Air from a reservoir at 10 atm and 80°C is discharged through a convergent–divergent nozzle fitted to the reservoir. The thrust exerted by the jet issuing from the nozzle is 11.12 kN. If the backpressure is 1 atm, calculate the nozzle throat and exit areas and Mach number of the jet issuing from

Question

Air from a reservoir at 10 atm and [latex]80^{\circ} C[/latex] is discharged through a convergent–divergent nozzle fitted to the reservoir. The thrust exerted by the jet issuing from the nozzle is 11.12 kN. If the backpressure is 1 atm, calculate the nozzle throat and exit areas and Mach number of the jet issuing from the nozzle. Assume the flow through the nozzle to be isentropic and correctly expanded.

Accepted Answer

Given, [latex]p_{0}=10 atm , T_{0}=80+273.15=353.15 K , ext { and } p_{b}=1 atm .[/latex] The flow is correctly expanded; therefore, [latex]p_{b}=p_{e}.[/latex]

The pressure ratio across the nozzle is

[latex]\frac{p_{e}}{p_{0}}=\frac{1}{10}=0.1[/latex]

For [latex]p_{e} / p_{0}=0.1,[/latex] from the isentropic table we have the Mach number of the jet issuing from the nozzle as

[latex]M_{e}=2.16[/latex]

Also,

[latex]\frac{T_{e}}{T_{0}}=0.5173[/latex]

[latex]\frac{A_{e}}{A_{ ext {th }}}=1.935[/latex]

Therefore,

[latex]T_{e}=0.5173 imes 353.15=182.68 K[/latex]

[latex]a_{e}=\sqrt{1.4 imes 287 imes 182.68}=270.93 m s ^{-1}[/latex]

The velocity at the nozzle exit is

[latex]V_{e}=M_{e} a_{e}=2.16 imes 270.93=585.2 m s ^{-1}[/latex]

The thrust generated by the jet issuing from the nozzle is

[latex]F=11.12 imes 10^{3}=\dot{m} V_{e}[/latex]

Therefore,

[latex]\dot{m}=\frac{11120}{585.2}=19 kg s ^{-1}[/latex]

The mass flow rate [latex]\dot{m}[/latex] is also given by

[latex]\dot{m}=\frac{0.6847}{\sqrt{R T_{0}}} p_{0} A_{ ext {th }}[/latex]

Therefore, the throat area becomes

[latex]A_{ th }=\frac{19 \sqrt{287 imes 353.15}}{0.6847 imes 10 imes 101325}[/latex]

[latex]=0.0087188 m ^{2}=87.19 cm ^{2}[/latex]

The nozzle exit area is

[latex]A_{e}=1.935 imes 87.19[/latex]

[latex]=168.71 cm ^{2}[/latex]

Question 2.3: Air from a reservoir at 10 atm and 80°C is discharged throug...

Related Answered Questions