Compressible Fluid Flow

Transcription

1 Compressible Fluid Flow For B.E/B.Tech Engineering Students As Per Revised Syllabus of Leading Universities in India Including Dr. APJ Abdul Kalam Technological University, Kerala Dr. S. Ramachandran, M.E., Ph.D., Dr. A. Anderson, M.E., Ph.D., Professors - Mech Sathyabama Institute of Science and Technology Chennai (Near All India Radio) 80, Karneeshwarar Koil Street, Mylapore, Chennai Ph.: , aishram006@gmail.com, airwalk800@gmail.com

2 First Edition : ISBN: / Cell: ,

3 Compressible Fluid Flow Course Plan Module Contents Hours I Introduction to Compressible Flow Concept of continuum system and control volume approach conservation of mass, momentum and energy stagnation state compressibility Entropy relations. Wave propagation Acoustic velocity Mach number effect of Mach number on compressibility Pressure coefficient physical difference between incompressible, subsonic, sonic and supersonic flows Mach cone Sonic boom Reference velocities Impulse function adiabatic energy equation representation of various flow regimes on steady flow adiabatic ellipse. II One dimensional steady isentropic flow Adiabatic and isentropic flow of a perfect gas basic equations Area Velocity relation using ID approximation nozzle and diffuser mass flow rate chocking in isentropic flow flow coefficients and efficiency of nozzle and diffuser working tables charts and tables for isentropic flow operation of nozzle under varying pressure ratios over expansion and under expansion in nozzles. FIRST INTERNAL EXAM III Irreversible discontinuity in supersonic flow one dimensional shock wave stationary normal shock governing equations Prandtl Meyer relations Shock strength Rankine Hugoniot Relation Normal Shock on T-S diagram working formula curves and tables Oblique shock waves supersonic flow over compression and expansion corners (basic idea only). Sem. Exam Marks 8 15% 7 15% 7 15%

4 IV V Flow in a constant area duct with friction (Fanno Flow) Governing Equations - Fanno line on hs and Pv diagram - Fanno relation for a perfect gas Chocking due to friction working tables for Fanno flow Isothermal flow (elementary treatment only) SECOND INTERNAL EXAM Flow through constant area duct with heat transfer (Rayleigh Flow) Governing equations Rayleigh line on hs and Pv diagram Rayleigh relation for perfect gas maximum possible heat addition location of maximum enthalpy point thermal chocking working tables for Rayleigh flow 6 15% 6 0%

5 Contents C1 Table of Contents Module - I: Introduction To Compressible Flow 1.1 Introduction to Compressible Flow Concept of Continuum Control Volume Approach Conservation of Mass Momentum Equation Energy Equation Energy equation for a flow process Adiabatic energy equation Adiabatic energy transfer and energy transformation Stagnation State And Stagnation Properties Stagnation enthalpy [ h 0 ] Stagnation temperature (or) Total temperature [ T 0 ] Stagnation pressure [ P 0 ] (or) Total pressure Stagnation velocity of sound [ a 0 ] Stagnation density Compressibility 1/k Acoustic Velocity (or) Sound Velocity Mach Number And Effect of Mach Number on compressibility Regions of Flow on Steady Flow Adiabatic ellipse (i) Incompressible flow region (ii) Subsonic flow region (iii) Transonic flow region (iv) Supersonic flow region (v) Hypersonic flow region Mach Angle and Mach Cone Sonic Boom Impulse Function Reference Velocities Non-dimensional Mach Number M Relationship between M and M

6 C Compressible Fluid Flow Crocco Number [ C r ] Bernoulli s Equation Module - : Isentropic Flow with Variable Area.1 Introduction Comparison of Isentropic and Adiabatic Processes... Mach Number Variation Stagnation (0) and Critical States (*) Impulse Function Mass Flow Rate Mass flow rate in terms of Pressure ratio Mass flow rate in terms of area ratio Mass flow rate in terms of Mach number Flow Through Nozzles Under Varying Pressure Ratios Convergent nozzles Convergent-divergent nozzles Over expanding and under-expanding in nozzles Nozzle efficiency Flow Through Diffusers Diffuser efficiency Module - 3: Flow with Normal and Oblique Shock Waves 3.1 Introduction Development of a Normal Shock Wave Governing Equations Fanno Line Rayleigh Line Prandtl-meyer Relation Mach Number Downstream of the Normal Shock Wave Static Pressure Ratio Across the Shock Temperature Ratio Across the Shock Density Ratio Across the Shock (or) Rankine-hugoniot Equation Stagnation Pressure Ratio Across the Shock Change in Entropy Across the Shock

7 Contents C Impossibility of Rarefaction Shock Wave Strength of a Shock Wave Supersonic Wind Tunnels Oblique Shock Wave Development of oblique shock wave Fundamental Relations Prandtl Meyer Equation Rankine Hugoniot Equation (or) Density Ratio across the shock: Trigonometrical Relations Density Ratio across the Shock Pressure Ratio Across the Shock Temperature Ratio across the Shock Stagnation Pressure Ratio Change in Entropy across the Shock Deflection Angle Module - 4: Flow in Constant Area Ducts with Friction [Fanno Flow] 4.1 Fanno (Curves) Lines on h-s Diagram Fanno Flow - Governing Equations Solution of Fanno Flow Equations Variation of Flow Properties Temperature Pressure Velocity and Density Stagnation Pressure Impulse Function Change of Entropy Variation of Mach Number with Duct Length Isothermal Flow in a Constant Area Duct with Friction Isothermal Flow Equations Variation of Flow Properties

