This PowerPoint presentation explains the basics abt presseure

This PowerPoint presentation explains the basics abt presseure. This PowerPoint presentation explains the basics abt presseure. This PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint presentation explains the basics abt presseureThis PowerPoint pres

1. Chapter 2 Fluid Statics

2. Lecture Topics on 14-09-2022 Chapter 2 Fluid Statics Learning Objectives 2.1 Pressure at a Point ME F212 Fluid Mechanics

3. Learning Objectives After completing this chapter, you should be able to: Determine the pressure at various locations in a fluid at rest Explain the concept of manometers and apply appropriate equations to determine pressure Calculate the hydrostatic pressure force on a plan or curved submerged surface Calculate the buoyant force and discuss the stability of floating or submerged objects ME F212 Fluid Mechanics

4. Pressure at a Point How does the pressure at a point vary with orientation of the plane passing through the point? The term pressure is used to indicate the normal force per unit area at a given point acting on a given plane within the fluid mass of interest. ME F212 Fluid Mechanics To answer this question, consider the free body diagram, that was obtained by removing a small triangular wedge of fluid from some arbitrary location within a fluid mass.

5. Wedged Shaped Fluid Mass Pressure at a Point ME F212 Fluid Mechanics

6. Pressure at a Point: Analysis No shearing stresses present in the fluid. For simplicity the forces in x-direction are not shown. External forces acting on the fluid are due to pressure and the weight. To make the analysis general, allow the fluid element to have accelerated motion. The assumption of zero shearing stresses will still be valid as long as the fluid element moves as a rigid body. ME F212 Fluid Mechanics

7. The equations of motion (Newton’s second law, F = ma ) in the y and z directions are: Pressure at a Point: Analysis It follows from the geometry that Where p s , p y , and p z are the average pressures on the faces, γ and ρ are the fluid specific weight and density and a y , a z the accelerations. ME F212 Fluid Mechanics

8. The equations of motion can be rewritten as Pressure at a Point: Analysis Since we are really interested in what is happening at a point, we take the limits as δ x , δ y, and δ z approach zero (while maintaining the angle θ ) ME F212 Fluid Mechanics

9. Pressure at a Point: Pascal’s Law p s d x d s p 1 d x d s p 2 d x d s The pressure at a point in a fluid at rest, or in motion, is independent of direction as long as there is no shearing stresses present. This important result is known as Pascal’s Law . Blaise Pascal (1623-1662) ME F212 Fluid Mechanics

10. 2.3 Pressure Variation in a Fluid at Rest 2.4 Standard Atmosphere ME F212 Fluid Mechanics Lecture Topics on 16-09-2022

11. Basic Equation for Pressure Field How does the pressure vary in a fluid or from point to point when no shear stresses are present? ME F212 Fluid Mechanics

12. Basic Equation for Pressure Field Let the pressure at the center of the element be designated as p . The average pressure on the various faces can be expressed in terms of p and its derivatives. ME F212 Fluid Mechanics Forward Taylor series: Backward Taylor series:

13. Pressure Field Equation The resultant surface forces in the y -direction: Similarly, the resultant surface forces in the x and z -dire The resultant surface forces acting on the element in vector form: ME F212 Fluid Mechanics

14. Pressure Field Equation The “del” vector operator or gradient is the following : Then, Now, rewriting the surface force equation, we obtain the following: Now, we return the body forces, and we will only consider weight: ME F212 Fluid Mechanics

15. Pressure Field Equations Newton’s Second Law, applied to the fluid element: δ m is the mass of the fluid element, and a is acceleration. Then summing the surface forces and the body forces: This is the general equation of motion for a fluid in which there is no shearing stresses. ME F212 Fluid Mechanics

16. Pressure Variation in a Fluid at Rest In components form The equations show that the pressure does not depend on x or y . This is the fundamental equation for fluids at rest. It is valid for fluids with constant specific weight, such as liquids, as well as fluids whose specific weight may vary with elevation, such as air or other gases. ME F212 Fluid Mechanics

17. Hydrostatic Condition: Incompressible Fluids The variation in g is negligible. For liquids the variation in ρ is negligible over large vertical distance, thus most liquids will be considered incompressible. We can immediately integrate since γ is a constant: This type of pressure distribution (varies linearly with depth) is commonly called a hydrostatic distribution . ME F212 Fluid Mechanics

18. Hydrostatic Condition: Incompressible Fluids h is known as the pressure head . It is the height of a column of fluid of specific weight γ required to give a pressure difference p 1 - p 2 . ME F212 Fluid Mechanics

19. When one works with liquids there is often a free surface and it is convenient to use this surface as a reference plane. The reference pressure p o would correspond to the pressure acting on the free surface (atmospheric pressure) Hydrostatic Condition: Incompressible Fluids The pressure p at any depth h below the free surface is given by ME F212 Fluid Mechanics

20. The pressure in a homogenous, incompressible fluid at rest depends on the depth of the fluid relative to some reference and is not influenced by the size or shape of the container. Hydrostatic Condition: Incompressible Fluids The pressure p at any depth h below the free surface is given by ME F212 Fluid Mechanics

21. Application: Transmission of Fluid Pressure The required equality of pressure at equal elevations throughout a system is important for the operation of hydraulic jacks, lifts, and presses, as well as hydraulic controls on aircraft and other types of heavy machinery. The fundamental idea behind such devices and systems is demonstrated below. ME F212 Fluid Mechanics

22. Application: Transmission of Fluid Pressure A piston located at one end of a closed system filled with a liquid, such as oil, can be used to change the pressure throughout the system Thus transit an applied force F 1 to a second piston where the resulting force is F 2 Since the pressure p acting on the faces on both pistons is the same (the effect of elevation changes is usually negligible for this type of hydraulic devices) ME F212 Fluid Mechanics

23. Application: Transmission of Fluid Pressure The piston area A 2 can be made much larger than A 1 and therefore a large mechanical advantage can be developed That is, a small force applied at the smaller piston can be used to develop a larger force at the larger piston ME F212 Fluid Mechanics

24. Application: Transmission of Fluid Pressure Hydraulic lift Hydraulic jack ME F212 Fluid Mechanics

25. Hydrostatic Condition: Compressible Fluids Gases such as air, oxygen and nitrogen as being compressible, we must consider the variation of density in the hydrostatic equation: The equation of state for an ideal gas is and by separating variables ME F212 Fluid Mechanics

26. Atmosphere layers ME F212 Fluid Mechanics

27. Atmosphere layers ME F212 Fluid Mechanics

28. Atmosphere layers ME F212 Fluid Mechanics

29. Standard Atmosphere One would like to have measurements of pressure versus altitude over the specific range for the specific conditions (temperature, reference pressure). This type of information is usually not available. Thus a standard atmosphere has been determined. The concept of standard atmosphere was developed in the 1920s. The currently accepted standard atmosphere is based on a report published in 1962 and updated in 1976. The U.S. standard atmosphere is an idealized representation of middle-latitude, year-round mean conditions of the earth’s atmosphere. ME F212 Fluid Mechanics

30. Standard Atmosphere The below figure shows the temperature profile for the U.S. standard atmosphere. Linear Variation, T = T a - b z Isothermal, T = T o Standard Atmosphere is used in the design of aircraft, missiles and spacecraft. Stratosphere: Troposphere: ME F212 Fluid Mechanics

31. Standard Atmosphere ME F212 Fluid Mechanics

32. U.S. Standard Atmosphere: Troposphere Starting from, For the troposphere, which extends to an altitude 11 km β is known as the lapse rate, 0.00650 K/m, and T a is the temperature at sea level, 288.15 K . p a is the pressure at sea level, 101.33 kPa, R is the gas constant, 286.9 J/kg.K ME F212 Fluid Mechanics

33. U.S. Standard Atmosphere: Contrails The main combustion products of hydrocarbon fuels are CO 2 and water vapor. At high altitudes this water vapor emerges into a cold environment. The vapor then condenses into tiny water droplets and/or deposits into ice. ME F212 Fluid Mechanics At the outer edge of troposphere, where the temperature is -56.5 o C, the absolute pressure is 23 kPa.

