# Design and Analysis of a Floor Radiant Heating System Based on Energy Substitution Technology
## 6 Actual System Energy Savings Calculation and Analysis
### Calculation
\begin{array}{c}q_{\mathrm{PV}}(j)=q_{\mathrm{PVr}}(j)+q_{\mathrm{PVc}}(j) \\ =4.98\times 10^{-8}[(t_{\mathrm{P}}(j)+273)^{4}-(T_{\mathrm{r}}+273)^{4}]+2.17 \times[t_{p}(j)-t_{\mathrm{r}}]^{1.31}\end{array}  \tag{11}\]  
where:  
\(q_{\mathrm{PV}}(j)\) is the instantaneous heat provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system, \(\mathrm{W}/\mathrm{m}^{2}\); \(q_{\mathrm{PVr}}(j)\) is the instantaneous radiation heat provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system, \(\mathrm{W}/\mathrm{m}^{2}\); \(q_{\mathrm{PVc}}(j)\) is the instantaneous convection heat provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system, \(\mathrm{W}/\mathrm{m}^{2}\); \(t_{\mathrm{P}}(j)\) is the floor surface temperature, which is measured by an infrared thermometer to calculate the heat from the photovoltaic floor radiant heating system, \({}^{\circ}\mathrm{C}\). The total heat \(q_{\mathrm{PV}}\) provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system is calculated as follows (Equation (12)):  
\[ \begin{array}{c}q_{\mathrm{PV}}=\sum\limits_{j=0}^{m}q_{\mathrm{PV}}(j)= \sum\limits_{j=0}^{m}\left[q_{\mathrm{PVr}}(j)+q_{\mathrm{PVc}}(j)\right]\\ =\sum\limits_{j=0}^{m}\left[4.98\times 10^{-8}[(t_{\mathrm{P}}(j)+273)^{4}-(T_{ \mathrm{r}}+273)^{4}]+2.17\times[t_{p}(j)-t_{\mathrm{r}}]^{1.31}\right]\end{array}  \tag{12}\]  
where:  
\(m\) is the total operation time of the photovoltaic floor radiant heating system in one year, s; \(q_{\mathrm{PV}}\) is the total heat provided by the photovoltaic floor radiant heating system of the proposed floor radiant heating system, \(\mathrm{W}/\mathrm{m}^{2}\). The total operation time is defined as the sum of the operating time of the photovoltaic system and its energy storage system. Calculation of the total heat provided by the proposed floor radiant heating system  
The total heat provided by the proposed floor radiant heating system involves the total heat provided by the photothermal floor radiant heating system, the total heat provided by the photovoltaic floor radiant heating system and the supplementary heat provided by the auxiliary heat source in extreme cases. The total heat \(q\) is (Equation (13)):  
\[ \begin{array}{c}q=q_{\mathrm{PT}}+q_{\mathrm{PV}}+q_{\mathrm{S}}\\ =\sum\limits_{i=0}^{n}\left[4.98\times 10^{-8}[(t_{\mathrm{P}}(i)+273)^{4}-(T_{ \mathrm{r}}+273)^{4}]+2.17\times[t_{p}(i)-t_{\mathrm{r}}]^{1.31}\right]\\ +\sum\limits_{j=0}^{m}\left[4.98\times 10^{-8}[(t_{\mathrm{P}}(j)+273)^{4}-(T_{ \mathrm{r}}+273)^{4}]+2.17\times[t_{p}(j)-t_{\mathrm{r}}]^{1.31}\right]+q_{ \mathrm{S}}\end{array}  \tag{13}\]  
where:  
\(q\) is the total heat provided by the proposed floor radiant heating system, \(\mathrm{W}/\mathrm{m}^{2}\); \(q_{\mathrm{S}}\) is the supplementary heat provided by the auxiliary heat source in extreme cases, \(\mathrm{W}/\mathrm{m}^{2}\). Revised calculation of the total heat provided by the proposed floor radiant heating system  
The proposed floor radiant heating system involves two different heating systems that have different loads, and the auxiliary heating system is also considered in extreme cases. Therefore, it is necessary to use the load factors to revise the total heat calculation of the proposed floor radiant heating system. The total load provided by the proposed floor radiant heating system includes the load provided by the photothermal floor radiant heating system, the load provided by the photovoltaic floor radiant heating system and auxiliary heat load in extreme cases. The load factors are \(L_{\text{PT}}\), \(L_{\text{PV}}\), \(L_{\text{S}}\).