Gas Flow through a Converging Nozzle Nitrogen enters a duct with varying flow area at T1 = 400 K, P1 = 100 kPa, and Ma1 = 0.3. Assuming steady isentropic flow, determine T2, P2, and Ma2 at a location where the flow area has been reduced by 20 percent.

Question

Gas Flow through a Converging Nozzle

Nitrogen enters a duct with varying flow area at [latex] T_{1}=400 K , P_{1}=100 kPa [/latex], and [latex] M a_{1}=0.3 [/latex]. Assuming steady isentropic flow, determine [latex] T_{2}, P_{2}, ext { and } Ma _{2} [/latex] at a location where the flow area has been reduced by 20 percent.

Accepted Answer

Nitrogen gas enters a converging nozzle. The properties at the nozzle exit are to be determined.
Assumptions 1 Nitrogen is an ideal gas with k = 1.4. 2 Flow through the nozzle is steady, one-dimensional, and isentropic.
Analysis The schematic of the duct is shown in Fig. 12–25. For isentropic flow through a duct, the area ratio A/A* (the flow area over the area of the throat where Ma = 1) is also listed in Table A–13. At the initial Mach number of Ma = 0.3, we read

[latex] \frac{A_{1}}{A^{*}}=2.0351 \quad \frac{T_{1}}{T_{0}}=0.9823 \quad \frac{P_{1}}{P_{0}}=0.9395 [/latex]

With a 20 percent reduction in flow area, [latex] A_{2}=0.8 A_{1} [/latex], and

[latex] \frac{A_{2}}{A^{*}}=\frac{A_{2}}{A_{1}} \frac{A_{1}}{A^{*}}=(0.8)(2.0351)=1.6281 [/latex]

For this value of [latex] A_{2} / A^{*} [/latex] from Table A–13, we read

[latex] \frac{T_{2}}{T_{0}}=0.9703 \quad \frac{P_{2}}{P_{0}}=0.9000 \quad Ma _{2}=0.391 [/latex]

Here we chose the subsonic Mach number for the calculated [latex] A_{2} / A^{*} [/latex] instead of the supersonic one because the duct is converging in the flow direction and the initial flow is subsonic. Since the stagnation properties are constant for isentropic flow, we can write

[latex]\begin{aligned}&\frac{T_{2}}{T_{1}}=\frac{T_{2} / T_{0}}{T_{1} / T_{0}} ightarrow T_{2}=T_{1}\left(\frac{T_{2} / T_{0}}{T_{1} / T_{0}} ight)=(400 K )\left(\frac{0.9703}{0.9823} ight)= 3 9 5 K \&\frac{P_{2}}{P_{1}}=\frac{P_{2} / P_{0}}{P_{1} / P_{0}} ightarrow P_{2}=P_{1}\left(\frac{P_{2} / P_{0}}{P_{1} / P_{0}} ight)=(100 kPa )\left(\frac{0.9000}{0.9395} ight)= 9 5 . 8 k P a\end{aligned}[/latex]

which are the temperature and pressure at the desired location.
Discussion Note that the temperature and pressure drop as the fluid accelerates in a converging nozzle.

Question 12.6: Gas Flow through a Converging Nozzle Nitrogen enters a duct ...

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