# Effect of Electric Ducted Fans Structural Arrangement on Their Performance Characteristics
## 2 Materials and Methods  
The objective of the study was to determine the most suitable EDF design solution for vertical thrust generation with the criterion of obtaining the highest thrust value with the lowest possible electric consumption. For the purposes of the study, two reference EDFs propulsion units were used for comparison with the proposed EDF design modifications. The use of reference EDFs enabled to monitor the effect of the modifications on the performance parameters. The reference EDFs were from QX-Motor (QX-Motor CO., Ltd., Kowloon, Hong Kong) with diameter of 50 mm and 64 mm, both with 12 blade fan and constant cross section of the flow channel. Both EDFs were driven by the same 5000 kV QF2611 electric motor. The EDFs were chosen mainly due to the declared maximum thrust value, which reaches up to 0.95 kgf (approx. 9.32 N) and 5000 kV value of the supplied electric motor, allowing testing at a up to 64,000 RPM. Figure 1 shows dimension of used fans and electric motor. Please note, for practical reasons, kilogram-force (kgf) as a unit of thrust is primarly used in this article. The relationship between units is given as 1 kgf = 9.80665 N.  
The design changes included mainly the flow channel, as changes in the ratio between the outlet cross section and fan area (\(a_{w}\)). The effect of a change in this parameter was determined by theoretical calculations based on the Bernoulli equation [20] and the continuity equation [21] in simplified form and does not include real conditions parameters. The reason for this is complexity and lack of parameters that are related to real conditions,  
Figure 1: Dimensions of (**A**) \(\diameter50\) and (**B**) \(\diameter64\) reference electric ducted fans and (**C**) used electric motor. such as roughness of the cowling surface or pressure changes through the flow channel [20]. However, this has no impact to the results presented below in the article as these parameters are same for all tested prototypes. Therefore, the results presented are still relevant. In the case of EDF, the total pressure before the fan and the total pressure behind the fan are important. The total pressure before the fan (\(p_{T1}\)) for static testing is [20]:  
\[p_{T1}=p_{1}+\frac{1}{2}\rho v_{1}^{2}=p_{2a}+\frac{1}{2}\rho v_{2}^{2}, \tag{1}\]  
where \(p_{1}\) is the static pressure before entering the EDF, \(\rho\) is the air density, \(v_{1}\) is the air velocity before entering the EDF, \(p_{2a}\) is the static pressure at the location just before the fan, and \(v_{2}\) is the air velocity at the location just before the fan [9]. Since it is a static test, the speed \(v_{1}=0\), and, therefore:  
\[p_{T1}=p_{1}=p_{2a}+\frac{1}{2}\rho v_{2}^{2}. \tag{2}\]  
The total pressure behind the fan (\(p_{T2}\)) is then [20]:  
\[p_{T2}=p_{2b}+\frac{1}{2}\rho v_{2}^{2}=p_{3}+\frac{1}{2}\rho v_{3}^{2}, \tag{3}\]  
where \(p_{2b}\) represents the static pressure just behind the fan. The velocity \(v_{2}\) is the same as in Equations (1) and (2). At the output of the flow channel, there is the static pressure \(p_{3}\) and velocity \(v_{3}\). The described equations are then valid for the case of constant cross section [9]. As mentioned, the velocity (\(v_{2}\)) of the air stream is the same immediately before and behind the fan. However, behind the fan, there is an increase in static pressure (\(p_{2b}>p_{2a}\)) due to the accumulation of air in the closed tube as the expansion of the air toward the exit is slower than the speed of the air supply by the fan. In an ideal constant flow tube, the air is no longer affected in any way and the static pressure is still the same toward the outlet. The fan thrust \(T_{v}\) is thus defined only by the increase of the static pressure behind the fan compared to the static pressure in front of the fan [9], that is:  
\[T_{v}=(p_{2b}-p_{2a})A_{2}. \tag{4}\]  
EDF thrust generally consists of the fan thrust \(T_{v}\) and the cowling thrust, which is the result of the adhesion forces on the cowling surfaces [9].