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

Question

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 S_B = 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.

Accepted Answer

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

[latex]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} [/latex]

[latex]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[/latex]

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

[latex]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} [/latex]

[latex]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[/latex]

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

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