GATE 2022 ME (ME1) – Question 44
Consider steady, one-dimensional compressible flow of a gas in a pipe of diameter 1 m. At one location in the pipe, the density and velocity are 1 kg/m$^3$ and 100 m/s, respectively. At a downstream location in the pipe, the velocity is 170 m/s. If the pressure drop between these two locations is 10 kPa, the force exerted by the gas on the pipe between these two locations is ____________ N.
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Correct answer: (B) $750\pi$
Explanation
The mass flow rate is $\dot{m} = \rho VA = 1 \times 100 \times \frac{\pi}{4} = 25\pi$ kg/s. The momentum balance on the gas between the two sections is $(P_1 - P_2)A + F_{pipe\ on\ gas} = \dot{m}(V_2 - V_1)$. With $(P_1 - P_2)A = 10000 \times \frac{\pi}{4} = 2500\pi$ N and $\dot{m}(V_2 - V_1) = 25\pi \times 70 = 1750\pi$ N, the force of the pipe on the gas is $-750\pi$ N. By the third law the gas exerts $750\pi$ N on the pipe, in the flow direction.