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GATE 2023 ME – Question 20

Fluid Mechanics · One-dimensional compressible flow, nozzles · 1 mark · Multiple choice

Consider an isentropic flow of air (ratio of specific heats = 1.4) through a duct as shown in the figure.
The variations in the flow across the cross-section are negligible. The flow conditions at Location 1 are given as follows: $P_1 = 100$ kPa, $\rho_1 = 1.2$ kg/m$^3$, $u_1 = 400$ m/s
The duct cross-sectional area at Location 2 is given by $A_2 = 2A_1$, where $A_1$ denotes the duct cross-sectional area at Location 1. Which one of the given statements about the velocity $u_2$ and pressure $P_2$ at Location 2 is TRUE?

a duct whose area increases from Location 1 to Location 2, with the flow from left to right.
  1. $u_2 < u_1$, $P_2 < P_1$
  2. $u_2 < u_1$, $P_2 > P_1$
  3. $u_2 > u_1$, $P_2 < P_1$
  4. $u_2 > u_1$, $P_2 > P_1$

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Correct answer: (C) $u_2 > u_1$, $P_2 < P_1$

Explanation

The speed of sound at Location 1 is $a_1 = \sqrt{\gamma P_1/\rho_1} = \sqrt{1.4 \times 100000/1.2} = 341.6$ m/s, so the Mach number is $M_1 = \frac{400}{341.6} = 1.17 > 1$ and the flow is supersonic. In supersonic flow a growing area makes the flow speed up and the pressure fall (the opposite of subsonic flow). So $u_2 > u_1$ and $P_2 < P_1$.