P10.5 Water flows down a rectangular channel that is 4 ft wide and 3 ft deep. The flow rate is 15,000 gal/min. Estimate the Froude number of the flow.

Transcription

1 P10.5 Water flows down a rectangular channel that is 4 ft wide and ft deep. The flow rate is 15,000 gal/min. Estimate the Froude number of the flow. Solution: Convert the flow rate from 15,000 gal/min =.4 ft /s. The water velocity and wave speed may be calculated: V = Q/ A = (.4 ft / s)/( 4 ft ) =.78 ft/ s ; c = g y = (. ft/ s )( ft) = 9.8 ft/ s Thus Fr = V / c =.78/ 9.8 = 0.8 (subcritical) Ans.

2 10.11 A rectangular channel is m wide and contains water m deep. If the slope is 0.85 and the lining is corrugated metal, estimate the discharge for uniform flow. Solution: For corrugated metal, take Manning s n 0.0. Get the hydraulic radius: Fig. P10.11 A () 1 1 R 0.75 m; Q AR S (6)(0.75) [tan(0.85 )] P + + n 0.0 / 1/ / 1/ h = = = h o = or: Q 7 m /s Ans.

3 10. A trapezoidal aqueduct has b = 5 m and θ = 40 and carries a normal flow of 60 m /s when y =. m. For clay tile surfaces, estimate the required elevation drop in m/km. Fig. P10. Solution: For clay tile, take n The geometry leads to these values: A = by + y cotθ = 8. m ; P = b + y cscθ = m, R = A/P = m 1.0 / 1/ Q = 60 m /s = (8.)(1.886) S o, solve for So = =0.8 m/ km h Ans.

4 10.5 In flood stage a natural channel often consists of a deep main channel plus two floodplains, as in Fig. P10.5. The floodplains are often shallow and rough. If the channel has the same slope everywhere, how would you analyze this situation for Fig. P10.5 the discharge? Suppose that y1 = 0 ft, y = 5 ft, b1 = 40 ft, b = 100 ft, n1 = 0.00, n = 0.040, with a slope of Estimate the discharge in ft /s. Solution: We compute the flow rate in three pieces, with the dashed lines in the figure above serving as water walls which are not counted as part of the perimeter: " 5 40 # 1/ (a) Deep channel: Q 1 = (5 40) % (0.000) 5659 ft /s 0.0 ' & ( " 1.486# " # 1/ (b) Flood plains: Q = % (5 100) (0.000) 1487 ft /s ' & 0.04 ( % ' & ( Total discharge Q = Q + Q = 7150 ft / s Ans. 1 / /

5 10.59 Uniform water flow in a wide brick channel of slope 0.0 moves over a 10-cm bump as in Fig. P A slight depression in the water surface results. If the minimum depth over the bump is 50 cm, compute (a) the velocity over the bump; and (b) the flow rate per meter of width. Fig. P10.59 Solution: For brickwork, take n Since the water level decreases over the bump, the upstream flow is subcritical. For a wide channel, Rh = y/, and Eq holds: q V y E y + = 0, q= Vy, E = + y Δh, Δ h= 0.1 m, y = 0.5 m g g 1 / 5/ Meanwhile, for uniform flow, q = y 1(y 1/) sin 0.0 = 0.785y Solve these two simultaneously for y1 = m, V1 = 0.56 m/s Ans. (a), and q = 0.4 m /s m. Ans. (b) [The upstream flow is subcritical, Fr1 0..]

6 10.85 In Prob the exit velocity from the sluice gate is 4. m/s. If there is a hydraulic jump at just downstream of section, determine the downstream (a) velocity; (b) depth; (c) Froude number; and (d) percent dissipation. Solution: If V = 4. m/s, then y = V1y1/V = (1.0)(0.)/4. = m, and the Froude number is Fr = V/[gy] 1/ = 6.4. Now use hydraulic jump theory: h f y y = (6.4) = 17., or: y = 0.98 m Ans. (b) Fr V q (1.0)(0.) m = = = 0.50 Ans. (a) y 0.98 s V 0.50 = = = 0.55 Ans. gy 9.81(0.98) ( ) hf 0.59 = = 0.59 m; = = 0.59 or 59% Ans. (d) 4(0.98)(0.046) E (4.) /g (c)

7 10.9 Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic jump, as in Fig. P10.9. If y1 = 1 m and y = 40 cm, estimate (a) V1; (b) V; (c) y4; and (d) the bump height h. Solution: Assume frictionless flow except in the jump. From point 1 to point : Fig. P10.9 V1 V 1 = + = = + 1 = Energy: E 1 E 0.4 ; Continuity: V (1.0) V (0.4) (9.81) (9.81) m m Solve simultaneously by iteration for V Ans. (a) V =.74 Ans. (b) s s The flow after the bump is supercritical: Fr =.74/ [9.81(0.4)] For the jump, y4 1 =! 1+ 8(1.89) 1" =., y4 =.(0.4) 0.89 m Ans. (c) y % ' &( Finally, use energy from 1 to and note that the bump flow must be critical:

8 1 = = = + V = = = E E y h; Vy 1.50 Vy ; and: V 9.81y g Solve simultaneously for y 0.61 m; V.45 m/s; h 0.0 m Ans. (d)