## 3 Lattice Boltzmann Method and Comprehensive Incremental Benefit Model
### Comprehensive Incremental Benefit Model  
Compared to the traditional buildings, the green buildings with PCM or SIM have a larger incremental cost during the construction period. However, with their better energy saving characteristics, the green buildings will generate an economic benefit, an environmental benefit, and a social benefit during their lifetime. The comprehensive incremental benefit is defined as [34]:  
\[\Delta E=\Delta S-\Delta C \tag{25}\]  
where \(\Delta E\) is the comprehensive incremental benefit, \(\Delta S\) is the incremental benefit, and \(\Delta C\) is the incremental cost for the extra expense of constructing the green building compared with the traditional building, which is expressed as:  
\[\Delta C=\Delta C_{i}+\Delta C_{p} \tag{26}\]  
\(\Delta C_{i}\) is the cost of integrating extra materials such as PCM or SIM:  
\[\Delta C_{i}=(C_{u}+C_{o}+C_{l})\times L\times H\times D \tag{27}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\boldsymbol{\rho}\) (\(\mathbf{kg/m^{3}}\)) & \(\boldsymbol{c_{p}}\) (\(\mathbf{J/kg\cdot K}\)) & \(\boldsymbol{k}\) (\(\mathbf{W/m\cdot K}\)) & \(\boldsymbol{C_{u}}\) \\ \hline Cement & 1807 & 840 & 0.854 & 850 \\ Concrete & 2500 & 920 & 1.74 & - \\ Paraffin & 775 & 534 & 0.148 & 8800 \\ Perlite & 400 & 1170 & 0.14 & 350 \\ Lime & 1600 & 1050 & 0.81 & 850 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties and prices of materials in a building’s roof (Data resource: China Chemical Engineering Website). Figure 1: CUDA code calibration for conjugate heat transfer and solid-liquid phase change. where \(C_{u}\) is the price of PCM or SIM as shown in Table 1, \(C_{o}\) is the price of other adhesive materials, which was chosen to be \(C_{o}=25\) yuan/m\({}^{3}\) and \(C_{l}=15\) yuan/m\({}^{3}\) was the labor fee. \(L\) was the length of the roof set to be 6 m, \(H\) is the width of the roof set to be 6 m, and \(D\) is the thickness of PCM or SIM layer. \(\Delta C_{p}\) is the cost for numerical modelling and consultation for the building design, which is \(\Delta C_{p}=0.1\%\times\Delta C_{i}\). The incremental benefit \(\Delta S\) is calculated as:  
\[\Delta S=\Delta S_{y}\cdot P\big{(}\Delta S_{y},i_{0},n_{y}\big{)} \tag{28}\]  
\(P\) is the annual income discount rate coefficient given as:  
\[P=1/i_{0}-1/\big{[}i_{0}\times(1+i_{0})^{n_{y}}\big{]} \tag{29}\]  
\(n_{y}\) is the year of the building life cycle, which is 50 years according to the standard of the Chinese government; \(i_{0}\) is the discount rate, which is set to be \(i_{0}=11\%\) for the Shaanxi province in China. The annual benefit \(\Delta S_{y}\) is defined as:  
\[\Delta S_{y}=\Delta S_{e}+\Delta S_{h}+\Delta S_{s} \tag{30}\]  
\(\Delta S_{e}\) is the economic benefit of green building:  
\[\Delta S_{e}=\frac{(\Delta W_{s}*70+\Delta W_{w}*70)\times L\times H}{3600\times \eta}\times EP_{unit} \tag{31}\]  
where \(\Delta W_{s}\) (kJ/m\({}^{2}\)) is the energy saving amount during the summer AC cooling time, and \(\Delta W_{w}\) (kJ/m\({}^{2}\)) is the energy saving amount during the winter AC heating time. In this paper, the period of summer cooling and winter heating was chosen to be 70 days for each. \(\eta\) is the Energy Efficiency Ratio (EER) or Coefficient of Performance (COP) of an air conditioner, which was set as 3.3 in this work. \(EP_{unit}\) is the electricity price in the city of Hanzhong, Shaanxi Province which is \(EP_{unit}=0.4983\) (yuan/kW\(\cdot\)h). \(\Delta S_{h}\) is the environmental benefit of a green building and it includes the benefits generated by the reduction of CO\({}_{2}\), SO\({}_{2}\), and NOx emissions. Using an effective equivalent method, the coefficient for transferring the saved electricity of the green building to the reduced combustion amount of coal was 0.0004 ton/(kW\(\cdot\)h). The emission coefficients of CO\({}_{2}\), SO\({}_{2}\), NOx, and smoke dust were 2.4567, 0.0165, 0.0156, 0.096 respectively.