## 3 Fundamentals of CFD in the Context of Mechanical Processes  
### Governing Equations  
CFD offers a robust mathematical and numerical framework for simulating fluid flow, heat transfer, and mass transport in food processing systems. At the heart of CFD lie the Navier-Stokes equations, which express the fundamental conservation principles of mass and momentum. However, the inherent complexity of food matrices and their transformation during mechanical processes often necessitates the integration of additional submodels, including those for turbulence, multiphase flow, and non-Newtonian rheology, to accurately capture real process behavior [22]. The continuity equation, which governs mass conservation, is expressed for incompressible fluids as:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho u)=0\]For systems with constant density \(\rho\), this simplifies to:  
\[\
abla\cdot u=0\]  
where \(\rho\)--fluid density [kg/m\({}^{3}\)], \(u\)--velocity vector [m/s], \(t\)--time [s]. The conservation of momentum is described by the incompressible Navier-Stokes equation:  
\[\rho\left(\frac{\partial u}{\partial t}+u\cdot\
abla u\right)=-\
abla p+\
abla \cdot\tau+\rho g\]  
where \(p\)--pressure [Pa], \(\tau\)--stress tensor [Pa], \(g\)--gravitational acceleration vector [m/s\({}^{2}\)]. For Newtonian fluids, the stress tensor is defined as:  
\[\tau=\mu(\
abla u+\
abla u^{T})\]  
with \(\mu\)--dynamic viscosity [Pa\(\cdot\)s], \(\
abla u^{T}\)--transpose of the velocity gradient tensor. In non-Newtonian food systems, viscosity is often a function of shear rate, temperature, or time, requiring the use of rheological models such as the power-law or Herschel-Bulkley formulations to describe flow resistance under process-specific conditions. When thermal effects are important--such as in extrusion, pasteurization, or high-shear mixing--the energy conservation equation is incorporated:  
\[\rho c_{p}\left(\frac{\partial u}{\partial t}+u\cdot\
abla T\right)=\
abla \cdot(k\
abla T)+\Phi\]  
where  
\(c_{p}\)--specific heat capacity at constant pressure [J/(kg\(\cdot\)K)]  
\(T\)--temperature [K]  
\(k\)--thermal conductivity [W/(m\(\cdot\)K)]  
\(\Phi\)--viscous dissipation term [W/m\({}^{3}\)]. Turbulence, a key feature in many mechanical food operations--including high-speed mixing, extrusion preconditioning, and pneumatic transport--is generally modeled using approximation methods due to the computational expense of Direct Numerical Simulation (DNS). The most widely used approach is Reynolds-Averaged Navier-Stokes (RANS), which introduces time-averaged equations and requires turbulence closure models. The k-\(\varepsilon\) model, known for its robustness and computational efficiency, is widely applied under the assumption of isotropic turbulence. The \(k\)-\(\omega\) model is a two-equation turbulence model that solves transport equations for turbulent kinetic energy (\(k\)) and specific dissipation rate (\(\omega\)). It provides improved accuracy in predicting near-wall flows and boundary layer behavior, making it suitable for confined or wall-bounded geometries commonly encountered in food processing equipment. For greater fidelity, especially in capturing transient or large-scale eddies, Large Eddy Simulation (LES) resolves the larger turbulent structures while modeling the smaller scales, at the cost of higher computational demand. Detached Eddy Simulation (DES) offers a hybrid approach that combines RANS and LES, providing a balance between resolution and cost for geometrically complex systems like twin-screw extruders or baffled tanks. The choice of turbulence model depends on the flow regime, domain complexity, and mesh resolution. It is important to note that RANS models, although computationally efficient, may exhibit limited accuracy in food systems characterized by strong phase separation, such as emulsions or aerated structures, or those with complex non-Newtonian rheology.