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Chapter 3: Pressure and Fluid Statics

University of Palestine College of Engineering & Urban Planning Applied Civil Engineering. Chapter 3: Pressure and Fluid Statics. Lecturer: Eng. Eman Al.Swaity. Fall 2009. FLUID STATICS.

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Chapter 3: Pressure and Fluid Statics

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  1. University of Palestine College of Engineering & Urban Planning Applied Civil Engineering Chapter 3: Pressure and Fluid Statics Lecturer: Eng. Eman Al.Swaity Fall 2009

  2. FLUID STATICS • Hydrostatics is the study of pressures throughout a fluid at rest and the pressure forces on finite surfaces. As the fluid is at rest, there are no shear stresses in it. Hence the pressure at a point on a plane surface always acts normal to the surface, and all forces are independent of viscosity. • The pressure variation is due only to the weight of the fluid. As a result, the controlling laws are relatively simple, and analysis is based on a straightforward application of the mechanical principles of force and moment. Solutions are exact and there is no need to have recourse to experiment. EGGD3109 Fluid Mechanics

  3. Pressure • Pressure is defined as a normal force exerted by a fluid per unit area(even imaginary surfaces as in a control volume). • Units of pressure are N/m2, which is called a pascal (Pa). • Since the unit Pa is too small for pressures encountered in practice, kilopascal (1 kPa = 103 Pa) and megapascal (1 MPa = 106 Pa) are commonly used. [ML-1T-2] • Other units include bar, atm, kgf/cm2, lbf/in2=psi. EGGD3109 Fluid Mechanics

  4. Pressure • 1 bar = 105 Pa = 0.1 MPa = 100 kPa • 1 atm = 101,325 Pa = 101.325 kPa = 1.01325 bars • 1 kgf/cm2 = 9.807 N/cm2 = 9.807  104 N/m2 = 9.807  104 Pa = 0.9807 bar = 0.9679 atm • 1 atm = 14.696 psi. • 1 kgf/cm2 = 14.223 psi. EGGD3109 Fluid Mechanics

  5. Pressure at a Point By considering the equilibrium of a small triangular wedge of fluid extracted from a static fluid body, one can show that for any wedge angle θ, the pressures on the three faces of the wedge are equal in magnitude: EGGD3109 Fluid Mechanics

  6. Pressure at a Point • Pressure at any point in a fluid is the same in all directions. • Pressure has a magnitude, but not a specific direction, and thus it is a scalar quantity. • Proof on the Blackboard EGGD3109 Fluid Mechanics

  7. Pressure at a Point EGGD3109 Fluid Mechanics

  8. Pressure at a Point • This result is known as Pascal's law, which states that the pressure at a point in a fluid at rest, or in motion, is independent of direction as long as there are no shear stresses present. • Pressure at a point has the same magnitude in all directions, and is called isotropic . EGGD3109 Fluid Mechanics

  9. Variation of Pressure with Depth Therefore, the hydrostatic pressure increases linearly with depth at the rate of the specific weight of the fluid. EGGD3109 Fluid Mechanics

  10. Variation of Pressure with Depth • In the presence of a gravitational field, pressure increases with depth because more fluid rests on deeper layers. • To obtain a relation for the variation of pressure with depth, consider rectangular element • Force balance in z-direction gives • Dividing by Dx and rearranging gives • ∆z is called the pressure head EGGD3109 Fluid Mechanics

  11. Variation of Pressure with Depth EGGD3109 Fluid Mechanics

  12. Variation of Pressure with Depth • Pressure in a fluid at rest is independent of the shape of the container. • Pressure is the same at all points on a horizontal plane in a given fluid. EGGD3109 Fluid Mechanics

  13. Absolute, gage, and vacuum pressures • Actual pressure at a given point is called the absolute pressure. • Most pressure-measuring devices are calibrated to read zero in the atmosphere, and therefore indicate gage pressure, Pgage = Pabs - Patm. • Pressure below atmospheric pressure are called vacuum pressure, Pvac=Patm - Pabs. EGGD3109 Fluid Mechanics

