Thermodynamic Analysis of Glycerin Steam Reforming - Energy

Oct 24, 2008 - The increasing demand for energy and the consequent high consumption of natural resources has resulted in the environmental degradation...
12 downloads 24 Views 3MB Size
Energy & Fuels 2008, 22, 4285–4291

4285

Thermodynamic Analysis of Glycerin Steam Reforming Xiaodong Wang, Shuirong Li, Hao Wang, Bo Liu, and Xinbin Ma* Key Laboratory for Green Chemical Technology, School of Chemical Engineering and Technology, Tianjin UniVersity, Tianjin 300072, People’s Republic of China ReceiVed June 20, 2008. ReVised Manuscript ReceiVed September 11, 2008

Thermodynamic properties of glycerin steam reforming have been studied with the method of Gibbs free energy minimization for hydrogen and/or synthesis gas production. Equilibrium compositions including the coke-formed and coke-free regions were determined as a function of water/glycerin molar ratios (1:1-12:1) and reforming temperatures (550-1200 K) at different pressures (1-50 atm). Optimum conditions for hydrogen production are temperatures between 925 and 975 K and water/glycerin ratios of 9-12 at atmospheric pressure, whereas temperatures above 1035 K and water/glycerin ratios between 2 and 3 at 20-50 atm are suitable for the production of synthesis gas that favors both methanol synthesis and low-temperature Fischer-Tropsch synthesis. However, synthesis gas obtained from glycerin steam reforming is not feasible for direct use in high-temperature Fischer-Tropsch synthesis. Under these optimum conditions, carbon formation can be thermodynamically inhibited.

1. Introduction The increasing demand for energy and the consequent high consumption of natural resources has resulted in the environmental degradation and the depletion of fossil energy reserves. Use of biomass resources is imperative for industrial society because they have been considered renewable and carbon dioxide neutral.1 Indeed, biodiesel has been largely produced and used as automotive fuels in Europe and the U.S. A total of 1 ton of biodiesel yields approximate 0.11 ton of crude glycerin or about 0.1 ton of pure glycerin.2 Therefore, there will be a large amount of byproduct glycerin in the world market with the increasing production of biodiesel. One key issue that may affect the further use of biodiesel is the full use of byproduct glycerin, which can potentially reduce the production costs of biodiesel.3 One promising way to use diluted glycerin aqueous solution is to produce hydrogen or synthesis gas via steam reforming.4 Hydrogen is mostly used in fuel cells and refinery hydrotreating operations.5 Synthesis gas is an essential intermediate for many primary chemicals production. It can also be used as a feedstock of Fischer-Tropsch synthesis (FTS) for liquid fuels production (see Figure 1). Methanol, which is one of the reactants for biodiesel production, on the other hand, can be produced by synthesis gas with a H2/CO ratio of 2, and thus, a real biodiesel * To whom correspondence should be addressed. Telephone: +86-2227406498. Fax: +86-22-27890905. E-mail: [email protected]. (1) Soares, R. R.; Simonetti, D. A.; Dumesic, J. A. Glycerol as a source for fuels and chemicals by low-temperature catalytic processing. Angew. Chem., Int. Ed. 2006, 45, 3982–3985. (2) Behr, A.; Eilting, J.; Irawadi, K.; Leschinski, J.; Lindner, F. Improved utilisation of renewable resources: New important derivatives of glycerol. Green Chem. 2008, 10, 13–30. (3) Haas, M. J.; McAloon, A. J.; Yee, W. C.; Foglia, T. A. A process model to estimate biodiesel production costs. Bioresour. Technol. 2006, 97, 671–678. (4) Zhang, B. C.; Tang, X. L.; Li, Y.; Xu, Y. D.; Shen, W. J. Hydrogen production from steam reforming of ethanol and glycerol over ceriasupported metal catalysts. Int. J. Hydrogen Energy 2007, 32 (13), 2367– 2373. (5) Rapagna, S.; Jand, N.; Foscolo, P. U. Catalytic gasification of biomass to produce hydrogen rich gas. Int. J. Hydrogen Energy 1998, 23, 551–557.

Figure 1. Use of synthesis gas derived from glycerin.

Figure 2. Moles of hydrogen produced at selected pressures and WGR ) 3.

on the exclusive basis of renewable feedstocks can be produced. Operation of glycerin steam reforming at certain pressure (i.e., 20-50 atm) provides opportunities to couple this process with FTS and methanol synthesis. This integrated process can potentially improve the economics of FTS or methanol synthesis by reducing costs associated with synthesis gas compression. Additionally, combination of endothermic steam reforming of glycerin with exothermic FTS or methanol synthesis can provide energy efficient routes for chemicals and fuels production. Recently, experimental investigations of hydrogen production from glycerin aqueous-phase reforming (APR) and steam

10.1021/ef800487r CCC: $40.75  2008 American Chemical Society Published on Web 10/25/2008

4286 Energy & Fuels, Vol. 22, No. 6, 2008

Figure 3. Moles of H2 as a function of WGR and temperature at atmospheric pressure.

reforming were reported by different researchers.4,6-13 Synthesis gas production from pyrolysis, steam gasification, and catalytic conversion of glycerin were studied too.1,14-16 Meanwhile, thermodynamic analysis for hydrogen production from different glycerin reforming processes has been investigated.17-20 Xiao et al. reported thermodynamic features of glycerin aqueous (6) Cortright, R. D.; Davad, R. R.; Dumesic, J. A. Hydrogen from catalytic reforming of biomass-derived hydrocarbons in liquid water. Nature 2002, 418, 964–966. (7) Huber, G. W.; Shabaker, J. W.; Dumesic, J. A. Raney Ni-Sn catalyst for H2 production from biomass-derived hydrocarbons. Science 2003, 300, 2075–2077. (8) Czernik, S.; French, R.; Feik, C.; Chornet, E. Hydrogen by catalytic steam reforming of liquid byproducts from biomass thermoconversion processes. Ind. Eng. Chem. Res. 2002, 41, 4209–4215. (9) Hirai, T.; Ikenaga, N. O.; Miyake, T.; Suzuki, T. Production of hydrogen by steam reforming of glycerin on ruthenium catalyst. Energy Fuels 2005, 19, 1761–1762. (10) Slinn, M.; Kendall, K.; Mallon, C.; Andrews, J. Steam reforming of biodiesel by-product to make renewable hydrogen. Bioresour. Technol. 2008, 99, 5851–5858. (11) Douette, A. M. D.; Turn, S. Q.; Wang, W. Y.; Keffer, V. I. Experimental investigation of hydrogen production from glycerin reforming. Energy Fuels 2007, 21, 3499–3504. (12) Adhikari, S.; Fernando, S.; Haryanto, A. Hydrogen production from glycerin by steam reforming over nickel catalysts. Renewable Energy 2008, 33, 1097–1100. (13) Adhikari, S.; Fernando, S.; Haryanto, A. Production of hydrogen by steam reforming of glycerin over alumina-supported metal catalysts. Catal. Today 2007, 129, 355–364. (14) Valliyappan, T.; Bakhshi, N. N.; Dalai, A. K. Pyrolysis of glycerol for the production of hydrogen or syn gas. Bioresour. Technol. 2008, 99, 4476–4483. (15) Valliyappan, T.; Ferdous, D.; Bakhshi, N. N.; Dalai, A. K. Production of hydrogen and syngas via steam gasification of glycerol in a fixed-bed reactor. Top. Catal. 2008, 49, 59–67. (16) Simonetti, D. A.; Kunkes, E. L.; Dumesic, J. A. Gas-phase conversion of glycerol to synthesis gas over carbon-supported platinum and platinum-rhenium catalysts. J. Catal. 2007, 247, 298–306. (17) Luo, N. J.; Zhao, X.; Cao, F. H.; Xiao, T. C.; Fang, D. Y. Thermodynamic study on hydrogen generation from different glycerol reforming processes. Energy Fuels 2007, 21, 3505–3512. (18) Luo, N. J.; Cao, F. H.; Zhao, X.; Xiao, T. C.; Fang, D. Y. Thermodynamic analysis of aqueous-reforming of polylols for hydrogen generation. Fuel 2007, 86, 1727–1736. (19) Adhikari, S.; Fernando, S.; Gwaltney, S. R.; To, S. D. F.; Bricka, R. M.; Steele, P. H.; Haryanto, A. A thermodynamic analysis of hydrogen production by steam reforming of glycerol. Int. J. Hydrogen Energy 2007, 32, 2875–2880.

