# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 2 Numerical Simulations
### Governing Equations  
In both simulations, under laminar and turbulent regime, the continuity equation (Equation (2)) was formulated as follows:  
\[\frac{\partial\left(\rho\,u_{i}\right)}{\partial x_{i}}=0 \tag{2}\]  
For the laminar simulation, the momentum (Equation (3)) and energy (Equation (4)) equations were given as follows:  
\[\frac{\partial}{\partial t}\left(\rho\,u_{i}\right)+\frac{\partial}{\partial x _{j}}\left(\rho\,u_{i}\,u_{j}\right)=-\frac{\partial p}{\partial x_{i}}+\frac {\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{\partial x_{j} }+\frac{\partial u_{j}}{\partial x_{i}}\right)\right] \tag{3}\]  
\[\frac{\partial}{\partial t}\left(\rho\,T\right)+\frac{\partial}{\partial x_{i }}\left(\rho\,u_{i}\,T\right)=\frac{\partial}{\partial x_{j}}\left[\frac{ \lambda}{c_{p}}\left(\frac{\partial T}{\partial x_{j}}\right)\right] \tag{4}\]  
With regard to the turbulent simulations, the equations were expressed in a Reynolds averaged Navier-Stokes (RANS) manner [23; 25], with the momentum (Equation (5)) and heat transport (Equation (6)) equations defined as follows.  
\[\frac{\partial}{\partial t}\left(\rho\,u_{i}\right)+\frac{\partial}{\partial x _{j}}\left(\rho\,u_{i}\,u_{j}\right)=-\frac{\partial p}{\partial x_{i}}+\frac {\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{\partial x_{j} }+\frac{\partial u_{j}}{\partial x_{i}}-\frac{2}{3}\,\delta_{ij}\,\frac{ \partial u_{k}}{\partial x_{k}}\right)\right]+\frac{\partial}{\partial x_{j}} \left(-\rho\,\overline{u_{i}^{\prime}\,u_{j}^{\prime}}\right) \tag{5}\]  
Figure 2: Mesh section for the turbulent numerical simulation using water of: Grid 1 (**a**); Grid 2 (**b**); and Grid 3 (**c**).  
\[\frac{\partial}{\partial t}\left(\rho\,T\right)+\frac{\partial}{\partial x_{i}} \left[u_{i}\left(\rho\,E+p\right)\right]=\frac{\partial}{\partial x_{j}}\left[ \left(\lambda+\frac{c_{p}\,\mu_{t}}{Pr_{t}}\right)\frac{\partial T}{\partial x _{j}}+\mu_{eff}\,u_{i}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\,\delta_{ij}\,\frac{\partial u_{k }}{\partial x_{k}}\right)\right] \tag{6}\]  
In addition, for the turbulent simulation, the Realizable \(k-\varepsilon\) model was used to model the turbulence [26], under which the modelled transport equations for \(k\) and \(\varepsilon\) were  
\[\frac{\partial}{\partial t}\left(\rho\,k\right)+\frac{\partial}{\partial x_{j }}\left(\rho\,k\,u_{j}\right)=\frac{\partial}{\partial x_{j}}\left[\left(\mu +\frac{\mu_{t}}{\sigma_{k}}\right)\,\frac{\partial k}{\partial x_{j}}\right] +G_{k}+G_{b}-\rho\,\varepsilon+S_{k} \tag{7}\]  
and  
\[\frac{\partial}{\partial t}\left(\rho\,\varepsilon\right)+\frac{\partial}{ \partial x_{j}}\left(\rho\,\varepsilon\,u_{j}\right)=\frac{\partial}{\partial x _{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\,\frac{ \partial\varepsilon}{\partial x_{j}}\right]+\rho\,C_{1}\,S_{\varepsilon}- \rho\,C_{2}\,\frac{\varepsilon^{2}}{k+\sqrt{\
u\,\varepsilon}}+C_{1\varepsilon }\,\frac{\varepsilon}{k}\,C_{3\varepsilon}\,G_{b}+S_{\varepsilon} \tag{8}\]  
According to Fluent [21], the coefficients can be defined as follows: \(C_{1}=\max(0.43,\,\eta/(\eta+5))\), \(\eta=Sk/\varepsilon\), \(S=\sqrt{2\,S_{ij}\,S_{ij}}\), \(S_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\right)\) and \(G_{k}=-\rho\,\overline{u_{i}^{\prime}\,u_{j}^{\prime}}\frac{\partial u_{j}}{ \partial x_{i}}\). The default values of the model constants were defined (\(C_{2}=1.9\), \(C_{1\varepsilon}=1.44\), \(\sigma_{k}=1.0\), \(\sigma_{\varepsilon}=1.2\)) and \(G_{b}\), \(C_{3\varepsilon}\), \(S_{k}\) and \(S_{\varepsilon}\) were not considered because gravitational force was neglected.