# The Optimization of the Geometry of the Centrifugal Fan at Different Design Points
## 2 Materials and Methods  
Virtual experiments require a 3D model of the test object, so the fan is modeled using the SolidWorks computer simulation software (version 2019; manufacturer: Dassault Systemes SE; location: 3 DS Paris Campus, 10 Rue [PERSON], 78140 Velizy-Villacoublay, France). The geometry consists of two parts: the fan impeller and the housing. The dimensions of the model are based on the dimensions of the Ziehl-Abegg product [22]. The centrifugal fan housing has an adjustable radius with dimensions given in [PERSON]'s Fan Handbook [23; 24]. There is a diagram in the manual that describes the relationship between the diameter of the impeller (d\({}_{1}\)) and the quarter radii of the fan housing. The centers of curvature of the envelope helix are located more than 0.0625-d\({}_{1}\) from the origin of the coordinates. The radii of convexity of the quarters (moving away from the housing outlet) are 0.712-d\({}_{1}\), 0.837-d\({}_{1}\), and 0.962-d\({}_{1}\). The outer perimeter smoothly transitions. The height of the fan body is taken to be 86 mm, and the width of the outlet channel is 94 mm, paying attention to the dimensions of products common on the market (Figure 1). Figure 1 shows a comparison between a typical impeller body and the one used in this work. [PERSON]'s Fan Handbook provides a schematic diagram describing the relationship between the impeller diameter and the quarter radii of the fan casing. The impeller model consists of two lower and upper discs, with 33 blades arranged between them. In conformity with the genuine fan design, the overall impeller height measures 60 mm. During geometry optimization, angles \(\beta_{1}\) and \(\beta_{2}\) (which represent the vane's leading and trailing angle, respectively) are considered parametric model elements that can be adjusted. The values for D\({}_{1}\) and D\({}_{2}\) are considered fixed at 116 mm and 140 mm, respectively. As these values remain constant, the model possesses an adequate number of parameters to effectively define the blade's shape (Figure 2). A model of an EC (electronically commuted) motor is built using the dimensions provided in the Ziehl-Abegg manufacturer's catalog [22]. This model accurately simulates  
Figure 1: The geometry of (**a**) is a typical example, and (**b**) represents the fan used in this work. (**a**) Typical 36.5-inch impeller body (Bleier [23; 24]) (measurements presented in inches); (**b**) perimeter of the case model developed for the 140 mm impeller (measurements presented in mm). Figure 2: Fan blade geometry: (**a**) 3D model; (**b**) blade geometry design variables. the working conditions of an actual impeller attached to an exterior rotor. A general model of the centrifugal fan used in the study is created by combining the individual modeled parts. The primary factors determining the shape of the air blade are the angles at which the blade starts (\(\beta_{1}\)) and ends (\(\beta_{2}\)). To ensure a consistent flow on the blade's surface under specific working conditions, it is crucial to accurately select the values for these parameters. This study focuses on fans equipped with impellers that have forward-facing blades. This means that for angles \(\beta_{1}\) and \(\beta_{2}\), which represent the specific blade positions, a boundary condition is set, stating that angles must be less than \(90^{\circ}\). To determine the optimal blade parameters, separate evaluations are conducted for the first and second design points. The values of the \(\beta_{1}\) and \(\beta_{2}\) parameters are adjusted in increments to identify the most favorable outcomes. Over a thousand impellers, each with unique geometry parameters, are tested during this study. The efficiencies of impellers are calculated at specific design points, and the impellers with the highest total efficiency are discovered. The initial operating point reflects the peak power consumption as the fan operates at 1930 rpm with maximum air flow. This condition occurs because there is no static pressure resistance in the system.