8 C4 Compressible Fluid Flow - Module - 5: Flow in Constant Area Ducts with Heat Transfer and without Friction [Rayleigh Flow] 5.1 Rayleigh Line on h-s and P-v Diagram Slope of the Rayleigh Line Constant Entropy Lines Constant Enthalpy Lines General Equations in Rayleigh Flow Process Rayleigh Flow Relations for Perfect Gas Pressure Temperature Density and Velocity Stagnation Pressure Stagnation Temperature Change of Entropy Heat Transfer Impulse Function Variation of Flow Properties Maximum Heat Addition - Location of Maximum Enthalpy Point Module - 6: Compressible Flow Field Visualization and Measurement 6.1 Introduction Classification of The Flow Visualization Techniques I. Direct Injection II. Optical Method: Shadowgraph method Interferometer Pressure Measurements Liquid Manometers Ring Balance Manometer and Barometer Dial Type Pressure Gauges Pressure Transducers Strain Gauge Pressure Cell Flattened tube pressure cell Cylindrical Type pressure cells

9 Contents C5 6.4 Temperature Measurement Bimetallic thermometers Pressure Thermometers (Fluid expansion type) Resistance thermometer Thermistors (Thermal Resistors) Velocity Measurements Anemometers Cup Anemometers Vane Anemometer Hot Wire Anemometer Laser Doppler Velocimetery Measurement of flow velocity in Subsonic Flow and Supersonic Flow Shock tube Pitot tube Flow Direction Measurement Five hole truncated probe Cobra probe Wedge probe Stagnation Temperature Measurements Laminar Flow: Turbulent Flow: Stagnation Pressure Measurement Kiel probe Wind Tunnels Types of wind tunnels: Subsonic Wind Tunnel Supersonic Wind Tunnel Blow down wind Tunnel (Open circuit type) Blow down wind Tunnel (closed circuit type) Shock Tunnel

10 Module - I Introduction To Compressible Flow Introduction to Compressible Flow Concept of continuum system and control volume approach conservation of mass, momentum and energy stagnation state compressibility Entropy relations. Wave propagation Acoustic velocity Mach number effect of Mach number on compressibility Pressure coefficient physical difference between incompressible, subsonic, sonic and supersonic flows Mach cone Sonic boom Reference velocities Impulse function adiabatic energy equation representation of various flow regimes on steady flow adiabatic ellipse. 1.1 INTRODUCTION TO COMPRESSIBLE FLOW Gas dynamics deals with the study of compressible fluid flow when it is in motion. It analyses the high speed flows of gases and vapours with considering its compressibility. Applications The applications of Gas Dynamics are (i) used in steam and Gas turbines (ii) high speed aerodynamics (iii) Jet and Rocket propulsion (iv) high speed turbo compressors etc. The fluid dynamics of compressible flow problems which involves the relation between force, velocity, density and mass etc. Therefore, the following laws are frequently used for solving the Gas Dynamic problems. (i) Steady flow energy equation [derived from first law of Thermodynamics] (ii) Entropy relations [derived from second law of Thermodynamics] (iii) Continuity equation [derived from law of conservation of mass] (iv) Momentum equation [derived from Newton s second law of motion]

11 1. Compressible Fluid Flow - Compressible Flow (i) The fluid velocities are (i) appreciable compared with the velocity of sound. (ii) The fractional variation in density is significant i.e., the density is not constant. (iii) (iv) The fractional variations in temperature and pressure are all of significant magnitude Compressibility factor is greater than one. Incompressible Flow The fluid velocities are small compared with the velocity of sound. (ii) The fractional variation in density are so small as to be negligible i.e., the density is constant. (iii) (iv) The variation in pressure and temperature may be very large Compressibility factor is one 1. CONCEPT OF CONTINUUM The concept of continuum is a kind of idealization of the continuous description of matter where the properties of the matter are considered as continuous functions of space variables. Although any matter is composed of several molecules, the concept of continuum assumes a continuous distribution of mass within the matter or system with no empty space instead of actual conglomeration of separate molecules. Describing a fluid flow quantitatively makes it necessary to assume that flow variables (pressure, velocity etc.) and fluid properties vary continuously from one point to another. Mathematical description of flow on this basic has proved to be reliable and treatment of fluid medium as a continuum has firmly become established. For example, the density at a point is normally defined as, lim m V 0 V where V is the volume of the fluid element and m is the mass. If V is very large, is affected by the inhomogenities in the fluid medium. Considering another condition, if V is very small, random movement of atoms or molecules would change their number at different

12 Introduction to Compressible Flow 1.3 times. In the continuum approximation, point density is defined at the smallest magnitude of V, before statistical fluctuations become significant. This is called continuum limit and is denoted by V C. lim V V C m V One of the factors considered important in determining the validity of continuum model is molecular density. It is the distance between the molecules which is characterised by mean free path. It is calculated by finding statistical average distance the molecules travel between two successive collisions. If the mean free path is very small as compared with characteristic length in a flow domain, then the gas can be treated as continuous medium. If the mean free path is large in comparison to some characteristic length, the gas cannot be considered continuous and it should be analysed by molecular theory. A dimensionless parameter known as knudsen number k n /L, where is the mean free path and L is the characteristic length. It describes the degree of departure from continuum. Usually when k n 0.01, the concept of continuum does not hold good. Beyound this critical range of Knudsen number, the flow are known as Slip flow 0.01 k n 0.1 Transition flow 0.1 k n 10 Free molecule flow k n 10 However, for the flow regions considered, where k n is always less than 0.01 and it is usual to say that the fluid is a continuum. Other factor which checks the validity of continuum is the elapsed time between collisions. The time should be small enough so that the random statistical description of molecular activity holds good.

