Finite Element Method

Department of Mechanical Engineering, IIU Islamabad

Fluid Flow Through Porous Media

• One Dimensional• Two Dimensional

Group Members

Syed Atif Iqrar (332-FET/BSME/F13) Ali Hasnain (356-FET/BSME/F13)

Contents Introduction Fundamental Concept Derivation of basics differential equations Darcy’s Law Two-dimensional Fluid Flow Fluid Flow in Pipes and Around Solid Bodies One-Dimensional Finite Element Formulation

For Fluid Flow Through Porous Media Problem Finite Element Formulation of a Two-Dimensional

Fluid Flow Problem

Introduction In fluid mechanics, fluid flow through porous

media is the manner in which fluids behave when flowing through a porous medium, for example sponge or wood, or when filtering water using sand or another porous material. As commonly observed, some fluid flows through the media while some mass of the fluid is stored in the pores present in the media.

Derivation of basics differential equations

Fluid Flow through a Porous Medium

Let us first consider the derivation of the basic differential equation for the one-dimensional problem of steady-state fluid flow through a porous medium. The purpose of this derivation is to present a physical insight into the fluid-flow phenomena, which must be understood so that the finite element formulation of the problem can be fully comprehended.

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We begin by considering the control volume shown in Figure 14–1. By conservation of mass, we have

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Darcy’s Law Darcy's law is a constitutive equation that describes the

flow of a fluid through a porous medium Darcy's law, as refined by Morris Musket, at constant

elevation is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance.

 Dividing both sides of the equation by the area and using more general notation leads.

where q is the flux (discharge per unit area, with units of length per time, m/s)

Continue…. The fluid velocity would be

By Darcy’s law, we relate the velocity of fluid flow to the hydraulic gradient (the change in fluid head with respect to x) as

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Boundary conditions are

Two-dimensional Fluid Flow

Boundary Conditions

Fluid Flow in Pipes and Around Solid Bodies

Continue… Boundary Conditions

To clarify sign convention see Figure14-3 & 14-4

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We conclude that the boundary flow velocity is positive if directed into the surface (region), as at the left edge, and is negative if directed away from the surface, as at the right edge.

One-Dimensional Finite Element Formulation

For Fluid Flow Through Porous Media We can proceed directly to the one-dimensional finite

element formulation of the fluid-flow problem by now realizing that the fluid-flow problem is analogous to the heat-conduction problem

We merely substitute the fluid velocity potential function f for the temperature function T, the vector of nodal potentials denoted by {p} for the nodal temperature vector {t}, fluid velocity v for heat flux q, and permeability

coefficient K for flow through a porous medium instead of the conduction coefficient K.

If fluid flow through a pipe or around a solid body is considered, then K is taken as unity. The steps are as follows.

Step 1: Select Element Type

The basic two-node element is again used, as shown in Figure 14–5.

Nodal fluid heads, or potentials, denoted by p1 and p2.

Step 2 : Choose a Potential Function

We choose the potential function f similarly to the way we chose the displacement function in chapter 3.

where p1 and p2 are the nodal potentials to be determined, and

Again the same shape functions used for the temperature element. The matrix [N] is then

Define the Gradient/Potential and Velocity/Gradient Relationships

The hydraulic gradient matrix {g} is given by

Where [B] is derivative of Shape function and it is identical

to temperature shape function derivative [B], that is

And

Step 3 :

Continue… The velocity/gradient relationship based on Darcy’s law

is given by

where the material property matrix is now given by

Step 4 : Derive the Element Stiffness Matrix and Equations Consider the fluid element shown in Figure 14–6 with

length L and uniform cross-sectional area A. Recall that the stiffness matrix is defined in the

structure problem to relate nodal forces to nodal displacements

In the temperature problem to relate nodal rates of heat flow to nodal temperatures.

In the fluid-flow problem, we define the stiffness matrix to relate nodal volumetric fluid-flow rates to nodal potentials or fluid heads as {f}= [k]{p}.

Continue… Therefore

defines the volumetric flow rate f in units of cubic meters or cubic inches per second. Now, using Eqs. (14.2.7) and (14.2.8) in Eq. (14.2.9), we obtain

in scalar form; based on Eqs. (14.2.4) and (14.2.5), g is given in explicit form by

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Applying Eqs. (14.2.10) and (14.2.11) at nodes 1 and 2, we obtain

In Matrix Form,

Continue… In general, the basic element may be subjected to

internal sources or sinks, such as from a pump, or to surface-edge flow rates, such

as from a river or stream. To include these or similar effects, consider the element of Figure

14–6 now to include a uniform internal source Q acting over the whole element and a uniform surface flowrate source q acting over the surface, as shown in Figure 14–7. The force matrix terms are

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where q will have units of m/s or in./s. Equations (14.2.16) and (14.2.17) indicate that one-half of the uniform volumetric flow rate per unit volume Q (a source being positive and a sink being negative) is allocated to each node and one-half the surface flow rate (again a source is positive) is allocated to each node.

Step 5 Assemble the Element Equations to Obtain

the Global Equations and Introduce Boundary Conditions We assemble the total stiffness matrix [K], total force

matrix [F], and total set of equations as

The assemblage procedure is similar to the direct stiffness approach, but it is now based on the requirement that the potentials at a common node between two elements be equal.

The boundary conditions on nodal potentials are given by Eq. (14.1.15).

Step 6 Solve for the Nodal Potentials

We now solve for the global nodal potentials, {p} where the appropriate nodal potential boundary

conditions, Eq. (14.1.15), are specified.

Step 7 Solve for the Element Velocities and Volumetric Flow Rates

Finally, we calculate the element velocities from Eq. (14.2.7) and the volumetric flow rate Qf as:

Problem Statement

The Flow rates and velocities at nodes are:

Finite Element Formulation of a Two-DimensionalFluid Flow

many fluid-flow problems can be modeled as two-dimensional problems, we now develop the equations for an element appropriate for these problems. Examples using this element then follow.

Step 1: Element Type

The three-node triangular element in Figure 14–15 is the basic element for the solution of the two-dimensional fluid-flow problem.

Step 2:The Potential Function

• where pi; pj , and pm are the nodal potentials (for groundwater flow, f is the piezometric fluid head function, and the p’s are the nodal heads), • shape functions are again given by Eq. (13.5.2) as

with similar expressions for Nj and Nm. The a’s, b’s, and g’s are defined by Eq. (6.2.10).

Define the Gradient/Potential and Velocity/Gradient Relationships

Step: 3

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Step 4 : Derive the Element Stiffness Matrix and Equations The element stiffness matrix is given by

Assuming constant-thickness (t) triangular elements and noting that the integrand terms are constant, we have

which can be simplified to

Step 5 Assemble the Element Equations to Obtain

the Global Equations and Introduce Boundary Conditions We assemble the total stiffness matrix [K], total force

matrix [F], and total set of equations as

The assemblage procedure is similar to the direct stiffness approach, but it is now based on the requirement that the potentials at a common node between two elements be equal.

The boundary conditions on nodal potentials are given by Eq. (14.1.15).

Step 6 Solve for the Nodal Potentials

We now solve for the global nodal potentials, {p} where the appropriate nodal potential boundary

conditions, Eq. (14.1.15), are specified.

Step 7 Solve for the Element Velocities and Volumetric Flow Rates

Finally, we calculate the element velocities from Eq. (14.2.7) and the volumetric flow rate Qf as:

Example

Solution

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Any Query?