## 3 Numerical Model  
### Governing Equations  
Air flow is assumed to be steady. A general differential equation of flow and heat transfer is governed by the following equation [31]:  
\[\frac{\partial(\rho u\varnothing)}{\partial x}+\frac{\partial(\rho v\varnothing )}{\partial y}+\frac{\partial(\rho w\varnothing)}{\partial z}=\frac{\partial} {\partial x}\left[\Gamma\frac{\partial\varnothing}{\partial x}\right]+\frac{ \partial}{\partial y}\left[\Gamma\frac{\partial\varnothing}{\partial y}\right]+ \frac{\partial}{\partial z}\left[\Gamma\frac{\partial\varnothing}{\partial z }\right]+S \tag{5}\]  
where \(\varnothing\) is the general dependent variable, which can represent \(u\), \(v\), \(w\), \(k\), \(\varepsilon\), \(T\), and \(1\). \(\rho\) is the air density; \(\Gamma\) is the generalized diffusion coefficient; and \(S\) is the source term. The specific forms of theseequations are listed in Table 1. In Table 1, \(\mu_{t}\) is the turbulent viscosity coefficient, and \(G\) is the turbulent energy generation rate derived from the following equations. \[\mu_{t}=\frac{C_{\mu}k^{2}}{\epsilon} \tag{6}\]  
\[G=\mu_{t}\left\{2\left[\left[\frac{\partial u}{\partial x}\right]^{2}+\left[ \frac{\partial v}{\partial y}\right]^{2}+\left[\frac{\partial w}{\partial z} \right]^{2}\right]+\left[\frac{\partial u}{\partial y}+\frac{\partial v}{ \partial x}\right]^{2}+\left[\frac{\partial u}{\partial z}+\frac{\partial w}{ \partial x}\right]^{2}+\left[\frac{\partial v}{\partial z}+\frac{\partial w}{ \partial y}\right]^{2}\right\} \tag{7}\]  
\(\rho G-\rho\epsilon\) is the net source term. The constants of the turbulence model in Table 1 are as follows: \(C_{\mu}=0.09,\ C_{1}=1.44,\ C_{2}=1.92,\ \sigma_{k}=1.0,\ \text{and}\ \sigma_{ \epsilon}=1.3.\) where \(P_{r}\) is the Prandlt number of fluid and \(P_{T}\) is the turbulent Prandlt number. The treatment of buoyancy is based on the Boussinesq approximation of density variation and temperature change. The air flow in the computational domain is assumed to be incompressible. The variation in air density in the tower is mainly caused by air temperature according to the following equation:  
\[\rho=1.26\times 10^{-5}T^{2}-4.63\times 10^{-3}T+1.2926 \tag{8}\]  
It is difficult to draw a grid for the complex structure of the radiator in the cooling tower, especially the finned tube bundles. In FLUENT, the radiator model is an effective method to manage the structure. The resistance in air flow caused by the radiators is expressed as a drag source term in a momentum equation:  
\[\Delta p=\frac{1}{2}k_{v}\rho V_{n}^{2} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Equation Name** & \(\varnothing\) & \(\mathbf{\Gamma}\) & \(\mathbf{S}\) \\ \hline Continuous & 1 & - & - \\ X-momentum & \(\mu\) & \(\mu+\mu_{t}\) & \(-\frac{\partial p}{\partial x}+\frac{\partial}{\partial x}\left[\mu_{eff}\frac {\partial u}{\partial x}\right]+\frac{\partial}{\partial y}\left[\mu_{eff} \frac{\partial v}{\partial x}\right]+\frac{\partial}{\partial z}\left[\mu_{ eff}\frac{\partial w}{\partial x}\right]\) \\ Y-momentum & \(v\) & \(\mu+\mu_{t}\) & \(-\frac{\partial p}{\partial y}+\frac{\partial}{\partial x}\left[\mu_{eff}\frac {\partial u}{\partial y}\right]+\frac{\partial}{\partial y}\left[\mu_{eff} \frac{\partial v}{\partial y}\right]+\frac{\partial}{\partial z}\left[\mu_{ eff}\frac{\partial w}{\partial y}\right]\) \\ Z-momentum & \(w\) & \(\mu+\mu_{t}\) & \(-\frac{\partial p}{\partial z}+\frac{\partial}{\partial x}\left[\mu_{eff}\frac {\partial u}{\partial z}\right]+\frac{\partial}{\partial y}\left[\mu_{eff} \frac{\partial v}{\partial z}\right]+\frac{\partial}{\partial z}\left[\mu_{ eff}\frac{\partial w}{\partial z}\right]\) \\ Turbulent kinetic energy & \(k\) & \(\mu+\frac{\mu_{t}}{\sigma_{k}}\) & \(\rho G-\rho\epsilon\) \\ Turbulence dissipation rate & \(\epsilon\) & \(\mu+\frac{\mu_{t}}{\sigma_{\epsilon}}\) & \(\frac{\epsilon}{k}(C_{1}G-C_{2}\rho\epsilon)\) \\ Energy equation & \(T\) & \(\frac{\mu}{P_{r}}+\frac{\mu_{t}}{P_{T}}\) \\ \hline \hline \end{tabular}
\end{table}