# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 2 Numerical Simulations
### Numerical Procedure  
Software Fluent (v. 19.2) [21] was used to compute fluid flow and heat transfer. In both cases, a double-precision and pressure-based solver was used to carry out the simulations. The Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm was used to couple the velocity and pressure fields [25]. The least square cell based option was selected for the spatial discretisation of the gradient, and second order upwind for the solution of the governing equations. The convergence criterion for the continuity equation was normalised residual values of \(10^{-5}\) and \(10^{-6}\) for the remaining variables.  
To analyse the flow behaviour along the computational domain, several planes were defined (Figure 3). Moreover, the computational domain was divided into three sections, one section at the straight inlet pipe (between Planes 2 and 8), one at the straight outlet pipe (between Planes 12 and 18) and another at the elbow (between Planes 8 and 12).  
At several specified locations, the local Nusselt number (\(Nu_{x}\)) and local heat transfer coefficient (\(h_{x}\)) were computed as follows:  
\[Nu_{x}=\frac{h_{x}\,D}{\lambda_{x}} \tag{11}\]  
and  
\[h_{x}=\frac{\dot{q}}{T_{w_{x}}-T_{f_{x}}} \tag{12}\]  
Figure 3: Plane locations along the computational domain.  
where \(D\) is the inner tube diameter, \(\lambda_{x}\) is the fluid thermal conductivity at temperature \(T_{f_{x}}\), \(\dot{q}\) is the heat flux in the tube, \(T_{w_{x}}\) is the average inner wall temperature at position \(x\) and \(T_{f_{x}}\) is the average bulk fluid temperature at position \(x\).  
In addition, the pressure was determined at each plane location, so it was possible to determine the pressure drop (\(\Delta p\)) along the curved pipe as well as the Fanning friction factor (\(C_{f}\)) using the following equation [27]:  
\[C_{f}=\frac{\Delta p\,D}{2\,\rho\,L\,u^{2}} \tag{13}\]  
where \(C_{f}\) is the Fanning friction factor, \(\Delta p\) is the total pressure difference, \(\rho\) is the density, \(L\) is the length of the pressure section and \(u\) is the average fluid velocity.  
With regard to the turbulent regime with water, to compare the results obtained from the simulations, the Fanning friction factor (\(C_{f_{s}}\)) [27] and the Nusselt number (\(Nu_{s}\)) [28] for a straight smooth tube were computed as reference values using the following equations, respectively:  
\[C_{f_{s}}=\frac{0.3164\,Re^{-0.25}}{4} \tag{14}\]  
and  
\[Nu_{s}=\frac{\frac{f}{2}\,Re\,Pr}{1.07+12.7\,\sqrt{\frac{f}{2}\,\left(Pr^{2/3} -1\right)}} \tag{15}\]  
where the variable \(f\) for Equation (15) is given by:  
\[f=\frac{1}{4\left(1.82\log\left(Re\right)-1.64\right)^{2}} \tag{16}\]  
For the turbulent regime with water, Reynolds number (Re) of \(45\times 10^{3}\) was considered at the inlet, calculated as follows:  
\[Re=\frac{\rho\,u\,D}{\mu} \tag{17}\]  
where \(\rho\) is the density, \(u\) is the average fluid inlet velocity, \(D\) is the inner diameter and \(\mu\) is the dynamic viscosity.  
Regarding the laminar simulations, the Fanning friction factor numerically obtained was compared to the theoretical Fanning factor obtained under laminar regime, defined as follows.  
\[C_{f_{s}}=\frac{16}{Re_{g}} \tag{18}\]  
In this case, the generalised Reynolds number (\(Re_{g}\) = 476), which is commonly implemented for non-Newtonian fluids for duct geometries with an uniform cross-section [29; 30], was used at the inlet, and defined according to the following equation.  
\[Re_{g}=\frac{\rho\,u^{2-n}\,D^{n}}{K^{{}^{\prime}}\,8^{n-1}} \tag{19}\]  
and  
\[K^{{}^{\prime}}=K(\frac{3n+1}{4n})^{n} \tag{20}\]  
where \(\rho\) is the density, \(u\) is the average fluid inlet velocity, \(n\) is the flow behaviour index, \(D\) is the inner diameter, \(K^{{}^{\prime}}\) is a constant and \(K\) is the consistency index.