# Design and Material Optimization of Oil Plant Piping Structure for Mitigating Erosion Wear
## 2 CFD Modeling
### Turbulence Model  
In this study, due to the presence of bitumen particles in the fluid, the flow was inherently turbulent. While many methods exist, for CFD analysis, k-\(\epsilon\) and k-\(\omega\) are the common models. The turbulence epsilon model differs from the k-\(\omega\) with its focus on predicting the dissipation rate of turbulent kinetic energy through the solution of transport equations and provides a more detailed representation of turbulence. Therefore, the standard k-\(\epsilon\) turbulence model was employed to characterize the behavior of the particulate phase throughout the flow domain. The TDR, which denotes the scale of turbulence intensity, was also a focal point of this investigation as it is closely related to erosion wear; higher TDR values signify increased turbulence intensity, consequently augmenting the kinetic energy of particles suspended in the fluid. This elevated kinetic energy results in more vigorous impacts on the pipe walls, leading to escalated erosion wear. To monitor turbulence during the transit of the particle phase, the dissipation rate (\(\epsilon\)) and turbulent kinetic energy (k) equations were utilized. The turbulent kinetic energy (K) quantifies the energy within turbulence owing to mean velocity gradients and the effect of buoyancy. The standard k-\(\epsilon\) model effectively captures flow conditions, especially in the near-wall region of a particulate-liquid medium, with enhanced accuracy, rendering it an appropriate choice for the DPM in erosion wear prediction. To compute both the TDR and turbulent kinetic energy (\(k\)), the following equations, as expressed in Equations (4) and (5), were employed [11; 29; 31]:\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial_{xii}}(pku_{i})= \frac{\partial}{\partial_{xj}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}} \bigg{)}\frac{\partial k}{2x_{j}}\Bigg{]}+G_{k}+G_{b}-\rho_{e}-Y_{M}+S_{k} \tag{4}\]  
\[\frac{\partial}{\partial t}(\rho e)+\frac{\partial}{\partial_{xii}}(\rho eu_{i})= \frac{\partial}{\partial_{xj}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{e}} \bigg{)}\frac{\partial e}{2x_{j}}\Bigg{]}+C_{1e}\frac{e}{K}(G_{k}+C_{3e}G_{b}) -C_{2e}\rho\frac{e^{2}}{k}+S_{e} \tag{5}\]  
In Equation (4), \(Y_{M}\) represents the contribution of fluctuating turbulence to the overall dissipation rate, and \(C_{1e}\) and \(C_{2e}\) are constants, as shown in Equation (6), and are the turbulent Prandtl numbers for \(k\) and epsilon, respectively. \(S_{k}\) and \(S\) are user-defined source terms. The model constants have been calculated to ensure that the model works properly for undisputed flows. The Prandtl numbers \(\sigma_{k}\) and \(\sigma_{e}\) serve the purpose of linking the diffusivities of turbulence kinetic energy (\(k\)) and TDR (\(\epsilon\)) to the eddy viscosity (\(\mu_{t}\)), as shown in Equation (7). This connection is essential as it ensures that the impact of eddy viscosity on these turbulent transport processes is appropriately considered in the gradient diffusion term of the turbulence equations. Eddy viscosity are calculated by the equations shown below [11; 29; 31]:  
\[C_{1e}=1.44,\ C_{2e}=1.92,\ \sigma_{k}=1.0,\ \sigma_{e}=1.2 \tag{6}\]  
\[\mu_{t}=\rho C_{\mu\frac{e^{2}}{k^{2}}} \tag{7}\]