Question 15.6: Flow through a Converging-Diverging Nozzle Air enters a conv...
Flow through a Converging-Diverging Nozzle
Air enters a converging-diverging nozzle (Fig. 15.26) with a velocity of 50 m/s at 400 kPa, 400 K. The nozzle has an exit-to-throat area ratio of 2. Determine (a) the maximum back pressure that will still choke the flow and (b) the design back pressure for isentropic supersonic flow.
Evaluate the total pressure from the inlet conditions. The maximum pressure to choke the flow is the back pressure that corresponds to curve C in Figure 15.24. Curve G corresponds to the supersonic design condition. Obtain the design Mach number from the area ratio of the nozzle.
Assumption
Isentropic, one-dimensional flow of a perfect gas.
Analysis
The inlet Mach number is:
M_{i}=\frac{V_{i}}{\sqrt{(1000 N / kN ) k R T_{i}}}=\frac{50}{\sqrt{(1000)(1.4)(0.287)(400)}}=0.125
Using the isentropic table, the total pressure can be obtained:
p_{t}=p_{t i}=p_{i} \frac{p_{t i}}{p_{i}}=p_{i}\left[\left(\frac{p}{p_{t}}\right)_{@ M_{i}=0.125}\right]^{-1}=400(0.9892)^{-1}=404.4 kPa
When the nozzle is choked—M = 1 occurs at the throat—the throat area becomes the critical area. Additionally, the flow must be subsonic and decelerating in the diverging section to maximize the exit pressure. The subsonic branch of the isentropic table (or the gas dynamics TESTcalc) for A_{e} / A_{*}=2 yields:
M_{e}=0.306 ; \quad \text { and } \quad p_{e}=p_{b}=p_{t}\left(\frac{p}{p_{t}}\right)_{@ M_{e}=0.306}=(404.4)(0.937)=379 kPa
The same area ratio produces the following supersonic isentropic solution:
M_{e}=2.197 ; \quad \text { and } \quad p_{e}=p_{b}=p_{t}\left(\frac{p}{p_{t}}\right)_{@ M_{e}=2.197}=(404.4)(0.0939)=37.98 kPa
TEST Analysis
Launch the gas dynamics TESTcalc and select air as the working fluid. Evaluate State-1 as the inlet state from the given properties p1, T1, and Vel1, and State-2 as the critical throat state with M2 = 1, T_t2 = T_t1, p_t2 = p_t1, and an arbitrarily chosen A2 = 1 m² . For the exit state, State-3, enter A3 = 2*A2, p_t3 = p_t1, T_t3 = T_t1, and Astar3 = A2. When you click the Calculate button the subsonic solution is displayed. At the same time, the Mach number for the alternative supersonic solution is displayed in the message panel as 2.197. Evaluate State-4 with M4 = 2.197, A4 = 2*A2, p_t4 = p_t1, and T_t4 = T_t1. The manual solution is now verified.
Discussion
The supersonic exit Mach number is called the design Mach number of a converging-diverging nozzle. It is a function of the exit-to-throat area ratio for a given gas.