13 1.4 Compressible Fluid Flow - In continuum approach, fluid properties such as density, viscosity, thermal conductivity, temperature etc., can be expressed as continuous functions of space and time. 1.3 CONTROL VOLUME APPROACH Control volume is defined as a region in space across the boundaries of which matter, energy and momentum may flow and it is a region within which source or sink of the same quantities may prevail. It is a region on which external forces may act. The control volume is located within a moving fluid. There are two approaches in control volume. (a) Eulerian approach Here the control volume is fixed in the space, and the transport of fluid across the fixed control volume is studied. Measurements made using stationary instruments (pilot tube, hot wire, laser dopler) can be directly compared with the solutions of differential equations obtained by the eulerion approach. Except when dealing with certain types of unsteady flows, the eulerian approach in generally used for its notional implicitly. (b) Lagrangian approach Here the control volume is moving in space with respect to time. 1.4 CONSERVATION OF MASS This derivation is derived by cartesian frame work (cartesian co - ordinates system). A cartesian coordinate system is used to describe the cartesian control volume in three dimensions. Consider a parallel piped as a control volume of sides x, y, z in x, y and z directions respectively. When we consider the conservation of mass, we assume a fixed control volume and study the amount of mass entering and leaving the control volume.

14 Introduction to Compressible Flow 1.5 y y x z x z Fig. 1.1 Consider the mass flow entering the control volume in x direction, u y z and the mass flow leaving the control volume in x direction, considering the mass flow entering the control volume by means by taylors series expansion, u y z u y z x x Similarly, the mass flow rate entering the control volume in y direction, v x z and the mass flow rate leaving the control volume in y direction, v x z v x z y y Similarly, the mass flow rate entering the control volume is z direction, w x y and the mass flow rate leaving the control volume in z direction, w x y w x y z z

15 1.6 Compressible Fluid Flow - From law of conservation mass, The change of mass inside the control volume Mass entering the control volume Mass leaving the control volume... (1.1) Net change in mass from the system, along x direction Similarlly u y x z t x Net change in mass from the system, along y direction also v x z y t y Net change in mass from the control volume in z direction w x y z t z But, change of mass in the control volume is given by x y z t Substituting in equation 1.1 t x y z t x u y v z w x y z t and we get t t x u y v z w 0... (1.) In a co-ordinate free representation, t V 0... (1.3)

16 Introduction to Compressible Flow 1.7 Rearranging 1. t u x v y w z u x v y w z 0 Let is the function of time and position (i.e.) t, x, y, z where dx dt, dy dt, dz dt D Dt t dt dt dx dx dt y dy dt z z t is velocity in x, y, z directions respectivity. D Dt t u x v y w z D Dt u x v y w z 0 using co-ordinate free representation, For steady flow, D Dt V 0... (1.4) The velocity is divergence free (i.e.) D 0 V 0 Dt from eqn 1.3, for steady flow, for compressible flow, t 0 V 0

17 1.8 Compressible Fluid Flow MOMENTUM EQUATION y yz + yz y y yy + yy y y yx + yx y y zz zy zx xx + xx x x zz + xz xx zz z z zy z zy + z xy z yz yx yy Fig. 1. zx + xy + xz + x zx z z x x x x Applying the Newton s law, The force vector is equal to the rate of change of a velocity F m DV Dt considering the forces acting on the control volume. Body forces The force per unit volume in x, y, z direction is g x, g y, g z. The normal shear force acting in x direction in left hand side xx y z The normal shear force acting in x direction in right hand side.

18 Introduction to Compressible Flow 1.9 Similarly, xx xx x x y z The normal shear force acting in y direction in bottom side. yy x z and The normal shear force acting in y direction in top side. and yy yy y y x z Similarly, The normal shear force acting in z direction in one side is zz y x and the normal shear force acting along z direction in other side zz zz z z y x The shear stresses act in the direction of area normal, which point away from the surface. The pressure forces also act in the direction of area normal, but they are compressive in nature, which point towards the surface. xx P xx yy P yy zz P zz Pressure force Viscous shear The viscous shear acting tangential to the faces (i.e) x plane on one side xy y z

19 1.10 Compressible Fluid Flow - The viscous shear acting tangential to the faces (i.e) x plane on another side Similarly, xy xy x x y z The viscous shear acting on tangential to the faces (i.e.) y plane on bottom side yx x z and the viscous shear acting on tangential to the faces (i.e.) y planes on top side Similarly, yx yx y y x z the other viscous shear acting tangential to the faces are (i.e) y planes in z direction on top side yz z x and the viscous shear acting tangential to faces are (i.e.) y plane in z direction. Similarly, yz yz y y x z the viscous shear acting tangential to the faces are (i.e.) z planes in x direction one side zx x y and the viscous shear acting tangential to the faces are (i.e) z planes in x direction on another side zx zx z z x y

