*Fluid Mechanics & Hydraulic Machinery*

*Contents*

*Unit I:*

*Fluid Statics*

*Unit II:*

*Fluid Kinematics, Fluid Dynamics*

*Unit III:*

*Closed conduit flow*

*Unit IV:*

*Basics of turbo machinery*

*Unit V:*

*Hydro electric power stations*

*Unit VI:*

*Hydraulic turbines*

*Unit VII:*

*Performance of hydraulic turbines*

*Unit VIII:*

*Centrifugal pumps*

*UNIT I*

*FLUID STATISTICS**Units and dimensions:*

*Angular Velocity…………………….rad/g*

*Angular Acceleration……………….rad/g*

^{2}

*Discharge ……………………………m*

^{3}

*/g*

*Sp.mass(mass density)…………….kg/m*

^{3}

*Stress , elastic modulus……………N/m*

^{2}

*Sp.weight(weight density)………….N.S/m*

^{2}

*Dynamic viscosity*

*Kinematic viscosity ………………….m*

^{2}

*/g*

*Work, Energy,Torque………………..J(N-m)*

*Power …………………………………watt(J/g)*

*Surface Tension……………………..N/m*

*Momentum, momemt of momentum……kg.m/g*

*Entropy ……………………………..J/kg.k*

*Sp.heat, gas constant*

*Thermal conductivity………………..W/m.k*

*Dynamic viscosity………….poise(p)=1/10=N.S/m*

^{2}

^{ }

*Kinematic viscosity –stoke(s)= 10*

^{-4 }

*m*

^{2}

*/s*

*Pressure of fluid………………………..10*

^{5}

*pa*

*Mass density:(ρ)*

*Mass density or specific mass is defined as the mass of the fluid per unit volume denoted by ‘ρ’ units-kg/m*

^{3}

*Specific weight or weight density:(ω)*

*It is defined as the weight of the fluid per unit volume denoted by ‘ω’ –units N/m*

^{3}

*Sp.wt of water is 9810N/m*

^{3}

*or 981 dynes/cm*

^{3}

*ρ and ω are related as ω=ρ.g or ρ=ω/g*

*Specific volume:*

*It is defined as the volume of the fluid per unit weight i;e it is the reciprocal of specific weight of the fluid denoted by ‘v’ units – m*

^{3}

*/N*

*For ploblems involving gas flow is defined as volume per unit mass I;e m*

^{3}

*/kg*

*Specific gravity:*

*It is the ratio of sp.wt of a fluid to the sp.wt of a standard fluid. The standard fluid generally we use to measure is water.*

*δ = ρ of liquid/ρ of water*

*s.p gravity of gasses = ρ of gas/ ρ of H*

_{2}

*or air*

*Viscosity :*

*It is that property of a fluid by virtue of which it offers resistance to the movement of one layer of fluid over an adjacent layer. It is due to cohesion and molecular movement exchange between fluid layers and these effect as shear stress between the moving fluid layers*

*Figure 1*

*Consider two plates placed at a small distance ‘y’ apart the space between them is placed between them is filled with a fluid. The lower plate is a stationery plate and the upper one is a movable plate, which is parallel to the lower plate with a velocity ‘v’ by the application of force ‘F’ corresponding to area ‘A’ of moving plate in contact with liquid particles of the fluid in contact with each plate will adhere to it and if the dist. ‘Y’ and ‘v’ and not too great , the velocity will vary from zero at lower plate which is at rest.*

*F ~ Av/y*

*From fig the ratio v/y can be replaced by velocity gradient dv/dy which is the rate of angular deformation of the fluid. The shear stress ‘T’ is given by*

*T= F/A*

*= µ.v/y*

*= µ.dv/dy*

*The above eq is called Newtons eqn of viscosity and it is defines the proportional constant*

*µ= t/(dv/dy)*

*which is called coeff. Of viscosity or dynamic viscosity or simple viscosity of the fluid. Thus the dynamic viscosity µ, may be defined as the shear stress required to produce unit rate of angular deformation.*

*Units of ‘µ’ are N.s/m*

^{3}

*or kg/m.g*

*The ratio of dynamic viscosity ‘µ’ and the mass density ‘ρ’ is known as kine,atic viscosity and is denoted by ‘v’ I;e*

