Numerical Simulation of Slow Drying in Porous Media Using Pore Network Model

In our first model, an internal and external coupled solver is presented to simulate the slow drying of a porous medium placed adjacent to a laminar flow of air in a slit. The porous medium is represented by a 20×20 pore-network model: the invasion-percolation algorithm is employed to simulate moisture redistribution; water-vapor migration in empty network is estimated using the purely diffusive approach. The external flow-field, unchanged during the drying simulation, is computed in the beginning using the Navier-Stokes equations. Subsequent water-vapor transport is modeled using a convection-diffusion type transport equation. A unique pore-to-cell meshing method and a novel unified (implicit) computational framework coupling the outer and the inner processes are proposed. Multi-scale problems in both space and time appear when solving the internal and external field simultaneously. To accurately simulate this kind of problem by aiming to minimize computation effort, the following aspects of the simulation are studied: space discretization schemes, numerical algorithms, mesh microstructure, and time-step refinement. The space discretization schemes tested in this paper include the Hybrid and the Hayase QUICK schemes. The numerical algorithms tested to solve the drying process include the operator-splitting and a non-operator-splitting algorithm. Different mesh densities are tested along the directions parallel to and normal to the outer flow- porous-medium interface. Different time-steps are tested to find a suitable time-step for both the internal and the external computations. The external air velocity has some impact on the drying in the initial stages. Significantly, the microstructure of the pore-network is found to have a strong influence on drying. In our second pore network model (which is based on our first model), the film effect is included and a novel logistic equation is used to relate the pore network variables with the external field variables. For migration of water vapors, the model accounts for both advective and diffusive transport in the external flow field while including diffusion in the dry part of the pore network. By conducting a parametric study on the drying of a 40×40 square network placed next to a slit flow, it is discovered that (a) higher hydraulic diameter of the throats leads to higher drying rates and longer constant drying-rate periods; (b) the drying time increases and the drying rate decreases as the throat cross section changes from a triangle to a square to a hexagon to a circle, which can be correlated to the weakening of the film effect; (c) increasing the external flow velocity (that leads to changing the Peclet number from 1 to 1000) has little effect on the drying rates and times; (d) increasing the external air humidity from 30% to 70 % leads to a large decrease in the drying rates and the consequent increase in the overall drying times. The developed model is then used to simulate the drying of several thin porous media (40×40, 80×20, and 160×10) with different aspect ratios placed either aligned-with or perpendicular-to a uniform 2-D flow. Plots of drying rates and drying times against the network saturation are studied. The presence of film during most of the drying period ensures that the surface pores are at saturated vapor pressure. As a result, the sharpest concentration gradients, which also control the drying rates, lie adjacent to the exposed surfaces. Consequently, the concentration gradients in the outer flow fields are very mild and play insignificant role in the drying of porous media. Hence, we reach a surprising conclusion—the orientation of thin porous media in the outer flow field is found to be irrelevant for drying. But expectedly, the higher exposed-area versus total volume ratio leads to faster drying. However, these conclusions should be examined further by future 3D simulations since a 2D simulation may underestimate the influence of external flow field on the drying of porous media. Finally, the model is applied to the dual-porosity porous media. The drying simulation of a square-shaped and dual-porosity pore network is compared with a previously published experimental study. Two cases of small-pores side open and large-pore side open are considered. It is observed that though the simulation results of the 12×12 network fail to match the experimental drying curves completely, important features of the drying process (such as complete emptying of large pores before the onset of drying in the small pore region of the large-pores side open case) are achieved. Next the drying of the same square-shaped, dual-porosity domain using a much refined 100×100 network is carried out in a uniform air flow after keeping either the large-pores or the small-pores side open. The former leads to faster drying and complete emptying of the large pores before the small pores. The latter witnesses the phenomenon of capillary pumping. Using the same refined network, the case of all side open is also studied. Changing the throat cross-section from circle to square leads to much faster drying. Introduction of microstructural irregularity in the network by randomly changing throat diameter and changing the coordination number of pores does not affect the drying rate and drying time significantly

Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization

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