# CFD Analysis of a Tubular Heat Exchanger for the Conditioning of Olive Paste
## 2 Materials and Methods
### Turbulence Model
The Lam and Bremhorst's damping functions are calculated as  
\[f_{1}=1+\left(\frac{0.05}{f_{\mu}}\right)^{3} \tag{17}\]  
\[f_{2}=1-e^{-R_{T}^{2}} \tag{18}\]  
The damping functions \(f_{\mu}\), \(f_{1}\) and \(f_{2}\), in the case of a too small \(R_{y}\), decrease turbulent viscosity and turbulence energy and increase the turbulence dissipation rate. It is worth noting that in the case of damping functions equal 1 the standard k-\(\varepsilon\) model is obtained. In addition to turbulence modeling, when simulating flows, it is also necessary to simulate fluid boundary layer effects near solid bodies or walls that can be difficult to resolve due to high velocity and temperature gradients across these near-wall layers. Flow simulation uses the two-scale wall function (ZSWF) approach that consists of two methods for coupling the boundary layer calculation with the solution of the bulk flow [34]:1. A \"thin\" boundary layer treatment that is used when the number of cells across the boundary layer is not enough for direct, or even simplified, determination of the flow and thermal profiles;
2. A \"thick\" boundary layer approach when the number of cells across the boundary layer exceeds that required to accurately resolve the boundary layer. In intermediate cases, a combination of the two above approaches is used. Hence, the software chooses the specific approach based on the mesh refinement near the solid-fluid interface. In the thin boundary layer approach, the Prandtl boundary layer equations already integrated along the normal-to-the-wall (i.e., along the y ordinate) from 0 (at the wall) to the boundary layer thickness are solved along a fluid streamline near the wall. If the boundary layer is laminar, these equations are solved with a method of successive approximations based on the Shvetz trial functions technology [34]. If the boundary layer is turbulent or transitional (between laminar and turbulent), a generalization of this method to such boundary layers employing the [PERSON] hypothesis about the mixing length in turbulent boundary layers is used [35]. From the boundary layer calculation, the boundary layer thickness \(\delta\), the wall shear stress \(\tau\) and the heat flux from the fluid to the wall are obtained, which are used as boundary conditions for the Navier-Stokes equations. When the near-wall computational mesh cell's fluid mass center is located inside the boundary layer (thick boundary layer), instead of y, Flow Simulation uses the dimensionless longitudinal velocity \(u^{+}\), which depends on the dimensionless wall distance \(y^{+}\): \[y^{+}=\frac{\sqrt{\rho\tau_{w}}y}{\mu}\] (19) \[u^{+}=\frac{u}{\sqrt{\frac{\tau_{w}}{\rho}}}\int_{0}^{y^{+}}\frac{2d\eta}{1+ \sqrt{1+4K^{2}\eta^{2}\left(1-e^{-\frac{\eta}{\Delta\sigma_{c}}}\right)^{2}}}\] (20) where \(\tau_{w}\) is the shear stress near the wall, \(K=0.4504\) is the Karman constant and \(A_{v}=26\) is the Van Driest coefficient. As regards, the diffusive heat flux [W/m\({}^{2}\)], it is defined as: \[q_{i}=\left(\frac{\mu}{Pr}+\frac{\mu_{t}}{\sigma_{c}}\right)\frac{\partial h }{\partial x_{i}}\] (21) with the constant \(\sigma_{c}=0.9\), \(Pr\) is the Prandtl number and \(h\) is the thermal enthalpy [J/kg]. Finally, it is important to highlight that these equations are suitable to describe both turbulent and laminar flows regime. In fact, in the latter case, \(k\) and \(\mu_{t}\) are equal to zero.