Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

Combining Eqs. 17–35 and 17–36 gives the pressure ratio across the shock:

(17–37)

Equation 17–37 is a combination of the conservation of mass and energy

equations; thus, it is also the equation of the Fanno line for an ideal gas with

constant specific heats. A similar relation for the Rayleigh line can be

obtained by combining the conservation of mass and momentum equations.

From Eq. 17–32,

However,

Thus,

or

(17–38)

Combining Eqs. 17–37 and 17–38 yields

(17–39)

This represents the intersections of the Fanno and Rayleigh lines and relates

the Mach number upstream of the shock to that downstream of the shock.

The occurrence of shock waves is not limited to supersonic nozzles only.

This phenomenon is also observed at the engine inlet of a supersonic air-

craft, where the air passes through a shock and decelerates to subsonic

velocities before entering the diffuser of the engine. Explosions also pro-

duce powerful expanding spherical normal shocks, which can be very

destructive (Fig. 17–33).

Various flow property ratios across the shock are listed in Table A–33 for

an ideal gas with k
1.4. Inspection of this table reveals that Ma 2 (the

Mach number after the shock) is always less than 1 and that the larger the

supersonic Mach number before the shock, the smaller the subsonic Mach

number after the shock. Also, we see that the static pressure, temperature,

and density all increase after the shock while the stagnation pressure

decreases.

The entropy change across the shock is obtained by applying the entropy-

change equation for an ideal gas across the shock:

(17–40)

which can be expressed in terms of k,R, and Ma 1 by using the relations

developed earlier in this section. A plot of nondimensional entropy change

s 2 s 1 
cp ln

T 2

T 1

R ln

P 2

P 1

Ma^22 

Ma^21  2 >1k 12

2Ma^21 k>1k 12  1

P 2

P 1

1 kMa^21

1 kMa^22

P 111 kMa^212 
P 211 kMa^222

rV^2 
a

P

RT

b1Mac 22 
a

P

RT

b1Ma 2 kRT 22 
PkMa^2

P 1 P 2 

m

#

A

1 V 2 V 12 
r 2 V^22 r 1 V^21

P 2

P 1

Ma 121 Ma^211 k 1 2> 2

Ma 221 Ma^221 k 1 2> 2

848 | Thermodynamics

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