Fluid Mechanics and Machinery

Transcription

1 Fluid Mechanics and Machinery For III Semester B.E., Mechanical Engineering Students As per Latest Syllabus of Anna University - TN With Short Questions & Answers and University Solved Papers New Regulations 017 Dr. S. Ramachandran, M.E., Ph.D., Professor - Mech Sathyabama Institute of Science and Technology Chennai Dr. R. Venkatasubramani Professor and Head - Department of Civil Engineering Dr. Mahalingam College of Engineering and Technology Pollachi, Coimbatore (Near All India Radio) 80, Karneeshwarar Koil Street, Mylapore, Chennai Ph.: , aishram000@gmail.com, airwalk800@gmail.com

2 Fifth Edition: June /

3 Contents C.1 CONTENTS Unit I: Fluid Properties and Flow Characteristics 1.3 Introduction Fluid and Continuum Units and Dimensions in Fluid Mechanics Properties of Fluids Gas and Liquid Density (or) mass Density Specific weight (or) Weight density Specific Volume v Specific gravity (or) Relative density s Temperature Viscosity (Dynamic Viscosity) Compressibility 1 K Vapour Pressure Cavitation Gas and Gas laws Surface Tension Surface Tension on Droplet Surface Tension on a Hollow Bubble Surface Tension on a Liquid Jet Capillarity Thermodynamic Properties Newton s Law of Viscosity Types of Fluid Fluid Statics Concept of Fluid Static Pressure Pressure of Fluids P Atmospheric Pressure Absolute zero Pressure (or) Absolute pressure Gauge Pressure

4 C. Fluid Mechanics and Machinery Vacuum Pressure Pressure - Density - Height Relationship Manometry Measurement of Pressure Simple Manometers Differential Manometer Fluid Kinematics Concept of System: Control Volume Continuum & Free Molecular Flows Flow Characteristics Types of Fluid Flows Steady Flow and Unsteady Flow Uniform and Non-Uniform Flows Laminar Flow and Turbulent Flow Incompressible and Compressible Flow Rotational Flow and Irrotational Flow Subsonic Flow Sonic Flow Supersonic Flow Subcritical flow Critical flow Supercritical flow One Dimensional Flow Flow Visualization - Lines of Flow Stream Line Stream Tube Path Line Streak Line Mean Velocity of Flow Principles of Fluid Flow Principle of Conservation of mass

5 Contents C Types of Motion Or Deformation of Fluid Element Circulation and Vorticity Stream Function Velocity Potential Function Relation Between Stream Function and Velocity Potential Function Equipotential Line Flow Net Equations of Motion Euler s equation along a Stream Line Principle of Conservation of Energy Bernoulli s Equation Navier-stokes Equations Bernoulli s Equation: Applications Venturi Meter Orifice Meter Principle of Conservation of Momentum Moment of Momentum Equation Pitot-Tube Head Concept of Control Volume Hydraulic Co-efficients (a) Coefficient of Velocity (C v ) (b) Coefficient of contraction C c (c) Coefficient of discharge C d (d) Coefficient of resistance C r Pitot-static Tube (or Prandtl Tube) Unit II: Flow Through Circular Conduits.1 Introduction Reynolds Number Laminar Flow Through Circular Tubes (Circular Conduits and Circular Annuli) (Hagen Poiseullie s Equation)

6 C.4 Fluid Mechanics and Machinery - Law of Fluid Friction Head loss due to friction (for laminar flow) Hagen Poiseuille Equation (in terms of Discharge).13.5 Stoke s Law Turbulent Flow Hydrodynamical Smooth and Rough Surfaces Pipe Roughness Friction Factor - Resistance to Flow Through Smooth and Rough Pipe - Darcy-weisbach Equation Moody s Diagram Darcy- Weisbach s Equation - Expression for Loss of Head Due to Friction in Pipes Chezy s Formula for Loss of head due to friction in pipes Shear Stress in Turbulent Flow Velocity Distribution For Turbulent Flow in Pipes Energy Losses in Pipes Major energy (Head) losses Minor energy losses Hydraulic Gradient Line (H.G.L) and Energy Gradient Line (EGL) Total Energy Line (T.E.L) (or) Energy Gradient Line (E.G.L) Hydraulic Gradient Line (H.G.L) Flow Through Long Pipes Under Constant Head H Flow Through Pipes in Series (or) Flow Through Compound Pipes Equivalent Pipe Flow Through Parallel Pipes Flow Through Branched Pipes Siphon (Flow Through Pipeline With Negative Pressure) Power Transmission Through Pipes

7 Contents C.5. Important Note About Power Water Hammer Cavitation Boundary Layer Concepts Boundary Layer Theory Laminar boundary layer Turbulent Boundary layer Laminar Sub - layer Boundary layer thickness Displacement Thickness Momentum Thickness Energy thickness Von Karman Momentum Integral Equation For Boundary Layer Drag Force F D on Plate of Length L Velocity Profiles Turbulent Boundary Layer on A Flat Plate Drag and Lift Coefficient Total Drag on A Flat Plate Boundary Layer Separation High Lights and Important Formulae Unit III Dimensional Analysis 3.1 Dimensional Analysis Need For Dimensional Analysis Dimensional Homogeneity Methods of Dimensional Analysis Dimensionless Numbers (Non Dimensional Numbers) Similitude Types of Similitude Geometric Similarity Kinematic Similarity

8 C.6 Fluid Mechanics and Machinery Dynamic Similarity Specific Quantities Model Analysis Reynolds model law Froude model law Euler s model law Weber model law Mach model law Problems in model laws Model Testing of Partially Submerged Bodies Classification of Hydraulic Models Undistorted models Distorted models Scale Ratios for Distorted models Applications For Model Testing Limitations of Model Testing Unit IV: Impact of Jets and Hydraulic Pumps 4.1 Impact of Jets Hydrodynamic Thrust of Jet on A Fixed Surfaces Impact of Jet on A Hinged Plate Hydrodynamic Thrust of Jet on A Moving Surface (Flat and Curved Plates) Thrust of Jet of Water on Series of Vanes Workdone per second (or) Power of jet on a series of a radial curved vanes Efficiency of the Radial curved vane Roto Dynamic Machines Elementary cascade theory Theory of Rotodynamic (Turbo) Machines Roto dynamic machines classifications (i) Impulse and Reaction Turbines (ii) Axial, Radial and Mixed flow machines

9 Contents C.7 (iii) Backward, Radial and Forward Blade Impellers Euler s Equation Velocity components at the entry and exit of the rotor Velocity triangles Degree of reaction Pumps: Centrifugal Pump H-Q Characteristics of A Centrifugal Pump Typical Flow System Characteristics System characteristics Curve Pump characteristics curve Operating point Priming Cavitation Net Positive Suction Head (NPSH) NPSH Required NPSH R NPSH Available NPSH A Type Number Multi Stage Centrifugal Pumps Performance Curves Main characteristic curves (i) Q v/s H Curve (ii) Q v/s Curve (iii) Q v/s P Curve Operating Characteristics Model Testing of Centrifugal Pumps Specific Speed of Centrifugal Pump Shape numbers N q Reciprocating Pumps Working Principle of A Reciprocating Pump Discharge, Workdone and Power Required to Drive A Single Acting Reciprocating Pump

10 C.8 Fluid Mechanics and Machinery Discharge, Work Done and Power Required to Drive A Double Acting Pump Slip of Reciprocating Pump Negative Slip Indicator Diagram Effect of acceleration of piston in suction and delivery pipes on indicator diagram Effect of acceleration in the suction pipe and delivery pipe Effect of friction in the suction and delivery pipes on the Indicator Diagram Separation Maximum Speed of A Reciprocating Pump Air Vessels Pump Selection Various Types of Pumps Jet Pump Positive Displacement Pump Gear Pumps (Rotary Pumps) Working Principle of External Gear Pump Working Principle of Internal Gear Pump Lobe Pumps Screw Pump Vane Pump Piston Pumps Unit V: Hydraulic Turbines 5.1 Hydraulic Turbines - Introduction Classification of Hydraulic Turbines Heads and Efficiency of A Turbine Pelton Turbine (or) Pelton Wheel Governing of Pelton Wheel Reaction Turbines

11 Contents C Francis Turbine Working of a Francis Turbine Velocity Triangles and Work done by water in Francis Turbine Hydraulic Efficiency h for Francis Turbine Points to be remembered in Francis Turbine Solved Problems on Francis Turbine Axial Flow Reaction Turbines Working Principle of a Kaplan Turbine Velocity Diagram For Kaplan Turbine Specific Speed of Turbine Unit Quantities Draft Tube Cavitation in Reaction Turbines Performance Curves of Turbines Selection of Turbines Governing of Turbine Surge Tank

12 UNIT I FLUID PROPERTIES AND FLOW CHARACTERISTICS 1 Units and dimensions- Properties of fluids- mass density, specific weight, specific volume, specific gravity, viscosity, compressibility, vapor pressure, surface tension and capillarity. Flow characteristics concept of control volume - application of continuity equation, energy equation and momentum equation. UNIT II FLOW THROUGH CIRCULAR CONDUITS 1 Hydraulic and energy gradient - Laminar flow through circular conduits and circular annuli- Boundary layer concepts types of boundary layer thickness Darcy Weisbach equation friction factor- Moody diagram- commercial pipes- minor losses Flow through pipes in series and parallel. UNIT III DIMENSIONAL ANALYSIS 1 Need for dimensional analysis methods of dimensional analysis Similitude types of similitude - Dimensionless parameters- application of dimensionless parameters Model analysis. UNIT IV PUMPS 1 Impact of jets - Euler s equation - Theory of roto-dynamic machines various efficiencies velocity components at entry and exit of the rotor- velocity triangles - Centrifugal pumps working principle - work done by the impeller - performance curves - Reciprocating pumpworking principle Rotary pumps classification. UNIT V TURBINES 1 Classification of turbines heads and efficiencies velocity triangles. Axial, radial and mixed flow turbines. Pelton wheel, Francis turbine and Kaplan turbines- working principles - work done by water on the runner draft tube. Specific speed - unit quantities performance curves for turbines governing of turbines. TOTAL: 60 PERIODS

