Fluid Flow In Porous Media 5q3724

FLUID FLOW IN POROUS MEDIA Christian Paolo S. Asequia November 12, 2015

Flow of Discussion  Definitions  Flow

Regime  Darcy’s Law and Kozeny-Carman Equations  Carman and Ergun Correlations  Concentrations by mass and volume  Summary

A porous mediumor is a material Definitions Voidage Bed containing pores (voids) typically Interstitial Velocity, U, is porosity is the void filled with a fluid. the velocity obtained by Pressure drop is defined as the fraction expressed as  Porous Medium Fluid flow through a Superficial velocity, U , difference in pressure between flow through the column o  Fluid Flow through a porous medium volume of solids in bed two points in a fluid carrying medium describes the the between particles. It is Superficial Velocity is the velocity in which divided by the total bed network. The resistance of fluid  Interstitial manner in which the Velocity related to the Superficial the medium is assumed flow gives rise to the PRESSURE volume.  Porosity/Voidage fluid that flows in a Velocity through the void DROP (ΔP) in the fluid. empty.  Pressure Drop / Pressure Difference fraction. media behaves.

Definitions

Flow Regimes  Reynolds

 Modified

Number

Reynolds Number

Flow Regimes  Modified

Reynolds Number ▪ Used to determine the flow regime of the fluid within the porous medium. ▪ Interstitial velocity is considered. Hence velocity is U which can be related to superficial through the void fraction. ▪ Characteristic linear dimension was deduced by Kozeny and is the volume open to the fluid flow divided by the surface area over which it must flow.

Flow Regimes  Modified

Reynolds Number ▪ Conventionally applied threshold to indicate significant turbulence is 2. ▪ Density term is the fluid density ▪ Sv = 6/x (for a single spherical particle)  specific surface area per unit volume

Darcy’s Law and the Kozeny-Carman Equation  Darcy’s

Law Darcy’s Law states that the flow is ▪ The basic law governing the flow proportional to the pressure drop and of fluids through porous media. inversely proportional to viscosity. ▪ Formulated by Henry Darcy on the basis of his experiments on vertical water filtration through sand beds.

Darcy’s Law and the Kozeny-Carman Equation  Kozeny-Carman

Equation:

In of particle size:

▪ Equation is derived from the hagenpoiseuille equation for a laminar flow of a fluid in

xSv is the Sauter mean awhere circular channel. ▪ Only applies laminar flows for porous medium diameter for to the particle distribution. (<2) ▪

where K, is Kozeny constant, has a value close to 5 however, it is not universally true.

Friction Factor   Friction

Factor, f

▪ Defined as the drag force per wetted surface unit area divided by the product of density and velocity head.

▪ Analogy of friction factor to cover flow regions in porous media was deduced by Carman.

Friction Factor

Friction Factor  Force

Balance

at

the

particle

surface: Let R = shear stress or drag force per unit area DragForce = R*particle surface area surface area of particle = Sv *(1-ε)*L*A Pressure Drop on fluid = ΔP Force by fluid = ΔP*A*ε Combining:

Friction Factor  Rearranging

 Expanding

 The

the previous equation;

into friction factor

pressure drop per unit length is:

Carman and Ergun Correlations  Carman

Correlation is generally used for solid objects forming a bed:

 Ergun

Correlation is used for hollow objects such as packing rings:

Concentrations by mass and volume  If

a sample resulted in a concentration by mass, Cw and the total mass is M, then the volumes can be calculated if the densities of the solid and fluid are known through the equation:

Concentrations by mass and volume  Dividing

the previous equation by the volume of solids gives rise to the equation:

SUMMARY  Under

laminar flow (NRe1 < 2)

Where K is the Kozeny constant.

 Modified

Reynolds Number:

SUMMARY  For

turbulent flow, (NRe1 > 2)

 Carman Correlation ( Solids forming a

bed)

 Ergun Correlation (Hollow Solids)

Example Problem