Fractional Calculus as a Mathematical Tool to Improve the Modeling of Mass Transfer Phenomena in Food Processing

Development of an Apple Snack Enriched with Probiotic Lacticaseibacillus rhamnosus: Evaluation of the Refractance Window Drying Process on Cell Viability

The objective of this study was to develop a dried apple snack enriched with probiotics, evaluate its viability using Refractance Window (RWTM) drying, and compare it with conventional hot air drying (CD) and freeze-drying (FD). Apple slices were impregnated with Lacticaseibacillus rhamnosus and dried at 45 °C using RWTM and CD and FD. Total polyphenol content (TPC), color (∆E*), texture, and viable cell count were measured, and samples were stored for 28 days at 4 °C. Vacuum impregnation allowed for a probiotic inoculation of 8.53 log CFU/gdb. Retention values of 6.30, 6.67, and 7.20 log CFU/gdb were observed for CD, RWTM, and FD, respectively; the population in CD, RWTM remained while FD showed a decrease of one order of magnitude during storage. Comparing RWTM with FD, ∆E* was not significantly different (p < 0.05) and RWTM presented lower hardness values and higher crispness than FD, but the RWTM-dried apple slices had the highest TPC retention (41.3%). Microstructural analysis showed that RWTM produced a smoother surface, facilitating uniform moisture diffusion and lower mass transfer resistance. The effective moisture diffusion coefficient was higher in RWTM than in CD, resulting in shorter drying times. As a consequence, RWTM produced dried apple snacks enriched with probiotics, with color and TPC retention comparable to FD.

Fractional calculus as a generalized kinetic model for biochemical methane potential tests

This study presents a fractional calculus model as a generalized kinetic model for estimating the maximum methane yield and degradation kinetics in biomethane potential (BMP) assays, a key analytical method in anaerobic digestion research and application. The fractional model outperformed common first-order kinetic models by yielding superior data fitting and properly managing substrate heterogeneity. The fractional model showed robust performance in mono-digestion, co-digestion and pre-treatment BMP assays with or without presence of large tailing or sigmoidal patterns in the BMP curve. The main advantage of the fractional model over other models is its ability to capture the complexities of the methane production process without losing model accuracy. Assessment of the mathematical model revealed that for fractional orders greater than 0.8 the Mittag-Leffler sequence could be transformed into a more computationally efficient exponential function.

Critical Analysis of the Use of Semiempirical Models on the Dehydration of Thin-Layer Foods Based on Two Study Cases

Moisture transport during food drying can be phenomenologically described by Fick’s second law and by the so-called anomalous diffusion model. However, in the literature, many studies have shown the extensive use of empirical/semiempirical models (EMs/SEMs) to adjust experimental data for the drying of thin-layer foods. This research aims to perform a critical analysis of the most commonly used EMs/SEMs and compare them with Fick’s second law and an anomalous diffusion model using two different sets of hot-air drying data. Two waste byproducts from the food industry, spent coffee grounds and passion fruit peels, were selected for analysis. The selected EMs/SEMs were found to be mathematically interrelated (i.e., some are a subset of others), and their appropriateness was incorrectly justified mainly by their statistical goodness-of-fit. As shown, it is highly recommended that researchers start analyzing drying data with phenomenological models. The extensive use of EMs and SEMs can be replaced by the anomalous diffusion model, which has a high capacity to adjust empirical data and a sound phenomenological description of the process.

Refractance window drying of food and biological materials: Status on mechanisms, diffusion modelling and hybrid drying approach

