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LOCK CULVERTS, MINIMUM BEND PRESSURE, RECTANGULAR SECTION, Sheet 534-2

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HYDRAULIC DESIGN CRITERIA
SHEETS 534-2 AND 534-2/1
LOCK CULVERTS
MINIMUM BEND PRESSURE
RECTANGULAR SECTION

1. Laboratory flow studies have shown that, for a rectangular conduit section, the minimum pressure in circular bends of 90 to 300 deg occurs on the inside of the bend 45 deg from the point of curvature. Experimental turbulent flow pressure data, at this location, closely approximate values computed for two-dimensional potential flow. McPherson and Strausser (1) have suggested an analytical procedure for determining the magnitude of the minimum pressure in a circular bend of rectangular section.
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2. Theory. 

The minimum bend pressure head can be computed from the equation

Cp = ( H - Hi ) / ( V^2 / (2g) )


where

Cp = pressure-drop parameter

H = average pressure head, in ft, at the 45-deg point computed as a straight-line extension of the upstream pressure gradient

Hi = minimum pressure head, in ft, at the 45-deg point on inside of bend

V = average culvert velocity in ft per sec

g = acceleration, gravitational, in ft per sec^2

Equation 1 is similar to the bend coefficient equation developed by Lansford (reference 4, Sheet 228-3). Based on equation 3 of reference 1, it can also be shown that

Cp = [ 2 / (( R/C - 1 ) * ln((R/C + 1) / (R/C - 1))) ]^2 - 1


where

R = center-line radius of the bend

C = one-half the culvert width

3. Application.

Hydraulic Desire Chart 534-2 shows the relation between the theoretical pressure-drop parameter and ratio of the radius of curvature to one-half the conduit dimension in the direction concernd. Values of Cp computed from experimental results reported by Silberman (2) and Yarnell and Woodward (3) are also shown. These data indicate the effects of Reynolds nunibers between 6.7 x 10^4 and 8.2 x 10^5 . Points computed from data summarized by McPherson and Strausserl from tests by Addison (4), Lell (5), Wattendorf (6), and Nippert (7) and on the Waynesboro and Mt. Alto model studies at Lehigh University are included on the chart. The indicated Reynolds number is about 10^5 to 10^6. The chart is considered applicable to bends of 45 to 300 deg.

4. Cavitation occurs when the instantaneous pressure at any point in a flowing liquid drops to the vapor pressure. Vapor pressure varies with temperature of the liquid (see Sheet 000-2). Since turbulence in flow causes pressure fluctuations, an estimate should be made of the maximum expected fluctuation from the minimum computed bend pressure. The sum of the estimated pressure fluctuation, the vapor pressue, and a few feet of water for a margin of safety should be computed. The local barometric pressure (see Chart 000-2) should be subtracted from this total to obtain the minimum permissible bend pressure. This pressure can then be used to determine the necessary average conduit pressure or the permissible average conduit velocity to prevent cavitation. Cavitation damage has been found where the average pressure is relatively high but violent negative pulsations reach cavitation pressures. Such criteria as indicated here should therefore be used conservatively.

5. Chart 534-2/1 is a sample computation showing the application of Chart 534-2 to the minimum bend pressure problem. Computations to indicate the minimum permissible average conduit pressure and the maximum permissible average conduit velocity to prevent cavitation are included. Chart 534-2 can also be used for the design of bends in rectangular sluices and siphons and in circular conduits. Its application to the latter is shown in Chart 228-3.

6. References

(1) McPherson, M. B., and Strausser, H. S., ¡°Minimumpressures inrectangular bends.¡± Proceedings, ASCE, vol 81, Separate Paper No. 747 (July 1955); vol 82, Separate Paper No. 1092 (October 1956), p 9, Closure.

(2) Silberman, E., The Nature of Flow in an Elbow. Project Report No. 5, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis, prepared for David Taylor Model Basin, December 1947.

(3) U. S. Department of Agriculture, Flow of Water Around 180-Degree Bends, by D. L. Yarnell, and S. M. Woodward. Technical Bulletin NO. 526, Washington, D. C., October 1936.

(4) Addison, H., ¡°The use of bends as flow meters.¡± Engineering, vol 145 (4 March 1938), pp 227-229 (25 March 1938), p 324.

(5) Len, J., ¡°Contribution to the Knowledge of Secondary Cwrents in Curved Channels (Beitrag zur Kenntnis der Sekundastromungen in gekrummten Kanalen).¡± Dissertation, R. Oldenbourg, Muchen, 1913. Also Zeitschrift fur das gesamte Turbinenwesen, Heft 11, July 1914, pp 129-135, 293-298, 313-317, and 325-330.

(6) Wattendorf, F. L., ¡°A study of the effects of curvature on fully developed turbulent flow.¡± Proceedings, Royal Society of London, Series A, vol 148 (February 1935), pp 565-598.

(7) Nippert, H., ¡°Uber den Stromungsverlust in gekrummten Kanalen.¡± VDI, orschungsarbeiten, Heft 320, Berlin (1929).
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USACE, Hydraulic Design Criteria, SHEET 534-2, LOCK CULVERTS, MINIMUM BEND PRESSURE, RECTANGULAR SECTION



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