13 A Absolute zero Pressure, 1.64 Air Vessels, Atmospheric Pressure, 1.64 Axial Flow Reaction Turbines, 5.76 B Bellows, 1.79 Bernoulli s Equation, Boundary Layer Concepts,.19 Boundary Layer Theory,.19 Boundary Layer Separation,.16 Bourdon gauge, 1.76 C Capillarity, 1.1 Cauchy Number (C a), 3.46 Cavitation, 1.15,.17, Centrifugal Pump, 4.5 Chezy s formula,.50 Circulation And Vorticity, 1.1 Co-efficients, Coefficient of contraction (C c), Coefficient of resistance (C r), Coefficient of discharge (C d), Coefficient of Velocity (C v), Compressibility 1 K, 1.14 Continuum Flow, Critical Reynolds Number,.3 Critical flow, D Darcy - Weisbach formula,.49 Dimensional Analysis, 3.1 Dimensionless Numbers (Non Dimensional Numbers), 3.43 INDEX Index I.1 Displacement Thickness,.13 Distorted models, 3.80 Draft Tube, 5.10 Drag And Lift Coefficient,.154 Drag Force (F D) on Plate of Length (L),.141 Dynamic Similarity, 3.49 E Elementary cascade theory, 4.38 Energy Losses In Pipes,.49 Energy thickness,.136 Equations of Motion, Equipotential Line, Equivalent Pipe,.94 Euler Number (E u), 3.45 Euler s model law, 3.58 Euler s Equation, 4.44 F Flow Through Branched Pipes,.107 Flow Characteristics, 1.10 Flow Through Parallel Pipes,.96 Flow Net, Fluid Kinematics, Fluid Statics, 1.63 Francis Turbine, 5.41 Free Molecular Flow, Froude model law., 3.55 Froude model law (gravity force is predominant), 3.55 Froude Number (F r), 3.44 G Gauge Pressure, 1.65 Gear Pumps (Rotary Pumps), 4.01 Geometric Similarity, 3.48

14 I. Fluid Mechanics and Machinery - Governing of Turbine, Governing of Pelton Wheel, 5.9 H Head, Hydraulic Gradient Line (H.G.L),.67 I Impact of Jets, 4.1 Indicator Diagram, J Jet Pump, K Kinematic Similarity, 3.48 Kinematic Viscosity (), 1.1 L Laminar Sub - layer,.131 Laminar boundary layer,.130 Law of Fluid Friction,.10 Lobe Pumps, 4.05 M Mach Number (M), 3.46 Mach model law, 3.60 Main characteristic curves, Major energy (Head) losses,.49 Manometry, 1.68 Measurement of Pressure, 1.75 Minor energy losses,.50 Model Analysis, 3.54 Momentum Thickness,.134 Moody s Diagram,.41 Multi Stage Centrifugal Pumps, N Navier-stokes Equations, Net Positive Suction Head (NPSH), Newton s Law of Viscosity, 1.5 NPSH Required (NPSH R), NPSH Available (NPSH A), O One Dimensional Flow, Operating Characteristics, Orifice Meter, 1.16 P Path Line, Pelton Turbine (or) Pelton Wheel, 5.5 Performance Curves, Piezometer, 1.83 Piston Pumps, 4.08 Pitot-static Tube (or Prandtl Tube), 1.00 Pitot-tube, Positive Displacement Pump, 4.00 Power Transmission Through Pipes,.119 Priming, Properties of Fluids, 1.6 Pump Selection, Pumps, 4.5 R Reaction Turbines, 5.38 Reciprocating Pumps, Reynold s Number (R e),., 3.44 Reynolds Experiment,.3 Reynolds model law, 3.54 Roto Dynamic Machines, 4.38 S Scale Ratios for Distorted models, 3.80 Selection of Turbines, 5.11 Separation, Servomotor (or) Relay cylinder, 5.113

15 Index I.3 Shear Stress In Turbulent Flow,.47 Similitude, 3.47 Simple Manometers, 1.83 Siphon,.110 Slip of Reciprocating Pump, Sonic Flow, Specific Speed of Turbine, 5.88 Specific weight (or) Weight density, 1.8 Specific Quantities, 3.50 Specific Speed, 3.51 Stoke s Law,.37 Strain Gauge Pressure Transducer, 1.8 Streak Line, Stream Tube, Stream Line, Stream Function, 1.16 Subcritical flow, Subsonic Flow, Supercritical flow, Supersonic Flow, Surface Tension, 1.18 Surge Tank, T Thermodynamic Properties, 1.3 Total Energy Line (T.E.L) (or) Energy Gradient Line (E.G.L),.66 Turbulent Bodary Layer on a Flat Plate,.153 Turbulent Flow,.38 Turbulent Boundary layer,.131 Type Number, U Unit Discharge or Unit flow, 5.90 Unit Speed, 5.90 Unit Power, 5.91 Unit Quantities, 5.90 V Vacuum Pressure, 1.65 Vane Pump, 4.07 Vapour Pressure, 1.15 Velocity Diagram For Kaplan Turbine, 5.78 Velocity Profiles,.150 Velocity Potential Function, 1.17 Venturi Meter, Viscosity (Dynamic Viscosity), 1.10 Von Karman Momentum Integral Equation For Boundary Layer,.140 W Water Hammer,.16 Weber Number (W e), 3.45 Weber model law, 3.59

