# Effects of an Inclined Blade on the Performance of a Sirocco Fan
## 2 Numerical Methods
### Numerical Simulation Method  
The pattern of fluid flow in centrifugal fans is governed by the Navier-Stokes equation, expressed as:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0, \tag{1}\]  
\[\frac{\partial\Big{(}\rho u_{j}u_{i}\Big{)}}{\partial x_{j}}=f_{i}-\frac{ \partial P^{*}}{\partial x_{i}}+\frac{\partial\Big{[}\mu_{x}\Big{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)} \Big{]}}{\partial x_{j}}, \tag{2}\]  
where \(x_{i}\) and \(x_{j}\) are the components of the Cartesian coordinate system in three directions \(x\), \(y\), and \(z\); \(u_{i}\) is the velocity component \(u\), \(v\), \(w\); \(P^{*}\) represents the conversion pressure (including turbulent energy and centrifugal force); the volume force component is \(f_{i}\); the effective viscosity coefficient is \(\mu_{\varepsilon}\). The RNG \(k\)-\(\varepsilon\) model was applied to consider the turbulent flow:  
\[\mu_{\varepsilon}=\mu+\mu_{t} \tag{3}\]  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}, \tag{4}\]  
in which the molecular viscosity coefficient is \(\mu\) and turbulent eddy viscosity coefficient is \(\mu_{t}\). The RNG \(k\)-\(\varepsilon\) turbulence model includes equations for turbulent energy and dissipation:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_ {i})=\frac{\partial}{\partial x_{j}}\bigg{(}\alpha_{k}\mu_{eff}\frac{ \partial k}{\partial x_{j}}\bigg{)}+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k}, \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i} }(\rho\varepsilon u_{i})=\frac{\partial}{\partial x_{j}}\bigg{(}\alpha_{ \varepsilon}\mu_{eff}\frac{\partial\varepsilon}{\partial x_{j}}\bigg{)}+C_{1 \varepsilon}\frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon }\rho\frac{\varepsilon^{2}}{k}-R_{\varepsilon}+S_{\varepsilon}, \tag{6}\]  
\[R_{\varepsilon}=\frac{C_{\mu}\rho\eta^{3}(1-\eta/\eta_{0})}{1+\beta\eta^{3}} \frac{\varepsilon^{2}}{k}, \tag{7}\]  
where \(G_{k}\) is the turbulent energy generated by the laminar velocity gradient; \(G_{b}\) is the turbulent flow energy generated by buoyancy; \(Y_{M}\) is fluctuations in transitions in compressible turbulence; \(C_{1\varepsilon}\), \(C_{2\varepsilon}\) and \(C_{3\varepsilon}\) are constant; \(\alpha_{k}\) is the turbulent Prandtl number of the \(k\)-equation; \(\alpha_{\varepsilon}\) is the turbulent Prandtl number of the \(\varepsilon\)-equation; \(S_{k}\) and \(S_{\varepsilon}\) are user-defined quantities.  
A steady-state simulation adopting ANSYS-Fluent was performed for the fan. The mass flow rate and the pressure boundary conditions were separately set at the inlet and outlet boundary of the computational domain. Air at 25 \({}^{\circ}\)C was used as the working fluid. Therefore, the aerodynamic loads were the main sources of loading. The RNG \(k\)-\(\varepsilon\) model was used with the standard wall function adopted near the wall, which could provide a reliable prediction for the wall-bounded flow, and the wall distance y\({}^{+}\) for the first layer node at the wall was around 15. The governing equations were discretized by the finite volume method; the convection term was treated by the second-order upwind scheme and the diffusion term by the second-order central difference scheme. No slip boundary condition was applied to the wall. The pressure and velocity coupling method was the SIMPLE algorithm. The multiple reference frame model (MRF) was set up to exchange of information between the rotating and the stationary domains. The residual of the iteration was set to \(10^{-5}\) for the mass, momentum, and energy equations.