1.50 × 1014 1.00 × 1014 7.00 × 1012 2.80 × 1013 6.17 × 1041 6.00 × 1013 4.10 × 108 1.81 × 1013 3.20 × 1011

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.14 -1.15 0.00 -0.08 0.00 -0.19 0.00 -1.80 0.46 -1.27 0.00 -4.90 -0.57 -1.98 -0.13 -2.72 0.29 0.00 -2.22 0.07 -0.03 -5.58 -4.47 0.00 0.00 2.10 2.10 0.00 0.00 2.85 1.97 0.00 -2.48 -2.40 -5.15 0.00 0.00 0.31 1.60 0.00 0.00 2.00 0.00 0.00 -7.68 -7.84 -4.82 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -2.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -6.63 0.00 1.50 0.00 0.00

3.70 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.70 10.15 30.85 0.00 0.13 0.00 13.74 0.00 2.07 0.03 3.81 13.06 14.05 0.81 1.10 0.71 3.86 3.27 0.00 2.36 1.11 0.57 111.80 3.77 56.91 11.00 7.62 2.46 18.00 0.00 22.51 1.49 15.00 7.46 95.40 109.70 54.00 7.62 11.90 2.07 21.66 6.00 1.18 10.70 9.40 86.73 84.99 3.81 7.10 15.30 2.09 18.27 2.94 7.20 4.90 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.01 0.00 137.00 23.66 1.69 0.00 0.20

Roseler et al., 1993 Hunter et al., 1994 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Hunter et al., 1994 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Kiefer et al., 1992 Roseler et al., 1993 Hunter et al., 1994 Kiefer et al., 1992 Hunter et al., 1994 Hunter et al., 1994 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Kiefer et al., 1992 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993

688

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Table 2. (Continued) (k ) ATb exp(-E/RT)) no.

reactions

A

b

E

ref

155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 222 223 224 225 226 227 228 229 230 231 232

C2H2 + OH h C2H + H2O C2H2 + HO2 h CH2CO + OH C2H + HCO h C2H2 + CO C2H2 + Cl h C2H + HCl C2H + C2H h C2H2 + C2 C2H2 + CH3 h CH4 + C2H C2H3 h C2H2 + H C2H3 + O2 h C2H2 + HO2 C2H3 + O2 h HCO + CH2O C2H3 + H h C2H2 + H2 C2H3 + O h CH2CO + H C2H3 + OH h C2H2 + H2O C2H3 + CH3 h C2H2 + CH4 C2H3 + 1CH2 h C2H2 + CH3 C2H3 + C2H h C2H2 + C2H2 C2H2Cl + O2 h CHClO + HCO C2H2Cl + O2 h CH2O + CClO C2H4 h C2H2 + H2 C2H4 h C2H3 + H C2H4 + O2 h C2H3 + HO2 C2H3 + H2 h C2H4 + H 1CH + CH h C H + H 2 3 2 4 C2H4 + O h CH3 + HCO C2H4 + O h C2H3 + OH C2H4 + OH h C2H3 + H2O C2H4 + Cl h C2H3 + HCl C2H4 + ClO h CH2Cl + CH2Ob 1CH + CH Cl h C H + HCl 2 3 2 4 CH2Cl + CH3 h C2H4 + HCl C2H4 + CH3 h C2H3 + CH4 C2H3 + C2H3 h C2H4 + C2H2 C2H + C2H4 h C2H2 + C2H3 C2H3Cl h C2H2 + HCl C2H3Cl h C2H3 + Cl C2H3Cl + H h C2H4 + Cl C2H3Cl + H h C2H3 + HCl C2H3Cl + O h C2H2Cl + OHb C2H3Cl + O h CHClO + CH2 C2H3Cl + O h CH2O + CHCl C2H3Cl + OH h C2H2Cl + H2O C2H3Cl + Cl h C2H2Cl + HCl C2H3Cl + ClO h CH2Cl + CHClOb CH2Cl + CH2Cl h C2H3Cl + HCl CHCl2 + CH3 h C2H3Cl + HCl CH2CCl2 + H h C2H3Cl + Cl CH2Cl + CHCl2 h CH2CCl2 + HCl CHClCHCl + H h C2H3Cl + Cl CH2Cl + CHCl2 h CHClCHCl + HCl C2H5 h C2H4 + H C2H5 + O2 h C2H4 + HO2 C2H5 + H h CH3 + CH3 C2H5 + H h C2H4 + H2 1CH + CH h C H + H 2 4 2 5 C2H5 + O h CH2O + CH3 C2H5 + OH h C2H4 + H2O C2H5 + HO2 h C2H4 + H2O2 1CH + CH Cl h C H + Cl 2 3 2 5 CH2Cl + CH3 h C2H5 + Cl C2H5 + CH3 h C2H4 + CH4 C2H3 + C2H5 h C2H4 + C2H4 C2H5 + C2H h C2H2 + C2H4 C2H5 + 1CH2 h C2H4 + CH3 CHCl2 + CH3 h CH3CHCl + Cl CH2Cl + CH2Cl h CH2ClCH2 + Cl C2H6 h CH3 + CH3 C2H6 h C2H5 + H C2H6 + H h C2H5 + H2 C2H6 + O h C2H5 + OH C2H6 + OH h C2H5 + H2O C2H6 + Cl h C2H5 + HCl C2H6 + CH3 h C2H5 + CH4 C2H6 + C2H3 h C2H4 + C2H5 C2H3 + C2H5 h C2H2 + C2H6 C2H5 + C2H5 h C2H4 + C2H6 C2H6 + C2H h C2H5 + C2H2 C2H6 + 1CH2 h CH3 + C2H5 C2H5Cl h C2H4 + HCl

