1 Fluid Statics - Fluid Properties

Fluid Statics:Fluid Properties

What is a fluid? Something that flows Something that cannot be in static equilibrium when exposed to a shear stress

Types of fluids:

Liquid Gas PlasmaDifference between gas and liquid?

Gas LiquidExpands to fill container Conforms to container

Generally considered compressible Generally considered incompressible

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Fluid MechanicsRocketHydro DamSonic BoomWater Jet Pack

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Properties of fluids Density Specific Weight Specific Gravity Surface Tension Viscosity Pressure Temperature ...


DensityAmount of mass per unit volume

mV kg/m3 1.1Units: kgm3

Specific WeightAmount of weight per unit volume

wV N/m3 1.2Units: Nm3

RelationshipEliminating volume, V, from (1) and (2) gives:

w g mg 1.3


Specific GravityEquivalently: Ratio of density of substance to density of water a 4C Ratio of specific weight of substance to specific weight of water at 4Csg H2O@4C H2O@4C 1.4


PressureForce per unit area (Pascal, Pa)

p FAN/m2

p FAPa

Pressure in a fluid Acts uniformly in all directions (scalar quantity) Force caused by fluid pressure acts perpendicular to a solid boundary (vector quantity)


Absolute and Gage pressurePressure measurements in a fluid are relative to some reference pressure Atmospheric pressure as reference: Gage Pressure Perfect vacuum as reference: Absolute Pressure Specified pressure as reference: Differential Pressurepabs patm + pgageFig 3.1 shows comparison between absolute and gage pressures


Units of PressurePascal1 Pa = 1 Nm2Hectopascal1 hPa = 100 Pa1 hPa = 1 millibar

Standard Atmosphere (Sea level at 15C)1 atm = 101325 Pa1 atm = 1013.25 millibar = 1013.25 hPa

Torr760 Torr = 1 atm

Millimetre of mercury1 mmHg = 133.322387415 Pa


Relationship between Pressure and ElevationElevation is the vertical distance, h, from some reference point

Figure 3.2 illustrates different reference points

Change in pressure due to change in elevation:

p h g h 3.3For low density fluids (e.g. gasses), the change in pressure due to elevation is very low compared to other factors, and can often be assumed to be zero.The book assumes that pressure in a gas is uniform unless otherwise specified.


Free body diagram for fluid at restThe figure shows forces acting on a fluid volume in static equilibrium

The pressure in the cylinder is opposed by the pressure outside the cylinder The vertical forces acting on the cylinder are caused by pressure and gravityF1 A p1F2 A p2p2 dp + p1F1 w + F2w A dz g

3.4

Eliminate the forces, pressures and area from the above and solve for the change in pressure, dp

-dz g w dp wdp -dz g 3.5We can integrate this expression over a distance between z1 and z2

p z1

z2-g zp -g (-z1 + z2) 3.6This is sometimes known as hydrostatic pressure.


Pascals ParadoxThe pressure in a liquid system is not dependent on the volume of fluid, but only the elevation (relative to the free surface).The pressure at each elevation of all volumes is the same. However, the forces are not the same.

Useful in, e.g. a water tower:

Pascals Paradox

http://www.howstuffworks.com/water.htm Fig 3.4


ManometerA manometer measures pressure. Hydrostatic manometers use the relationship between change in pressure and change in elevation in a static liquid.The following figure shows the U-tube manometer

Each discrete volume of fluid with the same density can be solved as a separate problem, and the results joined together. This also applies to changes in atmospheric pressure.


Manometer EquationThe deflection of a manometer is given by the static head difference around the manometer tube:

We can write the relationships between pressures:

pB g h1 + pApC pBpD pC - g h2 man 4.1Eliminating intermediate pressures:

-g h1 + pD + g h2 man pApA - pD -g h1 + g h2 manpA - pD g (- h1 + h2 man) 4.2


Manomater Tube DiametersEquation (4.2) relates the relative displacements h1 and h2. In industrial processes, it is advanta-geous to only have to read one leg of the manometer. This can be accomplished by making the other leg with a much bigger diameter.


Exercise: Manometer

What is the gage pressure at point A?

Answer

Equations relating pressure (note the signs!)

pB 0.25 g HgpA pB - 0.4 g H2O 4.1Eliminate pB and solve for pA


g (5. Hg - 8. H2O) 20. pApA 0.05 g 5. Hg - 8. H2O 4.2Substitute densities (Hg=13540){Hg 13540, H2O 1000, g 9.81}pA 29282.9


Pressure Transmission

Pascals PrincipleA change in the pressure applied to an enclosed, incompressible fluid is transmitted undiminished to every point in the fluid.

Ballast is added to the top of the piston to pressurise the fluid inside. If more ballast is added, the pressure increase will be the same at all points.

p0 g h + pextp1 p + g h + pextp1 p + p0


BarometerA barometer is a manometer for measuring atmospheric pressure.The reference pressure is a near-perfect vacuum. (I.e. a barometer measures absolute pressure)

Near perfect vacuum

patm = g h = hh

Bert Bolle water barometer (was largest barometer in the world, located in Denmark, Western Australia)


Force TransmissionA hydraulic arrangement can be used to provide mechanical advantage.It can magnify a force.

For the piston arrangment shown above, assuming both pistons are at the same elevation:

P1 P2Fo P2 AoFi P1 Ai 5.1Eliminate pressures and solve for output force

Ai Fo Ao FiFo Ao FiAi 5.2Note, however, that the work done is not changed.

Starting from the work done

Wi di FiWo do Fo 5.3Substitute using (5.2)

Wi di FiWo Ao do FiAi 5.4Now we know the same volume of fluid is displaced at both pistons.

Ao do Ai di 5.5


Making this substitution into (5.4)

Wi di FiWo di Fi 5.6Showing that work in and out are the same.


Exercise: Force of fluidConsider the following containers

What is the force created by the fluid on the bottom of each drum?

We can use the hydrostatic pressure equation

poil g hoil oilpwater poil + g hwater water 5.11In terms of specific gravity

oil sg waterpoil g sg hoil water

pwater poil + g hwater water 5.12We can eliminate poil and solve for pwater

g (sg hoil + hwater) water pwaterpwater g (sg hoil + hwater) water 5.13Substituting known values

{g 9.81, water 1000, sg 0.9, hoil 2.4, hwater 1.5}pwater 35904.6 5.14Calculating force


fwater A pwaterA D2

4, D 3

fwater 9 pwater4fwater 253795.

What is the weight of the fluid in the cylindrical drum? What is the unit?

Weight is given by volume times density:

A D2 4Vwater A hwaterVoil A hoil

mwater Vwater watermoil Voil oil

wtotal g moil + g mwateroil sg water5.15

Eliminating variables not of interest and solving for total weight

D2 g (sg hoil + hwater) water 4 wtotalwtotal 14 D2 g (sg hoil + hwater) water 5.16Substituting known values

{D 3, g 9.81, water 1000, sg 0.9, hoil 2.4, hwater 1.5}wtotal 253795.Note that the weight force is the same as the pressure force, for the cylindrical drum.What about the tapered drum? How do you explain this?


1 Fluid Statics - Fluid Properties

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