# Study on Noise Model of an Automotive Axial Fan Based on Aerodynamic Load Force
## 2 Establishment of Axial Fan Blade Loading Force
### Steady Force  
Figure 1 shows the force diagram at a certain section of the fan blade element. It can be seen in Figure 1 that \(\theta\) is the installation angle of the blade element and \(\alpha\) and \(\varphi\) are the airfoil attack angle and the inlet angle, respectively. In the blade element (airfoil) rotation plane, the relationship between relative velocity \(v_{rel}\), axial-induced velocity \(v\) of air, annular-induced angular velocity \(\omega_{f}\), blade rotation angular velocity \(\omega_{FAN}\), and the inlet angle \(\varphi\) is  
\[v_{rel}=\sqrt{v^{2}+\left(\omega_{FAN}-\omega_{f}\right)^{2}r^{2}} \tag{4}\]  
\[tan\varphi=\frac{v}{\left(\omega_{FAN}-\omega_{f}\right)r} \tag{5}\]  
The blade installation angle is  
\[\theta=\varphi+\alpha \tag{6}\]  
where \(\alpha\) is the attack angle.  
The lift force \(L\) and resistance \(D\) are constant forces and the lift force \(L\) and resistance \(D\) per unit length are, respectively,  
\[L=\frac{1}{2}\rho v_{rel}^{2}c_{h}C_{L} \tag{7}\]  
\[D=\frac{1}{2}\rho v_{rel}^{2}c_{h}C_{D} \tag{8}\]  
where \(c_{h}\) is the chord length of the airfoil, \(C_{L}\) is the lift coefficient, and \(C_{D}\) is the resistance coefficient.  
Based on the \(LS\) blade profile lift resistance coefficient and the blade profile size of existing automotive fans, the relationships between the lift coefficient \(C_{L}\), resistance coefficient \(C_{D}\), and attack angle are as follows:  
\[C_{L}=0.5+0.08*(\alpha-4) \tag{9}\]  
\[C_{D}=\frac{0.5+0.08*(\alpha-4)}{-0.116071\alpha^{4}+0.407738a^{3}+0.5625a^{2} +14.592262\alpha+14.553571} \tag{10}\]  
Since only the vertical and tangential forces in the blade rotation plane are focused on, the lift force and resistance are projected in these directions, as shown in Figure 1.  
\[F_{N}=Lcos\varphi-Dsin\varphi \tag{11}\]  
\[F_{T}=Lsin\varphi+Dcos\varphi \tag{12}\]  
According to Equations (2), (3), (11), and (12) and combined with momentum theory, the following equations can be obtained:  
\[8\pi v^{2}r=Bc_{h}v_{rel}^{2}(C_{L}cos\varphi-C_{D}sin\varphi) \tag{13}\]  
\[8\pi v\omega_{f}r^{2}=Bc_{h}v_{rel}^{2}(C_{L}sin\varphi+C_{D}cos\varphi) \tag{14}\]  
The limitation of the BEM theory is that it does not consider the influence of tip vortices. At present, the Blount tip loss model is widely used to modify blade tip loss based on the BEM theory. In this model, the blade tip loss factor \(Lf\) is used to modify the induced velocity field, as shown in Equation (15) [12]:  
\[C_{\rm F}=\frac{2}{\pi}\arccos\left(e^{-f_{\omega}}\right) \tag{15}\]  
where \(f_{\omega}=\frac{\rm B}{2}\left(\frac{R-r}{sin\varphi}\right)\).  
Then, Equations (13) and (14) are, respectively, expressed as  
\[8\pi v^{2}rC_{F}=Bc_{h}v_{rel}^{2}(C_{L}cos\varphi-C_{D}sin\varphi) \tag{16}\]  
\[8\pi v\omega_{f}r^{2}C_{F}=Bc_{h}v_{rel}^{2}(C_{L}sin\varphi+C_{D}cos\varphi) \tag{17}\]  
Figure 1: Force of fan blade element.  
Based on the condition that the installation angle \(\theta\) and chord length \(c_{h}\) are known, the axial-induced velocity \(\
u\), annular-induced angular velocity \(\omega_{f}\), and \(\alpha\) of air can be calculated by combining Equations (16) and (17), thus the steady force \(F_{N}\) and \(F_{T}\) can be obtained.