2.71 × 1013 6.03 × 109 6.00 × 1013 1.58 × 1014 1.00 × 1013 1.80 × 1011 6.24 × 1029 1.21 × 1011 5.40 × 1012 9.26 × 1013 3.00 × 1013 4.00 × 1012 7.94 × 1011 1.80 × 1013 1.00 × 1014 3.00 × 1012 2.00 × 1012 8.52 × 1043 8.53 × 1030 4.22 × 1013 3.16 × 109 1.80 × 1013 5.50 × 106 7.11 × 108 2.05 × 1013 1.00 × 1014 0.00 × 1012 1.60 × 1018 3.50 × 1028 4.20 × 1011 9.64 × 1011 2.00 × 1013 1.62 × 1028 1.71 × 1038 1.55 × 1013 1.20 × 1012 0.00 × 1012 1.60 × 1012 3.15 × 1012 1.80 × 1013 5.00 × 1013 0.00 × 1012 3.75 × 1035 1.35 × 1030 7.21 × 1012 3.75 × 1036 3.44 × 1013 1.22 × 1037 1.83 × 1039 2.00 × 1012 1.35 × 1022 3.00 × 1013 9.43 × 1012 1.00 × 1013 2.41 × 1013 3.01 × 1011 3.09 × 107 9.27 × 1019 7.94 × 1011 4.82 × 1011 1.80 × 1012 9.00 × 1012 2.74 × 1025 9.34 × 1029 5.34 × 1054 6.22 × 1047 5.40 × 102 1.40 8.85 × 109 4.37 × 1013 5.50 × 101 6.02 × 102 4.82 × 1011 1.38 × 1012 3.61 × 1012 1.14 × 1014 7.81 × 1019

0.00 0.00 0.00 0.00 0.00 0.00 -5.29 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -8.32 -5.87 0.00 0.70 0.00 2.08 1.55 0.00 0.00 0.00 -1.47 -4.49 0.00 0.00 0.00 -4.29 -7.13 -0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -6.73 -4.96 0.00 -7.22 -0.03 -7.20 -7.75 0.00 -2.17 0.00 -0.13 0.00 0.00 0.00 1.70 -2.07 0.00 0.00 0.00 0.00 -3.45 -4.95 -11.10 -9.76 3.50 4.30 1.04 0.00 4.00 3.30 0.00 0.00 0.00 0.00 -2.00

10.50 7.95 0.00 16.90 0.00 17.30 46.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30 0.00 121.20 118.20 57.62 5.11 0.00 0.00 3.76 5.94 7.00 0.00 2.71 9.18 11.11 0.00 0.00 75.78 96.37 5.84 15.00 0.00 1.20 1.50 4.00 7.00 0.00 13.16 11.55 7.51 13.62 5.89 13.64 52.82 4.99 7.00 0.00 6.62 0.00 0.00 0.00 0.52 10.13 0.00 0.00 0.00 0.00 12.81 14.07 112.20 111.20 5.21 2.62 1.81 0.10 8.28 10.50 0.00 0.00 0.00 0.00 60.66

Hunter et al., 1994 Roseler et al., 1993 Tsang and Hampson, 1986 Roseler et al., 1993 Kiefer et al., 1992 Tsang and Hampson, 1986 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Hunter et al., 1994 Hunter et al., 1994 Tsang and Hampson, 1986 Kiefer et al., 1992 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Weissman and Benson, 1988 Hunter et al., 1994 Roseler et al., 1993 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Tsang and Hampson, 1986 Frenklach et al., 1984 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Tsang and Hampson, 1986 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Hunter et al., 1994 Roseler et al., 1993 Roseler et al., 1993 Hunter et al., 1994 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Tsang and Hampson, 1986 Roseler et al., 1993

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 689 Table 2. (Continued) (k ) ATb exp(-E/RT)) no.

