FLUID MECHANICS - DME UPDATED(1) - Copy.pptx

Fluid mechanics

1. DIPLOMA MARINE ENGINEERING BY MS. BERNICE DOGBEY REGIONAL MARITIME UNIVERSITY FACULTY OF ENGINEERING MARINE ENGINEERING DEPARTMENT FLUID MECHANICS

2. Mode of Assessment Continuous Assessment – 40% Mid-Semester – 20 Assignments – 15 Attendance – 5 End of Semester Exams – 60%

3. Course Outline: Introduction And Basic Concepts Basic Definitions Application Areas of Fluid Mechanics The No-Slip Condition - Fluid Properties Density / Specific Gravity / Specific Weight Surface Tension and Capillarity Compressibility (Bulk Modulus of Elasticity) Viscosity - Principles of Fluid Statics Concept of Pressure/ Manometers Hydrostatic forces on submerged surfaces Buoyancy and Archimedes Principle Derivation of Basic Equations for a Control Volume Analysis of Fluids in Motion Steady And Unsteady Flow

4. Course Outline: Uniform And Non-Uniform Flow Compressible and Incompressible Flow Streamlines And Stream Tubes The Continuity Principle The Bernoulli (Energy)Equation Coefficient of; Discharge/Velocity/Contraction of Area Momentum Equation And Principle of Conservation of Mass - Flow Through Pipes Laminar flow Transitional Flow Turbulent flow Velocity Distribution in Pipe Flow Head Loss in Fluid Flow Pumps Reciprocating And Centrifugal Pumps Cativation Efficiences of Centrifugal Pumps

5. INTRODUCTION AND BASIC CONCEPTS

6. INTRODUCTION AND BASIC CONCEPTS Basic Definitions; What is a fluid? What is mechanics?

7. INTRODUCTION AND BASIC CONCEPTS Solid Therefore, fluid mechanics deals with the study of liquids and gases in motion (fluid dynamics) or at rest (fluid statics) and their interaction with solids or other fluids at the boundaries.

8. INTRODUCTION AND BASIC CONCEPTS Application Areas of Fluid Dynamics Fluid mechanics is widely used both in everyday activities and in the design of modern engineering systems. Artificial Heart Artificial Lungs

9. INTRODUCTION AND BASIC CONCEPTS Application Areas of Fluid Dynamics

10. INTRODUCTION AND BASIC CONCEPTS Classes of fluids : Gases and Liquids Due to the strong cohesive forces between the molecules in a liquid, a liquid takes the shape of the container it is in. A gas, on the other hand , expands until it encounters the walls of the container and fills the entire available space. This is because the gas molecules are widely spaced, and the cohesive forces between them are very small. (a)- Solid (b)-Liquid (c) - Gas

11. INTRODUCTION AND BASIC CONCEPTS Temperature And Pressure Adding heat to a substance increases its temperature. But what is actually going on? The jiggling of the atoms or molecules in the substance become more energetic as temperature increases. This is usually referred to as ‘ Internal Energy ’ in thermodynamics. When the atoms bounce off the wall, they exert a force on the walls. The change in momentum ( m∆ ν )causes the pressure. In a sealed container, pressure increases as temperature increases.

12. INTRODUCTION AND BASIC CONCEPTS The No-Slip Condition Fluid flow is often confined by solid surfaces, and it is important to understand how the presence of solid surfaces affects fluid flow. All experimental observations indicate that a fluid in motion comes to a complete stop at the surface and assumes a zero velocity relative to the surface. That is, a fluid in direct contact with a solid “sticks” to the surface, and there is no slip. This is known as the no-slip condition .

13. INTRODUCTION AND BASIC CONCEPTS The No-Slip Condition The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity . The region between the solid surface and the fluid is called the boundary layer.

14. INTRODUCTION AND BASIC CONCEPTS Dimensions And Units Any physical quantity can be characterized by dimensions. The magnitudes assigned to the dimensions are called units. The four primary dimensions used in fluid mechanics are;

15. INTRODUCTION AND BASIC CONCEPTS Secondary Dimensions These are formed by the multiplication/division of primary dimensions. The most important secondary dimension we will be working with is Force = ma = kgm /  

16. FLUID PROPERTIES

17. PROPERTIES OF FLUIDS Density/Specific Gravity/ Specific Weight Density is defined as the mass per unit volume; Specific Volume, Generally, the density of a substance depends on Temperature and Pressure. Because liquids and solids are incompressible, the variation of their density with pressure is negligible. The density of gases, however, is directly proportional to pressure and inversely proportional to temperature.  

