## 5 CFD in Extrusion Processes
### Multiphase and Thermal Aspects in CFD Models  
Food extrusion processes are inherently multiphase and thermomechanically complex, involving simultaneous interactions between solids, liquids, gases, and superheated vapors under dynamically changing and often non-isothermal conditions. Capturing this complexity in CFD simulations requires the integration of advanced multiphase flow models and robust thermal analysis that reflect the real behavior of heterogeneous food matrices. Key modeling considerations include the treatment of phase interactions, energy dissipation, moisture transport, and strong nonlinearities associated with material properties and transformations. During extrusion, food systems typically consist of heterogeneous mixtures comprising starch, proteins, lipids, fibers, and water. These components interact through various phase phenomena, including solid-liquid dispersion, vaporization at the die exit, and bubble formation during sudden pressure drops. Modeling such systems necessitates appropriate multiphase CFD frameworks. The EE approach treats all phases as interpenetrating continua and is effective for dense mixtures or slurries. It is particularly suitable for predicting bulk pressure buildup and phase distribution along the screw channel. In contrast, the EL model tracks discrete particles or droplets within a continuous phase, making it appropriate for localized simulations of entrained air, vapor bubbles, or solid inclusions. The VOF method is employed when accurate tracking of sharp phase interfaces is essential, such as in the case of vapor-melt boundaries or product swelling at the die exit. Each of these modeling strategies requires appropriate interphase momentum exchange terms, including drag, lift, and virtual mass forces, as well as rheological descriptions tailored to the specific material phases involved. Thermal effects are central to extrusion, as the process is highly non-isothermal and governed by both externally applied heat and internally generated thermal energy. Heat is supplied by barrel heaters and augmented by frictional forces and viscous dissipation within the screw channel. CFD models must solve the energy conservation equation in conjunction with mass and momentum balances. The generalized form of the energy equation used in CFD for extrusion processes is:  
\[\rho C_{p}\left(\frac{\partial T}{\partial t}+u\cdot\
abla T\right)=\
abla\cdot(k \
abla T)+\Phi\]  
where \(\rho\) is the material density, \(C_{p}\) is the specific heat capacity, \(k\) is the thermal conductivity, and \(\Phi\) is the viscous dissipation term. The inclusion of viscous heating is essential in high-shear zones, where mechanical energy is rapidly converted into heat, altering local temperature profiles. Since thermal properties of food materials are highly dependent on temperature and moisture content, variable-property models or empirical correlations are often needed to ensure accurate simulation. A defining feature of extrusion is the phase change of water from liquid to vapor, particularly during the depressurization that occurs at the die exit. This transition drives product expansion and contributes to the formation of the porous structure characteristic of many extruded foods. Accurately modeling this phenomenon requires the incorporation of latent heat effects, evaporation kinetics, and coupled heat and mass transfer processes. Advanced CFD models implement phase-change source terms within the energy equation and simulate moisture transport through diffusion-convection equations. These models are capable of predicting critical process outcomes such as die pressure, expansion ratio, and residual moisture content, all of which influence product texture, density, and shelf stability. Another important aspect of extrusion modeling is the rheological complexity of food materials, which typically exhibit non-Newtonian, temperature-dependent, and often viscoelastic behavior.