## 2 Theoretical Foundations of CFD  
### Principles, Mathematical Models, and Numerical Methods in CFD  
CFD is a powerful numerical tool that facilitates the simulation of fluid flow, heat transfer, and other related phenomena through the solution of governing differential equations. These include the Navier-Stokes equations, continuity equation, and energy equation, which are fundamental to describing the conservation of mass, momentum, and energy in fluid systems. The Navier-Stokes equation (Equation (1)), for instance, is expressed as follows:  
\[\frac{\partial v}{\partial t}+(v\times\
abla)v=-\frac{1}{\rho}\
abla p+v\
abla ^{2}v+f \tag{1}\]  
where \(v\) represents the velocity vector field, \(\rho\) is the density, \(p\) is the pressure, \(v\) is the kinematic viscosity, and \(f\) accounts for external body forces [14]. The solution of these equations provides detailed insights into fluid flow behavior under various boundary conditions and geometries. Similarly, the energy equation governs the transfer of thermal energy (Equation (2)) as follows:  
\[\rho C_{p}\left(\frac{\partial T}{\partial t}+v\times\
abla T\right)=\
abla \cdot(k\
abla T)+\dot{Q} \tag{2}\]where \(C_{p}\) is the specific heat capacity at constant pressure, \(T\) is the temperature, \(k\) is the thermal conductivity, and \(Q\) represents the heat generation term, which may arise due to chemical reactions or other sources [14]. To solve these continuous equations in a computational environment, CFD employs discretization techniques such as the finite volume method (FVM), finite element method (FEM), and finite difference method (FDM). These methods convert the governing partial differential equations into algebraic equations, allowing their numerical solution. FVM, for example, is widely used due to its ability to conserve quantities such as mass, momentum, and energy across control volumes. The domain is divided into small discrete cells (volumes), and the integral form of the conservation equations is applied to each cell [15; 16; 17]. CFD methods are widely used for modelling and optimizing thermal processes in food engineering. Specifically, they are applied to simulate heat transfer, turbulent flows, and multiphase phenomena, enabling improvements in process efficiency. Various CFD methods, such as the finite difference method (FDM), Reynolds-averaged Navier-Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS), have specific applications depending on the problem characteristics. FDM is mainly used for simulating simple heat transfer processes where the model geometry is homogeneous and boundary conditions are well defined. For example, it is applied to model bread baking, analyzing heat conduction in homogeneous dough, and predicting temperature distribution during the pasteurization or cooking of homogeneous food products, such as cheesecakes or chocolate masses. Its advantages include straightforward implementation and low computational requirements, though it is limited to simple geometries and less complex processes [18]. The RANS method is particularly popular for analyzing turbulent flows in industrial systems. It is used, for instance, to simulate the air flow in industrial ovens, optimizing the temperature and turbulence distribution to improve baking uniformity. RANS is also applied in freezing chambers to analyze turbulent air flows, which are crucial for ensuring the uniform cooling of products. The main advantage of this method is its low computational requirements, though it may not always capture detailed turbulent structures [19]. LES allows for the more detailed modeling of turbulent structures by directly resolving large eddies and modeling smaller scales using subgrid models. This method is applied in dynamic processes, such as multiphase mixing in homogenization or emulsification, and in industrial drying, where modeling turbulent flows improves process uniformity. While LES offers greater precision than RANS, it requires significantly higher computational resources [20; 21].