18. PROPERTIES OF FLUIDS Density/Specific Gravity/ Specific Weight Density of fresh water = Density of air   The ratio of the density of a substance to the density of a standard reference (usually water) at a specified temperature is termed as the Specific Gravity or Relative Density . It has no unit.   The weight of a unit volume of a substance is called Specific weight/Weight Density and it is expressed as  

19. PROPERTIES OF FLUIDS Density of Ideal Gases Any equation that relates temperature, pressure and density(or specific volume) of a substance is known as the Equation of State . The equation of state for substances in the gas phase is the Ideal Gas Equation of State , expressed as Where, P = Absolute Pressure v = Specific Volume T = Temperature (K) R = Gas Constant  

20. Examples A reservoir of glycerin has a mass of 1,200kg and a volume of 300m3. Calculate its density, specific gravity and specific volume. Calculate the specific weight, density and specific gravity of 1L of petrol which weighs 7N. The density of a liquid is 2.93g/cm3. What is its specific gravity, specific volume and specific weight? Calculate the density, specific weight and weight of 1L liquid of specific gravity 0.8. The specific gravity of ice is 0.9, calculate the weight density of the ice. The mass of a fluid system is 4kg, its density is 2g/cm3 and g=9.81m/s2. Determine the Specific volume, Specific weight and total weight of the fluid. If 25L of an oil weighs 425g, what is the density and specific gravity of the oil.

21. Examples 8. If 0.5m3 of a liquid has a density of 1.8 g/cm3, what is the weight of the liquid? 9. What is the volume of a solution that weighs 45N and has a specific gravity of 0.78? 10. What is the specific weight of air at 48kPa and 21 C. R = 0.287 kPa.m / kg.K . 11. A mass of 150g of argon is maintained at 200 Pa and 100°F in a tank. What is the volume of the tank ? 12. A 100L container is filled with 1 kg of air at a temperature of 27°C. What is the pressure in the container ? R = 0.287 kPa.m / kg.K . 13. Determine the density, specific gravity, and mass of the air in a room whose dimensions are 4 m x 5 m x 6 m at 100 kPa and 25°C. R = 0.287 kPa.m / kg.K .

22. Surface Tension Surface tension is defined as the tensile(elastic) force acting on the surface of a liquid in contact with air (gas) or between two immiscible liquids. This is due to the cohesive(attractive) forces between the molecules in the liquid. It is denoted by ‘ σ ’ and measured in N/m . Surface tension decreases with temperature. Contaminants as well as detergents also decrease surface tension.

23. Surface Tension Surface tension is the force per unit length of a liquid, i.e. Water Droplet/Air Bubble;   The surface tension is 0.073 N/m for water and 0.440 N/m for mercury surrounded by atmospheric air (20 C)

24. Capillarity Another interesting consequence of surface tension is the capillary effect , which is the rise or fall of a liquid in a small-diameter tube inserted into the liquid . Such narrow tubes or confined flow channels are called capillaries . The rise of kerosene through a cotton wick inserted into the reservoir of a kerosene lamp is due to this effect. The capillary effect is also partially responsible for the rise of water to the top of tall trees. The curved free surface of a liquid in a capillary tube is called the meniscus .

25. Capillarity It is commonly observed that water in a glass container curves up slightly at the edges where it touches the glass surface but the opposite occurs for mercury; it curves down at the edges. This effect is usually expressed by saying that water wets the glass (by sticking to it) while mercury does not . The strength of the capillary effect is quantified by the contact (or wetting) angle , defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact .

26. Capillarity A liquid is said to wet the surface when Ø < 90 ° and not to wet the surface when Ø > 90 °. In atmospheric air , the contact angle of water with glass is nearly zero, Ø ≈ 0 °. Therefore, the surface tension force acts upward on water in a glass tube along the circumference, tending to pull the water up. The contact angle for mercury–glass is 130° and 26° for kerosene–glass in air . Note that the contact angle, in general, is different in different environments (such as another gas or liquid in place of  air).

27. Capillarity The phenomenon of the capillary effect can further be explained by considering cohesive forces (the forces between like molecules, such as water and water) and adhesive forces (the forces between unlike molecules, such as water and glass ). The liquid molecules at the solid–liquid interface are subjected to both cohesive forces by other liquid molecules and adhesive forces by the molecules of the solid. The relative magnitudes of these forces determine whether a liquid wets a solid surface or not. T he water molecules are more strongly attracted to the glass molecules than they are to other water molecules, and thus water tends to rise along the glass surface . The opposite occurs for mercury, which causes the liquid surface near the glass wall to be suppressed.

