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## Q. 15.16

A highway drainage ditch is to carry 10 ft3/s of water in a ﬁnished concrete channel during the worst expected rainstorm. If the slope is SB = 0.001, ﬁnd the dimensions of the optimum rectangular channel and compare with those of an optimum circular channel. Assume uniform ﬂow and that the water ﬁlls each channel to the brim at peak ﬂow.

## Verified Solution

No sketch is necessary here. From Table 15.1 the Manning coefficient for ﬁnished concrete is n = 0.012, and we can use the appropriate formulas in Figure 15.41 with C0 = 1.49, since we are working in BG units. For the rectangular channel we have

$y_N=2^{-1/8}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/8}=2^{-1/8}\left[\frac{(0.012)(10)}{(1.49)(0.001)^{1/2}} \right]^{3/8}=1.30\ \mathrm{ft}$

$A=2^{3/4}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/4}=2^{3/4}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/4}=3.39\ \mathrm{ft} ^2$

Since the channel is ﬁlled to the brim and yN = w/2 = b/2, the rectangular channel is 1.30 ft deep and w = A/yN = 3.39 ft2/1.30 ft = 2.6 ft wide. For the optimum circular channel, the calculations yield

$y_N=2^{5/8}\pi ^{-3/8}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/8}=2^{5/8}\pi ^{-3/8}\left[\frac{(0.012)(10)}{(1.49)(0.001)^{1/2}} \right]^{3/8}=1.42\ \mathrm{ft}$

$A=2^{1/4}\pi ^{1/4}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/4}=2^{1/4}\pi ^{1/4}\left[\frac{nQ}{C_0S^{1/2}_B} \right]^{3/4}=3.19\ \mathrm{ft} ^2$

We see that the ﬂow area (and required excavation) for the circular channel are indeed smaller than that of the rectangular channel. The diameter of this semicircular channel is 2.84 ft.
TABLE 15.1 Values of the Manning Roughness Coefficient, n

 Wetted Perimeter n A. Natural channels Clean and straight 0.030 Sluggish with deep pools 0.040 Major rivers 0.035 B. Floodplains Pasture, farmland 0.035 Light brush 0.050 Heavy brush 0.075 Trees 0.15 C. Excavated earth channels Clean 0.022 Gravelly 0.025 Weedy 0.030 Stony, cobbles 0.035 D. Artiﬁcially lined channels Glass 0.010 Brass 0.011 Steel, smooth 0.012 Steel, painted 0.014 Steel, riveted 0.015 Cast iron 0.013 Concrete, ﬁnished 0.012 Concrete, unﬁnished 0.014 Planed wood 0.012 Clay tile 0.014 Brick work 0.015 Asphalt 0.016 Corrugated metal 0.022 Rubble masonry 0.025