Fluid Mechanics PDF

FLUID MECHANICS Syllabus (Hand-Out) Fluid Mechanics Overview Characteristics of Fluids Measures of Fluid Mass and Weight Viscosity Compressibility Vapor Pressure Surface Tension  Fluid Mechanics Overview...

FLUID MECHANICS Syllabus (Hand-Out) Fluid Mechanics Overview Characteristics of Fluids Measures of Fluid Mass and Weight Viscosity Compressibility Vapor Pressure Surface Tension  Fluid Mechanics Overview Fluid Mechanics Gas Liquids Statics Dynamics F  0 i  F  0 , Flows i Water, Oils, Stability Air, He, Ar, Alcohols, Buoyancy N2, etc. Pressure Compressible/ etc. Incompressible Laminar/ Surface Turbulent Tension Steady/Unsteady Compressibility Density Viscosity Vapor Viscous/Inviscid Pressure Chapter 1: Introduction Fluid Dynamics: Chapter 2: Fluid Statics Rest of Course  Characteristics of Fluids Gas or liquid state “Large” molecular spacing relative to a solid “Weak” intermolecular cohesive forces Can not resist a shear stress in a stationary state Will take the shape of its container Generally considered a continuum Viscosity distinguishes different types of fluids  Measures of Fluid Mass and Weight: Density The density of a fluid is defined as mass per unit volume. m  v m = mass, and v = volume. Different fluids can vary greatly in density Liquids densities do not vary much with pressure and temperature Gas densities can vary quite a bit with pressure and temperature Density of water at 4° C : 1000 kg/m3 Density of Air at 4° C : 1.20 kg/m3 1 Alternatively, Specific Volume:   Measures of Fluid Mass and Weight: Specific Weight The specific weight of fluid is its weight per unit volume.   g g = local acceleration of gravity, 9.807 m/s2 Specific weight characterizes the weight of the fluid system Specific weight of water at 4° C : 9.80 kN/m3 Specific weight of air at 4° C : 11.9 N/m3 Measures of Fluid Mass and Weight: Specific Gravity The specific gravity of fluid is the ratio of the density of the fluid to the density of water @ 4° C.  SG   H 2O Gases have low specific gravities A liquid such as Mercury has a high specific gravity, 13.2 The ratio is unitless. Density of water at 4° C : 1000 kg/m3  Viscosity: Introduction The viscosity is measure of the “fluidity” of the fluid which is not captured simply by density or specific weight. A fluid can not resist a shear and under shear begins to flow. The shearing stress and shearing strain can be related with a relationship of the following form for common fluids such as water, air, oil, and gasoline: du   dy  is the absolute viscosity or dynamics viscosity of the fluid, u is the velocity of the fluid and y is the vertical coordinate as shown in the schematic below: “No Slip Condition”  Viscosity: Measurements A Capillary Tube Viscosimeter is one method of measuring the viscosity of the fluid. Viscosity Varies from Fluid to Fluid and is dependent on temperature, thus temperature is measured as well. Units of Viscosity are N·s/m2 or lb·s/ft2 Movie Example using a Viscosimeter:  Viscosity: Newtonian vs. Non-Newtonian Toothpaste Latex Paint Corn Starch Newtonian Fluids are Linear Relationships between stress and strain: Most common fluids are Newtonian. Non-Newtonian Fluids are Non-Linear between stress and strain  Viscosity: Kinematic Viscosity    Kinematic viscosity is another way of representing viscosity Used in the flow equations The units are of L2/T or m2/s and ft2/s  Compressibility of Fluids: Bulk Modulus dp E  d /  P is pressure, and r is the density. Measure of how pressure compresses the volume/density Units of the bulk modulus are N/m2 (Pa) and lb/in.2 (psi). Large values of the bulk modulus indicate incompressibility Incompressibility indicates large pressures are needed to compress the volume slightly It takes 3120 psi to compress water 1% at atmospheric pressure and 60° F. Most liquids are incompressible for most practical engineering problems. Compressibility of Fluids: Compression of Gases Ideal Gas Law: p  RT P is pressure, r is the density, R is the gas constant, and T is Temperature Isothermal Process (constant temperature): p Math  cons tan t E  p  Isentropic Process (frictionless, no heat exchange): p Math k  cons tan t E  kp  k is the ratio of specific heats, cp (constant pressure) to cv (constant volume), and R = cp – cv. If we consider air under at the same conditions as water, we can show that air is 15,000 times more compressible than water. However, many engineering applications allow air to be considered incompressible. Compressibility of Fluids: Speed of Sound A consequence of the