28. Capillarity Capillary Rise in a Tube The weight of the liquid column is; Equating the vertical component of the surface tension force to the weight; Solving for h gives the capillary rise to be

29. Capillarity Capillary Rise in a Tube T he capillary effect for water is usually negligible in tubes whose diameter is greater than 1cm. The capillary rise is also inversely proportional to the density of the liquid, therefore lighter liquids experience greater capillary rises .

30. Examples Water rises to a height of 4.5cm in a capillary tube of radius r. Find r, assuming the surface tension of water is 0.073 N/m. Take the angle of contact in the glass as A liquid of density , rises to a height of 7mm in a capillary tube of internal diameter 2mm. If the angle of contact of the liquid to the glass is , find the surface tension of the liquid. A capillary tube of radius 0.05cm is dipped vertically into a liquid of surface tension 0.04N/m and density 0.8 . Calculate the height of capillary rise, if the angle of contact is A capillary tube 0.12mm in diameter has its lower end immersed in liquid with density . Calculate the height of capillary rise if σ . Find the angle of contact to a capillary tube of radius 0.0005m, having a density of 680 Given that the liquid has a surface tension of 0.062 and a capillary rise is 5.2cm.  

31. Compressibility The volume (or density) of a fluid changes with a change in its temperature or pressure. Fluids usually expand as they are heated or depressurized and contract as they are cooled or pressurized. But the amount of volume change is different for different fluids. That is, fluids act like elastic solids with respect to pressure.

32. Compressibility Therefore, it is appropriate to define a coefficient of compressibility, k (also called the bulk modulus of elasticity ) for fluids as k =−V , Pa; T= constant   T he coefficient of compressibility represents the change in pressure corresponding to a fractional change in volume or density of the fluid while the temperature remains constant.

33. Compressibility Bulk Modulus is the measure of ability of a substance to withstand changes in volume when it undergoes compression on all sides. A large value of ‘k’ indicates that a large change in pressure is needed to cause a small fractional change in volume, and thus a fluid with a large k, is essentially incompressible. This is typical for liquids, and explains why liquids are usually considered to be incompressible.

34. Compressibility For an ideal gas , ; Therefore ,   Therefore, the coefficient of compressibility of an ideal gas is equal to its absolute pressure, and so k, of the gas increases with increasing pressure.

35. Viscosity When two solid bodies in contact move relative to each other, a friction force develops at the contact surface in the direction opposite to motion . The situation is similar when a fluid moves relative to a solid or when two fluids move relative to each other. We move with ease in air, but not so in water. There is a property that represents the internal resistance of a fluid to motion or the “fluidity,” and that property is the viscosity .

36. Viscosity The force a flowing fluid exerts on a body in the flow direction is called the drag force , and the magnitude of this force depends, in part, on viscosity.

37. Viscosity For liquids the viscosity decreases with temperature, whereas for gases the viscosity increases with temperature. Consider a fluid layer between two parallel plates immersed in a large body of a fluid separated by a distance , where the top part is moved by a shear force F, moving at a constant rate (velocity) of ν (m/s).

38. Viscosity The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same speed, and the shear stress acting on this fluid layer is;  

39. Viscosity The rate of deformation or change in velocity increases with distance above the fixed plate. Hence Where the constant of proprotionality is known as the dynamic/absolute viscosity. Therefore dynamic/absolute viscosity of a fluid is the measure of its internal resistance to flow when an external force is applied. Unit is Pa.s or cP  

40. Viscosity Fluids for which the rate of deformation is linearly proportional to the shear stress are called Newtonian fluids , named after Sir Isaac Newton. Most common fluids such as water, air, gasoline, mercury and oils are Newtonian fluids.

41. Viscosity For non-Newtonian fluids , the relationship between shear stress and rate of deformation is not linear. Examples : Blood, toothpaste, ketchup, some paints,liquid plastics, etc. The ratio of dynamic viscosity to density is referred to as Kinematic viscosity ( ) , expressed as; ,  

42. Principles of Fluid Statics

43. Concept of Pressure Pressure is defined as a normal force exerted by a fluid per unit area. P = We speak of pressure only when we deal with a gas or a liquid. The counterpart of pressure in solids is normal stress. Since pressure is defined as force per unit area, it has the unit of newtons per square meter ( ), which is called a Pascal (Pa).  

