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.

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 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.

scholarly journals Mathematical simulation of temperature influence on pressure drop at pump suction line

2019 ◽  
Vol 213 ◽  
pp. 02069
Author(s):  
Tomáš Polášek ◽  
Adam Bureček ◽  
Lumír Hružík
Keyword(s):  
Pressure Drop ◽  
Physical Properties ◽  
Mathematical Simulation ◽  
Flow Rate ◽  
Experimental Measurement ◽  
Mineral Oil ◽  
Temperature Influence ◽  
Oil Flow ◽  
Suction Line

The article is focused on mathematical simulation of the temperature influence on the pressure drop at the pump suction line. It is evaluated pressure drop depending on mineral oil flow rate. The courses of individual dependencies are determined by mathematical simulation using the ANSYS CFD computing software. The temperature affects significantly physical properties of the flowing mineral oil. In a paper is also described the experimental measurement of a mineral oil sample and determination of its physical properties depending on temperature.

An Approximate Method for Non -Darcy Radial Gas Flow

1964 ◽  
Vol 4 (02) ◽  
pp. 96-114 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Linear Equation ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Gas Flow ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Non Linear Equation ◽  
Approximate Analytical Solutions ◽  
Non Linear ◽  
Flow Law

Abstract Approximate analytical solutions for non-Darcy radial gas flow are derived for bounded and infinite reservoirs producing at either constant rate or constant pressure. These analytical solutions are compared with published results for non-Darcy flow obtained on digital and analogue computers, and the agreement is shown to be very good. Some observations on the interpretation of gas well tests are made. Introduction The flow of gases in porous media is a problem that has been the subject of much study in recent years, and many methods have been proposed for solving the non-linear equations associated with it. The assumption that the flow satisfies Darcy's Law (1) leads to a non-linear equation of the form (2) in a homogeneous medium, assuming an equation of state(3) It has been observed, however, that the linear relationship between the flow rate and pressure gradient is only approximately valid even at low flow rates, and that as the flow rate increases the deviations from linearity also increase. It has been suggested by a number of authors that Darcy's Law should be replaced by a quadratic flow law of the form (4) This form of equation was first suggested by Forchheimer and, later, Katz and Cornell, and Irmay, developed a similar equation. Houpeurt derived this form of equation using the concept of an idealized pore system in which each channel consists of sequences of truncated cones giving rise to successive restrictive orifices along the channel. This type of representation leads to a quadratic flow law of type, for all fluids, but it is found that the quadratic term is only significant in the case of gas flow. The methods of Houpeurt for solving gas flow problems will be discussed further in another section of this paper. Solutions of the non-linear equation for Darcy gas flow may be classified as either computer (digital and analogue), or approximate analytical ones. The former include the well-known solutions of Bruce et al., and Aronofsky and Jenkins, but the latter solutions, apart from the simple linearization of equation [2] to yield a diffusion equation in p2, are not so well-known. SPEJ P. 96ˆ

Comprehensive analysis of the thermohydraulic performance of cooling networks subject to fouling and undergoing retrofit projects

2020 ◽  
pp. 0958305X2094531
Author(s):  
Hebert Lugo-Granados ◽  
Lázaro Canizalez-Dávalos ◽  
Martín Picón-Núñez
Keyword(s):  
Pressure Drop ◽  
Water Flow ◽  
Flow Rate ◽  
Fluid Velocity ◽  
Water Flow Rate ◽  
Volumetric Flow Rate ◽  
Cooling Capacity ◽  
Point Of View ◽  
Rate Distribution

The aim of this paper is to develop guidelines for the placing of new coolers in cooling systems subject to retrofit. The effects of the accumulation of scale on the flow system are considered. A methodology to assess the interconnected effect of local fluid velocity and fouling deposition is developed. The local average fluid velocity depends on the water flow rate distribution across the piping network. The methodology has four main calculation components: a) the determination of the flow rate distribution across the piping network, b) the prediction of fouling deposition, c) determination of the hydraulic changes and the effect on fouling brought about by the placing of new exchangers into an existing structure, and d) the calculation of the total cooling load and pressure drop of the system. The set of disturbances introduced to the system through fouling and the incorporation of new coolers, create network responses that eventually influence the cooling capacity and the pressure drop. In this work, these interactions are analysed using two case studies. The results indicate that, from the thermal point of view, the incorporation of new heat exchangers is recommended in series. The limit is the point where the increase of the total pressure drop causes a reduction in the overall volumetric flow rate. New coolers added in parallel create a reduction of pressure drop and an increase in the overall water flow rate; however, this increase is not enough to counteract the reduction of fluid velocity and heat capacity removal.

