Thermodynamics

The ratio of the stagnation to static pressure is obtained by substitutingEq. 17–18 into Eq. 17–5:P 0P c 1 a k 12k>1k12b Ma 2 d(17–19)The ratio of the stagnation to static density is obtained by substitutingEq. 17–18 into Eq. 17–6:T tPtrtChapter 17 | 835SubsonicnozzleThroatT *P *r *(if Ma t = 1)r 0r c 1 a k 12(17–20)Numerical values of T/T 0 , P/P 0 , and r/r 0 are listed versus the Mach numberin Table A–32 for k 1.4, which are very useful for practical compressibleflow calculations involving air.The properties of a fluid at a location where the Mach number is unity (thethroat) are called critical properties, and the ratios in Eqs. (17–18) through(17–20) are called critical ratios (Fig. 17–18). It is common practice in theanalysis of compressible flow to let the superscript asterisk (*) represent thecritical values. Setting Ma 1 in Eqs. 17–18 through 17–20 yieldsT* 2T 0 k 11>1k12b Ma 2 dk>1k12P* 2 aP 0 k 1 b1>1k12r* 2 ar 0 k 1 b(17–21)(17–22)(17–23)These ratios are evaluated for various values of k and are listed in Table17–2. The critical properties of compressible flow should not be confusedwith the properties of substances at the critical point (such as the criticaltemperature T c and critical pressure P c ).T oProoThroatT * , P * , r *(Ma t = 1)SupersonicnozzleFIGURE 17–18When Ma t 1, the properties at thenozzle throat become the criticalproperties.TABLE 17–2The critical-pressure, critical-temperature, and critical-density ratios forisentropic flow of some ideal gasesP*P 0T*T 0r*r 0Superheated Hot products Monatomicsteam, of combustion, Air, gases,k 1.3 k 1.33 k 1.4 k 1.6670.5457 0.5404 0.5283 0.48710.8696 0.8584 0.8333 0.74990.6276 0.6295 0.6340 0.6495