14 Advanced topic: Close-contact melting
Complex phase-change processes occur, whenever additional physical effects couple to the phase-change. Whenever a heat source is forced onto a phase-change material (PCM), such that phase change occurs in a micro-scale channel between both, we speak of close-contact melting (CCM). CCM is relevant for specific manufacturing processes, e.g., hot wire cutting, yet also model nuclear melt down or the dynamics of certain cryo-robots.
14.1 Physical situation
We distinguishe between internal and external close-contact melting, see Figure 14.1. External CCM refers to a situation, in which the heat source exerts a force on a PCM, and consequently sinks into the melting material. Internal CCM refers to a situation, in which the PCM exerts its own weight onto a heat source. In both cases, the weight leads to a micro-scale melt film, in which heat transfer and phase-change occur.
In this section, we will focus on external CCM also all arguments are easily transferable. We consider a situation, in which the heat source is sinking at constant speed \(V\) into the PCM. With respect to a reference frame moving at speed \(V\) (Lagrangian perspective), the temperature profile will be stationary. The position of the phase interface will consequently be constant at \(x=\delta\), see Figure 14.2.
14.2 First modeling attempt
The temperature profile within the liquid domain referred to as the melt film will hence be subject to the stationary heat equation
\[ 0 = \alpha \partial_{xx} T \quad \text{for} \quad 0 \leq x \leq \delta, \]
whereas the ice is moving towards the interface at speed \(V\):
\[ -V \partial_x T = \alpha \partial_{xx} T \quad \text{for} \quad \delta \leq x < \infty. \]
Considering the boundary conditions \(T(0,t)=T_0 > T_m\), \(T(\delta,t)=T_m\) and \(T(\infty,t)=T_{ini} \leq T_m\) yields the following solutions
\[ \begin{align} T(x) &= \frac{x}{\delta} (T_0 - T_m) \quad &\text{for} \quad 0 \leq x \leq \delta \\ T(x) &= (T_m - T_{ini}) e^{- \frac{V}{\alpha} (x - \delta)} + T_{ini} \quad &\text{for} \quad \delta \leq x < \infty \end{align} \]
Similar to the Stefan problem discussed in Chapter 12, we have the temperature profile fully specified except for the position of the interface. This time, however, interface position is constant over time. As before, we formulate the Stefan condition, which derives from local energy balance at the interface.
\[ \rho_s L V = - \left. \kappa_l \partial_x T \right|_{x=\delta^{(-)}} + \left. \kappa_s \partial_x T\right|_{x=\delta^{(+)}} \]
Note, that subscripts indicate phases, which reflects that material parameters, such as thermal conductivity \(\kappa\) and density \(\rho\) might change across the interface. Note also, the superscripts indicate at which side of the interface the temperature gradient is evaluated. Substitution of the parametrized temperature profiles yields
\[ \rho_s L V = \frac{\kappa_l}{\delta} (T_0 - T_m) - \kappa_s \left( (T_m - T_{ini}) \frac{V}{\alpha} + T_{ini} \right) \]
which can be solved for the interface position:
\[ \delta = \frac{\kappa_l (T_0 - T_m)}{\rho_s L V (1 + Ste_s) + \kappa_s T_{ini}} \]
Here, \(Ste_s\) denotes the Stefan Number with respect to the material parameters of the solid phase. This means, that given an experimental obsveration of the sinking velocity \(V\), we can determine the melt film thickness \(\delta\). An example can be found in the following plot:
14.3 Force-aware CCM model
Inspection of the situation leads to the conclusion that the modeling quality is bad, since the forces haven’t been considered. We need to consider the force equilibrium between exerted weight by the heat source \(F\) and the force due to hydrodynamic pressure within the melt film
\[ \oint p \, d \sigma = F. \]
In order to access the pressure, however, we need to resolve micro-scale fluid dynamic processes in the melt film, see Figure 14.4.
Processes within the melt film are subject to the incompressible Navier-Stokes-Fourier system
\[ \begin{eqnarray*} \nabla \cdot \mathbf u &=& 0\\ \partial_t \mathbf u + (\mathbf u \cdot \nabla) \mathbf u &=& -\frac{1}{\rho} \nabla p + \nu \triangle \mathbf u\\ \partial_t T_l + (\mathbf u \cdot \nabla) T_l &=& \alpha_l \triangle T_l \end{eqnarray*} \]
whereas in the solid phase it suffices to consider the heat equation
\[ \begin{eqnarray*} - \mathbf V \partial_z T_s &=& \alpha_s \partial_{zz} T_s. \end{eqnarray*} \]
The water-ice interface is subject to no-slip boundary conditions, as well as an inflow according to the phase-change. It is furthermore at melting temperature \(T_m\) and subject to the Stefan condition.
The heat source surface likewise is subject to a no-slip boundary condition. Furthermore, we are facing one of two conditions
- a given temperature at the surface, or
- a given heat flux at the surface.
All in all this means that the unknowns in the system are given by velocity field \(\mathbf u\), temperature \(T\), and pressure \(p\) in the melt film, as well as temperature field in the solid material, sinking velocity \(\mathbf V\) and melt film thickness \(\delta\). We need two additional informations in order to solve this system, which will be
- the Stefan condition, and
- the force equilibrium.
The second part of this section will follow soon. For now, we refer to your hand-written lecture notes.