Question 15.16: A highway drainage ditch is to carry 10 ft³/s of water in a ...
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.
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 |
