Question

A stream of air enters a 7.00-cm ID pipe at a velocity of 30.0 m/s at 27.0°C and 1.80 bar (gauge). At a point downstrream, the air flows through a 5.50-cm ID pipe at 60.0°C and 1.63 bar (gauge). What is the velocity of the gas at the downstream point

Answer 1

Answer:

Velocity of the gas at downstream point is = 57.607m/s

Explanation:

Entering conditions V1 = 30m/s, T1 = 27 C, D1 = 0.07m, P1 = 1.80 bar

At downstream v2 = ?, T2 = 60 C, D2 = 0.055m P2 = 1.63 bar

Convert pressure to absolute pressures, P1 + 1 bar = 1.8 + 1 = 2.8 bar

Use Ideal Gas Equation and replace Volume with Density using the simple equation

Volume = Mass/Density

PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the Plank's constant and T is the temperature.

P x Mass/Density = n x R x T

Re- arrange to solve for Density

Density = P x Mass/n x R x T

Density = (P/RT).(M/n), where M/n = Mr

Mr of air = 28.97 g/mol or 29 x 10-3 kg/mol

Density1 = (2.8 x 10⁵ x 29 x 10⁻³)/8.314 x (273 + 27) = 3.26 kg/m3

Density2 = P2 x Mr/R x T2

Density2 = {(1.63+1) x 29 x 10⁻³/8.314 x (273 + 60) = 2.75 kg/m3 , 1.63+1 to convert to absolute case as in the previous case

Apply continuity equation

Density1 x A1 x V1 = Density2 x A2 x V2

Where A = π x D²/4, where D is the diameter of the pipe(converted to meters)

Density1 x D1² x V1 = Density2 x D2² x V2

V2 = Density1 x D1² x V1/ Density2 x D2²

V2 = 3.26 x 0.07² x 30/2.75 x 0.055² = 57.607m/s