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

263

9

Adiabatic Flow in a Duct with Friction

Introduction

In the discussion of ow through ducts given up to this point, it was assumed, in almost

all cases, that the effects of viscosity were negligible. This is often an adequate assump-

tion when dealing with ow through nozzles or short ducts. For long ducts, however, the

effects of viscosity, i.e., the effects of uid friction at the walls, can in fact be dominant.

This is illustrated in Figure 9.1, which shows typical Mach number and pressure varia-

tions in a constant area duct with and without friction. In incompressible ow through a

duct of constant cross-sectional area, the friction only affects the pressure, which drops

in the direction of ow. The velocity in such a situation remains constant along the duct.

In compressible ow, however, friction effects all of the ow variables, i.e., the changes in

pressure cause changes in density, which lead to changes in velocity.

In some cases, the effects of viscosity may be negligible over part of the ow but then be

very important in other parts of the ow. This is illustrated in Figure 9.2.

In this chapter, consideration will be given to the effects of viscosity on steady gas ows

through ducts under such conditions that compressibility effects are important. In the

analyses given in this chapter it will be assumed that the ow is adiabatic, i.e., that the duct

is well insulated (a discussion of the effects of friction in the presence of heat exchange is

given in the next chapter). Attention will, in this chapter, mainly be restricted to ow in a

constant area duct although a brief discussion of the effects of area change will be given at

the end of this chapter.

Flow in a Constant Area Duct

Attention will here be given to the effects of wall friction on adiabatic ow through a duct

whose cross-sectional area does not change. This type of ow, i.e., compressible adiabatic

ow in a constant area duct with frictional effects, is known as “Fanno” ow.

Consider the momentum balance for the small portion of the duct shown in Figure 9.3.

Since steady ow is being considered, this gives

Netpressure forceForce duetowallshear stre− sss

Mass flow rate Velocity outVelocityin=×−

()

264 Introduction to Compressible Fluid Flow

Compressible

Compressible

Frictionless

Incompressible

Incompressible

and frictionless

Distance along ductDistance along duct

M

p

FIGURE 9.1

Effect of friction on Mach number and pressure in a duct.

Isentropic Viscosity important

FIGURE 9.2

Effect of friction in different portions of ow.

Control volume

V

p

dx

p + dp

V + dV

x

FIGURE 9.3

Control volume used in analysis of frictional ow in a duct.

265Adiabatic Flow in a Duct with Friction

i.e., since the force due to the wall shear stress is equal to the product of the shear stress

and the surface area of the portion of the duct being considered,

pA − (p + dp)A − τ

w

(perimeter) dx = ρVA(V + dV − V)

where τ

w

is the wall shear stress. Therefore,

−− =dp

P

A

dx VdV

w

τρ

(9.1)

where P is the perimeter of the duct and A is its cross-sectional area. In the case of a circu-

lar duct, P = πD and A = πD

2

/4 so P/A = 4/D. For this reason, even for noncircular ducts,

the ratio P/A is usually expressed in terms of an equivalent diameter, called the “hydraulic

diameter,” D

H

, dened by

D

A

P

H

==

44

Area

Perimeter

()

(9.2)

Dividing Equation 9.1 by ρV

2

then gives

−− =

dp

VV

P

A

dx

dV

V

w

ρ

τ

ρ

22

(9.3)

Next consider continuity of mass. This gives

ρAV = constant

However, in the present case, since ow in a constant area duct is being considered, this

reduces to

ρV = constant

Hence, for the portion of the duct being considered,

ρV = (ρ + dρ)(V + dV)

From this, it follows that to rst order of accuracy, i.e., by assuming that the term dρ × dV

is negligible because dρ and dV are very small.

ρ dV + V dρ = 0 (9.4)

Dividing this equation through by ρV then gives

dV

V

d

+=

ρ

ρ

0

(9.5)

266 Introduction to Compressible Fluid Flow

Next, consider the energy equation

cT

V

cT dT

VdV

pp

+= =++

+

22

22

constant ()

()

This gives to rst order of accuracy

c

p

dT + V dV = 0 (9.6)

Also, the equation of state gives

p

RT

pR

T

ρ

ρ

==

,i.e.,

and

pdp

d

RT dT pdpdRT dT

+

+

=+ +=

++

ρρ

ρρ(),()( )i.e.,

Hence, to rst order of accuracy,

dp = RT d ρ + ρR dT (9.7)

Further, since

M

V

a

V

RT

2

2

2

2

==

γ

and

()

()

()

()

()

MdM

VdV

ada

VdV

RT dT

+=

+

+

=

+

+

2

2

2

2

γ

i.e.,

M

dM

M

VdVV

RT dT T

2

2

22

1

1

1

+













=

+

+

()

()

/

/γ

i.e., to rst order of accuracy,

M

dM

M

V

RT

dV

V

dT

T

2

2

12 12 1+













=+













−













γ