## 2 Materials and Methods
### Mathematical Theory of Transfer  
The following assumptions are being made:  
1. In the cylindrical coordinate system, mass transfer from EO happens through one-dimensional diffusion along the (Oz) axis. 2. Under transient conditions, the diffusion process takes place with a constant diffusivity and flux. 3. Once the EO-PG formulation is mixed with the material, the concentration of EO on the surface of the material reaches an equilibrium value. 4. The concentration of EO within the material remains constant at the beginning of desorption. [PERSON]'s second law for diffusion in cylindrical coordinates can be formulated as [22]: \[\frac{\partial C}{\partial t}=\frac{1}{r}\times\left\{\frac{\partial}{\partial r }\left(r\times D_{r}\times\frac{\partial C}{\partial r}\right)+\frac{\partial }{\partial\theta}\left(\frac{D_{\theta}}{r}\times\frac{\partial C}{\partial \theta}\right)+\frac{\partial}{\partial z}\left(r\times D_{z}\times\frac{ \partial C}{\partial z}\right)\right\}\] (1) where \(C\) is the concentration of the diffusing species, \(t\) is time, \(r\) is the radial distance from the axis of the cylinder, \(\theta\) is the azimuthal angle, \(z\) is the axial coordinate, and \(D\) is the diffusivity of the species. As the dominant diffusion in the cylinder occurs in the height direction, which is significantly smaller than the diameter and lateral surface, the transfer of diffusion can be simplified and considered as one-dimensional. This simplifies the problem, and the diffusion equation can be expressed using [PERSON]'s second law:  
\[\frac{\partial C}{\partial t}=D_{z}\times\frac{\partial^{2}C}{\partial z^{2}} \tag{2}\]  
or  
\[D_{z}=D=cte \tag{3}\]  
According to the chronological conditions of monitoring at the following limits:  
\[ \begin{cases}t=0&C=C_{0}\quad 0\leq z\leq h\\ t>0&C=C_{t}\quad\quad z=h\end{cases}  \tag{4}\][PERSON] (1979) [23] derived an analytical solution to Equation (2) while taking into account the conditions specified in Equation (4). The resulting solution can be expressed as follows:  
\[C(z,t)=\frac{4}{\pi}\times\sum_{n=0}^{\infty}exp\left(-D\times\frac{\left(2n+1 \right)^{2}}{h^{2}}\times\pi^{2}\times t\right)\times cos\left(\frac{\left(2n+ 1\right)\pi}{2h}\times z\right) \tag{5}\]  
The total mass of essential oils in the PG at an instant t can be obtained by integrating the concentration \(C\) over the volume of the material. Since the diffusion is assumed to be one-dimensional, the concentration \(C\) is only a function of the axial coordinate \(z\), and the integration can be simplified to  
\[M_{t}=S\times\int_{0}^{h}C(z,t)dz \tag{6}\]  
where \(S\) is the cross-sectional area of the cylinder. The solution to these equations has an analytical solution that takes into account the boundary conditions and assumes an infinite desorption time, which leads to equilibrium. This solution can be expressed as follows:  
\[\frac{M_{t}-M_{\infty}}{M_{0}-M_{\infty}}=\frac{8}{\pi^{2}}\times\sum_{n=0}^{ \infty}\frac{1}{\left(2n+1\right)^{2}}exp\left(-D\times\frac{\left(2n+1 \right)^{2}}{h^{2}}\times\pi^{2}\times t\right) \tag{7}\]  
The variables used in the analytical solution include \(M_{0}\), which represents the initial mass of absorbed EO-PG; \(M_{t}\), which represents the mass of desorbed EO-PG at a given aging time; and \(M_{\infty}\), which represents the final mass at equilibrium. If we assume that \(M_{\infty}=0\) and take the first term of the series solution, Equation (7) can be simplified to  
\[\frac{M_{t}}{M_{0}}=\frac{8}{\pi^{2}}\times exp\left(\frac{-D\times\pi^{2} \times t}{h^{2}}\right) \tag{8}\]  
and if we assume that the term \((8/\pi^{2})\) is equal to 1, then Equation (8) simplifies to  
\[\frac{M_{t}}{M_{0}}=exp\left(\frac{-D\times\pi^{2}\times t}{h^{2}}\right) \tag{9}\]  
As part of our modeling study and taking into account the theoretical considerations already mentioned, we will focus on two parameters that are relevant to the desorption process: diffusivity (\(D\)) and flux (\(F\)).