AKM 205 AKIŞKANLAR MEKANİĞİ Yrd.Doç.Dr. Onur Tunçer İstanbul Teknik Üniversitesi “AKIŞKAN STATİĞİ”

Main Topics The Basic Equations of Fluid Statics Pressure Variation in a Static Fluid Hydrostatic Force on Submerged Surfaces Buoyancy

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 ductPipe or tube Heat exchanger Dam Pressure is a Normal Force (acts perpendicular to surfaces) It is also called a Surface Force

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:

Units for Pressure UnitDefinition or Relationship 1 pascal (Pa)1 kg m -1 s -2 1 bar1 x 10 5 Pa 1 atmosphere (atm)101,325 Pa 1 torr1 / 760 atm 760 mm Hg1 atm pounds per sq. in. (psi) 1 atm

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

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

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

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

The Basic Equations of Fluid Statics Body Force

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Total Force

The Basic Equations of Fluid Statics Newton’s Second Law

The Basic Equations of Fluid Statics Pressure-Height Relation

Pressure Variation in a Static Fluid Incompressible Fluid: Manometers

Pressure Variation in a Static Fluid Compressible Fluid: Ideal Gas Need additional information, e.g., T(z) for atmosphere

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 P 2 =P 3.

Inclined Manometer To measure small pressure differences need to magnify R m some way.

Compressible Flow: Natural gas well Tall Mountains

Compressible Linear Temperature Gradient

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

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 The upper surface of the body is subjected to a smaller force than the lower surface  A net force is acting upwards F1F1 F2F2 h1h1 h2h2 H

Buoyancy The net force due to pressure in the vertical direction is: F B = F 2 - F 1 = (P bottom - P top ) (  x  y) The pressure difference is: P bottom – P top =  g (h 2 -h 1 ) =  g H Combining: F B =  g H (  x  y) Thus the buoyant force is: F B =  g V

Buoyancy

For example, for a hot air balloon (Example Problem 3.8):

Intoduction Chee Any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Archimedes of Syracuse (287 BC – 212 BC ) Archimedes’ Principle Any floating object displaces its own weight of fluid. Archimedes of Syracuse (287 BC – 212 BC)

Compressible fluid Gases are compressible i.e. their density varies with temperature and pressure  = 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:

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface We can find F R, and y´ and x´, by integrating, or …

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface –Algebraic Equations – Total Pressure Force

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface –Algebraic Equations – Net Pressure Force

Hydrostatic Force on Submerged Surfaces Curved Submerged Surface

Hydrostatic Force on Submerged Surfaces Curved Submerged Surface –Horizontal Force = Equivalent Vertical Plane Force –Vertical Force = Weight of Fluid Directly Above (+ Free Surface Pressure Force)