44. Concept of Pressure That is , The pressure unit Pascal is too small for most pressures encountered in practice . Therefore, its multiples are commonly used; i.e. 1 KiloPascal ( kPa ) = 1 MegaPascal (MPa ) = 1 bar = 1 atm = 101,325  

45. Concept of Pressure The air above the earth’s surface is a fluid , which exerts a pressure on all points on the earth’s surface. This pressure is called atmospheric pressure . The actual pressure at a given position is called the absolute pressure , and it is measured relative to absolute zero/vacuum. Most pressure-measuring devices, however, are calibrated to read zero in the atmosphere and so they indicate the difference between the absolute pressure and the local atmospheric pressure . This difference is called the gauge pressure .

46. Concept of Pressure Absolute Pressure( ) = Gauge Pressure ( ) + Atmospheric pressure ( ) i.e. = For a fluid at rest, F = W =mg The density of the fluid ρ = The volume, V = A*h(depth) Therefore, =   The pressure exerted by a fluid at equilibrium at any point of time due to the force of gravity

47. Concept of Pressure Pascal’s Principle I n a fluid at rest in a closed container, a  pressure change in one part is  transmitted without loss to every portion of the fluid and to the walls of the container. This is the principle behind many inventions in our daily lives such as the hydraulic brakes, lifts and hydraulic press.  

48. Concept of Pressure Variation of Pressure With Depth Pressure in a fluid increases with depth because more fluid rests on deeper layers, and the effect of this “extra weight” on a deeper layer is balanced by an increase in pressure. Pressure in a fluid at rest is independent of the shape or cross section of the container. It changes with the vertical distance, but remains constant in a horizontal plane. 3 2 1

49. Concept of Pressure Variation of Pressure With Depth Therefore,   3 2 1

50. Concept of Pressure   The pressure exerted by a fluid at equilibrium at any point in time due to the force of gravity is referred to as hydrostatic pressure . Hydrostatic pressure is proportional to the depth measured from the surface as the weight of the fluid increases when a downward force is applied.

51. Barometers Atmospheric pressure is measured by a device called a barometer ; thus, the atmospheric pressure is often referred to as the barometric pressure . The atmospheric pressure at a location is simply the weight of the air above that location per unit surface area. Therefore , it changes not only with elevation but also with weather conditions.

52. Manometers The simplest pressure gauge is the open-tube manometer . It consists of a U-shape glass tube which is filled with mercury or some other liquid. Traditionally one end of the manometer tube is left open, susceptible to atmospheric pressure, while a manometer hose is connected via a gas tight seal to an additional pressure source. While normally associated with gas pressures a manometer gauge can also be used to measure the pressure exerted by liquids.

53. Manometers Initially, one end of the tube is open so that pressure exerted on both sides will be same.  If one end of the U-tube is left open to the atmosphere and the other connected to an additional gas/liquid supply this will create different pressures . If the pressure from the additional gas/liquid supply is greater than the atmospheric pressure this will exert a downward pressure on the measuring liquid. Therefore, the liquid will be pushed down on one side with the greater pressure causing the liquid to rise on the side with the lesser pressure. The opposite would occur if the additional gas/liquid supply creates a lesser pressure than the atmospheric pressure.

54. Manometers Usually, there is number indicated on the U tube, from which displacement of liquid inside the tube can be evaluated and that will accurately provide pressure.

55. Manometers  

56. Hydrostatic Forces on Submerged Surfaces Fluid statics deals with problems associated with fluids at rest . Fluid statics is generally referred to as hydrostatics when the fluid is a liquid and as aerostatics when the fluid is a gas . Hydrostatics is the branch of physics that deals with the characteristics of fluids at rest, particularly with the pressure in a fluid or exerted by a fluid on an immersed body.

57. Hydrostatic Forces on Submerged Surfaces The design of many engineering systems such as water dams and liquid storage tanks requires the determination of the forces acting on their surfaces using fluid statics . The complete description of the resultant hydrostatic force acting on a submerged surface requires the determination of the magnitude, the direction, and the line of action of the force.  

