Abstract
Mass or heat transfer may cause volume variation, and the food hydration model is one of them that undergoes hydration (or drying) conveying volume change. In this paper, the numerical approximate solution based on an integral method has been presented for soybean hydration model. Trace of the moving boundary and unknown moisture content at the center of the grain have been determined. The obtained results are well matched with numerical solutions in the literature.
References
[1] P. Geervani and F. Theophilus, Effect of home processing on the protein quality of selected legumes, J. Food Sci. 45 (1980), 707–710.10.1111/j.1365-2621.1980.tb04138.xSearch in Google Scholar
[2] W. Y. Lo, K. H. Steinkraus, D. B. Hand, L. R. Hackler and W. F. Wilkens, Soaking soybeans before extraction as it affects chemical composition and yield of soymilk, Food Technol. 22 (1968), 1188–1190.Search in Google Scholar
[3] H. L. Wang, E. W. Swain, C. W. Hesseltine and H. D. Heath, Hydration of whole soybeans affects solids losses and cooking quality, J. Food Sci. 44 (1979), 1510–1513.10.1111/j.1365-2621.1979.tb06474.xSearch in Google Scholar
[4] M. R. Coutinho, W. A. dos, S. Conceição, P. R. Paraˊso, C. M. G. Andrade, E. S. Omoto, R. M. M. Jorge, R. M. Filho, L. M. M. Jorge, Application of the Hsu model to soybean grain, Food Sci. Technol. 30(1) (2010), 19–29.10.1590/S0101-20612010005000019Search in Google Scholar
[5] M. Peleg, An empirical model for the description of moisture sorption curves, J. Food Sci. 53(4) (1988), 1216–1219.10.1111/j.1365-2621.1988.tb13565.xSearch in Google Scholar
[6] B. P. N. Singh and S. P. Kulshrestha, Kinetics of water sorption by soybean and pigeonpea grains, J. Food Sci. 52(6) (1987), 1538–1542.10.1111/j.1365-2621.1987.tb05874.xSearch in Google Scholar
[7] P. A. Sopade and J. A. Obekpa, Modelling water absorption in soybean, cowpea and peanuts at three temperatures using Peleg’s equation, J. Food Sci. 55(4), 1084–1087.10.1111/j.1365-2621.1990.tb01604.xSearch in Google Scholar
[8] Z. Pan and W. Tangratanavalee, Characteristic of soybeans as affected by soaking conditions, LWT-Food Sci. Technol. 36(1) (2003), 143–151.10.1016/S0023-6438(02)00202-5Search in Google Scholar
[9] A. Gowen, N. Abu-Ghannam, J. Frias and J. Oliveira, Influence of the pre-blanching on the water absorption kinetics of soybeans, J. Food Eng. 78(3) (2007), 965–971.10.1016/j.jfoodeng.2005.12.009Search in Google Scholar
[10] M. R. Coutinho, W. A. S. Conceição, E. S. Omoto, C. M. G. Andrade and L. M. M. Jorge, New model of lumped parameters applied to grain hydration, Ciencia e Technologia de Alimentos 27(3) (2007), 451–455.10.1590/S0101-20612007000300005Search in Google Scholar
[11] D. J. Nicolin, R. M. Neto, P. R. Paraˊso, R. M. M. Jorge and L. M. M. Jorge, Analytical solution and experimental validation of a model for hydration of soybeans with variable mass transfer coefficient, J. Food Eng. 149 (2015), 17–23.10.1016/j.jfoodeng.2014.09.044Search in Google Scholar
[12] K. H. Hsu, A diffusion model with a concentration-dependent diffusion coefficient for describing water movement legumes during soaking, J. Food Sci. 48(2) (1983), 618–622.10.1111/j.1365-2621.1983.tb10803.xSearch in Google Scholar
[13] M. R. Coutinho, E. S. Omoto, W. A. D. S. Conceição, C. M. G. Andrade and L. M. M. Jorge, Modeling of the soybean grains hydration by a distributed parameters approach, Int. J. Food Eng. 5(3) (2009). Article 11.10.2202/1556-3758.1654Search in Google Scholar
