Fluid Mechanics lecture CHAPTER TWO.pptx

Fluid Dynamics

1. Fluid Mechanics BMAE 210 BY: BERNICE DOGBEY REGIONAL MARITIME UNIVERSITY FACULTY OF ENGINEERING & APPLIED SCIENCES MARINE ENGINEERING DEPARTMENT

2. COURSE OUTLINE Chapter I: Concept of Fluid s Chapter II: Principles of Fluid Statics Chapter III: F orces on Submerged Surfaces Chapter IV: Fluid Dynamics Chapter V: Flow Through Pipes Chapter VI: Pumps Chapter VI: Dimensional Analysis and Similitude.

3. Reference Books 1. Durgaiah , D.R., Fluid Mechanics and Machinery 2. Massey B.S., Mechanics of Fluids 3. Dr. Fogiel , M., Fluid Mechanics: A complete solution guide to any textbooks 4. Douglas J. S., Solving Problems in Fluid Mechanics Vol 1 5. Douglas J. S., Solving Problems in Fluid Mechanics Vol 2 6. Khurmi R. J., A textbook of hydraulics and hydraulic machines 7. Hannah J. & Hillier M. J., Applied Mechanics

4. CHAPTER - 2 Principles of Fluid Statics

5. Principles of Fluid Statics Concept of Pressure Pascal’s Law Pressure Variation With Depth (Hydrostatic Law) Manometers and Pressure Determination

6. Concept of Pressure Fluids are generally found in contact with surfaces. Fluids in contact with surfaces exert a force on the surfaces. The force is mainly due to the specific weight of the fluid in the case of liquids. In the case of gases molecular activity is the main cause of force exerted on the surfaces of the containers. This chapter also deals with pressure exerted by fluids due to the weight and due to the acceleration/deceleration of the whole mass of the fluid without relative motion within the fluid. Liquids held in containers may or may not fill the container completely. When liquids partially fill a container a free surface will be formed. Gases and vapours always expand and fill the container completely.

7. Concept of Pressure Pressure is a measure of force distribution over any surface associated with the force. Pressure is a surface phenomenon and it can be physically visualised or calculated only if the surface over which it acts is specified. Pressure may be defined as the force acting along the normal direction on unit area of the surface. Mathematically,  

8. Concept of Pressure Pressure on a fluid is measured in two different systems. In one system, it is measured above absolute zero (complete vacuum) known as the absolute pressure; and in the other system, pressure is measured above the surrounding pressure (atmosphere) known as the gauge pressure. Thus; The absolute pressure is defined as the pressure which is measured with reference to absolute zero (vacuum) pressure. Gauge pressure is defined as the pressure which is measured with the help of a pressure measuring device, in which the atmospheric pressure is marked as zero. Absolute pressure = gauge pressure + surrounding pressure The surrounding pressure is usually the atmospheric pressure.

9. Concept of Pressure The vacuum pressure is defined as the pressure below the atmospheric pressure. The relationship between absolute, gauge, vacuum and atmospheric pressure is given mathematically as; Absolute pressure( = gauge pressure( ) + surrounding pressure( ) Absolute pressure ( = surrounding pressure( ) – gauge pressure ) Thus:  

10. Concept of Pressure

11. Concept of Pressure The atmospheric (surrounding) pressure is measured using a mercury barometer (Fortin) or a bellows type meter called Aneroid barometer. The mercury barometer and bellow type meter have zero as the reference pressure. The other side of the measuring surface in these cases is exposed to vacuum. Hence these meters provide the absolute pressure value. Aneroid Barometer

12. Examples A gauge indicates 12 kPa as the fluid pressure while, the outside pressure is 150 kPa . Determine the absolute pressure of the fluid. A vacuum gauge fixed on a steam condenser indicates 80 kPa vacuum. The barometer indicates 1.013 bar. Determine the absolute pressure inside the condenser. Convert this pressure into head of mercury. An open cylindrical vertical container is filled with water to a height of 30 cm above the bottom and over that an oil of specific gravity 0.82 for another 40 cm. The oil does not mix with water. If the atmospheric pressure at that location is 1 bar, determine the absolute and gauge pressures at the oil water interface and at the bottom of the cylinder. The gauge pressure at the surface of a liquid of density 900 kg/m3 is 0.4 bar. If the atmospheric pressure is 1 × 105 Pa, calulate the absolute pressure at a depth of 50 m.

