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

Property Relations for Rayleigh Flow

It is often desirable to express the variations in properties in terms of the Mach

number Ma. Noting that and thus ,

(17–57)

since P
rRT. Substituting into the momentum equation (Eq. 17–51) gives

P 1 kP 1 Ma 12 
P 2 kP 2 Ma^22 , which can be rearranged as

(17–58)

Again utilizing V
Ma , the continuity equation r 1 V 1 
r 2 V 2 can be

expressed as

(17–59)

Then the ideal-gas relation (Eq. 17–56) becomes

(17–60)

Solving Eq. 17–60 for the temperature ratio T 2 /T 1 gives

(17–61)

Substituting this relation into Eq. 17–59 gives the density or velocity ratio as

(17–62)

Flow properties at sonic conditions are usually easy to determine, and thus

the critical state corresponding to Ma 
1 serves as a convenient reference

point in compressible flow. Taking state 2 to be the sonic state (Ma 2 
1, and

superscript * is used) and state 1 to be any state (no subscript), the property

relations in Eqs. 17–58, 17–61, and 17–62 reduce to (Fig. 17–56)

(17–63)

Similar relations can be obtained for dimensionless stagnation tempera-

ture and stagnation pressure as follows:

(17–64)

which simplifies to

(17–65)

T 0

T* 0

1 k 12 Ma^232  1 k 12 Ma^24

11 kMa^222

T 0

T* 0

T 0

T

T

T*

T*

T* 0

a 1 

k 1

2

Ma^2 bc

Ma 11 k 2

1 kMa^2

d

2

a 1 

k 1

2

b

 1

P

P*

1 k

1 kMa^2

¬

T

T*

c

Ma 11 k 2

1 kMa^2

d

2

¬and¬

V

V*

r*

r

11 k 2 Ma^2

1 kMa^2

r 2

r 1

V 1

V 2

Ma^2111 kMa^222

Ma^2211 kMa^212

T 2

T 1

c

Ma 211 kMa^212

Ma 111 kMa^222

d

2

T 2

T 1

P 2

P 1

r 1

r 2

a

1 kMa^21

1 kMa^22

ba

Ma 21 T 2

Ma 11 T 1

b

r 1

r 2

V 2

V 1

Ma 22 kRT 2

Ma 12 kRT 1

Ma 22 T 2

Ma 12 T 1

1 kRT

P 2

P 1

1 kMa^21

1 kMa^22

rV^2 
rkRTMa^2 
kPMa^2

Ma
V>c
V> 1 kRT V
Ma 1 kRT

866 | Thermodynamics

V

V*

r*


r


(1( 1 k)M)Ma^2

1 kMaMa^2

P

P*

1 k

1 kMaMa^2

T

T*
a

Ma(Ma( 1 k)

1 kMaMa^2 b

2

P 0

P 0 *

k 1

1 kMaMa^2 a

2 (k1)Ma 1 )Ma^2

k 1 b

k/(/(k1) 1 )

T 0

T 0 *

(k1)Ma 1 )Ma^2 [2[ 2 (k1)Ma 1 )Ma^2 ]

(1( 1 kMaMa^2 )^2

FIGURE 17–56

Summary of relations for Rayleigh

flow.

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