ENVIRONMENTAL ENGINEERING SCIENCE Volume 21, Number 3, 2004 © Mary Ann Liebert, Inc. Simple Formulae for Velocity, Depth of Flow, and Slope Calculations in Partially Filled Circular Pipes Ömer Akgiray * Department of Environmental Engineering Faculty of Engineering Marmara University Istanbul, Turkey ABSTRACT 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 perime- ter. (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 wet- ted 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 for- mulas are derived for each of the three approaches. These equations are accurate enough to be used in de- sign and sufficiently simple to be used with a hand calculator. Key words: hydraulic radius; Manning equation; roughness coefficient; sewer design 371 * Corresponding author: Faculty of Engineering, Department of Environmental Engineering, Marmara University, Goztepe 81040, Istanbul, Turkey. Phone/Fax: (90) 216-3481369; E-mail: akgiray@eng.marmara.edu.tr INTRODUCTION S EWERS ARE COMMONLY DESIGNED to flow full only un- der maximum conditions. In sewer design, therefore, it is necessary to be able to predict the velocity and dis- charge when a sewer is partly filled. Manning’s equation has been the most commonly used formula in sewer de- sign because of its simplicity and the generally satisfac- tory results it has given. In metric units, the Manning equation can be written as follows: V 5 R h 2/3 S 1/2 (1) or Q 5 R h 2/3 S 1/2 (2) where V 5 the velocity (m/s), S 5 the slope of the en- ergy grade line, R h 5 the hydraulic radius defined as the flow area divided by the wetted perimeter (m), A 5 the cross-sectional area of flow (m 2 ), Q 5 the discharge (m 3 /s), and n 5 Manning’s coefficient of roughness. The A } n 1 } n