58. Hydrostatic Forces on Submerged Surfaces The center of pressure is the point where the total sum of a pressure field acts on a body, causing a force to act through that point.  Where A = area of the horizontal surface  

59. Hydrostatic Forces on Submerged Surfaces ( Vertical wall )  

60. Hydrostatic Forces on Submerged Surfaces ( Slanted wall )   y L Ø H

61. Hydrostatic Forces on Submerged Surfaces ( Slanted wall )   y L Ø H

62. BUOYANCY A n object feels lighter and weighs less in a liquid than it does in air. This can be demonstrated easily by weighing a heavy object in water by a waterproof spring scale. Also , objects made of wood or other light materials float on water. These and other observations suggest that a fluid exerts an upward force on a body immersed in it.

63. BUOYANCY This force that tends to lift the body is called the buoyant force . Buoyancy is the tendency of an object to float in a fluid. The buoyant force is the upward force exerted on an object wholly or partly immersed in a fluid. This upward force is also called Upthrust . Due to the buoyant force, a body submerged partially or fully in a fluid appears to lose its weight, i.e. appears to be lighter

64. BUOYANCY Buoyancy results from the differences in pressure acting on opposite sides of an object immersed in a static fluid. If , object sinks. If , object floats. Therefore, If , object sinks. If , object floats.  

65. BUOYANCY Therefore, Where, = volume of object immersed  

66. BUOYANCY Buoyancy results from the differences in pressure acting on opposite sides of an object immersed in a static fluid . The following factors affect buoyant force : the density of the fluid the submerged volume/volume of fluid displaced the acceleration due to gravity An object whose density is greater than that of the fluid in which it is submerged tends to sink . Buoyancy( upthrust /thrust force) makes it possible for swimmers, fishes, ships, hand icebergs to stay afloat.

67. Applications of Buoyancy Hot Air Balloon The atmosphere is filled with air that exerts buoyant force on any object. A hot air balloon rises and floats because hot air is less dense than cool air. Therefore the buoyant force is able to displace the weight of the hot air ballon . Ship/Boat A ship floats on the surface of the sea because the volume of water displaced by the ship is enough to have a weight equal to the weight of the ship. A ship is constructed in a way so that the shape is hollow to make the overall density of the ship lesser than the seawater. Therefore, the buoyant force acting on the ship is large enough to support its weight.

68. ARCHIMEDES PRINCIPLE Archimedes Principle states that; T he buoyant force on an object is equal to the weight of the fluid displaced by the object. OR Every object is buoyed upwards by a force equal to the weight of the fluid the object displaces .

69. ARCHIMEDES PRINCIPLE This means that if you want to know the buoyant force on an object, you only need to determine the weight of the fluid displaced by the object . Apparent weight =  

70. ARCHIMEDES PRINCIPLE Archimedes principle helps us to determine the volume of an irregular object. Therefore if the object is completely submerged in the fluid, the volume of the displaced fluid equals the volume of the object.  

71. ARCHIMEDES PRINCIPLE

72. Principle of Floatation The floatation principle states that  when an object floats in a liquid, the buoyant force acting on the object is equal to the object's weight . This means;  

73. Examples A block of wood with length 50cm and width 30cm is placed in water. If the 10cm of the total height of the wood is immersed in the water, calculate the buoyant force on the wood. [ ] A cube with a side length of 5cm is submerged in oil. What is the buoyant force on the cube if the density of the oil is 896 kg/ . The weight of an object in air is 10N. When it is submerged in a liquid of relative density 1.15, the volume of the liquid increased from 15cm to 20cm. What is the weight of the object in water? A concrete slab weighs 150 N. When it is fully submerged under the sea, its apparent weight is 102 N. Calculate the density of the sea water if the volume of the sea water displaced by the concrete slab is 4800 cm 3 , [g = 9.8 m/s2 ] You plunge a basketball beneath the surface of a swimming pool until half the volume of the basketball is submerged. If the basketball has a radius of 12 cm, what is the buoyancy force on the ball due to the water ? [ ]  

74. Examples Contd The volume of a 500g sealed packet is 350cm3. Will the packet sink or float? What is the mass of displaced by the packet. [ ] A block of wood with the dimensions 0.12 by 0.34 by 0.43 cubic meters floats along a river with the broadest face facing down. The wood is submerged to a height of 0.053 meters. What is the mass of the piece of wood? [ ] Gold, whose mass is 193g is fully submerged in kerosene having an upward force of 8N. If the density of kerosene is 0.8kg/ , find the density of the gold. A boat is loaded with some goods floating on the sea with water displacement of 1.5m3. If the density of the seawater is 1020kg/m3, calculate the additional weight of goods to be added to displace 4.5m3 of seawater. A piece of aluminium with a mass of 1kg and relative density of 2.7 is suspended from a string and then completely immersed in water. Determine the volume of the piece of aluminium and the tension in the string after immersion. [ ]  

75. FLUID KINEMATICS

76. Analysis of Fluids in Motion Fluid Dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids (i.e. liquids and gases). The study of air and other gases in motion is called aerodynamics . The study of liquids in motion is referred to as hydrodynamics . Some of the important technological applications of fluid dynamics include meteorology, rocket engines, wind turbines, oil pipelines, and air conditioning systems.

