## 3 Thermal Balance and Time Constant of a Solid Slab Containing Phase Change Materials (PCM)
By integrating Equation (13) with respect to time, over the time interval [\(t_{1}\),\(t_{2}\)], and interchanging the integration order, one obtains:  
\[Q(0,t_{1},t_{2})=\int\limits_{t_{1}}^{t_{2}}q(0,t)dt \tag{14}\]  
\[Q(0,t_{1},t_{2})=\frac{1}{R_{u}}\int\limits_{t_{1}}^{t_{2}}[T_{i}-T_{e}]dt+ \iint\limits_{0}^{L}\limits_{t_{1}}^{t_{2}}\rho\frac{\partial t_{t}}{\partial t }\cdot\frac{R_{x-e}}{R_{u}}dtdx \tag{15}\]  
The time integral in the second term represents the enthalpy difference \(\Delta t\); it can be expressed as follows:  
\[\Delta h(x,t_{1},t_{2})=\int\limits_{t_{1}}^{t_{2}}\frac{\partial h}{\partial t }dt=\int\limits_{t_{1}}^{t_{2}}\frac{\partial h}{\partial T}\cdot\frac{ \partial T}{\partial t}dt=\int\limits_{T(x,t_{1})}^{T(x,t_{2})}c_{eff}(x,T)dT \tag{16}\]  
Integration of \(c_{eff}(x,T)\) over temperature can be conducted when the final temperature distribution in the slab is known. For particular heat transfer processes, it is possible to determine the enthalpy difference \(\Delta t\) without performing detailed simulations. Taking into account Equation (15), the total heat flow in time interval [\(t_{1}\),\(t_{2}\)] across the surface \(x=0\) is given by:  
\[Q(0,\ t_{1},t_{2})=\frac{1}{R_{u}}\int\limits_{t_{1}}^{t_{2}}\Delta T_{i-e}(t )dt+\Delta H_{s0} \tag{17}\]  
\[\Delta T_{i-e}(t)=T_{i}(t)-T_{e}(t) \tag{18}\]  
\[\Delta H_{s0}(t_{1},t_{2})=\int\limits_{0}^{L}\rho\Delta h(x,t_{1},t_{2}). \frac{R_{x-e}}{R_{u}}dx \tag{19}\]  
\(\Delta H_{s0}\) represents that part of the total enthalpy difference in a slab which is due to the heat flow across the surface \(x=0\). For cyclic processes, time integration is over an integral number of cycles, \(\Delta H_{s0}=0\), and total heat flow depends on the mean ambient temperature difference. Time constant expresses the delay of the transient component of the heat flux in a building envelope assembly (generated by a rapid thermal excitation) [34; 38; 39]. In the case of PCM-enhanced building assemblies, the knowledge of time constants helps for optimizing their design from a whole building thermal response perspective and in the management of the energy demand, which are essential for building interaction with a power grid. Time constants can be utilized in the whole-building energy simulations as well. To estimate the time constant of a slab containing PCM, we start from the theoretical determination of the heat flux, \(q(t)\), across a unit surface area of the slab at the cold side, after a step change in temperature on the opposite hot side (as a function of time t). As derived below, it can be approximately described as a simple relaxation process:  
\[q(t)=-\frac{\Delta T}{R}\bigg{[}1-exp\bigg{(}-\frac{t}{\tau_{s}}\bigg{)} \bigg{]} \tag{20}\]  
The total heat flow in time interval [0,\(t\)] is given by:  
\[Q(t)=-\frac{\Delta T}{R}\bigg{[}t-\tau_{s}\bigg{[}1-exp\bigg{(}-\frac{t}{\tau _{s}}\bigg{)}\bigg{]}\bigg{]} \tag{21}\]For time \(t\) sufficiently long, as compared with \(\tau_{s}\), the steady state of heat flow is attained, and \(Q(t)\) can be approximated by:  
\[Q(t)\approx\frac{\Delta T}{R}(t-\tau_{s}) \tag{22}\]  
A comparison of Equations (17) and (22) yields:  
\[Q(0,t_{1},t_{2})=\int\limits_{t_{1}}^{t_{2}}q(0,t)dt \tag{14}\]0  
In earlier works of [PERSON] and [PERSON] (2018), a similar analysis was used for estimating the time constants for conventional non-PCM assemblies, for analysis of hot box test measurements. However, in the case of a PCM specimen, the time constant is strongly dependent on its enthalpy in the temperature interval. The thermophysical properties of the materials being discussed in this section are contained in Table 1, for a simple opaque building envelope that contains as a core a 0.14 m thick thermal insulation layer. The considered envelope contained the following, in series: (i) 0.013 m thick gypsum board; (ii) 0.140 m thick layer of cellulose insulation, or PCM-enhanced cellulose insulation; (iii) 0.013 m thickplywood board; and (iv) 0.013 m thick layer of exterior finish material.