AKM 205 AKIKANLAR MEKAN AKIKAN STAT Yrd Do

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

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

Definition of 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. Ø Usual pressure guages record guage pressure. To calculate absolute pressure:

Units for Pressure Unit 1 pascal (Pa) Definition or Relationship 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 14. 696 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

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:

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 Rm some way.

Compressible Flow: Tall Mountains Natural gas well

Compressible Linear Temperature Gradient

Atmospheric Equations • Assume constant • Assume linear 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 The upper surface of the body is subjected to a smaller force than the lower surface h 1 A net force is acting upwards Free liquid surface F 1 H h 2 F 2

Buoyancy The net force due to pressure in the vertical direction is: FB = F 2 - F 1 = (Pbottom - Ptop) (Dx. Dy) The pressure difference is: Pbottom – Ptop = r g (h 2 -h 1) = r g H Combining: FB = r g H (Dx. Dy) Thus the buoyant force is: FB = r g V

Buoyancy

Buoyancy For example, for a hot air balloon (Example Problem 3. 8):

Archimedes’ Principle • Any floating object displaces its own weight of fluid. Archimedes of Syracuse (287 BC – 212 BC) • 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 ) Intoduction Chee 223 1. 30

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:

Hydrostatic Force on Submerged Surfaces • Plane Submerged Surface

Hydrostatic Force on Submerged Surfaces • Plane Submerged Surface We can find FR, 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)