Flow Through a Converging Nozzle Air is discharged from a reservoir, at 500 kPa and 500 K, through an isentropic converging nozzle with an exit area of 10 cm². Determine the mass flow rate for a back pressure of (a) 0 kPa, (b) 100 kPa, (c) 200 kPa, and (d) 300 kPa.

Question

Flow Through a Converging Nozzle

Air is discharged from a reservoir, at 500 kPa and 500 K, through an isentropic converging nozzle with an exit area of 10 cm². Determine the mass flow rate for a back pressure of (a) 0 kPa, (b) 100 kPa, (c) 200 kPa, and (d) 300 kPa.

Accepted Answer

Determine if the flow is choked for a given back pressure and use Eq. (15.28) to determine the mass flow rate.

[latex] \dot{m}=A_{*} p_{t} \sqrt{\frac{(1000 J / kJ ) k}{R T_{t}}}\left(\frac{2}{k+1} ight)^{(k+1) /[2(k-1)]} ;\left[\frac{ kg }{ s } ight] [/latex] (15.28)

Assumption
Isentropic, one-dimensional flow of a perfect gas.
Analysis
From the isentropic table of air, Table H-1, or the table panel of the gas dynamics TESTcalc with air selected as the working fluid, the critical pressure ratio [latex] p_{*} / p_{t} ext { for } M=1 [/latex] can be obtained as 0.5283 so that:

[latex] p_{*}=\frac{p_{*}}{p_{t}} p_{t}=\left(\frac{p}{p_{t}} ight)_{@ M=1} p_{r}=(0.5283)(500)=264.15 kPa [/latex]

For back pressures less than the critical pressure 264.15 kPa—0 kPa, 100 kPa, and 200 kPa—the flow is choked.

For choked flow: [latex] T_{ th }=T_{*}=\left(\frac{T_{*}}{T_{t}} ight) T_{t}=\left(\frac{T}{T_{t}} ight)_{@ M=1} T_{t}=(0.8333)(500)=416.7 K [/latex]

The mass flow rate for the choked flow can be directly calculated:

[latex]\begin{aligned}\dot{m} &= ho_{*} A_{*} V_{*}= ho_{*} A_{*} c_{*}=\frac{p_{*}}{R T_{*}} A_{ th } \sqrt{(1000 N / kN ) k R T_{*}} \&=\frac{(264.15)(0.001) \sqrt{1000(1.4)}}{\sqrt{(0.287)(416.7)}} \&=0.904 \frac{ kg }{ s }\end{aligned}[/latex]

The same answer can be obtained by application of Eq. (15.28).
For a back pressure of 300 kPa, the flow is subsonic and the back pressure must be equal to the exit pressure. For [latex] p_{ th } / p_{t}=300 / 500=0.6 [/latex], the isentropic table produces [latex] M_{ ext {th }}=0.886 [/latex] and [latex] T_{ ext {th }}=(0.864)(500)=432 K [/latex]. Therefore:

[latex]\begin{aligned}\dot{m} &= ho_{ th } A_{ th } V_{ th }=\frac{p_{ th }}{R T_{ th }} A_{ th } M_{ th } \sqrt{(1000 N / kN ) k R T_{ ext {th }}} \&=\frac{(300)(0.001)(0.886) \sqrt{1000(1.4)}}{\sqrt{(0.287)(432.1)}} \&=0.893 \frac{ kg }{ s }\end{aligned}[/latex]

TEST Analysis
Launch the gas dynamics TESTcalc. Select Air as the working fluid. Evaluate State-1 with p1 = 500 kPa, T1 = 500 K, and Vel1 = 0 as the reservoir (total) state. For the choked throat state, evaluate State-2, with M2 = 1, T_t2 = T1, p_t2 = p1, and A2 = 10 cm2. For the nonchoked state, State-3, use p3, T_t3, p_t3, and A3 = A2. Mass flow rates are calculated as part of the complete flow states. For more details, see TEST-code (posted in TEST > TEST-codes module).

Discussion
By gradually increasing p3, and recalculating State-3, mdot3 can be plotted as a function of the back pressure, as shown in Figure 15.23. A similar analysis can be used to determine the exit area required to create a desired mass flow rate for a given total pressure and temperature. We could use Eq. (15.28) to obtain the mass flow rates, since [latex] p_{t} ext { and } T_{t} [/latex] are known from the reservoir conditions and [latex] A_{*} [/latex] can be obtained from Table H-1 for a given Mach number.

Question 15.5: Flow Through a Converging Nozzle Air is discharged from a re...

Related Answered Questions

Verified Answer:

Use of an Isentropic Table In Example 15-2, assume pr = 150 kPa and Tr = 500 K. If the exit area is 10 cm² , determine (a) the exit Mach number, (b) exit velocity, (c) critical pressure, and (d) critical area. Use the isentropic table. ...

Verified Answer:

Velocity of Sound Steam flows through a duct with a velocity of 300 m/s at 100 kPa and 200°C. Determine the velocity of sound using (a) the PG model and (b) PC model for steam. ...

Verified Answer:

Verified Answer:

Supersonic Diffuser with Normal Shock Air at 10 kPa approaches a supersonic diffuser at Mach 3.0. A normal shock stands at the inlet, as shown in Figure 15.44. If Ae/Ai = 3, determine (a) the exit Mach number, (b) the exit pressure, and (c) the pressure recovery factor. Assume the flow is ...

Verified Answer:

Non-isentropic Diffuser Analysis Air, at 100 kPa, 300 K, enters a 90% efficient adiabatic diffuser with a velocity of 300 m/s. If the exit velocity is 50 m/s, determine (a) the exit pressure, (b) pressure recovery factor, (c) pressure rise coefficient, and (d) exit-to-inlet area ratio. What-if ...

Verified Answer:

Verified Answer:

Flow through a Converging-Diverging Nozzle In Example 15-6, determine the back pressure for which a normal shock stands (a) in the diverging section at an area 50% larger than the throat area and (b) at the exit. ...

Verified Answer:

Thrust Calculation for a Rocket A rocket nozzle with an exit area of 0.1 m² is designed to work supersonically with a chamber pressure of 2.5 MPa and a chamber temperature of 1500 K at sea level, where the ambient pressure is 101 kPa. (a) Determine the thrust at sea level. Assume the chamber ...

Verified Answer:

Verified Answer: