GATE 2026 ME – Question 53
Water with a density of 1000 kg/m$^3$ comes out of an industrial condenser through a horizontal pipe of 15 cm radius at the flow rate of 4.5 m$^3$/min. The outlet of the pipe is connected to a coaxial diffuser of 0.5 m length using a flange to raise the pressure of water to atmospheric condition without any backflow. The inner radius ($r$, in m) of the diffuser cross-section is expressed as
$$r = 0.15 + 0.4x^2$$
where, $x$ represents the axial distance of the diffuser in m, from its inlet. Considering frictionless flow, the magnitude of the force exerted by the diffuser on the flange is ________ N (*rounded off to 2 decimal places*).
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Correct answer: 16.22 to 16.38
Explanation
The flow rate is $Q = \frac{4.5}{60} = 0.075$ m$^3$/s. At the inlet $A_1 = \pi(0.15)^2 = 0.07069$ m$^2$ and $V_1 = 1.061$ m/s. At the outlet $x = 0.5$ m, so $r = 0.25$ m, $A_2 = 0.19635$ m$^2$ and $V_2 = 0.382$ m/s. The outlet is at atmospheric pressure, so by Bernoulli's equation the inlet gauge pressure is $p_1 = \frac{\rho}{2}(V_2^2 - V_1^2) = -489.9$ Pa. The momentum equation for the diffuser gives the force from the walls on the fluid as $\rho Q(V_2 - V_1) - p_1A_1 = -50.93 + 34.63 = -16.30$ N. So the force exerted by the diffuser on the flange has a magnitude of 16.30 N.