# Numerical Calculation and Experimental Study of the Axial Force of Aero Fuel Centrifugal Pumps
## 4 Theoretical Calculation Formula of Axial Force
### Calculation Method Modification  
Referring to experimental data and simulation results, the pressure correction coefficient was introduced to correct the calculation formula of pressure distribution in the pump chamber [8; 30]. The formula for calculating the static pressure distribution of the cover plate can be modified, as follows:  
\[p=k_{1}\cdot p_{2}-k_{2}\cdot\frac{\rho u_{2}^{2}}{8}(1-\frac{R^{2}}{R_{2}^{2 }}) \tag{5}\]  
where \(p_{2}\) is the static pressure at the impeller outlet, Pa; \(k_{1}\) is the correction coefficient of static pressure term in the rear pump chamber; and \(k_{2}\) is the correction factor of pressure drop in the rear pump chamber. Then, the liquid pressure of the rear pump chamber on the rear cover plate is, as follows:  
\[F_{1}=\pi(R_{2}^{2}-R_{\rm h1}^{2})\left[k_{1}p_{2}-k_{2}\frac{\rho\omega^{2 }}{8}(\frac{R_{2}^{2}-R_{\rm h1}^{2}}{2})\right] \tag{6}\]  
where \(R_{\rm h1}\) is the radius of the driving shaft, m, and \(R_{2}\) is the impeller radius, m.  
Similarly, the pressure of the front pump chamber on the front cover plate of the impeller is, as follows:  
\[F_{2}=\pi(R_{2}^{2}-R_{\rm m}^{2})\left[k_{3}p_{2}-k_{4}\frac{\rho\omega^{2}} {8}(\frac{R_{2}^{2}-R_{\rm m}^{2}}{2})\right] \tag{7}\]  
where \(R_{\rm m}\) is the radius of the mouth ring; \(k_{3}\) is the correction coefficient of the static pressure item of the front pump chamber; and \(k_{4}\) is the pressure drop correction factor of the front pump chamber. The multiple linear regression function in Matlab 2016 is used to fit the pressure distribution curve of the pump chamber and the static pressure correction coefficient and pressure drop correction coefficient of the front and rear pump chamber are obtained, as shown in Table 2.  
Let the static pressure at the outlet of the inducer (an inlet of the centrifugal impeller) be \(p_{3}\) and the axial force on the inducer can be approximated as the product of the static pressure difference at both ends of the inducer and the cross-sectional area of the impeller runner. In order to verify the accuracy of the simplified calculation method, under different flow conditions at rated speed, the axial force on the inducer impeller surface was directly integrated by the post-processing software and the axial force calculated by the pressure difference at the inducer inlet and outlet surface was compared, as shown in Figure 14. As can be seen from the figure, at a certain speed, the axial force of the induction wheel gradually decreases with the increase in fuel flow. When the flow rate increases from 13 m\({}^{3}\)/h to 55 m\({}^{3}\)/h, the axial force induction wheel obtained by numerical simulation \begin{table}
\begin{tabular}{c c c c} \hline \hline \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(k_{4}\) \\ \hline
1.016 & 0.746 & 1.048 & 1.435 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correction factor of the pressure in the front and rear pump chamber. decreases from 446 N to 329 N, which is reduced by 117 N. At the same time, it can be seen that the induction wheel axial force obtained by the simplified calculation method is basically consistent with the numerical simulation value and the maximum error is 6.9%. Thus, the induction wheel axial force can be calculated more accurately by this method.