77. Types Of Flow Steady And Unsteady Flow A steady flow is one in which the conditions/parameters (velocity, pressure, density, acceleration, etc ) at a point do not change with time . If at any point in the fluid, the conditions ( velocity, pressure, density, acceleration, etc ) change with time, the flow is described as unsteady .

78. Uniform And Non-Uniform Flow If the velocity at a given instant of time is same in both magnitude and direction at all points in the flow, the flow is said to be uniform flow . When the velocity changes from point to point in a flow at any given instant of time, the flow is described as non-uniform flow .

79. Compressible and Incompressible Flow The flow in which   density   of the fluid   varies during the flow is called compressible fluid flow . ( i . e.   ). This is applicable in gas flow. Incompressible fluid flow is when the density of the fluid remains the constant during the flow (i.e. ρ =constant). Practically,all liquids are treated as incompressible .

80. Streamlines And Streamtubes An imaginary line drawn in a fluid such that its tangent at each point is parallel to the local fluid velocity is called a  streamline . The streamlines drawn through each point of a closed curve constitute a  streamtube . Streamtube flow

81. The Continuity Principle When fluids move through a full pipe, the volume of fluid that enters the pipe must equal the volume of fluid that leaves the pipe, even if the diameter of the pipe changes .

82. The Continuity Principle The volume of fluid passing by a given location through an area during a period of time is called flow rate  Q , or more precisely,  volume flow rate (discharge) . ; where   V  is the volume and  t   is the elapsed time.  

83. The Continuity Principle The rate of flow of a fluid can also be described by the  mass flow rate . This is the rate at which a mass of the fluid moves past a point . The mass can be determined from the density and the volume;  

84. The Continuity Principle The mass flow rate is then where   ρ  is the density,  A   is the cross-sectional area, and  ϑ  is the magnitude of the velocity.  

85. The Continuity Principle The continuity principle is based on the conservation of mass, which implies that the mass of fluid entering a pipe has to be equal to the mass of fluid leaving the pipe. For this reason the velocity at the outlet (v2 )  is greater than the velocity of the inlet (v1 ).    

86. The Continuity Principle Using the fact that the mass of fluid entering the pipe must be equal to the mass of fluid exiting the pipe, we can find a relationship between the velocity and the cross-sectional area by taking the rate of change of the mass in and the mass out: This is known as the continuity equation.

87. The Continuity Principle If the density of the fluid remains constant through the constriction, that is the fluid is incompressible, then the density cancels from the continuity equation. The equation reduces to show that the volume flow rate into the pipe equals the volume flow rate out of the pipe.  

88. The Bernoulli Principle Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases, the pressure within the fluid decreases. Energy can neither be produced nor destroyed but only transformed . Therefore, Bernoulli’s Principle based on conservation of energy states that in a steady ideal flow of incompressible fluid, the sum of pressure energy, kinetic energy and potential energy remains constant at every section provided no energy is added or taken out by an external source.

89. The Bernoulli Principle Pressure energy + Kinetic energy + Potential energy = constant   Where; p = the pressure exerted by the fluid, ϑ = the velocity of the fluid, ρ = the density of the fluid h = the height of the container. Bernoulli’s equation formula is a relation between pressure,  kinetic energy, and gravitational potential energy of a fluid in a container, written as;

90. The Bernoulli Principle For Bernoulli’s equation to be applied, the following assumptions must be met : The flow must be steady. (Velocity, pressure and density cannot change at any point ). The flow must be incompressible – even when the pressure varies, the density must remain constant along the streamline . Friction by viscous forces must be minimal.

91. Bernoulli equation application in fluid mechanics The Bernoulli equation is applied to all incompressible fluid flow problems. The Bernoulli equation can be applied to devices such as the orifice meter,  Venturi meter, and Pitot tube  and its applications for measuring flow in open channels and inside tubes .