20 Introduction to Compressible Flow 1.11 Similarly, The viscous shear acting tangential to the faces are (i.e.) x plane in z direction on one side and xz z y the viscous shear acting tangential to the faces are (i.e.) x plane in z direction on another side xz xz x x z y Similarly, The viscous shear acting tangential to the faces are (i.e.) z plane in y direction on one side zy y x and the viscous shear acting tangential to the faces are (i.e) z plane in y direction one another side zy zy z z y x Putting all the stresses together in a tensor form, and expand the tensor form, we get the matrix form as, xx yx zx Net force in x direction xy yy zy xz yz zz F x g x xx x yx y zx x y z z Similarly, Net force in y direction F y g y yy y xy x zy z x y z

21 1.1 Compressible Fluid Flow - Similarly, Net force in z direction F z g z zz z xz x yz x y z y Assuming there is no rotation to this control volume caused by the imbalanced forces xy yx ; xz zx ; yz zy This assumption limits the unknown from 9 to 6. The forces are expanded and equated to the total derivative. Hence, work. D u Dt u t u u x v u y w u z This leads to conversion of Lagrangian frame work to Eulerian frame Similarly, and Du Dt Dv Dt g x P x xx x yx y zx z g y P y yy y xy x zy z D w Dt g z P z zz z xz x yz y By assuming Newtonian fluid, relating stresses to strain rates, xx P u x v y w z yy P u x v y w z zz P u x v y w z u x v y w z

22 Introduction to Compressible Flow 1.13 Where absolute viscosity (or) shear viscosity and bulk viscosity (or) second coefficient of viscosity Similarly, for shear stresses in a Newtonian fluid, xy yx v x u y xz zx w x u z yz zy v z w y To relate, bulk viscosity and shear viscosity, stokes hypothesis is used Here replacing stresses in terms of strain rates. for x momentum Similarly, Du Dt for y momentum Dv Dt g y P y x g x P x x u x 3 V v y x u w y z x u z v x u y y v y 3 V z w y v z

23 1.14 Compressible Fluid Flow - Similarly, for z momentum, Dw Dt g z P z x w x u y y z v x w y w z 3 V This equations are the momentum equations for compressible fluid flows. 1.6 ENERGY EQUATION The first law of thermodynamics states that when a system executes a cyclic process, the algebric sum of work transfers is proportional to the algebric sum of heat transfers. i.e., O dw Od Q O dw J O dq When heat and work terms are expressed in the same units O dq O dw 0 The quantity dq and dw will follow the path function, but the quantity dq dw does not depend on the path of the process. Therefore, the change in quantity dq dw is a property called Energy (E). Thus de dq dw 1 de 1 E E 1 Q W dq 1 Q W E E 1... (1.5) In the above equation, the energy term E may includes kinetic energy, internal energy, gravitational potential energy, strain energy, magnetic energy, etc., By ignoring magnetic energy and strain energy the energy term E may be written as dw

24 Introduction to Compressible Flow 1.15 E U mg Z 1 mc... (1.6) The differential form of Equation (1.6) is de du mg dz 1 m CdC Integrating the above equation between the limits 1 and, then the equation becomes, E E 1 U U 1 mg Z Z 1 1 m C C1... (1.7) substituting (1.7) in equation (1.5) a general form of energy equation is obtained. i.e., Q W U U 1 mg Z Z 1 1 m C C 1... (1.8) Energy equation for a flow process A change or a series of changes in an open system is known as a flow process. Examples are (i) (ii) (iii) i.e., Flow through nozzles, diffusers and ducts etc, Expansion of steam and gas in turbines Compression of air and gases in turbo compressors etc. In such flow processes the work term W includes the flow work also substituting this in equation (1.8), we get W W s P V P 1 V 1 Shaft work Flow work... (1.9) Q W s P V P 1 V 1 U U 1 mg Z Z 1 1 m C C1 but, we know that the enthalpy H U PV

25 1.16 Compressible Fluid Flow - Q W s H H 1 mg Z mg Z 1 1 m C 1 m C 1 H 1 mg Z 1 1 mc 1 Q H mg Z 1 1 mc W s... (1.10) h 1 gz 1 C 1 q h gz C w s... (1.11) [per unit kg mass] This is a steady flow energy equation which is generally used in flow problems of gases and vapours Adiabatic energy equation Compared to other quantities, the change in elevation g z z 1 is negligible in flow problems of gases and vapours. In a reversible adiabatic process, the heat transfer q is negligibly small and can be ignored. Expansion of gases and vapours in nozzles and diffusers are examples of such process. For such processes equation (1.11) is reduced to h 1 C 1 h C... (1.1) [ w s 0 ] Adiabatic energy transfer and energy transformation In an adiabatic energy transfer process, the shaft work will present (eg) expansion of gases in turbines and compression in compressors etc., In an adiabatic energy transformation process, the shaft work is zero. e.g. expansion of gases in nozzles and compression in diffusers etc., The adiabatic energy equation (1.11) is valid for processes involving both energy transfer and energy transformation. The energy equation for compressors and turbines is h 1 C 1 h C w s... (1.13) The energy equation for a nozzle and diffuser is h 1 C 1 h C... (1.14)

26 Introduction to Compressible Flow STAGNATION STATE AND STAGNATION PROPERTIES The state of a fluid attained by isentropically decelerating it to zero velocity at zero elevation is referred to as the stagnation state. It is often used as a reference state. The properties of the fluid at stagnation state are the stagnation properties of the fluid. eg. stagnation temperature, stagnation pressure, stagnation enthalpy etc., Stagnation enthalpy [ h 0 ] Stagnation enthalpy of a gas or vapour is its enthalpy when it is adiabatically decelerated to zero velocity at zero elevation. As per the definition, At the initial state h 1 h : C 1 C and At the final state h h 0 : C 0 By substituting this in equation (1.14), we get where h 0 stagnation enthalpy and h static enthalpy h 0 h C... (1.15) In an adiabatic ener gy tr ansfor mation pr ocess, the stagnation enthalpy remains constant Stagnation temperature (or) Total temperature [ T 0 ] Stagnation temperature of a gas or vapour is defined as the temperature when it is adiabatically decelerated to zero velocity at zero elevation. For a perfect gas, the equation (1.15) can be written as C p T 0 C p T C divide by C p throughout the equation where, T 0 Stagnation temperature T Static temperature C C p Velocity temperature T 0 T C C p...(1.16)