16 Unit I Fluid Properties and Flow Characteristics Units and dimensions - Properties of fluids - mass density, specific weight, specific volume, specific gravity, viscosity, compressibility, vapor pressure, surface tension and capillarity. Flow characteristics - concept of control volume - application of continuity equation, energy equation and momentum equation. 1.1 INTRODUCTION Fluid mechanics is the science which deals with the mechanics of liquids and gases. Fluid is a substance capable of flowing. Fluid mechanics may be divided into three branches. 1. Fluid Statics. Fluid Kinematics 3. Fluid Dynamics Fluid statics is the study of mechanics of fluids at rest. Fluid kinematics is the study of mechanics of fluids in motion. Fluid Kinematics deals with velocity, acceleration and stream lines without considering the forces causing the motion. Fluid dynamics is concerned with the relations between velocities, accelerations and the forces exerted by (or) upon fluids in motion Distinction between solid and fluid Table 1.1 Solid Fluid 1. Solid is a substance which undergoes a finite deformation depending upon elastic limit on application of a force. 1. A fluid is a substance which undergoes continuous deformation under application of a shear force, no matter how small the force might be.. Atoms (molecules) are usually closer together in solid.. Atoms are comparatively loosely packed in fluid.

17 1. Fluid Mechanics and Machinery - Solid Fluid 3. Intermolecular attractive forces between the molecules of a solid are large 3. Inter molecular forces are not so large enough to hold the various elements of the fluid together and hence fluid will flow under the action of slightest stress. 4. A solid has a definite shape. 4. A fluid has no definite shape of its own but it conforms to the shape of the container vessel. 1. FLUID AND CONTINUUM A fluid is a substance that deforms continuously when subjected to even an infinitesimal shear stress. This continuous deformation under the application of shear stress constitutes a flow. Solids can resist tangential stress at static conditions undergoing a definite deformation while a fluid can do it only at dynamic conditions undergoing a continuous deformation as long as the shear stress is applied. The concept of continuum assumes a continuous distribution of mass within the matter or system with no empty space. In the continuum approach, properties of a system such as density, viscosity, temperature, etc can be expressed as continuous functions of space and time. The continuum concept is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum concept can lead to results which are not of desired accuracy. However, under the right circumstances, the continuum concept produces extremely accurate results. A dimensionless parameter known as knudsen number K n /L, where is the mean free path and L is the characteristic length, aptly describes the degree of departure from continuum. The continuum concept usually holds good when k n 0.01.

18 Fluid Properties and Flow Characteristics 1.3 Continuum Mechanics (The study of the physics of continuous materials) Solid M echanics (The study of the physics of continuous materials with a defined rest shape) Fluid Mechanics (The study of the physics of continuous materials which deform when subjected to a force) Fluid mechanics is a sub discipline of continuum mechanics as illustrated here. 1.3 UNITS AND DIMENSIONS IN FLUID MECHANICS Generally all physical quantities are measured in certain units either fundamental units or derived units. Physical quantities expressed in terms of the Length L, mass (M) and time T are called fundamental quantities and units are called fundamental units. Some units, derived from the above fundamental units like pressure, velocity, acceleration etc. are called derived units System of Units There are four internationally accepted system of units namely 1. F.P.S unit system: In F.P.S. unit System Length is Measured in Foot, mass is measured in pound and time is measured in seconds.. C.G.S unit system: In C.G.S unit system the Length, Mass and Time are measured in centimeter, gram and seconds respectively. 3. M.K.S unit system: In M.K.S unit system the length, mass and time are measured in Meter, Kilogram and second respectively.

19 1.4 Fluid Mechanics and Machinery S.I. unit system: It is also called as International System of units. It has six basic units (Length - Meter, Mass - Kilogram, Time - Second, Current - Ampere, Temperature - Kelvin, Luminous intensity - Candela), two supplementary units (plane Angle - radians, Solid angle - steradian) and twenty seven derived units (some are density - kg/m 3, Force - Newton, Power - Watts etc). In fluid mechanics, the basic dimensions are 1. Length L. Mass M 3. Time T 4. Temperature K. The above units ae fundamental units and physical quantities are expressed in terms of above units. Now According to Newton s second law Where Force F ma m mass in kg a acceleration in meter s or m s L T LT So F ma MLT [ here m mass ] So it is a derived unit. [ here m meter, s second ] Unit of force is Newton N, So the dimension for force N MLT. Similarly, a list of derived quantities used in fluid mechanics and their dimensions in terms of L, M, T are tabulated.

