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

17–6 ■ DUCT FLOW WITH HEAT TRANSFER AND

NEGLIGIBLE FRICTION (RAYLEIGH FLOW)

So far we have limited our consideration mostly to isentropic flow, also

called reversible adiabatic flowsince it involves no heat transfer and no

irreversibilities such as friction. Many compressible flow problems encoun-

tered in practice involve chemical reactions such as combustion, nuclear

reactions, evaporation, and condensation as well as heat gain or heat loss

through the duct wall. Such problems are difficult to analyze exactly since

they may involve significant changes in chemical composition during flow,

and the conversion of latent, chemical, and nuclear energies to thermal

energy (Fig. 17–50).

The essential features of such complex flows can still be captured by a

simple analysis by modeling the generation or absorption of thermal energy

860 | Thermodynamics

Solution We are to calculate the Mach number and pressure downstream of

a sudden expansion along a wall.

Assumptions 1 The flow is steady. 2 The boundary layer on the wall is

very thin.

Properties The fluid is air with k
1.4.

Analysis Because of assumption 2, we approximate the total deflection

angle as equal to the wall expansion angle (i.e., u
d
10°). With Ma 1 


2.0, we solve Eq. 17–49 for the upstream Prandtl–Meyer function,

Next, we use Eq. 17–48 to calculate the downstream Prandtl–Meyer function,

Ma 2 is found by solving Eq. 17–49, which is implicit—an equation solver is

helpful. We get Ma 2 
2.385.There are also compressible flow calculators

on the Internet that solve these implicit equations, along with both normal

and oblique shock equations; e.g., see http://www.aoe.vt.edu/~devenpor/aoe3114/calc

.html.

We use the isentropic relations to calculate the downstream pressure,

Since this is an expansion, Mach number increases and pressure decreases,

as expected.

Discussion We could also solve for downstream temperature, density, etc.,

using the appropriate isentropic relations.

P 2 

P 2 >P 0

P 1 >P 0

P 1 

c 1 a

k 1

2

bMa^22 d

k>1k 12

c 1 a

k 1

2

bMa^21 d

k>1k 12 1 230 kPa^2 
126 kPa

u
n 1 Ma 22 n 1 Ma 12 Sn 1 Ma 22 
un 1 Ma 12 
10°26.38°
36.38°

B

1.4 1

1.4 1

tan^1 c

B

1.4 1

1.4 1

1 2.0^2  12 dtan^1 a^2 2.0^2  1 b
26.38°

n 1 Ma 2 


B

k 1

k 1

tan^1 cB

k 1

k 1

1 Ma^2  12 dtan^1 a (^2) Ma^2  1 b
Fuel nozzles or spray bars
Flame holders
Air inlet
FIGURE 17–50
Many practical compressible flow
problems involve combustion, which
may be modeled as heat gain through
the duct wall.
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