Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: Coupled fluid transport and stress equations Pawan S. Takhar ⇑,1 Department of Animal and Food Sciences, International Center for Food Industry Excellence, Texas Tech University, Box 42141, Lubbock, TX 79409, USA article info Article history: Received 6 December 2010 Received in revised form 29 March 2011 Accepted 30 March 2011 Available online 12 April 2011 Keywords: Multiscales Drying Stress-cracking Moisture transport Hybrid mixture theory abstract In first part, a solution scheme for the multiscale fluid transport equation of Singh et al. (2003a) was developed, which could be easily implemented in a commercial finite elements package for solving swell- ing/shrinking problems. The solution scheme was applied to drying of corn kernels. Laplace transforma- tion was used to convert the integral part of the fluid transport equation to a set of differential equations. In materials undergoing volume change during drying, a moving mesh is needed in Eulerian coordinates, which is tedious to implement. An alternate method was used that involved transforming the equation to stationary Lagrangian coordinates. Once the equations were solved, the data was transformed back to the moving Eulerian coordinates during post-processing. Corn heterogeneity was taken into account by using different coefficient of diffusivity values for various corn components. To predict stresses, it was assumed that the shape of a kernel remains geometrically similar to its initial shape during drying. Using this assumption a relation between the strain tensor and volume fraction of water was developed. The visco- elastic stress–strain constitutive equation was used to calculate the magnitude of stresses at different spatial locations inside a corn kernel. In the companion paper (part 2), the experimental validation of the solution, simulation results for various drying conditions and the practical implications are discussed. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Food grains are routinely subjected to fluid transport processes such as drying and sorption during various stages of processing and storage. If the process conditions are not optimized, the grains may develop cracks due to complex interaction of several factors— swelling or shrinkage of the matrix; stresses exerted by the flowing moisture over the solid matrix over a hierarchy of spatial scales; hydration stresses at the solid–fluid interphase; spatial variation in material properties due to glass transition effects; random pro- cesses such as thermofluctuations; stress concentration around initially present flaws; and heterogeneity in the porous matrix. Cracked grains are generally considered of inferior quality and are more prone to insect and microbial damage. Cracked grains also result in more dust in processing plants, which is an explosion hazard. Although precise monetary impact is difficult to quantify, the phenomenon is expected to cause millions of dollars worth of damage to the food grains worldwide. Polymer science research has shown that the fluid flow in bio- polymers exhibits non-Fickian characteristics in the vicinity of glass-transition (Kim et al., 1996; Thomas and Windle, 1982). The flow becomes Fickian in rubbery and glassy states sufficiently from the glass transition regime. Since, corn kernels also exhibit glass- transition in the range of drying temperatures (Hundal and Takhar, 2009; Yang et al., 2002), it is important to use a fluid transport equation that exhibits non-Fickian behavior near glass-transition and Fickian behavior in glassy and rubbery states. Singh et al. (2003b) used hybrid mixture theory to develop a general transport theory for biopolymers and in subsequent work (Singh et al., 2003a) utilized it to develop a multiscale fluid transport equation. This equation is able to predict the fluid transport behavior discussed above in the glassy, rubbery and transition states of foods (Singh et al., 2003a; Takhar, 2008). In the previous modeling studies with corn kernels, non-Fickian behavior was not accounted (Fortes and Okos, 1981; Litchfield and Okos, 1988; Parti and Dugmanics, 1990; Irudayaraj and Haghighi, 1993; Jia et al., 2000). In addition, different corn components (germ, hard endosperm, soft endosperm and pericarp) were as- sumed to have the same form for the coefficient of diffusivity equa- tion due to lack of information on properties. In the current study, we included the heterogeneous diffusion in corn by utilizing the diffusivity equations of Chen et al. (2009). 0260-8774/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2011.03.033 ⇑ Tel.: +1 806 742 2805; fax: +1 806 742 4003. E-mail address: pawan.takhar@ttu.edu 1 Author has previously published as Pawan P. Singh. Journal of Food Engineering 105 (2011) 663–670 Contents lists available at ScienceDirect Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng