# CFD Analysis of a Tubular Heat Exchanger for the Conditioning of Olive Paste
## 2 Materials and Methods
### Turbulence Model  
While the flow regime of the olive paste is certainly laminar, with regard to the water in the thin jacket the motion becomes turbulent (\(Re>\) 10,000). In this case, to solve the system, the turbulence model transport equations for the turbulent kinetic energy and its dissipation rate must be included. Flow Simulation uses the Favre-averaged Navier-Stokes equations, and to close the problem, transport equations for the turbulent kinetic energy and its dissipation rate are used (k-\(\epsilon\) model). In particular, thanks to the modified k-\(\epsilon\) model with damping functions proposed by [33], this system of equations are employed to describe both laminar and turbulent flows, and transition from a laminar to turbulent state and/or vice versa is also possible [34]. Turbulent kinetic energy:  
\[\frac{\partial}{\partial x_{i}}(\rho u_{i}k)=\frac{\partial}{\partial x_{i}} \bigg{(}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{ \partial x_{i}}\bigg{)}+S_{k} \tag{6}\]  
Turbulence dissipation energy:  
\[\frac{\partial}{\partial x_{i}}(\rho u_{i}\varepsilon)=\frac{\partial}{ \partial x_{i}}\bigg{(}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}} \bigg{)}\frac{\partial\varepsilon}{\partial x_{i}}\bigg{)}+S_{\varepsilon} \tag{7}\]  
where the source terms \(S_{k}\) and \(S_{\varepsilon}\) are  
\[S_{k}=\tau_{ij}^{R}\frac{\partial u_{i}}{\partial x_{j}}-\rho\varepsilon+\mu _{t}P_{B} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Fluid** & \(\rho\) **[kg/m\({}^{3}\)]** & \(K\) **[Pa s\({}^{n}\)]** & \(n\) & \(c_{p}\) **[k]/kgK** & \(k\) **[W/mK]** & \(\mu\) **[Pa s]** \\ \hline Olive paste & 1100 & 1200 & 0.1 & 3.31 & 0.46 & - \\ Water & 992.17 & - & - & 4.18 & 0.63 & \(6.53\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Thermo-physical and rheological parameter of the olive paste and water. \[S_{\varepsilon}=C_{\alpha 1}\frac{\varepsilon}{k}\Bigg{(}f_{1}\tau_{ij}^{R}\frac{ \partial u_{i}}{\partial x_{j}}+\mu_{t}C_{B}P_{B}\Bigg{)}-C_{\epsilon 2}f_{2}\frac{ \rho\varepsilon^{2}}{k} \tag{9}\]  
where \(P_{B}\) represents the turbulent generation due to the buoyancy force  
\[P_{B}=-\frac{g_{i}}{\sigma_{B}}\frac{1}{\rho}\frac{\partial\rho}{\partial x_{i}} \tag{10}\]  
with \(\tau_{ij}^{R}\) the Reynolds stress tensor [Pa], \(g_{i}\) is the gravitational acceleration [m/s\({}^{2}\)] in direction \(x_{i}\) and \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(C_{\mu}\), \(C_{\epsilon 1}\) and \(C_{\epsilon 2}\) are constants determined empirically. In SOLIDWORKS Flow Simulation, these constants assume the following values: \(\sigma_{k}=1\), \(\sigma_{\varepsilon}=1.3\), \(C_{\mu}=0.09\), \(C_{\epsilon 1}=1.44\) and \(C_{\epsilon 2}=1.92\). The constant \(\sigma_{B}\) is equal to 0.9, and \(C_{B}\) is 1 when \(P_{B}>0\), and 0 otherwise. Following Boussinesq assumption, the Reynolds stress tensor has the following form:  
\[\tau_{ij}^{R}=\mu_{t}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{ \partial x_{k}}\Bigg{)}-\frac{2}{3}\delta_{ij}\rho k \tag{11}\]  
with \(\delta_{ij}\) Kronecker delta function equals to  
\[\delta_{ij}=\left\{ \begin{array}{ll}1\ for\ i=j\\ 0\ for\ i\
eq j\end{array} \right. \tag{12}\]  
and \(\mu_{t}\) the turbulent eddy viscosity coefficient, defined as  
\[\mu_{t}=f_{\mu}\frac{c_{\mu}\rho k^{2}}{\varepsilon} \tag{13}\]  
where \(f_{\mu}\) is the turbulent viscosity factor expressed as  
\[f_{\mu}=\Big{(}1-e^{-0.0165R_{y}}\Big{)}\Big{(}1+\frac{20.5}{R_{T}}\Big{)} \tag{14}\]  
\[R_{y}=\frac{\rho\sqrt{ky}}{\mu} \tag{15}\]  
\[R_{T}=\frac{\rho k^{2}}{\mu\varepsilon} \tag{16}\]  
with \(y\) representing the distance from point to the wall, \(R_{y}\) is the Reynolds number depending on the avarge velocity of the fluctuations and the distance from the wall and \(R_{T}\) is the turbulent Reynolds number.