# Numerical Study on Temperature Distribution of Steel Truss
## 2 Theory on Thermal Boundary Condition and Thermal Parameters  
For structures exposed to solar radiation, the thermal boundary conditions, such as solar radiation, heat convection, and longwave radiation, should be taken into consideration. The ASHRAE clear-sky model [31; 32; 33; 34], which was proposed by the American Society of Heating, Refrigerating, and Air-Conditioning Engineers, is adapted to calculate the solar radiation on the steel aqueduct. The solar radiation received by the aqueduct consists of direct radiation, diffuse radiation, and reflected radiation from the surrounding ground. Before introducing the ASHRAE clear-sky model, several parameters about the solar radiation should be clarified. Solar altitude \(\beta\) (\({}^{\circ}\)) is the angle between sunlight and its projection on the horizontal plane and is calculated using Equation (1)  
\[sin\ \beta=cos\ l\ cos\ h_{a}\ cos\ \delta+sin\ l\ sin\ \delta \tag{1}\]  
among which, \(l\) (\({}^{\circ}\)) is the geographic latitude; \(h_{a}\) (\({}^{\circ}\)) is solar hour angle; \(\delta\) (\({}^{\circ}\)) is solar declination angle. Solar azimuth angle \(\varphi\) (\({}^{\circ}\)) is the angle between sunlight's projection on horizontal plane and north direction along clockwise. \[cos\ \varphi=\frac{sin\ \delta\ cos\ l-cos\ \delta\ sin\ l\ cos\ h_{a}}{cos\ \beta} \tag{2}\]  
The direct solar radiation intensity \(G_{ND}\) (W/m\({}^{2}\)) on earth with a clear sky is defined as Equation (3). \[G_{ND}=\frac{A}{exp(B/sin\ \beta)}C_{N}\ cos\ \theta \tag{3}\]  
among which, \(A\) (W/m\({}^{2}\)) is the solar radiation at the exoatmosphere of the Earth; \(B\) is the atmospheric extinction coefficient; \(C_{N}\) is the clearness number of atmosphere, and \(\theta\) (\({}^{\circ}\)) is the angle of incidence. The diffuse radiation intensity \(G_{d\theta}\) (W/m\({}^{2}\)) on nonvertical surfaces on a clear day is defined as Equation (4). \[G_{d\theta}=CG_{ND}F_{ws} \tag{4}\]  
in which \(C\) is the ratio of diffuse irradiation on a surface to the direct normal radiation, and the angle factor between the surface and the sky \(F_{ws}\) is associated with surface tilt angle \(\alpha\), which can be written as Equation (5). \[F_{ws}=\frac{1+cos\ \alpha}{2} \tag{5}\]  
The ground reflected radiation intensity \(G_{R}\) (W/m\({}^{2}\)) on a clear day can be given with Equation (6). \[G_{R}=G_{tH}\rho_{g}F_{wg} \tag{6}\]  
where \(G_{tH}\) (W/m\({}^{2}\)) is the total radiation on horizontal plane or ground before striking the target wall; \(\rho_{g}\) is the ground radiation reflectance; the configuration or angle factor from surface wall to ground \(F_{wg}\) is also associated with surface tilt angle \(\alpha\), which can be given with Equation (7). \[F_{wg}=\frac{1-cos\ \alpha}{2} \tag{7}\]  
The total heat flux density of solar radiation \(q_{s}\) (W/m\({}^{2}\)) that is absorbed by the target surface can be calculated using the sum of three radiations intensity, as shown in Equation (8). \[q_{s}=a(G_{d}+G_{d\theta}+G_{R}) \tag{8}\]  
and \(a\) stands for solar absorption coefficient. The density of convectional heat flux \(q_{h}\) (W/m\({}^{2}\)) on the surface of a steel member can be computed referring to [PERSON] cooling formula as Equation (9)  
\[q_{h}=h(T_{0}-T) \tag{9}\]in which \(h\) is the heat convection coefficient and is calculated by Equation (10) [31]; \(T_{0}\) (\({}^{\circ}\)C) is the ambient air temperature, and \(T\) (\({}^{\circ}\)C) is the temperature of the surface. \[h=\sqrt{\left[C_{t}(\Delta T)^{1/3}\right]^{2}+\left[aV_{0}^{b}\right]^{2}} \tag{10}\]  
where \(C_{t}\) is the turbulent natural convection constant and sets as 0.84 W/(m\({}^{2}\)K\({}^{3/4}\)); \(\Delta T\) is the temperature difference between steel surface and surrounding air; \(V_{0}\) is the average wind speed, which is 3 m/s in the targeted region; \(a\) has a value of 2.38 W/(m\({}^{2}\)Km/s), and \(b\) is 0.89. The heat flux density of longwave radiation on a steel surface is calculated using the Stefan-Boltzmann equation (Equation (11)).