# Numerical Investigation of Latent Thermal Storage in a Compact Heat Exchanger Using Mini-Channels
## 2 Materials and Methods
### Simulation Model  
Three-dimensional modeling of the transient phase change is essential to capture the effect of the axial HTF temperature variations since such an effect is missing in two-dimensional modeling. The transient phase change simulations were carried out in three dimensions for the domain specified in Figure 1, using ANSYS Fluent 19.5. In the modeling of dry air flow, the following conditions were considered:  
* Incompressible laminar air flow;
* Constant air properties along the channel length;
* Negligible effects of viscous dissipation. In modeling solid-liquid phase change, the volume change was neglected and natural convection in the constrained melting was modeled via the Boussinesq approximation. The governing mechanism within the phase change is expressed in Equations (1)-(3):  
\[\
abla.\mathrm{V}\;=\;0 \tag{1}\]  
\[\rho\bigg{(}\frac{\partial\mathrm{V}}{\partial\mathrm{t}}+(\
abla.\mathrm{V}) \mathrm{V}\bigg{)}\;=\;\;\rho\mathrm{g}-\
abla\mathrm{P}+\
abla.(\tau)+ \mathrm{S} \tag{2}\]  
\[\rho\bigg{(}\frac{\partial\mathrm{H}}{\partial\mathrm{t}}+\
abla.(\mathrm{V} \mathrm{H})\bigg{)}\;=\;\
abla.(\mathrm{k}\
abla\;\mathrm{T}) \tag{3}\]  
An enthalpy-porosity method is used to model the phase change, adding a source term to the momentum equation, defined by the Carmen-Kozeny term shown by Equation (4). \[\mathrm{S}\;=\;-\frac{(1-\lambda)^{2}}{\bigg{(}\lambda^{3}+\varepsilon\bigg{)} }\mathrm{A}_{\mathrm{mushy}}(\mathrm{V}) \tag{4}\]  
Here, \(\lambda\) is the porosity factor ranging from zero to one and \(\mathrm{A}_{\mathrm{mushy}}\) is the mushy parameter, which is an arbitrary number ranging from \(10^{5}\) to \(10^{8}\), representing the convective intensity of the flow within the mushy zone. The lower the mushy numbers, the stronger convective flow is predicted within the mushy zone. In this study, the mushy parameter was set to \(10^{8}\) for both the constrained melting and solidification simulations since it was previously shown to give an appropriate fit to experimental data [25; 26; 27]. The porosity is incorporated into the sensible and latent energy through Equations (5)-(7). \[\mathrm{H}\;=\;\mathrm{h}+\;\lambda\mathrm{L} \tag{5}\]  
\[\mathrm{h}\;=\;\;\mathrm{h}_{\mathrm{ref}}+\int_{\mathrm{T}_{\mathrm{ref}}}^{ \mathrm{T}}\mathrm{C}_{\mathrm{p}}\mathrm{d}\mathrm{T} \tag{6}\]  
\[\Delta\mathrm{H}\;=\;\lambda\mathrm{L}\rightarrow\left\{ \begin{array}{cc} \lambda\;=\;0&\mathrm{if}\;\mathrm{T}<\mathrm{T}_{\mathrm{solidus}}\\ \lambda\;=\;\frac{\mathrm{T}-\mathrm{T}_{\mathrm{solidus}}}{\mathrm{T}_{ \mathrm{liquidus}}-\mathrm{T}_{\mathrm{solidus}}}&\mathrm{if}\;\mathrm{T}_{ \mathrm{solidus}}<\mathrm{T}<\mathrm{T}_{\mathrm{liquidus}}\\ \lambda\;=\;1&\mathrm{if}\;\mathrm{T}>\mathrm{T}_{\mathrm{liquidus}}\end{array} \right. \tag{7}\]  
A simple algorithm was used in the coupling of pressure and velocity. The second-order upwind scheme for discretization of momentum and energy equations and the PRESTO' scheme for pressure correction were used. The first-order implicit scheme was employed with fixed time steps for time discretization. The convergence criterion for the continuity and momentum equations was set to \(10^{-5}\) and \(10^{-8}\), respectively. A mesh independency study was carried out to select the element size for the final grids of air and PCM. In the beginning, the simulations were carried out for the hydrodynamic laminar air flow in a single channel, and the pressure drops along the channel length in different grids were compared. The element size ranged from 0.4 to 0.1 mm and 4 to 1 mm in the cross-sectional plane and along the channel length, respectively. Figure 4a shows the pressure drop along the length channel-2 for the cases 1-4 with the aforementioned ranging element size. As observed in Figure 4a, the simulated pressuredrop values vary insignificantly from case 3 to case 4. The mesh independency analysis proceeded for the PCM domain with the element sizes specified in cases 3 and 4 for the air domain.