Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

399

12

High-Temperature Flows

Introduction

Chapter 1 contains a discussion of some of the assumptions that are commonly adopted in

the analysis of compressible gas ows, i.e., of the assumptions that are commonly used in

modeling such ows. It was explained in Chapter 1 that most such analyses are based on

the assumptions that

• The specic heats of the gas are constant

• The perfect gas law, p/ρ = RT, applies

• There are no changes in the physical nature of the gas in the ow

• The gas is in thermodynamic equilibrium

However, as discussed in the previous chapter, if the temperature in the ow becomes

very high, it is possible that some of these assumptions will cease to be valid. To investigate

again (a discussion of this was also given in the previous chapter) whether it is possible

to get such high temperatures in a ow, consider the ow of air through a normal shock

wave. The air will be assumed to have a temperature of 216.7 K (i.e., –56.3°C) ahead of the

shock. This is the temperature in the so-called standard atmosphere between an altitude

of ~11,000 and ~25,000 m. The situation considered is therefore as shown in Figure 12.1a.

If the assumptions discussed above apply, it was shown in Chapter 5 that

TM M

M

21

2

1

2

2

1

2

216 7

21

21

1

.

[()][ ()]

()

=

−− +−

+

γγ γ

γ

Hence, since γ = 1.4 for air, the temperature downstream of the shock will be given by

TM M

M

21

2

1

2

1

2

216 7

28 04 204

576

.

(. .)(.)

.

=

−+

(12.1)

The variation of T

2

with M

1

given by this equation is shown in Figure 12.2.

If instead of passing through a normal shock, the ow is brought to rest isentropically

(see Figure 12.1b), the temperature attained, i.e., the stagnation temperature, is given by

T

M

2

1

2

216 7

1

1

2.

=+

−γ

400 Introduction to Compressible Fluid Flow

i.e., since the case of γ = 1.4 is being considered

T

M

2

1

2

216 7

102

.

.=+ (12.2)

The variation of T

2

with M

1

given by the equation is also shown in Figure 12.2. It will be

seen from Figure 12.2 that at Mach numbers of roughly 5 or greater, T

2

exceeds 1000°C in

both types of ow considered. When temperatures as high as this exist in a ow, it seems

prudent to investigate the applicability of the assumptions on which the analysis of the

ow is based, i.e., in particular, to consider whether at such temperatures the specic heats

can still be assumed to be independent of temperature, whether the perfect gas law is still

applicable, and whether dissociation of the gas molecules and, perhaps, even ionization of

the atoms is likely to occur. These effects will be examined in this chapter. However, only

a brief introduction to the very important topic of high temperature gas ows can be given

Normal

shock wave

(a) (b)

T

2

T

1

T

1

T

2

M

1

M

0

M

1

= 216.7 K

= 216.7 K

Isentropic flow

(= T

0

)

FIGURE 12.1

Flow situations considered: (a) normal shock; (b) isentropic deceleration.

2000

1500

1000

500

Stagnation

temperature

Behind normal

shock wave

0

1

2

3

4

5

6

M

1

γ = 1.4

T

1

= 216.7 K

T

1

(K)

FIGURE 12.2

Mach number dependence of temperature behind normal shock wave and in isentropic ow considered.

401High-Temperature Flows

in this chapter despite the fact that such ows occur in a number of situations of great

practical importance.

Effect of Temperature on Specific Heats

The rst high temperature gas effect considered here is the possibility of changes in the

specic heats of the gas at high temperatures. The increase of internal energy that results

when the temperature of a gas is increased is associated with an increase in the energy

possessed by the gas molecule. Now the increase in the energy of a molecule can be asso-

ciated with an increase in the translational kinetic energy or with an increase in the rota-

tional kinetic energy or with an increase in the vibrational kinetic energy of the molecule.

This is illustrated in Figure 12.3.

In addition, at high temperatures, changes in the energy associated with the electron

motion can occur. Therefore,

Δe = Δe

trans

+ Δe

rot

+ Δe

vib

+ Δe

el

(12.3)

where Δe is the change in internal energy, Δe

trans

is the change in translational energy,

Δe

rot

is the change in rotational energy, Δe

vib

the change in vibrational energy, and Δe

el

is

the change in electron energy. Measuring e from 0 at absolute zero temperature this gives

e = e

trans

+ e

rot

+ e

vib

+ e

el

(12.4)

Translational

motion of

center of mass

Center

of mass

Rotational

motion about

center of mass

Simple dumbbell

model of diatomi

c

molecule

Vibrational

motion along

axis

Electron

motion

FIGURE 12.3

Excitation modes of a diatomic molecule.