*v= µ/ρ*

*units of v is m*

^{2}

*/g*

*Effect of temperature on Viscosity:*

*The dynamic viscosity ‘µ’ of either a liquid or a gas is practically independent of the pressure for the pressure for the range that is ordinarily encountered in practice. However it varies widely with temperature for gasses , viscosity increases. For gasses viscosity increases with increase in temperature while for liquids it decreases with increase in temp this is because in liquids the viscosity is governed by the cohesive forces between the molecules of the liquid, where as in gasses the molecular activity plays a dominant role.*

*Vapour pressure:*

*All liquids possess a tendency to evaporate or vaporize to change from liquid to gases state. So, when a liquid is kept in a closed vessel. The liquid molecules get evaporated out the surface and make a vapour molecules exert a pressure on the liquid surface . so the pressure exerted by the liquid vapour molecules on the surface of the liquid is called vapour pressure.*

*Surface Tension*

*:*

*Due to molecular attraction, liquids posses certain properties such as cohesion and adhesion . Cohesion means inter molecular attraction between molecules of the same liquid to remain as an assemblage of particles. Adhesion means attraction between the molecules of a liquid and the solid boundary surface in contact with the liquid. Surface tension is due to cohesion bet liquid particles at the surface.*

*Consider two molecules A and B, A is inside the liquid, which has liquid molecules in all directions, so that the forces of attraction are in equilibrium and the molecule is equally attracted on all sides. Where as the liquid molecule B has some part of the molecule exposed to air and remaining part inside the liquid, so there is a net downward force on the molecule due to the attraction of the molecules below it. This force on the molecules at the surface is normal to the liquid surface. Due to these attractive forces ,a film or special layer seems to form on the liquid at the surface ,which is in tension and small loads can be supported over it.For ex; a small needle placed gently up on the water surface will not sink but will be supported by the tension at the water surface. The property of liquid surface to exert a tension is called the surface tension, denoted by ‘σ’ and it is the force required to maintain unit length of the film in equilibrium. The effect of surface tension is illustrated in case of a liquid droplet as well as a liquid jet.*

*Pressure intensity inside a droplet:*

*Consider a spherical droplet of radius r having inter pressure intensity p in excess of the outside pressure intensity. If the droplet cut into two halves then the forces acting on the one haif will be those due to pressure intensity p on the projected area and the tensile force due to surface tension acting around circumference.*

*P(πr*

^{2}

*) = σ(2πr) or p= 2σ/r*

*Pressure inside a soup bubble;*

*p(πr*

^{2}

*) = 2 σ(2πr) or p= 4σ/r*

*Pressure inside a liquid jet;*

*p(2rl) = σ(2l) or p=σ/r*

*Manometer:*

*Manometers are the devices used for the measurement of pressure of any liquids.*

*Simple manometers:*

*These are those which measure pressure at a point in a fluid contained in a pipe or a vessel.*

*Piezometer:*

*A piezometre is the simplest form of manometer which can be used for measuring moderate pressures of a liquid. It consists of a glass tube inserted in yhe wall of a pipe or a vessel containing a liquid whose pressure is to be measured. The tube extends vertically upward direction to such a height that liquid can freely rise in it without over flowing. Pressure at any point in liquid is indicated by the height of the liquid in the tube above that point , which can be read on the scale attached to it . Thus if w is the specific wt of the liquid, then the pressure at point m is p*

_{m}

*= wh*

_{m }

*, h*

_{m }

*is the pressure head at m.*

*Piezometres are also used to measure pressure heads in pipes where the liquid is in motion. Such tubes should enter the pipe in a direction at right angles to the direction of flow and the connecting end should be flush with the inner surface of the pipe. To prevent the capillary action from affecting the height of the column of the liquid in a piezometre , the glass tube having an internal diameter less than 12mm should not be used.*

*U-tube Manometre:*

*A tube manometer consists of a glass tube bent in U-shape, one end of which is connected of the gauge point and the other end remains open to the atmosphere. The tube contains a liquid of sp. Gravity greater than that of the liquid of which pressure is to be measured. Starting from A, if p*

_{A}

*is unknown pressure intensity at A, w is sp.wt. of water and s*

_{1}

*is sp.gravity of liquid in container, then pressure head at A = p*

_{A}

*/ws*

_{1}

*Since all points lying at the same horizontal level in the same continuous static mass of liquid have same pressure. Pressure at A = pressure at A*

^{1}

*. From A*

^{1}

*to*

^{ }

*B*

^{1}

*there being increase in elevation , pressure head decreases, so that pressure head at B*

