Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 2, Problem 2.14P
To determine

The height of mercury column in water tube.

The height of oil column.

Expert Solution & Answer
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Explanation of Solution

Calculation:

Write the expression for density of the oil.

ρoil=(SG)oil(ρwater) …… (I)

Here, the density of oil is ρoil, the specific gravity of oil is (SG)oil and the density of water is (ρwater).

Write the expression for pressure at bottom in mercury column.

pb,mercury=(ρmercury)(g)(hmercury) …… (II)

Here, the pressure at bottom in mercury column is pb,mercury, the density of mercury is ρmercury, the acceleration due to gravity is g and the height of mercury column is hmercury.

Write the expression for pressure at the bottom of the water column.

pb,water=(ρwater)(g)(hwater)+(ρmercury)(g)(h1) …… (III)

Here, the pressure at bottom in water column is pb,water, the height of water column is hwater and height of mercury column in water tube is h1.

Write the expression for pressure at bottom in oil column.

pb,oil=(ρoil)(g)(h2)+ρmercury(g)(h3) …… (IV)

Here, the pressure at bottom in oil column is pb,oil, the density of oil is ρoil, the height of oil column is h2 and the height of mercury column in oil tube is h3.

Substitute (SG)oil(ρwater) for ρoil in Equation (IV).

pb,oil=((SG)oil(ρwater))(g)(h2)+ρmercury(g)(h3) …… (V)

The pressure in the liquid varies according to the depth of fluid. But, the hydrostatic pressure of the fluid is same at same depth.

The pressure at bottom in mercury column is equal to pressure at bottom in water column.

pb,mercury=pb,water …… (VI)

The pressure at bottom in mercury column is equal to pressure at bottom in oil column.

pb,mercury=pb,oil …… (VII)

Conclusion:

Substitute 13550kg/m3 for ρmercury and 8cm for hmercury in Equation (II).

pb,mercury=(13550kg/m3)(g)(8cm)=(13550kg/m3)(g)(8cm)(1m100cm)=(13550kg/m3)(g)(0.08m) …… (VIII)

Substitute 998kg/m3 for ρwater, 27cm for hwater and 13550kg/m3 for ρmercury in Equation (III).

pb,water=(998kg/m3)(g)(27cm)+(13550kg/m3)(g)(h1)=(g)[(998kg/m3)(27cm)(1m100cm)+(13550kg/m3)(h1)]=(g)[(998kg/m3)(0.27m)+(13550kg/m3)(h1)] …. (IX)

Substitute 0.78 for (SG)oil, 998kg/m3 for ρwater, 13550kg/m3 for ρmercury and 5cm for h3 in Equation (IV).

pb,oil=[((0.78)(998kg/m3))(g)(h2)+(13550kg/m3)(g)(5cm)(1m100cm)]=(g)[(778.44kg/m3)(h2)+(677.5kg/m2)] …… (X)

Substitute (13550kg/m3)(g)(0.08m) for pb,mercury and (g)[(998kg/m3)(0.27m)+(13550kg/m3)(h1)] for pb,water in Equation (VI).

(13550kg/m3)(g)(0.08m)=(g)[(998kg/m3)(0.27m)+(13550kg/m3)(h1)](1084kg/m2)(269.46kg/m2)=(13550kg/m3)(h1)(h1)=(814.54kg/m2(13550kg/m3))=0.06011m

Further simplify.

(h1)=(0.06011m)(100cm1m)6.0cm

Thus, the height h1 of the mercury liquid column is 6.0cm.

Substitute (13550kg/m3)(g)(0.08m) for pb,mercury and (g)[(778.44kg/m3)(h2)+(677.5kg/m2)] for pb,oil in Equation (VII).

(13550kg/m3)(g)(0.08m)=(g)[(778.44kg/m3)(h2)+(677.5kg/m2)](1084kg/m2)(677.5kg/m2)=(778.44kg/m3)(h2)(406.5kg/m2)=(778.44kg/m3)(h2)h2=(406.5kg/m2778.44kg/m3)

Further simplify.

h2=(406.5kg/m2778.44kg/m3)=0.52219m(100cm1m)52.219cm

Thus, the height h2 of oil liquid column is 52.219cm.

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Chapter 2 Solutions

Fluid Mechanics

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