Water at 70^◦ F flows in a prototype 1700-ft-long concrete rectangular open channel with a width of 5 ft, a uniform flow depth of 3 ft, and an absolute channel roughness of 0.01 ft, as illustrated in Figure EP 11.13. A smaller model of the larger prototype is designed in order to study the flow

Question

Water at [latex] 70^{\circ } F [/latex] flows in a prototype 1700-ft-long concrete rectangular open channel with a width of 5 ft, a uniform flow depth of 3ft, and an absolute channel roughness of 0.01 ft, as illustrated in Figure EP 11.13. A smaller model of the larger prototype is designed in order to study the flow characteristics of turbulent open channel flow. The model fluid is also water at [latex] 70^{\circ } F [/latex], the velocity of water in the smaller model channel is 60 ft/sec, and the model scale, λ is 0.25. The flow resistance is modeled by the Manning’s roughness coefficient, [latex]n = f^n ( C_{D} ).[/latex] (a) Determine the friction slope (and head loss) in the flow of the water in the model. (b) Determine the velocity flow of the water in the prototype open channel flow in order to achieve dynamic similarity between the model and the prototype. (c) Determine the friction slope (and head loss) in the flow of the water in the prototype in order to achieve dynamic similarity between the model and the prototype.

Accepted Answer

(a) In order to determine the friction slope (and head loss) in flow of the water in the model, the major head loss equation, Equation 11.141 [latex] h_{f} = \frac{ au _{w}L}{\gamma R_{h}} = S_{f}L = C_{D} ho v^{2} \frac{L}{\gamma R_{h}} = \frac{v^{2}L}{C^{2}R_{h}} = f \frac{L}{(4R_{h})} \frac{v^{2}}{2g} = \left(\frac{vn}{R_{h}^{2/3}} ight)^{2}L [/latex], is applied as follows:

[latex] h_{f} = S_{f}L = \left(\frac{vn}{R_{h}^{2/3}} ight)^{2}L [/latex]

where the Manning’s roughness coefficient, n is used to model the flow resistance. Empirical calibration of the Manning’s roughness coefficient, n assumes turbulent flow, thus it is independent of R, and is only a function of ɛ/y. The absolute channel roughness, ɛ is indirectly modeled by the type of channel material, as illustrated in Table 8.6, which presents the Manning’s roughness coefficient, n for various channel materials. It important to note that although the derivation/formulation of the Manning’s equation assumes no specific units (SI or BG), the Manning’s roughness coefficient, n has dimensions of [latex] L^{-1/3}T [/latex]. Furthermore, because the Manning’s roughness coefficient, n in Table 8.6 has been provided/calibrated in SI units [latex] m^{-1/3} [/latex] sec, it must be adjusted when using BG units. In order to convert the Manning’s roughness coefficient, n with units of [latex] m^{-1/3} [/latex] sec to units of [latex] ft^{-1/3} [/latex] sec, note that 3.281 ft = 1m; thus:

[latex] \frac{\left[\frac{3.281 ft}{1m} ight] ^{1/3} }{n [m^{-1/3}s]} = \frac{1.486}{n[ft^{-1/3}sec ]} [/latex]

Thus, for a concrete channel, assume n = 0.012 [latex] m^{-1/3} [/latex] sec, which is converted to BG units as follows:

[latex] n_{m}: = 0.012 m^{\frac{-1}{3} } sec [/latex] [latex] m: = 3.281 ft[/latex] [latex] n_{m} = 8.076 imes 10^{-3} \frac{s}{ft^{0.333}} [/latex]

Furthermore, in order to determine the length, depth of flow, and the absolute channel roughness of the model channel, the model scale, λ (inverse of the length ratio) is applied. The fluid properties for water are given in Table A.2 in Appendix A.

[latex] b_{p}: = 5 ft [/latex] [latex] y_{p}: = 3 ft [/latex] [latex] L_{p}: = 1700 ft [/latex] [latex] \varepsilon _{p} : = 0.01 ft [/latex] [latex] \lambda : = 0.25 [/latex]

Guess value: [latex] b_{m}: = 1 ft [/latex] [latex] y_{m}: = 1 ft [/latex] [latex] L_{m}: = 1 ft [/latex] [latex] \varepsilon _{m}: = 0.01 ft [/latex]

