# The Optimization of the Geometry of the Centrifugal Fan at Different Design Points
## 2 Materials and Methods
The fan that has the highest energy consumption is the one with blades facing forward. The most efficient fan at this operating point, where the blade angles range from \(0^{\circ}\) to \(90^{\circ}\), is regarded as the most optimal. Calculating the velocity triangles at the front of the blade enables narrowing down of the parameter \(\beta_{1}\) identification range. The assumption is made that the ideal angle of the leading edge of the blade is aligned with the direction of the flow velocity vector. Determining angle \(\beta_{2}\) proves impossible due to the dependence of the outgoing flow vector on the blade's rear angle. Consequently, it is only be feasible to calculate a portion of the flow component, leading to insignificant results. Additionally, to calculate the value of the trailing angle (\(\beta_{2}\)), it is essential to assess the coefficients of flow losses, which can only be accurately determined through testing Cory [25]. Air is sucked out of the central zone and routed between the two blades. Centrifugal force then pushes air towards the outer zone of the centrifugal fan. From a single blade's perspective, air flow reaches its geometric basis and flows through the blade channel. The assumption made in this work is that the flow field is tangential to a blade's profile at the initial point. The purpose is to define the blade equation in a Cartesian coordinate system for further optimization of the process [26; 27; 28]. A Cartesian coordinate system \(\{x,y\}\) is defined with the y-axis containing the initial geometry point of the blade (Figure 2b). The profile of the blade can be assumed as a smooth function (\(x=F(y)\)); the velocity field of the flow is tangential, i.e., physical effects such as boundary layer or flow separation are ignored. The acceleration of the flow along the blade's side is the sum of the normal and tangential accelerations [26; 27; 28]. \[a_{n}=\omega^{2}\cdot\rho\cdot(\cos\alpha,\sin\alpha)=\frac{v^{2}}{\rho}\cdot( \cos\alpha,\sin\alpha) \tag{1}\]  
\[a_{t}=\varepsilon\cdot\rho\cdot(\cos\alpha,\sin\alpha)=\dot{v}\cdot(\cos \alpha,\sin\alpha) \tag{2}\]  
\[a=a_{n}+a_{t} \tag{3}\]  
\[\ddot{x}=\dot{v}\sin\alpha+\frac{v^{2}}{\rho}\cdot\cos\alpha \tag{4}\]  
where \(v\) is the instantaneous velocity in a defined coordinate system, \(\rho\) is the radius of the curvature, \(\omega\) is the angular velocity, and \(\varepsilon\) is the angular acceleration. Derivatives of smooth function \(x=F(y)\) have to be defined to express velocity and acceleration [26; 27; 28]. \[\dot{x}=\dot{F}(y)\dot{y} \tag{5}\]  
\[\ddot{x}=\ddot{F}(y)\cdot y^{2}+F(y)\cdot\ddot{y} \tag{6}\]where the first and second derivatives are velocity and acceleration, respectively [26; 27; 28]. \[v^{2}=\dot{x}^{2}+\dot{y}^{2}=(1+F^{2}(y))\cdot\dot{y}^{2} \tag{7}\]  
\[2\cdot v\cdot v^{2}=2\cdot\dot{F}(y)\cdot\ddot{F}(y)\cdot\dot{y}^{3}+2\cdot(1+ \dot{F}^{2}(y))\cdot\dot{y}\cdot\ddot{y} \tag{8}\]  
If \(\tan\alpha=-F(y)\), then simple trigonometric expression of angle \(\alpha\) is as follows [26; 27; 28]:  
Figure 2:0  
Figure 2:1  
These definitions and the assumption that \(\ddot{F}(y)>0\) allow Equation (4) to be rearranged as follows [26; 27; 28]:  
Figure 2:2  
At the points where velocity is equal to zero or \(\dot{F}(y)=0\), Equation (11) is satisfied; therefore, at the points where \(\dot{F}(y)\
eq 0\), the equation becomes the following [26; 27; 28]:  
Figure 2:3  
Equation (12) is a nonlinear differential equation and can be used in two cases. In the first case, if the blade profile is assigned by function \(F(y)\), then using the last equation, i.e., the profile itself, the law of motion (\(y=y(t)\)) along the blade side can be calculated. The second case involves particular motion (\(y(t)\)) assignment; then, the equation allows for the calculation of the blade profile (\(F(y)\)) itself.