scholarly journals The Use of Nonlinear Constitutive Equations to Evaluate Draw Resistance and Filter Ventilation

2001 ◽  
Vol 19 (4) ◽  
pp. 177-188
Author(s):  
B Eitzinger ◽  
G Ederer
Keyword(s):  
Pressure Drop ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Constitutive Equations ◽  
Darcy’S Law ◽  
Stationary Flow ◽  
Darcy's Law ◽  
Nonlinear Behaviour ◽  
Cigarette Paper ◽  
Drop Flow

AbstractThis study investigates by nonlinear constitutive equations the influence of tipping paper, cigarette paper, filter, and tobacco rod on the degree of filter ventilation and draw resistance. Starting from the laws of conservation, the path to the theory of fluid dynamics in porous media and Darcy's law is reviewed and, as an extension to Darcy's law, two different nonlinear pressure drop-flow relations are proposed. It is proven that these relations are valid constitutive equations and the partial differential equations for the stationary flow in an unlit cigarette covering anisotropic, inhomogeneous and nonlinear behaviour are derived. From these equations a system of ordinary differential equations for the one-dimensional flow in the cigarette is derived by averaging pressure and velocity over the cross section of the cigarette. By further integration, the concept of an electrical analog is reached and discussed in the light of nonlinear pressure drop-flow relations. By numerical calculations based on the system of ordinary differential equations, it is shown that the influence of nonlinearities cannot be neglected because variations in the degree of filter ventilation can reach up to 20% of its nominal value.

Numerical modelling of nonlinear effects in laminar flow through a porous medium

1988 ◽  
Vol 190 ◽  
pp. 393-407 ◽  
Author(s):  
O. Coulaud ◽  
P. Morel ◽  
J. P. Caltagirone
Keyword(s):  
Porous Medium ◽  
Pressure Drop ◽  
Nonlinear Term ◽  
Numerical Solutions ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Quadratic Term ◽  
Inertial Effects ◽  
Operator Splitting Methods ◽  
Flow Through

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

Pressure-driven flow across a hyperelastic porous membrane

2019 ◽  
Vol 871 ◽  
pp. 742-754 ◽  
Author(s):  
Ryungeun Song ◽  
Howard A. Stone ◽  
Kaare H. Jensen ◽  
Jinkee Lee
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Darcy’S Law ◽  
Applied Pressure ◽  
Soft Materials ◽  
Porous Membrane ◽  
Darcy's Law ◽  
Membrane Thickness ◽  
Pressure Driven Flow ◽  
Pressure Driven

We report an experimental investigation of pressure-driven flow of a viscous liquid across thin polydimethylsiloxane (PDMS) membranes. Our experiments revealed a nonlinear relation between the flow rate $Q$ and the applied pressure drop $\unicode[STIX]{x0394}p$, in apparent disagreement with Darcy’s law, which dictates a linear relationship between flow rate, or average velocity, and pressure drop. These observations suggest that the effective permeability of the membrane decreases with pressure due to deformation of the nanochannels in the PDMS polymeric network. We propose a model that incorporates the effects of pressure-induced deformation of the hyperelastic porous membrane at three distinct scales: the membrane surface area, which increases with pressure, the membrane thickness, which decreases with pressure, and the structure of the porous material, which is deformed at the nanoscale. With this model, we are able to rationalize the deviation between Darcy’s law and the data. Our result represents a novel case in which macroscopic deformations can impact the microstructure and transport properties of soft materials.

scholarly journals Generalized Darcy’s Law in Filtration Theory

2018 ◽  
Vol 173 ◽  
pp. 02017 ◽  
Author(s):  
Yuri P. Rybakov ◽  
Natalya V. Semenova
Keyword(s):  
Darcy’S Law ◽  
Grain Filling ◽  
Stationary Flow ◽  
Darcy's Law ◽  
Axially Symmetric ◽  
Filtration Theory ◽  
Filtration Process ◽  
Filtration Equation ◽  
Transverse Diffusion ◽  
Lattice Approximation

We study the hydrodynamics of flow in a porous medium modeling the grain filling in filters. Using the lattice approximation, we derive the structure of the current in porous media and obtain the transverse diffusion coefficient D which proves to be proportional to the diameter d of the grains as constituents of the medium. We consider the axially-symmetric stationary flow in a cylindrical filter and show that the vertical velocity takes its maximal value at the wall, this effect being known as the “near-wall” one. We analyze the solution to the Euler equation with the modified Darcy force, which depends not only on the velocity but also on the gradient of the pressure included in the Darcy coefficient. Finally, within the scope of the perturbation method, we derive the main filtration equation and discuss the influence of modifying the Darcy’s law on the efficiency of the filtration process.

Determination of the critical pressure drop for filtration of super-compactible cakes

2001 ◽  
Vol 44 (10) ◽  
pp. 171-176 ◽  
Author(s):  
F.M. Tiller ◽  
W.P. Li ◽  
J.B. Lee
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Critical Pressure ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Pressure Filtration ◽  
Porous Bed ◽  
Average Percentage

In accord with Darcy's law, the flow rate through a porous bed depends upon the pressure drop Δpc. In general, increasing Dpc leads to increased values of flow rate and average percentage solids in filtration operations. When cakes become super-compactible, their behavior undergoes an unexpected change in which both the flow rate and the percentage solids reach maximum values and thereafter are unaffected by increasing Δpc. The critical pressure drop ΔpcR is defined as that value at which the flow rate reaches 90% of its ultimate value. When Δpc is greater than DpcR and is doubled or tripled, the cake resistance approximately doubles or triples leaving the rate virtually unchanged. The super-compactibility problem is analyzed theoretically, and is verified by stepped pressure filtration experiments on different materials from Houston and Korea.

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