## 7 Applications to polar-symmetric problems
### Melting of iced-water sphere  
Moulds for ice spheres are advertised for serious on-the-rock drinkers by asserting that the minimum surface-to-volume ratio means that the ice melts at a slower pace, preventing the drinks from getting watery too quickly. This seems questionable and, indeed, the problem is not well posed, as the amount of melted ice depends mainly on the heat transferred from the container to the environment. Here, we present a simpler problem, where a sphere of water ice melts in liquid water. We assume the heat transferred by convection to be negligible. The present problem with spherical symmetry, schematised in Figure 28, is analogous to the cylindrical one presented in Section 7.1. The sphere is initially at the melting temperature and the essential boundary condition \(T\left(R\right)=T_{1}>T_{\text{{\tiny{max}}}}\) is imposed, so that the initially solid sphere starts melting. Due to the spherical symmetry of the problem, it can be studied by considering only the coordinate \(r\) (we use the same symbol \(r\) for the radial coordinate in both cylindrical and spherical coordinate systems, the context ensuring the disambiguation). Figure 27: Interface position within the pipe. The energy balance in spherical coordinates and for isotropic heat conduction is  
\[\rho\frac{\partial}{\partial t}\big{(}cT\big{)}=k\bigg{(}\frac{\partial^{2}T}{ \partial r^{2}}+\frac{2}{r}\frac{\partial T}{\partial r}\bigg{)}, \tag{74}\]  
whereas [PERSON] law, phase-interface temperature condition, boundary and initial conditions read as in Eqs from (66) to (69). Since no analytical solution is available. We develop then the PSS approximation. Eq. (74) becomes  
\[\frac{\partial^{2}T_{\rm PSS}}{\partial r^{2}}+\frac{2}{r}\frac{\partial T_{ \rm PSS}}{\partial r}=0\,. \tag{75}\]  
By integrating and considering the boundary conditions, the temperature field  
Figure 28: Melting ice sphere. Figure 29: Interface position in the spherical melting example. \[T_{\text{PSS}}\left(r,t\right)=\frac{1-\frac{r_{\text{s}}^{\text{PSS}} \left(t\right)}{r}}{1-\frac{r_{\text{s}}^{\text{PSS}}\left(t\right)}{R}}\left(T _{1}-T_{\text{tran}}\right)+T_{\text{tran}},\quad r\in\Omega_{\text{iiq}} \tag{76}\]  
is obtained. Substituting in Eq. (66) and integrating it with the initial condition of Eqs (69) leads to the implicit relation for the time evolution of the phase-interface,  
\[\frac{k\left(T_{1}-T_{\text{tran}}\right)}{\rho L}t=\left(r_{\text{s}}^{\text{ PSS}}\right)^{2}\left(\frac{r_{\text{s}}^{\text{PSS}}}{3R}-\frac{1}{2}\right)-a^{2} \left(\frac{a}{3R}-\frac{1}{2}\right). \tag{77}\]  
For the numerical analysis, the same data used in the cylindrical example in Section 7.1 are adopted (see Table 1), so that the same value of Stefan number is obtained, high enough for considering the PSS approximation as a suitable reference for comparing the numerical results. Figure 29 shows the phase-interface position calculated through the PSS approximation and the XFEM simulations with 10- and 40-element discretisations. As for the cylindrical problem discussed in Section 7.1, the approximation with the finer spatial discretisation improves especially in the vicinity of the sphere centre. In this case, such behaviour is even more pronounced, as it can be evinced by comparing Figure 29 and Figure 22. In fact, the accuracy depends on the ratios between the areas of the surfaces (cylindrical or spherical) at the two nodes of one element; the cylindrical case has then a better behaviour than the spherical one, as the areas are proportional to the radii, instead of the squares of the radii. The temperature field at \(t=2400\) s, 4800 s, 7200 s, 9600 s is shown in Figure 30. Temperature fields evaluated through PSS approximation and XFEM simulation notably differ, if they are evaluated at the same time instant, because of the error in the phase-interface position. Still, the  
Figure 30: Temperature field within the spherical domain.