92. Hydraulic Coefficients Hydraulic coefficients are defined in the study of fluid flow through orifices, nozzle, etc.. There are three main hydraulic coefficients; Coefficient of contraction ( ) - defined as the ratio of the area of jet at Vena contracta  to the area of orifice (theoretical area ). Coefficient of velocity ( ) - defined as the ratio of actual velocity of jet of fluid at vena- contracta to the theoretical velocity of jet . Coefficient of discharge ( ) - defined as the ratio of actual discharge of fluid to the theoretical discharge .  

93. The Momentum Equation We have all seen moving fluids exerting forces. For instance, a jet of water from a hose exerts a force on whatever it hits . For a rigid body of mass m, Newton’s second law is expressed as, F = ma . Newton’s 2nd Law can be written as: The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force. Momentum = m* ϑ This statement is more in line with Newton’s original statement of the second law, and it is more appropriate for use in fluid mechanics when studying the forces generated as a result of velocity changes of fluid streams.

94. The Momentum Equation Therefore , in fluid mechanics, Newton’s second law is usually referred to as the linear momentum equation .

95. The Momentum Equation Lets consider fluid flow in a tube, assuming the flow is steady and non-uniform.  

96. The Momentum Equation Volume of fluid entering the tube with time = however, the velocity, = * mass of fluid entering the tube = Momentum of fluid entering tube = mass * velocity = Similarly, at the exit, we obtain the momentum of fluid leaving the tube = This means;  

97. The Momentum Equation We know from continuity, And if ; Then  

98. Examples Calculate the flow rate of fluid if it is moving with the velocity of 20 m/s through a tube of diameter 0.03 m .  A pipe has an initial cross-sectional area of 2 cm 2  that expands into a 5 cm 2  area. Initially, the velocity of the water through the smaller cross-sectional area of the pipe is 20 m/s . Determine the velocity of the water through the larger cross-sectional area section of the pipe . Water flows through a storage tank of radius 15cm with a velocity of 1.5m/s into a storage tank. Calculate the mass flow rate. Determine the resultant force needed to move a fluid of relative density 0.87 from 2m/s to 5m/s at a rate of 3 m 3 /s

99. Flow Through Pipes

100. Flow Patterns Through Pipes Laminar flow is a type of flow pattern of a fluid in which all the particles are flowing in parallel lines.In this type of fluid flow, particles move along well defined paths or steam lines . Turbulent flow is a type of flow pattern in which the paths of the fluid flow are irregular or flow in chaotic or random directions. Transitional flow is a mixture of laminar and turbulent flow, with turbulence in the center of the pipe, and laminar flow near the edges .

101. Flow Patterns Through Pipes The flow pattern can be determined using the Reynolds Number, Re. Where = density of fluid = velocity of flow D = diameter of pipe = dynamic viscosity of fluid Re < 2100 = laminar flow 2100 < Re < 4000 = transitional flow Re > 4000 = turbulent flow  

102. Velocity Distribution in Pipe Flow Not all fluid particles travel with the same velocity within a pipe. The shape of the velocity curve depends on whether the flow is laminar or turbulent. If the flow in a pipe is laminar, the velocity distribution at a cross section will be parabolic in shape with the maximum velocity being t the center which is twice the average velocity in the pipe. In turbulent flow, a fairly flat velocity distribution exists across the section of pipe with result that the entire fluid flows at a given single value.

103. Velocity Distribution in Pipe Flow Pumps , and especially centrifugal pumps, work most efficiently when the fluid is delivered in a surge-free, smooth, laminar flow. Any form of turbulence reduces efficiency, increases head loss and exacerbates wear on the pump’s bearings, seals and other components .

104. Head Loss In Fluid Flow The term pipe flow is generally used to describe flow through round pipes, ducts, nozzles, sudden expansions and contractions, valves and other fittings . When a gas or a liquid flows through a pipe, there is a loss of pressure in the fluid, because energy is required to overcome the viscous or frictional forces exerted by the walls of the pipe on the moving fluid. In addition to the energy lost due to frictional forces, the flow also loses energy (or pressure) as it goes through fittings, such as valves, elbows, contractions and expansions.

