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<br />Without piers, the special bridge solution would indicate that no losses. <br /> <br />would occur. For a bridge with piers, the program goes through a momen- <br /> <br />turn balance for cross sections just outside and inside the bridge to <br />determine the class of flow. The momentum calculations are handled by <br /> <br />employing the following momentum relations based on the equations proposed <br />by Koch and Carstanjen (references b and c). <br /> <br />ml - mp1 + <br /> <br />where, <br /> <br />ml, m2, m3 <br /> <br />mpl' IIlp3 <br />Al' A3 <br /> <br />A2 <br /> <br />Apl' Ap3 <br /> <br />- - - <br />Y1' Y2' Y3 = <br /> <br />Co <br /> <br />- - <br />Ypl' Yp2 <br /> <br />Q <br /> <br />g <br /> <br />Q2 <br />g(Al)2 <br /> <br />Co <br />(Al - -ZApl) <br /> <br />2 <br />-m +--L= <br />- 2 <br />gA2 <br /> <br />m3 - mp3 + <br /> <br />Q2 <br />gA3 <br /> <br />,. A1Yl' AzY2 and A;Y3' respectively <br /> <br />= AplYpl and Ap;YP3' respectively <br /> <br />= unobstructed (gross) area at upstream and downstream <br />sections, respectively <br /> <br />= flow area (gross area - area of piers) at a section <br />within constricted reach <br /> <br />= obstructed areas at upstream and downstream sections, <br />respectively <br /> <br />vertical distance from water surface to center of <br />gravity of Al' A2' AJ' respectively <br /> <br />= drag coefficient equal to 2 for square pier ends <br />and 1.J3 for piers with semicircular ends. <br /> <br />= vertical distance from water surface to center of <br />gravity of Apl and ApJ' respectively <br /> <br />= discharge <br /> <br />= gravitational acceleration <br /> <br />4 <br />