1. Pressure & it’s Measurement

2. Pressure & it’s Measurement  Pressure & Measurement -Pascal’s law -Piezo-meter & Manometer -Atmospheric - Absolute - Gauge - Negative  Buoyancy force -Meta Center - Equilibrium

3. Section I

4. Pressure: It is define as the ration of Force to area unit: kgf/m 2, kgf/cm 2, N/m 2, N/mm 2 1 pa = 1 N/m 2 Pascal’s Law: p x. dy. 1 p y. dx. 1 p z. ds. 1 dy dx ds B A C θ θ px, py and pz are the pressure or intensity of pressure acting on the face AB, AC and BC. Let ∟ABC = θ

5. Then the forces acting on the element are: 1.Pressure force normal to the surfaces. 2.Wight of element in the vertical direction. Force on the face AB = px. Area of the face AB = px. dy. 1 face AC = py. dx. 1 face BC = pz. ds. 1 Weight of element Resolving the forces in x-direction, we have (1) p x. dy. 1 p y. dx. 1 p z. ds. 1 dydy dxdx dsds B A C θ θ

6. Similarly, resolving the force in y-direction we get But and also elements is very small and hence weight is negligible. (2) 1 & 2

7. Pressure variation in a fluid at rest: (Hydrostatic Law) Z A C D B Z ΔZΔZ pΔApΔA P+ dp/dz. ΔZ) ΔA Hydrostatic Law: the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at That point. ΔA = Area of Element ΔZ = Height of fluid element p = Pressure on face AB Z = Dis. Of fluid element from free Surface

8. Then the forces acting on the element are: 1.Pressure force on AB = p. ΔA 2. Pressure force on CD = 3. Weight of fluid element 4. Pressure force on surface BC and AD are equal and opposite For equilibrium of fluid elements,

9. Ex 1 A hydraulic pressure has a ram of 30 cm diameter And a plunger of 4.5 cm diameter. Find the weight lifted by the hydraulic press when the force applied At the plunger is 500N. Ans: Pressure = 314465.4 N/m 2, W = 22222N

10. Ex 2 Calculate the pressure due to a column of 0.25 m a)Water b)Oil of Sp.gr 0.8 c)Mercury of Sp. Gr. 13.6 Take density of water = 1000kg/m 3. Ans: a) 2452.5 N/m 2, b) 1962 N/m 2 c) 33354 N/m 2

11. Ex 3 The pressure intensity any point in a fluid is given 3.924 N/cm 2. Find the corresponding height of fluid when the fluid is a)Water b)Oil of Sp.gr 0.9. Ans: a) 4 m, b) 4.44m

12. 1.Absolute Pressure: it is define as the pressure which is measured with reference to absolute vacuum pressure. Various Types of Pressure:

13. 2. Gauge pressure: it is defined as the pressure which is measured with help of a pressure measuring instrument, in which the atmospheric pressure is taken as datum. 3. Vacuum pressure: it is defined as the pressure below the atmospheric pressure Vacuum pressure = Atmospheric pressure- Absolute pressure

14. Measurement of Pressure: 1)Manometers 2)Mechanical Gauge 1)Manometers: It is define as devices used for measurement the pressure at a point in fluid by balancing the column of fluid by the same or another column of fluid. 1)Simple Manometer 2) Differential Manometers Simple manometer consists of a glass tube having one of its ends connected to point where pressure is to be measured and other end remains open to atmosphere Differential manometers are the Devices used for measuring the difference of pressure between two points in the pipe

15. 1)Simple Manometer - Piezometer - U-tube Manometer - Single Column Manometer 1. Piezometer

16. 2. U-tube Manometera) For Gauge pressure Pressure above A-A left = Pressure above A-A Right =

17. b) For Vacuum pressure Pressure above A-A left = Pressure above A-A Right = 0

18. Ex 1 The right limb of a simple U-tube manometer containing Mercury is open to the atmosphere while the left limb is connected To a pipe in which a fluid of Sp.gr. 0.9 is flowing. The centre of the pipe is 12 cm below the level of mercury in the Right limb. Find the pressure of fluid in the pipe if the difference of mercury Level in the two limbs is 20cm. Ans: p = 2.587 N/cm 2

19. Ex 2 A simple U-Tube manometer containing mercury is connected To a pipe in which a fluid of Sp.Gr. 0.8 and having vacuum pressure Is flowing. The other end of the manometer is open to atmosphere. Find the vacuum pressure in pipe, if the difference of mercury Level in the two limbs is 40 cm and the height of fluid in the left From the centre of pipe is 15 cm below. Ans: p = - 5.454 N/cm 2

