# Design and Analysis of a Floor Radiant Heating System Based on Energy Substitution Technology
## 6 Actual System Energy Savings Calculation and Analysis
### Calculation
The instantaneous heat \(q_{\rm PT}(i)\) provided by the photothermal floor radiant heating system of the proposed floor radiant heating system is as follows (Equations (4)-(7)):  
\[q_{\rm PT}(i)=q_{\rm PTr}(i)+q_{\rm PTc}(i) \tag{4}\]  
\[q_{\rm PTr}(i)=4.98\times 10^{-8}[(t_{\rm P}(i)+273)^{4}-(T_{\rm r}+273)^{4}] \tag{5}\]  
\[q_{\rm PTc}(i)=2.17\times[t_{\rm P}(i)-t_{\rm r}]^{1.31} \tag{6}\]  
\[q_{\rm PT}(i)=4.98\times 10^{-8}[(t_{\rm P}(i)+273)^{4}-(T_{\rm r}+273)^{4}]+2.17 \times[t_{\rm P}(i)-t_{\rm r}]^{1.31} \tag{7}\]  
where:  
\(q_{\rm PT}(i)\) is the instantaneous heat provided by the photothermal floor radiant heating system of the proposed floor radiant heating system, W/m\({}^{2}\); \(q_{\rm PTr}(i)\) is the instantaneous radiation heat provided by the photothermal floor radiant heating system of the proposed floor radiant heating system, W/m\({}^{2}\); \(q_{\rm PTc}(i)\) is the instantaneous convection heat provided by photothermal floor radiant heating system of the proposed floor radiant heating system, W/m\({}^{2}\); \(t_{\rm P}(i)\) is the floor surface temperature, which is measured by an infrared thermometer to calculate the heat from the photothermal floor radiant heating system, \({}^{\circ}\)C; \(t_{\rm r}\) is the indoor temperature, which is measured by the indoor temperature and humidity sensor, \({}^{\circ}\)C; \(T_{\rm r}\) is the average radiant temperature of non-heated surfaces, \({}^{\circ}\)C, \(T_{\rm r}\) = \(\sum\limits_{k=1}^{n}t_{sk}A_{k}/\sum\limits_{k=1}^{n}A_{k}\) (\(t_{sk}\) is the surface temperature of each part (\({}^{\circ}\)C); \(A_{k}\) is the area of the corresponding enclosure structure (m\({}^{2}\))). The total heat \(q_{\rm PT}\) provided by the photothermal floor radiant heating system of the proposed floor radiant heating system is as follows (Equation (8)):  
\[ \begin{array}{c}q_{\rm PT}=\sum\limits_{i=0}^{n}q_{\rm PT}(i)=\sum\limits_ {i=0}^{n}[q_{\rm PTr}(i)+q_{\rm PTc}(i)]\\ =\sum\limits_{i=0}^{n}[4.98\times 10^{-8}[(t_{\rm P}(i)+273)^{4}-(T_{\rm r}+273)^{4 }]+2.17\times[t_{P}(i)-t_{\rm r}]^{1.31}]\end{array}  \tag{8}\]  
where:  
\(n\) is the total operation time of the photothermal floor radiant heating system in one year, s; \(q_{\rm PT}\) is the total heat provided by the photothermal floor radiant heating system of the proposed floor radiant heating system, W/m\({}^{2}\). The total operation time is defined as the sum of the operating time of the photothermal system and its energy storage system. * The heat provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system is calculated. For the photovoltaic floor radiant heating system of the proposed floor radiant heating system, the heat transfer process includes the following: the heating cables heat the floor and then the heat from the floor is transferred to the walls and indoor air by convection and radiation. The heat transfer process of the photovoltaic floor radiant heating system is different from that of the photothermal floor radiant heating system. The equations are as follows (Equation (9)):  
\[q_{\rm L}=I^{2}\times R^{\prime} \tag{9}\]  
where:  
\(q_{L}\) is the heat per unit length of the heating cable provided by the photovoltaic floor radiant heating system, W/m; \(I\) is the heating cable current of the photovoltaic floor radiant heating system, A; \(R^{\prime}\) is the correctional resistance of heating cables of the photovoltaic floor radiant heating system, \(\Omega\)/m (Equation (10)). \[R^{\prime}=R.\frac{[1+\alpha_{e}(t_{d}-20)]}{[1+\alpha_{0}(t_{d}-20)]} \tag{10}\]where:  
\(R\) is the resistance of the heating cables, \(\Omega/\mathrm{m}\); \(\kappa_{e}\) is the thermal coefficient of material resistance, \({}^{\circ}\mathrm{C}^{-1}\); \(\alpha_{0}\) is the coefficient of thermal expansion, \({}^{\circ}\mathrm{C}^{-1}\); \(t_{d}\) is the surface temperature of heating cable, \({}^{\circ}\mathrm{C}\) (Equation (11)). \[