Homogenization based heating control for moist paperboard with evaporation on the pore surface

A control problem for the heating of moist paperboard with the evaporation of moisture from the inner pore surface is considered. The microstructure of the paperboard is known from the CT images. It is easily parametrized w.r.t. the volume fraction of the pores, the surface area of the pores related to the unit volume, averaged area of the contact surfaces between two fibers and the unsupported fiber length between two contact nodes, fiber thickness, and tortuosity. Simple averaging formulas are provided for the computation of the effective coefficients in the coupled diffusion-heat problem and the dissipation energetic balance was analyzed and we pass to the limit w.r.t. the small parameter, the relation between the pore size and the paper-board thickness. The limit model is similar to those models, recently available in the literature. However, it provides more understanding and different limiting models and the thresholds for them. Also, the limitation on the heating is derived from the solvability criteria, which means, that the evaporation should not overtake the heating. The model is validated by experimental and numerical results found in the literature.