3310-Chapter 3: Fluid statics

Pressure

A normal force exerted by a fluid per unit area

1 Pa

1 N/m²

1 bar = 10⁵ Pa =

0.1 Mpa = 100 kPa

1 atm = 101,325 Pa =

101.325 kPa = 1.01325 bars

1 atm

14.696 psi

psi

lbf/in²

Absolute pressure:

Actual pressure at a given position - measured relative to absolute vacuum (i.e., absolute zero pressure).

Gage pressure:

Difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure.

Vacuum pressures:

Pressures below atmospheric pressure.

Gage Pressure Calculations

P_gage = P_abs - P_atm

Vacuum Pressure Calculation

P_vac = P_atm - P_abs

Absolute Pressure Calculation

P_abs = P_atm - P_vac

Why is pressure a scalar quantity

• Pressure at any point in a fluid is the same in all directions.

• Pressure has magnitude but not a specific direction

• The pressure at a point in a fluid is the same in all directions

• Applicable to fluids in motion and fluids at rest

Pressure at a point on a wedge shaped fluid element

variation of pressure with depth

• P_below = P_above + ρg|∆z| = P_above + γ_s|∆z|

• P = P_atm + ρgh OR P_gage = ρgh

• ∆P = P₂ - P₁ = -∫²₁ ρgdz

Pressure in a room filled with gas

Variation of pressure with height is negligible

Pressure in a liquid

at rest increases linearly with distance from free surface

Assuming incompressible fluid

• Pressure change with elevation

• P_below - P_above = ρg∆z

Pressure depth relationship for compressible fluids

• Density of gases – temperature and pressure effects significant

• Pressure gradient for gases is small bcos specific weight is small (0.07063 vs 62.4 lb/ft3), (0.118 vs 9.81 kN/m3)

• Effect of elevation changes on pressure for small distances very small (tanks, pipes)

• when elevation difference is thousands of feet

  • Picture

Standard Atmosphere properties in SI units

• Temperature, T = 288.15 K (15oC)

• Pressure, p = 101.33 kPa (abs)

• Density, ρ = 1.225 kg/m3

• Specific weight, g = 12.014 N/m3

• Viscosity, µ = 1.789 x 10-5 N s/m2

Standard Atmosphere properties in BG Units

• Temperature, T = 581.67 oR (59.00 oF)

• Pressure, p = 2216.2 lb/ft2 (abs) 14.686 lb/in2 (abs)

• Density, ρ = 0.002377 slugs/ft3

• Specific weight, g = 0.07647 lb/ft3

• Viscosity, µ = 3.737 x 10-7 lb s/ft2

Simon Stevin – Dutch mathematician

Pressure is the same at all points on a horizontal plane in a given fluid regardless of geometry, provided that the points are interconnected by the same fluid. • Note that pressure at H and I are different bcos not interconnected by the same fluid. • The pressure force exerted by a fluid is always normal to the surface

Pascal’s Law

Pressure applied to a confined fluid increases the pressure throughout by the same amount. P₁ = P₂ → F₁/A₁ =F₂/A₂ → F₂/F₁ = A₂/A₁

Application of Pascal’s Law

• Area ratio A2/A1: Ideal mechanical advantage of hydraulic lift

• Lifting a large weight with a small force

• •Examples: hydraulic jacks, lifts, brakes, presses

Barometer

• P_atm = ρgh

• P_B = P_C + P_fgh

• The length or cross-sectional area of the tube does not affect the height of the fluid column of a barometer,

• The tube diameter is large enough to avoid surface tension (capillary) effects.

Aneroid Brometer

• Lower accuracy (+- 0.2 hPa)

• • Compact and portable

• • Easier to handle and use

• • Self recording

• • Airplanes as altimeters (height)

• • Require calibration

Atmospheric Pressure

• The weight of air above that location per unit surface area

• Changes with elevation and weather conditions

• At high altitudes, a car engine generates less power, and a person gets less oxygen because of the lower density of air

• ~ 15% drop in density will affect cooking, breathing, airplanes, cooling fans, etc. ~15% drop in pressure

Manometry - Piezometer tube

• P_A = P₁ = P_atm + ρgh₁

• P_A = ρgh₁

• vertical or inclined liquid columns

• fluid in container must be liquid

• Pressure in container > atmospheric

• Piezometer Tube: simplest and accurate (sphygmomanometer for blood pressure measurement)

Manometer

• P₂ = P₁ = P_atm + ρgh

• Used to measure small and moderate pressure differences

• Contains one or more fluids

• Overcomes difficulties with the use of a piezometer

• Stops at a change in fluid first

pressure difference in measurement - U-tube manometers

• Gauge fluids immiscible with other fluids in contact with it

• p_A + γ₁h₁ - γ₂h₂ - γ₃h₃ = p₅ = p_B

• p_A - p_B = γ₂h₂ + γ₃h₃ -γ₁h₁

Differential Manometers

• In stacked-up fluid layers, the pressure change across a fluid layer of density r and height h is rgh.

• Measuring the pressure drop across a flow section or a flow device

Hydrostatic pressure with variable density

• ρ = ρ₀ √1 + tan²(πs/4H)

• ρ₀ = density on waters surface

• s = vertical distance measure downward

• H = thickness of gradient zone

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Bourdon tube

hollow metal tube bent like a hook whose end is closed and connected to a dial indicator needle.

Pressure transducer

convert the pressure effect to an electrical effect such as a change in voltage, resistance, or capacitance. Pressure transducers are smaller and faster, and they can be more sensitive, reliable, and precise than their mechanical counterparts.

Strain-gage pressure transducers

Work by having a diaphragm deflect between two chambers open to the pressure inputs.

Piezoelectric transducers

solid-state pressure transducers, work on the principle that an electric potential is generated in a crystalline substance when it is subjected to mechanical pressure

Deadweight Tester

• Used primarily for calibration and can measur extremely high pressures

• measures pressure directly through application of a weight that provides a force per unit area

• Constructed with an internal chamber filled with a fluid (usually oil), along with a tight-fitting piston, cylinder, and plunger.

• Weights applied to the top of the piston exerts a force on the oil in the chamber. The total force F acting on the oil at the piston–oil interface is the sum of the weight of the piston plus the applied weights.

Fluid statics

• Deals with problems associated with fluids at rest.

• no relative motion between adjacent fluid layers, and thus there are no shear (tangential) stresses in the fluid trying to deform it.

Hydrostatics

When fluid is a liquid.

Aerostatics

When fluid is a gas