20 Fluid Properties and Flow Characteristics 1.5 Table 1. Quantity with symbol SI unit Dimension Velocity V m/s LT 1 Acceleration a m/s LT Area A m L Density kg/m 3 ML 3 Volume V m 3 L 3 Force F N MLT Specific volume v m 3 /kg L 3 M 1 Pressure P or p N/m MLT L ML 1 T Flow rate (Discharge) Q) m 3 /sec L 3 T 1 Viscosity (Dynamic viscosity) N s/m MLT T L ML 1 T 1 Kinematic viscosity m /s L T 1 Frequency Hz Energy, work (or) Quantity of heat hertz (Hz) cycles s T 1 N m MLT L ML T Power P N m/s ML T ML T 3 T Specific weight w N/m 3 MLT L 3 ML T

21 1.6 Fluid Mechanics and Machinery - Example: Bernoulli s equation for the flow of an ideal fluid is given as follows where P Pressure in N/m P w Z V g constant Z Elevation height in m V Mean flow velocity in m/s w Specific weight in N/m 3 g Acceleration due to gravity 9.81 m/s Demonstrate that this equation is dimensionally homogeneous i.e all terms have the same dimensions. Term 1: Dimensions of P w N/m N/m 3 ML 1 T ML T L Term : Dimensions of Z L Term 3: Dimensions of V g m/s m/s m /s m/s L T LT L So all the terms have the same dimensions 1.4 PROPERTIES OF FLUIDS Gas and Liquid A fluid may be either a gas or a liquid. The molecules of a gas are much farther apart than those of a liquid. Hence a gas is highly compressible and a liquid is relatively incompressible. A vapour is a gas whose temperature and pressure are very closely nearer to the liquid phase. So steam is considered as vapour. A gas may be defined as a highly super heated vapour, i.e its state is far away from the liquid phase. So air is considered as a gas. Fluids are having the following properties:

22 Fluid Properties and Flow Characteristics 1.7 Table 1.3 Quantity Symbol Unit 1. Density (or) mass density kg/m 3. Specific weight (or) weight density w N/m 3 3. Specific volume v m 3 /kg 4. Specific gravity s No unit 5. Compressibility 1 K 6. Vapour pressure P N/m 7. Cohesion and Adhesion 8. Surface tension N/m 9. Capillary rise (or) fall h m 10. Viscosity-Dynamic viscosity (or) viscosity Ns/m 11. Kinematic viscosity m /s The above properties are discussed in detail Density (or) mass Density The density of a fluid is its mass per unit volume. Density mass volume kg m 3 So the unit of density is kg/m 3 and dimension is ML 3 [M for mass in kg and L for length in m] The density of liquids is normally constant while that of gases changes with the variation of pressure and temperature. Density of water at 4C is 1000 kg/m 3

23 1.8 Fluid Mechanics and Machinery Specific weight (or) Weight density Specific weight is the weight per unit volume. Its symbol is w. Specific weight represents the force exerted by gravity on a unit volume of fluid. Specific weight w Specific weight and density are related Weight of fluid Volume of fluid N m 3 Weight of fluid Mass of fluid g w Volume of fluid Volume of fluid g.. mass. Volume So w g...(1.1) Where g Acceleration due to gravity. The specific weight of water at 4 C is 9810 N/m 3. Density is absolute since it depends on mass and independent of location. Specific weight, on the other hand, is not absolute, since it depends on value of g which varies from place to place. Density and specific weight of fluids vary with temperature Specific Volume v Specific volume is the volume occupied by a unit mass of fluid. Its symbol is v. Its unit is m 3 /kg v Volume of fluid Mass of fluid V Mass m3 kg Specific volume is the reciprocal of density v 1...(1.) Note: v Specific volume; V Volume; u, v and V Velocity of flow.

24 Fluid Properties and Flow Characteristics Specific gravity (or) Relative density s Specific gravity of a liquid is the ratio of its density to that of pure water at a standard temp. 4C. Its symbol is s. It has no unit. where w 1000 kg/m 3 s Density of liquid Density of water w...(1.3) Density of liquid s w Specific gravity can also be defined in terms of specific weight. s Specific weight of liquid Specific weight of water w w w Specific weight of liquid w s w w...(1.4) where w w 9810 N/m 3 The specific gravity of gas is the ratio of its density to that of air Temperature It is intensive thermodynamic property which determines the hotness or the level of heat intensity of a body. A body is said to be hot if it is having high temperature indicating high level of heat intensity. Similarly a body is said to be cold if it is at low temperature indicating low level of heat intensity. The temperature of a body is measured by an instrument called thermometer. Temperatures are measured in well known two scales: (i) Centigrade scale C (ii) Fahrenheit scale F Both the scales are inter convertible as follows F 1.8C 3 C F 3 1.8

25 1.10 Fluid Mechanics and Machinery Viscosity (Dynamic Viscosity) Viscosity is the resistance offered to the movement of one layer of fluid by another adjacent layer of the fluid. Top layer u Velocity profile Y dy y u+du u du Lower layer Fig. 1.1 Velocity Variation Refer Fig Fluid is divided into different layers one over the other. Consider two layers of fluid. One is moving with velocity u. Another layer is moving with u du The distance between the layer is dy. The top layer causes a shear stress on the adjacent lower layer while the lower layer causes a shear stress on the adjacent top layer. This shear stress (denoted by ) is proportional to rate of change of velocity with respect to y. i.e du dy... (1.5) du dy... (1.6) The proportionality constant is and is known as coefficient of viscosity (or) absolute viscosity (or) dynamic viscosity (or) simply viscosity. du Velocity gradient (or) rate of shear strain (or) rate of shear dy deformation. The equation (1.6) can be rearranged as du/dy