20. Single Column Manometer: a) Vertical Single Column Manometer

21. Rise of heavy liquid level in Right Pressure in the right limb above Y-Y Pressure in the Left limb above Y-Y

22. Ex 1 A Single column manometer is connected to a pipe containing A liquid of Sp.gr. 0.9 as shown in figure. Find the pressure in the Pipe if the area of reservoir is 100 times the area of the tube for the Manometer reading shown in figure. The specific gravity of Mercury is 13.6. Ans: p = 5.12 N/cm 2 A/a =100

23. b) Inclined Single column Manometer: L = Length of heavy liquid moved in right limb from X-X Θ = Inclination of limb h 2 = L sin θ

24. Ex 1 An inclined single column manometer is connected to a pipe Containing a liquid of sp.gr. 0.9 as shown in Fig. The area of the Reservoir is 100 times the area of the tube. Calculate pressure in pipe. The manometric liquid is mercury of sp. Gr. 13.6. Ans: P = 35.43 kN/m 2

25. II) Differential Manometers: 1) U tube diff. Manometer h = Difference of Mercury level y = Dis. Of centre B from the Mercury Right limb x = Dis. Of centre A, From Mercury, Left limb ς 1 = Density of liquid A ς 2 = Density of liquid B ς g = Density of Mercury a) Two pipes at different level: Pressure X-X left limb= Pressure X-X Right limb= = Equating the press.

26. b) A and B at Same level: Pressure X-X left limb= Pressure X-X Right limb= =

27. Ex 1 A pipe contains an oil of Sp. Gr. 0.9. A differential manometer Connected same level at the two points A and B shows a difference in mercury Level as 15 cm. Find the difference of pressure at the two points Ans: p = 18688 N/m 2

28. Ex 2 A differential manometer is connected at the two points A and B of two pipes as shown in figure. The pipe A contains a liquid of Sp. gr. 1.5 while pipe B contains a liquid of Sp.gr. 0.9. The pressure at A and B are 1 kgf/cm 2 and 1.80 kgf/cm 2 respectively. Find the difference in mercury level in the differential manometer. Ans: h = 18.1 cm

29. II) Differential Manometers: 2) Inverted U tube diff. Manometer h 1 = Height of liquid in left limb below the datum line X-X h 2 = Height of liquid in right limb h = Difference of light liquid ς 1 = Density of liquid A ς 2 = Density of liquid B ς g = Density of Mercury p A = Pressure at A p B = Pressure at B Pressure X-X left limb= Pressure X-X Right limb=

30. Ex 1 Water is flowing through two different pipes to which an Inverted differential manometer having an oil Sp.gr. 0.8 is connected. The pressure head in the pipe A is 2 m of water. Find the pressure in the pipe B for the manometer readings as shown In figure. Ans: P b = 1.8599 N/cm 2

31. Ex 2 In figure an inverted differential manometer is connected to Two pipes A and B which convey water. The fluid in manometer Is oil of Sp.gr. 0.8. for the manometer readings shown in figure, Find the pressure difference between A and B. Ans: P A - P B = 4512.6-2943=1569.6 N/cm 2

32. Section II

33. Total Pressure: Total pressure is defined as the force exerted by static fluid on a surface either plane or curved when the fluid comes in contact with surface - This force always acts normal to the surface Centre of Pressure: CP is define as the point of application of the total pressure on the surface. 4 cases Submerged surfaces : -Vertical plane surface -Horizontal plane surface -Inclined plane surface -Curved surface

34. 1. Vertical Plane Surface Submerged in Liquid: A = Total area of the Surface = Dis. Of C.G. of the area from free surface of liquid G = Center of gravity of plane surface P = Center of Pressure h* = Dis. Of Centre of pressure from free surface a) Total pressure: Consider a strip Thickness dh and width b at a depth of h Pressure intensity on strip p = ς g h Area of the strip dA = b. dh Total pressure force on strip dF = p. Area = ς g h. b. dh Total pressure force on the whole surface,

35. Total pressure b) Centre of pressure (h*): by the “Principal of Moment” Moment of the force about free surface = F. h* Moment of the force dF, acting on strip =dF. h (dF = ς g h. b. dh) = ς g h. b. dh. h Sum of moments of all such forces about free surface of liquid

36. Sum of moments about free surface Parallel axis

37. Ex 1 A rectangular plane surface is 2 m wide and 3 m deep. It lies in Vertical plane in water. Determine the total pressure and position Of centre of pressure on the plane surface when its upper edge is Horizontal and a)Coincides with water surface b) 2.5 m below the free water surface. Ans: F = 88290 N, h* = 2.0 m B) F = 235440 N, h* = 4.1875 m

38. Ex 1 Determine the total pressure on a circular plate of diameter 1.5 m which is placed vertically in water in such a way that the Centre of the plate is 3 m below the free surface of water. Find the position of centre of pressure also.