13. Pascal’s Law In fluids under static conditions, pressure is found to be independent of the orientation of the area. This concept is explained by Pascal’s law which states that the pressure at a point in a fluid at rest is equal in magnitude in all directions . This is possible only if the pressure at a point in a fluid at rest is the same in all directions so that the resultant force at that point will be zero. This means that when a change in pressure is applied to a fluid in a closed container, that pressure is transmitted equally without loss to every part of the fluid and the walls of the container.

14. Pascal’s Law This is the principle behind many inventions in our daily lives such as the hydraulic brakes, lifts and hydraulic press .   This principle is the basis for many hydraulic systems, where a small force applied at one point can be used to generate a much larger force at another point.

15. Pressure Variation In Static Fluid ( Hydrostatic Law ) The hydrostatic paradox explains the concept in fluid mechanics that pressure exerted by a fluid at rest at any depth is determined only by the height of the fluid column above that point and does not depend on the shape or volume of the container.

16. Pressure Variation In Static Fluid ( Hydrostatic Law ) This means that regardless of whether the container is wide or narrow, the pressure at the same depth in the fluid will be the same. This means that if you were to measure the pressure at different points in a horizontal plane at the same depth, you would get the same value. For a static fluid, the only stress acting on the fluid is the normal stress. The pressure in the fluid increases with depth due to the weight of the fluid above it, but at the same depth (same height in a horizontal plane), there is no reason for the pressure to vary in any horizontal direction. This is because the fluid is not moving, and the forces in the horizontal direction are balanced.

17. Pressure Variation In Static Fluid ( Hydrostatic Law ) The pressure at any point in a fluid at rest is obtained by the Hydrostatic law which states that the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point.

18. Pressure Variation In Static Fluid ( Hydrostatic Law ) Mathematically, Integrating the above eqn : Therefore, ρ gZ If , ρ gZ Where, = Atmospheric pressure = Pressure above atmospheric pressure Z =the height of the point from free surfaces  

19. Pressure Variation In Static Fluid ( Hydrostatic Law ) From the above eqn. Here, Z is called the pressure head . The pressure head is  therefore, the height of the liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container . It gives a physical sense of how much height of fluid is "behind" the pressure you're measuring. It may also be called static pressure head or simply static head.  

20. Examples A hydraulic press has a ram of 30cm diameter and a plunger 4.5cm diameter. Find the weight lifted by the hydraulic press when the force applied at the plunger is 500N. The pressure intensity at a point in a fluid is given as 3.924N/cm2. Find the corresponding height of fluid when the fluid is i . Water ii. Oil with S.G. of 0.9 An open tank contains water up to a depth of 2m and above it an oil of S.G. 0.9 for a depth of 1m. Find the pressure intensity at the interface of the two liquids and at the bottom of the tank. The diameters of a small piston and a large piston of a hydraulic jack are 3cm and 10cm respectively. A force of 80N is applied on the small piston. Find the load lifted by the large piston when i . the pistons are at the same level ii. The small piston is 40cm above the large piston The density of the liquid in the jack is given as 1000kg/m3

21. Measurement of Pressure Pressure is generally measured using a sensing element which is exposed on one side to the pressure to be measured and on the other side to the surrounding atmospheric pressure or other reference pressure. The pressure of a fluid is measured by the following devices; Manometers - They are devices used for measuring the pressure at a point in a fluid by balancing the column of fluid by the same or by another column of fluid. Manometers can be classified as; Simple Manometers Differential Manometers

22. Measurement of Pressure Mechanical Gauges - They are devices used for measuring the pressure by balancing the fluid column by a spring or dead weight. The commonly used mechanical pressure gauges are Diaphragm pressure gauge; Bourdon Tube pressure gauge; Dead-weight pressure gauge; and Bellows pressure gauge.

