For the one-dimensional fluid flow through the stepped - Mechanical Engineering

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For the one-dimensional fluid flow through the stepped porous medium shown in Figure P14â3, determine the potentials at the junction of each area. Also determine the velocities in each element. Let K_xx = 1 in./s.

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Free Chegg Answer

The following figure represents one-dimensional flow through the porous media:

From the preceding figure the following is the element stiffness matrix for the element-1:

â¦â¦ (1)
Here

is the permeability of the porous medium of element 1,

is length of the element and

is area of the element.
Substitute

for

and

for

in the equation (1):

From the preceding figure the following is the element stiffness matrix for the element-2:

â¦â¦ (2)
Here

is the permeability of the porous medium of element 2,

is length of the element and

is area of the element.
Substitute

for

and

for

in the equation (2):

From the preceding figure the following is the element stiffness matrix for the element-3:

â¦â¦ (3)
Here

is the permeability of the porous medium of element 3,

is length of the element and

is area of the element.
Substitute

for

and

for

in the equation (3):

Calculate the global Stiffness matrix by using the following formula:

â¦â¦ (4)

The following matrix is the global nodal potential:

â¦â¦ (5)

The following matrix is the force matrix:

â¦â¦ (6)

Calculate the elemental force of the element-1 by using the following formula:

â¦â¦ (7)
Here

is potential at node 1,

is potential at node 2,

is the permeability of the porous medium of element 1 and A is area of the element.
Substitute

for

and

for A in the equation (7):

The volume flow rate at the remaining elements 2, 3 and 4 is equal to zero.

The following are the potential boundary conditions:
The potential of element 1 is of the following: