# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 2 Numerical Simulations
### Boundary Conditions  
In all the simulations under turbulent and laminar regime (for water and fruit juice, respectively), a steady-state model with three-dimensional incompressible flow was used. Regarding the boundary conditions, in both cases, velocity inlet was defined at the inlet section, considering uniform axial velocity and temperature, while the outlet condition was gauge pressure (\(P_{\text{outlet}}=0\)). For the turbulent simulations, at the inlet, the turbulent intensity \(I=5\,\%\) and hydraulic diameter (\(D_{h}=18\) mm) were defined. At the wall, no slip boundary condition (\(u=v=w=0\)) and a constant heat flux conditions of \(\dot{q}=20,000\,\text{W}/\text{m}^{2}\) were defined. In both cases, the working fluid had an inlet temperature of \(30\,^{\circ}\text{C}\).  
In the turbulent simulations with water, the thermophysical properties were assumed to be constant and are summarised in Table 1. In the case of the fruit juice, the regime was laminar and its density, thermal conductivity and specific heat were also defined as constant (see Table 1).  
Regarding the commercially available fruit juice, rheological properties were measured using a Modular Compact Rheometer (MCR-302, [PERSON], Ostfildern-Scharnhausen, Germany). All mechanical and electrical motor components are incorporated into one unit. The unit had a Peltier modular temperature device type C-PTD 200 linked to a counter-cooling by fluid circulator, which allowed controlling the temperature of the product to be analysed within a range of between \(-30\) and \(200\,^{\circ}\text{C}\). The viscometer had a high rotor speed with a powerful drive motor, which allowed a wide range of shear rates to be used. The measuring system was based on concentric cylinder geometry. The remaining fluid properties, such as density (\(\rho\)), thermal conductivity (\(\lambda\)) and specific heat (\(c_{p}\)) were known. Using the MCR system, the rheological behaviour of the fruit juice samples was measured at seven temperatures, ranging from \(24\) to \(55\,^{\circ}\text{C}\), considering a shear rate from \(1\) to \(800\) s\({}^{-1}\). The error expected for the temperature measurements and viscosity is \(0.2\,^{\circ}\text{C}\) and \(1.5\%\), respectively.  
The fluid behaviour of the fruit juice was modelled using the Power-Law model.  
\[\sigma=K(\gamma)^{n} \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fluid** & **Density (kg/m\({}^{3}\))** & **Thermal Conductivity (W/m K)** & **Specific Heat (J/kg K)** \\ \hline Water & 998.2 & 0.618 & 4175 \\ \hline Fruit juice & 1016.5 & 0.550 & 3910 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Water and fruit juice properties at the inlet.  
where \(\sigma\) is the shear stress exerted by the fluid, \(K\) is the consistency index, \(\gamma\) is the shear rate and \(n\) is the flow behaviour index.  
To analyse the effect of the temperature on the rheological model, the dependence of the consistency index was modelled using the Arrhenius equation, determining the activation energy (Ea)  
\[\eta=Ae^{(Ea/R_{u}T)} \tag{10}\]  
where \(\eta\) is the apparent viscosity, \(A\) is a constant, \(Ea\) is the activation energy, \(R_{u}\) is the universal gas constant and \(T\) is the absolute temperature.