2. Iowa Tests. 

Kalinske and Robertson(1) have published the results of tests on the air demand of a hydraulic jump in a circular conduit. They found the ratio of air demand to water discharge (β) to be a function of the Froude number minus one. The formula which was developed is indicated in HDC 050-1.

3. Prototyye Tests. 

A number of prototype tests on existing outlet works have been analyzed and compared graphically with the Kalinske and Robertson formula in HDC 050-1. In some of the prototype tests, gate openings varied from small to”full opening where pressure flow existed throughout the entire system. The maximum air demand is found at some int-ermediate gate opening. The ratios of this gate opening (Gm) to full gate opening (Gf) are shown in table 1 together with other pertinent information.

4. Extensive Corps of Engineers air-demand tests were made at Pine Flat Dam from 1952 to 1956. These tests included heads up to 370 ft although gates are not normally operated under such high heads. The Pine Flat test data are in good agreement with other field data, as shown by the
plots in HDC 050-1.

5. Reconunendations. 
A straight line in HDC 050-1 indicates a suggested design assumption. It is suggested that the maximum air demand be assumed to occur at a gate opening ratio of 80 percent in sluices through
concrete dams. A gate lip with a k5-degree angle on the bottom can be expected to have a contraction coefficient of approximately 0.80. The Froude number should be based on the effective depth at the vena contracta which, with the above-mentioned factors, would be 64 percent of the sluice depth. The suggested design curve can be used to determine the ratios of air demand to water discharge. It is further suggested that air vents be designed for velocities of not more than 150 ft per sec. The disadvantage of excessive air velocities is a high head loss in the air vent which causes subatmospheric pressures in the water conduit. Outlet works with well-streamlined water passages can tolerate lower pressures without cavitation trouble than those with less effectively streamlined water passages. The suggested design assumptions for sluices will result in area ratics of air vent to sluice of approximately 12 percent for each 150 ft of head on
a 4- by 6-ft sluice, and 12 percent for each 200 ft of head on a 5-ft-8-in. by 10-ft sluice. In applying the curve to circular tunnels controlled by one or more rectangular gates, the effective depth should be based on flow in 64 percent of the area of the tunnel for maximum air demand. These are general design rules which have been devised until additional experimental data are available.

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(1) A. A. Kalinske and J. W. Robertson, “Entrainment of air in flowing water--closed conduit flow.” Transactions, American Society of Civil Engineers, VOL 108 (1943), Pp 1435-1447.
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COMPUTATION SHEET

GIVEN:

- Sluice size : Width (B) = 4 ft, Height (D) = 9 ft, 45˚ gate lip
- Elevation sluice invert at gate (ELsi) : 127.0 ft
- Design pool elevation (ELp) : 352.0 ft

FROM HYDRAULIC DESIGN SHEET 050-1 AND 320-1

Assume maximum air discharge (Qa) at 80% gate opening (GO).
Discharge coefficient (C) for 45˚ gate lip = 0.80

Then:

Depth of water at vena contracta (y) = GO / 100 * C * D

Effective head (H),

Water discharge (Qw), (Qw) = C*A*V = B*y*(2*g*H)^(1/2)

Velocity of water at vena contracta (V) = Qw / A = Qw / (B*y)

Froude Number at vena contracta (F) = V / (g*y)^(1/2), (F-1) = V / (g*y)^(1/2) - 1

FROM HYDRAULIC DESIGN SHEET 050-1. β=0.28, Air Demand (Qa),

FROM HYDRAULIC DESIGN SHEET 050-1. Maximum Air Velocity (Va) = 150 ft/sec, 
Area for Air Demand (Av),

Diameter for circular vent (Dv) = (4*Av / π)^(1/2)