# Simulation of the Influence of Wing Angle Blades on the Performance of Counter-Rotating Axial Fan
## 4 Analysis of Optimization Results
### Entropy Production Analysis of Fan  
For axial fans, the existence of a vortex structure in the internal flow field will inevitably increase the entropy production of the system, which in turn will cause the internal energy loss of the fan. Entropy generation theory can evaluate the energy dissipation inside the fan, so more and more scholars use entropy generation analysis to study the internal efficiency of the fan [22; 23; 24]. As the internal temperature change of the axial flow fan is very small during operation, the entropy production caused by the temperature change is ignored. Therefore, the total entropy production rate S is composed of two parts, namely, the time-averaged entropy production rate \(S_{PRO,D}\) produced by the turbulent dissipation of the time-averaged flow field, and the pulsating entropy production rate \(S_{PRO,D}\) caused by the pulsating velocity. Among them, the pulsating entropy generation rate \(S_{PRO,D}\) cannot be directly calculated due to the use of the RANS equation in the numerical calculation, but [PERSON] found that \(S_{PRO,D}\) is related to the turbulent dissipation rate \(\epsilon\) through verification [25]. Therefore, the final time average entropy production rate \(S_{PRO,D}\)' the pulsating entropy production rate \(S_{PRO,D}\)' and the total entropy production formula are shown as Equations (5)-(7), respectively. \[ \begin{array}{l}S_{PRO,D}=\frac{\mu}{\hbar}\Big{\{}2{[{{{(\frac{\partial \overline{\mu}}{\partial x})}^{2}}+{{(\frac{\partial\overline{\mu}}{\partial y })}^{2}}+{{(\frac{\partial\overline{\mu}}{\partial z})}}}]}\\ +{{(\frac{\partial\overline{\mu}}{\partial y}+\frac{\partial\overline{\mu}}{ \partial x})}^{2}}+{{(\frac{\partial\overline{\mu}}{\partial x}+\frac{\partial \overline{\mu}}{\partial x})}^{2}}+{{(\frac{\partial\overline{\mu}}{\partial x }+\frac{\partial\overline{\mu}}{\partial y})}^{2}}\Big{\}}\end{array}  \tag{5}\]  
Figure 8: Distribution of plane A, B, and C positions. \[S_{PRO,D^{\prime}}=\frac{\rho\varepsilon}{\overline{T}} \tag{6}\]  
\[\hat{S}=\int_{V}S_{PRO,D}\text{d}V+\int_{V}S_{PRO,D^{\prime}}dV \tag{7}\]  
In the formula above, \(\overline{u}\), \(\overline{v}\), and \(\overline{w}\) are the time-average velocity components of velocity in the \(x\), \(y\), and \(z\) directions, m/s; \(\overline{T}\) is the time-average temperature, K, because the numerical calculation ignores the influence of temperature changes on the flow field, \(\overline{T}=300\) K; \(\rho\) is the density, kg/m\({}^{3}\); \(\varepsilon\) is the turbulent dissipation rate, the formula is \(\varepsilon=1.5\) (\(0.16\)U\(\cdot\)Re \(-\) 0.125) 1.5, m\({}^{-2}\cdot\)s\({}^{-3}\), where U is the average velocity of the target fluid, m/s, Re is the Reynolds number; V is the volume of the control body, m\({}^{3}\). Figure 10 shows the distribution of the total entropy production rate S in the primary and secondary blades. From the figure, it can be seen that the total entropy production rate S of the secondary leaves is greater than that of the primary leaves, and the entropy production rate of the tip part is the highest. This is because after the fluid enters the  
Figure 9: Static pressure distribution on plane A, B, and C. (**a**) Static pressure distribution at the section of the first-stage impeller; and (**b**) static pressure distribution at the section of the second-stage impeller. secondary impeller through the pressurization of the primary impeller, its flow velocity must be greater than that of the primary impeller, and the fluid velocity at the tip of the blade is the largest. It can be seen from Formula (4) and Formula (5) that the time-averaged entropy production rate \(S_{\text{PRO,D^{\prime}}}\) and the pulsating entropy production rate \(S_{\text{PRO,D^{\prime}}}\) are both positively correlated with the flow velocity, and the flow field at the tip clearance is relatively complicated.