Water enters a wide, flat channel which has a height of 2h, where h is 1 meter a
ID: 2993185 • Letter: W
Question
Water enters a wide, flat channel which has a height of 2h, where h is 1 meter as shown below, with a uniform inlet velocity of U = g m/s. The flow exits the channel with a parabolic velocity profile as shown. The channel has a width of 3 m (w) into the page and y is measured from the centerline of the channel. Assuming steady, incomrpessible flow, determine: a) An equation for the mass flow rate at the inlet. b) An equation for the mass flow rate at the exit (you can leave in intergral form). c) The exit centerline velocity (y=0), u max.Explanation / Answer
a) Mass flow at inlet m = AU = 1000*(2h*w)U
b) Mass flowrate at exit m = Integral [w*u(y)*dy]y=-hy=h
c) Assuming u = Umax[1-(y/h)^2]
m = Integral [w*Umax[1-(y/h)^2]*dy]y=-hy=h
m = *w*Umax Integral [(1-(y/h)^2)*dy]y=-hy=h
m = *w*Umax [(y-(y^3/3h^2)]y=-hy=h
m = wUmax* [(h - h^3/3h^2) - (-h - (-h)^3/3h^2)]
m = wUmax*[h - h/3 + h - h/3]
m = wUmax*4h/3
Equating it with part a), wUmax*4h/3 = (2h*w)U
Umax = 3/4*2U = 3U/2
Umax = 3*5/2 = 7.5 m/s
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