FUNDAMENTALS OF GAS DYNAMICS
CONTENTS REVIEW OF ELEMENTARY PRINCIPLES INTRODUCTION UNITS AND NOTATION SOME MATHEMATICAL CONCEPTS THERMODYNAMIC CONCEPTS FOR CONTROL MASS ANALYSIS REVIEW QUESTIONS REVIEW PROBLEMS CONTROL VOLUME ANALYSIS—PART I OBJECTIVES FLOW DIMENSIONALITY AND AVERAGE VELOCITY TRANSFORMATION OF A MATERIAL DERIVATIVE TO A CONTROL VOLUME APPROACH CONSERVATION OF MASS CONSERVATION OF ENERGY SUMMARY PROBLEMS CHECK TEST CONTROL VOLUME ANALYSIS—PART II COMMENTS ON ENTROPY PRESSURE–ENERGY EQUATION THE STAGNATION CONCEPT STAGNATION PRESSURE–ENERGY EQUATION CONSEQUENCES OF CONSTANT DENSITY MOMENTUM EQUATION INTRODUCTIONTO COMPRESSIBLE FLOW SONIC VELOCITY AND MACH NUMBER WAVE PROPAGATION EQUATIONS FOR PERFECT GASES IN TERMS OF MACH NUMBER H–S AND T –S DIAGRAMS VARYING-AREA ADIABATIC FLOW GENERAL FLUID—NO LOSSES PERFECT GASES WITH LOSSES THE ?REFERENCE CONCEPT ISENTROPIC TABLE NOZZLE OPERATION NOZZLE PERFORMANCE DIFFUSER PERFORMANCE (OPTIONAL) BEYOND THE TABLES STANDING NORMAL SHOCKS SHOCK ANALYSIS—GENERAL FLUID WORKING EQUATIONS FOR PERFECT GASES NORMAL-SHOCK TABLE SHOCKS IN NOZZLES SUPERSONIC WIND TUNNEL OPERATION MOVING AND OBLIQUE SHOCKS NORMAL VELOCITY SUPERPOSITION: MOVING NORMAL SHOCKS TANGENTIAL VELOCITY SUPERPOSITION: OBLIQUE SHOCKS OBLIQUE-SHOCK ANALYSIS: PERFECT GAS OBLIQUE-SHOCK TABLE AND CHARTS BOUNDARY CONDITION OF FLOW DIRECTION BOUNDARY CONDITION OF PRESSURE EQUILIBRIUM CONICAL SHOCKS PRANDTL–MEYER FLOW ARGUMENT FOR ISENTROPIC TURNING FLOW ANALYSIS OF PRANDTL–MEYER FLOW PRANDTL–MEYER FUNCTION OVEREXPANDED AND UNDEREXPANDED NOZZLES SUPERSONIC AIRFOILS FANNO FLOW ANALYSIS FOR A GENERAL FLUID REFERENCE STATE AND FANNO TABLE APPLICATIONS CORRELATION WITH SHOCKS FRICTION CHOKING RAYLEIGH FLOW REFERENCE STATE AND THE RAYLEIGH TABLE THERMAL CHOKING DUE TO HEATING REAL GAS EFFECTS WHAT’S REALLY GOING ON SEMIPERFECT GAS BEHAVIOR, DEVELOPMENT OF THE GAS TABLE REAL GAS BEHAVIOR, EQUATIONS OF STATE AND COMPRESSIBILITY FACTORS VARIABLE ? —VARIABLE-AREA FLOWS VARIABLE ? —CONSTANT-AREA FLOWS PROPULSION SYSTEMS BRAYTON CYCLE PROPULSION ENGINES GENERAL PERFORMANCE PARAMETERS, THRUST, POWER, AND EF?CIENCY AIR-BREATHING PROPULSION SYSTEMS PERFORMANCE PARAMETERS AIR-BREATHING PROPULSION SYSTEMS INCORPORATING REAL GAS EFFECTS ROCKET PROPULSION SYSTEMS PERFORMANCE PARAMETERS SUPERSONIC