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

(1886–1975) and develop relationships for the flow properties before and

after the shock. We do this by applying the conservation of mass, momen-

tum, and energy relations as well as some property relations to a stationary

control volume that contains the shock, as shown in Fig. 17–29. The normal

shock waves are extremely thin, so the entrance and exit flow areas for the

control volume are approximately equal (Fig 17–30).

We assume steady flow with no heat and work interactions and no

potential energy changes. Denoting the properties upstream of the shock

by the subscript 1 and those downstream of the shock by 2, we have the

following:

Conservation of mass: (17–29)

or

Conservation of energy: (17–30)

or

(17–31)

Conservation of momentum: Rearranging Eq. 17–14 and integrating yield

(17–32)

Increase of entropy: (17–33)

We can combine the conservation of mass and energy relations into a sin-

gle equation and plot it on an h-sdiagram, using property relations. The

resultant curve is called the Fanno line,and it is the locus of states that

have the same value of stagnation enthalpy and mass flux (mass flow per

unit flow area). Likewise, combining the conservation of mass and momen-

tum equations into a single equation and plotting it on the h-sdiagram yield

a curve called the Rayleigh line.Both these lines are shown on the h-sdia-

gram in Fig. 17–31. As proved later in Example 17–8, the points of maxi-

mum entropy on these lines (points aand b) correspond to Ma 
1. The

state on the upper part of each curve is subsonic and on the lower part

supersonic.

The Fanno and Rayleigh lines intersect at two points (points 1 and 2),

which represent the two states at which all three conservation equations are

satisfied. One of these (state 1) corresponds to the state before the shock,

and the other (state 2) corresponds to the state after the shock. Note that

the flow is supersonic before the shock and subsonic afterward. Therefore

the flow must change from supersonic to subsonic if a shock is to occur.

The larger the Mach number before the shock, the stronger the shock will

be. In the limiting case of Ma 
1, the shock wave simply becomes a sound

wave. Notice from Fig. 17–31 that s 2 s 1. This is expected since the flow

through the shock is adiabatic but irreversible.

s 2 s 1  0

A 1 P 1 P 22 
m# 1 V 2 V 12

h 01 
h 02

h 1 

V^21

2

h 2 

V^22

2

r 1 V 1 
r 2 V 2

r 1 AV 1 
r 2 AV 2

846 | Thermodynamics

Control

volume

Flow

Ma 1 > 1 P

V 1

s

Shock wave

P

V 2

12

12

12

12

h h

s

rr

Ma 2 < 1

FIGURE 17–29

Control volume for flow across a

normal shock wave.

FIGURE 17–30

Schlieren image of a normal shock in

a Laval nozzle. The Mach number in

the nozzle just upstream (to the left) of

the shock wave is about 1.3. Boundary

layers distort the shape of the normal

shock near the walls and lead to flow

separation beneath the shock.

Photo by G. S. Settles, Penn State University. Used

by permission.

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