[14] D. J. Nicolin, M. R. Coutinho, C. M. G. Andrade and L. M. M. Jorge, Hsu model analysis considering grain volume variation during soybean hydration, J. Food Eng. 111(3) (2012), 496–504.10.1016/j.jfoodeng.2012.02.035Search in Google Scholar
[15] D. J. Nicolin, R. M. M. Jorge and L. M. M. Jorge, Stefan problem approach applied to the diffusion process in grain hydration, Transp. Porous Media. 102(3) (2014), 387–402.10.1007/s11242-014-0280-0Search in Google Scholar
[16] D. J. Nicolin, R. M. M. Jorge and L. M. M. Jorge, Evaluation of distributed parameters mathematical models applied to grain hydration with volume change, Heat Mass Transfer. 51(1) (2015), 107–116.10.1007/s00231-014-1404-3Search in Google Scholar
[17] D. J. Nicolin, G. E. C. da Silva, R. M. M. Jorge and L. M. M. Jorge, Numerical solution of a nonlinear diffusion model for soybean hydration with moving boundary, Int. J. Food Eng. 12(5) (2015), 587–595.10.1515/ijfe-2015-0035Search in Google Scholar
[18] D. J. Nicolin, R. M. M. Jorge and L. M. M. Jorge, Moving boundary modeling of conventional and transgenic soybean hydration: Moisture profile and moving front experimental validation, Int. J. Heat Mass Transfer. 90 (2015), 568–577.10.1016/j.ijheatmasstransfer.2015.07.014Search in Google Scholar
[19] C. Engels, M. Hendrickx, S. de Samblanx, I. de Gryze and P. Tobback, Modeling water diffusion during long-grain rice soaking, J. Food Eng. 5(1) (1986), 55–73.10.1016/0260-8774(86)90019-1Search in Google Scholar
[20] S. Sayar, M. Turhan and S. Gunesekaran, Analysis of chickpea soaking by simultaneous water transfer and water–starch reaction, J. Food Eng. 50(2) (2001), 91–98.10.1016/S0260-8774(00)00196-5Search in Google Scholar
[21] A. Dutta, A. Chanda and R. Chakraborty, A linear driving force (LDF) approximation of moisture diffusion kinetics in white rice, Int. J. Food Eng. 4(8) (2008). Article 2.10.2202/1556-3758.1439Search in Google Scholar
[22] S. H. Lin, Water uptake and gelatinization of white rice, LebensmittelWissenschaft und – Technologie 26(3) (1993), 276–280.10.1006/fstl.1993.1057Search in Google Scholar
[23] S. Gulen and T. Ozis, Solution of Hsu model by Crank-Nicolson method and Splitting technique, Bull. Int. Math. Virtual Inst. 8 (2018), 431–437.Search in Google Scholar
[24] S. Gulen and T. Ozis, Fourth order compact finite difference scheme for soybean hydration model with moving boundary, Bull. Int. Math. Virtual Inst. 6 (2016), 227–239.Search in Google Scholar
[25] S. Gulen and T. Ozis, Compact finite difference schemes for soybean hydration model as Stefan Problem, NTMSCI 6(2) (2018), 184–199.10.20852/ntmsci.2018.284Search in Google Scholar
[26] M. R. Coutinho, E. S. Omoto, C. M. G. Andrade and L. M. M. Jorge, Modeling and validation of soya bean hydration, Ciencia e Tecnologia de Alimentos 25(3) (2005), 603–610.10.1590/S0101-20612005000300034Search in Google Scholar
[27] M. R. Coutinho, W. A. D. S. Conceição, E. S. Omoto, C. M. G. Andrade and L. M. M. Jorge, New model of lumped parameters applied to grain hydration, Ciencia e Tecnologia de Alimentos 27(3), 451–455.10.1590/S0101-20612007000300005Search in Google Scholar
[28] A. K. Verma, S. Chandra and B. K. Dhindaw, An alternative fixed grid method for solution of the classical one-phase Stefan problem, Appl. Math. Comput. 158(2) (2004), 573–584.10.1016/j.amc.2003.10.001Search in Google Scholar
[29] S. L. Mitchell and M. Vynnycky, Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. Math. Comput. 215(4) (2009), 1609–1621.10.1016/j.amc.2009.07.054Search in Google Scholar