DIFFUSERS APPENDIXES A SUMMARY OF THE ENGLISH ENGINEERING (EE) SYSTEM OF UNITS B SUMMARY OF THE INTERNATIONAL SYSTEM (SI) OF UNITS C FRICTION-FACTOR CHART D OBLIQUE-SHOCK CHARTS (? = 1 4) (TWO-DIMENSIONAL) E CONICAL-SHOCK CHARTS (? = 1 4) (THREE-DIMENSIONAL) F GENERALIZED COMPRESSIBILITY FACTOR CHART G ISENTROPIC FLOW PARAMETERS (? = 1 4) (INCLUDING PRANDTL–MEYER FUNCTION) H NORMAL-SHOCK PARAMETERS (? = 1 4) I FANNO FLOW PARAMETERS (? = 1 4) J RAYLEIGH FLOW PARAMETERS (? = 1 4) K PROPERTIES OF AIR AT LOW PRESSURES L SPECI?C HEATS OF AIR AT LOW PRESSURES SELECTED REFERENCES ANSWERSTO PROBLEMS CHAPTER UNITS AND NOTATION DIMENSION: A QUALITATIVE DE?NITION OF A PHYSICAL ENTITY (SUCH AS TIME, LENGTH, FORCE) REVIEW OF ELEMENTARY PRINCIPLES UNIT: AN EXACT MAGNITUDE OF A DIMENSION (SUCH AS SECONDS, FEET, NEWTONS) IN THE UNITED STATES MOST WORK IN THE AREA OF THERMO-GAS DYNAMICS (PARTICULARLY IN PROPULSION) IS CURRENTLY DONE IN THE ENGLISH ENGINEERING (EE) SYSTEM OF UNITS HOWEVER, MOST OF THE WORLD IS OPERATING IN THE METRIC OR INTERNATIONAL SYSTEM (SI) OF UNITS THUS, WE SHALL REVIEW BOTH SYSTEMS, BEGINNING WITH TABLE 1 1 FORCE AND MASS IN EITHER SYSTEM OF UNITS, FORCE AND MASS ARE RELATED THROUGH NEWTON’S SECOND LAW OF MOTION, WHICH STATES THAT F ?D( ???? MOMENTUM) DT (1 1) THE PROPORTIONALITY FACTOR IS EXPRESSED AS K = 1/GC, AND THUS F = UNITS AND NOTATION (1 4A) NOTE THAT GC IS NOT THE STANDARD GRAVITY (CHECK THE UNITS) IT IS A PROPORTIONALITY FACTOR WHOSE VALUE DEPENDS ON THE UNITS BEING USED IN FURTHER DISCUSSIONS WE SHALL TAKE THE NUMERICAL VALUE OF GC TO BE 32 2 WHEN USING THE ENGLISH ENGINEERING SYSTEM IN OTHER ENGINEERING ?ELDS OF ENDEAVOR, SUCH AS STATICS AND DYNAMICS, THE BRITISH GRAVITATIONAL SYSTEM (ALSO KNOWN AS THE U S CUSTOMARY SYSTEM) IS USED THIS IS VERY SIMILAR TO THE ENGLISH ENGINEERING SYSTEM EXCEPT THAT THE UNIT OF MASS IS THE SLUG IN THIS SYSTEM OF UNITS WE FOLLOW THE DE?NITION: A 1-POUND FORCE WILL GIVE A 1-SLUG MASS AN ACCELERATION OF (1 4C) SINCE GC HAS THE NUMERICAL VALUE OF UNITY (AND USES THE DYNAMICAL UNIT OF MASS, I E , THE KILOGRAM) MOST AUTHORS OMIT THIS FACTOR FROM EQUATIONS IN THE SI SYSTEM HOWEVER, WE SHALL LEAVE THE SYMBOL GC IN THE EQUATIONS SO THAT YOU MAY USE ANY SYSTEM OF UNITS WITH LESS LIKELIHOOD OF MAKING ERRORS DENSITY AND SPECI?C VOLUME DENSITY