Thermodynamics

Nozzles whose flow area decreases in the flow directionare called converging nozzles. Nozzles whose flow area firstdecreases and then increases are called converging–divergingnozzles. The location of the smallest flow area of a nozzle iscalled the throat. The highest velocity to which a fluid can beaccelerated in a converging nozzle is the sonic velocity.Accelerating a fluid to supersonic velocities is possible onlyin converging–diverging nozzles. In all supersonic converging–diverging nozzles, the flow velocity at the throat is the speedof sound.The ratios of the stagnation to static properties for idealgases with constant specific heats can be expressed in termsof the Mach number asandT 0T 1 a k 12P 0P c 1 a k 12r 0r c 1 a k 12b Ma 2k>1k12b Ma 2 d1>1k12b Ma 2 dWhen Ma 1, the resulting static-to-stagnation propertyratios for the temperature, pressure, and density are calledcritical ratios and are denoted by the superscript asterisk:T* 2k>1k12T 0 k 1 P* 2 aP 0 k 1 b1>1k12r* 2and ar 0 k 1 bThe pressure outside the exit plane of a nozzle is called theback pressure. For all back pressures lower than P*, the pres-Chapter 17 | 873sure at the exit plane of the converging nozzle is equal to P*,the Mach number at the exit plane is unity, and the mass flowrate is the maximum (or choked) flow rate.In some range of back pressure, the fluid that achieved asonic velocity at the throat of a converging–diverging nozzleand is accelerating to supersonic velocities in the divergingsection experiences a normal shock, which causes a suddenrise in pressure and temperature and a sudden drop in velocityto subsonic levels. Flow through the shock is highlyirreversible, and thus it cannot be approximated as isentropic.The properties of an ideal gas with constant specificheats before (subscript 1) and after (subscript 2) a shock arerelated byT 01 T 02 Ma 2 B1k 12Ma 2 1 22kMa 2 1 k 1T 2 2 Ma2 1 1k 12T 1 2 Ma 2 2 1k 12P 2and 1 kMa2 1 2kMa2 1 k 1P 1 1 kMa 2 2 k 1These equations also hold across an oblique shock, providedthat the component of the Mach number normal to theoblique shock is used in place of the Mach number.Steady one-dimensional flow of an ideal gas with constantspecific heats through a constant-area duct with heat transferand negligible friction is referred to as Rayleigh flow. Theproperty relations and curves for Rayleigh flow are given inTable A–34. Heat transfer during Rayleigh flow can be determinedfromq c p 1T 02 T 01 2 c p 1T 2 T 1 2 V 2 2 V 2 12REFERENCES AND SUGGESTED READINGS1. J. D. Anderson. Modern Compressible Flow with HistoricalPerspective. 3rd ed. New York: McGraw-Hill, 2003.2. Y. A. Çengel and J. M. Cimbala. Fluid Mechanics:Fundamentals and Applications. New York: McGraw-Hill, 2006.3. H. Cohen, G. F. C. Rogers, and H. I. H. Saravanamuttoo.Gas Turbine Theory. 3rd ed. New York: Wiley, 1987.4. W. J. Devenport. Compressible Aerodynamic Calculator,http://www.aoe.vt.edu/~devenpor/aoe3114/calc.html.5. R. W. Fox and A. T. McDonald. Introduction to FluidMechanics. 5th ed. New York: Wiley, 1999.6. H. Liepmann and A. Roshko. Elements of Gas Dynamics.Dover Publications, Mineola, NY, 2001.7. C. E. Mackey, responsible NACA officer and curator.Equations, Tables, and Charts for Compressible Flow.NACA Report 1135, http://naca.larc.nasa.gov/reports/1953/naca-report-1135/.8. A. H. Shapiro. The Dynamics and Thermodynamics ofCompressible Fluid Flow. vol. 1. New York: Ronald PressCompany, 1953.9. P. A. Thompson. Compressible-Fluid Dynamics. NewYork: McGraw-Hill, 1972.10. United Technologies Corporation. The Aircraft GasTurbine and Its Operation. 1982.11. Van Dyke, 1982.12. F. M. White. Fluid Mechanics. 5th ed. New York:McGraw-Hill, 2003.