Fluid Mechanics

What is a fluid?• Liquids and gases have the ability to flow• They are called fluids.

• Liquids are incompressible, assume the form of their containers, and have a fixed volume.

• Gasses are compressible, and assume the shape and volume of their containers.

Definitions

Density

• Regardless of form (solid, liquid, gas) we can define how much mass is squeezed into a particular space

density mass

volume*Sometimes we use “weight density” = weight/volume or mg/V

ρw=ρg

Densities of Common Stuff

• A measure derived by finding the ratio of the density of some material to the ratio of the density of water.

Weight Density of water = 62.4 lb/ft3

Specific Gravity (SG)

Pressure• A measure of the amount of force exerted

on a surface area, measured in pounds/in2.

pressure forcearea

Also: P = ρwh

Pressure in a Fluid

• The pressure is just the weight of all the fluid above you

• Atmospheric pressure is just the weight of all the air above on area on the surface of the earth

• In a swimming pool the pressure on your body surface is just the weight of the water above you (plus the air pressure above the water)

Pressure in a Fluid

• So, the only thing that counts in fluid pressure is the gravitational force acting on the mass ABOVE you

• The deeper you go, the more weight above you and the more pressure

• Go to a mountaintop and the air pressure is lower

Pressure in a Fluid

Pressure acts perpendicular to the surface and increases at greater depth.

Pressure in a Fluid

Hydraulic Lift

Pressure is the same everywhere.From Kuphaldt’s book Liii.pdf

Displacement of Water

The amount of water displaced is equal to the volume of the rock.

BuoyancyNet upward force is called the buoyant force!!!

Easier to lift a rock in water!!

Archimedes’ Principle

• An immersed body is buoyed up by a force equal to the weight of the fluid it displaces.

• If the buoyant force on an object is greater than the force of gravity acting on the object, the object will float

• The apparent weight of an object in a liquid is gravitational force (weight) minus the buoyant force

Flotation

• A floating object displaces a weight of fluid equal to its own weight.

Flotation

Gases

• The primary difference between a liquid and a gas is the distance between the molecules

• In a gas, the molecules are so widely separated, that there is little interaction between the individual molecules

• IDEAL GAS pressure drops 1/273 for each degree Celsius.

• Independent of what the molecules are.

The Gas LawsThe Gas Laws

Charles’s and Gay-Lussac’s law, (or simply Charles’s Law) states that the volume of a gas maintained at constant pressure is directly proportional to the absolute temperature of the gas.

V TV VT T

1 2

1 2

at constant pressure

Charles’s Law

• The Volume of a gas is directly proportional to the Temperature (Kelvin) at constant pressure and # moles.

2

1

2

1

T

T

V

V

Boyle’s Law

• Pressure depends on density of the gas

• Pressure is just the force per unit area exerted by the molecules as they collide with the walls of the container

• Double the density, double the number of collisions with the wall and this doubles the pressure

Boyle’s Law

Boyle’s Law

Density is mass divided by volume.

Halve the volume and you double the density and thus the pressure.

Boyle’s Law

• At a given temperature for a given quantity of gas, the product of the pressure and the volume is a constant

P1V1 P2V2

25#

IDEAL GAS LAWAn Ideal Gas or perfect gas is a hypothetical gas con-sisting of identical particles with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container. Real gases act like ideal gases at low pres-sures and high temperatures.

Real Gases do not exhibit these exact properties, although the approximation is often good enough to describe real gases. The properties of real gases are influenced by compressibility and other thermodynamic effects.

Ideal Gas Law• Pressure = kNT/V• Where k is the Boltzmann’s Constant

– K = 1.38 x 10-23 Nm/moleculesºK

• Where N is the Number of molecules

• Where T is Temperature

• Also PV=nRT where n is # of moles and R is the universal gas constant.

• .082 L*atmosphere/(mol*K)

27#

IDEAL GAS LAWPV = nRT

Where: P = Pressure (psia)

V = Volume (FT3)

n = Number of Moles of Gas

(1 mole = 6.02 x 1023 molecules)

R = Gas Constant (10.73 FT3 PSIA / lb-mole oR)

T = Temperature (oR)

28#

REAL GASES

• Compressibility Factor (Z) - The term "compressibility" is used to describe the deviance in the thermodynamic properties of a real gas from those expected from an ideal gas.

