# 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
 \tag{8}\]  
where indexes by the \(j\pm 0.5\) type denote the average value of the parameter between the cell with coordinates \(i,j\), and the adjacent cell (an example is shown in Figure 5); index \(k\) denotes the value calculated at the previous time step; index \(k+1\) denotes the value at the current time step; \(h_{r}\) is the radius step, m; \(h_{z}\) is the height step, m; \(\tau\) is the time step, s.  
Equation (8) is restricted by  
\[f_{ij}=\left\{ \begin{array}{l}0\ if\ T_{ij}<T_{mp}\\ 0\ldots 1\ if\ T_{ij}=T_{mp}\,,\text{relative units}\\ 1\ if\ T_{ij}>T_{mp}\end{array} \right. \tag{9}\]  
\[T_{ij}\left\{ \begin{array}{l}<T_{mp}\ if\ f_{ij}=0\\ T_{mp}\ if\ f_{ij}=0\ldots 1\,,\,^{\circ}\text{C}\\ >T_{mp}\ if\ f_{ij}=1\end{array} \right. \tag{10}\]  
where \(T_{mp}\)--melting point of the PCM, \({}^{\circ}\)C.  
Boundary conditions:  
\[T_{ij}=\left\{ \begin{array}{l}T_{tube}\ if\ i=0\\ T_{i-1,j}\ if\ i=K_{R}\\ T_{i,j-1}\ if\ j=K_{Z}\\ T_{i,j+1}\ if\ j=0\end{array} \right.\,,\,^{\circ}\text{C} \tag{11}\]  
where \(T_{tube}\)--temperature of the tube wall, \({}^{\circ}\)C.  
System (10) sets the temperature or conditions at the boundaries of the computational domain: the left boundary is the wall temperature; the right, lower, and upper boundaries are adiabatic walls. The values of \(T_{ij}\) and fij at step \(k\) are the initial values for calculating the values of \(T_{ij}\) and \(f_{ij}\) at a step k + 1. Values \(T_{ij}\) and fij at step \(k\) = 0 are the initial data of the simulation. Adjustment for heat exchange by convection inside the PCM was made using the coefficient of effective thermal conductivity:  
\[\lambda_{eff}=\lambda_{l}cRa^{m},W/(\text{m}\cdot\text{K}) \tag{12}\]  
where \(\lambda_{l}\) is the thermal conductivity of the PCM in the liquid state, \(W/(\text{m}\cdot\text{K})\); \(c^{\prime}=0.15\) and \(n^{\prime}=0.25\) are empirical dimensionless coefficients; \(Ra\) is the dimensionless Rayleigh number. This kind of estimation has been used successfully in several works [87; 88; 89]. In Article [87], to determine the Nusselt number (characterizing the ratio between heat transfer due to convection and thermal conductivity) for a vertical cylinder, the following values were described: \(c=0.1\) and \(n=1/3\) for \(Ra=10^{9}...10^{13}\); \(c=0.53\) and \(n=1/4\) for \(Ra=10^{4}...10^{9}\). Article [88] gives the values \(c=0.05\); \(n=0.25\); the range of Rayleigh numbers is not specified. In Article [89], the values for the vertical cylinder are as follows: \(c=0.15\) and \(n=0.34\) for Ra = \(10^{2}...10^{6}\); \(c=0.15\) and \(n=0.25\) for \(Ra=10^{2}...10^{7}\); \(c=0.28\) and \(n=0.25\) for \(Ra=10^{9}...10^{12}\). In our case, the \(Ra\) is in the range \(10^{6}...10^{7}\). Indicator \(n\) should be taken equal to 0.25--this statement can be considered unambiguous for all ranges of Rayleigh numbers because all the authors of the listed works converge in it (the exception is the range of \(Ra\) numbers equal to \(10^{9}...10^{13}\) in Article [87]). This situation is ambiguous about the value of indicator \(c\), about which the listed authors offer various options. It was decided to use \(c=0.15\) for the \(Ra=10^{2}...10^{6}\) and \(10^{2}...10^{7}\), according to the authors of the paper [89]. This paper most accurately describes a case similar to our study, namely, the natural convection in the PCM around a vertical pipe with a heat transfer fluid. Heat exchange between the wall and the heat transfer fluid was determined using the heat transfer coefficient from the tube wall to water (expressions (13) and (14))  
\[q=(T_{tube}-T_{w})a_{H2O} \tag{13}\]  
where \(T_{tube}\)--tube wall temperature, \({}^{\circ}\)C; \(T_{w}\)--water temperature, \({}^{\circ}\)C; \(a_{H2O}\)--coefficient of heat transfer from the tube wall to water, W/(m\({}^{2}\cdot\)K).