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<br />20 <br /> <br />gradually-varied water and sediment flows, as well as width changes, of <br />erodible channels. The principal assumptions incorporated into this model are <br />as follows: <br />(1) Flow is one dimensional, and hydrostatic pressure prevails at any <br />point in the channel. <br />(2) Channels lope is sma 11. <br />(3) The Manni ng equat i on and the sediment-transport fonnul a are <br />applicable to gradually-varied flow. <br />(4) Storage effect due to unsteady flow is negligible in the backwater <br />computation. <br /> <br />The following basic equations are employed: <br /> <br />(1) Flow-continuity equation: <br /> <br />an aA <br />=+--q=O <br />ax at <br /> <br />.... (2-26) <br /> <br />(2) Sediment-continuity equation: <br /> <br />aA aQ <br />(1 -A) ~ + ..2 - q = 0 <br />at ax s <br /> <br />.... (2-27) <br /> <br />(3) Flow-momentum equation: <br /> <br />2 <br />g ~~ + * ~ + * ~x (f) + gS - -\ q = 0 <br />A <br /> <br />.. .. (2-28) <br /> <br />where <br />Q & Qs = water and sediment discharges <br />A = cross-section area of flow <br />Ac = channel cross-section area within some reference frame <br />q = lateral water inflow <br />q = lateral sediment inflow <br />s <br />H = water-surface elevation <br />S = energy slope <br />A = porosity of bed material <br />