Modeling of soybean hydration as a Stefan problem: Boundary immobilization method

Abstract

In the present work, the mathematical modeling of the soybean hydration process is presented, considering the increase in the grain radius by Stefan problem approach. The boundary immobilization method was used in the model to take into account the spatial mesh variation caused by the diffusion of moisture in the grains. The results provided by the model show its quality when compared with experimental data, both for the adequacy of the predictions for moisture data over time and for the adequacy of grain radius increase predictions over time, for five different temperatures. The presented methodology showed improvements on results already reported in the literature. Practical applications The approach presented in this article allows one to fit diffusivity values for soybean hydration by taking into account a phenomenon commonly observed in this process: the increase of the grains during water absorption. Thus, it is expected the diffusivities to be more realistic and precise, improving the project of equipment. Since the model proposed provide the increase of the grains along the hydration process, an estimation of the increase of a higher amount of the grains could be obtained as well. Such estimation is relevant due to the considerable increase that grains suffer after absorbing water.

Fractional calculus to control transport phenomena in food engineering: A systematic review of barriers and data agenda

“…Simpson et al (2017) serve to guide research on FCs in drying and hydration in general since it explains the models used for diffusion and convection in these situations and relates these models to Fick's law, attributing and explaining the physical meaning. It is worth mentioning that most authors defend Simpson's theory since this author has the largest collaboration network as shown in Figure 3; some authors do not attribute physical meaning to fractional order models from the mass balance, among authors who do not comment on physical meaning can be mentioned in Figure 3, Rossoni, Jorge and Nicolin, this does not necessarily indicate that the authors disagree with Simpson et al (2017), but these authors make no mention of the physical meaning of the models as seen in the following works of their co‐authorship: Souza Matias et al (2019); Nicolin, Balbinoti, et al (2018) Nicolin, da Silva, et al (2018). It should also be noted that the authors Rossoni, Jorge, and Nicolin, as Souza Matias et al (2019); Nicolin, Balbinoti, et al (2018); Nicolin, da Silva, et al (2018), they attribute that their semi‐empirical model, having a fractional adjustment, indicates anomalous diffusion, because according to these authors the whole order model is an exponential model, and conventional diffusion is an exponential process.…”

“…The authors Rojas and Augusto, which can be seen in Figure 3, end up adopting methods similar to Simpson and Ramirez so that their works make several considerations adopted by Simpson et al (2017). Nicolin, da Silva, et al (2018) generalized the order of a kinetic model, based on the mass balance for drying, the model considers that the variation in humidity is proportional to a constant time the humidity, other authors such as Souza Matias et al (2019) propose models analogous to the law Newton's cooling method where the humidity variation is equal to a constant time.…”

“…As occurring in the integer order calculus, the fractional calculus presents a range of functions associated with it, among them the Mittag–Leffler function () stands out for having an equivalent meaning in the fractional calculus that exponential function role on Integer Order Calculus (Nicolin, Balbinoti, et al, 2018; Nicolin, et al, 2018). Thus, the Mittag–Leffler function is the solution of the linear differential equation (Nicolin, Balbinoti, et al, 2018; Nicolin, et al, 2018). Equation (5) presents the Mittag–Leffler Function: …”

“…The use of FC has increased in recent years but the interpretation regarding the real physical meaning of fractional‐order models is still an on‐going debate (Castro et al, 2014; Simpson et al, 2013; Simpson & Sroufe, 2014) and some authors consider that such models have no physical meaning (Matias et al, 2018; Nicolin, Balbinoti, et al, 2018; Nicolin, et al, 2018). Nonetheless, there is a lack of studies in the academic and gray literature aimed at determining the conditions using fractional‐order models.…”

Manifestation of neuro‐fuzzy simulation environment for prognostication of water absorption kinetics of soybean grains in thermo‐ultrasonication‐assisted soaking process

“…In the realm of soaking process of soybean grains, there are many works conducted by researchers for studying of water absorption kinetics of the process. Some of these works can be chronologically addressed herein (Bayram, Oner, & Kaya, 2004; Bayram, Kaya, & Oner, 2004a; Borges, Jorge, & Jorge, 2017; Borges, Jorge, & Jorge, 2019; Deshpande, Bal, & Ojha, 1994; Fracasso, Perussello, Haminiuk, Jorge, & Jorge, 2014; Gowen, Abu‐Ghannam, Frias, & Oliveira, 2007; Hsu, 1983; Lima, Kurozawa, & Ida, 2014; Mukherjee, Chakraborty, & Dutta, 2019; Nicolin, Coutinho, Andrade, & Jorge, 2012; Nicolin, Neto, Paraiso, Jorge, & Jorge, 2015; Nicolin, Rossoni, & Jorge, 2016; Nicolin, Silva, Jorge, & Jorge, 2018; Pan & Tangratanavalee, 2003; Singh & Kulshrestha, 1987; Sopade & Obekpa, 1990; Vasudeva & Vishwanathan, 2010; Wang, Swain, Hesseltine, & Heath, 1979; Wardhani, Vazquez, & Pandiella, 2008). The researchers who carried out the works employed major nonlinear mathematical models to investigate water absorption kinetics of soybean grains.…”