GATE PSU. Chemical Engineering. Fluid Mechanics. For. The Gate Coach 28, Jia Sarai, Near IIT Hauzkhas, New Delhi 16 (+91) ,

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1 For GATE PSU Chemical Engineering Fluid Mechanics GATE Syllabus Fluid statics, Newtonian and non-newtonian fluids, Bernoulli equation, Macroscopic friction factors, energy balance, dimensional analysis, shell balances, flow through pipeline systems, flow meters, pumps and compressors, elementary boundary layer theory, packed and fluidized beds. 8, Jia Sarai, Near IIT Hauzkhas, New Delhi 16 (+91) ,

2 TABLE OF CONTENTS Chapter 1 FLUIDS AND THEIR PROPERTIES 1.1 Introduction 1. Fluids 1.3 Newton s Law of Viscosity The Continuum Concept of a Fluid Types of Fluid Ideal Fluid Real Fluid Compressible Incompressible Fluid Physical Properties Density Mass Density Specific Weight Specific Gravity Viscosity Dynamic Viscosity Kinematic Viscosity Surface Tension Capillary Vapor Pressure Compressibility and Bulk Modulus Difference Between Solids And Fluids Types of Fluid Behavior Newtonian Fluids Non Newtonian Fluids Plastic Fluids Dilatants Fluids Bingham Plastic Fluids Pseudo Plastic Fluids Time Dependent Fluids Thixotropic Fluids Rheopectic Fluids Temperature Dependency of Viscosity Liquids Gases 13 Chapter DIMENSIONAL ANALYSIS AND SCALE UP 16.1 Introduction 16. Dimensions And Units 16.3 Dimensional Homogeneity 18.4 Rayleigh s Method 18.5 Buckingham s π Theorem 19 i Visit us at,

3 Chapter 3 FLUID STATICS Pressure 1 3. Pascal s Law For Pressure At a Point 3.3 Hydrostatic Law Classification of Pressure Pressure Measuring Devices Piezo Meter Barometer Manometer U Tube Manometer Tilted Manometer U- Tube Manometer With One Leg Enlarged Pressure Gauge Buoyancy Centre of Buoyancy Type of Equilibrium of Floating Bodies Stable Equilibrium Un-stable Equilibrium Neutral Equilibrium 30 Chapter 4 FLUID KINEMATICS Introduction 3 4. Types of Fluid Flow Steady And Unsteady Flow Uniform And Non Uniform Flow One, Two And Three Dimensional Flow Rotational And Irrotational Flow Laminar And Turbulent Flow Compressible And Incompressible Flow Types of Flow Lines Path Line Stream Line Stream Tube Streak Line Continuity Equation Continuity Equation in Cartesian Coordinates Velocity Potential Stream Function Properties of Stream Function Cauchy Riemann Equations Relation Between Stream Function And Velocity Potential 4 Chapter 5 FLUID DYNAMICS Introduction 45 ii Visit us at,

4 5. Types of Heads of Liquid in Motion Potential Head Velocity Head Pressure Head Bernoulli s Equation Euler s Equation of Motion Bernoulli s Equation For Real Fluid 49 Chapter 6 FLOW OF INCOMPRESSIBLE FLUID IN PIPES Flow Regimes Navier Stokes Equation of Motion Relation Between Shear Stress And Pressure Gradient Flow of Viscous Fluid in Circular Pipes Hagen Poiseuille Equation Laminar Uni Directional Flow Between Stationary Parallel Plates Laminar Uni Directional Flow Between Parallel Plates Having Relative Motion Flow Losses in Pipes Darcy Equation For Head Loss Due to Friction Empirical Correlations For Coefficient of Friction Minor Head Losses Sudden Enlargement Sudden Contraction Pipes in Series And in Parallel Concept of Equivalent Pipe Prandtl Mixing Length Theory For Shear Stresses in Turbulent Flow 7 Chapter 7 BOUNDARY LAYER THEORY Boundary Layer Theory Boundary Layer Characteristics Boundary Layer Thickness Displacement Thickness Momentum Thickness Energy Thickness Energy Loss Laminar Boundary Layer Turbulent Boundary Layer 76 Chapter 8 FLOW MEASUREMENT Introduction Units of Flow Measurement of Flow Mechanical Flow Meters Variable Head Flow Meters Flow of Incompressible Fluid in Pipe 81 iii Visit us at,

