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Fluid Statics

Definition of Pressure
Pressure is defined as the amount of force exerted on a unit area of a substance:

Direction of fluid pressure on boundaries
Furnace duct Pipe or tube Heat exchanger Pressure is a Normal Force (acts perpendicular to surfaces) It is also called a Surface Force Dam

Absolute and Gauge Pressure
Absolute pressure: The pressure of a fluid is expressed relative to that of vacuum (=0) Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere. Usual pressure guages record guage pressure. To calculate absolute pressure:

Definition or Relationship
Units for Pressure Unit Definition or Relationship 1 pascal (Pa) 1 kg m-1 s-2 1 bar 1 x 105 Pa 1 atmosphere (atm) 101,325 Pa 1 torr 1 / 760 atm 760 mm Hg 1 atm pounds per sq. in. (psi)

Pressure distribution for a fluid at rest
Let’s determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity The sum of the forces acting on the fluid must equal zero

What are the z-direction forces?
Let Pz and Pz+Dz denote the pressures at the base and top of the cube, where the elevations are z and z+Dz respectively. z y x

Pressure distribution for a fluid at rest
A force balance in the z direction gives: For an infinitesimal element (Dz0)

Incompressible fluid Liquids are incompressible i.e. their density is assumed to be constant: When we have a liquid with a free surface the pressure P at any depth below the free surface is: Po is the pressure at the free surface (Po=Patm) By using gauge pressures we can simply write:

Example In deep water oil fields offshore Louisiana one occasionally encounters a reservoir pressure of 10,000 psig at a depth of 15,000 ft. If this pressure is greater than the hydrostatic pressure of the drilling fluid in the well bore the result may be a well “blowout” which is dangerous to life and property. Assuming that you are responsible for selecting the drilling fluid for an area where such pressures are expected, what is the minimum density drilling fluid you can use ? Assume a surface pressure of 0 psig.

Static pressure - 10 minute problem:
Rainwater collects behind the concrete retaining wall shown below. If the water-saturated soil (specific gravity = 2.2) acts as a fluid, determine the force on a one-meter width of wall. Water 1m Soil 3m

Buoyancy A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces A floating body displaces its own weight in the fluid in which it floats Free liquid surface F1 The upper surface of the body is subjected to a smaller force than the lower surface  A net force is acting upwards h1 H h2 F2

Buoyancy The net force due to pressure in the vertical direction is:
FB = F2- F1 = (Pbottom - Ptop) (DxDy) The pressure difference is: Pbottom – Ptop = r g (h2-h1) = r g H Combining: FB = r g H (DxDy) Thus the buoyant force is: FB = r g V

Practical Application of Buoyancy
Nimitz Class Aircraft Carrier (100,000 tons)

Measurement of Pressure
Manometers are devices in which one or more columns of a liquid are used to determine the pressure difference between two points. U-tube manometer Inclined-tube manometer

Measurement of Pressure Differences
Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.

Manometer - 10 minute problem
A simple U-tube manometer is installed across an orifice plate. The manometer is filled with mercury (specific gravity = 13.6) and the liquid above the mercury is water. If the pressure difference across the orifice is 24 psi, what is the height difference (reading) on the manometer in inches of mercury ?

Compressible Flow Tall Mountains Natural gas well

Compressible fluid Gases are compressible i.e. their density varies with temperature and pressure r =P M /RT For small elevation changes (as in engineering applications, tanks, pipes etc) we can neglect the effect of elevation on pressure In the general case start from:

Compressible Linear Temperature Gradient

Atmospheric Equations
Assume constant Assume linear Temperature variation with altitude for the U.S. standard atmosphere

Compressible Isentropic

Compressible Fluid – 10 minute problem
The temperature of the earth’s surface drops about 5 C for every 1000 m of elevation above the earth’s surface. If the air temperature at ground level is 15 C and the pressure is 760 mm Hg, what is the air pressure on top of Mt. Everest at 8847 m ? Assume air behaves as an ideal gas.

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