# Modeling Heat Consumption of an Office Building During COVID-19 Restrictions
## 2 Materials and Methods
In fact, with the tools and machine capacity available today, numerical modeling of additional heat losses/savings due to indoor temperature feedback cannot be completed under reasonable efforts, i.e., the problem of interest is (computationally) intractable or takes too much time. Therefore, we apply an empirical formula, which includes adjustment factors to calculate additional heat losses/savings due to indoor temperature feedback [kWh] [24]:  
\[\mathrm{t_{1 TP}}=t_{in}+0.5\Big{(}\mathrm{t_{1}^{p}-t_{2}^{p}}\Big{)}\frac{ \overline{\mathrm{Q}}}{K}+\left(\frac{t_{1}^{p}+t_{2}^{p}}{2}-t_{in}\right) \left(\frac{\overline{\mathrm{Q}}}{K}\right)^{\frac{1}{1+m}}, \tag{2}\]0  
Here, \(L_{\text{{e}ent}}\) is the flow rate of incoming (heated) air [m\({}^{3}\)/h]:  
* \(L_{\text{{e}ent}}\) = 4 A, if the difference between the target indoor temperature and the actual one is less than 3\({}^{\circ}\)C;
* \(L_{\text{{e}ent}}\) = 5 A, if the difference between the target indoor temperature and the actual one ranges between 3 and 3.9\({}^{\circ}\)C;
* \(L_{\text{{e}ent}}\) = 7 A, if the difference between the target indoor temperature and the actual one ranges between 4 and 4.9\({}^{\circ}\)C;
* \(L_{\text{{e}ent}}\) = 10 A, if the difference between the target indoor temperature and the actual one is greater than 5\({}^{\circ}\)C.  
A: gross internal floor area [m\({}^{2}\)];  
\(p_{\text{{a}}}\): air density for indoor temperature [kg/m\({}^{3}\)],\(\rm n_{vent}\): AHU operating hours during a week [h];  
\(\rm G_{inf}\): the amount of infiltrating air for a single hour [kg/h];  
\(\rm n_{inf}\): hours of non-air-tightness during a week [h], typically 168 h;  
\(\rm c_{a}\): specific heat capacity of air [k]/kgC];  
\(\rm 0.04=0.28\cdot 24/168\). Here, 0.28 is the experimental factor, with 24 h a day and 168 h in a week. Equation (12) is applied to calculate the amount of infiltrating air for a single hour [kg/h] [24]:  
\[\mathrm{t_{1 TP}}=t_{in}+0.5\Big{(}\mathrm{t_{1}^{p}-t_{2}^{p}}\Big{)}\frac{ \overline{\mathrm{Q}}}{K}+\left(\frac{t_{1}^{p}+t_{2}^{p}}{2}-t_{in}\right) \left(\frac{\overline{\mathrm{Q}}}{K}\right)^{\frac{1}{1+m}}, \tag{2}\]1  
Here, \(\Delta P\) is the pressure difference between outdoor and indoor environments [Pa] [24]. Internal heat gains are calculated as follows [24]:  
\[\mathrm{t_{1 TP}}=t_{in}+0.5\Big{(}\mathrm{t_{1}^{p}-t_{2}^{p}}\Big{)}\frac{ \overline{\mathrm{Q}}}{K}+\left(\frac{t_{1}^{p}+t_{2}^{p}}{2}-t_{in}\right) \left(\frac{\overline{\mathrm{Q}}}{K}\right)^{\frac{1}{1+m}}, \tag{2}\]2  
Here, \(\rm f\) is the occupancy factor:  
* 0.7 for 80% of employees working;
* 0.6 for 70% of employees working;
* 0.5 for 60% of employees working, etc. \(\rm q_{int}\): specific internal heat gains [W/m\({}^{2}\)]. For the sake of simplicity, we exclude non-quadratic constrained components, and to add the influence of insolation, we use linear equations only. Heat gains due to solar insolation [kWh] are thus [24]:  
\[\mathrm{t_{1 TP}}=t_{in}+0.5\Big{(}\mathrm{t_{1}^{p}-t_{2}^{p}}\Big{)}\frac{ \overline{\mathrm{Q}}}{K}+\left(\frac{t_{1}^{p}+t_{2}^{p}}{2}-t_{in}\right) \left(\frac{\overline{\mathrm{Q}}}{K}\right)^{\frac{1}{1+m}}, \tag{2}\]3  
Here, \(\rm A_{\rm wN}\), \(\rm A_{\rm wS}\), \(\rm A_{\rm wW}\), and \(\rm A_{\rm wE}\) are glazing areas, oriented North, South, West, and East, respectively [m\({}^{2}\)];  
\(\rm A_{sky}\): surface area of skylights [m\({}^{2}\)];  
\(\rm I_{N}\), \(\rm I_{S}\), \(\rm I_{W}\), and \(\rm I_{E}\): average annual solar radiation on vertical surfaces, oriented North, South, West, and East, respectively [kWh/m\({}^{2}\)];  
\(\rm I_{hor}\): average annual solar radiation on a horizontal surface [kWh/m\({}^{2}\)]. To compare with the closest methodology, [PERSON] et al. [2] used the Dwelling Energy Assessment Procedure (DEAP), the official Irish method for calculating the energy performance of dwellings.