  14. Absolute, gage, and vacuum pressures EGGD3109 Fluid Mechanics

  15. Hydrostatic Pressure Difference Between Two Points • For a fluid with constant density, • If you can draw a continuous line through the same fluid from point 1 to point 2, then p1 = p2 if z1 = z2. • By this rule p1 = p2 and p4 = p5 • p2 does not equal p3 even though they are at the same elevation, because one cannot draw a line connecting these points through the same fluid. In fact, p2 is less than p3 since mercury is denser than water. EGGD3109 Fluid Mechanics

  16. Hydrostatic Pressure Difference Between Two Points Any free surface open to the atmosphere has atmospheric pressure, p0. The shape of a container does not matter in hydrostatics EGGD3109 Fluid Mechanics

  17. Hydrostatic Pressure Difference Between Two Points Pressure in layered fluid. EGGD3109 Fluid Mechanics

  18. Pascal’s Law • Two points at the same elevation in a continuous fluid at rest are at the same pressure, called Pascal’s law, • Pressure applied to a confined fluid increases the pressure throughout by the same amount. • In picture, pistons are at same height: • Ratio A2/A1 is called ideal mechanical advantage EGGD3109 Fluid Mechanics

  19. Pascal’s Law EGGD3109 Fluid Mechanics

  20. Pascal’s Law EGGD3109 Fluid Mechanics

  21. Pressure Measurement and Manometers Piezometer tube The simplest manometer is a tube, open at the top, which is attached to a vessel or a pipe containing liquid at a pressure (higher than atmospheric) to be measured. This simple device is known as a piezometer tube. This method can only be used for liquids (i.e. not for gases) and only when the liquid height is convenient to measure. It must not be too small or too large and pressure changes must be detectable. EGGD3109 Fluid Mechanics

  22. Pressure Measurement and Manometers U-tube manometer This device consists of a glass tube bent into the shape of a "U", and is used to measure some unknown pressure. For example, consider a U-tube manometer that is used to measure pressure pA in some kind of tank or machine. Finally, note that in many cases (such as with air pressure being measured by a mercury manometer), the density of manometer fluid 2 is much greater than that of fluid 1. In such cases, the last term on the right is sometimes neglected. EGGD3109 Fluid Mechanics

  23. Pressure Measurement and Manometers Differential manometer A differential manometer can be used to measure the difference in pressure between two containers or two points in the same system. Again, on equating the pressures at points labeled (2) and (3), we may get an expression for the pressure difference between A and B: EGGD3109 Fluid Mechanics

  24. Pressure Measurement and Manometers Inverted U-tube Differential manometers EGGD3109 Fluid Mechanics

  25. Pressure Measurement and Manometers Inverted U-tube Differential manometers-Example EGGD3109 Fluid Mechanics

  26. Pressure Measurement and Manometers Inclined-tube manometer As shown above, the differential reading is proportional to the pressure difference. If the pressure difference is very small, the reading may be too small to be measured with good accuracy. To increase the sensitivity of the differential reading, one leg of the manometer can be inclined at an angle θ, and the differential reading is measured along the inclined tube. EGGD3109 Fluid Mechanics

  27. Pressure Measurement and Manometers The Manometer • An elevation change of Dz in a fluid at rest corresponds to DP/rg. • A device based on this is called a manometer. • A manometer consists of a U-tube containing one or more fluids such as mercury, water, alcohol, or oil. • Heavy fluids such as mercury are used if large pressure differences are anticipated. EGGD3109 Fluid Mechanics

  28. Pressure Measurement and Manometers Mutlifluid Manometer • For multi-fluid systems • Pressure change across a fluid column of height h is DP = rgh. • Pressure increases downward, and decreases upward. • Two points at the same elevation in a continuous fluid are at the same pressure. • Pressure can be determined by adding and subtracting rgh terms. EGGD3109 Fluid Mechanics

  29. Pressure Measurement and Manometers Example:U-tube manometer containing mercury was used to find the negative pressure in the pipe, containing water. The right limb was open to the atmosphere. Find the vacuum pressure in the pipe, if the difference of mercury level in the two limbs was 100 mm and height of water in the left limb from the centre of the pipe was found to be 40 mm below. EGGD3109 Fluid Mechanics

  30. Pressure Measurement and Manometers EGGD3109 Fluid Mechanics

  31. Pressure Measurement and Manometers General Example • The atmospheric pressure is 755 mm of mercury • (sp. Gravity = 13.6), calculate • Absolute pressure of air in the tank, • Pressure gauge reading at L. EGGD3109 Fluid Mechanics

