The Laval nozzle in Fig. 13–22 is connected to a large chamber containing air at an absolute pressure of 350 kPa. Determine the backpressure in the pipe at B that will cause the nozzle to choke and yet produce isentropic subsonic flow through the pipe. Also, what backpressure is needed to cause

Question

The Laval nozzle in Fig. 13–22 is connected to a large chamber containing air at an absolute pressure of 350 kPa. Determine the backpressure in the pipe at B that will cause the nozzle to choke and yet produce isentropic subsonic flow through the pipe. Also, what backpressure is needed to cause isentropic supersonic flow?

Accepted Answer

Fluid Description. We assume steady isentropic flow through the nozzle.

Analysis. Here we must find the two backpressures, [latex] p_3 ext { and } p_4 [/latex], in Fig. 13–19b, required to produce M = 1 at the throat. The area ratio for the nozzle between the exit and the throat is

[latex] \frac{A_B}{A^*}=\frac{\pi(0.05 m )^2}{\pi(0.025 m )^2}=4 [/latex]

If this ratio is used in Eq. 13–36, two roots for M at the exit can be determined, Fig. 13–15. However, we can also solve this problem using Table B–1. With [latex] A_B / A^*=4 ext {, we get } M _1=0.1467 [/latex] (subsonic flow) and [latex] p_B / p_0=0.9851 [/latex]. Thus, the higher backpressure at B that will cause subsonic flow is

[latex] \frac{A}{A^*}=\frac{1}{ M }\left[\frac{1+\frac{1}{2}(k-1) M ^2}{\frac{1}{2}(k+1)} ight] \frac{k+1}{2(k-1)} [/latex] (13-36)

[latex] \left(p_B ight)_{\max }=0.9851(350 kPa )=345 kPa [/latex]

Further in the table, the alternative solution, where [latex] A_B / A^*=4 [/latex], gives [latex] M _2 [/latex] = 2.940 (supersonic flow) and [latex] p_B / p_0=0.02979 [/latex]. Thus, the lower backpressure at B is for supersonic flow. It is

[latex] \left(p_B ight)_{\min }=0.02979(350 kPa )=10.4 kPa [/latex]

note: Both of these pressures will produce the same mass flow through the nozzle, and its value can be calculated using [latex] \dot{m}= ho^* V^* A^* [/latex]

Question 13.10: The Laval nozzle in Fig. 13–22 is connected to a large chamb...

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