Wang et al.

reforming and hydrogen generation from aqueous reforming of polylols.17,18 Fernando and co-workers carried out thermodynamic analysis of glycerin steam reforming with the method of direct minimization of Gibbs free energy; the best conditions for the production of hydrogen is at a temperature > 900 K, atmospheric pressure, and a molar ratio of water/glycerin of 9:1.19,20 The reported thermodynamic analysis focused on hydrogen production from aqueous reforming of glycerin, as well as glycerin steam reforming at nearly atmospheric pressures. However, to our knowledge, detailed thermodynamic analysis of glycerin steam reforming for the production of synthesis gas has not been reported yet. In terms of further thermodynamic properties of glycerin steam reforming at 20-50 atm, there is no literature reported. The aim of this work is to carry out thermodynamic analysis of glycerin steam reforming for hydrogen production at low pressures (1, 3, and 5 atm were used here), as well as for the production of synthesis gas at pressures between 20 and 50 atm. The total Gibbs free energy minimization method was employed, while the Soave-Redlich-Kwong equation of state was used to calculate the fugacity coefficient of each component in the gas mixture. Our calculation results attempt to illustrate the effects of the process variables [pressure, temperature, and water/ glycerin ratio (WGR)] and the carbon formation in glycerin steam reforming. Additionally, optimum conditions for both hydrogen and synthesis gas production are discussed. 2. Methodology Gibbs free energy is the most commonly used function to identify the equilibrium state. A minimization of total Gibbs free energy is a suitable method to calculate the equilibrium compositions of any reacting system.21 The total Gibbs function for a system is given as follows: N

Gt )



ji) niG

i)1

N

∑n µ )∑nG i i

i

ˆfi

∑ n ln f

0 i + RT

i)1

i

0 i

(1)

For reaction equilibria in gas phase, ˆfi ) φˆ iyiP, f 0i ) P0, and because G0i is set equal to zero for each chemical element in its standard state, ∆G0 ) ∆Gf0i for each component is assumed. The minimum Gibbs free energy of each gaseous species and that of the total system can be expressed as eqs 2 and 3, with Lagrange’s undetermined multiplier method

∆Gf0i + RT ln N

∑n

i

i)1

(

φˆ iyiP P0

∆Gf0i + RT ln

+

∑λ a

k ik ) 0

k

φˆ iyiP P0

+

∑λ a

k ik

k

)

)0

(2)

(3)

with the constraints of elemental balances N

∑na

i ik ) Ak

(4)

i)1

When solid carbon (graphite) is considered in the system, Gibbs energy of carbon is usually considered as eq 5 by researchers22,23

j C(s) ) GC(s) = ∆Gf0 ) 0 j C(g) ) G G C(s)

(5)

However, for a temperature-steady process (20) Adhikari, S.; Fernando, S.; Haryanto, A. A comparative thermodynamic and experimental analysis on hydrogen production by steam reforming of glycerin. Energy Fuels 2007, 21, 2306–2310. (21) Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York, 1997; p 4.33.

Thermodynamic Analysis of Glycerin Steam Reforming

Energy & Fuels, Vol. 22, No. 6, 2008 4287

dGC(s) ) VCdP

(6)

VC is the mole volume of solid carbon, and it can be considered as a constant because it is less affected by temperature and pressure. Then, eq 6 can be expressed as eq 7

GC(s)(T, P) - GC(s)(T, P0) ) VC(P - P0)

(7)

where GC(s)(T, P0) is assumed to be zero and VC ) 4.58 × 10-6 m3/mol from Knovel.24 Equation 7 is further expressed as eq 8

GC(s)(T, P) ) 4.58 × 10-6(P - P0)

(8)

The minimization function of Gibbs energy as following eq 9 is obtained by substituting eq 1 by eq 2 for gaseous species and by eq 8 for solid species N-1

∑n

i

i)1

(

∆Gf0i + RT ln

φˆ iyiP P0

+

∑λ a

)

k ik

k

+ nCGC(s)(T, P) ) 0 (9)

The fugacity coefficient φˆ i of each component in the gas mixture can be calculated according to the Soave-Redlich-Kwong equation of state;21 that is

a(T) RT P) V - b V(V + b) a(T) ) aR(T) ) 0.42748 b ) 0.08664

(10)

R2TC2 PCR(T)

(11)

RTC PC

(12)

R(T) ) [1 + m(1 - Tr0.5)]2

(13)

T TC

Tr )

Figure 4. Moles of CO as a function of WGR and temperature at atmospheric pressure.

(14)

m ) 0.480 + 1.574ω - 0.176ω2

(15)

Consequently, the fugacity coefficient φˆ i can be calculated from the following equation:

ln φˆ )

bi P(V - bm) + (Z - 1) - ln bm RT

(

)

(

N am bi bm 2 y a ln 1 + bmRT bm am j)1 j ij V



)

(16)

The mixture parameters used above in eq 16 are defined by the mixture rules

am )

∑∑yya

(17)

∑yb

(18)

i j ij

i

bm )

j

i i

i

aij ) (aiaj)0.5(1 - kij)

(19)

Matlab language and Lagrange’s undetermined multiplier method are used to calculate the equilibrium compositions. We consider the products from glycerin steam reforming to be hydrogen, carbon monoxide, carbon dioxide, methane, carbon, unreacted glycerin, and water, because they are primary products in steam reforming of (22) Vasudeva, K.; Mitra, N.; Umasankar, P.; Dhingra, S. C. Steam reforming of ethanol for hydrogen production: Thermodynamic analysis. Int. J. Hydrogen Energy 1996, 21 (1), 13–38. (23) Lwin, Y.; Daud, W. R. W.; Mohamad, A. B.; Yaakob, Z. Hydrogen production from steam-methanol reforming: Thermodynamic analysis. Int. J. Hydrogen Energy 2000, 25 (1), 47–53. (24) KnoVel Critical Tables, 2nd ed.; Knovel, 2003. Online version available at http://www.knovel.com/knovel2/Toc.jsp?BookID)761& VerticalID)0 (accessed Aug 25, 2008).

Figure 5. Moles of carbon as a function of WGR and temperature at atmospheric pressure.

glycerin.1,4,9 The WGR is varied in a range of 1-12. For the thermodynamic analysis of hydrogen, the temperatures and pressures studied are 550-1000 K and 1, 3, and 5 atm, respectively. To couple with the methanol synthesis and FTS, the pressures of synthesis gas production are chosen between 20 and 50 atm. The boiling point of glycerin and water mixture increases with increasing pressure. To make sure that the reactions take place in the gas phase, the temperatures for synthesis gas production are selected between 750 and 1200 K. Thermodynamic data were obtained from Yaws and Daubert.25,26

3. Results and Discussion The effects of the process variables (pressure, temperature, and WGR) and the production of hydrogen and synthesis gas have been investigated on the basis of thermodynamic analysis. Over the temperature, pressure, and WGR ranges considered in this study, the conversion of glycerin was always 100% in both hydrogen production and synthesis gas production. Glycerin steam reforming can produce H2, CH4, CO, CO2, and C, accompanied with unreacted H2O. 3.1. Glycerin Steam Reforming for Hydrogen Production. 3.1.1. Hydrogen Production. Figure 2 shows the production of hydrogen at different temperatures and pressures. The number

4288 Energy & Fuels, Vol. 22, No. 6, 2008

Wang et al.

Figure 6. (a) Moles of synthesis gas as a function of WGR and temperature at 20 atm. (b) Ratio of H2/CO at different temperatures and WGRs at 20 atm. (c) Moles of carbon as a function of WGR and temperature at 20 atm.

of moles of hydrogen increases with a decrease in pressure. It can be attributed to the overall reaction of glycerin steam

reforming, of which the number of moles increases during the process (see eq 20). Apparently, low pressure favors the reaction,

Thermodynamic Analysis of Glycerin Steam Reforming

Energy & Fuels, Vol. 22, No. 6, 2008 4289

Figure 7. H2/(2CO + 3CO2) at different temperatures and WGRs at 20 atm.

and high pressure can reduce the capacity of hydrogen production significantly. Therefore, we selected atmospheric pressure (best one with respect to hydrogen production) all through the following discussions. C3H8O3 + 3H2O f 3CO2 + 7H2

(20)

