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<br />EM 1110-2-1601 <br />1 Jul 91 <br /> <br />may also be used for class A flow, the energy method <br />usually gives better results. <br /> <br />(2) Classes B and C flows (momentum method), <br /> <br />(a) A graph (example shown in Plate 12) co~ <br />from the equation proposed by Koch and CarstanJen <br />(Chow 1959) and based on the momentum relation can be <br />used for detennining graphically the flow classiilC3lion at <br />constrictions due 10 bridge piers. In addition, the graph <br />can be used for estimating unknown flow depths. A <br />swnm:uy of the equation derivation follows. <br /> <br />(b) In a given channel section the momentum per <br />unit time of the flow can be expressed by <br /> <br />M . P ("(~V) <br /> <br />(2-12) <br /> <br />where <br /> <br />M = momentum per unit time, pounds (lb) <br />(from pounds-second per second <br />(Ib-sec/sec)) <br /> <br />P = momentum correction coefficient <br /> <br />"( = specific weight of water, pounds per <br />cubic fOOl (pef) <br /> <br />Q = total discharge, cfs <br /> <br />v = averoge channel velocity, feet per <br />second (fps) <br /> <br />g = acceleration of gravity, fl/=.2 <br /> <br />In Equation 2-12 p is generally assumed 10 be equal 10 <br />1.0. Since <br /> <br />Q . AV <br /> <br />(2-13) <br /> <br />Equation 12 can be written <br /> <br />2 <br />M = yQ <br />gA <br /> <br />(2-14) <br /> <br />2.6 <br /> <br />(c) The lotal hydrostatic fOICe m (in pounds) in the <br />channel section can be expressed as <br /> <br />m = i'JA <br /> <br />(2-15) <br /> <br />where y is the distance from the water surface to the <br />center of gravity (centroid) of the flow section. <br /> <br />(d) Combining Equations 14 and 15 results in <br /> <br />2 <br />m+M=i'JA+ ~A <br /> <br />(2-16) <br /> <br />By the momentum principle in an unconstricted channel <br /> <br />2 yQ2 <br />m + yQ .mb+~ <br />a gx;: g"b <br /> <br />(2-17) <br /> <br />where m and mb are the lOlal hydroslatic forces of <br />. , <br />water in the upstream and downstream =.nons, <br />respectively, lb. <br /> <br />(e) Based on experiments under all conditions of <br />open-ch:utnel flow where <br />the channel was constricted by shan. flat surfaces perpen- <br />dicular 10 the flow, such as with bridge piers, Koch and <br />Carstanjen (Koch 1926) found that the upstream momen- <br />tum fOICe had 10 be reduced by (A.jA1)("{i/gAj) 10 <br />balance the IOtaI force in the constriction. <br /> <br />(f) Equating the swnmation of the external fOICes <br />above and below the structures with those within the con. <br />tracted =.tion yields <br /> <br />yQ2 [ (Ap) ( "(Q2 ] <br />ml + gA'i - lAi" 19A'i <br /> <br />(2-18) <br /> <br />+ yQ2 <br />a m2 + mp <br />'?Ai <br />