## 4 Results and Discussion  
### Correction Factors and New Empirical Correlation for the Equilibrium Constant  
The correction factors based on the correlations in Table 5 were calculated, resulting in \(f_{3}\), \(f_{4}\), \(f_{5}\), and \(f_{6}\) for each model M1-M6. However, the correction factor \(f_{3}\) obtained from M3 and M6 resulted in much different values than \(f_{3}\) obtained from the other models. Therefore, the correction factors used for further study were those obtained from M1, M2, M4, M5. A combination of the correction factors (\(f_{3}\)-\(f_{6}\)) obtained from M1, M2, M4, and M5 were named as factor models A, B, C, and D, respectively. Table 8 displays the factor models under the condition of \(\Delta\)H = 0. The correction factors in each factor model were computed to Equation (42) to produce the new equilibrium constants \(K^{\star}\). Combinations between M1-M6 and A-D (M1A-M6D) were applied during this computation step. Table 9 displays four new empirical correlations for the reaction equilibrium constant that corresponds to each factor model. These four correlations were found to have the lowest RMSE among 24 correlations developed from M1A-M6D. Figure 2: The flow diagram of developing the stoichiometric models. (a) the first stage was developing correlations for the correction factors and the new empirical correlation for the reaction equilibrium constants \(c\), (b) the second stage was validating the developed models and testing their accuracy. Correction factors for the reaction equilibrium constant have been identified for some reactions. The correction factor for hydrogasification reaction (R3) is reported in the range of 0.2 [58] and 8.5216 [35], depending on the values of ER and/or T [20; 59]. The correction factor for the water gas shift reaction (R4) ranges from 0.58 to 0.9677 [15], depending on the values of ER and/or T [14; 59]. The correction factor for the methane decomposition reaction (R5) ranges from 0.0006 to 0.003 [60], and is dependent on the values of ER or T [14]. There is no correction factor reported for the steam reforming reaction (R6). Therefore, the empirical correlations of \(K_{i}^{*}\) listed in Table 9, which also covers R6 were generalized as a temperature-dependent function as follows. \[K_{i}^{*}\ =\ a_{o}\text{exp}(a_{1}T) \tag{46}\]  
The coefficient values of \(a_{o}\) and \(a_{1}\) in Equation (46) were determined by the linear regression method. Table 10 presents the coefficients for calculating the new reaction equilibrium constant of hydrogasification processes (R3), water gas shift (R4), methane formation (R5), and steam reforming (R6), which are applicable within the temperature range of 800 to 1400 K.  
In an equilibrium reaction, the constant of \(K=1\) means that the number of moles of reactant and product remains constant. If \(K<1\), the reaction is on the reactant side, implying that the number of reactants exceeds the number of products at equilibrium. Under conditions of \(K>1\), the reaction occurs on the product side, implying that the number of products exceeds the number of reactants at equilibrium. Figure 3 displays the relationship between \(K_{i}\) and \(K_{i}^{*}\) as function of temperature between 800 K and 1400 K, which was plotted using the empirical correlations presented in Tables 2 and 10. The graph of \(K_{i}^{*}\) plotted in Figure 3 was obtained by using factor model C. Figure 3 indicates an evident shift in the functions of \(K_{i}\) and \(K_{i}^{*}\). Consequently, the equilibrium temperatures of the following reactions also shift. The equilibrium temperature of R3 shifts from 850 K to 890 K, the temperature of R4 shifts from 1070 K to 960 K, the temperature of R5 shifts from 880 K to 1110 K, and R6 is shown to never reach equilibrium. The change of the equilibrium constant of the relevant reaction was further determined by the number of gas molecules produced during the gasification.