Modeling Transport Phenomena in Porous Media with Applications

The chapter describes the geometric structure of a porous medium and the mechanisms of fluid flow and energy transport through its pores. It defines the notion of a representative elementary volume at the scale of which the mass, momentum, and energy equations are written. The starting point of flow analysis is Darcy’s law that introduces properties such as porosity and permeability as relevant parameters. Exceptions arising from pore sizes approaching the micron size are discussed. A variety of engineering applications where flow and transport in porous media is encountered is discussed. These include fuel cells, regenerators, gas reservoirs, and coils used in the biomedical context. Terminology commonly found in the literature on mathematical modeling of flow in porous media is summarized. The chapter closes with an exhaustive list of references.

Derivation of equations governing flow, heat, and mass transfer in porous media is discussed. The starting point is Darcy’s law that can be gradually extended to higher flow rates and porosity, leading to the non-Darcy form of the momentum equation. Complex pore geometries and the appearance of interfaces of immiscible fluids can be treated in this framework. A second starting point is the system of equations valid for a homogeneous fluid medium. It can be generalized to a multiphase system such as a porous medium by introducing source terms and effective medium properties. In each approach, the model carries a large number of parameters that are sensitive to the pore structure, though to a lesser extent on the thermophysical properties of the constituent media. Success in modeling transport in porous media is linked to careful parameter estimation from experiments. This step is expected to become critical in multiscale porous media where the pore scales span several orders of magnitude. The one-equation model and two-equation model of convective heat transfer and transport phenomena with chemical reactions are subsequently discussed in the chapter.

Transport in porous media can be analyzed at various scales. Mesoscale formulations, such as lattice Boltzmann method (LBM), play an important role in deciphering the pore-scale flow, heat and mass transfer. The present chapter uses LBM to quantify the mass and momentum transport within the gas diffusion layer (GDL) of polymer electrolyte membrane fuel cells. The chapter presents stochastic reconstruction of the GDL microstructure, followed by LBM simulation of two-phase flow and electrochemistry through the GDL pores. The chapter paves the way for connecting the mesoscopic information with the macroscopic physics of fuel cells.

In recent years, lithium ion batteries are identified as a promising way for storing electrical energy. Electrochemical phenomena as well as thermal management of lithium ion batteries are greatly influenced by heat and mass transport in the porous electrodes. Present chapter uses direct numerical simulation (DNS) to quantify the species transport through the composite electrodes of lithium ion batteries. The simulations also capture the irreversibilities within the batteries leading to heat generation and consequent heat transfer. The chapter emphasizes the importance of DNS in understanding the transport phenomena and electrochemistry within a lithium ion battery.

Transport through natural and artificial porous media plays an important role in biological systems. Examples include flow in plants through xylem and phloem as well as transport in extracellular space in the central nervous system of animals. Present chapter deals with the dynamics of blood flow through porous media. Focus of this chapter includes the hemodynamic modeling and simulations through coil-embolized blood vessels. While the mathematical model incorporates realistic blood rheology and the pulsatile nature of blood flows, the simulations are conducted over a patient-specific, three-dimensional domain. The study shows that the coil embolization reduces the wall shear considerably, leaving the wall pressure largely unaffected.

A stack of meshes assembled to form a regenerator of a Stirling cycle has been analyzed for determining flow distribution and heat transfer. A non-Darcy, thermal nonequilibrium model is driven by pulsatile flow with hot and cold fluid alternately going past the mesh in opposite directions. The characteristic frequency of pulsation is determined with reference to the time constant of the gas-wire system. The flow is shown to reach dynamic steady state rapidly, when compared to the thermal field. With this result, the flow field is computed by a harmonic analysis technique. The energy equations for the fluid and the solid phases are numerically integrated in time. Calculations continue for a total of 10⁴ cycles. Dense meshes reported in the literature are evaluated against coarse meshes. Results have been presented for two Reynolds numbers, 100 and 10,000, the latter being closer to a practically attainable value in applications. Velocity profiles in dense meshes are seen to be flatter in comparison with coarse meshes. In addition, thermal nonequilibrium effects are quite significant in coarse meshes, pointing to a need for preferring dense meshes in cryocooler applications.

Flow through geological porous media plays important role in transporting natural resources such as water, oil, gas, and methane hydrate. In recent years, geological porous media has also identified as a promising media for CO₂ sequestration. Present chapter deals with the recovery of methane from marine hydrate reservoir and the disposal of anthropogenic CO₂ in the same media. Focus of this chapter includes the modeling and simulations of methane recovery via depressurization and CO₂ injection. The chapter primarily incorporates single and multiphase simulations over a one-dimensional marine hydrate reservoir. The study outlines the advantages and disadvantages of simultaneous depressurization-injection technique for methane recovery from hydrate reservoir.

The book is an attempt toward connecting time-honored as well as contemporary theories of porous media with rapidly growing engineering applications. In recent decades, applications of transport in porous media have grown to such an extent that no book or monograph can claim exhaustive coverage of all relevant topics. In this context, the present work may be treated as a starting point for a conversation between the classical and contemporary theories, followed by a contextualization of theories useful for engineering applications. The relevance of the present book, therefore, lies in exploring the rigor of theory of transport in porous media and its utility for real-life analysis.