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Hydraulics Chapter - 2- ppt

Hydrostatics of fluids

Mekash
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Hydraulics Chapter - 2- ppt

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  1. CHAPTER TWO HYDROSTATICS OF FLUIDS 12/16/2022 Hydraulics 1

  2. 2. HYDROSTATICS • Hydrostatics is the study of fluid body in that are: At rest (no tangential force, ?=0) Move slowly with uniform velocity i.e. no relative motion b/n fluid element. • For fluid at rest or no relative motion The shear stress is zero b/n fluid element since it results from rate of deformation. The only force acting will be normal pressure force (that act perpendicular to the surface) The sum of the force in any direction zero and the sum of moments or forces about any point is zero. 12/16/2022 Hydraulics 2

  3. Cont.. • Engineering application of hydrostatics principles includes: Study of forces on submerged bodies such as dam faces, gates, submarines For analysis of stability of floating bodies e.g. Ships 12/16/2022 Hydraulics 3

  4. What is the other application area of hydrostatic force?  In addition, there is the various application of the hydrostatics • A Mercury barometer is used for measuring the local atmosphere by indicating the height raised of the mercury column. It works on the concept of hydrostatics. • The hydraulic system equipped in various automation machines (like used as hydraulic brakes in vehicles) is one of the best applications of hydrostatics. • The mechanical equipment present in the laboratories is equipped with a hydraulic system for the movement of objects (like the movement of the piston due to incompressible fluid in piston- cylinder assembly). 12/16/2022 Hydraulics 4

  5. Hydrostatic condition – Each fluid particle is in a force equilibrium with the net force due to pressure balancing the weight of fluid particle. Fluid pressure Fluid exerts a normal force on any boundary it is contact with. The force applied per unit area is the pressure. If the total force, F is uniformly distributed over an area A, then  P= F/A [N/m2], ML-1T-2 • If the force is not uniformly distributed, the expression of P will be the average value. 12/16/2022 Hydraulics 5

  6. Cont..  If the pressure varies from point to point, the pressure at any point is given by:  P = dF/dA  Where, dF = the force acting on an elementary area, dA.  Note:  A mass of fluid kept in a container/solid boundary exerts forces against the boundary surfaces.  The forces are always act in a direction normal to the surface. Because the fluid is at rest, there are no shear stresses in it. 12/16/2022 Hydraulics 6

  7. Principle of Hydrostatics • The principle of hydrostatics that works on the mechanics of fluid at rest is Pascal’s law. • Pascal's law is the fundamental principle that states that the pressure applied to the surface of the fluid is transmitted uniformly throughout the fluid in all directions. • Example of Pascal's law: when the force is applied on one piston, then the height of the other piston is raised due to the transmission of force through fluid that connects both of them. 12/16/2022 Hydraulics 7

  8. Pressure distribution PASCAL’s Law Pressure at point (Pascal law)  The pressure at a point in a fluid at rest is the same in all directions – Pascal’s Law.  Consider a finite but small element (small triangular prism) of a liquid at rest, acted up on by the fluid around it.  The values of average unit pressures on the three surfaces are P1, P2 and P3.  In the Z direction, the forces are equal and opposite and cancel each other. 12/16/2022 Hydraulics 8

  9. Basic equation of hydrostatics • The pressure variation throughout a fluid at rest can be obtained by applying Newton’s second law to a differential element such as shown below. • Consider a fluid element of rectangular parallelepiped shape below, the rate of change of pressure in all direction is the same. ? = ??????? 12/16/2022 Hydraulics 9

  10. Cont.. • For fluid element at rest, ΣFX = 0, ΣFy = 0, ΣFz = 0, the pressure force in the opposite vertical faces must be equal. • σ??= 0, ?????− ? +?? ???? ????= 0 • ?????− ?????-?? ????????= 0 −?? ????????= 0 ?? ??= 0………………. (1) • σ??= 0, ?????− ? +?? • • ???? ????= 0 • ?????− ?????-?? ????????= 0 −?? ????????= 0 ?? ??= 0…………….. (2), The above two equation shows that the pressure does not change or constant along horizontal plane. • • 12/16/2022 Hydraulics 10

  11. Cont.. • σ??= 0, ?????− ? +?? ???? ????− ???????= 0 • ?????− ?????−?? ????????− ???????= 0 ?? ????????= −??????? ?? ??= −? • This shows pressure is only function of z and this equation is basic equation of hydrostatics. • The negative sign indicates that as z gets higher up ward, the pressure gets smaller. • This equation can now be integrated to give the actual pressure variation in the vertical direction. • ׬?? = −? ׬?? • P + ?? = constant • • 12/16/2022 Hydraulics 11