27 1.18 Compressible Fluid Flow - From equation (1.16) T 0 T 1 C C p T C 1 1 RT 1 1 C a C p 1 R a RT M C a T 0 T T 0 T M... (1.17) M Stagnation pressure [ P 0 ] (or) Total pressure Stagnation pressure is the pressure of the gas when it is adiabatically decelerated to zero velocity at zero elevation. The adiabatic relation for a perfect gas is T 0 T 1 P 0 P P 0 P T 0 1 T 1 1 M 1... (1.18) Stagnation velocity of sound [ a 0 ] We know that the acoustic velocity of sound a RT. For a given value of stagnation temperature, the stagnation velocity of sound a 0 RT

28 Introduction to Compressible Flow Stagnation density 0 For the given values of stagnation pressure and temperature, the stagnation density is given by 0 P 0 RT 0 or from adiabatic relation 0 T 0 T P 0 P T 0 T M 1... (1.19) 1.8 COMPRESSIBILITY 1/k Compressiblity is the inverse of bulk modulus of elasticity k. The bulk modulus of elasticity of a fluid is defined by k increase in pressure relative change in volume dp k dv v... (1.0) (minus sign is because, increase in pressure takes place by decreasing the volume) But, v 1 By differentiating the equation a... (a) dv 1 d substituting this in equation (1.16) we get,

29 1.0 Compressible Fluid Flow - k v dp d k a dp d Hence, Compressibility 1 k dp d 1 a dp d a 1.9 ACOUSTIC VELOCITY (or) SOUND VELOCITY It is the velocity of sound in a fluid medium or the speed with which a small disturbance is transmitted through the fluid. Constant area duct A=constant P + dp + d h + dh dc P a h C = O P + dp + d h + dh a P h Wave front Wave front a Velocity dc Velocity a - dc Distance Distance Pressure P + dp Wave p Pressure P + dp p Distance Distance Fig 1.3 Fig 1.4

30 Introduction to Compressible Flow 1.1 Consider a stationary fluid in an insulated cylinder fitted with a frictionless piston. The piston and gas in the tube are at rest originally at a pressure P. Let the parameters across the wave front (is a plane across which pressure and density changes suddenly and there will be a discontinuity in pressure, temperature and density) be as shown in Fig.1.3. If a small impulse is given to the piston the gas immediately adjacent to the piston will experience a slight rise in pressure dp or in other words it will be compressed. The change in density d takes place because the gas is compressible and therefore, there is a lapse of time between the motion of the piston and the time. This (motion of piston) is observed at the far end of the tube. Thus it will take certain time to reach far end of the tube or in otherwords there is a finite velocity of propagation which is acoustic velocity. It is shown in Fig.1.3. In this case, the stagnant gas at pressure p on the right side moving with a velocity a towards left and thus its pressure is raised to p dp and its velocity lowered to a dc. This is because of the velocity of the piston dc acts opposite to the movement of gas a. Before deriving the equation, the following assumptions are made. 1. The fluid velocity is assumed to be acoustic velocity. There is no heat transfer in the pipe and the flow is through a constant area pipe. 3. The changes across an infinitesimal pressure wave can be assumed as reversible adiabatic (or) isentropic. Applying momentum equation between the two sides of the wave; pa p dp A m [ a dc a ] Pressure force Impulse force... [ m A a ] A [ p p dp ] A a [ a dc a ] dp a dc... A constant dp a dc... (1.0) From continuity equation for the two sides of the wave

31 1. Compressible Fluid Flow - m A a d A a dc a a a d dc d dc... (1.1) The product of d dc is very small. it is ignored. The equation (1.1) becomes adp dc Substituting this in equation (1.0), we get dp a d a dp d s cisentropic [or]... (1.) For isentropic flow, pv C or p constant (or) p constant Differentiating the above equation p 1 d e dp 0 p 1 d dp 0 ; dp d p dp d RT Substituting this in equation (1.) dp p d p R T p RT a dp d RT... (1.3) This is an important equation for solving gas dynamic problems.

32 Introduction to Compressible Flow MACH NUMBER AND EFFECT OF MACH NUMBER ON COMPRESSIBILITY Mach number M It is a non-dimensional number and is defined as M inertia force elastic force From Bernoulli equation for incompressible flow, the value of pressure co-efficient (or) compressibility factor is unity. i.e., P 0 P C P 0 P C 1... (1.4) For compressible flow, the value of pressure co-efficient deviates from unity and the magnitude of deviation increases with the mach number of the flow. where x We know that, T 0 T 1 P 0 1 P 1 M P 0 P 1 1 M 1 This can be expanded by Binomial expansion i.e., 1 x n 1 nx 1 P 0 P 1 M and n n n 1! 1 1 M 1 x n n 1 n 3! x M M 6 6 8

33 1.4 Compressible Fluid Flow - P 0 P P P 0 P 1 M M4 M Divide both sides by M M M4 M P 0 P P M 1 M 4 M6 48 M P 0 P 1 P 0 P P We know that, P RT 1 M 4 M (1.5) M C a C RT M C RT P M P 0 P C 1 M 4 RT C RT M4 4 C... (1.6) For 1.4 P 0 P C 1 M 4 M (1.7) Equation (1.7) gives the percentage deviation of the pressure co-efficient from its incompressible flow value with the Mach number. By substituting different values of M, we get the following table.

