## 2 Heat Transfer in PCM-Enhanced Envelopes  
Modern, dynamically operating roofs and attics typically include high levels of thermal insulation, advanced ventilation, radiant barriers, and latent heat storage components that boost thermal performance in a variety of environmental conditions. In addition to roof and attic geometries, optical characteristics of roof surface material and eventual location of radiant barrier, material thermal conductance, thickness, density, and heat-storage capacity of PCM-enhanced products are key design factors used by roof designers today. The scale of thermal behavior improvements attributable to PCM-enhanced building envelopes depends on the amount of PCM used and the physical properties of the PCM, climate conditions, and the design of the building. Therefore, a comprehensive numerical analysis of thermal and energy effects in buildings using PCMs is essential for an evaluation of their potential energy benefits. Several numerical models, developed for system-scale thermal simulations of PCM-enhanced envelopes, are available today to capture details of the heat transfer process [29; 30; 31]. However, more streamlined algorithms, are also frequently preferred, because they can rapidly generate estimates of PCM performance. Simplified theoretical models of materials with temperature-dependent specific heat can be used for analysis of phase transition processes in most common materials [32; 33; 34]. In homogeneous materials with temperature-dependent specific heat, the one-dimensional heat transport process can be described by the following equation, where h is the enthalpy density, which represents a heat term that includes energy associated with the phase-change process:  
\[\frac{\partial}{\partial t}(\rho h)=\frac{\partial}{\partial x}\left[k\frac{ \partial T}{\partial x}\right] \tag{1}\]  
The terms \(\rho\) and \(k\) are material density and thermal conductivity and \(T\) is temperature. The heat flux, \(q\), across a material thickness \(x\) is given by Equation (2):  
\[q(x,t)=-k\frac{\partial T(x,t)}{\partial x} \tag{2}\]  
After consideration of a constant pressure, the enthalpy derivative over the temperature represents the effective heat capacity, with phase change energy being one of the components:  
\[c_{eff}(T)=\frac{\partial h}{\partial T} \tag{3}\]For most PCM-enhanced insulations and composite materials, variations of enthalpy with temperature depend (to some extent) on the direction of the considered heat flow process and are different for melting and solidification. Therefore, in the case of materials containing PCM, theoretical models utilizing the temperature-dependent specific heat, represented by a unique function of temperature, are only approximations of a real material thermal capacitance. PCMs can be used in building components either as condensed applications, usually as arrays of plastic or metallic foil containers, or mixed with carrier materials such as thermal insulation, plastics, gypsum, or concrete. For blends of thermal insulation with PCM, the effective heat capacity, \(c_{\mathit{eff}}\), can be expressed as follows, using a simplified formula:  
\[c_{\mathit{eff}}=(1-\alpha)c_{\mathit{ins}}+\alpha c_{\mathit{effPCM}} \tag{4}\]  
where \(\alpha\) denotes the weight fraction of PCM, \(c_{\mathit{ins}}\) the specific heat of the carrier insulation without PCM, and \(c_{\mathit{effPCM}}\) is the effective heat capacity of the PCM product. In the melted liquid phase, the effective heat capacity of PCM does not show significant temperature dependence. Therefore, it can be a good approximation represented as the sum of the two following terms:  
\[c_{\mathit{effPCM}}(T)=c_{l}+\left[c_{\mathit{effPCM}}(T)-c_{l}\right] \tag{5}\]  
where \(c_{l}\) represents the temperature independent specific heat in the liquid state. For a material to be used as a thermal insulation, the apparent thermal conductivity, \(k_{a}\), is the most important property. For a given insulation material, \(k_{a}\) is a function of density, \(\rho\), and temperature, \(T\).