# Thermal Characteristics and Parametric Analysis of an Improved Solar Wall
## 3 Thermal Characteristics Analysis of the Improved Solar Wall
### Thermal Characteristic of the Photoelectric Fan  
The mass flow rate of the air gap is an important factor to determine the thermal efficiency of the color-changing solar wall. The mass flow is influenced by the air velocity, damper area and the position of the control damper. In this case, the position of the control damper is ignored and the damper is in a fixed position, with the damper open area of 0.008 m\({}^{2}\). The influence of air temperature is also ignored. As the fan is driven by photovoltaic panels, the wind speed is an unsteady value, which is varied with the solar irradiance. Therefore, the primary factor to characterize the thermal properties of the solar wall is to determine the mass flow rate and velocity in the air gap. The relationship between variables of fluid flow can be analyzed by dimensional analysis. The representational correlation of the average velocity of the air layer is determined based on the \(\pi\) theorem. Figure 8: Annual heating demand under solar walls heating. The heat transfer involving motion in the air gap is caused by the photoelectric fan and the temperature difference. Factors influencing the air velocity u include thickness of the air gap \(\delta\), height of the air gap H, the solar irradiance qi, the coefficient of thermal expansion \(\beta\), thermal conductivity \(\lambda\), the kinematic viscosity coefficient \(\upsilon\), air density \(\rho\), the thermal diffusivity a and the gravitational acceleration g (Equation (3)) [36]. \[u=f(\delta,H,q_{\rm{i}},\beta,\lambda,\upsilon,\rho,a,\rm{g}) \tag{3}\]  
In the above 10 variables, the fundamental dimension number is 4. Four repeated variables are selected, respectively:  
\[[\rho]=ML^{-3}\;[\beta]=\Theta^{-1}\;[\upsilon]=L^{2}T^{-1}\;[H]=L \tag{4}\]  
The dimensionless parameter \(\pi_{i}[i:1\sim 6]\) is written by the unknown exponent, and the dimension formula is as follows. \[\left\{ \begin{array}{l}[\tau_{1}]=(ML^{-3})^{w_{1}}(\Theta^{-1})^{x_{1}}(L^ {2}T^{-1})^{y_{1}}(L)^{z_{1}}(LT^{-1})=1\\ [\tau_{2}]=(ML^{-3})^{w_{2}}(\Theta^{-1})^{x_{2}}(L^{2}T^{-1})^{y_{2}}(L)^{z_ {2}}(L)=1\\ [\tau_{3}]=(ML^{-3})^{w_{3}}(\Theta^{-1})^{x_{3}}(L^{2}T^{-1})^{y_{3}}(L)^{z_ {3}}(MT^{-3})=1\\ [\tau_{4}]=(ML^{-3})^{w_{4}}(\Theta^{-1})^{x_{4}}(L^{2}T^{-1})^{y_{4}}(L)^{z_ {4}}(MLT^{-3}\Theta^{-1})=1\\ [\tau_{5}]=(ML^{-3})^{w_{5}}(\Theta^{-1})^{x_{5}}(L^{2}T^{-1})^{y_{5}}(L)^{z_ {5}}(L^{2}T^{-1})=1\\ [\tau_{6}]=(ML^{-3})^{w_{6}}(\Theta^{-1})^{x_{6}}(L^{2}T^{-1})^{y_{6}}(L)^{z_ {6}}(LT^{-2})=1\\ \end{array} \right. \tag{5}\]  
By the dimensional harmonious Equations, the value of each coefficient is solved by:  
\[\left\{ \begin{array}{l}\pi_{1}=\upsilon^{-1}H^{1}u^{1}=uH/\upsilon=\rm{Re} \\ \pi_{2}=H^{-1}\delta^{1}=\delta/H\\ \pi_{3}=\rho^{-1}\upsilon^{-3}H^{3}q_{i}^{1}=q_{i}H^{3}/\rho\upsilon^{3}\\ \pi_{4}=\rho^{-1}\beta^{-1}\upsilon^{-3}H^{2}\lambda^{1}=\lambda H^{2}/\rho \beta\upsilon^{3}\\ \pi_{5}=\upsilon^{-1}a^{1}=a/\upsilon=1/\rm{Pr}\\ \pi_{6}=\upsilon^{-2}H^{3}g^{1}=gH^{3}/\upsilon^{2}\\ \end{array} \right. \tag{6}\]  
According to Equation (6), the mathematical expression of the air velocity in the air gap is as follows,  
\[Re=\frac{uH}{\upsilon}=f\bigg{(}\frac{\delta}{H^{\prime}},\frac{q_{i}H^{3}}{ \rho\upsilon^{3}},\frac{\lambda H^{2}}{\rho\beta\upsilon^{3}},Pr,\frac{gH^{3} }{\upsilon^{2}}\bigg{)}=f(\frac{\delta}{H^{\prime}},\frac{g\beta q_{i}H^{4}}{ \lambda\upsilon^{2}},Pr)=f\bigg{(}\frac{\delta}{H},Ra^{*}\bigg{)} \tag{7}\]  
where the modified Rayleigh number \(Ra^{*}\) is described as follows,  
\[Ra^{*}=\frac{g\beta q_{i}l^{4}}{\lambda\upsilon^{2}}\cdot Pr \tag{8}\]  
Thus, the dependence of mass flow rate on the solar irradiance and channel depth can be expressed by the relationship Equation (9),  
\[Re=a(Ra^{*})^{b}(\frac{\delta}{H})^{c} \tag{9}\]  
where, \(Re\) is the Reynolds number, \(a\), \(b\), \(c\) is the weighted coefficient, \(\delta\) is the channel depth, m; \(H\) is the channel height, m.  
For the experimental device, the ratio is a fixed value, \(\delta/H=0.133\).