26 Fluid Properties and Flow Characteristics 1.11 To find unit of : Shear stress Change in velocity Change of distance Force/Area Length Time 1 Length Force Time Length In MKS Unit System, Force is measured in kgf So unit of kgfsec m In CGS System, Force is Measured in dyne N/m m/s/m Ns m dyne sec So unit of cm or poise [... dyne sec 1 cm 1 poise] In SI System Force is represented in Newton N So unit of N s m Pa s [... N/m Pascal Pa] Numerical Conversion From MKS unit to CGS unit. We know that 1 kgf 9.81 N So, 1 1 kgf Sec m kgf Sec m 9.81 N S m kg 1 m/sec sec m [... Force N m a] g 10 cm/sec sec 10 4 cm 1 kgf Sec m g cm/sec sec 10 4 cm 98.1 dyne sec cm [... 1 g cm sec 1 dyne]

27 1.1 Fluid Mechanics and Machinery - In S.I Units Also 1 kgf sec m kgf Sec 1 m 98.1 poise [... dyne sec 1 cm 1 poise] 1 kgf sec m 9.81 N sec m 1 N Sec m 98.1 poise 10 poise poise 98.1 poise So, 1 N sec m 10 poise [ in S.I unit sec is represented as S] So 1 Ns 10 poise m Sometimes unit of viscosity is given at centipoise 1 1 Centipoise CP 100 poise A widely known metric unit for viscosity is the poise (p) 1 poise 0.1 Ns/m in S.I units 1 centipoise 0.01 poise Ns/m [... 1 kgf 9.81 N] Kinematic Viscosity () What is the importance of kinematic viscosity? (Nov/Dec AU) Kinematic viscosity is the ratio of dynamic viscosity to the density of the fluid. The symbol for kinematic viscosity is...(1.7) Unit for kinematic viscosity N s/m kg/m 3

28 Fluid Properties and Flow Characteristics 1.13 kg m s s 1 m m3 kg m s In metric system, is in stoke 1 stoke 1 cm /s 10 4 m /s in S.I units 1 centistoke 10 6 m /s... N kg m s Variation of Viscosity with temperature Write down the effect of temperature on viscosity of liquids and gases. (Nov/Dec 016, Nov/Dec 015, Nov/Dec AU) The viscosity of liquids decreases with the increase of temperature while the viscosity of gases increases with the increase of temperature. This is due to the reason that in liquids the cohesive forces predominates the molecular momentum transfer, due to closely packed molecules and with the increase in temperature, the cohesive forces decreases with the result of decreasing viscosity. But in case of gases, the cohesive forces are small and molecular momentum transfer predominates. With the increase in temperature, molecular momentum transfer increases and hence viscosity increases. The relationship between viscosity and temperature for (a) Liquids: t t...(1.8) (b) Gases: 0 t t...(1.9) Where, Viscosity of liquid/gas at tc, in poise 0 Viscosity of liquid/gas at 0C, in poise, are constants for liquid/gas.

29 1.14 Fluid Mechanics and Machinery Compressibility 1 K Compressibility of a liquid is inverse of its bulk modulus of elasticity. Compressive Stress Bulk modulus K Volumetric strain Consider a cylinder piston mechanism Let P 1 Initial pressure inside the cylinder; P Final pressure inside the cylinder V 1 Initial volume; V Final volume K Increase in Pressure Volumetric strain dp dp dv dv V V...(1.10) Since rise in pressure reduces the volume by d V, the strain is indicated as dv V C ylinder Piston dv V Fig. 1. Compressibility 1 K Relationship between Bulk Modulus K and pressure P of a Gas for Isothermal and Isentropic Process For Isothermal Process We know that for Isothermal process PV constant Partial diffirenting the above equation we get PdV VdP 0 P VdP dv From equation 1.10 we know that K P K For Isothermal process. dp dv/v...(1.11) dp dv/v

30 Fluid Properties and Flow Characteristics 1.15 For Adiabatic (or) isentropic process We know that for Adiabatic process PV constant Ratio of Specific heat Partial differentiating the above equation we get P V 1 dv V dp 0 dividing above equation by V 1, we get P dv VdP 0 P VdP dv K We get P K, for adiabatic or isentropic Process Vapour Pressure When the liquid is kept in a closed vessel, it evaporates into vapour and this vapour occupies the space between the free surface of the liquid and top of the vessel. This vapour exerts a partial pressure on the free surface of the liquid. This pressure is known as vapour pressure of liquid Cavitation What is cavitation? What causes it? (Nov/Dec AU) Now consider a flow of liquid in a system. If the pressure at any point in this flowing liquid becomes equal to the vapour pressure, the vaporization of the liquid begins and bubbles are formed. When these bubbles are carried by flowing liquid into region of high pressure, these bubbles collapse creating very high pressure. The metallic surface above which this liquid is flowing is subjected to these high pressures causing pitting action on surface. This process is called cavitation.