34 Introduction to Compressible Flow 1.5 Table 1.1 Effect of Compressibility Sl.No. Mach Number M % deviation From the table it is observed that, percentage deviation increases when Mach number increases. When the mach number is unity, the percentage deviation is 7.5% 1.11 REGIONS OF FLOW ON STEADY FLOW ADIABATIC ELLIPSE The adiabatic energy equation for a perfect gas is derived in terms of fluid velocity (c) and sound velocity (a). Then it is plotted graphically in the x axis C and y axis a respectively. From adiabatic energy equation, We know that, h 0 h C constant h C p T 1 R T a 1 a RT a RT By substituting this in equation (1.15), we get

35 1.6 Compressible Fluid Flow - a h 0 1 C constant... (1.8) At T 0 h 0 a 0 and C C max Therefore equation (1.8) becomes At C 0 ; a a 0 Therefore, from equation (1.8) h 0 C max... (1.9) a 0 h 0 1 constant... (1.30) h 0 a 1 C C max a 0 1 constanṭ.. (1.31) Equation (1.31) is an another form of adiabatic energy equation. By substituting different values of (c) and (a) in the above equation and by plotting the values, a steady flow ellipse is obtained. It is shown in Fig.1.15 and there are five different regions on the ellipse. a=a O I II M < 1 III a M=1 Sonic M > 1 IV V o C Fig 1.5 Various regions of flow C=C max

36 Introduction to Compressible Flow 1.7 (i) Incompressible flow region The region of flow close to the axis a is an incompressible flow region. The fluid velocity is much smaller than the sound velocity which is shown in Fig Therefore the Mach number M 1 and is very close to zero. (ii) Subsonic flow region The region on the right of the incompressible region and upto a mach number is less than unity. (iii) Transonic flow region When the Mach number of flow is unity, the flow is sonic flow. A small region slightly less than unity and just above the sonic point is referred as transonic flow region. The mach number in this region is in between 0.8 to 1.. (iv) Supersonic flow region The region is on the right side of the transonic flow region. The mach number in this region is always above unity and up to 5. (v) Hypersonic flow region In this region, the flow velocity is very high compared to the sound velocity and hence the mach number is very high i.e., above 5. The region close to the c axis is called hypersonic region. 1.1 MACH ANGLE AND MACH CONE When a body moves through a fluid or when fluid flows past a body or with in the walls of a duct, each element of solid surface tends to divert the fluid from its direction of flow. For example, in case of projectile moving through air, each element of the projectile s surface area pushes the neighbouring air out of the way, and this local disturbance creates a pressure pulse which propagates in to the exterior air. The pressure field created by the most elementary type of moving disturbance is called Point source of disturbance. Point source may be imagined to emit infinitesimal pressure wave which spreads spherically from the point of emission with the speed of sound relative to the fluid.

37 1.8 Compressible Fluid Flow - Sound waves a O a 3a Fig 1.6 (a) Incompressible flow (C/a ~ o; M ~ o) Point source 3a C t a a Sound waves Wave front Fig 1.6 (b) Subsonic flow (C < a) Fig. 1.6 (a, b, c and d) show the movement of a source of disturbance O at a velocity c in a fluid from right to left. The disturbance travels distances of a, a and 3a meters when time is 1, and 3 unit times respectively. C=a 0 Sound Waves a 3a 1 3 a In an incompressible flow Fig. 1.6 (a) the velocity of source of disturbance C is negligibly small compared to the velocity of Zone of action Fig. 1.6 (c.) Sonic flow (M = 1) c = a

38 Introduction to Compressible Flow 1.9 Mach cone Wave front Zone of silence O C>a a 1 a 3 3a Point source Zone of action Fig. 1.6 (d) Supersonic flow (C > a) (M > 1) sound a. The sound waves generated which travel at a velocity a in all directions. The wave propagation will be a set of concentric circles as shown in Fig. 1.6(a). Mach angle at In a subsonic flow, the point source travelling with a velocity C a shown in Fig. 1.6 (b). At the reference time, the point of disturbance is assumed to be at O. At unit time later, the point source will have moved to 1 and the distance ct. At unit time units later, the source will have moved to ct and so on. It is observed that the wave fronts move ahead of the point source and the intensity is not symmetrical. The practical use of this in the case of automobiles, which move with C a. The horn is heard before the vehicle reaches a person standing on the road. In a sonic flow [ m 1 ], the point source travels with the same velocity as that of the wave. The wave fronts are always coincides with the point source and cannot move ahead of it. We won t hear any sound at the upstream side is called Zone of silence and the downstream is zone of action. In a supersonic flow, all the pressure disturbances are included in a cone which has the point source at its apex and the effect of the disturbance is not felt upstream of the source of disturbance. i.e., the point source is always ahead of the wave fronts. The cone with in which the disturbances O Ct

