# CFD Analysis of a Tubular Heat Exchanger for the Conditioning of Olive Paste
## 2 Materials and Methods
### Mesh Generation, Mesh Independence Study and Validation  
In the mesh generation process, the computational domain is divided into uniform rectangular parallelepiped-shaped cells (basic mesh). Then, by means of various refinements criteria, the basic mesh cells are split into smaller cells, thus better representing the model and fluid regions. The resulting computational mesh consists of cells, which are divided into: fluid cells located entirely in the fluid, solid cells located entirely in the solid and partial cells, which are partly in the solid and partly in the fluid.  
To choose the grid number elements, a mesh independence study was conducted by comparing the pressure drop of the olive paste obtained with four different grids (Table 3). The difference of the obtained results were very small, about 5%, exception made for the mesh 1 at 2000 kg/h (about 10%). Mesh 2 (Figure 2) resulted to be the best choice in terms of both obtained results and computation time.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline  & **Mesh 1** & **Mesh 2** & **Mesh 3** & **Mesh 4** \\ \hline N\({}^{\circ}\) elements & 324,301 & 569,322 & 964,659 & 2,163,502 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Number of elements of each mesh.  
Figure 2: Particular view of the generated mesh (Mesh 2).  
To validate the model, as reported in [29] and [36], it is possible to compare the results of the simulation with those obtained by a theoretical analysis. In particular, the pressure drop for a pseudoplastic fluid in laminar flow can be determined through the Fanning equation:  
\[\Delta P=2f\frac{L}{D}\rho v^{2} \tag{23}\]  
where \(f=16/Re_{G}\) is the Fanning friction faction coefficient (with \(Re_{G}\) calculated according to Equation (5)). To take into account the pressure drop due to the two elbows (r/D of about 1.75), it is possible to use the following expression:  
\[\Delta P_{f}=k_{f}\rho\frac{v^{2}}{2} \tag{24}\]  
where the resistance factor \(k_{f}\) depends on the type of fitting present in the pipe. In the case of non-Newtonian fluids, it can be used the relationship provided by [37] for an elbow:  
\[k_{f}=191(Re_{G})^{-0.896} \tag{25}\]  
Regarding the heat transfer coefficient, it can be calculated by means of the Nusselt number (\(Nu\)) that for a non-Newtonian fluid in laminar flow is expressed as:  
\[Nu=1.615\bigg{[}RePr\bigg{(}\frac{D}{L}\bigg{)}\bigg{]}^{0.33}\bigg{(}\frac{ \mu}{\mu_{w}}\bigg{)}^{0.33} \tag{26}\]  
where \(\mu\) is the apparent viscosity of the bulk fluid and \(\mu_{w}\) the fluid apparent viscosity at the wall. It is worth noting that Equation (26) is based on the velocity profile at boundary layer; thus, \(Re\) number must be calculated with \(\mu_{w}\). On the contrary, \(Pr\) depends on viscous dissipation and conduction in the bulk fluid; thus, \(\mu\) must be used.  
Figure 3a shows the comparison of the simulated results of pressure drop and the theoretical values calculated by means of Equations (23)-(25). The model results deviate less then 5% from theoretical results at lower paste flow rate, and a decrease in this difference is appreciable with the increase in the flow rate. Figure 3b reports the comparison between the simulated heat transfer coefficients and theoretical ones, obtained according to Equation (26). The model slightly underestimates the heat transfer for lower values of mass flow rate, while slightly overestimates it when the mass flow rate increases, reaching a maximum difference of about 11.5%.