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<br />- 78 - <br /> <br />N dM <br />o dt <br /> <br />c <br />= f (Y d - Ym)W <br /> <br />(S.79) <br /> <br />From eqns. (3.72) and (3.79) <br /> <br />dr <br />dt <br /> <br />c w <br />= N L~ b-l (Yd - Ym) <br />o I' <br /> <br />(S.80) <br /> <br />This is integrated to give <br /> <br />r{t + ~t) = [r{t)b + Cp(Yd - Ym) ;~~a ]l/b <br />o <br /> <br />(S.8l) <br /> <br />The total vertical velocity of the particles is <br /> <br />dz <br />= <br />dt <br /> <br />w - V <br />C <br /> <br />(3.82) <br /> <br />and use of eqn. (S.73) gives <br /> <br />dz <br />dr <br /> <br />d <br />= w-cr <br />dr/dt <br /> <br />(S.8S) <br /> <br />Since dr/dt is a known function of r (eqn. 3.80) this may be integrated to <br /> <br />yield <br /> <br />z(t.+ At) <br /> <br />abcNoL b+d b+d <br />= z{t) + w~t - (b+d)c (Y _ y}w [r(t + At) - r(t) ] <br />p d m <br /> <br />The horizontal position is simply <br /> <br />(3.84) <br /> <br />I~ <br />I <br />I. <br />I <br />I <br />1 <br />I <br />I. <br />I <br />,~ <br />I <br />I' <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />