## Chapter 15

## Q. 15.13

A culvert made of unﬁnished concrete and measuring 2 m × 2 m, as shown in Figure 15.38, carries a small stream. If the culvert drops 0.5 m over its 100 m length, and is designed for uniform ﬂow, what is the expected normal depth when the culvert is carrying a ﬂow of 4.7 m³/s?

## Step-by-Step

## Verified Solution

The culvert is sketched in Figure 15.38. From Table 15.1 the Manning coefficient for unﬁnished concrete is n = 0.014. Since we know that the slope of the channel is described in terms of an elevation change over its length, we can use Eq. 15.34 to calculate the bed slope as S_{B} = (h_{1} − h_{2})/L = 0.5/100 = 0.005.

S_{B} = tan θ = \frac{h_1-h_2}{L} (15.34)

The volume ﬂowrate is given by Eq. 15.53b, Q = (C_{0}/n)AR^{2/3} _H S^{1/2} _B, with C_{0} = 1, since we are working in SI units. Next we use Figure 15.6 to write the formulas A(y) = wy and R_{H}(y) = wy/(w + 2y). Inserting these into the formula for Q and rearranging yields

(wy)\left(\frac{wy}{w+2y} \right)^{2/3} =\frac{nQ}{S^{1/2}_B} (A)

The desired depth satisﬁes this equation. Inserting Q = 4.7 m^{3}/s, n = 0.014, S_{B} = 0.005, and w = 2 m, we obtain

2y\left(\frac{2y}{2+2y} \right)^{2/3} =0.931

as the equation to be solved. Using a symbolic math code, we ﬁnd y = 0.8 m.

We see that a ﬂow of 4.7 m³/s in this culvert occurs at a depth of 0.8 m. The ﬂow area in that case is A = wy = 2 m (0.8 m ) = 1.6 m², and the velocity is V = Q/A = 4 .7 m³/s/1.6 m² = 2.94 m/s. The Froude number is Fr = V/\sqrt{gy} = 2.94 m/s/\sqrt{(9.81\ m/s^2)(0.8\ m)} = 1.05. Thus the ﬂow is supercritical.

Note that to iterate this problem by hand we can recognize that the answer must be a depth of 2 m or less. Several iterations are shown:

for y = 0.6:

2y\left(\frac{2y}{2+2y} \right)^{2/3} =0.624

for y = 0.9:

2y\left(\frac{2y}{2+2y} \right)^{2/3} =1.33

for y = 0.7:

2y\left(\frac{2y}{2+2y} \right)^{2/3} =0.77

for y = 0.8:

2y\left(\frac{2y}{2+2y} \right)^{2/3} =0.932

**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 |