# Energy Saving Potential of PCMs in Buildings under Future Climate Conditions
## 3 Materials and Methods
### The Simulation Model
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **RubBitHERMPCM** & **DuPonEnergainPCM** & **BioPCM** \\ \hline Conductivity (W/m k) & 0.20 & 0.18 & 0.20 \\ Thickness (m) & 0.01 & 0.0053 & 0.010186 \\ Density (Kg/m\({}^{3}\)) & 800 & 855 & 860 \\ Specific heat (J/kg k) & 2000 & 2500 & 1970 \\ Latent heat (kJ/kg) & 170 & 130 & 219 \\ Melting point (\({}^{\circ}\)C) & 25 & 21.6 & 29 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Physical properties of the PCMs. Dynamic energy simulations were conducted using EnergyPlus (8.0, USA Department of Energy, Washington, DC, USA)software for three climate zones (Seoul, Tokyo, and Hong Kong) in two different climatic conditions: humid tropical and subtropical. The simulations were run for one year during the present (2017), 2020, 2050, and 2080 using the conduction finite difference algorithm (CondFD), which discretizes the building envelope (ceilings, floors, and walls) into different nodes (the user can select Crank-Nicholson or fully implicit discretization) and numerically solves the appropriate heat transfer equations using the finite difference method [18; 48; 49]. The current study used fully implicit discretization. To consider specific heat changes during the PCM simulation, the CondFD method is paired with the enthalpy temperature function, which reads user inputs of the enthalpies at different temperatures. The enthalpies in each node get updated in each iteration and are then used to develop the equivalent specific heat at each time step. Assuming fixed state conditions (Figure 3), thermal conduction for the control volume \"\(i-1\)\" to control volume \"\(i\)\" can be computed by [PERSON]'s equation. During a time-step, the increase in the enthalpy by the control volume \"\(i\)\" is summed by Equation (2). Equation (1) is included in Equation (2) to obtain Equation (3) and to expression about the specific changes in enthalpy as a temperature change multiplied by the specific heat capacity. Based on the user input data and Equation (4) the specific heat can be updated in each iteration using the Enthalpy-Temperature function. Through these processes, the node enthalpies are updated at each time step, and the changing characteristics of PCM can be accurately analyzed. In the CondFD algorithm, all elements are divided or discretized using Equation (5), which depends on a space discretization constant c, the thermal diffusivity of the material \(\alpha\), and the time step. The default space discretization value of three (equivalent to a Fourier number (Fo) of \(1/3\)) has been used [19; 48]. \[\underset{i}{\text{q}}^{j+1}=-\frac{k\left( \begin{array}{ccc}j+1&j+1\\ T&-&T\\ i&&i-1\end{array} \right)}{\Delta x}-\frac{k\left( \begin{array}{ccc}j+1&j+1\\ T&-&T\\ i+1&&i\end{array} \right)}{\Delta x}=-k\ \frac{\left( \begin{array}{ccc}j+1&j+1\\ T&-&T\\ i+1&&i-1\end{array} \right)}{\Delta x} \tag{1}\]  
\[\left( \begin{array}{ccc}\frac{\rho\text{Ar}\ \left(\begin{array}{ccc}j+1&j\\ h&-&h\\ i&i\end{array}\right)}{\Delta x}\\ \\ \end{array} \right)=-\left( \begin{array}{ccc}j+1&j\\ q&-&q\\ i&i\end{array} \right) \tag{2}\]  
Figure 1: Building model (DOE Department Of Energy medium office building). \[C_{p}=\frac{\left(h_{i}^{j+1}-h_{i}^{j}\right)}{\left(T_{i}^{j+1}-T_{i}^{j}\right)} \tag{4}\]  
\[\Delta x=\sqrt{C\times\alpha\times\Delta t}=\sqrt{\frac{\alpha\times\Delta t}{ \text{Fo}}} \tag{5}\]  
To ensure model accuracy with the CondFD and simulation process, the simulation time step was set to \(1\) min and the space discretization to \(3\) min. The CondFD algorithm and PCM were verified and validated analytically, through comparative testing, and empirically by [PERSON] et al. (48,49). The PCM algorithm in EnergyPlus was validated against observed data from other researchers [28,40,42] and the experimental data of [PERSON] and [PERSON], who investigated the thermal performance of a PCM copolymer composite wallboard in a lightweight building in a full scale test [50].