Given

[latex] \lambda = \frac{b_{m}}{b_{p}} [/latex] [latex] \lambda = \frac{y_{m}}{y_{p}} [/latex] [latex] \lambda = \frac{L_{m}}{L_{p}} [/latex] [latex] \lambda = \frac{\varepsilon _{m}}{\varepsilon _{p}} [/latex]
[latex] \left ( \begin{matrix} b_{m} \ y_{m} \ L_{m} \ \varepsilon _{m} \end{matrix} ight ) : = Find (b_{m}, y_{m}, L_{m}, \varepsilon _{m}) = \left ( \begin{matrix} 1.25 \ 0.75 \ 425 \ 2.5 imes 10^{-3} \end{matrix} ight ) ft [/latex]
[latex] slug: = 1 lb \frac{sec^{2}}{ft} [/latex] [latex] ho _{m} : = 1.936 \frac{slug}{ft^{3}} [/latex] [latex] \mu _{m} : = 20.5 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

[latex] V_{m}: = 60 \frac{ft}{sec} [/latex] [latex] R_{m}: = \frac{ ho _{m} .V_{m} .y_{m}}{\mu _{m}} = 4.25 imes 10^{6} [/latex]

[latex] A_{m}: = b_{m} . y_{m} = 0.938 ft^{2} [/latex] [latex] P_{m}: = 2.y_{m} + b_{m} = 2.75 ft [/latex] [latex] R_{hm}: = \frac{A_{m}}{P_{m} } = 0.341 ft [/latex]

Guess value: [latex] h_{fm}: = 1 ft [/latex] [latex] S_{fm}: = 0.01 \frac{ft}{ft} [/latex]

Given

[latex] h_{fm} = \left(\frac{V_{m}n_{m}}{R_{hm}^{\frac{2}{3} }} ight)^{2} L_{m} [/latex] [latex] S_{fm} = \frac{h_{fm}}{L_{m}} [/latex]

[latex] \left ( \begin{matrix} h_{fm} \ S_{fm} \end{matrix} ight ) : = Find ( h_{fm}, S_{fm} ) [/latex]

[latex] h_{fm} = 418.998 ft [/latex] [latex] S_{fm} = 0.986 \frac{ft}{ft} [/latex]

It is important to note that for uniform flow, the channel bottom slope, [latex] S_{o} [/latex] is equal to the friction slope, [latex] S_{f} [/latex] . Furthermore, in order to satisfy the geometric similarity requirement when applying the model scale (and the dynamic similarity requirement), the prototype channel bottom slope, [latex] S_{op} [/latex] must equal the model channel bottom slope, [latex] S_{om} [/latex] ; thus, the prototype friction slope, [latex] S_{fp} [/latex] must equal the model friction slope, [latex] S_{fm} [/latex] (see (b), (c) below).
(b)–(c) To determine the velocity flow of the water in the prototype open channel flow in order to achieve dynamic similarity between the model and the prototype for turbulent open channel flow, and to determine the friction slope (and head loss) in the flow of the water in the prototype in order to achieve dynamic similarity between the model and the prototype, for turbulent open channel flow, the ɛ/y must remain a constant between the model and prototype as follows:

[latex] \left(\frac{\varepsilon}{y} ight)_{p} = \left(\frac{\varepsilon }{y} ight)_{m} [/latex]
[latex]\frac{\varepsilon _{p}}{y_{p}} = 3.333 imes 10^{-3} [/latex] [latex]\frac{\varepsilon _{m}}{y_{m}} = 3.333 imes 10^{-3} [/latex]

However, because the Manning’s roughness coefficient, n is independent of R, R does not need to remain a constant between the model and the prototype.
(b)–(c) To determine the velocity flow of the water in the prototype open channel flow in order to achieve dynamic similarity between the model and the prototype for turbulent open channel flow, and to determine the friction slope (and head loss) in the flow of the water in the prototype, in order to achieve dynamic similarity between the model and the prototype for turbulent open channel flow, the Manning’s roughness coefficient, n must remain a constant between the model and the prototype (which is a direct result of maintaining a constant ɛ/y between the model and the prototype, and applying the “gravity model” similitude scale ratio; specifically the velocity ratio, [latex] v_{r} [/latex] given in Table 11.2) as follows:

[latex] \underbrace{\left[\frac{\frac{h_{f}}{v^{2}L} }{R_{h}^{4/3}} ight]_{p} }_{n^{2}_{p}} = \underbrace{\left[\frac{\frac{h_{f}}{v^{2}L} }{R_{h}^{4/3}} ight]_{m} }_{n^{2}_{m}} [/latex]
[latex] v_{r} = \frac{v_{p}}{v_{m}} = \frac{\left(\sqrt{gL} ight) _{p} }{\left(\sqrt{gL} ight) _{m}} = L_{r}^{\frac{1}{2} } [/latex]

[latex] ho _{p} : = 1.936 \frac{slug}{ft^{3}} [/latex] [latex] \mu _{p} : = 20.5 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex] [latex] g: = 32.174 \frac{ft}{sec^{2}} [/latex]

[latex] A_{p}: = b_{p} . y_{p} = 15 ft^{2} [/latex] [latex] P_{p}: = 2.y_{p} + b_{p} = 11 ft [/latex] [latex] R_{hp}: = \frac{A_{p}}{P_{p} } = 1.364 ft [/latex]

Guess value: [latex] V_{p}: = 1 \frac{ft}{sec} [/latex] [latex] h_{fp}: = 1 ft [/latex] [latex] S_{fp}: = 0.01 \frac{ft}{ft} [/latex]
[latex] n_{p}: = 0.01 ft^{\frac{-1}{3} }sec [/latex]

Given

[latex] n_{p}^{2} = \frac{h_{fp}}{\left(\frac{V^{2}_{p}.L_{p}}{R_{hp}^{\frac{4}{3} }} ight) } [/latex] [latex] \frac{V_{p}}{\sqrt{g.y_{p}} } = \frac{V_{m}}{\sqrt{g.y_{m}} } [/latex]

[latex] n_{p} = n_{m} [/latex] [latex] S_{fp} = \frac{h_{fp}}{L_{p}} [/latex] [latex] S_{fp} = S_{fm} [/latex]
[latex] \left ( \begin{matrix} V_{p} \ h_{fp} \ S_{fp} \ n_{p} \end{matrix} ight ) : = Find (V_{p}, h_{fp}, S_{fp}, n_{p} ) [/latex]

[latex] V_{p} = 120 \frac{ft}{s} [/latex] [latex] h_{fp} = 1.676 imes 10^{3} ft [/latex] [latex] S_{fp} = 0.986 \frac{ft}{ft} [/latex]
[latex] n_{p} = 8.077 imes 10^{-3} \frac{s}{ft^{0.333}} [/latex]

Furthermore, the Froude number, F remains a constant between the model and the prototype as follows:

[latex] F_{m}: = \frac{V_{m}}{\sqrt{g.y_{m}} } = 12.214 [/latex] [latex] F_{p}: = \frac{V_{p}}{\sqrt{g.y_{p}}} = 12.214 [/latex]

Therefore, although the similarity requirements regarding the independent π term,[latex] \varepsilon /y ((\varepsilon /y)_{p} = (\varepsilon /y)_{m} = 3.333 imes 10^{-3})[/latex], the dependent π term, F (“gravity model”) ([latex] F_{p} = F_{m} = 12.214 [/latex]), and the dependent π term, friction slope, [latex] S_{f} (S_{fp} = S_{fm} = 0.986 ft/ft ) [/latex] are theoretically satisfied, the dependent π term (i.e., the friction factor, f ) will actually/practically remain a constant between the model and its prototype ([latex] n_{p} = n_{m}= 8.076 imes 10^{-3} ft^{-1/3}sec [/latex]) only if it is practical to maintain/attain the model velocity, slope, fluid, scale, and cost. Furthermore, because the Manning’s roughness coefficient, n is independent of R, R does not need to remain a constant between the model and the prototype as follows:

[latex]R_{m} = 4.25 imes 10^{6} [/latex] [latex] R_{p} : = \frac{ ho _{p} . V_{p}. Y_{p}}{\mu _{p}} = 3.4 imes 10^{7} [/latex]