reactions

A

b

E

ref

233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309

CH2Cl + CH3 h C2H5Cl 1CH + CH Cl h C H Cl 2 3 2 5 C2H5Cl + Cl h HCl + CH2ClCH2 C2H4Cl2 h C2H3Cl + HCl CH2Cl + CH2Cl h C2H4Cl2 CH2ClCHCl2 h CH2ClCH2 CHCl2 + CH3 h CH3CHCl2 CH3 + C2H4 h C3H7 C3H7 h C3H6 + H C2 + C2H h C4 + H C2 + C2 + M h C4 + H C4 + H + M h C4H + M C2H + C2H h C4H + H C4H + C2H h C2H2 + C4 C2 + C2H2 h C4H + H C4H + H h C4 + H2 C2 + C2H h C4H C2 + C4H h C2H + C4 C2H + C2H2 h C4H2 + H C2H2 + C2H2 h C4H2 + H2 C2H + C4H2 h C4H + C2H2 C4H + C4H h C4 + C4H2 C4H + H2 h C4H2 + H C2H3 + C4H h C2H2 + C4H2 C2H + C2H h C4H2 C4H2 + M h C4H + H + M C2H2 + C2H2 h HCCHCCH + H H + HCCHCCH h C4H2 + H2 HCCHCCH + C2H h C2H2 + C4H2 C4H + HCCHCCH h C4H2 + C4H2 H + C4H2 h HCCHCCH C2H2 + C2H2 h CH2CHCCH C2H3 + C2H2 h CH2CHCCH + H HCCHCCH + H + M h CH2CHCCH + M CH2CHCCH + H h HCCHCCH + H2 CH2CHCCH + C2H h C2H2 + HCCHCCH CH2CHCCH + Cl h HCCHCCH + HCl C2H3 + C4H2 h CH2CHCCH + C2H C2H3 + CH2CHCCH h C2H4 + HCCHCCH C2H + C2H4 h CH2CHCCH + H C2H3 + C2H2 h C4H5 C4H5 h CH2CHCCH + H C2H3 + C2H4 h C4H6 + H H2 + C4H5 h H + C4H6 C4H5 + HCl h C4H6 + Cl C2H3 + CH2CHCCH h C2H + C4H6 C2H3 + C2H4 h C4H7 C4H7 h C4H6 + H CH3 + C3H6 h C4H9 C4H9 h C4H7 + H2 C4H + C2H h C6H + H C2 + C4H2 h C6H + H C6H + H h C2H2 + C4 C2 + C4H h C6H C2H + C4H2 h C6H2 + H C2H2 + C4H2 h C6H2 + H2 C2H2 + C4H2 h C6H2 + H + H C4H + C2H2 h C6H2 + H C2H + C6H2 h C6H + C2H2 C4H + C6H2 h C6H + C4H2 C6H + C2H h C6H2 + C2 C4H + C6H h C4 + C6H2 C6H + H2 h C6H2 + H C6H + C2H3 h C6H2 + C2H2 C6H + HCCHCCH h C4H2 + C6H2 C4 + C2H2 h C6H2 C6H2 + M h C6H + H + M C2H + C4H h C6H2 C2H + C6H2 h C4H2 + C4H C2H3 + C4H2 h C6H5 H + C6H5 h C6H6 H + C6H6 h C6H5 + H2 C2H + C6H6 h C6H5 + C2H2 C4H5 + C2H2 h C6H7 C6H7 h C6H5 + H + H C6H7 h c-C6H7 c-C6H7 h C6H6 + H

3.27 × 1040 7.85 × 1031 1.12 × 1013 6.76 × 1019 7.84 × 1045 6.41 × 1033 2.28 × 1041 1.00 × 1012 6.31 × 1013 1.20 × 1014 1.00 × 1027 1.74 × 1037 1.00 × 1014 1.00 × 1013 1.20 × 1014 2.00 × 1013 2.19 × 1031 1.20 × 1014 1.20 × 1014 1.51 × 1013 1.00 × 1013 1.20 × 1014 4.07 × 105 1.00 × 1014 8.32 × 1021 4.47 × 1017 1.51 × 1014 2.00 × 1013 1.00 × 1014 1.00 × 1013 4.90 × 1018 1.51 × 105 1.58 × 1013 1.00 × 1015 1.51 × 107 3.98 × 1013 1.00 × 1014 1.00 × 1013 1.00 × 1012 6.00 × 1011 2.51 × 105 1.00 × 1014 5.00 × 1011 4.00 × 109 8.00 × 1012 1.00 × 1013 2.00 × 1011 3.16 × 1013 6.30 × 1011 1.00 × 1014 1.00 × 1014 1.20 × 1014 1.00 × 1014 2.09 × 1023 1.20 × 1014 1.51 × 1013 1.51 × 1014 1.20 × 1014 1.00 × 1013 1.00 × 1013 1.20 × 1014 1.20 × 1014 4.07 × 105 1.00 × 1013 1.00 × 1013 3.16 × 1026 5.00 × 1016 1.55 × 1020 1.00 × 1013 1.00 × 1010 1.00 × 1013 1.00 × 1014 2.00 × 1013 2.51 × 105 1.00 × 105 3.16 × 1011 1.00 × 1014

-8.49 -6.15 0.00 -1.93 -10.2 -10.2 -8.68 0.00 0.00 0.00 -3.00 -5.50 0.00 0.00 0.00 0.00 -5.50 0.00 0.00 0.00 0.00 0.00 2.40 0.00 -3.00 0.00 0.00 0.00 0.00 0.00 -2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 0.00 0.00 1.90 0.00 0.00 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -3.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.40 0.00 0.00 -4.00 0.00 -2.30 0.00 0.00 0.00 0.00 0.00 1.80 0.00 0.00 0.00

10.59 5.83 1.50 58.71 13.15 12.91 11.62 2.00 38.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 42.70 0.00 0.00 0.20 0.00 0.00 80.06 56.00 0.00 0.00 0.00 0.00 5.50 25.10 0.00 6.00 0.00 1.00 0.00 16.25 3.82 2.10 41.40 7.30 3.70 1.00 0.00 2.00 34.80 8.80 36.30 0.00 0.00 0.00 0.00 0.00 42.70 56.00 0.00 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 80.06 0.00 0.00 0.00 0.00 0.00 0.00 1.19 0.00 8.00 15.00

Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Roseler et al., 1993 Weissman and Benson, 1984 Weissman and Benson, 1984 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Tanzawa and Gardiner, 1980 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Tanzawa and Gardiner, 1980 Tanzawa and Gardiner, 1980 Kiefer et al., 1992 Tanzawa and Gardiner, 1980 Weissman and Benson, 1984 Frenklach et al., 1984 Frenklach et al., 1984 Frenklach et al., 1984 Weissman and Benson, 1988 Weissman and Benson, 1984 Tsang and Hampson, 1986 Weissman and Benson, 1988 Weissman and Benson, 1984 Frenklach et al., 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Tanzawa and Gardiner, 1980 Kiefer et al., 1992 Frenklach et al., 1984 Frenklach et al., 1984 Frenklach et al., 1984 Frenklach et al., 1984 Frenklach et al., 1984 Weissman and Benson, 1988 Frenklach et al., 1984 Weissman and Benson, 1984 Weissman and Benson, 1984