39 1.30 Compressible Fluid Flow - are confined is called Mach Cone and the half angle of this cone is known as Mach angle. The space (or) zone outside the Mach Cone is called as Zone of silence i.e., there is no effect of disturbance in this region. While the region inside the Mach Cone is called Zone of action. In this region the fluid properties are affected by the disturbance. From the Fig., sin at Ct a C 1 C/a 1 M Mach angle sin 1 1 M... (1.3) 1.13 SONIC BOOM A sonic boom is the sound associated with the shock wave created whenever an object travelling through the air travels faster than the speed of sound. (Speed of sound is approximately 33 m/s or 1195 km/hr). These speeds are called supersonic speeds. Hence this phenomena is sometimes called the supersonic boom. Sonic boom generate significant amounts of sound energy, sounding similar to an explosion or a thunderclap to the human ear. Normally for a plane that is going at subsonic speeds (lower than that of sound), the sound of the plane is radiated in all directions. However, the individual sound wavelets are compressed at the front of the plane, because of the forward speed of plane. This effect in known as Doppler effect and accounts for the change of the pitch of the plane s sound as it passes us. When the plane is approaching us it s sound has a higher pitch than of it is going away from us. Now if the plane is travelling at the supersonic speeds, it is going faster than the speed of its sound. As a result, a pressure wave is produced in the shape of the cone whose vertex at the nose of the plane, and whose base in behind the plane. The angle opening of the cone depends on the actual speed the plane is travelling at. All of the sound pressure is contained in this cone. The crack of a supersonic bullet passing overhead or the crack of bull whip are examples of sonic boom in miniature.

40 Introduction to Compressible Flow IMPULSE FUNCTION For solving jet propulsion problems, it is sometimes convenient to employ a quantity called impulse function. It is defined as the sum of pressure force and impulse force. i.e., F PA AC Pressure force Impulse force...(1.33) One dimensional flow through a control surface is shown in Fig The 1 P net thrust (or) side wall thrust produced P 1 A A by the stream is a result of changes in 1 Flow pressure and Mach number between C 1 sections 1 and. Applying momentum equation 1 between sections 1 and. The thrust exerted by the fluid Control surface Fig..6 m C C 1 P A P 1 A 1 m C P A m C 1 P 1 A 1 A C C P A 1 A 1 C 1 C 1 P 1 A 1 A C P A 1 A 1 C 1 P1 A 1 F F 1 change in impulse function For a perfect gas,...(1.34) C P RT C PC RT PC a PM F PA PM A PA 1 M...(1.35) By substituting this in equation (1.34), we get P A 1 M P 1 A 1 1 M 1...(1.36)

41 1.3 Compressible Fluid Flow - The above equation is convenient to find the thrust exerted by the flowing fluids using Mach number (M). To obtain a relation between the non-dimensional impulse functions and the Mach number the flow is assumed to be isentropic, At M 1 F F,the equation 1.35 becomes Now, F P A [1 ] F F PA 1 M P A [1 ]...(1.37) P P M P P A A M 1 1 M M r Therefore the above equation becomes PA P A 1 1 M M Substituting this in equation (1.37), we get F F 1 M 1 M M 1 1

42 Introduction to Compressible Flow 1.33 F F 1 M 1 M 1 1 M... (1.38) REFERENCE VELOCITIES In compressible fluid flow analysis, it is often convenient to express the fluid velocity in non-dimensional forms. The various reference velocities used are ii(i) Local velocity of sound, a i(ii) Stagnation velocity of sound, a 0 (iii) Maximum velocity of fluid, C max (iv) Critical velocity of fluid/sound, C a (i) Local velocity of sound a The local velocity of sound a RT dp d s Constant (c) Stagnation velocity of sound, a 0 It is a sound velocity at the stagnation conditions, and its value is constant. In an adiabatic flow for a given stagnation temperature. i.e., a 0 RT 0 (iii) Maximum velocity of fluid, C max From adiabatic energy equation, h 0 h C. It has two components one is static enthalpy h and the another is kinetic energy C, when the static enthalpy is zero (or) when the entire energy is made up of kinetic energy only. The above equation becomes,

43 1.34 Compressible Fluid Flow - h 0 and C C max h 0 C max C max h 0 C p T 0 1 RT 0 C max 1 a 0 C max a (1.39) (iv) Critical velocity of fluid/sound, C a It is the velocity of a fluid at which the Mach number is unity. where T critical temperature i.e., M critical C a 1 C a RT At the critical state, the adiabatic energy equation becomes h 0 h C C p T 0 C p T C Divide throughout by C p T 0 T C C p C C p T 0 T... (1.40)

44 Introduction to Compressible Flow 1.35 We know that, T M T T T... M 1 T T 0 By sustituting this in equation C 1 R T 0 T 0 1 RT a RT0 a 0 C C 1 a a 0 1 a 0 1 Divide equation (1.39) by (1.41), we get C max a 0 a 0... (1.41) C 1 1

45 1.36 Compressible Fluid Flow - We know that, h 0 C max 1 1 C max C... (1.4) Substituting equation (1.4), we get h 0 C Therefore equation (1.4) becomes h 0 a 1 C a (1.43) 1 C max a (1.44) Non-dimensional Mach Number M This is an another type of mach number and is defined as the ratio between the local velocity of fluid to the critical velocity of sound. i.e. M C C C a M C Multiply both sides by a C M C a a C a M a It is more convenient to use M instead of M because (i) At high fluid velocities M approaches infinity. But M max C max C 1. Therefore, for doing calculations it is very 1 difficult if M.