Permeability and Inertial Coefficients of Porous Media for Air Bearing Feeding Systems

2007 ◽  
Vol 129 (4) ◽  
pp. 705-711 ◽  
Author(s):  
G. Belforte ◽  
T. Raparelli ◽  
V. Viktorov ◽  
A. Trivella
Keyword(s):  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Flow Rates ◽  
Mass Flow Rates ◽  
Thrust Pad ◽  
Gas Bearing ◽  
Forchheimer’S Law

In porous resistances, Darcy’s law provides a good approximation of mass flow rate when the differences between upstream and downstream pressures are sufficiently small. In this range, the mass flow rates are proportional to the porous resistance’s permeability. For gas bearings, the pressure difference is normally higher, and it is known experimentally that the mass flow rates are lower than would result from Darcy’s law. Forchheimer’s law adds an inertial term to Darcy’s law and, when an appropriate coefficient is selected for this term, provides a good approximation of flow rates for the same applications even with the highest pressure differences. This paper presents an experimental and theoretical investigation of porous resistances used in gas bearing and thrust pad supply systems. The porous resistances considered in the investigation were made by sintering bronze powders with different grain sizes to produce cylindrical inserts that can be installed in bearing supply devices. The paper describes the test setup and experimental results obtained for: (i) mass flow rate through single porous resistances at different upstream and downstream pressures and (ii) mass flow rate and pressure distribution on a pneumatic pad featuring the same porous resistances. The theoretical permeability of the chosen porous resistances was calculated, and the results from setup (i) were then used to obtain experimental permeability and to determine the inertial coefficients. The results, which are expressed as a function of the Reynolds number, confirmed the validity of using Forchheimer’s law. The mass flow rates from setup (ii) were compared to those from setup (i) at the same pressure differentials across the resistance.

scholarly journals Determination of the Upper Limit Up To Which the Linear Flow Law (Darcy's Law) Can Be Applied

2021 ◽  
Author(s):  
Sudad H Al-Obaidi ◽  
Chang WJ ◽  
Falah H Khalaf
Keyword(s):  
Porous Media ◽  
Pressure Gradient ◽  
Darcy’S Law ◽  
Direct Determination ◽  
Darcy's Law ◽  
Linear Flow ◽  
Linear Character ◽  
Upper Limit ◽  
Flow Law

In the practice of hydrodynamic calculations the linear flow law, commonly called Darcy's law, is now widely used. It is well known that it is violated at large pressure gradients. This means that there is a certain limit value of the pressure gradient Δp* above which a deviation from the linear character of the flow law begins. This value of the pressure gradient is the upper limit of applicability.A method is presented for the direct determination of the upper limit of the validity of the linear flow law (Darcy's law) for any porous media. The method is based on the principles of percolation modelling of fluid flows in porous media. The influence of the structure of the pore space on the value of the boundary gradient is analysed. A qualitative comparison with the experimental data is performed.

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.

Feeding System of Aerostatic Bearings With Porous Media

2006 ◽  
Author(s):  
G. Belforte ◽  
T. Raparelli ◽  
V. Viktorov ◽  
A. Trivella
Keyword(s):  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Flow Rates ◽  
Gas Bearings ◽  
Mass Flow Rates ◽  
Set Up ◽  
Forchheimer’S Law

In porous resistances, Darcy’s law provides a good approximation of mass flow rate when the differences between upstream and downstream pressures are sufficiently small. In this range, the mass flow rates are proportional to the porous resistance’s permeability. For gas bearings, the pressure difference is normally higher, and it is known experimentally that the mass flow rates are lower than would result from Darcy’s law. Forchheimer’s law adds an inertial term to Darcy’s law and, when an appropriate coefficient is selected for this term, provides a good approximation of flow rates for the same applications even with the highest pressure differences. This paper presents an experimental and theoretical investigation of porous resistances used in gas bearing supply systems. Cylindrical sintered bronze inserts featuring lengths, diameters and particle sizes commonly used in gas bearings and thrust pads were examined. The paper describes the test set-up and experimental results obtained for: a) Mass flow rate through single porous resistances at different upstream and downstream pressures; and b) Mass flow rate and pressure distribution on a pneumatic pad featuring the same porous resistances. The theoretical permeability of the chosen porous resistances was calculated, and the results from set-up a) were then used to obtain experimental permeability and to determine the inertial coefficients. The results, which are expressed as a function of the Reynolds number, confirmed the validity of using Forchheimer’s law. The mass flow rates from set-up b) were compared with those from set-up a) at the same pressure differentials across the insert.

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