• Real Gas Behavior can be calculated as:

PV = nZRT

29#

STANDARD CONDITIONS

• P = 14.7 PSIA

• T = 520 deg R (60 deg F)

• Behavior of gases in a process can be equally compared by using standard conditions – This is due to the nature of gases.

30#

ACTUAL CONDITIONS• Standard conditions can be converted to Actual Conditions using the

Ideal Gas Law.

PSVS = nRTS PAVA = nRTA

=PSVS

TS

PAVA

TA

=PSTAVS PATS

VA

31

Dalton’s Law of Partial Pressures indicates that• pressure depends on the total number of gas

particles, not on the types of particles

• the total pressure exerted by gases in a mixture is the sum of the partial pressures of those gases

PT = P1 + P2 + P3 + .....

Dalton’s Law of Partial Pressures

32

Dalton’s Law of Partial Pressures (continued)

33

• For example, at STP, one mole of a pure gas in a volume of 22.4 L will exert the same pressure as one mole of a gas mixture in 22.4 L.

V = 22.4 L Gas mixtures

Total Pressure

0.5 mole O2

0.3 mole He0.2 mole Ar1.0 mole

1.0 mole N2

0.4 mole O2

0.6 mole He1.0 mole

1.0 atm 1.0 atm 1.0 atm

Guide to Solving for Partial Pressure

34

• Freezing is the phase change as a substance changes from a liquid to a solid.

• Melting is the phase change as a substance changes from a solid to a liquid.

• Condensation is the phase change as a substance changes from a gas to a liquid.

• Vaporization is the phase change as a substance changes from a liquid to a gas.

Generic Heating/Cooling Curve

Phase Diagram Definitions

• Sublimation is the phase change as a substance changes from a solid to a gas without passing through the intermediate state of a liquid.

• Deposition is the phase change as a substance changes from a gas to a solid without passing through the intermediate state of a liquid.

• TRIPLE POINT - The temperature and pressure at which the solid, liquid, and gas phases exist simultaneously.

• CRITICAL POINT - The temperature above which a substance will always be a gas regardless of the pressure.

• Freezing Point - The temperature at which the solid and liquid phases of a substance are in equilibrium at atmospheric pressure.

• Boiling Point - The temperature at which the vapor pressure of a liquid is equal to the pressure on the liquid.

• Normal (Standard) Boiling Point - The temperature at which the vapor pressure of a liquid is equal to standard pressure (1.00 atm = 760 mmHg = 760 torr = 101.325 kPa)

Atmospheric Pressure

• Just the weight of the air above you

• Unlike water, the density of the air decreases with altitude since air is compressible and liquids are only very slightly compressible

• Air pressure at sea level is about 105 newtons/meter2

Barometers

Buoyancy in a Gas

• An object surrounded by air is buoyed up by a force equal to the weight of the air displace.

• Exactly the same concept as buoyancy in water. Just substitute air for water in the statement

• If the buoyant force is greater than the weight of the object, it will rise in the air

Buoyancy in a Gas

Since air gets less dense with altitude, the buoyant force decreases with altitude. So helium balloons don’t rise forever!!!

Bernoulli’s Principle

Bernoulli’s Principle

• Flow is faster when the pipe is narrower• Put your thumb over the end of a garden hose• Energy conservation requires that the pressure be

lower in a gas that is moving faster• Has to do with the work necessary to compress a

gas (PV is energy, more later)

Bernoulli’s Principle• When the speed of a fluid increases,

internal pressure in the fluid decreases.

Bernoulli’s Principle

Bernoulli’s Principle

Why the streamlines are compressed is quite complicated and relates to the air boundary layer, friction and turbulence.

Bernoulli’s Principle

THE END

Relative Density

• Relative Density or Specific Gravity - the ratio of the density of a material to the density of water– Substances with a specific gravity of less than 1

are lighter than water so they float– Substances with a specific gravity of greater than

1 are heavier than water so they sink– Knowing the specific gravity is important for

planning spill cleanup and fire-fighting procedures

Viscosity

• The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the informal notion of "thickness". For example, honey has a higher viscosity than water.Viscosity is due to friction between neighboring parcels of the fluid that are moving at different velocities. When fluid is forced through a tube, the fluid generally moves faster near the axis and very little near the walls, therefore some stress (such as a pressure difference between the two ends of the tube) is needed to overcome the friction between layers and keep the fluid moving. For the same velocity pattern, the stress is proportional to the fluid's viscosity.