105. Head Loss In Fluid Flow The pressure loss in pipe flows is commonly referred to as  head loss . The frictional losses are referred to as  major losses ( H f )  while losses through fittings,valves etc , are called  minor losses ( H m ) . Together they make up the  total head losses   ( H T )  for pipe flows. The head loss due to the friction  ( H f ) in a given pipeline for a given discharge  is determined by the Darcy- Weisbach equation :

106. Head Loss In Fluid Flow where: f = friction factor ( unitless ) L = length of pipe ( ft ) D = diameter of pipe ( ft ) ϑ = fluid velocity ( ft /sec) g = gravitational acceleration ( ft /sec 2 )  

107. Head Loss In Fluid Flow The friction factor can be determined by the Moody Chart. The friction factor is characterized by; Flow regime (Reynolds Number) Relative roughness Pipe cross-section

108. Moody Chart Relative Roughness Renolds Number

109. Examples Water flows through a pipe 25 mm in diameter at a velocity of 6 m/s. Determine whether the flow is laminar or turbulent. Assume that the dynamic viscosity of water is 1.30 x 10 -3 kg/ ms. In a laboratory, the water supply is drawn from a roof storage tank 25 m above the water discharge point. If the friction factor is 0.008, the pipe diameter is 5 cm and the pipe is assumed vertical, calculate the velocity of flow if the head loss due to friction is 3.61m. If oil of specific gravity 0.9 and kinematic viscosity 1.2 x 10 -6 m 2 /s is pumped at a velocity of 12m/s through a pipe of 50mm, what type of flow will occur? Water flows in a steel pipe with a rate of 2 m 3 /s. Determine the head loss due to friction per meter length of the pipe ( d = 40mm, Re = 31500, RR=0.0011). Crude oil is flowing through a pipe of diameter 300mm at a rate of 400 litres per second. Find the head loss due to friction for a length of 50m of the pipe. (Re = 250000, RR = 0.004)

110. Pumping Systems

111. Pumps Pumps are used to transfer and distribute liquids in various industries. Pumps convert mechanical energy into hydraulic energy. Electrical energy is generally used to operate the various types of pumps.    Pumps have two main purposes. Transfer  of liquid from one place to another place (e.g. water from an underground into a water storage tank). Circulate liquid around a system (e.g. cooling water or lubricants through machines and equipment ).

112. Components of a Pumping System The basic components of a Pumping System are ; Pump casing and impellers Prime movers: electric motors, diesel engines or air system Piping used to carry the fluid Valves , used to control the flow in the system Other  fittings, controls and instrumentation End-use equipment, which have different requirements (e.g. pressure, flow) and therefore determine the pumping system components and configuration. Examples include heat exchangers, tanks and hydraulic machines.

113. Classification of Pumps

114. Reciprocating Pumps Pumping takes place by to and fro motion of the piston or diaphragm in the cylinder. It characterized by an operation that moves fluid by trapping a fixed volume, usually in a cavity, and then forces that trapped fluid into the discharge pipe. The Piston Pump operates by driving the piston down into the chamber, thereby compressing the fluid inside . When the piston is drawn back up, it opens the inlet valve and closes the outlet valve, thereby utilizing suction to draw in new fluid.

115. Centrifugal Pumps They   use a rotating impeller to increase the pressure of a fluid. Centrifugal pumps are commonly used to move liquids through a piping system. The fluid enters the pump impeller along or near to the rotating axis and is accelerated by the impeller, flowing radially outward into a diffuser or volute chamber (casing), from where it exits into the downstream piping system. Centrifugal pumps are used for large discharge through smaller heads.

116. Cativation It is a phenomenon caused as a result of vapor bubbles imploding. This is the result of bubble formation at the suction point due to pressure difference. Cavitation can have a serious negative impact on pump operation and lifespan. It can affect many aspects of a pump, but it is often the pump impeller that is most severely impacted. A relatively new impeller that has suffered from cavitation typically looks like it has been in use for many years; the impeller material may be eroded and it can be damaged beyond repair

117. Efficiencies of Centrifugal Pumps Pump efficiency, η (%) is a measure of the efficiency with which the pump uses the input power to convert the energy into useful output.  η % = P out /P in where  η = efficiency (%) P in  = power input P out  = power output   Pump input or brake horsepower (BHP) is the actual horsepower delivered to the pump shaft. Pump output or hydraulic or water horsepower (WHP) is the liquid horsepower delivered by the pump.

118. Efficiencies of Centrifugal Pumps These two terms are defined by the following formulas; where: BHP   is the brake horse power required ( Watts) WHP  is the water horse power (Watts ) ρ  is the fluid density (kg/m 3 ) g  is the standard acceleration of gravity (9.81 m/s 2 ) H  is the energy Head added to the flow (m) Q  is the flow rate (m 3 /s) η  is the efficiency of the pump (decimal)  

119. Examples

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