compressibility of fluids is that small disturbances introduced at a point propagate at a finite velocity. Pressure disturbances in the fluid propagate as sound, and their velocity is known as the speed of sound or the acoustic velocity, c. dp Ev c or c  d  Isentropic Process (frictionless, no heat exchange because): kp c  Ideal Gas and Isentropic Process: c  kRT  Compressibility of Fluids: Speed of Sound Speed of Sound in Air at 60 °F  1117 ft/s or 300 m/s Speed of Sound in Water at 60 °F  4860 ft/s or 1450 m/s If a fluid is truly incompressible, the speed of sound is infinite, however, all fluids compress slightly. Example: A jet aircraft flies at a speed of 250 m/s at an altitude of 10,700 m, where the temperature is -54 °C. Determine the ratio of the speed of the aircraft, V, to the speed of sound, c at the specified altitude. Assume k = 1.40 Ideal Gas and Isentropic Process: c  kRT c  1.40 * (286.9 J / kgK ) * 219 K c  296.6 m / s  Compressibility of Fluids: Speed of Sound Example (Continued): V Ratio  c 250 m / s Ratio  296.6 m / s Ratio  0.84  The above ratio is known as the Mach Number, Ma  For Ma < 1 Subsonic Flow  For Ma > 1 Supersonic Flow For Ma > 1 we see shock waves and “sonic booms”: 1) Wind Tunnel Visualization known as Schlieren method 2) Condensation instigated from jet speed allowing us to see a shock wave  Vapor Pressure: Evaporation and Boiling Evaporation occurs in a fluid when liquid molecules at the surface have sufficient momentum to overcome the intermolecular cohesive forces and escape to the atmosphere. Vapor Pressure is that pressure exerted on the fluid by the vapor in a closed saturated system where the number of molecules entering the liquid are the same as those escaping. Vapor pressure depends on temperature and type of fluid. Boiling occurs when the absolute pressure in the fluid reaches the vapor pressure. Boiling occurs at approximately 100 °C, but it is not only a function of temperature, but also of pressure. For example, in Colorado Spring, water boils at temperatures less than 100 °C. Cavitation is a form of Boiling due to low pressure locally in a flow.  Surface Tension At the interface between a liquid and a gas or two immiscible liquids, forces develop forming an analogous “skin” or “membrane” stretched over the fluid mass which can support weight. This “skin” is due to an imbalance of cohesive forces. The interior of the fluid is in balance as molecules of the like fluid are attracting each other while on the interface there is a net inward pulling force. Surface tension is the intensity of the molecular attraction per unit length along any line in the surface. Surface tension is a property of the liquid type, the temperature, and the other fluid at the interface. This membrane can be “broken” with a surfactant which reduces the surface tension.  Surface Tension: Liquid Drop The pressure inside a drop of fluid can be calculated using a free-body diagram: Real Fluid Drops Mathematical Model R is the radius of the droplet, s is the surface tension, Dp is the pressure difference between the inside and outside pressure. The force developed around the edge due to surface tension along the line: F  2 R  Applied to Circumference surface This force is balanced by the pressure difference Dp: 2 Applied to Area F pressure  DpR  Surface Tension: Liquid Drop Now, equating the Surface Tension Force to the Pressure Force, we can estimate Dp = pi – pe: 2 Dp  R This indicates that the internal pressure in the droplet is greater that the external pressure since the right hand side is entirely positive. Is the pressure inside a bubble of water greater or less than that of a droplet of water? 4 Prove to yourself the following result: Dp  R  Surface Tension: Capillary Action Capillary action in small tubes which involve a liquid-gas-solid interface is caused by surface tension. The fluid is either drawn up the tube or pushed down. “Wetted” “Non-Wetted” Adhesion Cohesion Adhesion Cohesion Adhesion > Cohesion Cohesion > Adhesion h is the height, R is the radius of the tube, q is the angle of contact. The weight of the fluid is balanced with the vertical force caused by surface tension.  Flow In Circular Pipes Objective ä To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate. ä To correlate this in terms of the friction factor and Reynolds number. ä To compare results with available theories and correlations. ä To determine the influence of pipe fittings on pressure drop ä To show the relation between flow area, pressure drop and loss as a function of flow rate for Venturi meter and Orifice meter.  APPARATUS Pipe Network Rotameters Manometers  Theoretical Discussion Fluid flow in pipes is of considerable importance in process. Animals and Plants circulation systems. In our homes. City water. Irrigation system. Sewer water system  Fluid could be a single phase: liquid or gases Mixtures of gases, liquids and solids  NonNewtonian fluids such as polymer melts, mayonnaise  Newtonian fluids like in your experiment (water)  Theoretical Discussion Laminar flow To describe any of these flows, conservation of mass and conservation of momentum equations are the most general forms could be used to describe the dynamic system. Where the key issue is the relation between flow rate and pressure drop. If the flow fluid is: a. Newtonian b. Isothermal c. Incompressible (dose not depend on the pressure) d. Steady flow (independent on time). e. Laminar flow (the velocity has only one single component)  Laminar flow Navier-Stokes equations is govern the flow field (a set of equations containing only velocity components and pressure) and can be solved exactly to obtain the Hagen-Poiseuille relation. Pz Flow If the principle of conservation of momentum is applied to a fixed volume element through which fluid Vz(r) is flowing and on which forces are acting, then the forces must be Pz+dz balanced (Newton second law)  r In r  Body force due to gravity r r+dr r Pz+dz  Laminar flow Forces balance Continue Sum of forces  Rate of change of momentum       in the z - direction   in the z - direction  dFz r  2r zr r dz Pz dFz r  dr  2 (r  dr) zr r dr dz 1…Shear forces Vz(r)  p z 2  rdr 2….Pressure Pz+dz   p z  dz 2 rdr r r+dr r 3…..Body force g2 rdrdz  Laminar flow Continue Momentum is Mass*velocity (m*v) Momentum per unit volume is *vz Rate of flow of momentum is *vz*dQ dQ=vz2πrdr but vz = constant at a fixed value of r  v z (v2rdr) z  v z (v2rdr) z dz  0 Laminar flow  Laminar flow Continue 2 r zr r dz  2 (r  dr) zr r dr dz p z 2 rdr  p z dz 2rdr  g2rdrdz  0 dvz Dp  pz 0  pz L  gL   dr R R 4 Dp Hagen-Poiseuille Q  0 2vz dr  8 L  Turbulent flow When fluid flow at higher flowrates, the streamlines are not uz steady and straight and the flow is Uz averag not laminar. Generally, the flow úz e field will vary in both space and time with fluctuations that ur comprise "turbulence Ur averag For this case almost all terms in úr e the Navier-Stokes equations are p important and there is no simple p averag solution P’ e P = P (D, , , L, U,) Time  Turbulent flow All previous parameters involved three fundamental dimensions, Mass, length, and time From these parameters, three dimensionless groups can be build DP L 2  f (Re, ) U D UD inertia Re    Viscous forces  Friction Factor for Laminar Turbulent flows From forces balance and the definition of Friction Factor DP  Ac    S  L Ac: cross section area of the pip  DP R Ac 1  rh  D S: Perimeter on which T acts (wetted f 2 f S 4 perimeter) 1/2 U U 2 L DP Rh hydraulic radius   R 2L  r 4 DP For Laminar flow Q (Hagen - Poiseuill eq) 8 L DP R 8 16 DP 8U f 2    2 L U  UR Re L R DP D 0.25 For Turbulent Flow f  0.079Re L 2 U 2  Turbulence: Flow Instability  In turbulent flow (high Reynolds number) the force leading to stability (viscosity) is small relative to the force leading to instability (inertia).  Any disturbance in the flow results in large scale motions superimposed on the mean flow.  Some of the kinetic energy of the flow is transferred to these large scale motions (eddies).  Large scale instabilities gradually lose kinetic energy to smaller scale motions.  The kinetic energy of the smallest eddies is dissipated by viscous resistance and turned into heat. (=head loss)  Velocity Distributions Turbulence causes transfer of momentum from center of pipe to fluid closer to the pipe wall. Mixing of fluid (transfer of momentum) causes the central region of the pipe to have relatively constant velocity (compared to laminar flow) Close to the pipe wall eddies are smaller (size proportional to distance to the boundary)  Surface Roughness Additional dimensionless group /D need to be characterize Thus more than one curve on friction factor- Reynolds number plot Fanning diagram or Moody diagram Depending on the laminar region. If, at the lowest Reynolds numbers, the laminar portion corresponds to f =16/Re Fanning Chart or f = 64/Re Moody chart  Friction Factor for Smooth, Transition, and Rough Turbulent flow DP D f  L 2 U 2 1 Smooth pipe, Re>3000 f   4.0 * log Re*  f  0.4 f  0.079Re0.25 1 D  4.0 * log  2.28 Rough pipe, [ (D/)/(Re√ƒ)