# The Significance of the Sorption Isotherm on the Simulated Performance of Grain Driers
## 4 Speeds of Temperature and Drying Fronts  
The velocities of the drying and temperature fronts are given by solving Equations (5) and (6) across each of the heat and mass transfer fronts that propagate through the bed. We generally know the initial thermodynamic state of the bed, state B. It is defined by the moisture content, \(W_{initial}\), and temperature, \(T_{initial}\), of the grain that is to be dried; hence, the humidity, \(w_{initial}\), of the interstitial air can be determined from a sorption isotherm. Similarly, state A of the air entering the bed is defined by its temperature \(T_{inlet}\) and humidity \(w_{inlet}\) of the air entering the grain. The equilibrium moisture content, \(W_{inlet}\), of the grain initially in the bed is given by a suitable isotherm. Making use of Equations (6) and (7), we define a function as follows:  
\[\varphi\big{(}w_{p},T_{p}\big{)}=\big{|}\mathcal{V}_{w,d}-\mathcal{V}_{h,d}\big{|} +\big{|}\mathcal{V}_{w,t}-\mathcal{V}_{h,t}\big{|} \tag{15}\]  
where the subscripts \(w\) and \(h\) correspond to those defined in Equations (5) and (6), and \(d\) and \(t\) represent the drying and heating fronts, respectively. Recall that \(\mathcal{V}_{w,d}=\mathcal{V}_{h,d}\) and \(\mathcal{V}_{w,t}=\mathcal{V}_{h,t}\). The Nelder-Mead simplex method [17] is used to search for values of \(w_{p}\) and \(T_{p}\) that render \(\varphi\big{(}w_{p},T_{p}\big{)}=0\). However, searching for the unknown values in the plateau state is very sensitive to the initial values of \(w_{p}\) and \(T_{p}\). This difficulty is overcome by observing that the order of magnitude of the terms in the denominator of Equation (5) are  
\[\Big{\{}\varepsilon\rho_{\alpha},\ \varepsilon\Delta\rho_{\alpha},\ \varepsilon\rho_{ \alpha}\frac{w}{\Delta w}\Big{\}}=\mathcal{O}\Big{(}10^{0}\Big{)} \tag{16}\]  
and  
\[\rho_{s}(1-\varepsilon)\frac{\Delta W}{\Delta w}=\mathcal{O}\Big{(}10^{3} \cdot 10^{1}\Big{)}=\mathcal{O}\Big{(}10^{4}\Big{)} \tag{17}\]  
As a result, we can write  
\[\mathcal{V}_{w}\cong\frac{G}{\rho_{s}(1-\varepsilon)\frac{\Delta W}{\Delta w}} \tag{18}\]  
Similar reasoning yields  
\[\mathcal{V}_{h}\cong\frac{G}{\rho_{s}(1-\varepsilon)\frac{\Delta H}{\Delta h}} \tag{19}\]  
which suggests we can write  
\[\frac{\Delta W_{P-A}}{\Delta w_{P,A}}=\frac{\Delta H_{P-A}}{\Delta h_{P-A}} \tag{20}\]  
where the subscripts refer to the difference between the plateau state P and inlet state A where the con la is in equilibrium with the air that enters the bed. The thermodynamic state, \((W_{P},T_{P},\ w_{P})\), in the plateau state is unknown. Hence, we cross-multiply Equation (20) and define the function \(\phi(T_{P-A},w_{P-A})\) as  
\[\phi(T_{P-A},w_{P-A})=\Delta h_{P-A}\Delta W_{P-A}-\Delta H_{P-A}\Delta w_{P-A} \tag{21}\]  
in which  
\[\Delta h_{P-A}=c_{a}(T_{P}-T_{A})+c_{w}(w_{P}T_{P}-w_{A}T_{A})+w_{P}h_{v} \big{|}_{P}-w_{A}h_{v}\big{|}_{A} \tag{22}\]  
\[\Delta H_{P-A}=c_{s}(T_{P}-T_{A})+c_{w}(W_{P}T_{P}-W_{A}T_{A})+H_{W}\big{|}_{ P}-H_{W}\big{|}_{A} \tag{23}\]  
\[\Delta W_{P-A}=W_{P}-W_{A} \tag{24}\]  
and  
\[\Big{\{}\varepsilon\rho_{\alpha},\ \varepsilon\Delta\rho_{\alpha},\ \varepsilon\rho_{ \alpha}\frac{w}{\Delta w}\Big{\}}=\mathcal{O}\Big{(}10^{0}\Big{)} \tag{16}\]0  
We define a similar function \(\phi(T_{P-B},w_{P-B})\) that involves the differences between the enthalpies and moisture contents at the plateau state and the initial conditions, and we write  
\[\Big{\{}\varepsilon\rho_{\alpha},\ \varepsilon\Delta\rho_{\alpha},\ \varepsilon\rho_{ \alpha}\frac{w}{\Delta w}\Big{\}}=\mathcal{O}\Big{(}10^{0}\Big{)} \tag{16}\]1  
Each of the two terms on the right-hand side of Equation (26) must be zero; hence, a search procedure is used to determine the values of \(T_{P}\) and \(w_{P}\) that satisfy this condition.