# The Optimization of the Geometry of the Centrifugal Fan at Different Design Points
## 2 Materials and Methods
#### Optimization Problem Definition  
A particular law of motion (\(y=y(t)\)) is required in the optimization process to achieve the best possible fan efficiency; therefore, boundary conditions \(g\) and \(h\) are essential and have to be defined. It follows that [26; 27; 28]  
\[y=g(y) \tag{13}\]  
and  
\[\ddot{y}=\dot{g}(y)\cdot g(y) \tag{14}\]  
It has to be noted that for geometrical assumptions, the boundary condition can be defined as \(F(h)=0\), where \((0,h)\) are Cartesian coordinates of the initial point of the impeller blade. The derivative of the boundary condition \(\dot{F}(y,h)=c_{0}\) comes from the physical assumption about the tangential flow at the initial point of the blade. The slope of the flow field at the impeller inlet is noted as \(c_{0}\). These conditions allow for the formulation of the differential optimization problem as follows [26; 27; 28]:  
\[\left\{ \begin{matrix}\dot{g}(y)\cdot g(y)\cdot(1+\dot{F}^{2}(y)+g^{2}(y) \cdot\dot{F}(y)\cdot\ddot{F}(y)=0)\\ \dot{F}(h)=0\\ \dot{F}(y,h)=0\end{matrix} \right\} \tag{15}\]  
The boundary conditions can be easily revised, considering that the blade has zero curvature, that is, \(\ddot{F}(y)=0\) for every coordinate point. Then, according to Equation (15),  
\[\dot{g}(y)\cdot g(y)\cdot(1+\dot{F}^{2}(y))=0 \tag{16}\]It is obvious that \(\dot{g}(y)=0\) at every coordinate point, that is, \(y(t)=At+B\) is equal to \(\ddot{y}=0\). From the opposite point of view, if \(y(t)\) is linear in time, then boundary condition \(g(y)=0\) for every coordinate point, and the differential formulation becomes the following:  
\[g^{2}(y)\cdot\dot{F}(y)\cdot\ddot{F}(y)=0 \tag{17}\]  
As a result, \(\ddot{F}(y)=0\), and from Equation (5), it follows that \(x(t)\) is linear as well. This drives optimization considering constant speed along the airflow direction. Using the principle of superposition and fixing the blade, the flow at the leading edge of the blade (\(\omega_{1}\)) can be described in terms of the centrifugal (\(c_{1}\)) and tangential (\(u_{1}\)) velocity vectors Darvish [28]. If the impeller under study has relatively similar operating conditions to the product discussed by the Ziehl-Abegg manufacturer [22], then the fan air flow rate (\(v_{p}\)) in this duct is determined by Equation (18), where the area of the intake duct (\(A_{p}\)) and the amount of air blown by the Ziehl-Abegg fan at the design point (\(Q_{p}\)) are expressed as follows:  
\[v_{pi}=\frac{Q_{pi}}{A_{p}}\text{ or }v_{p1}=\frac{Q_{p1}}{A_{p}}=13.15\text{ m/s} \tag{18}\]  
When applying the flow continuity expression, the relationship between the velocity (\(v_{pi}\)) and the area (\(A_{p}\)) of the inlet channel and that between the centrifugal velocity (\(c_{i}\)) at the leading edge of the impeller and the area (\(A_{1}\)) on the perimeter of \(D_{1}\) are expressed. The components of the flow velocity vectors at the leading edge of the blade are then found as follows:  
\[\frac{A_{1}}{A_{p}}=\frac{v_{pi}}{c_{i}} \tag{19}\]  
\[c_{i}=A_{p}\cdot v_{pi}/A_{1} \tag{20}\]  
When calculating the area of the channel at \(D_{1}\), the part of the channel blocked by the blades is not considered, since the calculations are used only to reduce the range of the search field. Next, the component of the flow velocity vector in the tangential direction (\(u_{1}\)) with respect to \(D_{1}\) is calculated as follows:  
\[u_{1}=D_{1}\cdot\omega/2 \tag{21}\]  
Knowing the values of both velocity vector components, the leading edge of the blade can be expressed by \(\beta_{1}\) as follows:  
\[\beta_{i}=\tan^{-1}\frac{c_{i}}{u_{i}}\text{ or }\beta_{1}=\tan^{-1}\frac{c_{1} }{u_{1}}=29.3^{\circ} \tag{22}\]  
By conducting these calculations and making certain assumptions, the size of the search field for the initial simulations can be reduced. This is because there is no need to check the \(\beta_{1}\) parameter throughout the entire section if the assumption regarding the blade and flow angle coincidence is confirmed.