39. 2. Horizontal Plane Surface Submerge in liquid: h* = Depth of centre of pressure from free surface = h

40. 3. Inclined plane surface submerged in liquid: Total pressure Centre of pressure (h*)

41. Ex 1 A rectangular plane surface 2 m wide and 3 m deep lies in water in such a way that its plane makes an angle of 30 o with the surface of water. Determine the total pressure and position of centre of pressure when the upper edge is 1.5 m below the free water surface. Ans: F = 132435 N, h* = 2.33 m (h*)

42. Ex b) A rectangular plane surface 3 m wide and 4 m deep lies in water in such a way that its plane makes an angle of 30 o with the free surface of water. Determine the total pressure force and position of centre of pressure, when the upper edge is 2 m below the free water surface. Ans: F = 353.16 N, h* = 3.11 m (h*)

43. Ex A Circular plate 3 m dia. Is immersed in water in such a way that its greatest and least depth below the free surface are 4 m and 1.5 m respectively. Determine the total pressure on one face of the plate and position of the centre of pressure. (h*) Ans: F = 190621 N, h* = 2.89 m

44. 4. Curved surface submerged in liquid: Total pressure Centre of pressure (h*)

45. Section III

46. Buoyancy and Floatation Buoyancy Force: When a body is immersed in a fluid, an upward force is exerted by the fluid on the body. This upward forces is equal to the weight of the fluid displaced by the body and is called the buoyancy force Centre of Buoyancy: It is defined as the point, through which the force of buoyancy is supposed to act.

47. Meta-Centre : It is define as the point about which a body starts oscillating when the body is tilted by small angle. Or The meta-centre may also be define as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement

48. Meta-Centric height : The distance MG, i.re. the distance between the meta-centre of a floating body and the centre of gravity of the body is called meta-centric height.

49. Ex 1 Find the volume of the water displaced and position of centre of buoyance for a wooden block of width 2.5 m and of the depth 1.5 m, when it floats horizontally in water. The density of wooden block is 650 kg/m 3 and its length 6 m. W = ς g V Volume of water Disp = Weight/Weight den. Of water Ans: W = 143471 N, V = 14.625 m 3, h = 0.975 m, CB = 0.48 m from B. b h L = V

50. Analytical method for meta-centre height : Wedge shape prism BOB’ (Emerges in) Wedge shape prism AOA’ (Emerges out) Couple Due to Wedges: The height of strip

51. Area of strip = Height x thickness Volume of strip = Area x L Weight of strip Moment of the couple = Weight of each strip x Distance between these two waights Moment of the couple for the whole wedge (1)

52. Moment of the couple due to shifting of centre of buoyancy from B to B 1 (2) From Eq. (1) & (2)

53. W= Weight of the body = ς g x volume of the fluid displaced by the body = ς g x v

54. Ex 1 A rectangular pontoon is 5 m long, 3 m wide and 1.20 m high. The depth of immersion of the pontoon is 0.80 m in sea water. If the centre of gravity is 0.6 m above the bottom of the pontoon, determine the meta-centric height. The density for sea water = 1025 kg/m 3. Ans: GM= 0.7375 m. Section. Plan.

55. Condition of equilibrium of a floating and Sub merged bodies: 1)Stability of Sub-merge body: a)Stable Equilibrium: When W = F B and point B is above G, the body is said to be stable equilibrium b)Unstable Equilibrium: if W = F B, but the centre of buoyancy (B) is below centre of gravity (G), the body is in unstable equilibrium. c)Neutral Equilibrium: if W = F B and G are at the same point, the body is said to be Neutral Equilibrium

56. 1)Stability of Floating body: a)Stable Equilibrium: if the point M is above G, the floating body will be in stable equilibrium b)Unstable Equilibrium: if the point M is below G, the body will be in unstable equilibrium. c)Neutral Equilibrium: if the point M is at Centre of Gravity (CG) of the body, the floating body will be in neutral equilibrium.

58. Experimental method of determination of meta-centre height :

59. Prepared, Prepared by, Dr Dhruvesh Patel Prepared, Prepared by, Dr Dhruvesh Patel