## 3 Results
### Analytical Model  
As can be seen in Figure 5a,b, the relative mass changes of reduction and oxidation show almost a linear rise followed by flattening gradients down to zero. This temporal behavior can be approximated by the linear driving force (LDF) model [44], which is described by Equation (6).  
\[\frac{d\gamma_{\text{m}}}{dt}=k\cdot(\gamma_{\text{eq}}-\gamma_{\text{m}}) \tag{6}\]  
Herein, the rate constant \(k\left[1/\text{min}\right]\) depends, among other factors, on the concentration. The LDF model simply describes the averaged mass change rate \(d\gamma_{\text{m}}/dt\), resulting from the difference between the equilibrium mass \(\gamma_{\text{eq}}\) and the averaged mass \(\gamma_{\text{m}}\) over the entire pellet volume ([45]).  
After separating the variables in Equation (6) and subsequent integration, Equation (7) results.  
\[ln\left(\frac{\gamma_{\text{m}}}{\gamma_{\text{m},0}}\right)=-k\cdot t\Rightarrow \gamma_{\text{m}}=\gamma_{\text{m},0}\cdot e^{-k\cdot t} \tag{7}\]  
Thereby, \(\gamma_{\text{m},0}\) describes the initial relative mass change at \(t=0\), which shall decay with time according to the exponential Equation (7). In analogy to the parameterization of common adsorption models [46], the effective mass transfer coefficient \(k\) can be described with the reciprocal value of the characteristic time constant \(\tau\). Analyzing the obtained reaction kinetic curves of the investigated iron oxide pellets reveals that the oxidation kinetics do follow this rule quite well. Therefore, the effective mass transfer coefficient for the oxidation reactions is expressed as \(k_{\text{Ox}}=1/\tau_{\text{Ox}}\), and the temporal change of the oxidation fraction is accurately described by the exponential Equation (8), with the time constant \(\tau_{\text{Ox}}\) as the time to reach (\(1-1/e\)) or 0.63212 of the final oxidation fraction.  
\[\gamma_{\text{Ox,num}}(t)=1-e^{-k_{\text{Ox}}\cdot t} \tag{8}\]  
In order, however, to best fit the obtained temporal change of the reduction fraction, the exponential term should be modified as described in Equation (9). First, the effective mass transfer and reduction reaction coefficient \(k_{\text{Re}}\) is described by \(q_{\text{Re}}/\tau_{\text{Re}}\), where \(a\) is a concentration-dependent parameter. Here, \(\tau_{\text{Re}}\) is the time required for the temporal  
Figure 5: Effect of the gas concentration \(\text{ro}_{1}\) to \(\text{ro}_{4}\) (cf. Table 2) on the temporal reduction (**a**) and oxidation (**b**) fractions of S1 at a reactant flow rate of 250 mL/min and the reaction temperature of 700 \({}^{\circ}\)C.  
reduction fraction to reach \((1/e)\) or 0.3679 of its maximum fraction (one at zero time). Second, the time variable \(t\) is raised to the exponent \(\sqrt{2}\).  
\[\gamma_{\text{Re,num}}(t)=e^{-\text{k}_{\text{Re}}\cdot t^{\sqrt{2}}} \tag{9}\]  
The estimated parameters for Equations (8) and (9) to best fit the experimental results obtained for the redox experiments r0\({}_{1}\) to r0\({}_{4}\) (cf. Table 2) are listed in Table 3.  
The comparison between the experimental (E-ro\({}_{1-4}\)) and analytical (A-ro\({}_{1-4}\)) reduction and oxidation results is shown in Figure 6a,b. A very good agreement between the mathematical model in Equations (8) and (9) and the experimental results is achieved, which is indicated by an average deviation of less than 3%.