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

Another parameter sometimes used in the analysis of one-dimensional

isentropic flow of ideal gases is Ma*, which is the ratio of the local velocity

to the speed of sound at the throat:

(17–27)

It can also be expressed as

where Ma is the local Mach number,Tis the local temperature, and T* is

the critical temperature. Solving for Tfrom Eq. 17–18 and for T* from

Eq. 17–21 and substituting, we get

(17–28)

Values of Ma* are also listed in Table A–32 versus the Mach number for

k
1.4 (Fig. 17–23). Note that the parameter Ma* differs from the Mach

number Ma in that Ma* is the local velocity nondimensionalized with

respect to the sonic velocity at the throat,whereas Ma is the local velocity

nondimensionalized with respect to the localsonic velocity. (Recall that the

sonic velocity in a nozzle varies with temperature and thus with location.)

Ma*
Ma

B

k 1

2  1 k 12 Ma^2

Ma*

V

c

c

c*

Mac

c*

Ma 2 kRT

2 kRT*

Ma

B

T

T*

Ma*

V

c*

Chapter 17 | 839

Ma*

0 T 0

0.90

A*

T

P 0 r

A P r

Ma

..

.

..

.

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

.

1.00

1.10

0.9146

1.0000

1.0812

..

..

.

.

1.0089

1.0000

1.0079

..

..

.

.

0.5913

0.5283

0.4684

..

..

.

...

FIGURE 17–23

Various property ratios for isentropic

flow through nozzles and diffusers are

listed in Table A–32 for k
1.4 for

convenience.

Pb

At = 50 cm^2

Pi= 1 MPa

Vi = 150 m/s

Ti = 600°C

AIR

Converging

nozzle

FIGURE 17–24

Schematic for Example 17–5.

EXAMPLE 17–5 Effect of Back Pressure on Mass Flow Rate

Air at 1 MPa and 600°C enters a converging nozzle, shown in Fig. 17–24,

with a velocity of 150 m/s. Determine the mass flow rate through the nozzle

for a nozzle throat area of 50 cm^2 when the back pressure is (a) 0.7 MPa

and (b) 0.4 MPa.

Solution Air enters a converging nozzle. The mass flow rate of air through

the nozzle is to be determined for different back pressures.

Assumptions 1 Air is an ideal gas with constant specific heats at room

temperature. 2 Flow through the nozzle is steady, one-dimensional, and

isentropic.

Properties The constant-pressure specific heat and the specific heat ratio of

air are cp
1.005 kJ/kg 
K and k
1.4, respectively (Table A–2a).

Analysis We use the subscripts iand tto represent the properties at the

nozzle inlet and the throat, respectively. The stagnation temperature and

pressure at the nozzle inlet are determined from Eqs. 17–4 and 17–5:

These stagnation temperature and pressure values remain constant through-

out the nozzle since the flow is assumed to be isentropic. That is,

T 0
T 0 i
884 K¬and¬P 0
P 0 i
1.045 MPa

P 0 i
Pia

T 0 i

Ti

b

k>1k 12


 1 1 MPa2a

884 K

873 K

b

1.4>11.4 12


1.045 MPa

T 0 i
Ti

V^2 i

2 cp

873 K

1 150 m>s 22

21 1.005 kJ>kg#K 2

a

1 kJ>kg

1000 m^2 >s^2

b
884 K

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