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PRESSURE & FLUID STATICS

PRESSURE & FLUID STATICS. Objectives. Review concepts of pressure, and absolute and gage pressure. Calculate the forces exerted by a fluid at rest on plane or curved submerged surfaces . Analyze the stability of floating and submerged bodies. 68 kg. 136 kg. A feet =300cm 2. 0.23 kgf/cm 2.

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PRESSURE & FLUID STATICS

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  1. PRESSURE &FLUID STATICS

  2. Objectives • Review concepts of pressure, and absolute and gage pressure. • Calculate the forces exerted by a fluid at rest on plane orcurved submerged surfaces. • Analyze the stability of floating and submerged bodies.

  3. 68 kg 136 kg Afeet=300cm2 0.23 kgf/cm2 0.46 kgf/cm2 P=68/300=0.23 kgf/cm2 PRESSURE Pressure: A normal force exerted by a fluid per unit area The normal stress (or “pressure”) on the feet of a chubby person is much greater than on the feet of a slim person. Some basic pressure gages.

  4. Absolute pressure: The actual pressure at a given position. It is measured relative to absolute vacuum (i.e., absolute zero pressure). • Gage pressure: The difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure. • Vacuum pressures: Pressures below atmospheric pressure. Throughout this text, the pressure P will denote absolute pressure unless specified otherwise.

  5. Example 1 A vacuum gage connected to a chamber reads 40 kPa at a location where the atmospheric pressure is 100kPa. Determine the absolute pressure in the chamber.

  6. Variation of Pressure with Depth When the variation of density with elevation is known The pressure of a fluid at rest increases with depth (as a result of added weight). Free-body diagram of a rectangular fluid element in equilibrium.

  7. In a room filled with a gas, the variation of pressure with height is negligible. Pressure in a liquid at rest increases linearly with distance from the free surface. The pressure is the same at all points on a horizontal plane in a given fluid regardless of geometry, provided that the points are interconnected by the same fluid.

  8. Pascal’s law: The pressure applied to a confined fluid increases the pressure throughout by the same amount. The area ratio A2/A1 is called the ideal mechanical advantageof the hydraulic lift. Lifting of a large weight by a small force by the application of Pascal’s law.

  9. The Manometer It is commonly used to measure small and moderate pressure differences. A manometer contains one or more fluids such as mercury, water, alcohol, or oil. Measuring the pressure drop across a flow section or a flow device by a differential manometer. The basic manometer. In stacked-up fluid layers, the pressure change across a fluid layer of density  and height h is gh.

  10. Other Pressure Measurement Devices • Bourdon tube: Consists of a hollow metal tube bent like a hook whose end is closed and connected to a dial indicator needle. • Pressure transducers: Use various techniques to convert the pressure effect to an electrical effect such as a change in voltage, resistance, or capacitance. • Pressure transducers are smaller and faster, and they can be more sensitive, reliable, and precise than their mechanical counterparts. • Strain-gage pressure transducers: Work by having a diaphragm deflect between two chambers open to the pressure inputs. • Piezoelectric transducers: Also called solid-state pressure transducers, work on the principle that an electric potential is generated in a crystalline substance when it is subjected to mechanical pressure. Various types of Bourdon tubes used to measure pressure.

  11. THE BAROMETER AND ATMOSPHERIC PRESSURE • Atmospheric pressure is measured by a device called a barometer; thus, the atmospheric pressure is often referred to as the barometric pressure. • A frequently used pressure unit is the standard atmosphere, which is defined as the pressure produced by a column of mercury 760 mm in height at 0°C (Hg = 13,595 kg/m3) under standard gravitational acceleration (g = 9.807 m/s2). The length or the cross-sectional area of the tube has no effect on the height of the fluid column of a barometer, provided that the tube diameter is large enough to avoid surface tension (capillary) effects. The basic barometer.

