# Prediction of Temperature Distribution for Previous Cement Concrete Pavement with Asphalt Overlay
## 2 Mathematical Model of Pavement Temperature Field
### The Boundary Condition at the Pavement Surface  
It has been proven that the impact of cyclical climate factors (such as solar radiation fluxes (\(Q\)) and air temperature (\(T_{a}\))) on pavement materials' properties and their field performance is significant. Further, solar radiation fluxes (\(Q\)) and air temperature (\(T_{a}\)) are mathematically disposed to be continuous functions, considering the boundary condition of the surface, i.e., \(Q(t)\) and \(T_{a}(t)\), respectively. The \(m\) and \(t_{e}\) are assumed to be a positive integer and the maximum predicted time, respectively. Then, the predicted time interval \([0,t_{e}]\) (i.e., \(t\in(0,t_{e})\)) was assumed to be decided by the equal length (\(\frac{t_{e}}{2m}\)) into \(2m\) sub-intervals. Finally, the data of solar radiation fluxes (\(Q(t)\)) and air temperatures (\(T_{a}(t)\)) were assumed to be measured at two endpoints in each sub-interval except at the time (\(t_{e}\)). The data of the solar radiation fluxes (\(Q(t)\)) and measured air temperature (\(T_{a}(t)\)) are fitted by a discrete least-squares approximation of the interpolatory trigonometric polynomials as follows [18]:  
\[T_{a}(t)=\frac{a_{0}}{2}+\frac{a_{m}}{2}\cos(m\overline{t})+\sum_{k=1}^{m-1} \Bigl{[}a_{k}\cos(k\overline{t})+b_{k}\sin(k\overline{t})\Bigr{]} \tag{5}\]\[Q(t)=\frac{c_{0}}{2}+\frac{c_{m}}{2}\cos(m\overline{t})+\sum_{k=1}^{m-1}\Bigl{[}c _{k}\cos(k\overline{t})+d_{k}\sin(k\overline{t})\Bigr{]} \tag{6}\]  
with \(t\in[0,t_{e}-\frac{t_{e}}{2m}]\), \(\overline{t}=\pi\Bigl{[}\frac{2t}{t_{e}}-1\Bigr{]}\);  
\[a_{k} =\frac{1}{m}\sum_{l=0}^{2m-1}T_{l}\cos\Biggl{[}\frac{k\pi}{m}(l-m) \Biggr{]},\text{ for each }k=0,1,\ldots,m;\] \[b_{k} =\frac{1}{m}\sum_{l=0}^{2m-1}T_{l}\sin\Biggl{[}\frac{k\pi}{m}(l-m) \Biggr{]},\text{ for each }k=0,1,\ldots,m;\] \[c_{k} =\frac{1}{m}\sum_{l=0}^{2m-1}Q_{l}\cos\Biggl{[}\frac{k\pi}{m}(l-m) \Biggr{]},\text{ for each }k=0,1,\ldots,m;\] \[d_{k} =\frac{1}{m}\sum_{l=0}^{2m-1}Q_{l}\sin\[PERSON]{[}\frac{k\pi}{m}(l-m) \[PERSON]{]},\text{ for each }k=0,1,\ldots,m;\]  
where \(Q_{l}\) and \(T_{l}\) are the measured solar radiation flux and air temperature at the \(l\)th partitioning point of the time interval at time \(t=lt_{e}/(2m)\), for each \(l=0,1,2,\ldots,2m-1\), i.e., \([0,t_{e}]\), respectively. At the pavement surface, the energy balance can be described by the second boundary condition as shown in Equation (7):  
\[-\lambda_{1}\frac{\partial T_{1}(0,t)}{\partial z}=h_{f}[\frac{\alpha_{s}}{h_ {f}}Q(t)+T_{a}(t)-T_{1}(0,t)] \tag{7}\]  
where \(Q(t)\) is the solar radiation flux (W/m\({}^{2}\)); \(T_{a}(t)\) is the air temperatures (\({}^{\circ}\)C); \(h_{f}\) is the surface material convection coefficient (W/(m\({}^{2}\).\({}^{\circ}\)C)); \(T_{1}(0,t)\) is the pavement surface temperatures (\({}^{\circ}\)C); \(\lambda_{1}\) is the thermal conductivity of the 1 th layer (W/(m\({}^{\circ}\)C)); and \(\alpha_{s}\) represents the effective surface absorptivity relative to the total solar radiation (dimensionless). The convection coefficient of pavement can be affected by several main factors including the pavement surface temperature, the wind speed and surface roughness as well as the air temperature. Equation (8) can be used to approximately determine the surface material convection coefficient (\(h_{f}\)) [19]:  
\[h_{f}\approx 9.4+3.7v \tag{8}\]  
where \(v\) is the average speed of the wind during the time period of interest (m/s). Substituting Equations (5) and (6) into Equation (7), leads to  
\[ \begin{array}{l}-\lambda_{1}\frac{\partial T_{1}(0,t)}{\partial z}=h_{f} \biggl{\{}\frac{a_{0}}{2}+\frac{a_{s}c_{0}}{2h_{f}}+(\frac{a_{m}}{2}+\frac{a_ {s}c_{m}}{2h_{f}})\cos(m\overline{t})\\ +\sum\limits_{k=1}^{m-1}\bigl{[}(a_{k}+\frac{a_{s}c_{k}}{h_{f}})\cos(k \overline{t})+(b_{k}+\frac{a_{s}d_{k}}{h_{f}})\sin(k\overline{t})\bigr{]}-T_{ 1}(0,t)\biggr{\}}\end{array}