## 5 CFD in Extrusion Processes
### Coupling CFD with Reaction Kinetics and Product Texture Modelling  
Food extrusion is a complex process that combines mechanical, thermal, and chemical transformations. Alongside mixing and shaping, extrusion induces temperature- and shear-dependent reactions that significantly affect the final product's structure, nutritional quality, and sensory properties. To achieve predictive capability and process understanding, CFD can be coupled with reaction kinetics and texture formation models. This integrated approach enables the simulation of structure-process-property relationships and supports the design of extrusion conditions that deliver targeted product attributes, such as expansion, texture, and nutrient retention [73]. Among the key reactions occurring during extrusion are starch gelatinization, derivatization, protein denaturation and aggregation, lipid melting, Maillard browning, and thermal degradation of sensitive compounds such as enzymes and vitamins. These transformations can be described using classical Arrhenius-type kinetics, where the temperature-dependent rate constant \(k(T)\) is given by:  
\[k(T)=A\cdot exp\left(-\frac{E_{a}}{RT}\right)\]  
where \(A\) is the pre-exponential factor, \(E_{a}\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the absolute temperature, typically obtained from CFD simulation outputs. The extent of reaction or conversion, denoted as \(\alpha\), can then be modeled using a general rate equation:  
\[\frac{d\alpha}{dt}=k(T)\cdot f(\alpha)\]  
In this formulation, \(f(\alpha)\) reflects the reaction mechanism or kinetics model, which may be first-order, autocatalytic, or diffusion-controlled. When coupled with CFD-derived profiles of temperature, shear rate, and residence time, these equations allow prediction of starch gelatinization levels (relevant to expansion and digestibility), protein unfolding and network formation (affecting texture), color and flavor development via Maillard reactions, and nutrient losses due to thermal degradation [98; 103]. For example, [PERSON] and [PERSON] (2024) developed a coupled CFD-kinetics model to simulate the Maillard reaction during spray drying of dairy-based feed [104]. The model integrated local temperature and moisture profiles with first-order reaction kinetics and included both deterministic and stochastic modeling frameworks. Their predictions of browning intensity across the drying chamber correlated well with experimental Lab* color measurements, confirming the model's capacity to forecast spatial non-uniformities in product appearance. This work exemplifies how CFD models, when coupled with thermal reaction kinetics, can be used not only for process optimization but also for predicting quality attributes like browning [104]. In an industrial baking context, [PERSON] et al. (2004) developed a coupled 3D CFD-kinetic model to simulate starch gelatinization during the continuous baking of sandwich bread in a travelling-tray oven [105]. The model integrated simulated crumb temperature profiles with a first-order kinetic equation, allowing for time- and position-resolved predictions of gelatinization extent. Validation with experimental tray-scale variation confirmed the model's predictive capability for baking progression and identified heat distribution asymmetries as key drivers of structural inhomogeneity. The model was also applied to test modified operating conditions, successfully predicting complete baking while reducing energy usage [105]. Texture formation in high-moisture extrusion of plant-based meat analogues, such as soy or pea protein systems, strongly depends on shear gradients and thermal transitions within the die. CFD has been coupled with viscoplastic and thermal models to simulate fiber alignment and anisotropic structure development, mimicking the fibrous texture of realmeat. Studies by [PERSON] et al.