Question

Air flows into a commercial steel pipe of 5 cm ID at 70 m/s. The inlet pressure and temperature of the air are 1 MPa and 100?C, respectively.

a) Determine the Mach number at a distance 60 m down the pipe if the flow is adiabatic.

b) Calculate the temperature at the same point. How would you comment on the assumption of

constant Re, hence constant f, along the pipe by considering the temperature variation?

Answer 1

a)

At 100 deg C, Air kinematic viscosity = 22.97*10^-6 m²/s

Reynolds number Re = V*D /

= 70 * 0.05 / (22.97*10^-6)

Re = 1.52*10⁵

For commercial steel pipe, roughness = 0.046 mm

Relative roughness = 0.046 / D = 0.046 / 50 = 0.00092

From Moody diagram, for Re = 1.52*10⁵ and relative roughness = 0.00092, we get friction factor f = 0.022

Velocity of sound at inlet a = (kRT)^0.5 = (1.4 * 287 * (100 + 273))^0.5 = 387.13 m/s

Inlet Mach number Ma = V / a = 70 / 387.13 = 0.181

From adiabatic flow in a duct tables, for k = 1.4 and Ma = 0.181 we get fL* / D = 18.5427

We get sonic length L* = 18.5427 * 0.05 / 0.022 = 42.14 m

Since the given length L = 60 m > 42.14 m, the inlet Ma will be reduced until the exit flow is sonic and mass flow will be reduced by frictional choking.

Hence, exit Ma = 1.0

b)

fL* / D = 0.022 * 60 / 0.05 = 26.4

For k = 1.4 and fL*/D = 26.4, we get from adiabatic duct flow tables that Ma = 0.155

Hence, the inlet Ma will be reduced to 0.155.

From adiabatic flow in a duct tables, for k = 1.4 and Ma = 0.155 we get T / T* = 1.194

Putting T = 100 deg C = 373 K, we get T* = 373 / 1.194 = 312.4 K = 39 deg C

The temperature varies from 100 deg C at inlet to 39 deg C at outlet. This will impact the constant kinematic viscosity assumption. Hence, Re and therefore, f will vary considerably.