46 Introduction to Compressible Flow 1.37 (ii) M C a. Since a is constant for any process. Therefore M is proportional to the fluid velocity only. But M C a where M is not proportional to the fluid velocity alone Relationship between M and M From equation (1.44) a 1 C a 1 1 a C 1 1 a 1 1 a a C 1 1 a a C a C M 1 1 M M M 1 1 M M 1 1 M M 1 M 1 Comparison of M and M M 0 M 0 M M 1 M 1... (1.45) M 1 M 1 M 1 M 1 M 1 M 1

47 1.38 Compressible Fluid Flow - M M max 1 1 C max C Crocco Number [ C r ] Crocco number is a non-dimensional fluid velocity which is defined as the ratio of fluid velocity to its maximum fluid velocity. C r Multiply both Nr. and Dr. by C C fluid velocity C max Max.fluid velocity C r C C C C max M 1 1 C r M (1.46) By substituting equation (1.45) in (1.46) C r Cr M 1 M 1 M 1 1 C r M 1 1 C r M 1 Cr M 1 C r M 1 [ 1 Cr ] M C r 1 C r 1... (1.47)

48 Introduction to Compressible Flow 1.39 We know that, T 0 T T 0 T 1 C r Cr 1 C r M C r 1 C r C r T 0 T 1 1 C r Problem 1.1: An air jet at 400 K has sonic velocity Determine iii(i) Velocity of sound at 400 K ii(ii) Velocity of sound at stagnation conditions (iii) Maximum velocity of jet. (iv) Stagnation enthalpy i(v) Crocco number.... (1.48) (FAQ) Given Data: T 400 K, At sonic, condition M 1 and C a, 1.4 (air) (i) Velocity of sound a RT a m/sec (ii) T 0 T 1 1 M 1 T K. Velocity of sound at stagnation condition

49 1.40 Compressible Fluid Flow - (iii) From adiabatic energy equation, a 0 RT 0 C max (iv) Stagnation enthalpy, h 0 C max (v) Crocco number, C r m/sec a 0 1 C max a m/sec KJ/Kg C C max 98 Result (i) Velocity of sound a m/sec (ii) Velocity of sound at stagnation conditions a m/sec (iii) Maximum velocity of jet C max 98 m/sec (iv) Stagnation enthalpy h kj/kg (v) Crocco number Cr Problem 1.: The jet of gas at 593 K 1.3 and R 469 J/Kg K has a Mach number of 1.. Determine for static and stagnation conditions. (i) Velocity of sound (ii) Enthalpy (iii) What is the maximum attainable velocity of this jet? (FAQ)

50 Introduction to Compressible Flow 1.41 Given Data: T 593 K, 1.3, R 469 J/Kg K, M 1. C p R J/Kg K T M T K Velocity of sound a RT m/sec a 0 RT m/sec Enthalpy h C p T kj/kg Maximum attainable velocity h 0 C p T kj/kg C max h m/sec

51 1.4 Compressible Fluid Flow - Result Static Condition Stagnation condition (i) Velocity of sound (m/sec) (ii) Enthalpy (kj/kg) (iii) Maximum attainable velocity of jet C max m/sec Problem 1.3: (a) Determine the velocity of air 1.4 C p kj/kgk corresponding to a velocity of temperature of 1C. (b) Determine the Mach number of an aircraft at which the velocity temperature of air at the entry of the engine equals the static temperature. Given Data: (a) 1.4, C p KJ/Kg K T c C C p K T c C C p 74 C m/sec (b) T c T We know that, T 0 T T c T T 0 T 1 1 M 1 0. M M Result (a) Velocity of air C m/sec (b) Mach number of an aircraft M.36

52 Introduction to Compressible Flow 1.43 Problem 1.4: Air flows from a reservoir at 550 KPa and 70C. Assuming isentropic flow. Calculate the velocity, temperature, pressure and density at a section where M 0.6. (FAQ) Given Data: [In a reservoir the fluid is in a stagnation state i.e., the velocity of the fluid C 0] P KPa, T K since the flow is isentropic. From isentropic table 1.4 and M 0.6 T T , P P T K and P 4.31 bar M C a C M RT We know that m/sec P RT P RT kg/m Result (i) Velocity of flow C m/sec (ii) Temperature at the section T K (iii) Pressure at the section P 4.31 bar (iv) density at the section kg/m 3. Problem 1.5: An air stream at 1 bar and 400 K flowing with a velocity of 400 m/sec is brought to rest isentropically. Determine the stagnation pressure and temperature. (FAQ)

53 1.44 Compressible Fluid Flow - Given Data: P 1 bar, T 400 K, C 400 m/sec (i) Mach number M C a We know that T 0 T 1 1 C RT From adiabatic relations, M K. T 0 T 1 P 0 P P 0 T 0 T P bar P Result: (i) Stagnation Pressure P bar (ii) Stagnation temperature T K. Problem 1.6: Air enters a straight axisymmetric duct at 7C, 3.45 bar and 150 m/sec and leaves 4C,.058 bar and 60 m/sec. Under adiabatic flow conditions, for an inlet cross sectional area of 500 sq.cms, estimate the stagnation temperature, maximum velocity, mass flow rate and the exit area. (FAQ)

54 Introduction to Compressible Flow 1.45 Given Data: T K P bar, C m/sec T 77 K P.058 bar, C 60 m/sec A cm We know that, a 0 RT 0 C max M 1 C 1 RT 0.43 T M T 1 1 T T K a m/sec a 0 1 C max a 0 1 C max m/sec Mass flow rate, m 1 A 1 C kg/sec