Thermodynamics

828 | Thermodynamicswhich yieldsdh c dV 0where we have neglected the second-order term dV 2 . The amplitude of theordinary sonic wave is very small and does not cause any appreciablechange in the pressure and temperature of the fluid. Therefore, the propagationof a sonic wave is not only adiabatic but also very nearly isentropic.Then the second Tdsrelation developed in Chapter 7 reduces toT ds ¡0 dh dP r(b)ordh dP r(c)Combining Eqs. a, b, and c yields the desired expression for the speed ofsound asc 2 dP at s constantdrorc 2 a 0P(17–9)0r b sIt is left as an exercise for the reader to show, by using thermodynamicproperty relations (see Chap. 12) that Eq. 17–9 can also be written asc 2 k a 0P0r b T(17–10)where k is the specific heat ratio of the fluid. Note that the speed of soundin a fluid is a function of the thermodynamic properties of that fluid.When the fluid is an ideal gas (P rRT), the differentiation in Eq. 17–10can easily be performed to yieldAIRHELIUMorc 2 k a 0P0r b k c 0 1rRT2d kRTT 0r T284 m/s347 m/s634 m/s200 K300 K1000 K832 m/s1019 m/s1861 m/sc 2kRT(17–11)Noting that the gas constant R has a fixed value for a specified ideal gas andthe specific heat ratio k of an ideal gas is, at most, a function of temperature,we see that the speed of sound in a specified ideal gas is a function oftemperature alone (Fig. 17–9).A second important parameter in the analysis of compressible fluid flowis the Mach number Ma, named after the Austrian physicist Ernst Mach(1838–1916). It is the ratio of the actual velocity of the fluid (or an object instill air) to the speed of sound in the same fluid at the same state:FIGURE 17–9The speed of sound changes withtemperature and varies with the fluid.Ma V c(17–12)Note that the Mach number depends on the speed of sound, which dependson the state of the fluid. Therefore, the Mach number of an aircraft cruising