Question

2. (10 pts) A municipality is designing an open storm drain channel in the form of an in verted trapezoid. It has a width of L at its base, is symmetric about the vertical, and its sides are at an angle θ to the horizontal (Fig. 2). Assuming the Manning formula, i.e grating for pedestrians where O-v3o./n is fired for this prob- lem (because material and slope are already known), determine Q as a function of Co, L, 0, and the water depth, h Fig. 2: Stom drain channel 3. (10 pts) Let the height from the channel base to the pedestrian grating in Question 2 be T. Also, assume that final design decisions for other parts of the system are to be determined based on the maximum volume flow rate capacity of this channel, i.e. the absolute maximum value of Q that the channel can handle. The municipal water manager claims this is calculated by simply substituting h = T, i.e. assuming the channel is filled with water all the way up to the grating. You reply that, although it seems intuitively obvious that this should be true, the reality is that it is not in this particular case. Show via a simple plot of Q/Co vs. h that the maximum flow occurs for h< T. Specifically, assume values for this example of L 3 m, 0- 45°, and T- 2 m, i.e. plot 0sh2. From your plot, estimate the depth h at which the volume flow rate Q is a maximum (you do not need to calculate this maximum). Comment on the underlying physical aspects of this unexpected behavior. (Note: the problem could likewise be done purely analytically by taking dQ/dh 0 and finding the roots, but this is not trivial for this particular scenario.)

Answer 1

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