# The Significance of the Sorption Isotherm on the Simulated Performance of Grain Driers
## 5 Sorption Isotherms
### [PERSON] [19] presents the following expression for the relative humidity, \(r\), of air with grain that has a moisture content, \(W\):  
\[r=\left(\frac{p_{s}}{p_{0}}\right)^{\frac{h_{s}}{h_{v}}-1} \tag{52}\]  
in which \(p_{0}\) is an empirical constant and \(p_{s}\) is the saturation vapour pressure at the temperature, \(T\), of interest. \(h_{s}/h_{v}\) is an empirical function of the moisture content, \(W\), of the grains, given by  
\[\frac{h_{s}}{h_{v}}-1=\frac{c_{1}{\rm In}(c_{2}W)-\left(W/W_{0}\right)^{ \alpha}c_{3}{\rm In}(c_{4}W)}{1-\left(W/W_{0}\right)^{\alpha}} \tag{53}\]  
[PERSON] [19] provides the following equation for \(W_{0}\):  
\[W_{0}=\left(\frac{c_{2}c_{1}}{c_{4}c^{\alpha_{3}}}\right)^{\frac{1}{C_{3}- \alpha_{1}}} \tag{54}\]  
The mean values of the moisture contents of canola at the relative humidities, \(r\), measured by [PERSON] [8] were calculated. This enables the unknown parameters \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(c_{1}\), and \(\alpha\) to be estimated by minimising the sum of the errors between the measured values of the relative humidities and those given by Equation (52). Algebraically, when the relative humidity, \(r\), and temperature, \(T\), are known, then the unknown parameters are found by minimising \(\varphi_{min}\), which is given by  
\[\varphi_{min}=\left|r-\left(\frac{p_{s}}{p_{0}}\right)^{\frac{h_{s}}{h_{v}}-1}\right| \tag{55}\]The following values of the parameters that satisfy Equations (53) and (55) are found to be \(c_{1}=-0.29681\), \(c_{2}=15.633\), \(c_{3}=-0.00075041\), \(c_{4}=0.44649\), and \(\alpha=3.2919\). In Equation (52), \(p_{0}=44.040\times 10^{5}\).  
\(h_{s}/h_{v}\) as a function of the moisture content, \(W\), is given by Equation (53), and to simplify the algebra, the function is replaced by the following polynomial equations:  
\[\frac{h_{s}}{h_{v}}=\sum_{i=1}^{8}\beta_{i}W^{i-1}\hskip 28.452756 ptW\leq 0.2 \tag{56}\]  
and  
\[\frac{h_{s}}{h_{v}}=\sum_{i=1}^{7}\gamma_{i}W^{i-1}\hskip 28.452756 ptW>0.2 \tag{57}\]  
where \(\beta_{i}\) and \(\gamma_{i}\) are empirical constants.  
Using Equation (56) in Equations (13) and (14) yields  
\[H_{W}=h_{v}\left(W-\sum_{i=1}^{8}\frac{\beta_{i}W^{i}}{i}\right)\hskip 28.452756 ptW \leq 0.2 \tag{58}\]  
and Equations (13), (14), (56) and (57) give  
\[H_{W}=h_{v}\left\{\left(0.2-\sum_{i=1}^{8}\frac{\beta_{i}2^{i}}{i}\right)+ \left(W-0.2-\sum_{i=1}^{8}\frac{\gamma_{i}W^{i}}{i}\right)\right\}\hskip 14.226378 ptW >0.2 \tag{59}\]  
Values of the coefficients \(\beta\) and \(\gamma\) are given in Table 1.  
[PERSON]'s [19] isotherm faithfully fits sorption data that are experimentally determined. However, like [PERSON] and [PERSON] [12], [PERSON] [13] observed that anomalies arise when calculating the speeds of transfer waves through beds of aerated wheat.  
In this work, the saturation vapour pressure of water, \(p_{s}\), as a function of temperature is given by the expression presented by [PERSON] [20], namely  
\[p_{s}=\frac{e_{f}}{(T+105)^{1.59}}\text{exp}\left(34.494-\frac{4924.99}{T+237. 1}\right) \tag{60}\]  
where the enhancement factor, \(e_{f}\), assumes the value 1.005 at atmospheric pressure.