Discharge of a River In studying the flow of water in an open channel, such as a river

Chapter 6, Problem 1

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Discharge of a River In studying the flow of water in an open channel, such as a river in its bed, the amount of water passing through a cross section per secondthe discharge (Q)is of interest. The following formula is used to compute the discharge: Q = % B 0 v(b)h(b) db (6.18) In this formula, b is the distance from one bank of the river to the point where the depth h(b) of the river and the average velocity v(b) of the vertical velocity profile of the river at b were measured. The total width of the cross section is B. (See Figure 6.48.) b h(b) 0 B x Figure 6.48 The river for 1. To evaluate the integral in (6.18), we would need to know v(b) and h(b) at every location b along the cross section. In practice, the cross section is divided into a finite number of subintervals and measurements of v and h are taken at, say, the right endpoints of each subinterval. The following table contains an example of such measurements: Location h v 0 0 0 1 0.28 0.172 3 0.76 0.213 5 1.34 0.230 7 1.57 0.256 9 1.42 0.241 11 1.21 0.206 13 0.83 0.187 15 0.42 0.116 16 0 0 The location 0 corresponds to the left bank, and the location B = 16 to the right bank, of the river. The units of the location and of h are meters, and of v, meters per second. Approximate the integral in (6.18) by a Riemann sum, using the locations in the table, and find the approximate discharge, using the data from the table.