Computational Design of Fat Marbling Formation in Plant-Based Meat: Coupled CFD and Image Analysis of Oil Transport During Co-Extrusion

Featured Application

The integrated rheology, thermal transition, imaging, and computational fluid dynamics (CFD) framework developed in this study can be applied to design and scale cooling-die temperature profiles, residence times, and oil injection conditions for co-extrusion of plant-based meat analogs, supporting more consistent control of fat marbling distribution, texture development, and product-to-product consistency in industrial production lines.

Abstract

This study developed and evaluated an integrated experimental–computational framework to quantify coconut-oil transport and marbling stabilization in soy protein concentrate (SPC) during static holding and co-extrusion with a cooling die. Temperature-sweep rheology and Differential Scanning Calorimetry (DSC) identified the main gelation transition at 65–78 °C, with oil shifting gelation to higher temperatures and increasing enthalpy, supporting an exit/cooling target of 70–75 °C. Static drop tests at 100 °C for 60 s were analyzed by depth-resolved imaging and coupled with a single-phase CFD model to inversely calibrate an effective diffusion coefficient for coconut oil in SPC (D_ref = 4.86 × 10⁻¹⁸ m²/s). A viscosity-coupled fractional Stokes–Einstein relationship then gave temperature-dependent effective diffusivities of 1.89 × 10⁻¹⁸ to 4.86 × 10⁻¹⁸ m²/s over 60–100 °C, indicating reduced oil mobility during cooling. Additional static time-temperature comparisons suggested limited redistribution beyond ~50 s. Co-extrusion simulations and product imaging further indicated that staged hot-zone residence followed by rapid cooling can help stabilize oil domains into marbling-like structures. The framework can support selection of cooling-die temperatures, residence times, and oil-injection conditions. Future work should extend the framework by linking marbling microstructure with sensory performance, oxidative stability, and sensitivity analysis of key transport parameters.

1. Introduction

Marbling plays a decisive role in the sensory quality of meat, contributing to perceived juiciness, tenderness, flavor release, and visual appeal [1,2]. In conventional muscle foods, marbling arises from the spatial distribution of intramuscular fat embedded within aligned fibrous protein matrices, where lipid domains interact dynamically with muscle microstructure during cooking and mastication [3]. This hierarchical organization governs moisture retention, lubrication, and flavor transport, ultimately defining consumer acceptance and product value [4]. Replicating this complex microstructural architecture remains one of the most critical challenges in developing high-fidelity plant-based meat analogs.

Plant-based meat systems typically rely on structured plant proteins, such as soy protein concentrate (SPC), soy protein isolate (SPI), wheat gluten, or pea protein, combined with dispersed lipid phases to emulate the appearance and mouthfeel of animal fat [5]. Among these, SPC offers a favorable balance of functionality, cost, and nutritional profile, and readily undergoes heat-induced gelation and network formation during extrusion processing. However, unlike biological muscle tissue, where fat deposition occurs gradually during growth, oil distribution in plant-based systems must be engineered dynamically during high-temperature and high-shear processing [6]. Achieving stable marbling patterns, therefore, requires precise control over protein gelation kinetics, rheological evolution, and lipid transport mechanisms within the flowing matrix.

Co-extrusion and layered structuring have emerged as dominant manufacturing strategies for producing marbled or fat-embedded plant-based meat analogs. These approaches typically involve the simultaneous extrusion of a protein-rich continuous phase and an oil-rich dispersed phase through concentric or multi-channel dies, followed by thermal setting and cooling to lock in the structure [4]. While successful in generating visually appealing products, most reported studies emphasize formulation optimization, die design, or macroscopic texture outcomes, rather than the underlying transport phenomena governing oil migration, entrapment, and pattern stabilization. Consequently, the mechanistic relationship between protein gelation, temperature-dependent viscosity evolution, and lipid diffusion during co-extrusion remains insufficiently understood.

The gelation of SPC under thermal and shear fields involves protein unfolding, exposure of hydrophobic domains, aggregation, and the formation of a three-dimensional network stabilized by hydrophobic interactions, hydrogen bonding, and disulfide crosslinks. As temperature increases, the protein matrix transitions from a viscous sol to a viscoelastic gel, accompanied by rapid changes in modulus, permeability, and molecular mobility [7]. These transformations strongly influence the diffusion pathways available to lipid droplets or molten oil domains. During the early sol stage, relatively low viscosity and high mobility allow oil to migrate, coalesce, or deform under shear. As gelation proceeds, the developing network progressively restricts mass transfer, immobilizing the lipid phase and fixing the spatial marbling pattern.

Diffusion in polymeric and protein gels is governed by a combination of molecular mobility, hydrodynamic drag, and steric hindrance imposed by the evolving network structure [8]. In protein-rich food systems, transport is further influenced by gelation state, matrix density, water availability, and free-volume restriction, all of which change dynamically during heating and cooling. Similar transport limitations have also been reported in emulsion-filled gels, where droplet–matrix interactions, interfacial protein layers, and filler effects can reduce phase mobility and alter bulk network development [9]. In viscoelastic matrices, oil migration is therefore controlled not only by intrinsic diffusivity but also by the progressive increase in viscosity and structural setting, which together define a narrow processing window in which marbling can develop without excessive spreading, coalescence, or phase separation [4].

Several studies have examined the behavior of fat or oil in structured protein matrices, including emulsion-filled gels, oleogels, and extruded protein systems [9,10]. These studies generally show that lipid transport and retention depend strongly on the rheological state of the continuous phase, the extent of interfacial stabilization, and the degree of matrix confinement. However, many of these investigations were conducted under static or low-shear conditions and therefore do not fully capture the coupled thermal, shear, and flow fields characteristic of extrusion processing. High-moisture extrusion, in particular, introduces complex non-Newtonian flow behavior, strong temperature gradients, and anisotropic fiber formation that directly influence transport behavior and microstructure development [11,12]. As a result, diffusion and oil-migration behavior observed in simplified gel systems cannot be directly extrapolated to industrial co-extrusion environments without a process-relevant modeling framework.

Computational fluid dynamics (CFD) has increasingly been applied to model heat transfer, shear distribution, residence time, and phase flow in extrusion and food processing systems [13,14]. CFD enables the prediction of local temperature fields, velocity gradients, and viscosity evolution, providing a quantitative framework for linking processing conditions to microstructural outcomes. When coupled with experimentally derived rheological models and diffusion coefficients, CFD can offer predictive insights into oil migration behavior during co-extrusion [13]. Nevertheless, validation of such models requires robust experimental quantification of oil transport and spatial distribution under realistic processing conditions.

Recent advances in image analysis and digital microscopy have enabled quantitative tracking of phase migration, domain growth, and spatial heterogeneity in complex food matrices [15]. Image-based diffusion analysis allows direct extraction of oil front displacement, concentration gradients, and pattern evolution over time, providing experimentally grounded metrics that can be integrated with numerical simulations [16]. When combined with temperature-controlled processing and synchronized imaging, these techniques offer a powerful pathway for elucidating the coupling between gelation kinetics and lipid transport dynamics.

Coconut oil is particularly relevant for plant-based meat formulations because of its sharp melting transition near body temperature, relatively high saturated fat content, and ability to contribute characteristic mouthfeel and flavor release; under extrusion conditions, it becomes molten and behaves as a mobile lipid phase whose redistribution is highly sensitive to temperature, matrix viscosity, and interfacial interactions [17]. Understanding how coconut oil migrates and becomes retained within a thermally gelling SPC matrix is therefore essential for achieving reproducible marbling patterns and desirable product structure. Despite growing industrial interest in marbled plant-based products, a critical knowledge gap remains in quantitatively linking thermal history, protein gelation, effective oil transport, and final marbling morphology under realistic co-extrusion conditions. Existing studies rarely integrate rheology, transport modeling, and spatial image analysis within a unified framework for process-level prediction and design.

Therefore, the present study addresses this challenge by combining CFD modeling with image-based diffusion analysis to investigate the diffusion mechanism of coconut oil in high-temperature SPC during co-extrusion. Specifically, the study seeks to: (i) quantify the gelation kinetics and rheological evolution of SPC under controlled thermal and shear conditions; (ii) characterize oil diffusion behavior and marbling formation at different gelation stages using image analysis and a CFD model under static and co-extrusion conditions; (iii) integrate rheological data with CFD simulations to predict temperature-dependent diffusion dynamics during co-extrusion; and (iv) establish correlations between effective diffusion coefficient and visual marbling uniformity to support predictive process control and scalable manufacturing strategies for next-generation plant-based meat analogs.

2. Materials and Methods

2.1. Sample Preparation

Soy protein concentrate (SPC) was used as the primary structural matrix for investigating oil diffusion during thermal processing. The SPC was supplied as a spray-dried powder produced from defatted soybean flakes through aqueous extraction and protein recovery, yielding a protein-rich ingredient with reduced soluble carbohydrates and anti-nutritional components. According to the supplier’s specification, the SPC contained approximately 69% protein (dry basis) and less than 8% residual moisture, providing favorable water-holding capacity and heat-induced gelation functionality for high-moisture extrusion applications (Shandong Wonderful Biotech Co., Ltd., Dongying, China).

For sample preparation, all material quantities were normalized to 100 g of SPC melt. SPC powder (19.01 g) was dispersed in distilled water (78.45 mL) to achieve a final moisture content of 80% (wet basis), corresponding to an effective protein concentration of approximately 13% (w/w). To enhance optical contrast for image-based analysis of oil migration, a trace amount (<0.01%, w/w) of a food-grade lipophilic red colorant (FD&C Red No. 40 aluminum lake) was added. The formulation was mechanically homogenized using a high-shear Stephan cutter mixer (UMC-5 Electronic, Stephan Machinery GmbH, Hameln, Germany) until a uniform paste was obtained. Entrapped air was removed under mild vacuum, and the slurry was equilibrated at ambient temperature prior to loading into the extrusion cylinder.

2.2. Co-Extrusion Process

Co-extrusions were carried out using a laboratory-scale piston-driven extrusion assembly that enabled independent control of melt compression, thermal history, and oil injection (Figure 1). The SPC mixture was loaded into a cylindrical stainless-steel barrel (inner diameter 90 mm, height 175 mm) and compressed using a texture analyzer (TA.XT Plus, Stable Micro Systems, Surrey, UK) fitted with a matching piston. The piston displacement rate was maintained at a constant speed (1 mm/s) throughout extrusion to ensure steady volumetric flow of the protein melt and reproducible residence time within the barrel and die [14].

Thermal conditioning of the melt inside the barrel was achieved using integrated heating channels surrounding the barrel wall, allowing precise temperature regulation over the range of 25–100 °C. After thermal equilibration, the pressurized melt was extruded into a rectangular cooling die (20 × 10 mm cross-section). The die was divided into three independently controlled thermal zones to impose a defined axial temperature gradient. The upstream zone preserved melt fluidity and oil injection, the intermediate zone promoted gradual network development, and the downstream zone induced structural stabilization. A Peltier-based thermal control module regulated the exit temperature between 70 and 75 °C to ensure consistent gelation behavior at the die outlet.

Coconut oil was selected as the model lipid phase because its relatively high saturated-fat content produces a clear melt–solid transition near processing temperatures, which supports the formation and stabilization of discrete fat domains during cooling and facilitates marbling-like structure development. Molten coconut oil was delivered using a syringe mounted on a texture analyzer platform and driven at a constant displacement rate (1 mm/s) to provide continuous and reproducible oil injection. The oil reservoir was maintained at 50 °C to ensure complete melting and stable viscosity during injection. The injection port was positioned approximately 10 mm downstream from the entrance of the cooling die, enabling controlled interaction between the flowing protein melt and the injected oil stream. The extrudate exiting the die was collected immediately and prepared for sectioning and image-based characterization of oil distribution and marbling structure.

2.3. Rheological Characterization and Gelation Kinetics of SPC

2.3.1. Rheological Measurement

Rheological characterization was performed to quantify temperature-dependent viscosity, linear viscoelastic limits, and dynamic mechanical behavior of the SPC formulation using a Discovery Hybrid Rheometer HR-3 (TA Instruments, New Castle, DE, USA) equipped with a Peltier temperature control system and solvent trap to minimize moisture loss. Measurements were conducted using a parallel-plate geometry (40 mm diameter) with a fixed gap of 1.0 mm. Samples were carefully loaded to avoid pre-shearing and equilibrated at the initial test temperature for 3 min prior to measurement.

Apparent viscosity was measured during a continuous heating ramp from 25 to 100 °C at 5 °C/min under a constant low shear rate within the linear viscoelastic region (LVR), followed by cooling back to 25 °C at the same rate to assess thermal hysteresis and structural reversibility. Viscosity–temperature profiles were used to identify sol–gel transition behavior and to parameterize temperature-dependent viscosity functions for computational modeling [18].

Strain sweep tests were performed at an angular frequency of 1 rad/s with strain amplitude logarithmically increased from 0.01% to 100%. The LVR was defined as the strain range over which storage modulus (G′) and loss modulus (G″) remained within ±5% of their plateau values. Dynamic frequency sweeps were subsequently conducted within the LVR over 0.1–100 rad/s to characterize viscoelastic behavior, network stability, and relaxation dynamics relevant to oil transport and structural development. For each rheological condition, measurements were performed using independently prepared sample batches, and the reported values represent the mean ± standard deviation of these batch-based replicates.

2.3.2. Thermal Gelation Transition Characteristics by Differential Scanning Calorimetry

Thermal transitions associated with protein unfolding and gelation were evaluated using a Discovery Series DSC (TA Instruments, New Castle, DE, USA). Approximately 15 mg of the homogenized sample was weighed (±0.01 mg) and hermetically sealed in stainless-steel pans, with an empty sealed pan used as a reference. Samples were heated from 20 to 100 °C at 5 °C/min under a nitrogen purge (50 mL·min⁻¹) to minimize oxidative effects and moisture loss during scanning [19]. The temperature range was selected to cover the relevant thermal window from the initial ungelled state of the SPC mixture to the upper processing temperature used in the extrusion system, while the heating rate of 5 °C/min was chosen to provide sufficient thermal resolution of the transition peak without excessively prolonging the scan. Heat flow thermograms were analyzed to determine onset (T₀), peak (T_p), and end-set (T_e) transition temperatures. The apparent enthalpy change (ΔH) was obtained by integrating the endothermic peak area using TRIOS software (version 5.10, TA Instruments, New Castle, DE, USA) and is reported per unit wet sample mass (J/g) based on the sample mass entered for each pan.

Because a single heating rate was applied, these DSC outputs were interpreted as thermal transition descriptors. Specifically, T₀, T_p, and T_e were used to compare the temperature domain of unfolding/aggregation and gel-network formation, where shifts to higher temperatures indicate increased thermal stability and delayed structural setting of the SPC matrix. ΔH was used as an index of the overall extent of thermally driven structural change (e.g., unfolding/association) contributing to gel formation. Collectively, these parameters quantified the thermal sensitivity and gelation-related transition behavior of SPC and were used to support interpretation of temperature-dependent rheological changes and oil migration dynamics.

2.4. Oil Diffusion and Image Analysis Protocol

2.4.1. Static Diffusion Analysis Design

To isolate intrinsic oil diffusion behavior from flow-induced effects, static diffusion analysis was conducted using thermally conditioned SPC melts. The melt was extruded at controlled temperatures of 90, 80, 70, and 60 °C into a rectangular mold (30 × 10 × 20 mm). Immediately after shaping, 0.5 mL of molten coconut oil, which corresponds to approximately 2.5% (w/w), was injected at the geometric center of the sample at a depth of 1 mm below the surface (Figure 2). The injection time was defined as t = 0, and samples were allowed to undergo diffusion for predefined durations (5–60 s) under isothermal conditions. At the target time point, samples were rapidly cooled to 25 °C to arrest further molecular transport and induce oil solidification before sectioning and analysis.

Spatial diffusion profiles were quantified using orthogonal sectioning planes (Figure 2). Cross-sectional images were collected at the surface and at depths of 3 mm from the top surface to evaluate lateral spreading within the matrix. All slicing was performed after complete cooling to prevent oil redistribution during cutting. The exposed surfaces were immediately imaged for subsequent quantitative analysis.

2.4.2. Image Acquisition and Image-Based Quantification

Digital images of each sectioned surface were captured using a digital camera (Lumix DC-GX9, Panasonic Corporation, Osaka, Japan) mounted at a fixed height of 300 mm above the sample. Illumination was provided by uniform white LED lighting, and samples were placed on a matte black background to maximize contrast between the oil phase and the protein matrix. Camera positioning, focal length, and exposure settings were kept constant for all measurements to ensure consistency across datasets.

Image processing and quantitative analysis were performed using MATLAB (R2025a, MathWorks Inc., Natick, MA, USA). The analysis workflow consisted of color-channel separation, adaptive thresholding and binarization, background removal, and segmentation of oil-rich regions [20] (Figure 3). The projected oil area was extracted and normalized to the total cross-sectional area to obtain an apparent diffusion index, which was used as a two-dimensional proxy for the extent of oil redistribution visible on a given section. All measurements were performed on at least 10 samples and reported as mean ± standard deviation. Temperature-dependent diffusion trends obtained from the static analysis were used to identify optimal thermal windows for controlled oil migration, which was employed for the co-extrusion process.

2.5. Estimation of the Effective Diffusion Coefficient (D_eff)

The diffusion of coconut oil within the SPC matrix was modeled using a multi-scale approach that links molecular theory with experimentally calibrated process-scale behavior. Because the SPC melt was treated as a non-porous, fully compressed continuum, oil movement was modeled as transport through a highly viscous protein matrix subject to macromolecular obstruction and free-volume limitation, rather than hydraulic flow through pores [21]. In this context, the fitted diffusion coefficient should be interpreted as an effective transport parameter, rather than as a strict molecular diffusivity, because it may also reflect unresolved matrix–oil and interfacial effects.

First, a theoretical “ideal” diffusion coefficient (D₀) was calculated to estimate the upper bound of coconut-oil mobility in the molten state. The hydrodynamic radius (R_h) of the primary coconut oil triglycerides (trilaurin) was derived using the molecular volume approach [22,23]:

$${\mathrm{V}}_{\mathrm{m}}={\displaystyle \frac{{\mathrm{M}}_{\mathrm{w}}}{\mathsf{\rho }·{\mathrm{N}}_{\mathrm{A}}}}$$

(1)

where M_w is the molecular weight of trilaurin (639.5 g/mol), ρ is the density of the oil (915 kg/m³), and N_A is Avogadro’s number (6.022 × 10²³ mol⁻¹). From this, the hydrodynamic radius (R_h) was derived based on the Stokes-sphere assumption [21]:

$${\mathrm{R}}_{\mathrm{h}}={\left({\displaystyle \frac{{3\mathrm{V}}_{\mathrm{m}}}{4\mathsf{\pi }}}\right)}^{1/3}$$

(2)

The calculated R_h of 0.65 nm was then utilized in the Stokes-Einstein relation to define the baseline diffusion coefficient (D₀) at a specific temperature (T):

$${\mathrm{D}}_0(\mathrm{T})={\displaystyle \frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{6\mathsf{\pi }\mathsf{\mu }\left(\mathrm{T}\right){\mathrm{R}}_{\mathrm{h}}}}$$

(3)

where kB is the Boltzmann constant (1.3806 × 10⁻²³ JK⁻¹) [21]. The variable T represents the absolute temperature (K), and μ(T) is the temperature-dependent apparent viscosity of the SPC melt, obtained from rheological measurements (Section 2.3.1). The resulting D₀ value at 100 °C was 7.696 × 10⁻¹⁷ m²/s and was used as a theoretical upper-bound reference to confirm that the experimentally calibrated value remained physically realistic.

Secondly, to account for matrix-specific resistance under the peak processing condition (100 °C), a reference diffusion coefficient (D_ref) was estimated by inverse calibration. In this procedure, static drop tests were conducted in which molten coconut oil was applied to the SPC matrix and held at 100 °C for 60 s, followed by rapid cooling to preserve the oil-penetration profile. Depth-resolved oil fractions were then measured from digital images at 2 mm intervals over five depth points (0–8 mm). A range of candidate diffusion coefficients was subsequently tested in the CFD model, and the simulated depth profile for each candidate value was compared with the corresponding experimental profile. The final D_ref was selected as the value that minimized the root mean square error (RMSE) between simulated and measured depth-resolved oil fractions. This inverse calibration yielded a best-fit value of D_ref = 4.86 × 10⁻¹⁸ m²/s. At this selected value, the CFD-predicted depth-resolved oil profile agreed with the experimental profile within 5% across the fitted depth range. Based on the five measured depth points used in the fitting, an approximate 95% confidence range for D_ref was estimated from the RMSE profile as 6.11 × 10⁻¹⁹ to 3.86 × 10⁻¹⁷ m²/s. This calibrated parameter therefore represents a processing-relevant effective diffusion coefficient that captures obstruction effects within the SPC matrix beyond those predicted by the ideal Stokes–Einstein relation.

Finally, to simulate the co-extrusion process under changing thermal conditions, the fractional Stokes–Einstein (FSE) relationship was used to define the effective diffusion coefficient (D_eff) as a function of viscosity evolution during cooling [24]:

$${\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{T})={\mathrm{D}}_{\mathrm{r}\mathrm{e}\mathrm{f}}{\left[{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{r}\mathrm{e}\mathrm{f}}}{\mathsf{\mu }\left(\mathrm{T}\right)}}\right]}^{\mathrm{n}}$$

(4)

where μ_ref is the melt viscosity at the calibration temperature, and n is a coupling exponent set to 0.92 to represent the viscosity-dependent restriction of transport in highly viscous food systems [24,25]. This framework allows diffusion to decrease dynamically as the melt cools and viscosity rises in the die, thereby describing the progressive immobilization of oil domains as the protein matrix transitions from a viscous melt to a structured gel. In this way, the model provides a process-level framework for predicting marbling stabilization trends during co-extrusion.

2.6. CFD Numerical Model for Oil Diffusion

2.6.1. Computational Domain and Boundary Conditions

The CFD domains were developed for two configurations: a static transport model used for parameter calibration and validation, and a co-extrusion transport model used to simulate oil redistribution under processing-relevant flow and thermal conditions (Figure 4).

For the static model (Figure 4(a1–c1)), the SPC melt was represented as a 3D rectangular block (30 × 20 × 10 mm). Coconut oil was initialized as a localized cylindrical region (“oil patch”) on the top surface to replicate the drop-test condition (Figure 4(b1)). Using the Region Register tool in ANSYS Fluent 2025R2 (ANSYS Inc., Canonsburg, PA, USA), the oil region was positioned at the center of the top face and defined to match the experimental oil volume (0.5 mL). All boundaries were treated as stationary no-slip walls, and the top surface was assigned a zero-mass-flux boundary for the oil species to represent a closed system. This static framework was first used to calibrate the reference diffusion coefficient (D_ref) from the 100 °C, 60 s depth-resolved oil profile. After calibration, the same model was applied to independent static holding conditions at different temperatures and times to evaluate predictive performance against experimental surface and subsurface oil-fraction data.

For the co-extrusion model (Figure 4(a2–c2)), the domain included the barrel-to-die flow path to capture oil transport under processing-relevant advection–diffusion conditions. The inlet boundary was prescribed as a velocity inlet derived from the piston displacement rate (1 mm/s), and the outlet was set as a pressure-outlet (0 Pa gauge). All solid surfaces (barrel and die walls) were treated as stationary, no-slip walls. To reproduce the thermal management strategy of the fabricated cooling die, fixed wall temperatures were imposed: the barrel wall was maintained at 100 °C, and the cooling die was divided into three axial temperature zones to achieve controlled cooling of the extrudate. Specifically, Section 1 (0–100 mm) was set to 100 °C, Section 2 (>100–350 mm) to 50 °C, and Section 3 (>350–500 mm) to 10 °C. Coconut oil was introduced through an injection boundary located 10 mm downstream of the cooling-die inlet, specified as an oil inlet (species mass fraction = 1 for oil) with the corresponding injection flowrate used experimentally. This zoned thermal boundary condition ensured that, upon exiting the die, the strand reached an average temperature in the target range of 70–75 °C, thereby coupling oil mobility to the viscosity increase and progressive setting of the SPC matrix.

2.6.2. Governing Equations

Transport in the SPC–oil system was modeled in ANSYS Fluent as a three-dimensional, transient, incompressible flow in a single-phase continuum, in which coconut oil was represented as a transported species within the SPC melt (continuous phase). This approximation was adopted because the objective of the model was to describe bulk oil redistribution and depth-resolved transport at the process scale, rather than to resolve individual droplet-scale interfacial events. Under the highly viscous conditions of the SPC matrix and the short redistribution distances considered in this study, the experimentally observed oil migration profile was treated as an effective transport phenomenon. Accordingly, unresolved microscale effects such as interfacial retention, local droplet deformation, coalescence, and phase separation were incorporated implicitly into the calibrated effective diffusion coefficient, D_eff (Section 2.5), rather than being solved explicitly through a multiphase model.

Under this formulation, the model combines the conservation equations for mass, momentum, and energy with a species convection–diffusion equation for the coconut-oil mass fraction. The governing continuity and momentum equation are as follows [26]:

$$\nabla ·\mathrm{u}=0$$

(5)

$$\mathsf{\rho }\left({\displaystyle \frac{\mathsf{\partial }\mathrm{u}}{\mathsf{\partial }\mathrm{t}}}+(\mathrm{u}·\nabla )\mathrm{u}\right)=-\nabla \mathrm{p}+\nabla ·\left[\mathsf{\mu }\left(\mathrm{T}\right)\left(\nabla \mathrm{u}+{\left(\nabla \mathrm{u}\right)}^{\mathrm{T}}\right)\right]$$

(6)

where u is velocity, p is pressure, and ρ the effective mixture density of the SPC-oil continuum.

Because the rheological behavior of the SPC matrix was strongly temperature-dependent, viscosity was introduced as a temperature-dependent property using an Arrhenius-type relationship:

$$\mathsf{\mu }\left(\mathrm{T}\right)={\mathsf{\mu }}_{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}}\left({\displaystyle \frac{1}{\mathrm{T}}}-{\displaystyle \frac{1}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}\right)\right]$$

(7)

where E_a is the apparent activation energy for flow, R is the universal gas constant, and μ_ref is the apparent viscosity at a reference temperature T_ref. The parameters in Equation (7) were obtained by fitting the experimentally measured viscosity–temperature data (Section 2.3.1) and implemented in Fluent as a temperature-dependent material property.

Energy conservation:

$$\mathsf{\rho }{\mathrm{c}}_{\mathrm{p}}\left({\displaystyle \frac{\mathsf{\partial }\mathrm{T}}{\mathsf{\partial }\mathrm{t}}}+\mathrm{u}·\nabla \mathrm{T}\right)=\nabla ·(\mathrm{k}\nabla \mathrm{T})$$

(8)

where c_p is specific heat (4 × 10⁻¹³ T² + 0.362T + 3606) and k is thermal conductivity (−9 × 10⁻¹⁹T² − 0.0003T + 0.5265) measured using the KS-1 single-needle sensor (60 mm long, 1.3 mm diameter) of KD2 Pro thermal properties analyzer (METER Group, Inc., Pullman, WA, USA) at respective temperature and implemented via polynomial function to ensure energy conservation accuracy during the sol-to-gel transition [27,28].

Species transport (oil mass fraction): The evolution of the oil mass fraction (Y₀) is governed by the convection-diffusion equation:

$${\displaystyle \frac{\mathsf{\partial }(\mathsf{\rho }{\mathrm{Y}}_0)}{\mathsf{\partial }\mathrm{t}}}+\nabla ·(\mathsf{\rho }\mathrm{u}{\mathrm{Y}}_0)=\nabla ·(\mathsf{\rho }{\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\nabla {\mathrm{Y}}_0)$$

(9)

where Y₀ is the local mass fraction of coconut oil in the SPC continuum (0 ≤ Y₀ ≤ 1), ρ is the continuum density used consistently across all equations, and D_eff(T) is the temperature-dependent effective diffusion coefficient defined in Section 2.5 (Equation (4)). For the static drop-test validation, the velocity field was set to u = 0, such that the convective term vanishes and Equation (9) reduces to diffusion-driven transport governed by concentration gradients and D_eff(T). In contrast, during co-extrusion, both convection and diffusion contribute to oil redistribution. Therefore, D_eff should be interpreted as an effective process-scale transport coefficient rather than as a purely molecular diffusion constant.

2.6.3. Numerical Strategy and Solution Control

A structured hexahedral mesh was generated using ANSYS Meshing (version 2025R2, ANSYS Inc., Canonsburg, PA, USA), with local refinement applied to the oil-melt interface to resolve high concentration gradients (Figure 4(c1,c2)). Mesh independence was confirmed when further refinement yielded < 1.5% change in the predicted oil penetration depth at t = 60 s.

Spatial discretization utilized second-order upwind schemes for momentum, energy, and species equations to minimize numerical diffusion. Pressure-velocity coupling was handled via the SIMPLE algorithm. Time integration was performed using a first-order implicit scheme with a time step of 0.05 s, ensuring a Courant number < 1. Convergence was monitored through normalized residuals, with target criteria of 10⁻⁵ for continuity and momentum, and 10⁻⁶ for energy and species transport.

2.7. Statistical Analysis

All experimental measurements were reported as the mean ± standard deviation of at least triplicate independent replicates. The influence of coconut oil concentration and temperature on the measured physical properties was evaluated using a One-Way Analysis of Variance (ANOVA) after confirming homogeneity of variance. When the ANOVA indicated significant effects, Tukey’s Honest Significant Difference (HSD) post-hoc test was used to identify differences between group means. For CFD model validation, agreement between simulated and experimentally measured oil-depth profiles was quantified using the root mean square error (RMSE). All statistical analyses were performed at a significance level of p < 0.05 using IBM SPSS Statistics (version 27, IBM Corporation, Armonk, NY, USA).

3. Results and Discussion

3.1. Rheological Characterization and Sol–Gel Transition

The temperature-dependent rheological behavior of SPC matrices is shown in Figure 5. Three samples were evaluated to represent processing-relevant states: SPC (soy protein concentrate melt), CO (coconut oil), and SPC-O (SPC containing 2.5% w/w coconut oil). The 2.5% lipid level was selected to approximate the local oil fraction associated with the 0.5 mL droplet used for co-extrusion; therefore, SPC-O represents the effective “mixture rheology” governing oil-domain stability and marbling fidelity during flow and setting.

In the apparent-viscosity temperature sweep (Figure 5a; 5 °C/min), SPC and SPC-O displayed clear thermal hysteresis. During heating (25–100 °C), viscosity decreased progressively, consistent with thermal thinning as increased molecular mobility lowered resistance to flow. Upon cooling from 100 °C, viscosity increased sharply and nonlinearly, indicating sol–gel setting as thermally denatured protein chains progressively re-associated into a space-spanning network [29]. CO showed a pronounced viscosity decrease across its melting region and remained low at higher temperatures, as expected for a low-molecular-weight lipid phase with limited elastic structuring. Across the sweep, SPC-O exhibited lower apparent viscosity than SPC, indicating reduced resistance to flow in the oil-containing system and suggesting that the dispersed lipid phase altered the continuity and flow response of the protein matrix under shear. This trend is consistent with previous high-moisture SPC studies showing that stronger thermal treatment reduced complex viscosity and promoted structural changes relevant to anisotropic product formation; for example, Pietsch et al. [30] reported SPC structuring over 100–143 °C during extrusion and a pronounced decrease in complex viscosity after thermomechanical pretreatment from 100 to 160 °C. Review-level evidence in plant-protein high-moisture extrusion similarly indicates that added oil or fat can reduce apparent or complex viscosity and viscoelastic moduli of protein matrices, although the magnitude of the effect depends on formulation, moisture level, and measurement conditions [29].

Strain and frequency sweeps (Figure 5b,c) were performed after heating to 100 °C and cooling to 70 °C, capturing viscoelastic behavior at a processing-relevant setpoint. At 70 °C, SPC exhibited a broader linear viscoelastic region (LVR = 1.61 ± 0.12%) than SPC-O (1.01 ± 0.09%), corresponding to an approximately 37% reduction in LVR after oil incorporation. Likewise, the crossover strain (G′ = G″) decreased from 3.16 ± 0.34% for SPC to 1.68 ± 0.28% for SPC-O, a reduction of about 47%, indicating earlier yielding of the oil-containing system. Frequency sweeps confirmed gel-dominant behavior for both systems (G′ > G″ across frequency), while the lower moduli for SPC-O indicate reduced load-bearing connectivity of the continuous protein phase at 70 °C. These differences show that even a relatively small oil addition measurably altered the viscoelastic stability window of the SPC matrix under conditions relevant to co-extrusion and post-die setting.

Overall, these rheological results define the mixture’s flow–yield window at the intended co-extrusion setpoint and indicate that oil incorporation lowers steady-shear viscosity and deformation tolerance at 70 °C, conditions that directly influence oil-domain stability during extrusion and setting.

3.2. Thermal Gelation Transition of Gelation in SPC Mixture

Temperature-sweep oscillatory rheology and DSC provide complementary insight into the thermal transition and network development behavior of SPC with and without oil (Figure 6;

Table 1. Thermal transition parameters of SPC mixtures with and without oil: onset temperature, peak temperature, end-set temperature, and transition enthalpy.

Sample	Onset of Gelation (°C)	Peak Gelation Temperature (°C)	End of Gelation (°C)	Gelation Enthalpy (J/g)
SPC	64.52 ± 0.41 b	69.11 ± 0.75 b	75.52 ± 0.89 b	3.55 ± 0.24 b
SPC-O	65.71 ± 0.37 a	71.68 ± 0.91 a	77.95 ± 0.94 a	4.31 ± 0.28 a

Values within the same column having different superscripts (^a and ^b) differ significantly (p < 0.05).

). The DSC thermogram (Figure 6c) showed a dominant endothermic transition spanning approximately 65–78 °C. Incorporation of oil shifted the transition to higher temperatures (Table 1), with onset increasing from 64.52 to 65.71 °C, peak from 69.11 to 71.68 °C, and end temperature from 75.52 to 77.95 °C. At the same time, the enthalpy increased from 3.55 to 4.31 J/g, indicating that the SPC-O system required greater thermal energy to complete the structural transitions associated with protein unfolding and gel-network development. This suggests that oil incorporation modified not only the transition temperature range but also the energetic requirement of gelation, which is important for understanding melt-to-solid conversion during high-moisture extrusion and cooling-die solidification [29].

During heating (Figure 6a), G′ and G″ remained low until about 60–65 °C and then increased sharply, with the steepest rise occurring around 70–72 °C, in good agreement with the DSC peak region (Table 1). This behavior is consistent with heat-induced denaturation of the major soy globulins (7S β-conglycinin and 11S glycinin), where unfolding exposes hydrophobic regions and reactive sulfhydryl groups that promote intermolecular association and disulfide-mediated crosslinking, leading to the formation of a load-bearing elastic network [29]. In SPC-O, the upward shift in gelation temperature can be explained not only by protein adsorption and redistribution at the oil–water interface, but also by the increased enthalpy requirement observed in DSC. The higher ΔH suggests that additional thermal input was needed for the system to undergo comparable conformational and associative changes, likely because part of the protein participated in interfacial stabilization of oil droplets before contributing to bulk network formation. In this way, oil altered both protein availability in the continuous phase and the thermal energy demand of structuring, thereby delaying bulk gel-network percolation until a higher temperature was reached [31].

