# Parametric Analysis of Cylinder Drying Process in Association with Various Materials
## 2 Methodology
### Mathematical Model  
Normally, to dry material, especially clothes, the wet pressing process is first conducted, and the rest of the water will be removed in the drying process. During the drying process, the water contained in the wet material is heated until it is evaporated; the heated moisture on the top surface of the drying material is directly released to the ambient environment; this leads to the moisture concentration on the top surface to be lower than that in the inner layer; therefore, the heated moisture inner of the drying material will defuse  
Figure 1: Phases in a computational model [10]. (**a**) A 3D drum dryer [10]; (**b**) A schematic diagram of the drum dryer [10]; (**c**) A schematically phased map of the drying drum; (**d**) A computational model. up to the top surface due to the difference in moisture concentration. The process happens continuously until an equilibrium of moisture between the drying material and the ambient environment is set. Therefore, the water contained in the drying material is decreased during the drying process, and the rate of the loss of moisture,\(\frac{dM}{dt}\), can be calculated as follows [4]:  
\[\frac{dM}{dt}=-\xi(M-M_{e}) \tag{1}\]  
where \(M\), \(M_{e}\), and \(\xi\) denote the initial moisture amount, the equilibrium moisture amount, and the drying constant, respectively. The drying constant is calculated as follows [4]:  
\[\xi=\frac{(\pi^{2}\alpha)}{y^{2}} \tag{2}\]  
where \(\alpha\) and \(y\) are the thermal diffusivity and the thickness of the drying material, respectively. The energy equation of the drying process is expressed as follows:  
\[\frac{\partial^{2}T}{\partial y^{2}}=\frac{1}{\alpha}\frac{\partial T}{ \partial t} \tag{3}\]  
where \(\alpha\) is the thermal diffusivity of the drying material. \(T\), \(y\), and \(t\) stand for temperature, the thickness of the drying material, and drying time, respectively. For calculating the heat transfer characteristics in the computational model, the boundary conditions for the four phases in drying process are expressed as follows [8, 10, 17, 19, 20, 28, 29]:  
Phase-I  
\[-\frac{\partial T}{\partial y}\bigg{|}_{y=0}=\frac{h_{cs}}{k}(T_{c}-T_{y=0}) \tag{4}\]  
\[-\left.\frac{\partial T}{\partial y}\right|_{y=Y}=\frac{h_{sat}}{k}(T_{y=Y}-T _{a})+\frac{\lambda}{k}(\frac{\partial w}{\partial t})_{y=Y} \tag{5}\]  
Phase-II  
\[-\frac{\partial T}{\partial y}\bigg{|}_{y=0}=\frac{h_{cs}}{k}(T_{c}-T_{y=0}) \tag{6}\]  
\[-\left.\frac{\partial T}{\partial y}\right|_{y=Y}=\frac{h_{sf}}{k}(T_{y=Y}-T _{f})+\frac{\lambda}{k}(\frac{\partial w}{\partial t})_{y=Y} \tag{7}\]  
Phase-III  
\[-\frac{\partial T}{\partial y}\bigg{|}_{y=0}=\frac{h_{cs}}{k}(T_{c}-T_{y=0}) \tag{8}\]  
\[-\left.\frac{\partial T}{\partial y}\right|_{y=Y}=\frac{h_{sat}}{k}(T_{y=Y}-T _{a})+\frac{\lambda}{k}(\frac{\partial w}{\partial t})_{y=Y} \tag{9}\]  
Phase-IV  
\[\frac{dM}{dt}=-\xi(M-M_{e}) \tag{1}\]0  
\[\frac{dM}{dt}=-\xi(M-M_{e}) \tag{1}\]1  
where \(\frac{\partial T}{\partial y}\) denotes the change in temperature of the cloth along with its thickness, \(y=0\) denotes the base layer of the cloth, \(y=1\) denotes the topmost layer of the cloth, \(T_{c}\) denotes the cylinder temperature, \(T_{f}\) denotes the temperature of the felt, \(T_{a}\) denotes the ambient temperature, \(T_{y}\) (at \(y=Y\)) denotes the cloth temperature at the subsequent thickness, \(k\) denotes the thermal conductivity of the drying material, \(\lambda\) denotes the latent heat of water, \(\frac{\partial w}{\partial t}\) denotes the change of mass of water in the drying material on time, \(h_{cs}\) denotes the heat transfer coefficient of the cylinder to drying sheet (at the base layer), \(h_{sat}\) denotes the heat transfer coefficient of the drying sheet to air (at the top surface), \(h_{sf}\) denotes the heat transfer coefficient from sheet to felt. For more simplicity in the calculation, assumptions for the computational model are as follows:  
1. There is no heat loss between the cylinder and the ambient environment;
2.