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

Also,

(17–66)

which simplifies to

(17–67)

The five relations in Eqs. 17–63, 17–65, and 17–67 enable us to calculate

the dimensionless pressure, temperature, density, velocity, stagnation tem-

perature, and stagnation pressure for Rayleigh flow of an ideal gas with a

specified kfor any given Mach number. Representative results are given in

tabular form in Table A–34 for k
1.4.

Choked Rayleigh Flow

It is clear from the earlier discussions that subsonic Rayleigh flow in a duct

may accelerate to sonic velocity (Ma 
1) with heating. What happens if

we continue to heat the fluid? Does the fluid continue to accelerate to super-

sonic velocities? An examination of the Rayleigh line indicates that the fluid

at the critical state of Ma 
1 cannot be accelerated to supersonic velocities

by heating. Therefore, the flow is choked. This is analogous to not being

able to accelerate a fluid to supersonic velocities in a converging nozzle by

simply extending the converging flow section. If we keep heating the fluid,

we will simply move the critical state further downstream and reduce the

flow rate since fluid density at the critical state will now be lower. There-

fore, for a given inlet state, the corresponding critical state fixes the maxi-

mum possible heat transfer for steady flow (Fig. 17–57). That is,

(17–68)

Further heat transfer causes choking and thus the inlet state to change (e.g.,

inlet velocity will decrease), and the flow no longer follows the same Rayleigh

line. Cooling the subsonic Rayleigh flow reduces the velocity, and the Mach

number approaches zero as the temperature approaches absolute zero. Note

that the stagnation temperature T 0 is maximum at the critical state of Ma 
1.

In supersonic Rayleigh flow, heating decreases the flow velocity. Further

heating simply increases the temperature and moves the critical state further

downstream, resulting in a reduction in the mass flow rate of the fluid.

It may seem like supersonic Rayleigh flow can be cooled indefinitely, but it

turns out that there is a limit. Taking the limit of Eq. 17–65 as the Mach

number approaches infinity gives

(17–69)

which yields T 0 /T* 0 
0.49 for k
1.4. Therefore, if the critical stagnation

temperature is 1000 K, air cannot be cooled below 490 K in Rayleigh flow.

Physically this means that the flow velocity reaches infinity by the time the

temperature reaches 490 K—a physical impossibility. When supersonic flow

cannot be sustained, the flow undergoes a normal shock wave and becomes

subsonic.

LimMaS

T 0

T* 0

 1 

1

k^2

qmax
h* 0 h 01 
cp 1 T* 0 T 012

P 0

P* 0

k 1

1 kMa^2

c

2  1 k 12 Ma^2

k 1

d

k>1k 12

P 0

P* 0

P 0

P

P

P*

P*

P* 0

a 1 

k 1

2

Ma^2 b

k>1k 12

a

1 k

1 kMa^2

ba 1 

k 1

2

b

k>1k 12

Chapter 17 | 867

T 01

Rayleigh

flow

Choked

flow

qmax

T 02 
 T 01

T 1 T 2 
 T*

*

FIGURE 17–57

For a given inlet state, the maximum

possible heat transfer occurs when

sonic conditions are reached at the exit

state.

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