Question 12.6: Gas Flow through a Converging Nozzle Nitrogen enters a duct ...
Gas Flow through a Converging Nozzle
Nitrogen enters a duct with varying flow area at T_{1}=400 K , P_{1}=100 kPa , and M a_{1}=0.3 . Assuming steady isentropic flow, determine T_{2}, P_{2}, \text { and } Ma _{2} at a location where the flow area has been reduced by 20 percent.
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
\frac{A_{1}}{A^{*}}=2.0351 \quad \frac{T_{1}}{T_{0}}=0.9823 \quad \frac{P_{1}}{P_{0}}=0.9395
With a 20 percent reduction in flow area, A_{2}=0.8 A_{1} , and
\frac{A_{2}}{A^{*}}=\frac{A_{2}}{A_{1}} \frac{A_{1}}{A^{*}}=(0.8)(2.0351)=1.6281
For this value of A_{2} / A^{*} from Table A–13, we read
\frac{T_{2}}{T_{0}}=0.9703 \quad \frac{P_{2}}{P_{0}}=0.9000 \quad Ma _{2}=0.391
Here we chose the subsonic Mach number for the calculated A_{2} / A^{*} 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
\begin{aligned}&\frac{T_{2}}{T_{1}}=\frac{T_{2} / T_{0}}{T_{1} / T_{0}} \rightarrow T_{2}=T_{1}\left(\frac{T_{2} / T_{0}}{T_{1} / T_{0}}\right)=(400 K )\left(\frac{0.9703}{0.9823}\right)= 3 9 5 K \\&\frac{P_{2}}{P_{1}}=\frac{P_{2} / P_{0}}{P_{1} / P_{0}} \rightarrow P_{2}=P_{1}\left(\frac{P_{2} / P_{0}}{P_{1} / P_{0}}\right)=(100 kPa )\left(\frac{0.9000}{0.9395}\right)= 9 5 . 8 k P a\end{aligned}
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.
