## 4 Numerical Models for Phase Change Processes
### Latent Heat Modeling
#### 4.1.2 Temperature Transforming Methods (E-TTM)  
In the temperature methods, such as the one presented by [PERSON] [89], the energy equation assumes the basic form shown in Equation (2). Latent heat is modeled as an increase in the specific heat capacity within the temperature interval of the phase change. Such methods are stable in the convergence for continuous properties phase change processes, even if the convergence is difficult in the case of the reduction concerning the phase change interval [32]. As E-TTM needs a temperature interval to model the latent heat, the accuracy of the results is limited when applied to isothermal phase change [27]. The apparent heat capacity method uses the basic formulation for the specific heat capacity shown in Equation (4) [31], i.e.,  
\[C_{p}=C_{ap}=\left\{\begin{array}{ccc}C_{p,s}&if&T<T_{s}\\ \frac{\int_{T_{s}}^{T_{l}}C_{p}dT+L}{(T_{l}-T_{s})}&if&T_{s}<T<T_{l}\\ C_{p,l}&if&T>T_{l}\end{array}\right\} \tag{4}\]  
where \(C_{ap}\) is the apparent heat capacity and the subscripts \"s\" and \"1\" refer to the solid and liquid phases, respectively. The main issue existing in the apparent heat capacity method emerges when significant temperature changes occur (in a time step) in a control volume that is close to the phase change temperature. In such situations, latent heat is not accounted for if the temperature variation is greater than the phase change interval. Therefore, small-time steps are needed, with the implication of higher computational costs [31] to manage this condition. The effective heat capacity method [90] is an improvement on the apparent heat capacity method. The value of the effective heat capacity (\(\text{C}_{\text{eff}}\)) is obtained through the integration in the control volume, as shown in Equation (5), i.e.,  
\[C_{eff}=\frac{\int\text{C}_{ap}dV}{V} \tag{5}\]  
where V is the control volume. It is worth mentioning that the adoption of the effective heat capacity method allows the use of higher time steps and smaller phase change intervals when compared with the apparent heat capacity method. However, the integration necessary for the calculation of effective heat capacity represents an increase in computational cost, as pointed out in work [31]. In the heat integration or post-iterative method, the solution is achieved in two main steps. The first step consists of the resolution of the energy equation without considering the latent heat, and the second step consists of the addition of equivalent latent heat. In the control volumes where the temperature reaches or exceeds the phase change interval, the temperature is set back to the beginning of the phase change interval and the energy is adjusted. The modification consists of the addition of an amount of latent heat equivalent to the energy transferred from the beginning of the phase change interval [90]. As the latent heat is added, the temperature in the remaining control volumes is corrected from the cells in the phase change interval. When the amount of energy added in the control volume reaches the total latent heat, the solution is performed, usually considering only sensible heat [31].