## Chapter 15

## Q. 15.16

A highway drainage ditch is to carry 10 ft^{3}/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.

## Step-by-Step

## 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 C_{0} = 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 y_{N} = w/2 = b/2, the rectangular channel is 1.30 ft deep and w = A/y_{N} = 3.39 ft^{2}/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 |