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Flow and Momentum

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A hot gas stream enters a uniform diameter return bend as shown. The entrance velocity is 100 f/s, the gas density is .02lbm/cubic ft. and the mass flow rate is 1 lbm/cubic ft. The gas is cooled before exiting the bend and exits the bend at a density of .06 lbm/cubic ft. The pressures at the entrance and exit of the bend are equal and are equal to atmospheric pressure. Find the force required to hold the bend.

(See attached file for full problem description with diagram).

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This in-depth solution has an annotated diagram and step-by-step calculations and explanations to determine the force required to hold the bend. All workings and formulas are included.

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A hot gas stream enters a uniform diameter return bend as shown. The entrance velocity is 100 f/s, the gas density is .02lbm/cubic ft. and the mass flow rate is 1 lbm/cubic ft. The gas is cooled before exiting the bend and exits the bend at a density of .06 lbm/cubic ft. The pressures at the entrance and exit of the bend are equal and are equal to atmospheric pressure. Find the force required to hold the bend.x
Solution:
We will work in SI and finally we will convert in British System as well.
We denote:
V1 = inlet speed of the gas , V1 = 100 ft/s = 100*0.3048 m/s = 30.48 m/s
ρ1 = gas density at the inlet, ρ1 = 0.02 lbm/ft3 =
= 0.02*(0.4536 kg)/(0.3048m)3 = 0.3204 kg/m3.
ρ2 = gas density at the exit, ρ2 = 0.06 lbm/ft3 = 0.9612 kg/m3.
Mg = mass flow rate of gas, Mg = 1 lbm/s = 0.4536 kg/s

(note: in the text of problem is stated Mg = 1 lbm/cubic ft which is not correct, mass flow rate must be considered in mass units per second)
Q = volumetric ...

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