## 3 Thermal Balance and Time Constant of a Solid Slab Containing Phase Change Materials (PCM)  
Thermal design and optimization of building envelopes containing PCM require an in-depth performance analysis. This is usually the first step in the process of determining the location, amount, and active temperature region for the PCM. This analysis frequently includes decisions about the following: (i) sequence of materials, (ii) location of PCM, (iii) amount of PCM, (iv) expected time lag associated with heat storage, and their impact on energy use. The determination of the optimal phase transition temperature range is an important step in the optimization, since this leads to the selection of a specific PCM product and its placement. To illustrate dynamic thermal performance of a building component containing PCM, the temperature-dependent specific heat model is utilized. An integral formula, for the total heat flow in finite time interval, across the surface of a slab of a PCM enhanced material, is derived. Theoretical analysis is performed to investigate heat transfer through a light-weight envelope assembly with PCM-enhanced insulation under cyclic external temperature excitations. In this analysis, a plane slab of thickness, \(L\), composed of materials containing PCM was assumed. Linear boundary conditions at the internal (\(x=0\)), and external (\(x=L\)) surface can be expressed in the following form:  
\[q(0,t) =\frac{1}{R_{si}}[T_{i}(t)-T(0,t)] \tag{7}\] \[q(L,t) =\frac{1}{R_{se}}[T(L,t)-T_{e}(t)] \tag{8}\]  
where \(T_{i}\) and \(T_{e}\) are ambient temperatures and \(R_{si}\) and \(R_{se}\) are surface film resistances, assuming for simplification that \(R_{se}\) is constant (not dependent from exterior wind conditions). Let us denote, \(R_{i-x}\) and \(R_{x-e}\) as the resistances for heat transmission from the point \(x\) in the slab to the internal and external environment, respectively, and \(R_{u}\) as the total resistance for heat transmission through the envelope assembly. With conductivity along the assembly's thickness represented by the function of \(k(x)\), the resistances can be denoted as follows:  
\[R_{i-x}=R_{si}+\int_{0}^{x}\frac{dx}{k(x)} \tag{9}\]  
\[R_{x-e}=\int_{x}^{L}\frac{dx}{k(x)}+R_{se} \tag{10}\]  
\[R_{u}=R_{si}+\int_{0}^{L}\frac{dx}{k(x)}+R_{se} \tag{11}\]  
Multiplying Equation (1) by the function \(R_{x-e}\) and integrating with respect to \(x\) over thickness of the slab, we obtain the following formula:  
\[ \begin{array}{c}\int\limits_{0}^{L}\rho\cdot\frac{\partial h}{ \partial t}R_{x-e}dx=\int\limits_{0}^{L}\frac{\partial}{\partial x}\left[k \frac{\partial T}{\partial x}\right]R_{x-e}dx=\\ =\left[k\frac{\partial T}{\partial x}\right]R_{x-e}+\int\limits_{0}^{L}\frac {\partial T}{\partial x}dx=-R_{se}q(L)+R_{0-e}q(0)+\left[T(L)-T(0)\right]\end{array}  \tag{12}\]  
Temperatures \(T(0)\) and \(T(L)\) can be eliminated using Equations (7) and (8), giving the following expression for the heat flux at the internal surface:  
\[q(0)=\frac{1}{R_{u}}(T_{i}-T_{e})=\int\limits_{0}^{L}\rho\frac{ \partial h}{\partial t}\frac{R_{x-e}}{R_{u}}dx \tag{13}\]  
An analogous formula for \(q(L)\) is obtained upon multiplying Equation (1) by \(R_{i-x}\) and integrating by parts. The effect of time variations of enthalpy, \(h\), along thickness of the slab, on the heat flux across the surface \(x=0\), depends, thus, on the factor \(R_{x-e}/R_{u}\), which takes on comparatively high values, close to 1, in the vicinity of that surface and low values, close to 0, in the vicinity of the opposite surface \(x=L\). This means that only time variations of enthalpy near the surface considered, play an important role, whereas the effect of variations near the opposite surface may be small. Now, assume that densities and conductivities are temperature independent, and thus constant in time. The last assumption should be understood as an approximation; especially in the case of insulation materials (see Equation (6)).