At one point in a pipeline the water’s speed is 3.00 m/s and the gauge pressure is 5.00∙104 Pa. Find the gauge pressure at a second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.
Note: density of water is 1000 kg/m3.
Solution:
Given data
The speed of the water at point one,
and the Gauge pressure at point one,
We have to find the Gauge pressure at a second point which is 11.0 m (h1-h2) below the first point, where the diameter of the tube is twice the diameter of the first point.
From this we have
First, we calculate the velocity of the water at the second point using the equation of continuity
Where
Now, the gauge pressure at the second point can be calculated by using Bernoulli's equation.
Substitute the known values
The concepts used to solve this problem are Bernoulli’s theorem and equation of continuity.
First, calculate the speed of the water at second point in the line by using the continuity equation. Finally, calculate the gauge pressure at a second point in the line by using the Bernoulli’s equation.
The equation of continuity states that when the fluid is in motion, then the motion of the fluid must be in such a way that mass is conserved. The equation of rate of flow of liquid is given as follows:
Here, is the area of cross section at point 1, is the speed of liquid at point 1, is the area of cross at point 2 and is the speed of liquid at point 2.
The Bernoulli equation is the conservation of energy principle for the flowing fluids. The equation of the Bernoulli principle is given as follows:
Here, is the pressure at point 1 and 2, is the kinetic energy per unit volume at point 1 and 2, are the potential energies at point 1 and 2, is the density, g is the acceleration due to gravity, and h is the height.
Rearrange the above equation for .
The diameter of the pipe at the second point is twice that at the first. So, the radius of the pipe at the second point is also twice that at the first.
Here, is the radius of the pipe at the second point and is the radius of the pipe at the first point.
The cross-sectional area of the pipe at first point is,
The cross-sectional area of the pipe at second point is,
Substitute for .
The equation of continuity is given as follows:
Rearrange the above equation for .
Substitute for and for .
Substitute 3.00 m/s for in the above equation.
The expression for the gauge pressure at a second point in the line is,
Substitute for , for , 3.0 m/s for , 0.75 m/s for and 11.0 m for in the above equation.
Ans:
The gauge pressure at a second point in the line is .
At one point in a pipeline the water’s speed is 3.00 m/s and thegauge pressure...
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