23. Manometers and Pressure Determination Simple Manometers A simple manometer consists of a glass tube having one of its end connected to a point where pressure is to be measured and the other end remains open to the atmosphere. Common types of simple manometers are ; Piezometer U-tube Manometer Single Column Manometer

24. Manometers and Pressure Determination Piezometer It is the simplest form of manometers used in measuring pressure gauges. One end of this manometer is connected to the point where pressure is to be measured and the other end is open to the atmosphere as shown. The rise of liquid gives the pressure head at that point. If at point A, the height of liquid is h in the piezometer tube, then  

25. Manometers and Pressure Determination U-tube Manometer It consists of a glass tube bent in U-shape, one end of which is connected to a point at which pressure is to be measured and the other end open to the atmosphere as shown. The tube generally contains mercury or any other liquid whose specific gravity is greater than the specific gravity of the liquid whose pressure is to be measured.

26. Manometers and Pressure Determination In determining the Gauge pressure Let B be the point at which pressure is to be measured, whose pressure is . The datum line is A-A and = height of liquid(measured) above the datum line ‘A’. = height of mercury above the datum line ‘A’. = density of liquid(measured). = density of mercury.  

27. Manometers and Pressure Determination In determining the Gauge pressure As the pressure is the same for the horizontal surface, the pressure above the horizontal datum line A-A in the left column and in the right column of the U-tube manometer should be the same. That is; Pressure above A-A in the left column = Pressure above A-A in the right column For Gauge pressure, = 0 Therefore,  

28. Manometers and Pressure Determination In determining the Vacuum pressure For measuring vacuum pressure, the level of the heavy liquid(mercury) in the manometer will be as shown in the fig. Pressure above A-A in the left column = Pressure above A-A in the right column  

29. Examples 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 S.G. 0.9 is flowing. The centre of the pipe is 12cm below the level of mercury in the right limb. Fin the pressure of fluid in the pipe if the difference of mercury level in the two limbs is 20cm . A simple U-tube manometer containing mercury is connected to a pipe in which a fluid of S.G. 0.8 and having vaccum pressure is flowing. The other end of the manometer is open to atmosphere. Find the vacuum pressure in the pipe, if the difference of mercury level in the limbs is 40cm and the height of fluid in the left from the centre of pipe is 15cm.

30. Examples A U-tube manometer is used to measure the pressure of water in a pipeline, which is in excess of atmospheric pressure. The right limb of the manometer contains mercury and is open to atmosphere. The contact between water and mercury is in the left limb. Determine the pressure of the water in the main line, if the difference in level of mercury in the limbs of U-tube is 10cm and the free surface of mercury is in the centre of the pipe. If the pressure of water in the pipeline is reduced to 9810N/m2, calculate the new difference in the level of mercury. A manometer is fitted as shown in the figure. Determine the pressure at point A.

31. Manometers and Pressure Determination Single Column Manometers This is a modified form of the U-tube manometer in which a reservoir, having a large cross-sectional area (about 100 times) as compared to the area of the tube is connected to one of the limbs(say left limb) of the manometer. Due to the large cross-sectional area of the reservoir, for any variation in pressure, the change in the liquid level in the reservoir will be very small which may be neglected and hence the pressure is given by the height of the liquid in the other limb. The other limb may be vertical or inclined. Thus, there are two types; the Vertical and Inclined single column manometer .

32. Manometers and Their Various Types Vertical Single Column Manometer The fig. shows the vertical single column manometer, where X-X is the datum line in the reservoir and in the right limb of the manometer when it is not connected to a pipe. When the manometer is connected to a pipe, due to high pressure at ‘A’, the heavy liquid (mercury) in the reservoir will be pushed downward and will rise in the right limb.

33. Manometers and Their Various Types Vertical Single Column Manometer Where; = height of centre of pipe above line X -X. = rise of mercury in the right limb. = fall of heavy liquid in reservoir. = cross-sectional area of the reservoir. = cross-sectional area of the right limb. = density of liquid (measured). = density of mercury in reservoir.  