Refractance window (RW) dryer has an immense advantage in terms of final product quality (textural and color attributes, nutrient retention), energy consumption, and drying time over other conventional dryers. RW is a thin film drying system and a technologically evolving drying process. RW drying is an energy-efficient (re-circulation of water) short drying process as the drying of food materials occurs due to a combined mode of heat transfer conduction, radiation, and convection (hot air circulates over film). The high-quality dried product is obtained because the product temperature remains below 80 °C. RW dryer application is not only limited to drying food products, but it can also be further used for improving the gelling and emulsion properties, formation of leather and edible film, and can be used for handling high protein products, drying leafy vegetables or marine foods as this process does not change any functional properties. Due to these advantages over other drying techniques, RW drying has gained academic and industrial interest in recent years. The industrial application of this technology at large scale is becoming difficult due because of large surface area requirement for mass production. Researchers are trying to scale-up by combing this technology with others technology (Infrared, ultrasound, solar energy, and osmotic dehydration). RW dryer is now extending from the food sector to other sectors like pharmaceutical, cosmetic, pigment, edible film formation, and encapsulation. Majority of the reviews on RW drying focuses on the product quality aspects. This review paper aims to comprehend the RW drying system more mechanistically to understand better the principles, diffusion models explaining the transfer processes, and emerging novel hybrid drying approaches.

Effect of a moderate electric field on the salting of Atlantic Salmon (Salmo salar): An experimental study and phenomenological understanding

Salting is one of the oldest methods of preserving food. The main limitation of salting is its extended processing time due to slow salt diffusion. A moderate electric field (MEF) can improve the mass transfer rate through electroporation. Regularly, mass transfer processes are modeled with Fick's second law. However, due to the anisotropic nature of food microstructures, it might be more appropriate to use an anomalous model. The main objective of this study was to search for a phenomenological explanation for salt and water diffusion in the salmon brining process coupled with MEF. Salmon fillets were cut into finite cylinders (0.025 × 0.025 m) and brined in two salt concentrations (6 and 24% w/w NaCl) at 6 °C for 20 h. MEFs were applied in the range of 0 to 2 V/cm. The salt and water contents of the salmon were measured during the process. Fick's second law and anomalous model based on fractional calculus were used to describe the diffusion phenomena. The results showed that an MEF tended to reduce the brining processing time and increase the salt content of salmon. This effect is predominantly due to an increase in the equilibrium salt concentration in the salmon tissue. Mathematical analysis shows that the anomalous diffusion model is more suitable for representing the brining process, exhibiting superdiffusion behavior (α > 1). An MEF accelerates the salt mass transfer into salmon tissue even at lower temperatures, significantly reducing the processing time. In addition, the diffusion process can be characterized with an anomalous model.

Kinetic Model for Drying in Frame of Generalized Fractional Derivatives

In this article, the Lewis model was considered for the soybean drying process by new fractional differential operators to analyze the estimated time in 50 ∘ C , 60 ∘ C , 70 ∘ C , and 80 ∘ C . Moreover, we used dimension parameters for the physical meaning of these fractional models within generalized and Caputo fractional derivatives. Results obtained with generalized fractional derivatives were analyzed comparatively with the Caputo fractional, integer order derivatives and Page model for the soybean drying process. All results for fractional derivatives are discussed and compared in detail.

Modeling Anomalous Moisture Transport in Cement-Based Materials with Kinetic Permeability

The durability of reinforced concrete structures is closely related to moisture state in cement-based materials. Therefore, it is crucial to develop moisture models that can accurately predict moisture state in the materials. However, many studies reported anomalous moisture transport in cement-based materials that cannot be well simulated by the conventional models. Several reasons have been investigated in the literature, such as the complex pore structure, chemical reactions with water, dimensional changes of the tested specimen, etc. Nevertheless, only a few models are able to capture the anomaly of moisture transport. This study viewed the main moisture transport coefficient-permeability-as a kinetic variable that depends on both the degree of moisture saturation and the contact time. The time-dependence was formulated by the decay (for drying) or growth (for wetting) functions. The saturation-dependence was calculated by the van Genuchten-Mualem (VGM) model. These functions were then implemented into a moisture transport model that was developed in previous studies. The proposed model was validated by experimental data and showed a good agreement for cement pastes that were dried or wetted in the hygroscopic range. Numerical simulation results were also compared with the simplified solutions to a fractional derivative model (FDM) of anomalous diffusion and the empirical Weibull function. We found that the solutions to the FDM cannot provide appropriate results. Weibull function performs as well as the proposed model, but the empirical function lacks physical meanings.