Concept explainers
Helium expands in a nozzle from 0.8 MPa, 500 K, and negligible velocity to 0.1 MPa. Calculate the throat and exit areas for a mass flow rate of 0.34 kg/s, assuming the nozzle is isentropic. Why must this nozzle be converging–diverging?
The throat and exit area of the nozzle.
Answer to Problem 131RP
The throat area of nozzle is
The exit area is
Explanation of Solution
It is given that the initial velocity is negligible. Hence, the inlet properties are equal to the stagnation properties at inlet.
Consider the flow through the nozzle is isentropic. Hence, the stagnation properties at inlet and exit equal.
Write the formula for the critical temperature of the mixture.
Here, the critical temperature of mixture is
Write the formula for the critical pressure of the mixture.
Here, the critical pressure of mixture is
Write the formula for the critical density.
Here, the critical density of mixture is
Write the formula for critical velocity of helium gas through the nozzle.
Here, the superscript
Write the formula for mass flow rate of helium at throat region.
Here, the cross sectional area of the throat is
Rearrange the Equation (V) to obtain
Refer Table A-1, “Molar mass, gas constant, and critical-point properties”.
The gas constant
Refer Table A-2, “Ideal-gas specific heats of various common gases”.
The specific heat ratio
Write the formula of ratio of stagnation pressure to the static pressure at exit of the nozzle.
Here, the actual (static) pressure at the exit of nozzle is
Write the formula of ratio of stagnation temperature to the static temperature at exit of the nozzle.
Here, the actual (static) temperature at the exit of nozzle is
Write the formula for velocity of sound at the exit conditions.
Here, speed of sound at the exit condition is
Write formula for the velocity of helium at exit.
Write the formula for mass flow rate of helium at exit condition.
Here, the exit cross sectional area is
Rearrange the Equation (X) to obtain
Conclusion:
Substitute
Substitute
Substitute
Substitute 1.667 for
Substitute
Equation (VI).
Thus, the throat area of nozzle is
Substitute
Here, the downstream Mach number
Substitute
Substitute
Substitute
Thus, the exit area is
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