Modeling
As mentioned through the kinetic data and chemical analysis above, aging processes in dry air and humid air are controlled by the oxidation reaction. The aging experiments were conducted under a continuous flow system such that the concentrations of gas streams were constant during the aging processes. Accordingly, the pseudo reversible reaction model was applied to describe the kinetics of aging processes on Ag0Z. For the modeling work, the correlation coefficient (R2) was used to evaluate the goodness of fit between experimental data and model results. According to the modeling results, pseudo 1st order reversible reaction models for aged Ag0Z in dry air and humid air at different aging temperatures (100 oC, 150oC, and 200 oC) and different water vapor concentrations (d.p. -40 oC, -15oC, and +15 oC) for up to 2 weeks are placed in Figure 8 which are incorporated with the kinetic data shown in Figure 7.
Modeling results on the aged Ag0Z in dry air at different temperatures for different aging time are shown in Figure 8 (A). Eqn. (13) represents a reaction equation for a kinetic model on the aged Ag0Z in dry air shown in Figure 8 (A).\(C_{\text{Ag}}\) is the normalized concentration of silver which indicates the amount of I2 loaded on the aged Ag0Z, \(C_{O_{2}}\) is the concentration of oxygen in dry air, \(C_{\text{Ag}^{+}}\) is the concentration of silver oxidized by oxygen in dry air, \(n\) is the reaction order, and \(k_{1}\) is the forward reaction rate constant, and \(k_{-1}\) is the reverse reaction rate constant. Here, \(k_{1}\) and \(C_{O_{2}}\) are lumped into\({k_{1}}^{*}\) because the gas concentration was constant all the time during the aging experiment. \(C_{\text{Ag}^{+}}\) can be calculated by substracting the amount of I2 loaded on the aged Ag0Z from the amount of I2 loaded on the unaged Ag0Z. The model parameters of all reaction models and correlation coefficients (R2) between experimental data and model results are shown in Table 1. These results indicate a model is fitted to experimental data well when \(n\geq 1\)with strong correlation coefficient (for example, R2 = 0.988 at 150 oC) using Eqn. 2, which means the Pseudo 1st reversible reaction model fits experimental data (Figure 8 (A)). These results are consistent with experimental data (Figure 7) that the aging effects of gas streams on Ag0Z increase with increasing aging time and temperatures.
\begin{equation} {\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Ag}{\text{Ag}_{s}}^{+}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\backslash n}{v=\frac{\text{dC}_{\text{Ag}}}{\text{dt}}={-k}_{1}C_{O_{2}}{C_{\text{Ag}}}^{n}+k_{-1}{C_{\text{Ag}^{+}}}^{n}}\nonumber \\ \end{equation}\begin{equation} {k_{1}}^{*}=k_{1}C_{O_{2}}\nonumber \\ \end{equation}\begin{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v=\frac{\text{dC}_{\text{Ag}}}{\text{dt}}={{-k}_{1}}^{*}C_{\text{Ag}}+{k_{-1}C}_{\text{Ag}^{+}}\ \ \ \ (for\ n=1)\nonumber \\ \end{equation}
Modeling results on the aged Ag0Z in humid air with different temperatures, aging time, and water vapor concentrations are placed in Figure 8 (B) and (C). A reaction equation for the model is provided in Eqn. (14). Considering the results of correlation coefficient to each order model with the model results, when\(n\geq 1\), model results show good fitting to experimental data.\(C_{H_{2}O}\) and \(C_{O_{2}}\) are the concentrations of water vapor and oxygen in humid air, \(k_{1}\) is the reaction rate constant,\(k_{-1}\) is the reverse reaction rate constant, and \(n\) is the reaction order. Here, \(k_{1}\), \(C_{H_{2}O}\), and \(C_{O_{2}}\) are lumped into \({k_{1}}^{*}\), and m is the power of the water vapor concentration determined by a exponential curve expressed by\({k_{1}}^{*}\) versus \(C_{H_{2}O}\) as shown in Figure 9 (m = 0.20). These modeling results indicate that the pseudo 1streversible reaction model fits experimental data (For example, R2 = 0.985 at 150 oC) with similar approach to the model result of dry air aged Ag0Z.
\begin{equation} \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Ag}{\text{Ag}_{s}}^{+}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (14)\nonumber \\ \end{equation}\begin{equation} v=\frac{\text{dC}_{\text{Ag}}}{\text{dt}}={{-k}_{1}}^{*}{C_{\text{Ag}}}^{n}+{{k_{-1}C}_{\text{Ag}^{+}}}^{n}\nonumber \\ \end{equation}\begin{equation} {k_{1}}^{*}=k_{1}{C_{H_{2}O}}^{m}C_{O_{2}}\nonumber \\ \end{equation}\begin{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v=\frac{\text{dC}_{\text{Ag}}}{\text{dt}}={{-k}_{1}}^{*}C_{\text{Ag}}+{k_{-1}C}_{\text{Ag}^{+}}\ \ \ \ (for\ n=1)\nonumber \\ \end{equation}