[30] F. Yigit, One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition, Appl. Math. Comput. 217(14) (2011), 6541–6555.10.1016/j.amc.2011.01.033Search in Google Scholar
[31] S. L. Mitchell, M. Vynnycky, I. G. Gusev and S. S. Sazhin, An accurate numerical solution for the transient heating of an evaporating spherical droplet, Appl. Math. Comput. 217(22) (2011), 9219–9233.10.1016/j.amc.2011.03.161Search in Google Scholar
[32] T. E. Lee, M. J. Baines and S. Langdon, A finite difference moving mesh method based on conservation for moving boundary problems, J. Comput. Appl. Math. 288 (2015), 1–17.10.1016/j.cam.2015.03.032Search in Google Scholar
[33] J. V. Miller, K. W. Morton and M. J. Baines, A finite element moving boundary computation with an adaptive mesh, J. Inst. Math. Appl. 22(4) (1978), 467–477.10.1093/imamat/22.4.467Search in Google Scholar
[34] S. G. Ahmed, An approximate method for oxygen diffusion in a sphere with simultaneous absorption, Int. J. Numer. Methods Heat Fluid. Flow 9(6) (1999), 631–643.10.1108/09615539910276115Search in Google Scholar
[35] J. Caldwell and C. C. Chen, Spherical solidification by enthalpy method and heat balance integral method, Appl. Math. Modell. 24(1) (2000), 45–53.10.1016/S0307-904X(99)00031-1Search in Google Scholar
[36] S. Catal, Numerical approximation for the oxygen diffusion problem, Appl. Math. Comput. 145(2-3) (2003), 361–369.10.1016/S0096-3003(02)00493-9Search in Google Scholar
[37] N. Sadoun, E. K. Si-Ahmed and P. Colinet, On the refined integral method for the one-phase Stefan problem with time dependent boundary conditions, Appl. Math. Modell. 30(6) (2006) 531–544.10.1016/j.apm.2005.06.003Search in Google Scholar
[38] S. Kutluay, A. S. Wood and A. Esen, A heat balance integral solution of the thermistor problem with a modified electrical conductivity, Appl. Math. Modell. 30(4) (2006), 386–394.10.1016/j.apm.2005.05.002Search in Google Scholar
[39] N. Sadoun, E. K. Si-Ahmed and P. Colinet, On the Goodman heat-balance integral method for Stefan-like problems: Further considerations and refinements, Thermal Sci. 13(2) (2009), 81–96.10.2298/TSCI0902081SSearch in Google Scholar
[40] S. L. Mitchell and T. G. Myers, Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev. 52(1)(2010), 57–86.10.1137/080733036Search in Google Scholar
[41] J. Crank, Free and moving boundary problems, pp. 425, Clarendon Press, Oxford (1984).Search in Google Scholar
[42] P. E. Viollaz and C. Suarez, An equation for diffusion in shrinking or swelling bodies, J. Polymer Sci., Polym. Phys. Ed. 22(5) (1984), 875–879.10.1002/pol.1984.180220509Search in Google Scholar
[43] P. E. Viollaz, C. O. Rovedo and C. Suarez, Numerical treatment of transient diffusion in shrinking or swelling solids, Int. Commun. Heat Mass Transfer. 22(4) (1995), 527–538.10.1016/0735-1933(95)00038-ZSearch in Google Scholar
[44] P. E. Viollaz and C. O. Rovedo, A drying model for three-dimensional shrinking bodies, J. Food Eng. 52(2) (2002), 149–153.10.1016/S0260-8774(01)00097-8Search in Google Scholar
[45] M. J. Davey, K. A. Landman, M. J. McGuinness and H. N. Jin, Mathematical modeling of rice cooking and dissolution in beer production, AIChE J. 48(8) (2002), 1811–1826.10.1002/aic.690480821Search in Google Scholar
[46] M. J. McGuinness, C. P. Please, N. Fowkes, P. McGowan, L. Ryder and D. Forte, Modelling the wetting and cooking of a single cereal grain, IMA J. Manage. Math. 11(1) (2000), 49–70.10.1093/imaman/11.1.49Search in Google Scholar
