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Chapter 3 Fluid Statics

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Main Topics Introduction Pressure Variation in a Static Fluid Equilibrium of a fluid of constant density The measurement of Pressure

• The Barometer• Manometers• The Bourdon Gauge• Other types of Pressure Gauge

Uniform Rectilinear Acceleration Pressure Variation in a variable density

fluid and the Standard Atmosphere

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3.0 Introduction

Fluid Statics is a branch of mechanics of fluid which deals primarily with fluids at rest. As individual elements do not move relative to each other, shear stresses are not involved and all forces due to the pressure of the fluid are normal to the surfaces on which they acts.

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3.1 Pressure variation in a static fluid

Fig.3.1.1

Derivation of (eq.3.1.1)

Equation (3.1.1) is the basic equation of fluid statics and it states that the maximum rate of change of pressure occurs in the direction of the gravitational vector.

3.1 Pressure Variation in a Static Fluid:

For an infinitesimal fluid element as shown in Fig.3.1.1. The forces acting on the element are

the stresses from the surroundings and the gravity force. For equilibrium, we have

Dividing through by the volume Δx Δy Δz, we have

Taking the limit when Δx, Δy, Δz approaches to zero, then

which is

or (3.1.1)

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3.2 Equilibrium of a fluid of constant density

Assumptions: Constant density (liquid) Gravitational acceleration in negative z direction Relating to liquid surface

Deriving eq.3.2.1

Assumptions: Constant density Gravitational acceleration in negative z direction

Deriving eq.3.2.2

3.2 Equilibrium of a Fluid of Constant Density:

(3.2.1)

(3.2.2)

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3.3 The Measurement of Pressure

g P gy g P gxc A c atm B

The Barometer• Fig.3.3.1.1• The governing equation is

Manometers• Fig.3.3.2.1• The governing equation is

g P g P ghc atm c v

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3.3 The Measurement of Pressure

g P y x g g P gy gxc A c A B1 2 ( )

Applying manometer on a pipeFig.3.3.2.3The governing equation is

The Bourdon GaugeFig.3.3.3.1

Other types of Pressure Gaugee.g., strain-gage pressure transducer

g P P gxc B A( ) ( )1 2

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3.4 Uniform Rectilinear Acceleration

Fig.3.1.1*

Derivation of eq.3.4.1

Example 3.4.1

3.4 Uniform Rectilinear Acceleration:

(3.4.1)

Example 3.4.1: A fuel tank is shown in Fig.3.4.1. If the tank is given a uniform acceleration to

the right, what will be the pressure at point B?

From (eq.3.4.1), the pressure gradient is in the direction, therefore the surface of fluid

will be perpendicular to this direction.

Choosing the y axis parallel to , we find that (eq.3.4.1) may be integrated between point

B and the surface. The pressure gradient becomes

Integrating between y = 0 and y = d, yields

The depth of fluid d at point B is determined from the tank geometry, the fluid's volume and the

angle θ.

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3.5 Pressure Variation in a variable-density fluid and the Standard Atmosphere

For ideal gas under isothermal conditionderivation of (eq.3.5.3)

For ideal gas under adiabatic conditiongoverning equations (3.5.4) & (3.5.5)

U.S. Standard Atmosphere (Fig.3.5.1)in Troposphere – (eq.3.5.7)in Stratosphere – (eq.3.5.8)

3.5 Pressure Variation in a Variable- Density Fluid and the Standard Atmosphere:

At low pressure the densities of most gases are reasonably well approximated by the ideal gas

law.

(3.5.1)

For an ideal gas, considering with the normal Cartesian coordinate, as we mentioned that

gravitational acceleration is acting in the z direction, Eq. (3.1.1) becomes

Solving for the pressure,

(3.5.2)

To complete the integration on the right, we must know how the gas temperature varies with z.

If the temperature is constant, this can be separated and integrated as follows:

(eq.3.5.3)

For gas under adiabatic conditions, we have

(3.5.4)

(3.5.5)

B

-100 -80 -60 -40 -20 0 20

Temperature,oC

0

5

10

15

20

25

30

35

40

45

50

Alti

tude

,km

Fig. 3.5.1 Temperature - Altitude Curve

0 km (P = 101.325 kPa)

-56.

5o C

11 km (P = 22.6 kPa)

20.1 km (P = 5.5 kPa)

32.2 km (P = 0.9 kPa)

47.3 km(P = 0.1 kPa)

-44.5

o C

-2.5oC

Troposphere

Stratosphere

Ionosphere

The temperature variation in the troposphere is

T = T0 - B z (3.5.6)

where T0 is the surface temperature and B is the lapse rate. For the U.S. Standard Atmosphere,

T0 = 15oC = 288.15 K = 59oF = 518.67oR

and

B = 0.00650 K/m = 0.003566oR/ft

Substituting (eq.3.5.6) into (eq.3.5.2) and integrating we obtain

(3.5.7)

In the stratosphere, the temperature is

T = Tc = 216.7 K = -69.6oF

Solving (eq.3.5.2) for the stratosphere, we have

(eq.3.5.8)

where Pc is the pressure at the lower edge of the stratosphere zc. For the U.S. Standard

Atmosphere, zc = 11 km (36100 ft) and Pc = 22.6 kPa (3.28 psi).

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Points to remember

The governing equation for static fluid is

The above equation applies to static fluid where there is no relative motion between fluid particles.

For static fluid, the maximum rate of change of pressure is along the gravitational acceleration and the constant pressure line is perpendicular to the gravitational acceleration.

Constant pressure line is for fluid with continuous phase.

g P gc

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Tutorial

Link to Tutorial 2

Tutorial 2(Chapter 3)

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1. The vapour pressure of mercury is 2.5x10-5 psia at 70 oF. Calculate the error in barometer

height due to neglecting the vapour pressure of mercury. Would this be detectable for

engineering calculations? (1.701×10-4 %)

2. Determine the gage pressure in psig at point a in Fig.T.2.2, if liquid A has a specific gravity

of 0.75 and liquid B has a specific gravity of 1.20. The liquid surrounding point a is water,

and the tank on the left is open to the atmosphere. (1.18)

3. A rectangular tank, as shown in Fig.T.2.3, open to atmosphere, is filled with water to a depth

of 2.5 m as shown. A U-tube manometer is connected to the tank at a location 0.7 m above

the tank bottom. If the zero level of the manometer fluid, Meriam blue (specific gravity

1.75), is 0.2 m below the connection, determine the deflection R after the manometer is

connected and the air has been removed from the connecting leg. (1.6)

4. The inclined manometer as shown in Fig.T.2.4 has reservoir diameter, D, of 96 mm and

measuring tube diameter, d, of 8 mm. Determine the angle, θ, required to provide a 5 : 1

increase in liquid deflection, L, compared to the total deflection in a regular U-tube

manometer. (11.21)

5. If the variation in specific weight of atmospheric air between sea level and an altitude of

3600 ft were given by , where γ0 is the specific weight of air at sea level,

z is the altitude above sea level, and , determine the pressure

in psia at an altitude of 3600 ft when sea level conditions are 14.7 psia, 59oF. (12.99)

6. Calculate the pressures at points A, B, C and D in pascals for the set-up as shown in

Fig.T.2.6. (95.44, 107.21, 107.21, 123.99)

7. For Fig.T.2.7, A contains water, and the manometer fluid has a specific gravity of 2.94.

When the left meniscus is at zero on the scale, PA = 90 mm H2O. Find the reading of the right

meniscus for PA = 8 kPa with no adjustment of the U tube or scale. (0.3835)

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