(a) What are the best dimensions y and b for a rectangular brick channel designed to carry 5 m^3/s of water in uniform flow with S0 = 0.001? (b) Compare results with a half-hexagon and semicircle.

Question

(a) What are the best dimensions y and b for a rectangular brick channel designed to carry 5 [latex]m^3/s[/latex] of water in uniform flow with [latex]S_0[/latex] = 0.001? (b) Compare results with a half-hexagon and semicircle.

Accepted Answer

Part (a)

From Eq. (10.26), A = 2[latex]y^2[/latex] and [latex]R_h = \frac{1}{2}y[/latex]. Manning’s formula (10.19) in SI units gives, with n ≈ 0.015 from Table 10.1,

[latex]A = 2y^2 P = 4y R_h = \frac{1}{2}y b = 2y[/latex] (10.26)

[latex]Q = \frac{1.0}{n} AR^{2/3}_h S^{1/2}_0 or 5 m^3/s = \frac{1.0}{0.015} (2y^2) \left(\frac{1}{2}y\right)^{2/3} (0.001)^{1/2}[/latex]

Table 10.1 Experimental Values of Manning’s n Factor[latex]^*[/latex]		Average roughness height ε
	n	ft	mm
Artificial lined channels:
Glass	0.010 ± 0.002	0.0011	0.3
Brass	0.011 ± 0.002	0.0019	0.6
Steel, smooth	0.012 ± 0.002	0.0032	1
Painted	0.014 ± 0.003	0.008	2.4
Riveted	0.015 ± 0.002	0.012	3.7
Cast iron	0.013 ± 0.003	0.0051	1.6
Concrete, finished	0.012 ± 0.002	0.0032	1
Unfinished	0.014 ± 0.002	0.008	2.4
Planed wood	0.012 ± 0.002	0.0032	1
Clay tile	0.014 ± 0.003	0.008	2.4
Brickwork	0.015 ± 0.002	0.012	3.7
Asphalt	0.016 ± 0.003	0.018	5.4
Corrugated metal	0.022 ± 0.005	0.12	37
Rubble masonry	0.025 ± 0.005	0.26	80
Excavated earth channels:
Clean	0.022 ± 0.004	0.12	37
Gravelly	0.025 ± 0.005	0.26	80
Weedy	0.030 ± 0.005	0.8	240
Stony, cobbles	0.035 ± 0.010	1.5	500
Natural channels:
Clean and straight	0.030 ± 0.005	0.8	240
Sluggish, deep pools	0.040 ± 0.010	3	900
Major rivers	0.035 ± 0.010	1.5	500
Floodplains:
Pasture, farmland	0.035 ± 0.010	1.5	500
Light brush	0.05 ± 0.02	6	2000
Heavy brush	0.075 ± 0.025	15	5000
Trees	0.15 ± 0.05	?	?

[latex]^*[/latex]A more complete list is given in Ref. 2, pp. 110–113.

which can be solved for

[latex]y^{8/3} = 1.882 m^{8/3}[/latex]

y = 1.27 m

The proper area and width are

[latex]A = 2y^2 = 3.21 m^2 b = \frac{A}{y} = 2.53 m[/latex]

Part (b)

It is constructive to see what flow rate a half-hexagon and semicircle would carry for the same area of 3.214 [latex]m^2[/latex].

For the half-hexagon (HH), with [latex]\beta = 1/3^{1/2} = 0.577[/latex], Eq. (10.25) predicts

[latex]A = y^2[2(1 + \beta^2)^{1/2} -\beta] P = 4y(1 + \beta^2)^{1/2} - 2\beta y R_h = \frac{1}{2}y[/latex] (10.25)

[latex]A = y^2_{HH} [2(1 + 0.577^2)^{1/2} - 0.577] = 1.732y^2_{HH} = 3.214[/latex]

or [latex]y_{HH}[/latex] = 1.362 m, whence [latex]R_h = \frac{1}{2}y = 0.681 m[/latex]. The half-hexagon flow rate is thus

[latex]Q = \frac{1.0}{0.015}(3.214)(0.681)^{2/3}(0.001)^{1/2} = 5.25 m^3/s[/latex]

or about 5 percent more than that for the rectangle.

For a semicircle, [latex]A = 3.214 m^2 = \pi D^{2}/8, or D = 2.861 m, whence P = \frac{1}{2} \pi D = 4.494 m and R_h = A/P = 3.214/4.494 = 0.715 m[/latex]. The semicircle flow rate will thus be

[latex]Q = \frac{1.0}{0.015} (3.214)(0.715)^{2/3}(0.001)^{1/2} = 5.42 m^3/s[/latex]

or about 8 percent more than that of the rectangle and 3 percent more than that of the half-hexagon.

Question 10.4: (a) What are the best dimensions y and b for a rectangular b...

Related Answered Questions