# Modeling of the Drying Process of Apple Pomace
###### Abstract
Therefore, it is important to consider its individual changes in the mathematical model of kinetics of drying of such bodies. A mathematical model of drying kinetics of solid body molecules, with consideration to shrinkage of the particles, is described by the following system of equations:  
\[u_{I}(\tau)=u_{0}\left[\frac{1}{1-b}\left(1-\frac{1-b}{Nu_{0}}k_{0}\tau\right) ^{N}-\frac{b}{1-b}\right] \tag{1}\]  
where:  
\[k_{0}=\frac{A_{0}\alpha}{m_{s}L}(t-t_{A})=-\frac{du(\tau)}{d\tau} \tag{2}\]  
The value of exponent \(N\) in Equation (1) may be determined by the trial and error method. The values of coefficient b of drying shrinkage are determined experimentally or calculated from the formula [28]:  
\[b=\frac{\rho_{0}}{\rho_{s}(1+u_{0})}\cong\frac{0.85}{1+u_{0}} \tag{3}\]  
A factor that determines the further course of the process is the internal diffusion of water in a solid body being dried. It is indicated by a sudden increase in the relative error of the model of the drying process. It is visible that the process further occurs according to other laws. This change is not abrupt but continuous. At this stage, conditions of internal diffusion of water molecules to the surface of the body play a decisive role. The process of convective drying of solid bodies in such conditions is modeled with the diffusion Equation (4). \[u_{II}(\tau)=u_{r}+(u_{0}-u_{r})\exp^{-K\tau} \tag{4}\]The continuity of the process requires that when \(u=u_{cr}\) the drying rate during the transport of water molecules from the surface of the body being dried and during internal conditions of drying be equal. As this cannot be determined precisely, the drying rate coefficient, in the range of water content, which contains the border value, is denoted by the symbol \(K\) and determined from the following Equation (5):  
\[k_{0}\bigg{(}1-\frac{1-b}{Nu_{0}}k_{0}\tau_{cr}\bigg{)}^{N-1}=K(u_{cr}-u_{r}) \tag{5}\]  
Hence, the modeling equation takes the following form:  
\[u_{II}(\tau)=u_{r}+(u_{cr}-u_{r})\exp\left[-\frac{k_{0}(\tau-\tau_{cr})}{u_{cr }-u_{r}}\bigg{(}1-\frac{1-b}{Nu_{0}}k_{0}\tau_{cr}\bigg{)}^{N-1}\right] \tag{6}\]  
In practice, simplified equations are used, with the equation of average water content, which takes the following form, being used more often (4). Since critical water content in a body may be reached after critical time tau, Equation (4) may take the form of Equation (7):  
\[u_{II}(\tau-\tau_{cr})=u_{r}+(u_{cr}-u_{r})\exp\left[-K(\tau-\tau_{cr})\right] \tag{7}\]  
The models presented so far, scientifically explain the essence of the drying process. The exponential model successfully describes kinetics of the drying of certain porous materials such as clay [29; 30], Al-Ni catalyzer [31] and foodstuffs [32; 33]. In the literature, the Page model is a model that reflects the results of the experiment well, and is relatively simple, which makes it suitable for use in practice. Research conducted by numerous authors confirm the Page model to be a good fit for the experimental data [15; 34; 35; 36; 37]. For this reason, the Page model [38] was used to describe the process of drying apple pomace. \[MR=\exp(-K\tau^{n}), \tag{8}\]  
So  
\[u(\tau)=u_{r}+(u_{0}-u_{r})\exp(-K\tau^{n}), \tag{9}\]  
Fresh apple pomace from Energreen company was used in the experiment. Forced convection drying was carried out in the laboratory dryer shown in the Figure 1. Drying took place at an air temperature of 40 \({}^{\circ}\)C and 80 \({}^{\circ}\)C in forced convection, with the speed of the drying agent at the level of 0.8 m/s. Fountain-microwave drying was performed in the MP20 dryer of the Institute of Agricultural Engineering of the University of Life Sciences in Wroclaw.