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.
Similar content being viewed by others
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
References
Aguerre, R.J., Tolaba, M., Suarez, C.: Modeling volume changes in food drying and hydration. Lat. Am. Appl. Res. 38, 345–349 (2008)
Barry, S.I., Caunce, J.: Exact and numerical solutions to a Stefan problem with two moving boundaries. Appl. Math. Model. 32, 83–98 (2008)
Caldwell, J., Chan, C.: Spherical solidification by the enthalpy method and the heat balance integral method. Appl. Math. Model. 24, 45–53 (2000)
Coutinho, M.R.: Modeling, simulation and analysis of the hydration of soybeans. Thesis (Doctorate in Chemical Engineering), State University of Maringá, Brazil (2006)
Crank, J.: Free and Moving Boundary Problems, 1st edn. Oxford University Press, New York (1984)
Davey, M.J., Landman, K.A., McGuinness, M.J., Jin, H.N.: Mathematical modeling of rice cooking and dissolution in beer production. AIChE J. 48, 1811–1826 (2002)
Hsu, K.H.: A diffusion model with a concentration-dependent diffusion coefficient for describing water movement in legumes during soaking. J. Food Sci. 48(2), 618–622 (1983)
Javierre, E., Viuk, C., Vermolen, F.J., van der Zwaag, S.: A comparison of numerical models for one-dimensional Stefan problems. J. Comput. Appl. Math. 192, 445–459 (2006)
Kharab, A.: Numerical approximation to a moving boundary problem for the sphere. Comput. Chem. Eng. 21(6), 559–562 (1997)
McCue, S.W., Wu, B., Hill, J.M.: Classical two-phase Stefan problem for spheres. Proc. R. Soc. A 464, 2055–2076 (2008)
McGuinness, M.J., Please, C.P., Fowkes, N., McGowan, P., Ryder, L., Forte, D.: Modelling the wetting and cooking of a single cereal grain. IMA J. Manage. Math. 11, 49–70 (2000)
Mills, A.F., Chang, B.H.: Two-dimensional diffusion in a Stefan tube: the classical approach. Chem. Eng. Sci. 90, 130–136 (2013)
Mitchell, S.L., O’Brien, S.B.G.: Asymptotic, numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers. Appl. Math. Comput. 219, 376–388 (2012)
Nicolin, D.J., Coutinho, M.R., Andrade, C.M.G., Jorge, L.M.M.: Hsu model analysis considering grain volume variation during soybean hydration. J. Food Eng. 111(3), 496–504 (2012)
Purlis, E., Salvadori, V.O.: Bread baking as a moving boundary problem. Part 1: mathematical modelling. J. Food Eng. 91, 428–433 (2009a)
Purlis, E., Salvadori, V.O.: Bread baking as a moving boundary problem. Part 2. Mode validation and numerical simulation. J. Food Eng. 91, 434–442 (2009b)
Savovic, S., Caldwell, J.: Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. Int. J. Heat Mass Transf. 46, 2911–2916 (2003)
Voller, V.R.: An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Int. J. Heat Mass Transf. 53(23–24), 5622–5625 (2010)
Voller, V.R., Falcini, F.: Two exact solutions of a Stefan problem with varying diffusivity. Int. J. Heat Mass Transf. 58(1–2), 80–85 (2013)
Acknowledgments
We thank CAPES—Brazil for the support provided.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-014-0280-0