The results of a comprehensive computational evaluation of the Ergun and Forchheimer equations for the permeability of fibrous porous media are reported in this study. Square and hexagonal arrays of uniform fibers have been considered, as well as arrays in which the fiber size is allowed to change in a regular manner, expressed by a size variation parameter (δ). The range of porosity (φ) examined is from 0.30 to 0.60, the Reynolds number ranges between 0 and 160, and the size variation parameter (δ) between 0 (corresponding to the uniform array) and 0.90 (in which case the diameter of the large fibers in the array is 19 times that of the small ones). We obtain computational results for pressure drop and flow rate in a total of 440 cases mapping the (φ,δ,Re) space; these are presented in terms of a friction factor and are compared to the predictions of the Ergun and Forchheimer equations, both widely used models for the permeability of porous media. In the limit of creeping flow (Re<1), the Forchheimer equation is in excellent agreement with the computational results, while the Ergun equation is unable to capture the behavior of fiber arrays in which the flow has a strong contracting/expanding element. The Forchheimer equation, in its original form, is in closer agreement with the computational results. When the Forchheimer term (F) is expressed as a function of porosity, we obtain a modified form of the Forchheimer equation that is in excellent agreement with computational results for the entire range of (φ,δ,Re) examined.
The dynamics of capillary climb of a wetting liquid into a porous medium that is opposed by gravity force is studied numerically. We use the capillary network model, in which an actual porous medium is represented as a network of pores and throats, each following a predefined size distribution function. The liquid potential in the pores along the liquid interface within the network is calculated as a result of capillary and gravity forces. The solution is general, and accounts for changes in the climbing height and climbing velocity. The numerical results for the capillary climb reveal that there are at least two distinct flow mechanisms. Initially, the flow is characterized by high climbing velocity, in which the capillary force is higher than the gravity force, and the flow is the viscous force dominated. For this single-phase flow, the Washburn equation can be used to predict the changes of climbing height over time. Later, for longer times and larger climbing height, the capillary and gravity forces become comparable, and one observes a slower increase in the climbing height as a function of time. Due to the two forces being comparable, the gas-liquid sharp interface transforms into flow front, where the multiphase flow develops. The numerical results from this study, expressed as the climbing height as a power law function of time, indicate that the two powers, which correspond to the two distinct mechanisms, differ significantly. The comparison of the powers with experimental data indicates good agreement. Furthermore, the power value from the Washburn solution is also analyzed, where it should be equal to 1/2 for purely viscous force driven flow. This is in contrast to the power value of ∼0.43 that is found experimentally. We show from the numerical solution that this discrepancy is due to the momentum dissipation on the liquid interface.
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