As shown in Figure 3, when WGR is less than 5, the number of moles of hydrogen increases steady with an increasing temperature; however, when WGR is higher than 5, it increases with the increase in temperature, goes through a maximum around 925-975 K, and then begins to decrease at higher temperatures. Similarly, the number of moles of hydrogen increases with increasing WGR. The greatest quantity of hydrogen is observed at 925 K with WGR of about 9-12, which is slightly different from 960 K reported by Fernando.19,20 Because the largest amount of hydrogen is produced with excess WGR at each temperature, high temperature and WGR are necessary to maximize hydrogen production. The maximum hydrogen production per mole of glycerin is 6.2 at 925 K and atmospheric pressure with a WGR of 12 (note that the stoichiometric value is 7). 3.1.2. Carbon Monoxide Production. The formation of CO can poison the anode of proton exchange membrane (PEM) fuel cells. Therefore, the inhibition of CO production from thermodynamics was also studied in this work. Figure 4 shows that CO produced increases with increasing temperature but decreases with increasing WGR. Notably, nearly no CO is observed below 700 K, which is primarily due to the exothermicity of the water-gas shift reaction. To minimize CO formation, low temperatures and high WGRs are desired. Indeed, the CO concentration can be controlled below 100 ppm at temperatures lower than 550 K and WGRs higher than 5. However, as shown in Figure 3, less hydrogen is produced under the conditions that favor the inhibition of CO. The major products in the equilibrium are CH4 and CO2. Thus, with the purpose of hydrogen production, higher temperatures are considered even if there is more CO produced. However, removal of CO is needed for further application in fuel cell. 3.1.3. Carbon Formation. Carbon is an undesirable product in glycerin steam reforming, because the existence of carbon can poison the catalysts, especially using crude glycerin as a reactant.12,13 Thermodynamic analysis can provide useful information about the inhibition of carbon. Figure 5 shows the number of moles of carbon as a function of WGR and temperature. Generally, high WGR and temperature disfavor

carbon formation. It is found that carbon would not form in the entire considered temperature range if WGR is greater than 5. Carbon formation would occur when the temperature is lower than 975 K (WGR ) 1), 925 K (WGR ) 2), 825 K (WGR ) 3), and 575 K (WGR ) 4), respectively. The number of moles of carbon reached 1.2 at 550 K and WGR ) 1, which is the maximum in the whole range considered, and it is slightly higher than the figure reported by Fernando.19,20 3.2. Glycerin Steam Reforming for Synthesis Gas Production. 3.2.1. Properties of Synthesis Gas. The required properties of synthesis gas vary with different synthesis. In the synthesis of methanol, the desirable synthesis gas composition is best characterized by a H2/CO ratio of about 2.16 In lowtemperature Fischer-Tropsch synthesis (LTFT), the water-gas shift reaction is inactive and CO2 remains inert when cobaltbased FTS catalysts are used; therefore, the H2/CO usage ratio is about 2.15. For iron-based catalysts, the H2/CO usage ratio is typically approximate 1.7. In high-temperature FischerTropsch synthesis (HTFT), CO2 and CO are both reactants because the water-gas shift reaction is active in the reaction conditions; therefore, synthesis gas with a ratio of H2/(2CO + 3CO2) equal to about 1.05 is needed.27 3.2.2. Synthesis Gas Production. Figure 6a depicts moles of synthesis gas (sum of H2 and CO) at 20 atm as a function of WGR and temperature. The production of synthesis gas increases with increasing temperature and WGR, but this increasing trend is not significant at high temperatures. The moles of synthesis gas keep steady around 6.9 when the temperature and WGR are high enough. The H2/CO ratio decreases with a decrease in WGR but decreases with an increase in temperature (see Figure 6b). Synthesis gas with a H2/CO ratio of approximately 2 can only be produced over 1030 K and a WGR less than 3. Synthesis gas with a H2/CO ratio of 2 can be directly used in methanol synthesis, through which biodiesel can be produced from renewable feedstocks. Synthesis gas with a H2/CO ratio of about 2.15, coupled with cobalt-based catalysts in LTFT, can be produced over 1016 K and a WGR less than 3, and for LTFT iron-based catalysts, the ratio should be about 1.7, with the temperature higher than 1069 K, and a WGR between 0 and 2. However, as we can see from Figure 6c, carbon will be produced when the temperature is lower than 1100 K with a WGR of 1, as well as in the entire considered temperature range with a WGR of 0, suggesting that a WGR between 2 and 3 is feasible for the production of synthesis gas. Synthesis gas with a H2/CO ratio of about 2.15 can be produced over 1035 K with a WGR of 2. The composition of synthesis gas with H2/(2CO + 3CO2) equal to about 1.05 is suitable for HTFT, but the H2/(2CO + 3CO2) of synthesis gas is less than 1.05 (see Figure 7). An increase of hydrogen or removal of carbon oxides is necessary to increase the ratio of H2/(2CO + 3CO2), so that the synthesis gas from glycerin steam reforming can be suitable for HTFT. The information of synthesis gas obtained from glycerin steam reforming at 50 atm is shown in Figure 8. The trend of synthesis gas production is quite similar to that in Figure 6a; nevertheless, it is apparent that the capacity of synthesis gas production decreases when pressure increases to 50 atm. There are little changes in the ratio of H2/CO and carbon formation at 50 atm (25) Yaws, C. L. Chemical Properties Handbook; McGraw-Hill: New York, 1999; p 779. (26) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Hemisphere: New York, 1989; p iii. (27) Dry, M. E. The Fischer-Tropsch process: 1950-2000. Catal. Today 2002, 71, 227–241.

