## 3 Results and Discussion
### Model Parameter Determination and Statistical Analysis  
The predictive capability of the three models on NH\({}_{4}\)-N degradation was analyzed. This was done by simulating the predicted model data in comparison to experimental data and also by plotting parity plots to assess the relationship between predicted data with the observed experimental values. The selected models included the Rate law, Haldane and [PERSON]'s models. The experimental data was fitted into the models to estimate kinetic constant in each model, using Polymath 6.0 software. The Rate law and Haldane models adequately described the experimental data of the nitrification step with a determination coefficient (R\({}^{2}\)) of 0.91 and 0.92, respectively (Table 1). Furthermore, the adjusted determination coefficient (Adj R\({}^{2}\)) was 0.89 and 0.89 respectively, with such a high Adj R\({}^{2}\) signifying a high significance, thus suitability of the estimated model kinetic constants. Furthermore, the variance was 0.015 and 0.016 for the generic rate law and the Haldane model, respectively. This variance is very low and is evidence that the predicted (modelled) values generated using estimated kinetic parameters did not differ much from experimental data. Similarly, [PERSON] model generated data with substrate inhibition was also compared to experimental data. The model gave a better fit with a correlation coefficient of 0.94 and a linear correlation efficient of 0.98 and an Adj R\({}^{2}\) 0.90. The average standard deviations were compared for the selection of the better predicting model for NH\({}_{4}\)-N degradation as a limiting step, with standard deviation of [PERSON], Rate law and [PERSON]'s model being 0.041, 0.043, and 0.036, respectively. However, for the selection of the model that is adequate in predicting multiple nitrogenous pollutants (CN\({}^{-}\), NH\({}_{4}\)-N, NO\({}_{2}\)-N and NO\({}_{3}\)-N) removal, the prediction ability of the model towards CN\({}^{-}\) degradation needed to be considered. The parity plot indicated a satisfactory correlation between the experimental rate of NH\({}_{4}\)-N degradation and the predicted rates with R\({}^{2}\) being 0.96, 0.92, and 0.94 for Rate law, [PERSON] and [PERSON]'s model respectively. Moreover, the variance for [PERSON]'s model also proved the significance of the model with variance being 0.015. This variance was minute which indicated that there was a negligible difference between the model and experimental data (see Figure 2 and Table 2). [PERSON] and [PERSON] models were also assessed for their predictive ability of NH\({}_{4}\)-N (Figure S3 and Table S1, Supplementary file A) and CN\({}^{-}\) (Figure S4 and Table S2, Supplementary file A) degradation rates; however, they did not fit well into the experimental data. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Values of Kinetic Parameters (\(\pm\)95\% Confidence Interval)**} \\ \hline
**Model** & \(V_{m}\) (h\({}^{-1}\)) & \(K_{i}\) (mgL\({}^{-1}\)) & \(K_{s}\) (mgL\({}^{-1}\)) & \(n\) (-) & \(\mathbf{R^{2}}\) & **Variance** \\ \hline Rate law & & & & & & \\ \(-\frac{dC_{s}}{dt}=r=K_{i}C_{a}^{n}\) & - & 0.0000629 & - & 2.27 & 0.91 & 0.02 \\ [PERSON] & & & & & & \\ \(-\frac{dC_{s}}{dt}=\frac{V_{m}C_{c}}{C_{s}+\frac{C_{s}^{2}}{K_{s}}}\) & 0.45 & 23.94 & - & - & 0.92 & 0.02 \\ \hline \multicolumn{6}{c}{**Model with Substrate Inhibition**} \\ \hline
**[PERSON]** & & & & & & \\ \(\frac{dC_{s}}{dt}=-V_{m}[1-\frac{(K_{i})}{C_{s}}]\) & 0.36 & 27.54 & 13.22 & - & 0.94 & 0.02 \\ \hline \multicolumn{6}{c}{Equations (2)–(6) were obtained from [PERSON] et al. [13].} \\ \end{tabular}
\end{table} 
Table 1: Estimated kinetic parameter values for the models for NH\({}_{4}\)-N degradation, a rate limiting step in nitrification.