5 Ratio Discharge Coefficient Flow Coefficient Orifice Flow Meter Venturi Meter Pitot Tube Variable Area Flow Meter Types of Variable Area Flow Meter Basic Equation 88 Chapter 9 FLOW PAST IMMERGED BODIES Introduction 9 9. Lift And Drag Concept Drag Coefficient Stokes Law Terminal Velocity Applications of Stokes Law Stagnation Point 96 Chapter 10 FLOW THROUGH POROUS MEDIA Description of Porous Media Hydraulic Diameter Porous Medium Friction Factor Porous Medium Reynolds Number Friction in Flow Through Beds of Solids Kozney Carman Equation Burke Plummer Equation Ergun Equation Packed Column Fluidization Minimum Fluidization Velocity 104 Chapter 11 TRANSPORTATION OF FLUIDS Pumping Equipments of Liquids Positive Displacement Pumps Centrifugal Pumps Characteristics of Centrifugal Pumps Specific Speed Operating Characteristics Affinity Laws For Pumps Cavitations And NPSH Vapor Lock And Cavitations NPSH 111 iv Visit us at,

6 Chapter 1 LEVEL 1 17 LEVEL 138 Chapter 13 UNSOLVED QUESTIONS 150 Chapter 14 QUESTIONS (004 TO 015) Chapter 15 SOLUTIONS 174 v Visit us at,

7 CHAPTER 3 FLUID STATICS Fluid statics deals with the fluids at rest. There are following points should be noted: i. A static fluid can have no shearing force acting on it, and that ii. Any force between the fluid and the boundary must be acting at right angles to the boundary. iii. For an element of fluid at rest, the element will be in equilibrium - the sum of the components of forces in any direction will be zero. iv. The sum of the moments of forces on the element about any point must also be zero. 3.1 PRESSURE A fluid will exert a normal force on any boundary it is in contact with. Since these boundaries may be large and the force may differ from place to place it is convenient to work in terms of pressure, p, which is the force per unit area. If the force exerted on each unit area of a boundary is the same, the pressure is said to be uniform. p F p A Force Area over which the force is applied Units: Newton s per square metre, N m, kg m -1 s - (The same unit is also known as a Pascal, Pa, i.e. 1Pa = 1 N m ) (Also frequently used is the alternative SI unit the bar, where 1 bar = 10 5 N m ) Dimensions: ML -1 T - 1 Visit us at,

8 3. PASCAL S LAW FOR PRESSURE AT A POINT For a fluid having no shear forces, the direction of the plane over which the force due to pressure acts has no effect on the magnitude of the pressure at a point. This result is known as the Pascal s Law and its derivation is given below. By considering a small element of fluid in the form of a triangular prism which contains a point P, we can establish a relationship between the three pressures p x in the x direction, p y in the y direction and p s in the direction normal to the sloping face. The fluid is a rest, so we know there are no shearing forces, and we know that all force are acting at right angles to the surfaces.i.e. p s acts perpendicular to surface ABCD, p x acts perpendicular to surface ABFE and p y acts perpendicular to surface FECD. And, as the fluid is at rest, in equilibrium, the sum of the forces in any direction is zero. Summing forces in the x-direction: Force due to p x, Fxx px AreaABCD Sin Component of force in the x-direction due to p s, Fxs ps AreaABCD Sin y y ps s z Sin s s p y z s Component of force in x-direction due to p y, Fxy 0 To be at rest (in equilibrium) Fxx Fxs Fxy 0 Visit us at,

9 x x y 0 p x y p y z p p Similarly, summing forces in the y-direction. Force due to p y, s Fyy py AreaABCD py x z Component of force due to p s, Fys ps AreaABCD Cos x x ps s z Cos s s p x z s Component of force due to p x, Force due to gravity, To be at rest (in equilibrium) Fyx 0 Weight = specific weight volume of element 1 g xy z Fyx Fys Fyy weight 0 1 py x z ps x z g xyz 0 The element is small i.e. δx, δy and δz are small, and so δ δ δ x y z is very small and considered negligible, hence p y p s Thus px py ps The element is so small that it can be considered a point so the derived expression p p p, indicates that pressure at any point is the same in all directions. x y s Pressure at any point is the same in all directions. This is known as Pascal s Law and applies to fluids at rest. 3 Visit us at,

