HPGE: General Concept and Principles

Fluid Mechanics

Is a physical science dealing with the action of fluids at rest or in motion, and with applications and devices in engineering using fluids.

Fluid Static

Deals with fluids at rest.

Fluid dynamics/Hydrodynamics

Deals with the fluids in motion.

Ideal Fluids

Assumed to have no viscosity, incompressible, have uniform velocity when flowing, no friction between moving layers of fluids, and no eddy currents or turbulence.

Real Fluids

Exhibit infinite viscosities, non-uniform velocity distribution when flowing, compressible, and experience friction and turbulence in flow.

Mass Density

The density of a fluid is its mass per unit of volume.

Mass Density Formula

ρ = M / V, where ρ - Mass density, M - Mass of the fluid, V - Volume of the fluid.

Ideal Gas Density Formula

ρ = p / (RT), where ρ - Mass density, p - Absolute pressure of gas in Pa, R - Gas constant in J/kg-˚K or lb-ft/slug-˚R, T - Absolute Temperature in Kelvin or Rankine.

Approximate Room-Temperature Densities of Common Fluids

Air (STP) 1.29 kg/m3, Air (21 ˚F, 1 atm) 1.20 kg/m3, Alcohol 790 kg/m3, Ammonia 602 kg/m3, Gasoline 720 kg/m3, Glycerin 1260 kg/m3, Mercury 13600 kg/m3, Water 1000 kg/m3.

Specific Volume

The density of a fluid is its mass per unit of volume (reciprocal of Mass Density).

Specific Volume Formula

Vs = 1 / ρ, where Vs - Specific volume of the fluid, ρ - Mass density.

Specific Weight or Unit Weight

It is the weight of a fluid per unit volume.

Unit Weight Formula

γ = W / V or γ = Mg or γ = ρg, where γ - Unit Weight of the fluid, W - Weight of the Fluid, V - Volume of the fluid, g - Acceleration due to gravity, ρ - Mass density.

Specific Gravity

It is dimensionless ratio of a fluid's density to some standard reference density.

Specific Gravity Formula (Mass Density)

G = ρobject / ρreference, where G - Specific gravity, ρobject - Mass density of the specific object, ρreference - Mass density of the reference fluid.

Specific Gravity Formula (Unit Weight)

G = γobject / γreference, where γobject - Unit Weight of the specific object, γreference - Unit Weight of the reference fluid.

Common Water Density

γw = 62.4 lbs/ft3 = 9.81 kN/m3, ρw = 1.94 slugs/ft3 = 1000 kg/m3.

Viscosity

It is a property of a fluid that determines its resistance to shearing forces; a perfect fluid would have no viscosity.

Viscosity Formula

μ = τ / (V/y), where μ - absolute viscosity in lb-sec/ft2 (poises) or Pa-sec, τ - Shear stress in lb/ft2 or Pa, V - Velocity in ft/s or m/s, y - Distance between the plates in ft or m.

Kinematic Viscosity

It is the ratio of the dynamic viscosity of the fluid to its mass density.

Dynamic Viscosity (μ)

Absolute viscosity in lb-sec/ft2 (poises) or Pa-sec.

Mass Density (ρ)

Mass density of fluid in slugs/ft3 or kg/m3.

Kinematic Viscosity (v)

Kinematic viscosity.

Common Units for Viscosity (English)

Absolute μ in lb-sec/ft2 (slug/ft2-sec) and Kinematic v in ft2/sec.

Common Units for Viscosity (Metric)

Absolute μ in Dyne-sec/cm2 (poise) and Kinematic v in cm2/s (stoke).

Common Units for Viscosity (SI)

Absolute μ in Pa-sec (N-sec/m2) and Kinematic v in m2/s.

Poise

1 poise = 1 Dyne-sec/cm2 (1 dyne = 10^-5 N).

Stoke

1 stoke = 0.0001 m2/s.

Surface Tension

The member of 'skin' that seems to form on the free surface of a fluid due to intermolecular cohesive forces.

Pressure inside a droplet of liquid

p = 4σ/d, where p is Gage Pressure in Pa, σ is Surface tension in N/m, and d is Diameter of the droplet in m.

Capillarity

The behavior of the liquid in a thin-bore tube caused by surface tension, depending on the relative magnitudes of cohesion and adhesion.

Adhesion > Cohesion

Indicates that the liquid adheres more to the walls than it sticks to itself.

Adhesion < Cohesion

Indicates that the liquid sticks to itself more than it adheres to the walls.

Constant Angles for Mercury-Glass

140°.

Constant Angles for Water-Paraffin

107°.

Constant Angles for Water-Silver

90°.

Constant Angles for Kerosene-Glass

26°.

Constant Angles for Glycerin-Glass

19°.

Constant Angles for Water-Glass

0°.

Constant Angles for Ethyl Alcohol-Glass

0°.

Coefficient of Compressibility (β)

The fractional change in volume of a fluid per unit change in pressure at constant temperature.

Bulk Modulus of Elasticity (EB)

It expresses the compressibility of the fluid, defined as the ratio of the change in unit pressure to the corresponding volume change per unit of volume.

Pressure Disturbances or Celerity

Pressure waves known as acoustical or sonic velocity.

Pressure Equation for Rigid Pipes

c = √(EB/ρ).