690

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Table 2. (Continued) (k ) ATb exp(-E/RT)) no.

reactions

A

b

E

ref

310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338

C6H8 h C6H6 + H2 c-C6H8 h C6H6 + H2 C2H3 + C4H6 h C6H9 C6H9 h C6H8 + H C6H9 h c-C6H9 c-C6H9 h c-C6H8 + H C4H + C4H h C8H + H C2H + C6H h C8H + H C2 + C6H2 h C8H + H C8H + H h C4H2 + C4 C6H + C2 h C8H C4H + C4 h C8H C2H + C6H2 h C8H2 + H C2H2 + C6H2 h C8H2 + H2 C2H2 + C6H2 h C8H2 + H + H C4H + C4H2 h C8H2 + H C4H2 + C4H2 h C8H2 + H2 C4H2 + C4H2 h C8H2 + H + H C6H + C2H2 h C8H2 + H C2H + C8H2 h C8H + C2H2 C4H + C8H2 h C8H + C4H2 C2H + C8H h C8H2 + C2 C8H + C4H h C4 + C8H2 C8H + H2 h C8H2 + H C8H + C2H3 h C8H2 + C2H2 C4H + C4H h C8H2 C2H + C6H h C8H2 C4 + C4H2 h C8H2 C8H2 + M h C8H + H + M

3.16 × 1013 2.51 × 1012 3.16 × 1011 3.16 × 1013 2.00 × 1011 6.31 × 1010 1.00 × 1014 1.00 × 1014 1.20 × 1014 1.00 × 1014 2.00 × 1021 2.00 × 1021 1.20 × 1014 1.51 × 1013 1.51 × 1014 1.20 × 1014 1.51 × 1013 1.51 × 1014 1.20 × 1014 1.00 × 1013 1.00 × 1013 1.00 × 1013 1.00 × 1013 4.07 × 105 1.00 × 1013 1.66 × 1012 3.63 × 1012 1.58 × 1023 5.00 × 1016

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -2.30 -2.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.40 0.00 0.00 0.00 -3.00 0.00

40.00 43.80 3.00 42.20 8.00 34.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 42.70 56.00 0.00 42.70 56.00 0.00 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 0.00 80.00

Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Weissman and Benson, 1984 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Kiefer et al., 1992 Tanzawa and Gardiner, 1980

a Units are cm3 mol-1 s-1 kcal-1. The 1CH in these reactions indicates a singlet methane. Apparent rate constant is used for pressure2 dependent reactions not at their low/high pressure limit. See Ho et al. (1992) for a discussion. b These reactions were not in the original mechanism of Roseler et al. (1993) and were included here to investigate their possible role through sensitivity analysis.

Table 3. Reactor Parameters and Objective Function Coefficients for All Examples example example example example 1 2 3 4 pressure (atm) reactor length (cm) cross section (cm2) initial flow (g/s) initial mass fraction CH4 O2 Cl2 C2H6 initial temp (K)

Figure 1. Dominant reaction network for the chlorine-catalyzed oxidative pyrolysis system.

of heat flux alone to maximize the mass fraction of C2H4 in the output, while minimizing C6H6, C4H6, and C4H4 formation. The last example demonstrates the application of the proposed algorithm for the problem of maximizing the mass fraction of C2H2. The computer code employed for these simulations has also been used previously with various other mechanisms, including a much smaller mechanism (37 species) given in the work of Karra and Senkan (1988). Although different mechanisms gave slightly different results, the flux profiles and the mass fraction trajectories share the same characteristic shapes. The point to be emphasized here is that the algorithm is found to perform well with various methane pyrolysis mechanisms, and the present work serves to point out the potential significance of optimally controlling methane pyrolysis to C2 products rather than corroborating to any specific reaction mechanism. Further experimental

Reactor Parameters 1 1 100 100 40 40 1 1 0.98 0.01 0.00 0.01 1200

0.99 0.00 0.01 1000

1 100 40 1 0.96 0.02 0.01 0.01 1200

Objective Function Coefficients desired mass fraction xfi C2H4 0.5 0.36 0.5 C2H2 CH2CHCCH 0.0 0.0 0.0 C4H6 0.0 0.0 0.0 C6H6 0.0 0.0 0.0 weight factor Ci C2H4 1000 1000 1000 C2H2 CH2CHCCH 7 7 7 C4H6 5 5 5 C6H6 10 10 10

1 100 40 1 0.96 0.02 0.01 0.01 1200

0.7 0.0 0.0 0.0 1000 7 5 10

reactor work is needed to bring the results presented here to practical fruition. In all examples, the length of the PFR in which the reaction proceeds is 100 cm, with the cross section area being 40 cm2. The initial flow rate is 1 gm/s. The reactor is at a constant pressure of 1 atm. The initial conditions and control parameters for each example are given in Table 3. 4.1. Example 1, Methane Conversion to Ethylene: Control with Chlorine and Heat Flux. In this example, the initial composition of the feedstock was

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 691 Table 4. Percent of Carbon Remaining in Main Species in Product Stream in Each Example example 1 example 2 example 3 example 4

Figure 2. Results from example 1. The objective is to attain a C2H4 mass fraction level of 0.5 in the output stream, while minimizing the formation of C6H6, C4H6, and C4H4 by using Cl2 and heat fluxes. (a) Optimal Cl2 flux profile for example 1. (b) Optimal heat flux profile for example 1. (c) Mass fraction trajectories at the optimal solution of example 1. (d) Temperature profile at the optimal solution of example 1.

CH4

C2H2

C2H4

C6H6

CO

6.19 27.7 5.16 2.33

15.15 16.2 14.8 88.9

57.7 41.6 57.5 0.73

3.41 2.72 2.97 0.38

0.92 ≈0 1.93 1.56

chosen as 98 wt % CH4, 1 wt % C2H6, and 1 wt % O2, and the initial temperature was 1200 K. Ethane was added to the methane to simulate the kinetic effects of small amounts of higher molecular weight hydrocarbons typically found in natural gas. In particular, ethane (and other higher order hydrocarbons) assists the initiation of methane pyrolysis by thermally dissociating at lower temperatures and thus providing the initial radical for attacking the methane. A small amount of oxygen was added also for aiding the initiation and for minimizing production of hydrocarbons with more than two carbon atoms. The form of the objective function is given in eq 2 where Ci is a weight fraction that determines the relative importance and “trade off ratio” among each term. The stepwise approach was found to produce the best quality solutions. In the first run, xfi for C2H4 was set to a small number, and if the product mass fraction achieved the desired value, xfi was increased. The flux profiles from the previous run were used as starting points, and the optimal control procedure was repeated in subsequent runs until no better solutions could be formed. This approach yielded a much better quality solution than the case where xfi was set to 1.0 in the first run. In this example, the most demanding C2H4 set point, xCf 2H4, achieved by this system was 0.5. Besides obtaining high yields of C2H4, we also would like to minimize the formation of heavier hydrocarbons that could lead to soot. Since C6H6 was the hydrocarbon with carbon number greater than two that had the highest concentration in the uncontrolled reaction, the value of xCf 6H6 was put to zero. Moreover, without any additional control, it was found that, in the output stream, a considerable amount of carbon was accumulated in CH2CHCCH and C4H6. Hence, it was also desirable to minimize their concentration in the output. Therefore, their corresponding xfi values were also set to zero. The coefficients Ci with respect to C2H4, C6H6, CH2CHCCH, and C4H6 were set to 1000, 10, 7, and 5, respectively, since a high conversion to C2H4 is valued in this example as more “important” than the reduction of benzene, butadiene, and vinyl acetylene. Note that the summation is performed only over this set of “controlled” species, and others will not participate in the objective function. The values of xi and Ci as well as the initial conditions for the reactor are listed in Table 3. In the first iteration of the final run, the objective function value was 5.0 and the norm of the gradient was 2718. By the 15th iteration, the objective function value was reduced to 0.0059 and the gradient norm was 1.14. An average iteration for this example (74 species, 338 reactions, and 100 discretization points) takes about 1400 s of CPU time on an R4000 IRIS Indigo. Although global optimality could not be guaranteed, it is evident that good quality solutions were obtained using the proposed algorithm. The optimal flux profile and selected composition trajectories are shown in Figure 2, and the percent breakdown of carbon content in each of the key species at the end of the reactor is listed in Table 4. The carbon not accounted for in Table 4 was

692

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Figure 3. Normalized sensitivity coefficients of C2H4 mass fraction with respect to reaction rate constants d ln(xC2H4)/d ln(k) for the six most sensitive reactions at the optimal solution of example 1. The number given in front of each reaction refers to its order listed in Table 2. Solid lines correspond to the forward reaction, and dotted curves correspond to the reverse reaction.

typically distributed among C2H6, C4H4, and C4H6. Chlorinated hydrocarbons were insignificant. Through reaction flux analysis and sensitivity analysis, the major reaction pathways in this example are in agreement with Figure 1. It was found that the two

competing pathways, one involving CH3Cl and the other involving C2H6 are equally important. From Figure 2b, it is also evident that heat is fluxed in at the beginning of the reactor so that more CH3 radicals can be generated; nevertheless, at high temperature, C2H2 is a more

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 693 Table 5. Comparison Cases for Example 1a initial temp (K)

C2H4 mass fraction

C2H2 mass fraction

C6H6 mass fraction

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200

0.00070 0.0021 0.0066 0.0073 0.0074 0.0084 0.011 0.018 0.028 0.041 0.054 0.068 0.080

2.155 × 10-7 4.41 × 10-6 0.00020 0.0016 0.0085 0.033 0.065 0.097 0.13 0.16 0.18 0.22 0.25

1.01 × 10-14 1.48 × 10-12 1.33 × 10-9 3.26 × 10-8 5.01 × 10-7 5.98 × 10-6 4.41 × 10-5 0.00019 0.00055 0.0013 0.0025 0.0046 0.0078

a Chlorine flux is fixed at the optimal level found in that example, and the initial temperature is varied.