  12. Example 2 The gage pressure in a liquid at a depth of 3 m is read to be 28 kPa. Determine the gage pressure in the same liquid at a depth of 9 m.

  13. Example 3 The absolute pressure in water at depth of 5 m is read to be 145 kPa. Determine a) the local atmospheric pressure. b) the absolute pressure at a depth of 5 m if a liquid whose specific gravity is 0.85 at the same location.

  14. Example 4 (Manometer) A manometer is used to measure the pressure in a tank. The fluid used has a specific gravity of 0.85 and the manometer column height is 55cm. If the local atmospheric pressure is 96 kPa, determine the absolute pressure within the tank.

  15. Example 5 (Multi-fluid Manometer) The water in a tank is pressurized by air, and the pressure is measured by a multi-fluid manometer. The tank is located on a mountain at an altitude of 1400m where the atmospheric pressure is 85.6 kPa. Determine the air pressure in the tank if h1=0.1m, h2=0.2m and h3=0.35m. Take the densities of water, oil and mercury to be 1000kg/m3, 850kg/m3 and 13,600kg/m3 respectively.

  16. Fluid Statics • Fluid statics:Deals with problems associated with fluids at rest. • The fluidcan be either gaseous or liquid. • Fluid statics is generally referred to ashydrostaticswhen the fluid is a liquid and as aerostaticswhen the fluid is agas. • In fluid statics, there is no relative motion between adjacent fluid layers,and thus there are no shear (tangential) stresses in the fluid trying todeform it. • The only stress we deal with in fluid statics is the normal stress,which is the pressure, and the variation of pressure is due only to the weightof the fluid. • The topic of fluid statics has significance only ingravity fields. • The design of many engineering systems such as water damsand liquid storage tanks requires the determination of the forces acting onthe surfaces using fluid statics.

  17. HYDROSTATIC FORCES ONSUBMERGED PLANE SURFACES A plate, such as a gate valve in a dam, the wall of a liquid storage tank, orthe hull of a ship at rest, is subjected tofluid pressure distributed over its surfacewhen exposed to a liquid. On a planesurface, the hydrostaticforces form a system of parallel forces,and we often need to determinethe magnitude of the force and its point ofapplication, which is called thecenter of pressure. Hoover Dam. When analyzing hydrostatic forces onsubmerged surfaces, the atmosphericpressure can be subtracted forsimplicity when it acts on bothsides of the structure.

  18. Hydrostatic force on an inclined plane surface completely submerged in a liquid. The pressure at the centroid of asurface is equivalent to the averagepressure on the surface.

  19. The resultant force acting on a planesurface is equal to the product of thepressure at the centroid of the surfaceand the surface area, and its line ofaction passes through the center ofpressure. second moment of area(area moment of inertia) about the x-axis.

  20. The centroid and the centroidal moments of inertia for some common geometries.

  21. Special Case: Submerged Rectangular Plate Hydrostatic force acting on the top surface of a submerged tiltedrectangular plate.

  22. Hydrostatic force acting on the top surface of a submerged vertical rectangular plate.

  23. Hydrostatic force acting on the top surface of a submerged horizontalrectangular plate.

  24. Example 6 A heavy car plunges into a lake during an accident and lands at the bottom of the lake on its wheels. The door is 1.2m high and 1m wide. The top edge of the door is 8m below the free surface of the water. Determine the hydrostatic force on the door and the location of the pressure center.

  25. Example 7 A rectangular gate with 3m high and 6m wide is hinged at the top edge at A and is restrained by a fixed ridge at B as shown in Figure 2. Determine: • The hydrostatic force exerted by the water on the gate AB. • The location of the pressure center.

  26. HYDROSTATIC FORCES ONSUBMERGED CURVED SURFACES Determination of the hydrostatic force acting on a submerged curved surface.

  27. The hydrostatic force acting on acircular surface always passesthrough the center of the circle sincethe pressure forces are normal to thesurface and they all pass throughthe center. When a curved surface is above theliquid, the weight of the liquid and thevertical component of the hydrostaticforce act in the opposite directions.