Pressure Equation for Non-Rigid Pipes

c = √(EB/ρ[1 + EBd/Et]).

Ideal Gas Property Changes

p1V1/T1 = p2V2/T2.

Isothermal Condition for Ideal Gas

p1V1 = p2V2 when temperature is held constant.

Sample Problem Results

For a reservoir of glycerin: weight = 11.772 kN, unit weight = 12.366 kN/m3, mass density = 1260.5 kg/m3, specific gravity = 1.26.

Pressure Definition

The force per unit area exerted by a liquid or gas on a body or surface, acting at right angles to the surface uniformly in all directions.

Pressure Equation

p = F/A, where p is Pressure, F is Force, and A is Area.

Unit of Measurement for Pressure (SI)

N/m2 (Pascal, Pa).

Force

lbs (Pound/s)

Corresponding Area

in2 (square inch/es)

Fractional Unit

lbs/in2

Stress Unit (FullName)

Pounds per square inch

Stress Unit

Psi

Stress Unit (FullName)

Pounds per square foot

Stress Unit

Psf

Stress Unit (FullName)

kips per square inch

Stress Unit

ksi or x103Psi

Pascal's Law

Developed by a French mathematician Blaise Pascal, states that the pressure on a fluid is equal in all directions and in all parts of the container.

Gage pressure

Gauge pressure is the pressure relative to atmospheric pressure. Gauge pressure is positive for pressures above atmospheric pressure, and negative for pressures below it.

Atmospheric Pressure

Atmospheric pressure, also known as barometric pressure, is the pressure within the atmosphere of Earth.

Standard atmospheric pressure

1 atm = 2166 lbs/ft2 = 14.7 psi = 29.9 inches of mercury (Hg) = 760 mm Hg = 103.325 kPa

Absolute Pressure

It is the pressure above absolute zero pressure (vacuum). Absolute pressure is the sum of gauge pressure and atmospheric pressure.

Pressure measuring tool/equipment

Mercury Barometer - is an accurate and relatively simple way to measure changes in atmospheric pressure.

Aneroid Barometer

In an aneroid barometer, a partially evacuated metal drum expands or contracts in response to change in air pressure.

Pressure in a homogenous fluid

p = γh - eq.1

Pressure difference in a homogenous fluid

p2 - p1 = γh - eq.2

Pressure below layers of different fluids

p_bottom = p_a + ∑γh

Pressure head

Is the height of a column of a homogeneous fluid of unit weight that will produce an intensity of pressure.

Manometer

Manometer is a tube usually bent in a form of a U shape, containing a liquid of known specific gravity, the surface of which moves proportionally to changes of pressures.

Open Type Manometer

Has an atmospheric surface in one leg and is capable of measuring gage pressures.

Differential Type Manometer

Without an atmospheric surface and capable of measuring only differences of pressures.

Piezometer

The simplest form of an open manometer. It is a tube tapped into a wall of a container or conduit for the purpose of measuring pressures.

Specific Gravity Calculation

If a depth of liquid of 1m causes a pressure of 7kPa, the specific gravity of the fluid is 0.714.

Hydrostatic force

Force located at the center of pressure.

Unit weight of the fluid (γ)

Weight per unit volume of the fluid.

Vertical distance from liquid surface to center of gravity (h)

Distance from the liquid surface to the center of gravity of the submerged surface.

Area of the submerged surface (A)

Total surface area that is submerged in the fluid.

Distance from liquid surface to center of gravity (y)

Distance from the liquid surface to the center of gravity of the submerged surface, parallel to the line of action.

Distance from liquid surface to center of pressure (yp)

Distance from the liquid surface to the center of pressure of the submerged surface, parallel to the line of action.

Eccentricity (e)

Distance from the center of gravity to the center of pressure of the submerged surface.

Centroidal moment of inertia (Icg)

Moment of inertia of the submerged surface about its centroid.

Dams

Structures that block the flow of a river, stream or other waterway, used for various purposes including power generation and flood control.

Purpose of dams

Includes agricultural and domestic use, power supply (hydroelectric), navigation, flood control, and multi-purpose.

Types of Dams

Gravity dams, embankment dams, arch dams, and buttress dams.

Vertical Hydrostatic Forces

Forces acting vertically due to the weight of the fluid above the submerged surface.

Horizontal Hydrostatic Forces

Forces acting horizontally due to the pressure exerted by the fluid.

Weight of the Dam

Total weight of the dam structure itself.

Hydrostatic Uplift Pressure

Pressure acting upwards on the dam, calculated as unit weight of liquid multiplied by respective depth.

Uplift pressure formula

Uplift pressure = unit weight of liquid x respective depth.

Vertical Reaction

Summation of vertical forces acting on the dam, with upward forces considered positive.

Horizontal Reaction

Summation of horizontal forces acting on the dam, with rightward forces considered positive.

Righting Moment (RM)

Moment that causes rotation towards the upstream side of the dam.

Center of pressure

Point where the total hydrostatic force acts on the submerged surface.

Homogeneous liquid

Liquid with uniform density throughout.

Curved surfaces hydrostatic force

Hydrostatic force calculations for surfaces that are not flat.

Common Plane sections properties

Includes center of gravity and centroidal moment of inertia.

Analysis of Gravity Dams

Considerations and calculations necessary for evaluating the stability and performance of gravity dams.