  32. Measuring Pressure Drops • Manometers are well--suited to measure pressure drops across valves, pipes, heat exchangers, etc. • Relation for pressure drop P1-P2 is obtained by starting at point 1 and adding or subtracting rgh terms until we reach point 2. • If fluid in pipe is a gas, r2>>r1 and P1-P2r2gh (Mistyped on page 73) EGGD3109 Fluid Mechanics

  33. The Barometer • Atmospheric pressure is measured by a device called a barometer; thus, atmospheric pressure is often referred to as the barometric pressure. • PC can be taken to be zero since there is only Hg vapor above point C, and it is very low relative to Patm. • Change in atmospheric pressure due to elevation has many effects: Cooking, nose bleeds, engine performance, aircraft performance. EGGD3109 Fluid Mechanics

  34. The Barometer • Standard atmosphere is defined as the pressure produced by a column of mercury 760 mm (29.92 inHg or of water about 10.3 m ) in height at 0°C (rHg = 13,595 kg/m3) under standard gravitational acceleration (g = 9.807 m/s2). • 1 atm = 760 torr and 1 torr = 133.3 Pa EGGD3109 Fluid Mechanics

  35. Fluid Statics • Fluid Statics deals with problems associated with fluids at rest. • In fluid statics, there is no relative motion between adjacent fluid layers. • Therefore, there is no shear stress in the fluid trying to deform it. • The only stress in fluid statics is normal stress • Normal stress is due to pressure • Variation of pressure is due only to the weight of the fluid → fluid statics is only relevant in presence of gravity fields. • Applications: Floating or submerged bodies, water dams and gates, liquid storage tanks, etc. EGGD3109 Fluid Mechanics

  36. Hoover Dam EGGD3109 Fluid Mechanics

  37. Hoover Dam EGGD3109 Fluid Mechanics

  38. Hoover Dam • Example of elevation head z converted to velocity head V2/2g. We'll discuss this in more detail in Chapter 5 (Bernoulli equation). EGGD3109 Fluid Mechanics

  39. Pressure Distributions-Flat Surfaces EGGD3109 Fluid Mechanics

  40. Pressure Distributions-Flat Surfaces EGGD3109 Fluid Mechanics

  41. Pressure Distributions-Flat Surfaces EGGD3109 Fluid Mechanics

  42. Pressure Distributions-Curved Surfaces EGGD3109 Fluid Mechanics

  43. Hydrostatic Forces on Plane Surfaces • On a plane surface, the hydrostatic forces form a system of parallel forces • For many applications, magnitude and location of application, which is called center of pressure, must be determined. • Atmospheric pressure Patm can be neglected when it acts on both sides of the surface. EGGD3109 Fluid Mechanics

  44. Resultant Force The magnitude of FRacting on a plane surface of a completely submerged plate in a homogenous fluid is equal to the product of the pressure PC at the centroid of the surface and the area A of the surface EGGD3109 Fluid Mechanics

  45. Resultant Force EGGD3109 Fluid Mechanics

  46. Center of Pressure • Line of action of resultant force FR=PCA does not pass through the centroid of the surface. In general, it lies underneath where the pressure is higher. • Vertical location of Center of Pressure is determined by equation the moment of the resultant force to the moment of the distributed pressure force. • Ixx,C is tabulated for simple geometries. • Derivation of FR and examples on blackboard EGGD3109 Fluid Mechanics

  47. The centroidal moments of inertia for some common geometries EGGD3109 Fluid Mechanics

  48. Submerged Rectangular Plate What is the yp for case (a)? EGGD3109 Fluid Mechanics

  49. Submerged Rectangular Plate What is the yp for case (a)? EGGD3109 Fluid Mechanics

  50. Example: Hydrostatic Force Acting on the Door of a Submerged Car A heavy car plunges into a lake during an accident and lands at the bottom of the lake on its wheels. The door is 1.2 m high and 1 m wide, and the top edge of the door is 8 m below the free surface of the water. Determine the hydrostatic force on the door and the location of the pressure center, and discuss if the driver can open the door. Pave = PC =rghC =rg(s +b/2)= 84.4 kN/m2 FR =PaveA = (84.4 kNm2) (1 m  1.2 m) = 101.3 kN yP = 8.61 m EGGD3109 Fluid Mechanics

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