Please help Consider a two-dimensional duct. At location 1 exists a linear veloc

Question

Please help

Consider a two-dimensional duct. At location 1 exists a linear velocity profile that starts at zero on the bottom wall and is 10 ft/s at the top wall. The duct heights are 2 ft, 1 ft and 1.5 ft, respectively, at locations 1, 2, and 3. The angle of incline of location 3 is 600. The velocity profile at location 2 is uniform and equal to 15 ft/s. The velocity profile at location 3 is also uniform, but the velocity is unknown. Determine the Caj magnitude and (b) direction (into or out of the control volume) of the velocity profile at location 3 if the flow is incompressible and steady.

Explanation / Answer

since it is given flow is incompressible and steady.density of fluid is constant

but by conservation of mass principle mass which is flowing in to system is equal to mass flowing out of system

but density of fluid is constant.so, we can write volume flow in to system is equal to volume flow out of system

let depth of duct (in to paper is 1 unit)

volume flow in at section 1 = area of crossection*velocity = 2*1*((0+10)/2) = 2*1*5 = 10 ft3/s

volume flow out at section 2 = 1*1*15 = 15 ft3/s

since volume flow out at section 2 is more than volume flow at section 1, flow should go in at section 3

i,e flow should go in to the control volume at section 3

volume flow in at section 3 = 15-10 =5 ft3/s = 1.5*1*(velocity) (@ section 3)

velocity at section 3 = 5/1.5 = 3.33 ft/s

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