The pipe is sketched in Figure 15.37. From Table 15.1 the Manning coefficient for cast iron is n = 0.013. Since we know that the slope of the channel is 0.2°, we can use Eq. 15.34 to calculate the bed slope as S_B = tan θ = tan 0.2° = 0.00349.

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

Using the formulas in Figure 15.6 for a round pipe we can first calculate θ = 2 cos⁻¹(1 − 2y/D), where y is the depth, and D = 1 ft. Next we calculate the flow area A(y) = (D²/8)(θ – sin θ), wetted perimeter P(y) = Dθ/2, and hydraulic radius R_H(y) = (D/4)(1 – sin θ/θ). The results are shown in Table 15.2. The volume flowrate is calculated next, from Eq. 15.53b: Q = (C₀/n)AR^{2/3}_ H S^{1/2} _B , with C₀ = 1.49, since we are working in BG units. Velocity can then be obtained from V = Q/A, and we can calculate Fr = V/\sqrt{gy} . The results are included in Table 15.2.

TABLE 15.1 Values of the Manning Roughness Coefficient, n

TABLE 15.2 Solution Table for Example 15.12