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

Now we begin to reduce the back pressure and observe the resulting

effects on the pressure distribution along the length of the nozzle, as shown

in Fig. 17–20. If the back pressure Pbis equal to P 1 , which is equal to Pr,

there is no flow and the pressure distribution is uniform along the nozzle.

When the back pressure is reduced to P 2 , the exit plane pressure Pealso

drops to P 2. This causes the pressure along the nozzle to decrease in the

flow direction.

When the back pressure is reduced to P 3 (
P*, which is the pressure

required to increase the fluid velocity to the speed of sound at the exit plane

or throat), the mass flow reaches a maximum value and the flow is said to

be choked.Further reduction of the back pressure to level P 4 or below does

not result in additional changes in the pressure distribution, or anything else

along the nozzle length.

Under steady-flow conditions, the mass flow rate through the nozzle is

constant and can be expressed as

Solving for Tfrom Eq. 17–18 and for Pfrom Eq. 17–19 and substituting,

(17–24)

Thus the mass flow rate of a particular fluid through a nozzle is a function

of the stagnation properties of the fluid, the flow area, and the Mach num-

ber. Equation 17–24 is valid at any cross section, and thus m

.

can be evalu-

ated at any location along the length of the nozzle.

For a specified flow area Aand stagnation properties T 0 and P 0 , the maxi-

mum mass flow rate can be determined by differentiating Eq. 17–24 with

respect to Ma and setting the result equal to zero. It yields Ma 
1. Since

the only location in a nozzle where the Mach number can be unity is the

location of minimum flow area (the throat), the mass flow rate through a

nozzle is a maximum when Ma 
1 at the throat. Denoting this area by A*,

we obtain an expression for the maximum mass flow rate by substituting

Ma 
1 in Eq. 17–24:

(17–25)

Thus, for a particular ideal gas, the maximum mass flow rate through a

nozzle with a given throat area is fixed by the stagnation pressure and tem-

perature of the inlet flow. The flow rate can be controlled by changing

the stagnation pressure or temperature, and thus a converging nozzle can be

used as a flowmeter. The flow rate can also be controlled, of course, by

varying the throat area. This principle is vitally important for chemical

processes, medical devices, flowmeters, and anywhere the mass flux of a

gas must be known and controlled.

A plot of m

.

versus Pb/P 0 for a converging nozzle is shown in Fig. 17–21.

Notice that the mass flow rate increases with decreasing Pb/P 0 , reaches a

maximum at Pb
P*, and remains constant for Pb/P 0 values less than this

m#max
A*P 0

B

k

RT 0

a

2

k 1

b

1 k 1 2> 3 21 k 124

m

#

AMaP 02 k>1RT 02

31  1 k 12 Ma^2 > 241 k^1 2> 3^21 k^124

m#
rAV
a

P

RT

bA 1 Ma 2 kRT 2 
PAMa

B

k

RT

Chapter 17 | 837

/P 0

P*

0

1.0

P 0

Pb

Pe

543

2

1

P 0

P* 1.0

P 0

mmax

Pb

543

2

1

P 0

P* 1.0

P 0

m

⋅

⋅

FIGURE 17–21

The effect of back pressure Pbon the

mass flow rate and the exit pressure

Peof a converging nozzle.

m

#

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