34. U.S. Standard Atmosphere: Stratosphere Starting from, For the stratosphere, the temperature has a constant value T o over the range z 1 to z 2 (isothermal conditions), it then follows This equation provide the desired pressure-elevation relationship for an isothermal layer. ME F212 Fluid Mechanics

35. Measurement of Pressure Absolute Pressure : Pressure measured relative to a perfect vacuum (absolute zero pressure). or Gage Pressure : Pressure measured relative to local atmospheric pressure. A gage pressure of zero corresponds to a pressure that is at local atmospheric pressure. Absolute pressure is always positive. Gage pressure can be either negative or positive. Negative gage pressure is known as a vacuum or suction. The pressure at a point is designated as either an ME F212 Fluid Mechanics

36. Measurement of Pressure : Absolute, gage, and vacuum pressures ME F212 Fluid Mechanics

37. Measurement of Pressure : Barometers Evangelista Torricelli (1608-1647) The measurement of atmospheric pressure is usually accomplished with a mercury barometer. Often p vapor is very small, 0.16 N/m 2 at 20 °C, thus: For P atm =101.3 kPa 10.4 m of H 2 76 cm of Hg ME F212 Fluid Mechanics

38. Measurement of Pressure : Manometry A standard technique for measuring pressure involves use of liquid columns in vertical or inclined tubes are called Manometry . Pressure measuring devices based on this technique are called manometers . The three common types of manometers are The Piezometer Tube The U-Tube Manometer The Inclined Tube Manometer The fundamental equation for manometers is since they involve columns of fluid at rest. ME F212 Fluid Mechanics

39. Measurement of Pressure : Piezometer Tube p o The simplest type of manometer consists of a vertical tube, open at the top, and attached to the container in which the pressure is desired. Disadvantages: 1)The pressure in the container has to be greater than atmospheric pressure. 2) Pressure must be relatively small to maintain a small column of fluid. 3) The measurement of pressure must be of a liquid. ME F212 Fluid Mechanics

40. Measurement of Pressure : U-Tube Manometer Then the equation for the pressure in the container is If the fluid in the container is a gas, then the fluid 1 terms can be ignored : The fluid in the manometer is known as the gage fluid. ME F212 Fluid Mechanics

41. Measurement of Pressure : Differential U-Tube Manometer Used to measure the difference in pressure between two containers or two points in a given system. Moving from left to right: ME F212 Fluid Mechanics

42. Differential U-Tube Manometer Final notes: Capillarity due to surface tension at the various fluid interfaces are not considered. Capillarity can play a role, but in many cases each meniscus will cancel (ex: simple U-tube manometer). Making the Capillary rise negligible by using relative large bore tubes. Temperature must be considered in very accurate measurements, as the gage fluid properties can change. Common gage fluids are Hg and Water, some oils, and must be immiscible. ME F212 Fluid Mechanics

43. Measurement of Pressure: Inclined-Tube Manometer This type of manometer is used to measure small pressure changes. q h 2 l 2 If the pressure difference is between gases: ME F212 Fluid Mechanics

44. Measurement of Pressure: Mechanical and Electrical Devices ( a ) Liquid-filled Bourdon pressure gages for various pressure ranges. ( b ) Internal elements of Bourdon gages. The “C-shaped” Bourdon tube is shown on the left, and the “coiled spring” Bourdon tube for high pressures of 1000 psi and above is shown on the right. ME F212 Fluid Mechanics

45. Measurement of Pressure: Mechanical and Electrical Devices Pressure transducer which combines a linear variable differential transformer (LVDT) with a Bourdon gage. ME F212 Fluid Mechanics

46. Measurement of Pressure: Mechanical and Electrical Devices Two different sized strain-gage pressure transducers. Schematic diagram of the P23XL transducer with the dome removed ME F212 Fluid Mechanics

47. Two pipes are connected by a manometer, as shown below. Determine the pressure difference between the pipes.

48. For the incline-tube manometer given below, the pressure in pipe A is 4.1 kPa. The fluid in pipes A and B is water, and the gauge fluid in the manometer has a specific gravity of 2.6. What is the pressure in pipe B corresponding to the differential reading shown?

49. Determine the change in the elevation of the mercury in the left leg of the manometer for figure below as a result of an increase in pressure of 34.5 Kpa in Pipe A while the pressure in Pipe B remains constant.

50. Two chambers with the same fluid at their base are separated by a 30-cm-diameter piston whose weight is 25 N, as shown in Fig. Calculate the gage pressures in chambers A and B .

51. Consider a hydraulic jack being used in a car repair shop, as in Fig. The pistons have an area of A 1 =0.8 cm2 and A 2 = 0.04 m2. Hydraulic oil with a specific gravity of 0.870 is pumped in as the small piston on the left side is pushed up and down, slowly raising the larger piston on the right side. A car that weighs 13,000 N is to be jacked up. ( a ) At the beginning, when both pistons are at the same elevation ( h = 0), calculate the force F 1 in newtons required to hold the weight of the car. ( b ) Repeat the calculation after the car has been lifted two meters ( h = 2 m). Compare and discuss.