## 4 Results and Discussion
### Correction Factors and New Empirical Correlation for the Equilibrium Constant
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factor Model**} & \multicolumn{4}{c}{**Correction Factor**} \\ \cline{2-5}  & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) \\ \hline A & 14.7690 & 3.5143 & 0.0924 & 0.0092 \\ B & 66.4224 & 1.3221 & 0.0904 & 0.0513 \\ C & 33.9060 & 1.5202 & 0.0602 & 0.0054 \\ D & 305.3084 & 1.2181 & 0.0364 & 0.3595 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The correction factors obtained from M1, M2, M4, and M5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor Model** & \(K_{3}^{*}\) & \(K_{4}^{*}\) & \(K_{5}^{*}\) & \(K_{6}^{*}\) \\ \hline A & 2.2054\(K_{3}\) & 0.7462\(K_{4}\) & 0.1474\(K_{5}\)0.4325 & 0.5568\(K_{6}^{0.0125}T^{-2}\) \\ B & 3.2117\(K_{3}\) & 0.5844\(K_{4}\) & 0.1441\(K_{5}\)0.4325 & 0.6898\(K_{6}^{0.0125}T^{-2}\) \\ C & 2.7147\(K_{3}\) & 0.6052\(K_{4}\) & 0.0960\(K_{5}\)0.4325 & 0.5203\(K_{6}^{0.0125}T^{-2}\) \\ D & 4.7026\(K_{3}\) & 0.5726\(K_{4}\) & 0.0581\(K_{5}\)0.4325 & 0.8800\(K_{6}^{0.0125}T^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The new reaction equilibrium constants (\(K^{*}\)) for each factor model. Simulation, Validation, and Accuracy of the Composition of Producer Gas Using the Developed Stoichiometric Models  
The chemical composition of \(n_{1}\)-\(n_{8}\) can be determined by solving Equations (5)-(8) and (25)-(28). The equilibrium constant of each reaction (R3, R4, R5, and R6) is computed using Equation (46), and the results in terms of \(\Delta\)H, yield of tar, and RMSE of factor model A-D for each developed model M1-M6 are displayed in Table 11. The data in Table 7 were used as the model inputs. The data in Supplementary Table S2 were used to validate the predicted results by M1A-M6D. Based on Table 11, the original stoichiometric models (M1O, M2O, M3O, M4O, M5O, and M6O) have average \(\Delta\)H > 0, yield of C\({}_{6}\)H\({}_{6}\) < 0, and RMSE > 10% (Table 11). These three indications are the challenges for using the original model to simulate the gas composition of biomass gasification. The results of \(\Delta\)H, a yield of C\({}_{6}\)H\({}_{6}\), and RMSE were improved when the factor model was introduced into the developed models (M1A-M6D). The average \(\Delta\)H became less than 0, a yield of C\({}_{6}\)H\({}_{6}\) was more than 0, and RMSE was smaller than 3.00%. Based on these three, the developed stoichiometric models with factor models are deemed to be valid. RMSE of the stoichiometric models developed in this study (M1A-M6D) are in line with the RMSE of the stoichiometric models developed by [PERSON] et al. (2015). The models had RMSE values between 2.3018 and 2.7718. The stoichiometric models developed by [PERSON] et al. (2010) that were tested against 32 experimental cases had RSME ranges from 3.854 to 3.3373, which were more than 3%. Semi-empirical modified models by [PERSON] et al. (2018) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Factor Models** & **Coefficient and R\({}^{2}\)-adj** & \(K_{3}^{*}\) & \(K_{4}^{*}\) & \(K_{5}^{*}\) & \(K_{6}^{*}\) \\ \hline A & \(a_{0}\) & 4289.2370 & 46.2788 & \(1.1535\times 10^{-5}\) & \(2.0959\times 10^{-6}\) \\  & \(a_{1}\) & \(-0.009770\) & \(-0.003742\) & 0.010617 & \(-0.000864\) \\  & R\({}^{2}\)-adj & 0.9817 & 0.9725 & 0.9803 & 0.9998 \\ B & \(a_{0}\) & 6246.2697 & 36.2444 & \(1.1279\times 10^{-5}\) & \(2.5967\times 10^{-6}\) \\  & \(a_{1}\) & \(-0.009770\) & \(-0.003742\) & 0.010617 & \(-0.000864\) \\  & R\({}^{2}\)-adj & 0.9817 & 0.9725 & 0.9803 & 0.9998 \\ C & \(a_{0}\) & 5939.6868 & 37.5318 & \(7.5092\times 10^{-6}\) & \(1.9585\times 10^{-6}\) \\  & \(a_{1}\) & \(-0.009770\) & \(-0.003742\) & 0.010617 & \(-0.000864\) \\  & R\({}^{2}\)-adj & 0.9817 & 0.9725 & 0.9803 & 0.9998 \\ D & \(a_{0}\) & 10,289.1435 & 35.5092 & \(4.5442\times 10^{-6}\) & \(3.3126\times 10^{-6}\) \\  & \(a_{1}\) & \(-0.00977\) & \(-0.00374\) & 0.0106166 & \(-0.000864\) \\  & R\({}^{2}\)-adj & 0.9817 & 0.9725 & 0.9803 & 0.9998 \\ \hline \hline \end{tabular}
\end{table}