IS THE MASS PER UNIT VOLUME AND IS GIVEN THE SYMBOL ? IT HAS UNITS OF LBM/FT3, KG/M3, OR SLUG/FT3 SPECI?C VOLUME IS THE VOLUME PER UNIT MASS AND IS GIVEN THE SYMBOL V IT HAS UNITS OF FT3/LBM, M3/KG, OR FT3/SLUG THUS ? = (1 9B) THE VALUES 460 AND 273 WILL BE USED IN OUR CALCULATIONS VISCOSITY WE SHALL BE DEALING WITH ?UIDS, WHICH ARE DE?NED AS ANY SUBSTANCE THAT WILL CONTINUOUSLY DEFORM WHEN SUBJECTED TO A SHEAR STRESS SOME MATHEMATICAL CONCEPTS THE (EQUIVALENT) MOLECULAR MASS OF AIR IS 28 97 R = 28 97 = 53 3 FT-LBF/LBM-°R OR R = SOME MATHEMATICAL CONCEPTS VARIABLES THE EQUATION Y = F (X) (1 14) INDICATES THAT A FUNCTIONAL RELATION EXISTS BETWEEN THE VARIABLES X AND Y FURTHER, IT DENOTES THAT X IS THE INDEPENDENT VARIABLE, WHOSE VALUE CAN BE GIVEN ANYPLACE WITHIN AN AP- PROPRIATE RANGE Y IS THE DEPENDENT VARIABLE, WHOSE VALUE IS ?XED ONCE X HAS BEEN SELECTED IN MOST CASES IT IS POSSIBLE TO INTERCHANGE THE DEPENDENT AND INDEPENDENT VARIABLES AND WRITE X = F (Y) (1 15) FREQUENTLY, A VARIABLE WILL DEPEND ON MORE THAN ONE OTHER VARIABLE ONE MIGHT WRITE P = F (X,Y,Z) (1 16) 3! + (1 25) THERMODYNAMIC CONCEPTS FOR CONTROL MASS ANALYSIS 2 MATHEMATICAL: DE?NED FROM COMBINATIONS OF OTHER PROPERTIES (DENSITY, SPECI?C HEATS, ENTHALPY, ETC ) 3 DERIVED: ARRIVED AT AS THE RESULT OF ANALYSIS A INTERNAL ENERGY (FROM THE ?RST LAW OF THERMODYNAMICS) B ENTROPY (FROM THE SECOND LAW OF THERMODYNAMICS) STATE CHANGE: COMES ABOUT AS THE RESULT OF A CHANGE IN ANY PROPERTY PATH OR PROCESS: REPRESENTS A SERIES OF CONSECUTIVE STATES THAT DE?NE A UNIQUE PATH FROM ONE STATE TO ANOTHER SOME SPECIAL PROCESSES: ADIABATIC ? NO HEAT TRANSFER ISOTHERMAL ? T = CONSTANT ISOBARIC ? P = CONSTANT ISENTROPIC ? S = CONSTANT CYCLE: A SEQUENCE OF PROCESSES IN WHICH THE SYSTEM IS RETURNED TO THE ORIGINAL STATE POINT FUNCTIONS: ANOTHER WAY OF SAYING PROPERTIES, SINCE THEY DEPEND ONLY ON THE STATE OF THE SYSTEM AND ARE INDEPENDENT OF THE HISTORY OR PROCESS BY WHICH THE STATE WAS OBTAINED PATH FUNCTIONS: QUANTITIES THAT ARE NOT FUNCTIONS OF THE STATE OF THE SYSTEM BUT RATHER DEPEND ON THE PATH TAKEN TO MOVE FROM ONE STATE TO ANOTHER HEAT AND WORK ARE PATH FUNCTIONS THEY CAN BE OBSERVED CROSSING THE SYSTEM’S BOUNDARIES DURING A PROCESS LAWS OF CLASSICAL THERMODYNAMICS RELATION AMONG PROPERTIES THERMAL EQUILIBRIUM CONSERVATION OF ENERGY