favorable product. As a result, the temperature of the system must be lowered (see Figure 2b and d) in order to favor the conversion of C2H2 to C2H4. As the flux profile of Cl2 is large in this quenching region, Cl2 must play a role in the conversion process. This two-step process hypothesis is supported by the shape of the mass fraction trajectories shown in Figure 2c. In the first half of the reactor, while CH4 is being consumed, the reaction is allowed to pass through ethylene formation converting nearly all the ethylene to acetylene. When the consumption of CH4 virtually stops near the midway point of the reactor, reformation of C2H4 from C2H2 begins, and as seen from Figure 2, the mass fraction of C2H4 rises in proportion with the dip in C2H2 mass fraction. The chlorine added to the second stage of the reaction reacts with the molecular hydrogen produced from the first stage to form the necessary H-atoms for addition to C2H2 to yield C2H3 and eventually C2H4. Figure 3, which shows the results of the sensitivity analysis, further substantiates the aforementioned hypothesis. The six largest sensitivity coefficients of the C2H4 mass fraction with respect to each reaction rate constant d ln(xC2H4)/d ln(k) are shown. Reaction 175 (C2H4 + HdC2H3 + H2) and 161 (C2H3dC2H2 + H), which control the conversion between C2H2 and C2H4, are found to have significant effects on the mass fraction of C2H4. This is in agreement with the reaction flux analysis, which shows that the fluxes through reactions 175 and 161 are very significant in favor of C2H4. The role of C2H6 in the system is that of a CH3 and H-atom radical generator. In this example, the addition of C2H6 in the input was found to reduce the induction time (note the importance of reaction 226 only at the beginning of the reactor), thus allowing a higher yield of C2H4 for a given residence time. To compare roughly the quality of the solution obtained by the optimal control method with the so-called “intuitive” approach, the optimal flux chlorine profile of this example is used, and the initial temperature is varied in a series of runs. The reaction is started with the same initial composition. The final mass fractions of C2H2 and C2H4 along with various initial temperatures are listed in Table 5. From the results, we observe that the highest mass fraction level of C2H4 obtained by the method (0.080) is far less than the solution obtained from the optimal control method (0.5). It also is clear from Figure 2 that the optimal control solution may not be obtained from intuitive arguments. 4.2. Example 2, Methane Conversion to Ethylene: Control with Oxygen and Heat Flux. It was observed in example 1 that the two competing mecha-

Figure 4. Results from example 2. The objective is to attain a C2H4 mass fraction level of 0.36 in the output stream, while minimizing the formation of C6H6, C4H6, and C4H4 by using O2 and heat fluxes. (a) Optimal O2 flux profile for example 2. (b) Optimal heat flux profile for example 2. (c) Mass fraction trajectories at the optimal solution of example 2. (d) Temperature profile at the optimal solution of example 2.

nistic pathways are almost equally important. Since it is desirable to find an alternative to using chlorine for

694

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Figure 5. Results from example 3. The objective is to attain a C2H4 mass fraction level of 0.5 in the output stream, while minimizing the formation of C6H6, C4H6, and C4H4 by using only heat flux. (a) Optimal heat flux profile for example 3. (b) Mass fraction trajectories at the optimal solution of example 3. (c) Temperature profile at the optimal solution of example 3.

handling and environmental reasons, this second example will attempt to produce a high conversion of methane to ethylene when the control scheme utilizes oxygen and heat flux as controllers, and only a trace amount of chlorine is present in the initial input. The initial condition in the reactor is 99 wt % CH4 and 1 wt % Cl2, and the initial temperature is 1000 K. Oxygen is not present in the reactor initially because it will be used as a manipulated variable. Interestingly, it was found that this initial condition without C2H6 resulted in better yields of C2H4 than any other initial condition tested. The reason for this behavior has not been further investigated but shows that additional optimization with respect to the initial conditions (to the degree that they may be realized with readily available feedstocks) could substantially improve the quality of solutions. For natural gas, this could be especially important due to the variation in its composition.

Figure 6. Results from example 4. The objective is to attain a C2H2 mass fraction level of 0.7 in the output stream, while minimizing the formation of C6H6, C4H6, and C4H4 by using only heat flux. (a) Optimal heat flux profile for example 4. (b) Mass fraction trajectories at the optimal solution of example 4. (c) Temperature profile at the optimal solution of example 4.

The objective function was chosen to be of the same form as the previous example with the best solutions achieved with xCf 2H4 equal to 0.36 (Table 3). The results are shown in Figure 4, and the percent breakdown of carbon in each of the key species at the end of the reactor is listed in Table 4. It is evident from the mass fraction trajectories that the two-step process hypothesis still applies in this case; the flux profiles and mass fraction trajectories in the two examples look very similar. Nevertheless, the best yield obtained using this configuration is not as good as in the previous example. The reason for the difference may be attributed to the existence of different local solutions depending on the starting points used, or to the differences in the initial conditions of the reactor. 4.3. Example 3, Methane Conversion to Ethylene: Control with Heat Flux. Another case study was performed with the initial feed composition consist-

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 695

ing of (by weight) 96% CH4, 2% O2, 1% Cl2, and 1% C2H6, and an initial temperature equal to 1200 K (Table 3). The optimal solution in this example achieved a C2H4 mass fraction of 0.5. The flux profiles and mass fraction profiles of the key species are shown in Figure 5a-c. The high yield in this case suggests that there may be other solutions with an even higher C2H4 yield. However, the present high yield may be at the cost of running the reactor with a greater total amount of O2 and Cl2. An important conclusion from examples 1-3 is that high yields of C2H4 can be produced under a variety of conditions. 4.4. Example 4, Methane Conversion to Acetylene: Control with Heat Flux. From the kinetic processes discussed in the previous sections, a high C2H2 yield seems to be easier to achieve than that for C2H4 since the controller does not need to direct the reaction to generate C2H4 from C2H2. Hence, this example is explored to verify the ability to design the reactor for a high yield of C2H2 while still minimizing the C6H6, C4H6, and C4H4. The initial feed conditions for this example are the same as in example 3. The objective function coefficients as well as reactor parameters are given in Table 3. The optimal mass fraction yield of C2H2 was 0.7, and the results are reported in Figure 6. A very high C2H2 yield could be achieved because the stable product obtained from the mechanism at high temperature is C2H2. Also note that the amount of C6H6 and other heavier species are significantly less than the previous three cases where the objective was to maximize the yield of C2H4, since the formation of these species require acetylene and the vinyl radical as intermediates, and consequently, they are produced more rapidly during the second stage.