  12. Cont.. • The constant of integration can be absorbed by integrating between two elevations z1and z2with corresponding pressure P1and P2, • ׬?? = −? ׬?? ?2?? = −? ׬?1 ?2?? • ׬?1 • ?2− ?1= −?(?2− ?1) • ?2− ?1= −?? • Since the pressure at the surface is atmospheric it can be taken to be zero gage pressure, so ?2= 0. • So the above equation becomes, ?1= ?? • From this, pressure is proportional to the depth below the free surface i.e. pressure at a point in a stationary liquid is the product of the depth of the point and the specific weight of the fluid. 12/16/2022 Hydraulics 12

  13. Cont.. Note  The pressure in a homogeneous, incompressible (?= constant) fluid at rest depends on the depth of the fluid relative to some reference plane, and it is not influenced by the size or shape of the container in which the fluid is held.  Pressure is the same at all points at given horizontal plane in the fluid.  Pressure increases linearly with depth in the fluid.  Pressure acts normal to any surface in static fluid. 12/16/2022 Hydraulics 13

  14. ??= ??= ??= ??and ??= ??= ??≠ ?? 12/16/2022 Hydraulics 14

  15. 12/16/2022 Hydraulics 15

  16. Cont.. • Example: pressure distribution along the edge of swimming pool 1. Along the bottom the pressure is constant due to constant depth 2. Along the vertical wall the pressure varies linearly with depth and acts horizontal direction 3. Along the sloped wall the pressure again varies linearly with depth but acts normal to the surface 4. At the junctions of the walls and the bottom the pressure is the same. 12/16/2022 Hydraulics 16

  17. Cont.. Pressure head (h) • It is the pressure difference between two points or it’s the depth of the fluid that would exert an equivalent pressure deference. • ?2− ?1= −?(?2− ?1), h = ?2− ?1 • ℎ =?1−?2 ? • E.g. pressure difference of 60,000pa means pressure head of 6.12 of water or 0.45m of mercury. 12/16/2022 Hydraulics 17

  18. Pressure measurement Atmospheric, Absolute and gage pressures • The pressure at a point within a fluid mass can be designated as either an absolute pressure or a gage pressure. • Depending on level/datum used, pressure is described by; 1. Atmospheric pressure (????)  Is a pressure at any point in the atmosphere caused by the weight of atmospheric air above the measurement point.  E.g. at the sea level the standard ????is 101.325KPa or 10.3m of water column or 760mm of mercury.  ℎ =???? ? 0.76? = 760?? , ℎ?=101325 101325 13600∗9.81= 9810= 10.3? or ℎ?= 12/16/2022 Hydraulics 18

  19. Cont.. 2. Gauge pressure Is a pressure measurement relative to atmospheric pressure i.e. pressure measurement above or below ????. If the pressure of a fluid is below atmospheric pressure it is called vacuum pressure/suction pressure/negative gauge pressure/. E.g. 10 pa vacuum pressure = 10 pa below ????= -10 pa gauge pressure = 91.3 KPa absolute pressure 3. Absolute pressure (????) Is pressure measurement with respect to absolute vacuum/zero pressure/. ????= ????+ ?????? 12/16/2022 Hydraulics 19

  20. Cont.. 12/16/2022 Hydraulics 20

  21. Cont.. • Devices used for pressure measurement either of the following principles; 1. By balancing the column of liquid (whose pressure is to be found) by the same or another column of liquid, (known as tube gauges) e.g. Manometer, piezometer 2. By balancing column of liquid by spring or dead weight, (known as mechanical gauges) 12/16/2022 Hydraulics 21

  22. The following devices are used in pressure measurement; 1. Mercury barometer  It is instrument used to measure atmospheric pressure.  It consists of a glass tube closed at one end and the other open end immersed in in a container of mercury.  The column of mercury come to equilibrium position where it’s weight and vapor pressure balances ????. 12/16/2022 Hydraulics 22

  23. Cont.. • ????= ?? • ??= ??+ ???ℎ ,??=???????= 0 • ??= ???ℎ • ????= ?? • Limitation of barometer is it is widely used to measure atmospheric pressure. 12/16/2022 Hydraulics 23

  24. 2. Manometer  It is pressure measuring techniques in which a column of liquid in vertical or inclined tubes containing one or more liquid of different specific gravity (S) is used.  Measures pressure by raising and lowering column of liquid in manometer.  In using manometer, generally a known pressure (which may be atmospheric) is applied to one end of manometer tube and the unknown pressure to be determined is applied to the other end. 12/16/2022 Hydraulics 24