^{1}

*= P*

_{A}

*.z/w.s*

_{1}

*. From c to d, pressure head decreases . If s*

_{2 }

*represents the sp.gravity of manometrenliquid then from equation, pressure head equivalent to CD column of manometer liquid = y.s*

_{2}

*/s*

_{1. }

*Thus pressure head , D=(P*

_{A}

*/ w.s*

_{1}

*) – z – (y.s*

_{2}

*/s*

_{1}

*)*

*But at D pressure head =0, (P*

_{A}

*/ w.s*

_{1}

*) – z – (y.s*

_{2}

*/s*

_{1}

*) = 0*

*p*

_{A}

*/w = zs*

_{1}

*+ ys*

_{2}

*If A contains a gas then sp.wt. is small so, p*

_{A}

*/w = ys*

_{2}

*In other terms, P*

_{A}

*/ w.s*

_{1}

*= (h*

_{1}

*s*

_{2}

*/s*

_{1}

*) – h*

_{2 }

*In terms of water, p*

_{A}

*/w = h*

_{1}

*s*

_{2}

*– h*

_{2}

*s*

_{1}

*Again if A contains gas then, p*

_{A}

*/w = h*

_{1}

*s*

_{2}

*Now the pressure head at A is given by A = P*

_{A}

*/ w.s*

_{1}

*i.e, P*

_{A}

*/ w.s*

_{1 }

*= -h*

*In terms of water, P*

_{A}

*/ w = -s*

_{1}

*h*

*For measuring –ve pressure larger magnitude of manometric liquid having higher sp.gravityis employed.*

*Atmospheric pressure:*

*the atmospheric air exerts a normal pressure up on all surfaces with which it is in contact and known as atmospheric pressure. It is also known as barometric pressure.atmospheric pressure at sea level is called standard atmospheric pressure.*

*Guage pressure:*

*it is the pressure measured with the help of pressure measuring instrument in which the atmosphericis taken as datum. The atmospheric pressure on the scale is marked as zero.*

*If the pressure of the liquid is below the local atmospheri*

*UNIT-II FLUID KINEMATICS & DYNAMICS*

*Path line:*

*A path line is the path followed by a fluid particles in motion.a path line shows the direction of particular particle as it moves ahead.in general,this is the curve in 3-D space. If the flow is 2-D then the curve becomes 2-D.*

*Stream line:*

*A stream line is the imaginary line with in the flow so that the tangant at any point on it indicates the velocity at that point.*

*Eqn. of a stream line in 3-D flow is given as*

*1)*

*A stream line cannot intersect itself nor two stream lines can cross.*

*2)*

*There cannot be any movement of the fluid mass across the stream lines.*

*3)*

*Stream line spacing varies inversely as the velocity; converging of stream lines in any particular direction shows accelerated flow in that direction.*

*4)*

*A path line gives the path of one particle at successive instants of time , whereas stream line indicates the direction of a no. of particles at the same instant.*

*5)*

*The series of streamlines represent the flow pattern at an instant.*

*Ø*

*In steady flow, the pattern of stream-line remains invatiant with time. The path lines and stream lines will then be identical.*

*Ø*

*In unsteady flow, the pattern of stream lines may (or) may not remain the same at the next instant.*

*Stream tube:*

*A stream tube is a fluid mass bounded by a group of stream lines. The contents of a stream tube are known as “current-filament”. EX: pipes, nozzles.*

*1)*

*Stream tube has finite dimensions.*

*2)*

*As there is flow perpendicular to stream lines, therefore, there is noflow across the surface(called stream surface) of the stream tube. The stream surface functions as if it were a solid wall.*

*3)*

*The shape of a stream tube changes from one instant to another because of change in the position of stream lines.*

*Streak lines:*

*The streak line is a curve which gives an instantaneous picture of the location of the fluid particles, which have passed through a given point.*

*EX: path taken by smoke coming out of chimney.*

*Types of fluid flow :**Fluids may be classified as follows :*

*1)*

*Steady and unsteady flows.*

*2)*

*Uniform and non-uniform flows.*

*3)*

*One, two and three dimensional flows.*

*4)*

*Rotational and ir-rotational flows.*

*5)*

*Laminar and turbulent flows.*

*6)*

*Compressible and in-compressible flows.*

*I.*

*Steady and unsteady flow:**The type of flow in which the fluid characteristics like velocity ,pressure, density etc, at a point do not change with time is called steady flow.*

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