Table 8.6
Empirical Values for the Manning Roughness Coefficient, [latex] n (m^{−1/3} s) [/latex] for Various Boundary Surfaces
Boundary Surface	Manning Roughness Coefficient, n (Min)	Manning Roughness Coefficient, n (Max)
Lucite	0.008	0.010
Brass	0.009	0.013
Glass	0.009	0.013
Wood stave pipe	0.010	0.013
Neat cement surface	0.010	0.013
Plank flumes, planed	0.010	0.014
Vitrified sewer pipe	0.010	0.017
Concrete, precast	0.011	0.013
Cement mortar surfaces	0.011	0.015
Metal flumes, smooth	0.011	0.015
Plank flumes, unplaned	0.011	0.015
Common-clay drainage tile	0.011	0.017
Concrete, monolithic	0.012	0.016
Brick with cement mortar	0.012	0.017
Cast iron, new	0.013	0.017
Wood laminated	0.015	0.020
Asphalt, rough	0.016	0.016
Riveted steel	0.017	0.020
Cement rubble surfaces	0.017	0.030
Canals and ditches, smooth earth	0.017	0.025
Corrugated metal pipe	0.021	0.030
Metal flumes, corrugated	0.022	0.030
Gravel bottom with rubble	0.023	0.036
Canals (excavated or dredged)
Dredged in earth, smooth	0.025	0.033
In rock cuts, smooth	0.025	0.035
Rough beds and weeds on sides	0.025	0.040
Rock cuts, jagged and irregular	0.035	0.045
Dense weeds	0.050	0.120
Dense brush	0.080	0.140
Natural streams
Smoothest (clean, straight)	0.025	0.033
Roughest	0.045	0.060
Very weedy	0.075	0.150
Floodplains (brush)	0.035	0.070
streams	0.025	0.060

Table A.2
Physical Properties for Water at Standard Sea-Level Atmospheric Pressure as a Function of Temperature
Temperature (θ) [latex]^{\circ } F[/latex]	Density (ρ) [latex]slug/ft^{3} [/latex]	Specific Weight (γ) [latex]Ib/ft^{3}[/latex]	Absolute (Dynamic) Viscosity (μ) [latex]10^{-6} Ib-sec/ft^{3}[/latex]	Kinematic Viscosity (ν) [latex]10^{-6} ft^{2}/sec[/latex]	Surface Tension (σ) lb/ft	Vapor Pressure [latex]( ho _{ u } )[/latex] psia	Bulk Modulus of Elasticity [latex](E_{\upsilon } )[/latex] psi
32	1.940	62.42	37.46	19.31	0.00518	0.0885	293,000
40	1.940	62.43	32.29	16.64	0.00514	0.1220	294,000
50	1.940	62.41	27.35	14.10	0.00509	0.1780	305,000
60	1.938	62.37	23.59	12.17	0.00504	0.2560	311,000
70	1.936	62.30	20.50	10.59	0.00498	0.3630	320,000
80	1.934	62.22	17.99	9.30	0.00492	0.5070	322,000
90	1.931	62.11	15.95	8.26	0.00486	0.6980	323,000
100	1.927	62.00	14.24	7.39	0.00480	0.9490	327,000
110	1.923	61.86	12.84	6.67	0.00473	1.2750	331,000
120	1.918	61.71	11.68	6.09	0.00467	1.6920	333,000
130	1.913	61.55	10.69	5.58	0.00460	2.2200	334,000
140	1.908	61.38	9.81	5.14	0.00454	2.8900	330,000
150	1.902	61.20	9.05	4.76	0.00447	3.7200	328,000
160	1.896	61.00	8.38	4.42	0.00441	4.7400	326,000
170	1.890	60.80	7.80	4.13	0.00434	5.9900	322,000
180	1.883	60.58	7.26	3.85	0.00427	7.5100	318,000
190	1.876	60.36	6.78	3.62	0.00420	9.3400	313,000
200	1.868	60.12	6.37	3.41	0.00413	11.5200	308,000
212	1.860	59.83	5.93	3.19	0.00404	14.6900	300,000
[latex]^{\circ } C[/latex]	[latex]kg/m^{3} [/latex]	[latex]KN/m^{3} [/latex]	[latex]N-sec/m^{2} [/latex]	[latex]10^{-6} m^{2} /sec[/latex]	[latex]N/m[/latex]	[latex]KN/m^{2} abs[/latex]	[latex]10^{6} KN/m^{2} [/latex]
0	999.8	9.805	0.001781	1.785	0.0756	0.611	2.02
5	1000.0	9.807	0.001518	1.519	0.0749	0.872	2.06
10	999.7	9.804	0.001307	1.306	0.0742	1.230	2.10
15	999.1	9.798	0.001139	1.139	0.0735	1.710	2.14
20	998.2	9.789	0.001002	1.003	0.0728	2.340	2.18
25	997.0	9.777	0.000890	0.893	0.0720	3.170	2.22
30	995.7	9.765	0.000798	0.800	0.0712	4.240	2.25
40	992.2	9.731	0.000653	0.658	0.0696	7.380	2.28
50	988.0	9.690	0.000547	0.553	0.0679	12.330	2.29
60	983.2	9.642	0.000466	0.474	0.0662	19.920	2.28
70	977.8	9.589	0.000404	0.413	0.0644	31.160	2.25
80	971.8	9.530	0.000354	0.364	0.0626	47.340	2.20
90	965.3	9.467	0.000315	0.326	0.0608	70.100	2.14
100	958.4	9.399	0.000282	0.294	0.0589	101.330	2.07

Table 11.2
Similitude Scale Ratios for Physical Quantities for a Gravity Model
Physical Quantity	FLT System	MLT System	Primary Scale Ratios	Secondary/Similitude Scale Ratios for a Pressure Model
			[latex] F_{r} = \frac{F_{G_{p}}}{F_{G_{m}}} = \frac{F_{I_{p}}}{F_{I_{m}}} = constant [/latex]	[latex] \underbrace{\left[\left(\frac{ v}{\sqrt{gL} } ight)_{p} ight] }_{F_{p}} = \underbrace{\left[\left(\frac { v}{\sqrt{ gL} } ight)_{m} ight] }_{F_{m}} [/latex]
Geometrics Length, L	L	L	[latex] L_{r} = \frac{L_{p}}{L_{m}} [/latex]	[latex] L_{r} = \frac{L_{p}}{L_{m}} [/latex]
Area, A	[latex] L^{2} [/latex]	[latex] L^{2} [/latex]	[latex] L_{r}^{2} = \frac{L_{p}^{2}}{L_{m}^{2}} [/latex]	[latex] L_{r}^{2} = \frac{L_{p}^{2}}{L_{m}^{2}} [/latex]
Volume, V	[latex] L^{3} [/latex]	[latex] L^{3} [/latex]	[latex] L_{r}^{3} = \frac{L_{p}^{3}}{L_{m}^{3}} [/latex]	[latex] L_{r}^{3} = \frac{L_{p}^{3}}{L_{m}^{3}} [/latex]
Kinematics Time, T	T	T	[latex] T_{r} = \frac{L_{r}}{v_{r}} [/latex]	[latex] T_{r} = \frac{L_{r}}{v_{r}} = L_{r}^{1/2} [/latex]
Velocity, v	[latex] LT^{-1}[/latex]	[latex] LT^{-1}[/latex]	[latex] v_{r} = \frac{v_{p}}{v_{m}} [/latex]	[latex] v_{r} = \frac{v_{p}}{v_{m}} = \frac{\left(\sqrt{gL} ight) _{p} }{\left(\sqrt{gL} ight) _{m}} = L_{r}^{1/2} [/latex]
Acceleration, a	[latex] LT^{-2}[/latex]	[latex] LT^{-2}[/latex]	[latex] a_{r} = \frac{L_{r}}{T_{r}^{2}} = \frac{v_{r}^{2}}{L_{r}} [/latex]	[latex] a_{r} = \frac{v_{r}^{2}}{L_{r}} =1[/latex]
Discharge, Q	[latex] L^{3}T^{-1}[/latex]	[latex] L^{3}T^{-1}[/latex]		[latex] Q_{r} = v_{r}. L_{r}^{2} = L_{r}^{5/2} [/latex]
Dynamics Mass, M	[latex] FL^{-1}T^{2}[/latex]	M		[latex] M_{r} = F_{r}a_{r}^{-1} = ho _{r} L_{r}^{3} [/latex]
Force, F	F	[latex] MLT^{-2}[/latex]	[latex] F_{r} = \frac{F_{G_{p}}}{F_{G_{m}}} = \frac{F_{I_{p}}}{F_{I_{m}}} [/latex]	[latex] F_{r} = ho_{r} L_{r}^{3} g_{r} = ho _{r} v^{2}_{r} L_{r}^{2} [/latex]
Pressure, p	[latex] FL^{-2} [/latex]	[latex] ML^{-1}T^{-2}[/latex]		[latex] p_{r} = F_{r} L_{r}^{-2} = ho _{r} L_{r} [/latex]
Momentum, Mv or Impulse, FT	FT	[latex] MLT^{-1}[/latex]		[latex] F_{r} T_{r} = ho _{r} L_{r}^{7/2} [/latex]
Energy, E or Work, W	FL	[latex] ML^{2}T^{-2}[/latex]		[latex] W_{r} = F_{r} L_{r}= ho _{r} L_{r}^{4} [/latex]
Power, P	[latex] FLT^{-1} [/latex]	[latex] ML^{2}T^{-3}[/latex]		[latex] p_{r} = W_{r} T_{r}^{-1} = ho _{r} L^{7/2}_{r} [/latex]

Question 11.13: Water at 70^◦ F flows in a prototype 1700-ft-long concrete r...

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