Water flows down a long, straight, inclined pipe of diameter D and length L (Fig. 9-123). There is no forced pressure gradient between points 1 and 2; in other words, the water flows through the pipe by gravity alone, and
FIGURE P9-123
(a)
The expression for average velocity as the function of
Answer to Problem 123P
The expression for average velocity is
Explanation of Solution
Given Information:
The pressure at the points 1 and 2 is same and equal to the atmospheric pressure and the velocity of the flow is
Write the energy equation for the pipe.
Here, the kinetic energy correction factor at point 1 is
Calculation:
Substitute
Write the expression to calculate the frictional loss.
Here, the length of the pipe is
Write the expression for friction factor in laminar flow.
Substitute
Substitute
Conclusion:
Thus, the expression for average velocity is
(b)
The expression for average velocity as the function of
Answer to Problem 123P
The expression for average velocity is
Explanation of Solution
Given Information:
The pressure at the points 1 and 2 is same and equal to the atmospheric pressure.
Write the force balance equation in the direction of the flow for the volume element shown in Figure-(1).
Here, the fluid weight in the direction of the flow is
Write the equation for horizontal component of weight of the fluid.
Here, the angle of inclination is
Substitute
Write the expression to calculate the weight of the fluid for the volume element.
Here, the density of the fluid is
Calculation:
The figure below represents the free body diagram of the pipe and forces acting on the pipe.
Figure-(1)
Write the expression to calculate the volume of the element.
Substitute
Substitute
Divide the equation (V) by
Substitute
Since the pressure at point 1 and point 2 is same, therefore substitute zero for
Integrate the equation (X).
Write the expression for average velocity.
Substitute
Substitute
Write the expression to calculate the value of
Here, the length of the pipe is
Substitute
Conclusion:
The expression for average velocity of inclined pipe is
(c)
The dimensionless expression for velocity.
Answer to Problem 123P
The dimensionless equation for the velocity is
Explanation of Solution
Write the expression for the velocity in the pipe.
Calculation:
Rearrange the equation (XIV) to obtain the dimensionless expression for velocity.
Here, the first dimensionless parameter is
Conclusion:
The dimensionless equation for the velocity is
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Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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