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?

Question

Accepted Answer

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 = (h₁ − h₂)/L = 0.5/100 = 0.005.

S_B = tan θ = [latex]\frac{h_1-h_2}{L}[/latex] (15.34)

The volume ﬂowrate is given by Eq. 15.53b, Q = (C₀/n)A[latex]R^{2/3} _H S^{1/2} _B[/latex], with C₀ = 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

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

The desired depth satisﬁes this equation. Inserting Q = 4.7 m³/s, n = 0.014, S_B = 0.005, and w = 2 m, we obtain

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

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/[latex]\sqrt{gy}[/latex] = 2.94 m/s/[latex]\sqrt{(9.81\ m/s^2)(0.8\ m)}[/latex] = 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:

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

for y = 0.9:

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

for y = 0.7:

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

for y = 0.8:

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

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

Question 15.13: A culvert made of unﬁnished concrete and measuring 2 m × 2 m...

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