  25. 12/16/2022 Hydraulics 25

  26. Cont.. • The common types of manometer includes; A. Piezometer tube B. U-tube manometer C. Micro manometer D. Inclined-tube manometer A. Piezometer It is simple type of manometer consisting of vertical tube open at the top This tube is attached to the container in which pressure is desired to be determined. The height of fluid in the tube gives the difference between pressure in the container and atmosphere. Pressure is measured relative to ????, so its gauge pressure. For fluid at rest pressure will increase as we move downward and decrease as we move upward. 12/16/2022 Hydraulics 26

  27. Cont.. • ??= ?1= ?ℎ + ????, ????=0 • ??= ?1= ?ℎ https://www.youtube.com/watch?v=SR9rxSgLQEQ Disadvantage of piezometer • Fluid in container needs to be liquid rather than gas. • The pressure in the container has to be greater than ????, otherwise air would be sucked to the system (vacuum/suction pressure). ??????????> ???? • Not practical for large pressure /column of liquid becomes too large/, vacuum pressure (????< ????) and small diameter tube. 12/16/2022 Hydraulics 27

  28. Cont.. B. U-tube manometer  Used to measure pressure relative to the atmospheric pressure or pressure difference between two points.  The bottom of U-tube partially filled with other fluid called gage fluid or monomeric fluid (different fluid which do net mix) is used  One end of the tube is connected and the other end is open to atmosphere. Hydraulics 12/16/2022 28

  29. To determine pressure we use step by step (SS) procedure a. Start at one end and write pressure there b. Add the change in pressure there  Positive downward  Negative upward c. Continue until the other end of the gage and equate the pressure at that point Hydraulics 12/16/2022 29

  30. Cont.. • ??= ?1and ?2= ?3 • ?2= ?3= ??+ ?1ℎ1 • ??+ ?1ℎ1− ?2ℎ2= ????= 0, ????=0 • ??= ?2ℎ2− ?1ℎ1 • To measure larger pressure differences we can choose a manometer with higher density, and to measure smaller pressure differences with accuracy we can choose a manometer fluid which is having a density closer to the fluid density. https://www.youtube.com/watch?v=MpXhXVF9-HM 12/16/2022 Hydraulics 30

  31. Cont.. Negative pressure • Positive pressure ??= ?2ℎ2− ?1ℎ1 ??= −?1ℎ1− ?2ℎ1 12/16/2022 Hydraulics 31

  32. Cont.. • A U-tube manometer called differential manometer is used to measure the difference in pressure between two container of two points in the system when the actual pressure at any point in the system cannot be determined. • ??= ?1&?2= ?3&?5= ?? • ??+ ?1ℎ1− ?2ℎ2− ?3ℎ3= ?? • ??− ??= ?2ℎ2+ ?3ℎ3+ ?1ℎ1= 12/16/2022 Hydraulics 32

  33. Cont.. C. Inclined manometer  Is used to measure small pressure changes (for very small pressure).  ??+ ?1ℎ1− ?2ℎ2− ?3ℎ3= ??, but ℎ2= ?2????  ??− ??= ?2?2???? + ?3ℎ3− ?1ℎ1 12/16/2022 Hydraulics 33

  34. Cont.. Disadvantage of manometer i. Not used conveniently to measure large pressure differences ii. Some liquids are difficult/unsuitable/ for use because they do not form well defined menisci. iii. Surface tension can cause errors due to capillary rise (avoided by providing large diameter tube, >15mm) iv. Slow response which makes it unsuitable for measuring fluctuating pressures. 12/16/2022 Hydraulics 34

  35. Cont.. 3. Mechanical and electronic pressure measuring devices  Although manometer are widely used, they are not well suited for measuring very high pressures, pressure that are changing with time and it is time consuming.  To overcome this problem mechanical/electronic pressure measuring devices are developed.  This type of gauge can be used to measure a negative pressure (vacuum) as well as positive pressure. E.g. Bourdon gauge 12/16/2022 Hydraulics 35

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  38. Hydrostatic force on plane and curved surfaces • When a body is submerged in a fluid, forces develop on the surface due to the fluid. • For design of different hydraulic structures (dams, storage tanks, ships etc.) determination of these forces; their magnitude, direction and point of application is very important. 12/16/2022 Hydraulics 38

