Proof of Eq. 15-1 in "Open-channel hydraulics," by Ven T. Chow 1 E2/E1 = [(8F12 + 1)3/2 - 4F12 + 1]/[8F12(2 + F12)]
Froude numbers: • F1 = v1/(gy1)1/2 • F2 = v2/(gy2)1/2 From water continuity: • v1y1 = v2y2 • v12y12 = v22y22 • F12y13 = F22y23 • F22 = F12/(y2/y1)3 The hydraulic jump equation, Eq. 3-21 (Chow): • y2/y1 = (1/2)[(1 + 8F12)1/2 - 1] • N2 = 1 + 8F12 • y2/y1 = (1/2)[N - 1] • 2(y2/y1) = N - 1 • (y2/y1)3 = (1/8)[N - 1]3 • 2(y2/y1)3 = (1/4)[N - 1]3
• N = (1 + 8F12)1/2 • N3 = (1 + 8F12)3/2
• N2 - 1 = 8F12 • F12 = (N2 - 1)/8 • 4F12 = (N2 - 1)/2 Efficiency of the hydraulic jump: • E2/E1 = [y2 + v22/(2g)]/[y1 + v12/(2g)] • E2/E1 = [y2(1 + F22/2)]/[y1(1 + F12/2)] • E2/E1 = 2 (y2/y1){1 + F12/[2(y2/y1)3]}/(2 + F12) • E2/E1 = (N - 1){1 + (N2 - 1)/[2(N - 1)3]}/(2 + F12) • E2/E1 = (N2 - 1)(N - 1){1 + (N2 - 1)/[2(N - 1)3]}/[8F12(2 + F12)] • E2/E1 = [(N2 - 1)(N - 1) + (1/2)(N + 1)2]/[8F12(2 + F12)] • E2/E1 = {(N3 - N2 - N - 1) + [(N2/2) + N + (1/2)]}/[8F12(2 + F12)] • E2/E1 = {(N3 - [(N2 - 1)/2] + 1}/[8F12 (2 + F12)] • E2/E1 = [(8F12 + 1)3/2 - 4F12 + 1]/[8F12(2 + F12)]
1 Chow, V. T. 1959. "Open-channel hydraulics," McGraw-Hill, New York. | ||
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