Download - Ch 14-fluid flow through porous media
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.
Continue….
We begin by considering the control volume shown in Figure 14–1. By conservation of mass, we have
Continue….
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
Continue…
Continue…
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
Continue…
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
Continue…
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
Continue…
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
Continue…
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
Continue…
Any Query?