# Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
## 2 Methods
### Numerical Model
Thus:  
\[\left(r\,\overline{D}_{d}\frac{\overline{c}_{d}}{\partial\tau}\right)_{i,j+0.5} ^{f-1}=r_{i,j+0.5}\left(\overline{D}_{d}\right)_{i,j+0.5}^{f-1}\frac{(\overline {c}_{d})_{i,j+1}^{f-1}-(\overline{c}_{d})_{i,j}^{f-1}}{h}=\left(r_{ij}+\frac{h} {2}\right)\frac{(\overline{c}_{d})_{i,j+1}^{f-1}-(\overline{c}_{d})_{i,j}^{f- 1}}{\overline{W}_{i,j+1}^{f-1}} \tag{22}\]  
\[\left(r\,\overline{D}_{d}\frac{\overline{c}_{d}}{\partial\tau}\right)_{i,j-0.5 }^{f-1}=r_{i,j-0.5}\left(\overline{D}_{d}\right)_{i,j-0.5}^{f-1}\frac{( \overline{c}_{d})_{i,j}^{f-1}-(\overline{c}_{d})_{i,j-1}^{f-1}}{h}=\left(r_{ij} -\frac{h}{2}\right)\frac{(\overline{c}_{d})_{i,j}^{f-1}-(\overline{c}_{d})_{i,j-1}^{f-1}}{\overline{W}_{i,j-1}^{f-1}} \tag{23}\]  
where the interval diffusion coefficients were assumed in the form of harmonic means of values corresponding to the basic nodes:  
\[\left(\overline{D}_{d}\right)_{i,j+0.5}^{f-1}=\frac{2\left(\overline{D}_{d} \right)_{ij}^{f-1}\left(\overline{D}_{d}\right)_{i,j+1}^{f-1}}{(\overline{D}_{ d})_{i,j}^{f-1}+(\overline{D}_{d})_{i,j+1}^{f-1}},\qquad\left(\overline{D}_{d} \right)_{i,j-0.5}^{f-1}=\frac{2\left(\overline{D}_{d}\right)_{i,j}^{f-1}\left( \overline{D}_{d}\right)_{i,j-1}^{f-1}}{(\overline{D}_{d})_{i,j}^{f-1}+( \overline{D}_{d})_{i,j-1}^{f-1}} \tag{24}\]  
and the diffusion resistances between the central node and the adjoining ones were the following:  
\[\overline{W}_{i,j+1}^{f-1}=\frac{0.5h}{(\overline{D}_{d})_{i,j}^{f-1}}+\frac{ 0.5h}{(\overline{D}_{d})_{i,j+1}^{f-1}},\qquad\qquad\overline{W}_{i,j-1}^{f-1}= \frac{0.5h}{(\overline{D}_{d})_{i,j}^{f-1}}+\frac{0.5h}{(\overline{D}_{d})_{i,j -1}^{f-1}} \tag{25}\]  
Using the mean differential quotient again, the following formula was obtained:  
\[\left[\frac{1}{\tau}\frac{\partial}{\partial t}\left(r\,\overline{D}_{d}\frac {\overline{c}_{d}}{\partial\tau}\right)\right]_{i,j}^{f-1}=\frac{1}{r_{i,j}^{k }}\left[\left(r_{ij}+\frac{h}{2}\right)\frac{(\overline{c}_{d})_{i,j+1}^{f-1} -(\overline{c}_{d})_{i,j}^{f-1}}{\overline{W}_{i,j+1}^{f-1}}+\left(r_{ij}- \frac{h}{2}\right)\frac{(\overline{c}_{d})_{i,j}^{f-1}-(\overline{c}_{d})_{i,j- 1}^{f-1}}{\overline{W}_{i,j-1}^{f-1}}\right] \tag{26}\]  
The same approach was used for the second term on the right-hand side of Equation (5), and after transformations, we obtained:  
\[\left[\frac{\partial}{\partial z}\left(\overline{D}_{d}\frac{\overline{c}_{d}}{ \partial z}\right)\right]_{i,j}^{f-1}=\frac{1}{k}\left[\frac{(\overline{C}_{d}) _{i+1,j}^{f-1}-(\overline{C}_{d})_{i,j}^{f-1}}{\overline{W}_{i+1,j}^{f-1}}+ \frac{(\overline{C}_{d})_{i-1,j}^{f-1}-(\overline{C}_{d})_{i,j}^{f-1}}{ \overline{W}_{i-1,j}^{f-1}}\right] \tag{27}\]while:  
\[\overline{W}_{i+1,j}^{f-1}=\frac{0.5k}{(\overline{D}_{d})_{i,j}^{f-1}}+\frac{0.5k} {(\overline{D}_{d})_{i+1,j}^{f-1}},\qquad\qquad\overline{W}_{i-1,j}^{f-1}=\frac{0.5k}{(\overline{D}_{d})_{i,j}^{f-1}}+\frac{0.5k}{(\overline{D}_{d})_{i-1,j}^{f- 1}} \tag{28}\]  
The final approximate form of the interval mass Equation (5) for the internal nodes (_i, j_) and the transition \(t^{f-1}\to t^{f}\) was the following:  
\[ \begin{array}{c}\frac{(\overline{c}_{d})_{i,j}^{f}-(\overline{c}_{d})_{i,j}^ {f-1}}{\Delta t}=\frac{\Phi_{i,i-1}}{\overline{W}_{i-j-1}^{f-1}}\big{[}( \overline{C}_{d})_{i,j-1}^{f-1}-\big{(}\overline{C}_{d})_{i,j}^{f-1}\big{]}+ \frac{\[PERSON]_{i,j+1}}{\overline{W}_{i,j+1}^{f-1}}\big{[}\big{(}\overline{C}_{d} \big{)}_{i,j+1}^{f-1}-\big{(}\overline{C}_{d}\big{)}_{i,j}^{f-1}\big{]}\\ +\frac{\[PERSON]_{i-1,j}}{\overline{W}_{i-1,j}^{f-1}}\big{[}\big{(}\overline{C}_{d} \big{)}_{i-1,j}^{f-1}-\big{(}\overline{C}_{d}\big{)}_{i,j}^{f-1}\big{]}+\frac{ \Phi_{i+1,j}}{\overline{W}_{i+1,j}^{f-1}}\big{[}(\overline{C}_{d})_{i+1,j}^{ f-1}-\big{(}\overline{C}_{d}\big{)}_{i,j}^{f-1}\big{]}\end{array}