The flow Q [m3/s] in an open channel can be predicted with the Manning equation (recall Sec. 8.2)

Q = 1/n AcR2/3 S1/2

Where n = Manning roughness coefficient (a dimensionless number used to parameterize the channel friction), Ac = cross-sectional area of the channel (m2), S = channel slope (dimensionless, meters drop per meter length), and R = hydraulic radius (m), which is related to more fundamental parameters by R = Ac/P, where P = wetted perimeter (m). As the name implies, the wetted perimeter is the length of the channel sides and bottom that is under water. For example, for a rectangular channel, it is defined as P = B + 2H, where H = depth (m). Suppose that you are using this formula to design a lined canal (note that farmers line canals to minimize leakage losses).

(a) Given the parameters n = 0.03, S = 0.0004, and Q = 1 m3/s, determine the values of B and H that minimize the wetted perimeter. Note that such a calculation would minimize cost if lining costs were much larger than excavation costs.

(b) Repeat part (a), but include the cost of excavation. To do this minimize the following cost function,

C = c1Ac + c2P

Where c1 is a cost factor for excavation = $100/m2 and c2 is a cost factor for lining $50/m.

(c) Discuss the implications of your results.