On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders

Abstract

Effects of the bounding solid walls are examined numerically for slow flow overregular, square arrays of circular cylinders between two parallel plates. A local magnitudeof the rate of entropy generation is used effectively to determine the flow region affected bythe presence of the solid boundary. Computed axial pressure gradients are compared to thecorresponding solution based on the Darcy-Brinkman equation for porous media in whichthe effective viscosity appears as an additional property to be determined from the flowcharacteristics. Results indicate that, between two limits of the Darcian porous medium andthe viscous flow, the magnitude of μ (the ratio of the effective viscosity to the fluid ˆviscosity) needs to be close to unity in order to satisfy the non-slip boundary conditions atthe bounding walls. Although the study deals with a specific geometric pattern of the porousstructure, it suggests a restriction on the validity of the Darcy-Brinkman equation to modelhigh porosity porous media. The non-slip condition at the bounding solid walls may beaccounted for by introducing a thin porous layer with μ = 1 near the solid walls. ËÂ

Thermal dispersion effects on forced convection in a porous-saturated pipe

“…With non-zero C values, interestingly, the solution shows a linear increase of Nu with C. This trend is in-line with the observations made in [28][29][30]. In fact, recasting Eq.…”

Gas–liquid dynamics at low Reynolds numbers in pillared rectangular micro channels

“…The empirically determined model to calculate the single-phase pressure drop in the gas phase used by Krishnamurthy and Peles (2007) resulted in an error of 40% for pitches of 27 and 17 lm and an error of 85% for pitches of 7 lm. This is due to neglecting the expansion of the gas at different pressures over the length of the channel by the correlation used by Krishnamurthy and Peles (2007) and by the dependency of (Ergun 1952), DarcyBrinkman equation (Liu et al 2007) and the equation used by Krishnamurthy and Peles (2007) with the experimental data for the single-phase pressure drop in the pillared micro channel with pitches of 27, 17, 12, and 7 lm as a function of the superficial liquid velocity It can be concluded that the model used by Krishnamurthy and Peles (2007) can not be applied to calculate the pressure drop in a pillared micro channel with pillar diameters smaller than 100 lm.…”

“…The equations of motion reduce to the creeping flow case and the pressure drop can be described by the Darcy-Brinkman equation (Liu et al 2007): …”

An Overview on Extension and Limitations of Macroscopic Darcy’s Law for a Single and Multi-Phase Fluid Flow through a Porous Medium

“…15is valid for a fluid flow in the horizontal direction; and eqn. (16) provides the expression for a tilted flow for the same case.…”

“…Further, in describing fluid flow through both homogeneous and heterogeneous reservoirs, the geological formation of interest is assumed to be characterized by single-phase single-layered primary porosity, while in reality, it will also be characterized by a multi-phase multi-layered multiple-porosity systems. On top of it, for most of the applications, it is required to have the details at the microscopic scale, however, conceptually and mathematically simple macroscopic Darcy's law is widely used [Bradford and Leij, 1997;Diaz et al, 1987;Gray and Neill, 1976;Gray and Hassanizadeh, 1989;Hassanizadeh and Gray, 1990;Jerald and Salter, 1990;Kalaydjin, 1990;Liu et al, 2007;Muccino et al, 1998;Siddiqui et al, 2015;Teng and Zhao, 2000;Whitakar, 1986].Thus, there is a need to understand the fundamental concepts of the subsurface fluid flow at a scale lesser than Darcy's scale.For example, both groundwater and crude oil being the subsurface fluid flow, a litre of spilled crude oil in the subsurface environment can contaminate as much as one hundred thousand litres of groundwater; in addition, remediation of either onshore oil spill (contamination of groundwater by spilled crude oil) is not so easy as it will take several decades to address the same; and such field investigations require an understanding at the micro-or pore-scale processes and Darcy's law can only help us to address immediate concerns at a larger field scale; and any incorrect understanding of subsurface fluid flow processes will only land up with a marginal improvement in (enhanced) oil recovery factor. Since there is no mathematical model that can describe fluid flow through a porous medium at the microscopic-scale as on date, and since macroscopic Darcy's law is widely used till date by the oil and gas industries, it is necessary to understand the actual limitations associated with Darcy's law, when it is extended for various practical applications.…”