  28. Example 8 A 1m long solid cylinder of radius 0.8m hinged at point A is used as an automatic gate. When the water level reaches 5m, the gate opens by turning about the hinge at point A. Determine a)the hydrostatic force acting on the cylinder and its line of action when the gate opens b)the weight of the cylinder per m length of the cylinder.

  29. 45m 9m Example 9 • A semicircular tunnel with 9m diameter is to be built under a 45m deep, 240m long lake as shown. Determine the hydrostatic force exerted by the water on roof of the tunnel.

  30. BUOYANCY AND STABILITY Buoyant force:The upward force a fluid exerts on a body immersed in it. The buoyant force is caused by the increase of pressure with depth in a fluid. The buoyant force acting on the plate is equal to the weightof the liquid displaced by the plate. For a fluid with constant density, thebuoyant force is independent of the distance of the body from the free surface. It is also independent of the density of the solid body. A flat plate of uniform thickness hsubmerged in a liquid parallel to thefree surface.

  31. Archimedes’ principle:The buoyant force acting on a body immersed in a fluid is equal to the weightof the fluid displaced by the body, and it acts upward through the centroid ofthe displaced volume.

  32. For floating bodies, the weight of the entire body must be equal to the buoyant force, which is the weight of the fluid whose volume is equal to the volume of the submerged portion of the floating body: A solid body dropped into a fluid willsink, float, or remain at rest at any pointin the fluid, depending on its averagedensity relative to the density of thefluid.

  33. The altitude of a hot air balloon iscontrolled by the temperaturedifference between the air inside andoutside the balloon, since warm airis less dense than cold air. When theballoon is neither rising nor falling,the upward buoyant force exactlybalances the downward weight.

  34. Stability of Immersed and Floating Bodies Stability is easily understood byanalyzing a ball on the floor. For floating bodies such asships, stability is an importantconsideration for safety.

  35. A floatingbody possesses vertical stability, while an immersed neutrally buoyant bodyis neutrally stable since it does not return to its original position after adisturbance. An immersed neutrally buoyant bodyis (a) stable if the center of gravityG is directly below the center ofbuoyancy B of the body, (b) neutrallystable if G and B are coincident, and(c) unstable if G is directly above B.

  36. When the center of gravity G of animmersed neutrally buoyant body isnot vertically aligned with the centerof buoyancy B of the body, it is not inan equilibrium state and would rotateto its stable state, even without anydisturbance.

  37. A floating body is stable if the body isbottom-heavy and thus the center ofgravity G is below the centroid B ofthe body, or if the metacenter M isabove point G. However, the body isunstable if point M is below point G. Metacentricheight GM:The distance between the center of gravity G and themetacenter M—the intersection point of the lines of action of the buoyantforce through the body before and after rotation. The length of the metacentric height GM aboveG is a measure of thestability: the larger it is, the more stable is the floatingbody.

  38. Example 10 A crane is used to lower weights into the sea (density = 1025kg/m3) for an underwater construction project. Determine the tension in the rope of the crane due to the rectangular 0.4m x 0.4m x 3m concrete block (density = 2300kg/m3) when it is (a)suspended in the air (b)completely immersed in water.

  39. FB Wload Wboat Example 11 The hull of a boat has a volume of 150 m3, and the total mass of the boat when empty is 8560 kg. Determine how much load this boat can carry without sinking (a)in a lake and (b)in sea water with a specific gravity of 1.03.

  40. Summary • Pressure • Variation of pressure with depth • The manometer and the atmospheric pressure • Hydrostatic Forces on Submerged PlaneSurfaces • Special Case: Submerged Rectangular Plate • Hydrostatic Forces on Submerged CurvedSurfaces • Buoyancy and Stability • Stability of Immersed and Floating Bodies

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