# Heat Storage as a Way to Increase Energy Efficiency and Flexibility of NPP in Isolated Power System
## 2 Materials and Methods
### Mathematical Model of PCM Melting and Solidification Processes  
A mathematical model of heat exchange in the accumulator with the PCM was compiled. The mathematical model was used to determine the PCM parameters (temperature and fraction) and outlet temperature of heat transfer fluid at every moment in the processes of charge. The objective functions are the values of \(T(\tau)\)--PCM and heat transfer fluid temperature, \(f(\tau)\)--fraction of liquid phase of the PCM. Simulation was performed by the finite difference method. This method is widely used for modeling non-stationary thermal processes. It is the basis of software products for modeling, for example, ANSYS [86]. The objective functions were determined by the linear programming method. Using iterative calculation, adjusted values were determined at each iteration step. The following assumptions were accepted in the modeling process:  
* Heat transfer is carried out only by thermal conductivity (adjusted for convection using the effective thermal conductivity coefficient for the liquid phase);
* The properties of the PCM are isotropic throughout the considered volume;
* The phase transition occurs at a constant temperature (isothermal phase transition);
* The upper, lower, and right boundary of the computational domain--adiabatic walls;
* The density of the PCM remains constant with changes in temperature and during the melt-solidification process. * The physical features of the melting process of various PCMs are not taken into account. In the process of simulation, the studied area of space was considered as a grid of elementary cells. Each cell was assigned a coordinate: \(j\)--in height and \(i\)--in radius. The scheme of partitioning the studied area of space into elementary cells is shown in Figure 5.  
Figure 5: Grid of the calculation area. The studied area was the space around one straight section of one heat exchange element. Because the problem of non-stationary heat exchange was being considered, the values of the objective functions had to be determined at each time step. For each time step \(k\), a finite difference grid was constructed with the values of the objective functions accordingly. The calculation of heat exchange processes in a heat accumulator with the PCM was based on the solution to a Stefan problem of a phase transition. Therefore, the main equation in the mathematical model was the enthalpy formulation of a Stefan problem in cylindrical coordinates (in partial derivatives):  
\[\rho c\frac{\partial h}{\partial t}=\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)+\frac{1}{r}\ \frac{\partial}{\partial r}\left(\lambda\frac{\partial T}{\partial r}\right)+ \gamma\rho\frac{\partial f}{\partial t} \tag{7}\]  
where \(\rho\)--density of a PCM kg/m\({}^{3}\), \(c\)--mass heat capacity of a PCM, kJ/(kg\({}^{\circ}\)C), \(t\)--time step, s; \(h\)--enthalpy of a PCM, kJ/kg; \(z\)--coordinates of the height, m \(r\)--coordinate of the radius; \(\lambda\)--thermal conductivity of a PCM, W/(m\(\cdot\)K); \(T\)--temperature, \({}^{\circ}\)C;\(f\)--liquid phase of the PCM, relative units; \(\gamma\)--PCM melting heat, kJ/kg. Based on Equation (7), the parameters of all cells of the computational domain were calculated at each time step. As a result of approximation by finite differences, Equation (7) was transformed to the form:  
\[ \begin{array}{l}c\rho\left(T_{ij}^{k+1}-T_{ij}^{k}\right)+\gamma\rho\left(f_ {ij}^{k+1}-f_{ij}^{k}\right)\\ =r\frac{1}{r_{ij}^{2}}\left[\lambda_{i+0,5j}^{k+1}\left(T_{i+1j}^{k+1}-T_{i,j} ^{k+1}\right)+\lambda_{i-0,5j}^{k+1}\left(T_{i-1,j}^{k+1}-T_{i,j}^{k+1}\right) \right]\\ +r\frac{1}{h_{z}^{2}}\left[\lambda_{i+0,5j}^{k+1}\left(T_{i+1}^{k+1}-T_{i,j}^ {k+1}\right)+\lambda_{ij-0,5}^{k+1}\left(T_{i,j-1}^{k+1}-T_{i,j}^{k+1}\right) \right]\end{array}