Abstract
The Stefan problem approach was applied to the mathematical modeling of the diffusion process that occurs during hydration of soybean grains. This technique considers that the boundaries of the problem are unknown, and finding them is a task, which is also part of the problem. The problem has boundaries which move along time. The governing differential equations are numerically solved for the case of transient diffusion equation, and an analytical solution is obtained for the hydration fronts by considering the pseudo-steady-state hypothesis for the diffusion equation. The behavior of hydration fronts was analyzed, which are defined by the moving boundaries, and their velocities for both cases and the main differences were analyzed in terms of considering or not the transient term of the diffusion equation. The results show the main differences in the behavior of hydration fronts that arise when the transient term is taken into account or the pseudo-steady-state hypothesis is considered and compare the obtained behaviors with experimental data.
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Abbreviations
- a :
-
Model constant
- b :
-
Model constant
- \(C_{1}\) :
-
Integration constant
- \(C_{2}\) :
-
Integration constant
- D (m\(^{2}\)/s):
-
Diffusivity
- D*:
-
Nondimensional diffusivity
- \(D_{0}\) (m\(^{2}\)/s):
-
Pre-exponential factor
- \(k_{1}\) :
-
Exponential factor
- r (m):
-
Radial coordinate
- R (m):
-
External moving front (grain radius)
- S (m):
-
Internal moving front
- t (s):
-
Time coordinate
- \(\beta \) :
-
Exponential factor of nondimensional diffusivity
- \(\gamma \) :
-
Model constant
- \(\Theta \) :
-
Nondimensional volume fraction
- \(\varphi \) :
-
Liquid volume fraction
- \(\omega \) :
-
Integration variable
- 0:
-
Initial
- 1:
-
On the surface
- max:
-
Maximum
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We thank CAPES—Brazil for the support provided.
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Nicolin, D.J., Jorge, R.M.M. & Jorge, L.M.M. Stefan Problem Approach Applied to the Diffusion Process in Grain Hydration. Transp Porous Med 102, 387–402 (2014). https://doi.org/10.1007/s11242-014-0280-0
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DOI: https://doi.org/10.1007/s11242-014-0280-0