## 3 Simplifying the Models
### Simplified Model MPL1: Linear Heat Capacities  
Although we now take into account two preheaters in the overall model, this does not change the general approach to obtain the exact solution in case of linearly approximated heat capacities. Equations (5)-(8) stay the same. The equation for the temperature turns into  
\[\dot{\theta}=-c_{m}^{\rm inv}\cdot(k_{\rm const}(t)+k_{\rm lin}(t)\theta+k_{ \rm sq}(t)\theta^{2})\enspace, \tag{15}\]where \(k_{\text{const}}(t)\), \(k_{\text{lin}}(t)\) depend on time and \(\theta^{2}\) appears on the right-hand side of the equation with the constant coefficient \(k_{\text{sq}}(t)\):  
\[k_{\text{const}}(t) =-\theta_{A}k_{a}-(y_{3}(t)c_{\text{CG,0}}+y_{1}(t)c_{\text{N2,0}} )\ \ (k_{a}\text{ as in Section \ref{eq:k_const}})\ \,\] \[k_{\text{lin}}(t) =k_{+}\dot{m}_{\text{CG}}^{\text{in}}\cdot c_{\text{CG,0}}+\dot{m} _{\text{N2}}^{\text{in}}\cdot c_{\text{N2,0}}-(y_{3}(t)c_{\text{CG,1}}+y_{1}(t )c_{\text{N2,1}})\ \,\] \[k_{\text{sq}}(t) =\dot{m}_{\text{CG}}^{\text{in}}\cdot c_{\text{CG,1}}+\dot{m}_{ \text{N2}}^{\text{in}}\cdot c_{\text{N2,1}}\ \.\]  
Note that we have two additional parameters \(c_{\text{CG,1}}\), \(c_{\text{N2,1}}\) now, resulting from the fact that we represent heat capacities of gases as linear polynomials in temperature: \(c_{\text{N2}}(\theta)=c_{\text{N2,0}}+c_{\text{N2,1}}\cdot\theta\), \(c_{\text{CG}}(\theta)=c_{\text{CG,0}}+c_{\text{CG,1}}\cdot\theta\). Equation (15) is a Riccati equation. Since \(k_{\text{sq}}\) does not depend on time in our case because gas mass flows are constant, it can be transformed into a system of the following linear ODEs with non-constant coefficients:  
\[ \begin{pmatrix}y_{4}\\ y_{5}\end{pmatrix} ^{\prime}= \begin{pmatrix}0&1\\ -(c_{m}^{\text{inv}})^{2}k_{\text{const}}(t)k_{\text{sq}}&-c_{m}^{\text{inv}}k_ {\text{lin}}(t)\end{pmatrix} \cdot \begin{pmatrix}y_{4}\\ y_{5}\end{pmatrix} \ \, \tag{16}\]  
where the temperature \(\theta\) equals \(\frac{y_{5}}{c_{m}^{\text{inv}}k_{\text{sq}}y_{4}}\). It is probably impossible to find closed-form solutions to that set of equations in our case because the matrix in Equation (16) does not even necessarily commute with itself for given points of time \(t\). Therefore, we assumed that the preheaters behaved as piecewise constant functions similarly to externally specified control variables. Due to new sensor material (in comparison to [26]), it was possible to measure the values of \(y_{1}\) and \(y_{3}\) in all relevant points of time. The closed-form solution for the temperature equation with-now constant-coefficients \(k_{\text{const}}\), \(k_{\text{lin}}\) with \(y_{1}(t)=y_{1}^{\text{meas}}\), \(y_{3}(t)=y_{3}^{\text{meas}}\) is given in dependence on \(D=k_{\text{lin}}^{2}-4k_{\text{const}}k_{\text{sq}}\) according to  
\[D>0\text{: }\theta(t) =\frac{1}{k_{\text{sq}}}\left(\frac{\sqrt{D}}{1-e^{-c_{m}^{\text {inv}}(t-t_{0})\sqrt{D}}\cdot\left(1-\frac{2\sqrt{D}}{2k_{\text{sq}}\text{gl}^ {\text{ic}}+k_{\text{lin}}+\sqrt{D}}\right)}-\frac{k_{\text{lin}}+\sqrt{D}}{2}\right)\] \[D<0\text{: }\theta(t) =\frac{\sqrt{-D}\tan\left(-\frac{\sqrt{-D}}{2}c_{m}^{\text{inv}} (t-t_{0})+\theta^{\text{c}}\right)-k_{\text{lin}}}{2k_{\text{sq}}}\text{, }\theta^{\text{c}}=\text{atan}\left(\frac{2k_{\text{sq}}\theta^{\text{ic}}+k_{ \text{lin}}}{\sqrt{-D}}\right)\] \[D=0\text{: }\theta(t) =\frac{2\theta^{\text{ic}}+k_{\text{lin}}/k_{\text{sq}}}{2+c_{m}^ {\text{inv}}(t-t_{0})(2k_{\text{sq}}\theta^{\text{ic}}+k_{\text{lin}})}-\frac{k _{\text{lin}}}{2k_{\text{sq}}}\text{ ( improbable in our case).}\]  
From the definitions for \(k_{\text{const}}\), \(k_{\text{lin}}\), \(k_{\text{sq}}\) and the data, it is likely (but not certain) that \(D\) is positive. Nonetheless, it is necessary to check the sign in the corresponding parameter identification/simulation routines and choose the solution branch accordingly. The results of parameter identification using this multi-branch analytical expression to obtain new parameter sets are given in [31].