## 2 Simulation Models
### Detailed Radiant Floor Thermal Modelling  
Transient, two-dimensional modelling is required to simulate the thermal behaviour of embedded pipe structures [37]. The well-known response factor method [38] is applicable [39; 40] for these purposes, since it delivers high calculation accuracy while ensuring processing speed by minimising the number of computational operations required.  
The need to couple the heating to the building model and simulate control strategies entailed the use of a small time step in the simulations. A 15 min time step was defined for the response factor method used here to simulate the transient thermal response in radiant floorings. That the resulting radiant flooring model could be coupled to the simplified building model described in Section 2.2 with no need for iteration or matrix inversion also contributed to lowering computational costs.  
The response factor method as initially proposed by [PERSON] and [PERSON] [41; 42] is described in detail in [43]. In essence, it consists of obtaining the heat fluxes across three surfaces of a radiant floor (top, bottom, and pipe) over time in response to a unitary triangular temperature pulse on each surface, generating as output three sets of responses per surface excited. Such heat fluxes are known as response factors and must be determined analytically or numerically. Here, the widely used international ANSYS [44] software was applied to that end, performing the calculations with the finite element method, which is a detailed numerical method for solving the partial differential equations that govern transient two-dimensional heat conduction across radiant floors. ANSYS APDL derives  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Jan** & **Feb** & **Dec** \\ \hline Chengdu (Ch) & 451 & 359 & 400 \\ Istanbul (TK) & 444 & 402 & 363 \\ London (UK) & 467 & 422 & 429 \\ Madrid (SP) & 452 & 364 & 434 \\ Tokyo (JP) & 423 & 367 & 352 \\ \hline \end{tabular} \({}^{*}\)Source: author calculations using data drawn from Meteororm [35], a commercial database.  
\end{table}
Table 1: Cities with a number of heating-degree days at base T = 20 \({}^{\circ}\)C similar to Madrid’s \({}^{*}\).  
the surface distribution of heat flux on each surface of the radiant floor. It then internally calculates the averaged heat fluxes on each surface weighting the node values by the amount of element surface area associated with each node. These averaged heat fluxes are the response factors.  
The response factors calculated were then used to calculate the actual heat fluxes in each surface of the radiant floor based on the superposition principle. Although a more detailed explanation is given in Appendix A, Equation (1), the general expression applied to calculate the heat flux across the three surfaces of the radiant floor is shown below:  
\[\dot{q}_{s,k,floor}(n)=f_{1,k}(0)\ T_{s,1}(n)+f_{2,k}(0)\ T_{s,2}(n)+f_{3,k}( 0)\ T_{s,3}(n)+h_{k} \tag{1}\]  
where k denotes the top (1), bottom (2) or pipe (3) surface; n the simulation time step at issue; \(\dot{\mathrm{f}}_{\mathrm{j,k}}\) the response factor for surface k when a unitary triangular pulse is applied to surface j; and \(\mathrm{h}_{\mathrm{k}}\) a term grouping the information on the preceding time steps for surface k.