the differential algebraic equation approach by Biegler (1989) and Cuthrell and Biegler (1989), and the recent work of Vassiliadis (1993). In this presentation, we chose to implement the classical strategy belonging to the class known as gradient methods in function space, which utilizes discretization of the function space and an iterative conjugate gradient algorithm. More detailed information on the algorithm used can be found in the works of Hicks and Ray (1971) or Jones and Finch (1984). Given the system’s governing equation shown earlier in eq 1 and the objective function in the form J(ji, xi, T), the classical optimal control approach (Bryson and Ho, 1969) introduces the Lagrange multiplier λ to assure that eq 1 is satisfied. The Lagrange equation is formulated as follows:

dλi

dJ )-

dl

1 +

dxi

dλT dl

Acknowledgment The authors acknowledge support from the Department of Energy and the National Science Foundation. Appendix Optimal Control Formulation. There have been significant efforts in recent years attempting to treat the practical difficulties of optimal control problems that arise from the need to solve boundary value problems and algebraic equations simultaneously (Bryson and Ho, 1969). Various strategies have been proposed, including

dxi

k)1

+ λT

)

dwk dT

n

+ λi

∑ jk

k)1

)

(3a)

dwk′ Hfk′ + dT FC h p k)1 k′)1 dxi n n dwk dHfk + wk λT Hfk + λTC h pf jk (3b) dT dT k)1 k)1 n

1

+

∑

n

λk

(

)

∑

)

Here λi are Lagrange multipliers corresponding to the mass conservation equations of species i in the plug flow model, and λT is a Lagrange multiplier corresponding to the energy conservation equation. The final conditions of the adjoint equations can be formulated as

5. Conclusions We have considered the optimal control of an adiabatic PFR reactor that converts methane to ethylene and acetylene. The reaction mechanism revealed two dominant mechanistic routes from methane to ethylene: one involving CH3Cl or CH2Cl when chlorine is present and the other involving C2H6 or C2H5. Heat flux is found to be an important component in controlling the reaction, and the mechanistic path involving C2H6 can be utilized to obtain a high yield of C2H4 and C2H2 by using oxygen flux as a controller. The algorithm performed well in all examples presented, and convergence to high-yield solutions was obtained. Indications were found for the achievement of local optimal solutions, and even better results are likely possible. A basic conclusion from the work is that laboratory studies of methane conversion seem attractive for pursuit, especially including feedback control to get the best possible solutions.

λk

-

F

dJ

)-

( ∑( (∑ ∑

dwk

n

λi(L) )

dJ dxi(L)

(4a)

λT(L) )

dJ dT(L)

(4b)

where L is the total length of the reactor. The gradient of the objective function with respect to the control variables can be calculated by using the following expression:

λi

δJ(l) ) δji(l)

1

F

(To - T) Cpi FC hp

n

∑λixi + λT Fi)1 1

(

n

∫lL 2 ∑(λi(wi - jt(t)xi + ji(t))) + i)1 F

)

n 1 λT (jt(t)Cpf(To - T) - Hfiwi) dt (5a) C hp i)1

∑

λT δJ(l) ) hp δq(l) FC

(5b)

The optimal control algorithm used in this paper is described below: 1. Decide on the objective function and the choice of control flux(es). 2. Discretize l into N points lk, k ) 1 ... N, with l0 being 0 and lN being L. Then assign initial values for jik ) ji(lk), k ) 1 ... N, and qk ) q(lk). 3. Apply a cubic spline to jik and qk, k ) 1 ... N to get continuous representations of jik and qk. Knowing xi(0), j(l), and q(l), one can solve eq 1 to obtain xi(l) and T(l). 4. Knowing the x’s, one can calculate the initial condition for the adjoint equation by eqs 4a and 4b.

696

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Then eqs 3a and 3b can be integrated backward to obtain λi(l) and λT(l). 5. Knowing the x’s and λ’s, eqs 5a and 5b can be used to evaluate the derivative of the objective function with respect to the control variables at each discretization point. 6. If the norm of the gradient calculated in the previous step is less than a tolerance parameter , the algorithm terminates, otherwise the gradient is used in the conjugate gradient minimizer to update the values of jik and qk and go back to step 3. To ensure positive mass flux densities ji, we transform to a new control variable j′i where ji ) exp(j′i), in which case the gradient will be modified to

δJ(l) δJ ) exp(j′i) δj′i(l) δjl(l) and ji must be calculated from j′i at step 3. In the calculations presented, the following FORTRAN routines are used: the chemical kinetics package CHEMKIN II (Kee et al., 1989) is used to interface the thermodynamic and kinetics data, LSODA (Hindmarsh, 1983) is used as a differential equation integrator, CONMIN (Shanno and Phua, 1980) is used as a conjugate gradient minimizer, and AIM (Kramer et al., 1981) is used in the sensitivity analysis. Literature Cited Back, M. H.; Back, R. A. Thermal decomposition and reactions of methane. In Pyrolysis: Theory and Industrial Practice; Albright, L. F., Crynes, B. L., Corcoran, W. H., Eds.; Academic Press: New York, 1983; Chapter 1. Benson, S. W. Conversion of Methane. U.S. Patent 4,199,533, Apr 22, 1980. Benson, S. W. The mechanism of the reversible reaction: C2H2 a vinyl acetylene, and the pyrolysis of butadiene. Int. J. Chem. Kinet. 1989, 21, 233-243. Biegler, L. T. Strategies for simultaneous solution and optimization of differential-algebraic system. In Foundations of ComputerAided Process Design; Siirola, J. J., Grossmann, I. E., Stephanopoulus, G., Eds.; Elsevier Science Publishing, Inc.: New York, 1989; pp 155-180. Bryson, A. E.; Ho, Y.-C. In Applied Optimal Control; Blaisdell Publishing Company: Waltham, MA, 1969. Chammugathas, C.; Heicklen, J. Pyrolysis of acetylene-vinyl acetylene mixtures between 400 and 500 °C. Int. J. Chem. Kinet. 1986, 18, 701-718. Cuthrell, J. E.; Biegler, L. T. Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 1989, 13 (1/2), 49-62. Frenklach, M.; Wang, H. Detailed modeling of soot particle nucleation and growth. In Twenty-Third Symposium (International) on Combustion. The Combustion Institute: Pittsburgh, PA, 1990; pp 1559-1566. Frenklach, M.; Taki, S.; Durgaprasad, M. B.; Mutula, R. A. Soot formation in shock-tube pyrolysis of acetylene, allene and 1,3butadiene. Combust. Flame 1983, 54, 81-101. Frenklach, M.; Clary, D. W.; Gardiner, W. C., Jr.; Stein, S. E. Detailed kinetic modeling of soot formation in shock tube pyrolysis of acetylene. In Twentieth Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1984; pp 887-901.

Granada, A.; Karra, S. B.; Senkan, S. M. Conversion of CH2 into C2H2 and C2H4 by the chlorine-catalyzed pyrolysis process 1. Oxidative pyrolysis of CH3Cl. Ind. Eng. Chem. Res. 1987, 26, 1901-1905. Hicks, G. A.; Ray, W. H. Approximation methods for optimal control synthesis. Can. J. Chem. Eng. 1971, 49, 522-528. Hindmarsh, C. Odepack, A Systemized Collection of Ode Solvers in Scientific Computing; North-Holland: Amsterdam, 1983. Ho, W.; Yu, Q.; Bozzelli, J. W. Kinetic study on pyrolysis and oxidation of CH3Cl in Ar/H2/O2 mixtures. Combust. Sci. Technol. 1992, 85, 23-63. Hunter, T. B.; Wang, H.; Litzinger, T. A.; Frenklach, M. The Oxidation of Methane at Elevated Pressures: Experiments and Modeling. Combust. Flame 1994, 97, 201-224. Jones, D. I.; Finch, J. W. Comparison of optimization algorithms. Int. J. Control 1984, 40 (4), 747-761. Karra, S. B.; Senkan, S. M. A detailed chemical kinetic mechanism for the oxidative pyrolysis of CH3Cl. Ind. Eng. Chem. Res. 1988, 27, 1163-1168. Kee, R. J.; Rupley, F. M.; Miller, J. A. The CHEMKIN thermodynamic data base; Tecnical Report SAND 87-821513; Sandia National Laboratories: Livermore, CA, 1987. Kee, R. J.; Rupley, F. M.; Miller, J. A. CHEMKIN-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics; Technical Report SAND 89-8009B; Sandia National Laboratories: Livermore, CA, 1993. Kiefer, J. H.; Von Drasek, W. A. The mechanism of the homogeneous pyrolysis of acetylene. Int. J. Chem. Kinet. 1990, 22, 747-786. Kiefer, J. H.; Sidhu, S. S.; Kern, R. D.; Xie, K.; Chen, H.; Harding, L. B. The homogeneous pyrolysis of acetylene II: The high temperature radical chain mechanism. Combust. Sci. Technol. 1992, 82, 101-130. Kramer, M. A.; Calo, J. M.; Rabitz, H. An improved computational method for sensitivity analysis: Green’s function method with ‘AIM’. Appl. Math. Model. 1981, 5, 432-441. Rojnuckarin, A.; Floudas, C.; Rabitz, H.; Yetter, R. A. Optimal control of a plug flow reactor with complex reaction mechanism. J. Phys. Chem. 1993, 97, 11689-11696. Roseler, J. F.; Yetter, R. A.; Dryer, F. L. Perturbation of moist CO oxidation by trace quantities of CH3Cl. Combust. Sci. Technol. 1994, 101, 197-207. Senkan, S. M. Production of higher molecular weight hydrocarbon from methane. U.S. Patent 4,714,796, Dec 22, 1987. Shanno, D. F.; Phua, K. H. Minimization of unconstrained multivariate functions. ACM Trans. Math. Software 1980, 6, 618620. Tanzawa, T.; Gardiner, W. C., Jr. Reaction mechanisms of homogeneous thermal decomposition of acetylene. J. Phys. Chem. 1980, 84, 236-239. Tsang, W.; Hampson, R. F. Chemical kinetic data base for combustion chemistry. Part I. Methane and related compounds. J. Phys. Chem. Ref. Data 1986, 15, 1087-1279. Vassiliadis, V. Computational Solution of Dynamic Optimization Problems with General Differential/Algebraic Constraints. Ph.D. Dissertation, Imperial College of Science, Technology and Medicine, London, 1993. Weissman, M. A.; Benson, S. W. Pyrolysis of methyl chloride, a pathway in the chlorine catalyzed polymerization of methane. Int. J. Chem. Kinet. 1984, 16, 307-333. Weissman, M. A.; Benson, S. W. Rate parameters for the reactions of C2H3 and C4H5 with H2 and CH2. J. Phys. Chem. 1988, 92, 4080-4084. Yetter, R. A.; Dryer, F. L.; Rabitz, H. A Comprehensive Reaction Mechanism for Carbon Monoxide/Hydrogen/Oxygen Kinetics. Combust. Sci. Technol. 1991, 79, 97-128.

Received for review June 21, 1995 Accepted July 6, 1995X IE940542N X Abstract published in Advance ACS Abstracts, October 1, 1995.