## 2 Simulation Models
### Coupling the Building and Radiant Floor Thermal Models
The boundary conditions in the flooring model were the top (T\({}_{s,1}\)), bottom (T\({}_{s,2}\)), and piping (T\({}_{s,3}\)) surface temperatures and the responses to the heat fluxes across those surfaces (q\({}_{s,1}\), q\({}_{s,2}\) and q\({}_{s,3}\), respectively). The boundary conditions in the simplified building model were the indoor temperatures in the adjacent dwellings (T\({}_{i}\)), the outdoor sol-air temperature (T\({}_{sol-air}\)), the top and bottom flooring temperatures (T\({}_{s,1}\) and T\({}_{s,2}\), respectively), and the heat flux at the capacitance node (\(\dot{Q}_{cr}\)). The responses in this model were also the heat fluxes at the top and bottom surfaces of the flooring (q\({}_{s,1}\) and q\({}_{s,2}\), respectively), as well as the indoor temperature in the space to be heated, i.e., the temperature of the capacitance node (T\({}_{cr,i}\)). As the figure shows, the two models were coupled via the heat fluxes at the top and bottom surfaces on the radiant flooring. The results of this coupling, i.e., the indoor environmental temperature T\({}_{cr,1}\) and the radiant floor surface temperatures T\({}_{s,1}\), T\({}_{s,2}\), and T\({}_{s,3}\) were found with no need for iteration or other numerical procedures. The values adopted for each pre-defined building parameter are given in Section 3. The heat flux at the top surface of the flooring, q\({}_{s,1,\,build}\), was calculated with Equation (2):  
\[\dot{q}_{s,1,\,build}=h_{cr,1}(T_{cr,i}-T_{s,1}) \tag{2}\]  
for which the capacitance node temperature T\({}_{cr,1}\) needed to be found. The procedure to determine the temperature for the boundary conditions listed earlier is described in [60] and in Appendix A. The excitation was taken to vary linearly at every simulation time step, a reasonable premise inasmuch as excitation data are normally given as discrete values spaced at time-step intervals. Applying continuity across the top surface of the flooring, the conduction heat flux obtained with the flooring model (Equation (1)) is equal to the convective-radiative heat flux found with the RC building model (Equation (3)):  
\[\dot{q}_{s,1,floor}=\dot{q}_{s,1,\,build} \tag{3}\]  
\[f_{1,1}(0)\,\,T_{s,1}+f_{2,1}(0)\,\,T_{s,2}+f_{3,1}(0)\,\,T_{s,3}+h_{1}=h_{cr,1}(T_{cr,i}-T_{s,1}) \tag{4}\]  
Substituting the expression for temperature \(T_{cr,i}\) obtained with the RC model for the building (Appendix A) and re-grouping terms in Equation (2) yielded the following simple algebraic equation:  
\[a_{1}\,\,T_{s,1}+b_{1}\,\,T_{s,2}+c_{1}\,\,T_{s,3}+d_{1}=0 \tag{5}\]  
Figure 2: Radiant flooring model coupled to the building RC model. where: \(b_{1}=f_{2,1}(0)\), \(c_{1}=f_{3,1}(0)\), \(a_{1}=f_{1,1}(0)+h_{\sigma,1}\)\((1-A_{1})\) and \(d_{1}=h_{1}-h_{\sigma,1}\)\(B_{1}\); A1 and B1 represent the known values at the current time step that depend on boundary conditions and the model parameters that define the building, i.e., heat capacity, C, and the thermal resistance between the space and the flooring surface. Explicit equations for A1 and B1, given in Appendix A, are omitted here for the sake of simplicity. Similarly, applying continuity across the bottom and pipe surfaces yields:  
\[f_{1,2}(0)\ T_{s,1}+f_{2,2}(0)\ T_{s,2}+f_{3,2}(0)\ T_{s,3}+h_{2}=h_{\sigma,2} (T_{i}-T_{s,2}) \tag{6}\]  
\[f_{1,3}(0)\ T_{s,1}+f_{2,3}(0)\ T_{s,2}+f_{3,3}(0)\ T_{s,3}+h_{3}=h_{c,3}(T_{w} -T_{s,3}) \tag{7}\]  
These equations can be re-grouped as shown below:  
\[a_{2}\ T_{s,1}+b_{2}\ T_{s,2}+c_{2}\ T_{s,3}+d_{2}=0 \tag{8}\]  
\[a_{3}\ T_{s,1}+b_{3}\ T_{s,2}+c_{3}\ T_{s,3}+d_{3}=0 \tag{9}\]  
where \(a_{2}=f_{1,2}(0)\), \(c_{2}=f_{3,2}(0)\), \(b_{2}=f_{2,2}(0)+h_{\sigma,1}\) and \(d_{2}=h_{2}-h_{\sigma,2}\ T_{i}\) in Equation (8) and \(a_{3}=f_{1,3}(0)\), \(b_{3}=f_{2,3}(0)\), \(c_{2}=f_{3,3}(0)+h_{c,3}\), and \(d_{3}=h_{2}-h_{c,3}\ T_{w}\) in Equation (9). Equations (4), (7) and (8) form a system of equations where the unknowns, T\({}_{8,1}\), T\({}_{8,2}\), and T\({}_{8,3}\), can be represented in a matrix as follows:  
\[\left[