4290 Energy & Fuels, Vol. 22, No. 6, 2008

Wang et al.

Figure 8. (a) Moles of synthesis gas as a function of WGR and temperature at 50 atm. (b) Ratio of H2/CO at different temperatures and WGRs at 50 atm. (c) Moles of carbon as a function of WGR and temperature at 50 atm.

Thermodynamic Analysis of Glycerin Steam Reforming

compared to that at 20 atm (see parts b and c of Figure 8). The properties of synthesis gas obtained from glycerin steam reforming at 30 and 40 atm are similar to that at 20 and 50 atm. Synthesis gas can be produced with suitable H2/CO ratios for methanol synthesis and LTFT at 20-50 atm. For instance, synthesis gas with a H2/CO ratio of 2 can be produced at 1076 K and 50 atm with a WGR of 2, and a H2/CO ratio of 2.15 can be produced at 1163 K and 20 atm with a WGR of 3. Operation of glycerin steam reforming at these pressures can be coupled with industrial methanol and Fischer-Tropsch synthesis, and these integrated processes can improve energy efficiency and economics of chemicals and fuels production. However, the ratios of H2/(2CO + 3CO2) are less than 1.05 all of the time, suggesting that synthesis gas derived from glycerin is not suitable for HTFT directly. 4. Conclusion A thermodynamic analysis of glycerin steam reforming for hydrogen and synthesis gas production has been carried out with the total Gibbs free energy minimization method and the Soave-Redlich-Kwong equation of state. Lagrange’s undetermined multiplier method and Matlab language were used to calculate the equilibrium compositions. High temperature, low pressure, and high WGR favor hydrogen production. The optimum conditions for the production of hydrogen are a temperature of 925-975 K and a WGR of 9-12 at atmospheric pressure, under which no carbon formation occurs. The maximum number of moles of hydrogen produced per mole of glycerin is 6.2 at 925 K and atmospheric pressure with a WGR of 12. To couple with methanol synthesis or Fischer-Tropsch synthesis operating at certain pressures, the optimum conditions for the production of synthesis gas are temperatures above 1035

Energy & Fuels, Vol. 22, No. 6, 2008 4291

K and WGRs between 2 and 3 at 20-50 atm. However, because synthesis gas from glycerin steam reforming is not suitable for direct use in HTFT, an increase of supplied hydrogen or decarburization is necessary to increase the ratio of H2/(2CO + 3CO2). Acknowledgment. The financial support from the Program for New Century Excellent Talents in University (NCET-04-0242) and the Program of Introducing Talents of Discipline to Universities (B06006) is gratefully acknowledged.

Nomenclature aik ) number of atoms of the kth element present in each molecule of species i Ak ) total mass of kth element in the feed ˆfi ) fugacity of species i in system f 0i ) standard-state fugacity of species i GC(s) ) molar Gibbs free energy of solid carbon j i ) partial molar Gibbs free energy of species i G Gt ) total Gibbs free energy G0i ) standard Gibbs free energy of species i ∆Gf0i ) standard Gibbs function of formation of species i ∆Gf0c(s) ) standard Gibbs function of formation of solid carbon nC ) mole of carbon N ) number of species in the reaction system P ) pressure of system P0 ) standard-state pressure of 101.3 kPa R ) molar gas constant T ) temperature of the system yi ) mole fraction of each substance in gas products VC ) mole volume of solid carbon Greek Symbols λk ) Lagrange multiplier µ j i ) chemical potential of species i φˆ i ) fugacity coefficient of species i EF800487R