Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 3, Problem 26P

Water from a reservoir is raised in a vertical tube of internal diameter D =30 cm under the influence of the pulling force F of a piston. Determine the force needed to raise the water to a height of h = 1.5 m above the free surface. What would your response be for h = 3 m? Also, taking the atmospheric pressure to be 96 kPa, plot the absolute water pressure at the piston face as h varies from 0 to 3 m.

Chapter 3, Problem 26P, Water from a reservoir is raised in a vertical tube of internal diameter D =30 cm under the
FIGURE P3-26

Expert Solution & Answer
Check Mark
To determine

The force needed to raise the water to a height of h=1.5m above the free surface.

The force needed to raise the water to a height of h=3m above the free surface.

The force needed to raise the water to a height of h=0m above the free surface.

Answer to Problem 26P

The force needed to raise the water to a height of h=1.5m above the free surface is 1.04kN.

The force needed to raise the water to a height of h=3m above the free surface is 2.08kN.

The force needed to raise the water to a height of h=0m above the free surface is 0kN.

Explanation of Solution

Given information:

The internal diameter of the piston is 30cm, and atmospheric pressure is 96kPa.

Write the expression for the pressure exerted by a fluid

  P=ρgh   ...... (I)

Here, the pressure exerted by a fluid is P, and acceleration due to gravity is g, and density of the fluid is ρ, height of the column is h.

Write the expression for area.

  A=π4D2

Here, cross-section area of the piston is A, internal diameter of the piston is D.

  W=AP   ...... (II)

Here, weight exerted by the fluid column is W.

Substitute the π4D2 for A in Equation (II)

  W=(π4D2)P   ...... (III)

Write the expression for net vertical force.

  FW=0F=W   ...... (IV)

Here, force required to raise the water is F

Write the expression for absolute pressure acting on the piston face.

  Pabs=PatmP   ...... (V)

Here, absolute pressure is Pabs, atmospheric pressure is Patm.

Calculation:

Substitute 1000kg/m3 for ρ, 9.81m/s2 for g and 1.5m for h in Equation (I).

  P=(1000kg/ m 3)(9.81m/ s 2)(1.5m)=14.72×1000(kgm/ s 2)(1/ m 2)=14.72×1000Pa( 1kPa 1000Pa)

Substitute 30cm for D, 14.72kPa for P in Equation (III).

  W=π4(30cm)2(14.72kPa)=π4(( 30cm)( 0.01m 1cm ))2(14.72kPa)( 1kN/ m 2 1kPa)=1.04kN

Substitute 1.04N for W in Equation (IV).

  (F)h=1.5=1.04kN

Here, the force required to raise the water is at height of 1.5m is (F)h=1.5.

Substitute 1000kg/m3 for ρ, 9.81m/s2 for g, and 3m for h in Equation (I).

  P=(1000kg/ m 3)(9.81m/ s 2)(3m)=29.43×1000(kgm/ s 2)(1/ m 2)=29.43×1000Pa( 1kPa 1000Pa)=29.43kPa

Substitute 30cm for D, 29.43kPa for P in Equation (III).

  W=π4(30cm)2(29.43kPa)=π4(( 30cm)( 0.01m 1cm ))2(29.43kPa)( 1kN/ m 2 1kPa)=2.08kN

Substitute 2.08kN for W in Equation (IV).

  F=2.08kN

Here, the force required to raise the water is at height of 3m is (F)h=3.

Substitute 1000kg/m3 for ρ, 9.81m/s2 for g and 0m for h in Equation (I).

  P=(1000kg/ m 3)(9.81m/ s 2)(0m)=0kPa

Substitute 30cm for D, 0kPa for P in Equation (III).

  W=π4(30cm)2(0kPa)=0kN

Substitute 0kN for W in Equation (IV).

  F=0kN

Here, the force required to raise the water is at height of 0m is (F)h=0.

Substitute 96kPa for Patm and 14.72kPa for P in Equation (V).

  ( P abs)h=1.5=96kPa14.72kPa=81.28kPa

Here, absolute pressure at height 1.5m is (P abs)h=1.5.

Substitute 96kPa for Patm and 29.43kPa for P. in Equation (V).

  ( P abs)h=3=96kPa29.43kPa=66.57kPa

Here, absolute pressure at height 3m is (P abs)h=3.

Substitute 96kPa for Patm and 0kPa for P. in Equation (V).

  ( P abs)h=0=96kPa0kPa=96kPa

Here, absolute pressure at height 0m is (P abs)h=0.

The following Table-(1) shows the water pressure at different height of water level in the tube.

    Water level h(m)Absolute Pressure P(Pa)
    1.096000
    0.591095
    1.086190
    1.581285
    2.076380
    2.571475
    3.066570

The following figure variation of absolute water verse height of water in the tube.

  Fluid Mechanics: Fundamentals and Applications, Chapter 3, Problem 26P

Figure-(1)

Conclusion:

The force needed to raise the water to a height of h=1.5m above the free surface is 1.04kN.

The force needed to raise the water to a height of h=3m above the free surface is 2.08kN.

The force needed to raise the water to a height of h=0m above the free surface is 0kN.

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Chapter 3 Solutions

Fluid Mechanics: Fundamentals and Applications

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