## 2 Materials and Methods
### Effective Moisture Diffusion Coefficient  
The effective moisture diffusion coefficient (D\({}_{\text{eff}}\)), or the internal mass transfer coefficient of the apple slices, was calculated using the Crank and the Dincer (thin layer) method. The research showed that using the parameter of the effective moisture diffusion coefficient with assuming that the transfer of moisture takes place only through diffusion is a suitable method for predicting the process of removing moisture from the slice. For this purpose, according to Equation (1), [PERSON]'s second law is widely used to describe moisture diffusion in agricultural products [20]. \[\frac{\partial\text{M}}{\partial\text{t}}=\text{D}_{\text{eff}}\frac{\partial ^{2}\text{M}}{\partial\text{x}^{2}}, \tag{1}\]  
As can be observed from Equation (1), mass transfer is transient and is considered only in one direction (\(x\)-axis), and, due to the blade shape of the sample, mass transfer in the other directions is negligible. The Crank model was simplified to obtain the effective moisture diffusion coefficient with the following assumptions:  
Figure 1: Schematic picture of the combined hot air–infrared dryer: 1—centrifugal fan; 2—heating element; 3—temperature sensor; 4—drying tray; 5—drying chamber; 6—infrared lamps; 7—central controller; 8—wattmeter. 1. Mass transfer of moisture inside the product continues only through diffusion. 2. The exit of moisture from the product is uniform. 3. External resistance against mass transfer is negligible. 4. Sample volume is constant during the drying process, and the effects of shrinkage are not taken into account. 5. The diffusion coefficient is constant. 6. \(\mathrm{M}\left(\frac{\mathrm{L}}{2},\ 0\right)=\mathrm{M}_{0}\) 7. \(\mathrm{\frac{\partial M(0,t)}{\partial x}}=0\) 8. \(\mathrm{-D}_{\mathrm{eff}}\frac{\mathrm{M}\left(\frac{\mathrm{L}}{2},\mathrm{ t}\right)}{\partial x}=\mathrm{h}_{\mathrm{m}}(\mathrm{M}-\mathrm{M}_{\mathrm{e}})\) 9. \(\mathrm{M}\left(\frac{\mathrm{L}}{2},\ \mathrm{t}\right)=\mathrm{M}_{\mathrm{e}}\) 10. \(\mathrm{M}\left(\frac{\mathrm{L}}{2},\ \mathrm{t}\right)=\mathrm{M}_{\mathrm{e}}\) 11. \(\mathrm{M}\left(\frac{\mathrm{L}}{2},\ \mathrm{t}\right)=\mathrm{M}_{\mathrm{e}}\) 12. Therefore, [PERSON]'s solution of [PERSON]'s equation for a product with the assumption of shape as an infinite blade ends in Equation (2) [19]. \[\mathrm{MR}=\frac{\mathrm{M}-\mathrm{M}_{\mathrm{e}}}{\mathrm{M}_{0}-\mathrm{ M}_{\mathrm{e}}}=\frac{8}{\pi^{2}}\sum_{n=0}^{\infty}\frac{1}{\left(2n-1 \right)^{2}}\exp\left(-(2n-1)^{2}\pi^{2}\frac{\mathrm{D}_{\mathrm{eff}}t}{4 \mathrm{l}^{2}}\right), \tag{2}\]  
Because the drying time is long (\(\mathrm{Fo}>0.2\)), the above series converges very quickly, and only the first term can be used to approximate the above series with high accuracy, so Equation (2) converts as follows [17]:  
\[\mathrm{MR}=\frac{8}{\pi^{2}}\exp\left[-\frac{\pi^{2}\mathrm{D}_{\mathrm{eff}} \ \mathrm{t}}{4\mathrm{l}^{2}}\right], \tag{3}\]  
Moreover, the above equation can be written in the following logarithmic form:  
\[\mathrm{ln}\,\mathrm{MR}=-\left[\frac{\pi^{2}\mathrm{D}_{\mathrm{eff}}\ \mathrm{t}}{4\mathrm{l}^{2}}\right]\mathrm{ln}\left(\frac{8}{\pi^{2}}\right), \tag{4}\]  
By deriving the above equation, the value of \(\mathrm{D}_{\mathrm{eff}}\) is obtained from the slope of the logarithmic diagram of the moisture ratio [32]. \[\mathrm{k}=\frac{\pi^{2}\mathrm{D}_{\mathrm{eff}}\ \mathrm{t}}{4\mathrm{l}^{2}}, \tag{5}\]  
Another method for determining the effective moisture diffusion coefficient is the use of the Henderson and Pabis thin layer model, which was proposed by [PERSON] and [PERSON], 1996. For this purpose, the following equation, which is the Henderson and Pabis model, is used to predict the dimensionless moisture ratio:  
\[\mathrm{MR}=\frac{\mathrm{M}-\mathrm{M}_{\mathrm{e}}}{\mathrm{M}_{0}-\mathrm{ M}_{\mathrm{e}}}=\mathrm{G}\exp(-\mathrm{St}), \tag{6}\]  
In the above equation, \(\mathrm{G}\) represents the lag factor, and \(\mathrm{S}\) is the drying constant (\(1/\mathrm{s}\)) [16].