  39. Cont. • Hydrostatic force- is force due to the pressure of a fluid at rest. • For fluid at rest – – We know that the force must be perpendicular to the surface since there are no shearing stresses present. – We also know that the pressure will vary linearly with depth if the fluid is incompressible. 12/16/2022 Hydraulics 39

  40. Cont.. • The fluid may be in contact with – Plane surface (horizontal, vertical and inclined) – Curved surface 1. Forces on plane surface i. Horizontal plane surface A plane surface in horizontal position in a fluid at rest is subjected to constant pressure, Since every point on the surface is at the same depth from free surface. 40 12/16/2022 Hydraulics

  41. Cont.. • The pressure force acting on elemental area ?? on one side is; dF = PdA = γhdA • The resultant force acting on the surface is found by summing contribution of infinitesimal forces over the entire area, • To find line of action of resultant equate the moment of resultant to moment of distributed force system about any axis (X and Y axis). Take Y-axis 41 Hydraulics 12/16/2022

  42. Cont.. Taking X-axis ???and ???are distance from Y and X axis to resultant and ҧ ? and ത ? are distance to centroid. (for horizontal plane, തℎ=h i.e. c.g. coincides with c.p.), and Xcp,Ycp = ത x, ത y • So, for horizontal area subjected to static fluid pressure, resultant force passes through centroid of the area. And the magnitude of resultant hydrostatic force is product of pressure at the centroid of the area and area of the surface; ? = ?? . 12/16/2022 Hydraulics 42

  43. Cont.. ii. Vertical plane surface • Consider a plane vertical surface of area A immersed vertically in a liquid. • Since the depth from free surface to the various points on the surface varies, the pressure intensity on the surface is not constant and varies directly with depth. • Force on the elemental area dA is, dF = PdA = γydA • Total pressure force on the one side of the whole area dA is; 12/16/2022 Hydraulics 43

  44. Cont.. ??= γAത y • The position of CP can be determined by equating moment of resultant to moment of distributed force about any arbitrary axis; Determine ???; Mresultant force= Mdistributed force 12/16/2022 Hydraulics 44

  45. Cont.. Io Aഥ y, YCP= From parallel axis theorem, Io= Icg+ Aത y2 YCP=Icg+ Aത y2 Aത y Icg Aഥ y→ this shows CP lies below cg. YCP= ത y + For vertical plane surface the center of pressure does not coincide with centroid of the area, since pressure increases with depth CP lies below the centroid of the surface area. (No uniform P distn.) Determine ???; Mresultant= Mdistributed γAത y ∗ XCP= γ ∗ IXY From parallel axis theorem, IXY= ҧIXY+ ത xത yA XCP=IXY Aത y= ҧIXY+ ത xത yA Aത y ҧIXY Aത y = ത x + Hydraulics 12/16/2022 45

  46. Cont.. 12/16/2022 Hydraulics 46

  47. Cont.. iii. Inclined plane surface • Consider a body immersed in homogenous fluid which is inclined at some angle ? to free surface 12/16/2022 Hydraulics 47

  48. Cont.. Force on the elemental area dA is, dF = PdA = γhdA h = ysin θ, so dF = PdA = γhdA = γysin θdA Total pressure force on the one side of the whole area dA is; FR= ?????ത ?? = ?ത ??????, തℎ = ത ????? • FR= ?തℎ? • The net resultant force is the area multiplied by pressure at the centroid. 12/16/2022 Hydraulics 48

  49. Cont.. Center of pressure (Xcp,Ycp), can be determined as follows; Mresultant Force= Mdistributed Force Take moment about X-axis Io Aഥ y, YCP= From parallel axis theorem, Io= Icg+ Aത y2 Icg+Aഥ y2 Aഥ y e = eccentricity, cg. to cp., Io= centroidal moment of inertia, A = area of the plane surface, ത y = distance of cp from free surface along the axis of the plane surface. Icg Aഥ y= ത y +e • YCP= = ത y + 12/16/2022 Hydraulics 49

  50. Cont.. But, ത y = തh sinθ+ ഥh Icg Aഥ y hcp sinθ= hcp sinθ sinθand ycp= Icg തh sinθ) YCP= ത y + A( Icg∗sin2θ Aഥh hCP=തh + • This is the general equation for the depth of center of pressure for bodies immersed in the homogenous liquid weather it is horizontal, vertical or inclined. • For horizontal surface θ = 00, and ҧIXY Aഥ y for vertical surface θ = 900, XCP= ത x + • For plane surface with symmetry about an axis normal to free surface, ҧIXY= 0 and XCP= ത x. 12/16/2022 Hydraulics 50

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