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## 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?

## 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 SB = (h1 − h2)/L = 0.5/100 = 0.005.

SB = tan θ = $\frac{h_1-h_2}{L}$              (15.34)

The volume ﬂowrate is given by Eq. 15.53b, Q = (C0/n)A$R^{2/3} _H S^{1/2} _B$, with C0 = 1, since we are working in SI units. Next we use Figure 15.6 to write the formulas A(y) = wy and RH(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 m3/s, n = 0.014, SB = 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