Thermodynamics

Chapter 17 | 849FIGURE 17–33Schlieren image of the blast wave(expanding spherical normal shock)produced by the explosion of afirecracker detonated inside a metal canthat sat on a stool. The shock expandedradially outward in all directions at asupersonic speed that decreased withradius from the center of the explosion.The microphone at the lower rightsensed the sudden change in pressureof the passing shock wave andtriggered the microsecond flashlampthat exposed the photograph.Photo by G. S. Settles, Penn State University. Usedby permission.across the normal shock (s 2 s 1 )/R versus Ma 1 is shown in Fig. 17–34.Since the flow across the shock is adiabatic and irreversible, the second lawrequires that the entropy increase across the shock wave. Thus, a shockwave cannot exist for values of Ma 1 less than unity where the entropychange would be negative. For adiabatic flows, shock waves can exist onlyfor supersonic flows, Ma 1 1.EXAMPLE 17–8The Point of Maximum Entropyon the Fanno LineShow that the point of maximum entropy on the Fanno line (point b of Fig.17–31) for the adiabatic steady flow of a fluid in a duct corresponds to thesonic velocity, Ma 1.Solution It is to be shown that the point of maximum entropy on the Fannoline for steady adiabatic flow corresponds to sonic velocity.Assumptions The flow is steady, adiabatic, and one-dimensional.Analysis In the absence of any heat and work interactions and potentialenergy changes, the steady-flow energy equation reduces to(s 2 s 1 )/R0s 2 – s 1 > 0s 2 – s 1 < 0ImpossibleSubsonic flowbefore shockMa 1 = 1 Supersonic flow Ma 1before shockFIGURE 17–34Entropy change across the normalshock.Differentiating yieldsh V 22 constantdh V dV 0For a very thin shock with negligible change of duct area across the shock, thesteady-flow continuity (conservation of mass) equation can be expressed asrV constant