[47] S. I. Barry and J. Caunce, Exact and numerical solutions to a Stefan problem with two moving boundaries, Appl. Math. Modell. 32(1) (2008), 83–98.10.1016/j.apm.2006.11.004Search in Google Scholar
[48] T. R. Goodman, The heat balance integral and it’s application to problems involving a change of phase, Trans. - ASME J. Heat Transfer. 80 (1958), 335–345.10.1115/1.4012364Search in Google Scholar
[49] T. R. Goodman, Application of integral methods in transient non-linear heat transfer, in: T. F. Irvine Jr. and J. P. Hartnett (Eds.), Advances in heat transfer, vol.1, pp. 51–122, Academic Press, New York (1964).10.1016/S0065-2717(08)70097-2Search in Google Scholar
[50] W. C. Reynolds and T. A. Dolton, The use of integral methods in transient heat transfer analysis. Department of Mechanical Engineering Report No. 36, Stanford University, Stanford, California, Sept. 1 (1958).Search in Google Scholar
[51] D. Langford, The heat balance integral method, Int. J. Heat Mass Transfer. 16(12) (1973), 2424–2428.10.1016/0017-9310(73)90026-4Search in Google Scholar
[52] B. Noble, Heat balance method in melting problems, in: J. R. Ockendon and W. R. Hodgkins (Eds.), Moving boundary problems in heat flow and diffusion, pp. 208–209, Clarendon Press, Oxford (1975).Search in Google Scholar
[53] G. E. Bell, A refinement of the heat balance integral method applied to a melting problem, Int. J. Heat Mass Transfer. 21(11) (1978), 1357–1362.10.1016/0017-9310(78)90198-9Search in Google Scholar
[54] R. S. Gupta and N. C. Banik, Approximate method for the oxygen diffusion problem, Int. J. Heat Mass Transfer. 32(4) (1989), 781–783.10.1016/0017-9310(89)90227-5Search in Google Scholar
[55] J. Crank, The mathematics of diffusion, second ed., pp. 129–135, Clarendon Press, Oxford (1975).Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Solution of a Moving Boundary Problem for Soybean Hydration by Numerical Approximation
- Comparison between 2D and 3D Simulation of Contact of Two Deformable Axisymmetric Bodies
- Weak Solutions for Fractional Differential Equations via Henstock–Kurzweil–Pettis Integrals
- Soliton Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics Using He’s Variational Method
- Application of the Euler and Runge–Kutta Generalized Methods for FDE and Symbolic Packages in the Analysis of Some Fractional Attractors
- Dynamical Analysis of a Fractional-Order Hantavirus Infection Model
- An Avant-Garde Handling of Temporal-Spatial Fractional Physical Models
- Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses
- A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions
- Unique Solution for Multi-point Fractional Integro-Differential Equations
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Solution of a Moving Boundary Problem for Soybean Hydration by Numerical Approximation
- Comparison between 2D and 3D Simulation of Contact of Two Deformable Axisymmetric Bodies
- Weak Solutions for Fractional Differential Equations via Henstock–Kurzweil–Pettis Integrals
- Soliton Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics Using He’s Variational Method
- Application of the Euler and Runge–Kutta Generalized Methods for FDE and Symbolic Packages in the Analysis of Some Fractional Attractors
- Dynamical Analysis of a Fractional-Order Hantavirus Infection Model
- An Avant-Garde Handling of Temporal-Spatial Fractional Physical Models
- Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses
- A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions
- Unique Solution for Multi-point Fractional Integro-Differential Equations