34. Manometers and Their Various Types Vertical Single Column Manometer The fall of the mercury in the reservoir will cause a rise of mercury in the right limb . Therefore, Considering datum line Y-Y , The pressure in the right limb above Y-Y will be = The pressure in the left limb above Y-Y will be =  

35. Manometers and Their Various Types Vertical Single Column Manometer The pressure in the right limb = The pressure in the left limb Therefore, = However, Therefore , As area ‘A’, is very large as compared to area ‘a’, approaches zero. Hence,  

36. Manometers and Their Various Types Inclined Single Column Manometer The fig. shows the inclined single column manometer. This manometer is more sensitive. Due to inclination, the distance moved by the mercury in the right limb will be more. From the diagram, L = length of mercury moved in the right limb from line X-X. = inclination of right limb with horizontal = L*sin = vertical rise of mercury in right limb from X-X.  

37. Manometers and Their Various Types Inclined Single Column Manometer From Due to the inclination, Therefore,  

38. Examples A single column manometer is connected to a pipe containing a liquid of density 900kg/m3 as shown in the fig. Find the pressure in the pipe if the area of the reservoir is 100times the area of the tube of the manometer. The specific gravity of mercury is 13.6. Determine the fluid pressure at a tapping connected with an inclined manometer if the rise in fluid level is 10 cm along the inclined tube above the reservoir level. The tube is inclined at 20° to horizontal as shown in figure. The density of manometric fluid is 800 kg/m3.

39. Manometers and Their Various Types Differential Manometers They are the devices used for measuring the difference of pressures between two pints in a pipe or in two different pipes. A differential manometer consists of a U-tube, containing mercury whose two ends are connected to the points, whose difference of pressure is to be measured. Most common ones are; The U-tube differential manometer, and Inverted U-tube differential manometer.

40. Manometers and Their Various Types U-tube Differential Manometers The fig. shows different levels at points A and B c onnected by a U-tube differential manometer. The pressures at points A and B are . Where, difference of mercury level in the U-tube distance of the centre of B, from the mercury level in the right limb. distance of the centre of A, from the mercury level in the right limb. Density of liquid at A Density of liquid at B Density of mercury   z

41. Manometers and Their Various Types U-tube Differential Manometers Pressure above X-X in the left limb = Pressure above X-X in the right limb = Equating the two pressures, we have Therefore,   z

42. Manometers and Their Various Types U-tube Differential Manometers From the fig. points A and B are at the same level and contains the same liquid of density, . Pressure above X-X in the left limb= Pressure above X-X in the right limb = Equating the two pressures, we have Therefore,  

43. Manometers and Their Various Types Inverted U-tube Differential Manometers It consists of an inverted U-tube, containing a light liquid. The two ends of the tube are connected to the points whose difference of pressure is to be measured. It is used for measuring difference of low pressures. The fig. shows an inverted U-tube differential manometer connected to two points A and B, having pressures are where is greater than .  

44. Manometers and Their Various Types Inverted U-tube Differential Manometers Where, difference of light liquid in the U-tube height of liquid in left limb below X-X height of liquid in right limb below X-X density of liquid at A density of liquid at B density of light liquid  

45. Manometers and Their Various Types Inverted U-tube Differential Manometers Pressure in the left limb below X-X = Pressure in the right limb below X-X = Equating the two pressure  

46. Examples A differential manometer is connected at the two points A and B of two pipes . The pipe A contains a liquid of S.G 1.5 while pipe B contains a liquid of S.G. 0.9. The pressures at A and B are 98.1kN/m2 and 176.58kN/m2 respectively. Find the difference in the mercury level in the differential manometer. In the figure shown, an inverted differential manometer is connected to two pipes A and B which convey water. The fluid in the manometer is oil of S.G. 0.8. For the manometer readings shown in the figure, find the pressure difference between A and B.

47. Examples A differential manometer is connected at the two points A and B as shown. At B, air pressure is 9.81N/cm2(abs), find the absolute pressure at A. Water is flowing through two different pipes to which an inverted differential manometer having an oil of S.G. 0.8 is connected. The pressure head in the pipe A is 2m of water, find the pressure in the pipe B for the manometer readings.

48. END OF CHAPTER - 2