10 3.3 HYDROSTATIC LAW Consider a hypothetical differential cylindrical element of fluid of cross sectional area A and height (z - z 1 ). Upward force due to pressure P 1 on the element = P 1 A Downward force due to pressure P on the element = P A Force due to weight of the element = mg = ρ A(z - z 1 )g Equating the upward and downward forces, P 1 A = P A + ρ A (z - z 1 ) g P - P 1 = - ρ g (z - z 1 ) P g z Thus in any fluid under gravitational acceleration, pressure increases, with increasing depth z in the down-ward direction. 3.4 CLASSIFICATION OF PRESSURE Pressure can be classified on the basis of the atmospheric pressure, as positive or negative pressure. Positive pressure is termed as Gauge pressure i.e. pressure above atmospheric pressure. Negative pressure is termed as Vacuum Pressure i.e. Pressure below atmospheric pressure. Absolute Pressure = Atmospheric Pressure + Gauge (Vacuum) Pressure pabsolute patm gh pgauge gh As g is (approximately) constant, the gauge pressure can be given by stating the vertical height of any fluid of density ρ which is equal to this pressure. 4 Visit us at,

11 p gh This vertical height is known as head of fluid. 3.5 PRESSURE MEAUSRING DEVICES The relationship between pressure and head is used to measure the pressure of any fluid and pressure can be measure with the help of following devices: PIEZOMETER The simplest manometer is a tube, open at the top, which is attached to the top of a vessel containing liquid at a pressure (higher than atmospheric) to be measured. An example can be seen in the figure below. This simple device is known as a Piezometer tube. As the tube is open to the atmosphere the pressure measured is relative to atmospheric so is gauge pressure. Pressure at A = Pressure due to column of liquid above A pa gh 1 Pressure at B = Pressure due to column of liquid above B pb gh This method can only be used for liquids (i.e. not for gases) and only when the liquid height is convenient to measure. It must not be too small or too large and pressure changes must be detectable. 5 Visit us at,

12 3.5. BAROMETER A barometer is a device for measuring atmospheric pressure. A simple barometer consists of a tube more than 30 inch (760 mm) long inserted in an open container of mercury with a closed and evacuated end at the top and open tube end at the bottom and with mercury extending from the container up into the tube. Strictly, the space above the liquid cannot be a true vacuum. It contains mercury vapor at its saturated vapor pressure, but this is extremely small at room temperatures (e.g Pa at 0 0 C). The atmospheric pressure is calculated from the relation P atm = ρgh where ρ is the density of fluid in the barometer MANOMETER A somewhat more complicated device for measuring fluid pressure consists of a bent tube containing one or more liquid of different specific gravities. Such a device is known as manometer. In using a manometer, generally a known pressure (which may be atmospheric) is applied to one end of the manometer tube and the unknown pressure to be determined is applied to the other end. In some cases, however, the difference between pressures at ends of the manometer tube is desired rather than the actual pressure at the either end. A manometer to determine this differential pressure is known as differential pressure manometer U TUBE MANOMETER Using a U -Tube enables the pressure of both liquids and gases to be measured with the same instrument. The U is connected as in the figure below and filled with a fluid called the manometric fluid. The fluid whose pressure is being measured should have a mass density less than that of the manometric fluid and the two fluids should not be able to mix readily - that is, they must be immiscible. 6 Visit us at,

13 Pressure in a continuous static fluid is the same at any horizontal level so, Pressure at B = Pressure at C P B = P C For the left hand arm Pressure at B = Pressure at A + Pressure due to height h of fluid being measured p p gh c atm man As we are measuring gauge pressure we can subtract p atm giving Pressure at B = Pressure at C P B = P C p gh gh A man 1 If the fluid being measured is a gas, the density will probably be very low in comparison to the density of the manometric fluid i.e. ρ man >> ρ. In this case the term ρgh 1 can be neglected, and the gauge pressure given by p A man gh TILTED MANOMETER When the pressure to be measured is very small then tilting the arm provides a convenient way of obtaining a larger (more easily read) movement of the manometer. The above arrangement with a tilted arm is shown in the figure below. The pressure difference is still given by the height change of the manometric fluid but by placing the 7 Visit us at,

