Question

Air at 20°C, enters a large circular duct (diameter D=1m), with a velocity of 0.4m/s. Using the momentum integral relationship, what is the flow velocity 5m from the inlet? (if D >> 8, the boundary layer inside a circular pipe can be approximated to a flat plate boundary layer) a) 0.446 m/s b) 0.381 m/s c) 0.415 m/s d) 0.461 m/s e) 0.420 m/s f) 0.551 m/s

Please use the following table to solve.

The answer should be (a)

Answer 1

Given Temperature of air= 200. a 20 Density of air= 4-204 kg/m3 Dynamic viscosity of 1.825x10s air pas Velocity of air = v= 0.4 m/s Diameter of duct Delen Reynolds Number (Re) = Pura Re - 1 204 x 0-4X! 1.825 x 10-5 Re = 26389-041 For pipe flow, if Rex 2500, flow is turbulent. . At the entrance of Duct itself, the flow is two bulent. Now using momentum integral estimation from the table, let us find the boundary lager thickness is at sm from inlet 8 – z = 0-16 (he)

. S= 0.162 (Re) 117 S= (0.16)5 (26389.047)/7 8= 0.1868349 m Here saad Hence our assumption will be valid that we can use flat plate relations for this problem. Now Let us find displacement thickness. Again from the given table - 0.02 Here Re is same @ every because kea where average velocity & it remains throughout the flow ] point is Constant * = (0.02) (5) (26389-041)/7 Sensor = 0.023354

Now, by the definition of displacement thickness, it represents the last mass flow nate due to formation of Boundary layer for flat plate rate. Lost mass flow - x As a result, the flow rate is driven upward. ų that mass flow rate cost is represented in displacement thickers But incase of closed conduit flow, the lost mass flow rate is not able to move upward. Hence it is made to accelerate at the centre as a result, the velocity increases For Cincelar pipe flow mass lost nate Centre. accelerated at For circular pipe, displacement assumed to be around the pipe thickness is Cintum feresce. میں * Uool

Here Uo foi . Literally, the flow takes place inside the ge layer - . By continuity equation, flow rate should be same : flow rate & a flow rate @ entrane distance & Xox Up & I (0-28) x Us Tx 0.4 = {1-2(0-023354)] x Us 0.4² (0.953292) XUDI : [Wola 0.4401 mis Elence at a distance a=sm from entrance, the velocity = 0.4401 m/s = 0.44 m/s It is close to option @