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

designed by a Swedish engineer, Carl G. B. de Laval (1845–1913), and

therefore converging–diverging nozzles are often called Laval nozzles.

Variation of Fluid Velocity with Flow Area

It is clear from Example 17–3 that the couplings among the velocity, den-

sity, and flow areas for isentropic duct flow are rather complex. In the

remainder of this section we investigate these couplings more thoroughly,

and we develop relations for the variation of static-to-stagnation property

ratios with the Mach number for pressure, temperature, and density.

We begin our investigation by seeking relationships among the pressure,

temperature, density, velocity, flow area, and Mach number for one-

dimensional isentropic flow. Consider the mass balance for a steady-flow

process:

Differentiating and dividing the resultant equation by the mass flow rate, we

obtain

(17–13)

Neglecting the potential energy, the energy balance for an isentropic flow with

no work interactions can be expressed in the differential form as (Fig. 17–15)

(17–14)

This relation is also the differential form of Bernoulli’s equation when

changes in potential energy are negligible, which is a form of the conserva-

tion of momentum principle for steady-flow control volumes. Combining

Eqs. 17–13 and 17–14 gives

(17–15)

Rearranging Eq. 17–9 as (∂r/∂P)s
1/c^2 and substituting into Eq. 17–15 yield

(17–16)

This is an important relation for isentropic flow in ducts since it describes

the variation of pressure with flow area. We note that A,r, and Vare positive

quantities. For subsonicflow (Ma 1), the term 1 Ma^2 is positive; and

thus dAand dPmust have the same sign. That is, the pressure of the fluid

must increase as the flow area of the duct increases and must decrease as the

flow area of the duct decreases. Thus, at subsonic velocities, the pressure

decreases in converging ducts (subsonic nozzles) and increases in diverging

ducts (subsonic diffusers).

In supersonicflow (Ma 1), the term 1 Ma^2 is negative, and thus dA

and dPmust have opposite signs. That is, the pressure of the fluid must

dA

A

dP

rV^2

11 Ma^22

dA

A

dP

r

a

1

V^2



dr

dP

b

dP

r

V dV
 0

dr

r



dA

A



dV

V

 0

m#
rAV
constant

832 | Thermodynamics

Converging nozzle

Converging–diverging nozzle

Throat

Throat

Fluid

Fluid

FIGURE 17–14

The cross section of a nozzle at the

smallest flow area is called the throat.

0 (isentropic)

dP

CONSERVATION OF ENERGY

(steady flow, w = 0, q = 0, ∆pe = 0)

h 1 +

V^2

2

(^1) = h 2 +V
2
2
2
or
h +
V^2
2 = constant
Differentiate,
dh + V dV = 0
Also,
= dh –^ dP
dh = vdP r
r
v
=^1
Substitute,
dP (^) + V (^) dV = 0
T ds
FIGURE 17–15
Derivation of the differential form of
the energy equation for steady
isentropic flow.
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