14 scale along the line of the tilted arm and taking this reading large movements will be observed. The pressure difference is then given by p p gz gxsin U TUBE WITH ONE LEG ENLARGED The U -tube manometer has the disadvantage that the change in height of the liquid in both sides must be read. This can be avoided by making the diameter of one side very large compared to the other. In this case the side with the large area moves very little when the small area side move considerably more. Assume the manometer is arranged as above to measure the pressure difference of a gas of (negligible density) and that pressure difference is p 1 p. If the datum line indicates the level of the manometric fluid when the pressure difference is zero and the height differences when pressure is applied is as shown, The volume of liquid transferred from the left side to the z d /4 And the fall in level of the left side is Volume moved z1 Area of left side z z d /4 d D /4 D 1 z We know from the theory of the U tube manometer that the height different in the two columns gives the pressure difference so d p1 p g z z D 1 d p1 p gz D Clearly if D is very much larger than d then (d / D) is very small so p p gz LIMITATIONS TO MANOMETER 8 Visit us at,

15 The manometer in its various forms is an extremely useful type of pressure measuring instrument, but suffers from a number of limitations. While it can be adapted to measure very small pressure differences, it cannot be used conveniently for large pressure differences - although it is possible to connect a number of manometers in series and to use mercury as the manometric fluid to improve the range. (limitation) A manometer does not have to be calibrated against any standard; the pressure difference can be calculated from first principles. ( Advantage) Some liquids are unsuitable for use because they do not form well-defined menisci. Surface tension can also cause errors due to capillary rise; this can be avoided if the diameters of the tubes are sufficiently large - preferably not less than 15 mm diameter. (limitation) A major disadvantage of the manometer is its slow response, which makes it unsuitable for measuring fluctuating pressures.(limitation) It is essential that the pipes connecting the manometer to the pipe or vessel containing the liquid under pressure should be filled with this liquid and there should be no air bubbles in the liquid.(important point to be kept in mind) PRESSURE GAUGE The pressure to be measured is applied to a curved tube, oval in cross section. Pressure applied to the tube tends to cause the tube to straighten out, and the deflection of the end of the tube is communicated through a system of levers to a recording needle. This gauge is widely used for steam and compressed gases. The pressure indicated is the difference between that communicated by the system to the external (ambient) pressure, and is usually referred to as the gauge pressure. 3.6 BUOYANCY Whenever a body is immersed wholly or partially in a fluid it is subjected to an upward force which tend to lift (or buoy) it up. This tendency for an immersed body to be lifted up in the fluid, due to an upward force opposite to action of gravity is known as buoyancy. The force tending to lift up the body under such conditions is known as buoyant force or up thrust. The magnitude of the buoyant force can be determined with help of Archimedes Principle which states as follows: 9 Visit us at,

16 When a body is immersed in a fluid either wholly or partially, it is buoyed or lifted up by a force, which is equal to the weight of fluid displaced by the body CENTER OF BUOYANCY The point of application of the force of buoyancy on the body is known as center of buoyancy. It is always the center of gravity of the volume of fluid displaced TYPES OF EQUILIBRIUM OF FLOATING BODIEES The equilibrium of the floating bodies is of the following types: STABLE EQUILIBRIUM When a body is given a small angular displacement (i.e. tilted slightly), by some external force, and then it returns back to its original position due to the internal forces (the weight and the up thrust), such an equilibrium is called stable equilibrium UNSTABLE EQUILIBRIUM If the body does not return to its original position from the slightly displaced angular position and heels further away, when given an small angular displacement, such an equilibrium is called an unstable equilibrium NEUTRAL EQUILIBRIUM If a body, when given a small angular displacement, occupies a new position and remains at rest in this position it is said to possess a neutral equilibrium. Example 3.1 A simple U tube manometer is installed across an orifice meter. The manometer is filled with mercury (sp. Gravity = 13.6) and the liquid above the mercury is CCl 4 (sp. Gravity = 1.6). The manometer reads 00 mm. What is the pressure difference over the manometer in Newton per square meter? Solution: Specific gravity of heavier liquid, S hl = 13.6 Specific gravity of lighter liquid, S ll = 1.6 Reading of the manometer, y = 00 mm Pressure difference over the manometer: p 30 Visit us at,

17 Differential head, S h y S hl ll h h 1500 mm of CCl Pressure difference over the manometer, 1500 p wh p 3544 N / m 4 For details and full study material Contact us: 8, Jia Sarai, Near IIT Hauzkhas, New Delhi 16 (+91) Visit us at,

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