(from Kuphaldt’s book Liii.pdf)

Viscosity (2)

53#

LAMINAR FLOW• Laminar Flow - Is Characterized By Concentric Layers Of Fluid

Moving In Parallel Down The Length Of A Pipe. The Highest Velocity (Vmax) Is Found In The Center Of The Pipe. The Lowest Velocity (V=0) Is Found Along The Pipe Wall.

SIDE VIEW END VIEW

VMAX

CONCENTRIC FLUID LAYERSPARABOLIC FLOW PROFILE

54#

FLOW MEASUREMENT - TERMS

• DENSITY (rho) – A Measure Of Mass Per Unit Of Volume (lb/ft3 or kg/M3).

• SPECIFIC GRAVITY – The Ratio Of The Density Of A Material To The Density Of Water Or Air

Depending On Whether It Is A Liquid Or A Gas.

• COMPRESSIBLE FLUID – Fluids (Such As Gasses) Where The Volume Changes With Respect To Changes In

The Pressure. These Fluids Experience Large Changes In Density Due To Changes In Pressure.

• NON-COMPRESSIBLE FLUID – Fluids (Generally Liquids) Which Resist Changes In Volume As The Pressure

Changes. These Fluids Experience Little Change In Density Due To Pressure Changes.

55#

TURBULENT FLOW• Turbulent Flow - Is Characterized By A Fluid Motion That Has

Local Velocities And Pressures That Fluctuate Randomly. This Causes The Velocity Of The Fluid In The Pipe To Be More Uniform Across A Cross Section.

SIDE VIEW

VMAX ~ VAVG

56#

REYNOLDS NUMBER• The Reynolds number is the ratio of inertial forces (velocity and density that

keep the fluid in motion) to viscous forces (frictional forces that slow the fluid down) and is used for determining the dynamic properties of the fluid to allow an equal comparison between different fluids and flows. The Reynolds number of a fluid is a dimensionless quantity expressing the ratio between a moving fluid’s momentum and its viscosity, and is a helpful gauge in predicting how a fluid stream will move.

• Laminar Flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion

• Turbulent Flow occurs at high Reynolds numbers and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations.

• The Reynolds number is the most important value used in fluid dynamics as it provides a criterion for determining similarity between different fluids, flow rates and piping configurations.

57#

REYNOLDS NUMBER

Re =Dv rm

D

v

r

m

DIAMETER (FT)

VELOCITY (FT/SEC)

DENSITY (LB/FT3)

VISCOSITY (cp)

=

==

=

C

C CONSTANT (6.72X10-4 LB/FT SEC cp)=

0 2000 4000

LAMINAR TRANSITION TURBULENT

58#

REYNOLDS

NUMBER

from Kuphaldt’s book Liii.pdf

59#

BERNOULLI’S LAW

• Bernoulli's Law Describes The Behavior Of An Ideal Fluid Under Varying Conditions In A Closed System. It States That The Overall Energy Of The Fluid As It Enters The System Is Equal To The Overall Energy As It Leaves.

PE1 + KE1 = PE2 + KE2

PE = Potential Energy

KE = Kinetic Energy

60#

BERNOULLI’S EQUATION

• Bernoulli’s Law Is Described By The Following Equation For An Ideal Fluid.

V2 > V1

P2 < P1

Increased Fluid Speed Decrease Fluid Pressure

V1, P1

V2, P2

P1 + r gh1 r V12 +

21

= P2 + r gh2 r V22 +

21

Pressure Energy

Kinetic Energy Per Unit Volume

Potential Energy Per unit Volume

Energy Per Unit Volume Before = Energy Per Unit Volume After

61#

HEAD METER THEORY OF OPERATION

Beta Ratio b = d/D Should Be 0.3 – 0.75Meter Run – Dependent On PipingNormally 20 Diameters Upstream & 5 Diameters Downstream