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

a state function, and the entropy change of an ideal gas with constant

specific heats during a change of state from 1 to 2 is given by

(17–55)

The entropy of a fluid may increase or decrease during Rayleigh flow,

depending on the direction of heat transfer.

Equation of stateNoting that P
rRT, the properties P,r, and Tof an

ideal gas at states 1 and 2 are related to each other by

(17–56)

Consider a gas with known properties R,k, and cp. For a specified inlet

state 1, the inlet properties P 1 ,T 1 ,r 1 ,V 1 , and s 1 are known. The five exit

properties P 2 ,T 2 ,r 2 ,V 2 , and s 2 can be determined from the five equations

17–50, 17–51, 17–53, 17–55, and 17–56 for any specified value of heat

transfer q. When the velocity and temperature are known, the Mach number

can be determined from.

Obviously there is an infinite number of possible downstream states 2

corresponding to a given upstream state 1. A practical way of determining

these downstream states is to assume various values of T 2 , and calculate all

other properties as well as the heat transfer qfor each assumed T 2 from the

Eqs. 17–50 through 17–56. Plotting the results on a T-sdiagram gives a

curve passing through the specified inlet state, as shown in Fig. 17–52. The

plot of Rayleigh flow on a T-sdiagram is called the Rayleigh line,and sev-

eral important observations can be made from this plot and the results of the

calculations:

1.All the states that satisfy the conservation of mass, momentum, and

energy equations as well as the property relations are on the Rayleigh

line. Therefore, for a given initial state, the fluid cannot exist at any

downstream state outside the Rayleigh line on a T-sdiagram. In fact,

the Rayleigh line is the locus of all physically attainable downstream

states corresponding to an initial state.

2.Entropy increases with heat gain, and thus we proceed to the right on

the Rayleigh line as heat is transferred to the fluid. The Mach number is

Ma 
1 at point a, which is the point of maximum entropy (see Exam-

ple 17–13 for proof). The states on the upper arm of the Rayleigh line

above point aare subsonic, and the states on the lower arm below point

aare supersonic. Therefore, a process proceeds to the right on the Ray-

leigh line with heat addition and to the left with heat rejection regard-

less of the initial value of the Mach number.

3.Heating increases the Mach number for subsonic flow, but decreases it

for supersonic flow. The flow Mach number approaches Ma 
1 in both

cases (from 0 in subsonic flow and from ∞in supersonic flow) during

heating.

4.It is clear from the energy balance q 
cp(T 02 T 01 ) that heating

increases the stagnation temperature T 0 for both subsonic and super-

sonic flows, and cooling decreases it. (The maximum value of T 0 occurs

at Ma 
1.) This is also the case for the thermodynamic temperature T

Ma
V>c
V> 1 kRT

P 1

r 1 T 1

P 2

r 2 T 2

s 2 s 1 
cp ln

T 2

T 1

R ln

P 2

P 1

862 | Thermodynamics

Mab = 1/ k

Maa = 1

a

a

b

Ma < 1

Ma > 1

Cooling

(Ma S 0)

Cooling

(Ma S )

Heating

(Ma S 1)

Heating

(Ma S 1)

Tmax

smax

s

T

FIGURE 17–52

T-sdiagram for flow in a constant-area

duct with heat transfer and negligible

friction (Rayleigh flow).

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