31 1.16 Fluid Mechanics and Machinery Gas and Gas laws The gas is the term applied to the state of any substance of which the evaporation from the liquid state is complete. Substances like Oxygen, Air, Nitrogen and Hydrogen etc may be regarded as gases within the temperature limits. A vapour may be defined as a partially evaporated liquid and consists of pure gasesous state together with the particles of liquid in suspension. Examples of vapour are steam, ammonia, SO, CO etc. A perfect gas or an ideal gas is one which obeys all gas laws under all conditions of temperature and pressures. No gas is perfect i.e., no gas strictly obeys the gas laws but within the temperature limits of applied thermodynamics many gases like H, O, N and even air may be regarded as perfect gases. Gas laws: Three variables control the physical properties of a gas. The pressure exerted by gas P, the volume V occupied by it and its temperature T. If any of these two variable are known, then the third can be calculated by gas laws. Gas law does not apply to vapours. Boyle s law: Boyle s law states that The volume of a given mass of a gas varies inversely as its absolute pressure, provided the temperature remains constant. V 1 or PV constant P Charles laws: Charle s law states that The volume of a given mass of a gas varies directly as its absolute temperature, provided the pressure is kept constant. The variation is same for all gases. V T or V T Constant Perfect gas law (combination of boyle s and charle s law). Let us assume that we have a perfect gas at absolute pressure, volume and absolute temperature of P 1, V 1 and T 1 respectively. Suppose the gas expands or contracts at a constant temperature to its volume V 1 such that the corresponding volume of its new absolute pressure is P. According to boyle s law P 1 V 1 P V...(1)

32 Fluid Properties and Flow Characteristics 1.17 Now let the gas be expanded (or contracted) further such that the pressure remains constant and its volume and absolute temperature change from V 1 to V and T 1 to T respectively. According to charles law V 1 V T 1 T...() According to Gay-Lussac s Law P T constant P 1 P T 1 T... (3) Combining these three gas laws, we can get a new equation, P 1 V 1 P V constant T 1 T If, is the volume of unit mass of gas, then this constant is R (characteristic gas constant) i.e, Pv R Pv RT T If m is the mass of gas under consideration, then the equation becomes PV mrt, which is called the equation of perfect gas or characteristic gas equation. Law Equation Constant variable Boyle s Law P 1 V 1 T 1 P V T P 1 V 1 P V Temperature T 1 T Charle s Law P 1 V 1 T 1 P V T V 1 T 1 V T Gay-Lussac s Law P 1 V 1 T 1 P V T P 1 T 1 P T Pressure P 1 P Volume V 1 V

33 1.18 Fluid Mechanics and Machinery Surface Tension When a liquid is put inside a narrow tube, the free surface of the liquid displays either a rise (or) depression near the walls of the tube. This phenomena is attributed to a property of fluids known as surface tension. Soap bubbles, small droplets of water and dew on a dry solid surface also are attributed to surface tension. Surface Tension is also defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrane under tension. The magnitude of this force per unit length of the free surface will have the same value as the surface energy per unit area. Surface tension in a liquid is caused by (i) Cohesive forces i.e forces of attraction between molecules of the same material or fluid and (ii) Adhesive forces i.e forces of attraction between molecules of different materials, say, the attraction between molecules of liquid and the molecules of container (or) air. Example for Cohesive Force When mercury is poured on the floor, it does not wet the surface of floor and forms sphere. When two spheres of mercury are brought close together, they combine together to form a bigger sphere. This means that the mercury molecules have cohesive tendency and have no tendency to adhere (adhesion) to the floor (solid surface). Example for Adhesive Force When water is poured on the floor, the water molecules wet the surface. This means that water molecules have adhesive tendency to adhere to the floor (solid surface). At the interface between a liquid and a gas ie at the liquid surface, and at the interface between two immiscible (not mixable) liquids, the out of balance attraction force between molecules forms an imaginary film capable of resisting tension. This liquid property is known as surface tension. Because the tension acts on a surface, we compare such forces by measuring the tension per unit length of surface. The surface tension is denoted by the symbol. The unit of surface tension is N/m.

34 Fluid Properties and Flow Characteristics 1.19 Consider a liquid in a vessel as shown in Fig B Consider two molecules A A and B of a liquid in a mass of liquid. The molecule A is attracted in all directions equally by the surrounding molecules of the liquid. Thus the resultant force acting on the Fig. 1.3 Surface tension molecule A is zero. But the molecule B is situated near the free surface. This molecule B is acted upon by upward and downward forces which are unbalanced. Thus net resultant force on molecule B is acting in a downward direction. Like molecule B, all the molecules near the free surface experience a downward force. So the free surface of the liquid acts like a very thin film under tension of the surface of the liquid Surface Tension on Droplet Let us consider a spherical droplet of liquid of radius r. The surface tension on the surface of droplet in N m P Pressue intensity inside the droplet in excess of the outside pressure intensity r Radius of droplet Let us assume that the droplet is cut into two halves as shown in Fig The forces acting on one half (say left half) will be P Water Droplet SurfaceTension Pressure Forces Fig. 1.4 Forces on droplet

