Question 12.8: Shock Wave in a Converging–Diverging Nozzle If the air flowi...

Air flowing through a converging–diverging nozzle experiences a normal shock at the exit. The effect of the shock wave on various properties is to be determined.
Assumptions   1  Air is an ideal gas with constant specific heats at room temperature. 2  Flow through the nozzle is steady, one-dimensional, and isentropic before the shock occurs. 3  The shock wave occurs at the exit plane.
Properties   The constant-pressure specific heat and the specific heat ratio of air are c_p = 1.005  kJ/kg·K and k = 1.4. The gas constant of air is 0.287 kJ/kg⋅K.
Analysis   (a) The fluid properties at the exit of the nozzle just before the shock (denoted by subscript 1) are those evaluated in Example 12–6 at the nozzle exit to be

P_{01} = 1.0  MPa    P_1 = 0.1278  MPa     T_1 = 444.5  K     \rho_1 = 1.002  kg/m^3

The fluid properties after the shock (denoted by subscript 2) are related to those before the shock through the functions listed in Table A–14. For Ma_1 = 2.0, we read

Ma _2=0.5774 \quad \frac{P_{02}}{P_{01}}=0.7209 \quad \frac{P_2}{P_1}=4.5000 \quad \frac{T_2}{T_1}=1.6875 \quad \frac{\rho_2}{\rho_1}=2.6667

Then the stagnation pressure P_{02}, static pressure P_2, static temperature T_2, and static density \rho_2 after the shock are

P_{02} = 0.7209P_{01} = (0.7209)(1.0  MPa) = 0.721  MPa
P_2 = 4.5000P_1 = (4.5000)(0.1278  MPa) = 0.575  MPa
T_2 = 1.6875T_1 = (1.6875)(444.5  K) = 750  K
\rho_2 = 2.6667\rho_1 = (2.6667)(1.002  kg/m^3) = 2.67  kg/m^3

(b) The entropy change across the shock is

s_2-s_1=c_\rho \ln \frac{T_2}{T_1}-R \ln \frac{P_2}{P_1}

= (1.005 kJ/kg·K) ln (1.6875) − (0.287 kJ/kg·K) \ln (4.5000)
= 0.0942 kJ/kg·K

Thus, the entropy of the air increases as it passes through a normal shock, which is highly irreversible.
(c) The air velocity after the shock is determined from V_2 = Ma_2c_2, where c_2 is the speed of sound at the exit conditions after the shock:

V_2= Ma _2 c_2= Ma _2 \sqrt{k R T_2}

=(0.5774) \sqrt{(1.4)(0.287  kJ / kg \cdot K )(750.1  K )\left(\frac{1000  m ^2 / s ^2}{1  kJ / kg }\right)}

= 317 m/s

(d ) The mass flow rate through a converging–diverging nozzle with sonic conditions at the throat is not affected by the presence of shock waves in the nozzle. Therefore, the mass flow rate in this case is the same as that determined in Example 12–6:

\dot{m}=2.86  kg / s

Discussion   This result can easily be verified by using property values at the nozzle exit after the shock at all Mach numbers significantly greater than unity.