Question

Air flows subsonically in an adiabatic 2 cm diameter duct. The average friction factor is 0.024. What length of duct is necessary to accelerate the flow from M1 = 0.1 to M2=0.5? What additional length accelerate it to M3 = 1.0?

Answer 1

This is a case of Fanno Flow. Write the differential equation for Fanno flow

For air, k=1.4, f=0.024, D=2cm. Substituting these values,

Separate variables

Substitute

Separate to partial fractions on LHS

Integrate between limits

Substitute , and

This is the length required to accelerate from M=0.1 to M=0.5

Substitute , and (In the problem M_2=0.5, and M_3=1)

This is the additional length required to accelerate from M=0.5 to M=0.1