35 1.0 Fluid Mechanics and Machinery - (i) Tensile force acting around the circumference of the cut portion (This tensile force is due to surface tension) and is given as circumference r (ii) Pressure force on P r the area Water droplet Fig. 1.6 Pressure inside a water droplet Under equibibrium conditions, these two forces will be equal and opposite in direction. Equate both forces, we get, d p P r r P or 4 r d...(1.1) Where d Dia of droplet The equation shows that with the decrease of radius of the droplet, pressure intensity inside the droplet increases. The excess pressure P inside a bubble is known to be a function of the surface tension and the radius. By dimensional reasoning determine how the excess pressure will vary if we double the surface tension and the radius. (Nov/Dec AU) Surface Tension on a Hollow Bubble A hollow bubble (soap bubble) has two surfaces in contact with air, one inside and other outside. The above Soap bubble two surfaces are subjected to surface tension. In such case, Surface tension on both circumferences r We can equate two forces acting on bubble Fig. 1.7 Pressure inside a soap bubble d p

36 Fluid Properties and Flow Characteristics 1.1 P r r P 4 r or P 8 d...(1.13) Surface Tension on a Liquid Jet Consider a liquid jet of diameter d and length L as shown in Fig P Pressure intensity inside the liquid jet above the outside pressure d Surface tension of the liquid Consider the equilibrium of forces acting on the half of the liquid jet Force acting on jet P area of half of jet (rectangle). L P L d. Force due to surface tension L P L d L Fig. 1.8 Force on liquid jet P L L d d r P r Capillarity Capillarity is the property of exerting forces on fluids by fine tubes. It is due to cohesion and adhesion. Capillarity may be defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid. When a fine glass tube is partially immersed in water, the water will rise in the tube to a height of h m above the water level. This happens when cohesion is of less effect than adhesion.

37 1. Fluid Mechanics and Machinery - On the otherhand, if the same tube is partially immersed in mercury, the mercury level in tube will be lower than the adjacent mercury level. This happens because cohesion predominates than adhesion. The rise of the liquid surface is called capillary rise and the depression of the liquid surface is called capillary depression (or) capillary fall Expression for Capillary Rise Refer capillary rise in Fig 1.9 (a) Mercury h=capillary fall water (a) Capillary Rise (b) Capillary Depression Lifting force created by surface tension Vertical component of the surface tension force circumference cos d cos Fig. 1.9 Capillary Rise (or) Fall Weight of the liquid of height h in the tube mass g Volume g Area h g 4 d h g Under equilibrium condition, equate both forces, we get, d cos 4 d h g Capillary rise h 4 cos g d...(1.14)

38 Fluid Properties and Flow Characteristics 1.3 Where h: Height of liquid in the tube (or) capillary rise : Surface tension of liquid. : Angle of contact between liquid and glass tube. : Density of liquid Generally value is approximately equal to zero and hence cos 1 then Capillary rise h 4 gd...(1.15) Expression for Capillary Fall Refer Capillary Fall in Fig. 1.9 (b) Hydrostatic force acting upward P 4 d g h 4 d Downward force making depression due to surface tension d cos Under equilibrium condition, Capillary fall h 4 cos g d g h 4 d d cos Thermodynamic Properties The characteristic equation (or) equation of state for perfect gases is given as Where PV m RT V Volume of gas in m 3 ; M mass of gas in kg R Characteristic gas constant or simply gas constant... (i) R for air 0.87 kj/kg K T absolute temperature in K Kelvin T tc 73

39 1.4 Fluid Mechanics and Machinery - The equation can be written Pv RT P R T Where v specific volume in m 3 /kg ; density in kg/m 3 Also, we can write the equation as... (ii)... (iii) P 1 V 1 P V P 3 V 3 constant T 1 T T 3 For Isothermal process, temperature remains constant Pv Constant or PV Constant. For Isentropic process Pv Constant or PV Constant. For Polytropic process, Pv n constant [n Index of compression (or) expansion] [n ranges from 0 to ] Ratio of specific heats C p C v C p and C v Specific heats of a gas at constant pressure and constant volume respectively According to Avagadro s hypothesis, all the pure gases at the same temperature and pressure have the same number of molecules per unit volume. For any gas, PVm MRT... (iv) Where Vm Molar Volume. (Molar volume is the volume occupied by the molecular mass of any gas at standard temperature and pressure). T Absolute temp in K; M Molecular weight So the equation (iv) can be written as PV m R T

40 R MR Universal gas constant From Avagadro s law, When P N/m and T K Molar volume Vm.4 m3 /kg mol PV m R T R PV m T J/kg mol K kj/kg mol K R kj/kg mole K for any gas [Universal gas constant] R R for gas A M for gas A R R air M for air Fluid Properties and Flow Characteristics kj/kg K [... M for air 8.96 kg/kg mol] 1.5 NEWTON S LAW OF VISCOSITY Define Newton s law of viscosity. (Nov/Dec 01 - AU) A fluid is a substance that deforms continuously when subjected to a shear stress, no matter how small that shear stress may be. A shear force is the force component tangent to a surface, and this force divided by the area of the surface is the average shear stress over the area. Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point. Y b b c c V u F t a d y x Fig. 1.1 Deformation Due to Constant Shear Force