A horizontal water main with a cross-sectional area of 206 cm2 necks down to a pipe of area 42.4 cm2. Meters mounted in the flow on each side of the transition coupling show a change in gauge pressure of 71.0 kPa. Determine the flow rate through the system, taking the fluid to be ideal.
To determine the flow rate through the system, we can use the Bernoulli's equation, which relates the pressure, velocity, and elevation of a fluid in a system.
The Bernoulli's equation is given by:
P1 + 1/2 ρv1^2 + ρgh1 = P2 + 1/2 ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at two different points in the system,
v1 and v2 are the velocities at those two points,
ρ is the density of the fluid,
g is the acceleration due to gravity, and
h1 and h2 are the elevations at those two points.
In this case, we can assume the fluid is ideal, so there is no change in elevation (h1 = h2). Therefore, we can simplify the equation to:
P1 + 1/2 ρv1^2 = P2 + 1/2 ρv2^2
Since the fluid is incompressible, the density (ρ) is constant and can be canceled out:
P1 + 1/2 v1^2 = P2 + 1/2 v2^2
Now we can plug in the known values in the equation.
Let's assume the cross-sectional area of the first section (before the neck-down) is A1, and the cross-sectional area of the second section (after the neck-down) is A2. We are given:
A1 = 206 cm^2
A2 = 42.4 cm^2
P2 - P1 = 71.0 kPa = 71,000 Pa (since 1 kPa = 1000 Pa)
Using the continuity equation, which states that the product of the cross-sectional area and velocity is constant in an incompressible fluid:
A1v1 = A2v2
We can rearrange this equation to solve for v2:
v2 = (A1v1) / A2
We now have two equations: the Bernoulli's equation and the continuity equation.
Let's solve the continuity equation for v1:
v1 = (A2v2) / A1
Now substitute the expressions for v1 and v2 in the Bernoulli's equation:
P1 + 1/2 ((A2v2) / A1)^2 = P2 + 1/2 v2^2
Multiply the equation through by 2A1^2 to clear the fractions:
2A1^2P1 + A2^2v2^2 = 2A1^2P2 + A1^2v2^2
Simplify the equation further:
(A1^2 - A2^2)v2^2 = 2A1^2(P2 - P1)
v2^2 = (2A1^2(P2 - P1)) / (A1^2 - A2^2)
Finally, take the square root of both sides to solve for v2:
v2 = sqrt((2A1^2(P2 - P1)) / (A1^2 - A2^2))
Now we can calculate the flow rate (Q) using the formula:
Q = A2v2
Substitute the value of v2 we just calculated and the known value of A2:
Q = 42.4 cm^2 * sqrt((2*(206 cm^2)^2(P2 - P1)) / ((206 cm^2)^2 - (42.4 cm^2)^2))
Note: Make sure to convert all the units to the same system (e.g., cm to m) to get the final answer in the desired units.