Simple formulae for velocity, depth of flow, and slope calculations in partially filled circular pipes

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 less than or equal to h/D less than or equal to 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.