Fluid statics I

CHEE 3363 Spring 2014 Handout 5

�Reading: Fox 3.1, 3.3, 3.6

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Learning objectives for lecture

1. Apply the hydrostatic equation to manometers.�

2. Use the hydrostatic equation to calculate the air pressure. �

3. Calculate the buoyant force on a submerged object.

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Assumptions for hydrostatics problems

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Assumptions for derivation:

pressure , ( , , )p x y z

dy

dx

dz

2p dyp dxdzy

Plus similar pressureforces on the x and z faces

2p dyp dxdzy

zg dxdydz

ernate Approach:

Goal: calculate the force on surfaces due to static fluids

Recall: Taylor series expansion:

Derivation of hydrostatic equation

pressure , ( , , )p x y z

dy

dx

dz

2p dyp dxdzy

Plus similar pressureforces on the x and z faces

2p dyp dxdzy

zg dxdydz

ernate Approach:∑F = 0 :

Divide by dx dy dz and simplify:

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Specific weight and specific gravity

At sea level: g = 32.2 ft / s2 = 9.81 m / s2

Specific weight γ ≡ ρg

For water (4 C, 760 mm Hg)

- Density: ρH2O = 62.4 lbm/ft3 = 1000 kg/m3 = 1.94 slug/ft3

- Specific weight = 62.4 lbf / ft3 = 9810 N / m3

Specific gravity SG ≡ ρ / ρH2O

Atmospheric pressure at sea level ρatm

- 14.7 lbf / in2 = 101 kPa = 1.01 × 105 Pa = 760 mm Hg

Gage pressure ≡ ρ - ρatm

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Hydrostatics for manometers (liquids) 1For liquids, often take the origin at the free surface and measure distances as positive downward from that surface

For liquids, also assume that the density is constant

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Hydrostatics for manometers (liquids) 2Assume h measured

+ downward

This shows that the pressure difference between two points in a static fluid can be determined by measuring the elevation distance (devices are called manometers)

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ρ

Example: force applied to piston 1

Equation:

initial final

Initial:

Initial free-body diagram:

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Given: setup as shown, before and after an applied force F.

Calculate: F in terms of D, d, h, H, ρ.

ρ

Example: force applied to piston 2Given: setup as shown, before and after an applied force F.

Calculate: F in terms of D, d, h, H, ρ.

initial final

Final free-body diagram:

Final:

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Example: multifluid manometer 1A B

d2d3

d4

d5

ˆ

2

3

4

1

d1

Hydrostatic equation:

Calculate: pressure drop pB - pA in above diagram.

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Do sequentially, starting at left:

Example: multifluid manometer 2A B

d2d3

d4

d5

ˆ

2

3

4

1

d1

Hydrostatic equation:

Calculate: pressure drop pB - pA in above diagram.

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Sum equations:

Examine: inclined manometer 1

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Given: Inclined tube manometer of diameter D and tube diameter d filled with oil of specific gravity SG. Find: Angle θ that gives a deflection of length L for an applied gauge pressure Δp.

Basic equation:

Calculate Δh:

Examine: inclined manometer 2

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Given: Inclined tube manometer of diameter D and tube diameter d filled with oil of specific gravity SG. Find: Angle θ that gives a deflection of length L for an applied gauge pressure Δp.

Find x:

Applying hydrostatics to gases

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Additional points when applying the hydrostatic equation to gases:

Example: gases in hydrostaticsExample: Find pressure variation with altitude; assume that the temperature decreases linearly with altitude, T = T0 − cz

Integrate (pressure p0 at z = 0, p at z):

Exercise: solve for p15

BuoyancyDefinition of buoyancy:

Net vertical pressure on a cylindrical volume element:

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Example: bubble rising in fluid 1

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Given: A hydrogen bubble of diameter d rises in water because of buoyancy but eventually reaches a terminal velocity. The drag force of the water on bubble is F = 3πµVd, where µ is the water viscosity and V the bubble speed relative to water. Find: (a) Buoyancy force that acts on the bubble; (b) estimate terminal speed.

Basic equations:

Terminal speed condition:

Assume: