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The application of the Manning equation to partly filled circular channels is considered. The four types of problem that require iterative calculations are elucidated, and explicit solutions proposed in the literature are reviewed. New and improved equations are presented for one type of problem. Explicit analytical equations are developed for the two types of problem for which no explicit solutions could be found in the literature. For each type of problem, two cases are considered: (i) constant Manning roughness coefficient, and (ii) variable Manning roughness coefficient that depends on the depth of flow. Separate equations are presented for each case. The accuracy of each equation is demonstrated by calculating and reporting its maximum deviation from the exact solution within its range of applicability. In addition to obviating the need for iterative calculations, these equations facilitate the calculation of both solutions when the problem at hand has two distinct solutions (two possible flow depths). The proposed equations are accurate enough to be used in computer calculations and sufficiently simple to be used with a hand-held calculator.Résumé : Cet article considère l'application de la formule de Manning à des canaux circulaires partiellement remplis. Les quatre types de problèmes demandant des calculs interactifs sont éclaircis. Des solutions explicites proposées dans la littérature sont examinées. De nouvelles équations améliorées sont proposées pour l'un de ces types de problème. Des équations analytiques explicites sont développées pour les deux types de problème pour lesquels aucune solution explicite n'a été trouvée dans la littérature. Deux cas ont été considérés pour chaque type de problème : (i) le coefficient de rugosité constant de Manning et (ii) le coefficient de rugosité variable de Manning qui dépend de la profondeur de l'écoulement. Des équations séparées sont présentées pour chaque cas. L'exactitude de chaque équation est démontrée en calculant et en rapportant sa déviation maximale de la solution exacte dans sa plage d'application. En plus d'enlever le besoin de faire des calculs itératifs, ces équations facilitent le calcul des deux solutions lorsque le problème possède deux solutions distinctes (deux profondeurs d'écoulement possibles). Les équations proposées sont suffisamment exactes pour être utilisées dans les calculs informatiques et suffisamment simples pour être calculées à l'aide d'une calculatrice de poche.Mots clés : solution explicite, hydraulique, formule de Manning, écoulement à surface libre, coefficient de rugosité, conception des égouts.[Traduit par la Rédaction]

The application of the Manning equation to partially filled circular pipes is considered. Three different approaches based on the Manning equation are analyzed and compared: (1) using a constant value for the roughness coefficient n and defining the hydraulic radius as the flow area divided by the wetted perimeter. (2) Taking the variation of n with the depth of flow into account and employing the same definition of the hydraulic radius. (3) Defining the hydraulic radius as the flow area divided by the sum of the wetted perimeter and one-half of the width of the air-water surface and assuming n is constant. It is shown that the latter two approaches lead to similar predictions when 0.1 # h/D # 1.0. With any one of these approaches, tedious iterative calculations become necessary when diameter (D), slope (S), and flow rate (Q) are given, and one needs to find the depth of flow (h/D) and the velocity (V ). Simple explicit formulas are derived for each of the three approaches. These equations are accurate enough to be used in design and sufficiently simple to be used with a hand calculator.

A new approach to model media expansion during filter backwash is presented. The proposed approach is based on the assumption that the Ergun equation remains valid after fluidization. Mathematical formulas are derived for predicting expanded porosity for a given backwash velocity or backwash velocity for a given expanded porosity. These formulas can be easily used by the engineer. Values predicted using the proposed approach are in good agreement with experimental measurements.