During cooling (Figure 6b), G′ and G″ increased markedly as temperature decreased, reflecting consolidation of the gel network and growth of junction zones. In this stage, the role of oil became state-dependent: once a continuous protein matrix had formed, protein-coated oil droplets could become mechanically integrated into the network and act as active fillers, thereby enhancing elasticity through improved stress transfer within the composite structure [32]. This explains why SPC-O exhibited higher gelation temperatures during heating, yet could still develop greater stiffness during cooling once network formation was established.

From a processing standpoint, the sharp increase in G′ near 71 °C indicates that allowing the material to progress too far into gelation inside the cooling die would increase elasticity and flow resistance, resulting in higher pressure and poorer strand alignment, whereas discharging the material too early would risk insufficient structural set. Therefore, an exit temperature of 70–75 °C is justified as a practical balance between extrudability and rapid post-exit stabilization, consistent with previous cooling-die studies showing that die temperature gradients strongly influence modulus development and anisotropic texture in high-moisture meat analogs [33].

3.3. Estimation and Validation of the Diffusion Coefficient

The results of the depth-resolved oil redistribution and inverse diffusion calibration are shown in Figure 7. Figure 7a(1–5) presents representative cross-sectional images taken at increasing depths within the 10-mm SPC matrix after 100 °C, 60 s “drop test”, and rapid cooling. The images show that oil does not remain confined to the application surface; instead, distinct oil-rich domains are observed within the interior layers, with the most pronounced accumulation occurring in the near-subsurface region (4 mm depth, Figure 7b). This spatial pattern indicates that the applied droplet behaves as a finite-dose source, where the interfacial oil layer is progressively depleted while oil penetrates and becomes retained within the matrix [34].

In a highly viscous, protein-rich continuous phase, transport is governed not only by molecular diffusion but also by matrix resistance and interfacial structuring: adsorption and rearrangement of soy proteins at the oil–water interface can immobilize part of the oil at droplet surfaces and increase oil–matrix coupling, thereby promoting internal retention rather than continued surface pooling [35]. Consequently, the qualitative and quantitative analysis provided the physical basis for calibrating a processing-relevant diffusion coefficient in the subsequent CFD matching step.

To obtain a processing-relevant transport parameter, the diffusion coefficient was calibrated by iteratively matching the CFD-predicted depth profile to the experimental profile. The RMSE curve (Figure 7c) showed a clear minimum at D_ref = 4.86 × 10⁻¹⁸ m²/s, indicating the best agreement between simulated and measured oil fractions across the 0–8 mm domain. At this selected value, the CFD-predicted depth profile agreed with the experimental profile within 5% across the fitted depth range. Based on the RMSE profile from the five-point fitting procedure, an approximate 95% confidence range for D_ref was estimated as 6.11 × 10⁻¹⁹ to 3.86 × 10⁻¹⁷ m²/s. This empirically anchored D_ref is lower than the idealized Stokes–Einstein estimate (D₀) because the SPC matrix at 100 °C is highly viscous and exhibits macromolecular obstruction and free-volume restrictions that suppress oil mobility [36]. Thus, D_ref should be interpreted as an effective inverse-calibrated transport coefficient, rather than as a strict molecular diffusion constant.

This low magnitude is also consistent with the fact that transport in dense soy matrices is strongly restricted. In high-moisture soy protein extrudates, an apparent water diffusion coefficient of about 9.5 × 10⁻¹⁰ m²/s has been reported, while water self-diffusion in soy-protein-based tofu/SPI systems has been reported at 2.23 × 10⁻⁹ m²/s. Although those values were obtained for water rather than oil and under different structures and temperatures, they are still many orders of magnitude higher than the present D_ref, indicating that oil redistribution in the thermally setting SPC matrix was much more strongly constrained by viscosity increase, network development, and interfacial retention [37,38].

Temperature dependence was then introduced using the fractional Stokes–Einstein model with n = 0.92, linking diffusion directly to the measured viscosity rise during cooling. The resulting D_eff decreases from 4.86 × 10⁻¹⁸ m²/s at 100 °C to ~1.9 × 10⁻¹⁸ m²/s by 70–60 °C (Figure 7d), consistent with progressive loss of oil mobility as the matrix stiffened [29,39]. At the process level, this behavior helps explain why oil redistribution is more permissive in the hotter region of the system but becomes increasingly restricted as the strand passes through the cooling die and viscosity/structure develop, thereby promoting stabilization of oil domains before discharge. However, because interfacial-tension-driven effects, droplet deformation, coalescence, and explicit phase separation were not resolved separately, D_eff should be interpreted cautiously as an effective bulk transport parameter for marbling stabilization rather than as a fully mechanistic molecular diffusivity.

3.4. Static Drop Test Model Validation

Figure 8 and Figure 9 summarize the independent static drop-test datasets used to evaluate the predictive capability of the calibrated transport model prior to rapid cooling. Unlike the calibration step in Section 3.3, which used only the depth-resolved oil profile obtained at 100 °C after 60 s of holding to estimate D_ref, the results in Figure 8 and Figure 9 represent additional experimental conditions not used in the fitting procedure. These datasets therefore provide an independent basis for assessing whether the calibrated model can reproduce time- and temperature-dependent oil redistribution trends in SPC.

The surface oil mass fraction decreased significantly with waiting time at all temperatures (Figure 8a). The temperature effect was strongest at early times: at 90 °C, the surface fraction decayed more rapidly, indicating faster oil migration away from the application interface when the matrix was in a lower-resistance (more mobile) state. This is consistent with high-moisture protein melts exhibiting a steep viscosity–temperature dependence, where small decreases in temperature substantially increase resistance to molecular transport and suppress phase mobility, particularly as gelation or structuring initiates [29].

The measured surface oil fraction is governed not only by diffusion into the bulk but also by interfacial transport processes. A finite oil droplet on a hot viscoelastic substrate can undergo gravitational leveling and lateral spreading driven by wetting and surface/interfacial tension, and spreading can be further modified by surface-tension gradients (Marangoni stresses) arising from temperature or composition gradients at the oil–matrix interface [40]. These mechanisms can accelerate the apparent surface depletion beyond what would be expected from purely one-dimensional diffusion.

In contrast, Figure 9a reports the oil mass fraction at 3 mm depth, which increases with waiting time as oil penetrates into the interior. At short holding times, the temperature dependence is clear: the 90 °C condition shows the fastest increase in subsurface oil fraction, followed by 80 °C and 70 °C, whereas 60 °C increases most slowly. This trend indicates that higher temperatures promote faster inward transport because the matrix offers lower viscous resistance and the effective diffusivity is higher. As temperature decreases toward 60 °C, the melt approaches a more solid-like, partially set state, reducing mobility and slowing inward transport, which is consistent with the role of the cooling die as the critical zone where viscosity gradients and progressive structuring limit component migration and promote spatial “locking” of internal phases [29]. The CFD contour maps (Figure 8b and Figure 9b) reinforce this interpretation by showing faster early-time redistribution at 90 °C and more persistent localization at 60 °C, where the transport front advances more slowly. Below the surface, transport is primarily governed by concentration-driven diffusion through a crowded protein network (with obstruction and reduced free volume), rather than by free-surface forces; this distinction supports using depth profiles (e.g., 3 mm) as a more direct metric of bulk diffusional kinetics for calibration and validation.

A key processing outcome from both datasets is the emergence of a practical equilibration time. In Figure 8a, the reduction in surface oil fraction becomes insignificantly different between 50 and 60 s, and in Figure 9a the increase at 3 mm depth also shows diminishing gains over the same interval. This convergence indicates that, under the static hold condition, additional residence time beyond ~50 s provides limited incremental redistribution relative to the measurement variability and model resolution. Operationally, this defines an upper bound for the residence time in the hot injection/transport zone. Extending holding time beyond ~50 s is unlikely to materially increase bulk penetration but may increase the risk of unwanted lateral spreading at the interface and loss of spatial control.

The validated static kinetics suggest a staged die strategy for marbling formation and stabilization. The first die section (oil injection and initial transport) should be maintained at approximately 100 °C and dimensioned to provide a residence time of about 50 s, enabling controlled redistribution before the process transitions to the fibrous-formation zone where viscosity gradients and shear/elongational flow align the protein matrix. Subsequent rapid cooling to ~70–75 °C in the final section is then used to arrest mobility and preserve the developed oil-domain architecture. This linkage between residence time, temperature-driven viscosity, and phase mobility aligns with broader high-moisture extrusion evidence that die geometry and cooling profiles govern structure development and internal phase stability [41].

3.5. Numerical Simulation of Co-Extrusion Oil Transport and Marbling Formation

Figure 10 shows the CFD-predicted evolution of oil mass fraction during the co-extrusion stage and links these predictions to the experimentally observed surface appearance and internal marbling. The contour series (Figure 10a; oil mass fraction) shows that the injected oil enters as a concentrated stream near the inlet and is transported downstream primarily by advection, while transverse spreading into the surrounding SPC matrix is governed by diffusion that progressively weakens as the matrix viscosity increases and structure develops during cooling. This coupling between temperature, viscosity, and component mobility is a defining feature of high-moisture extrusion and cooling-die solidification: a temperature-induced viscosity gradient constrains mass transfer and contributes to stabilizing internal heterogeneities as the strand sets [29].

At early residence times (10–30 s), the simulation retains a localized high-oil region close to the injection zone, while the downstream channel remains oil-lean. With increasing residence time (>50 s), the oil field extends further along the flow direction and becomes more spatially distributed, indicating continued redistribution before the system approaches a quasi-steady pattern. This time dependence is consistent with the calibrated transport framework developed in Section 3.3 and Section 3.4, where oil mobility is relatively higher at the hot stage but collapses as the matrix stiffens, thereby limiting further redistribution [29]. Importantly, the simulation implies diminishing changes in the oil field at longer times, supporting the processing decision to cap the effective “diffusion/redistribution window” at approximately 50 s before transitioning to the next die section for fibrous structuring and rapid cooling.

The photographic panels are consistent with the model-based interpretation. The co-extrudate immediately after extrusion (Figure 10b) shows a continuous strand with visible oil-associated surface heterogeneity, whereas after cooling (Figure 10c) the strand exhibits improved shape retention and a more stable surface, consistent with reduced mobility following cooling-die solidification. The cross-sectional slice (Figure 10d) reveals discrete light domains distributed within a fibrous matrix, indicating that oil was retained as internal inclusions rather than fully coalescing or draining, which is an expected outcome when diffusion is arrested by rapid viscosity increase and network consolidation [42].

Collectively, Figure 10 supports a staged die strategy for marbling: a hot upstream section (~100 °C) that provides ~50 s residence time for controlled oil redistribution, followed by a fibrous-formation zone and a final rapid-cooling section (to ~70–75 °C) to “lock” the oil architecture into the final marbled structure.

4. Conclusions

This study employed a multi-modal framework integrating thermal–rheological characterization with image-derived oil profiling and CFD transport modeling to elucidate coconut-oil migration and marbling stabilization in SPC during static and co-extrusion processing. The results showed that oil incorporation modifies the thermal setting behavior of the SPC matrix and that oil mobility becomes progressively restricted as the system cools and the protein network develops. By linking experimentally measured oil profiles with inverse diffusion calibration, the study generated a processing-relevant transport parameter and extended it to temperature-dependent prediction of oil-domain stabilization during cooling. The main contribution of this work is not only the characterization of gelation and oil migration behavior, but also the development of a predictive design strategy for controlling marbling architecture in plant-based meat analogs. The combined imaging–CFD approach provides a practical basis for selecting process conditions such as thermal gradients, residence time, and oil injection location to achieve more stable and visually uniform fat distribution. In this way, the framework moves beyond descriptive observation toward process-guided design of structured plant-based products. From an application perspective, the findings support the use of staged thermal control during co-extrusion, where oil redistribution can occur under hotter upstream conditions and be stabilized through downstream cooling as the protein matrix sets. This offers a useful foundation for scaling marbling control in continuous manufacturing systems. Future work should extend this framework by linking marbling microstructure with sensory quality, texture perception, oxidative stability, and product performance during storage and cooking, thereby advancing from structural prediction to full product optimization.