Hydraulic Radius Calculators in Excel Spreadsheets
Posted on Wednesday, January 19, 2011 at 1:39 PM

 Introduction

The hydraulic radius is used a lot with Manning equation calculations for uniform open channel flow.  Excel spreadsheets can conveniently calculate hydraulic radius for open channel flow through common channel shapes, such as that of a rectangular, triangular, or trapezoidal flume.  This article describes the use of parameters such as rectangle, triangle and trapezoid area and perimeter for determining hydraulic radius.

The definition of hydraulic radius for open channel flow is the cross sectional area of flow divided by the wetted perimeter.  As an equation: R = A/P, where:

  • R is the hydraulic radius (ft for U.S. units, m for S.I. units)
  • A is the cross sectional area of flow (sq ft for U.S. units, sq m for S.I. units)
  • P is the wetted perimeter, that is the part of the cross sectional perimeter that is in contact with flowing liquid (ft for U.S. units, m for S.I. units)

Equations for calculating A, P, and R for common open channel shapes will be covered in the next several sections, followed by information about the use of Excel spreadsheets for the calculations.

Rectangular Channels

The rectangular channel is a common shape for man made channels.  Because of the simple calculations for area and perimeter of a rectangle, the calculation of hydraulic radius is quite straightforward.  The diagram at the left shows a rectangular channel cross section with the channel width represented by b and the depth of flow represented by y.  From the diagram it can be seen that A = by and P - 2y + b.  By substituting, the equation for hydraulic radius for open channel flow through a rectangular channel becomes: 

R = by/(2y + b)

 Trapezoidal Flume

Many man made and natural open channels have a trapezoidal or nearly trapezoidal cross sectional shape.  The diagram at the right shows a trapezoidal flume cross section with the parameters typically used to describe its size and shape and to calculate the trapezoid area and wetted perimeter.  The parameters shown in the diagram are:

  • y, the liquid depth in ft for U.S. & m for S.I. units,
  • b, the bottom width of the channel, ft for U.S. & m for S.I. units,
  • B, the width of the liquid surface, ft for U.S. & m for S.I. units,
  • λ is the wetted length measured along the sloped side (ft for U.S. & m for S.I.)
  • α is the angle of the sloped side from vertical. The side slope also often specified as horiz:vert = z:1.

 The formula typically used for the area of a trapezoid, when applied to the diagram, gives: A = y(b + B)/2.  By using B = b + 2zy, as you can see from the diagram, the trapezoid area becomes: A = (y/2)(b + b + 2zy).  This can be simplified to: A = by + zy2.   , which gives the area of the trapezoid in terms of y, b, and z, parameters that are often known.

An equation for the wetted perimeter is: P = b + 2λ. The sloped length, λ, is typically unknown, but can be eliminated using the Pythagoras Theorem:

 l2 = y2 + (yz)2, or l = [y2 + (yz)2]1/2 .  Thus the wetted perimeter is:

  P = b + 2y(1 + z2)1/2,  and the hydraulic radius for a trapezoid can be calculated from the equation:

R = (by + zy2)/[b + 2y(1 + z2)1/2]

Triangular Flumes

The triangular flume is another shape used in open channel flow.   The diagram at the left shows a triangular flume and the parameters used for its description. The side slope is the same on both sides of the triangle in the diagram, which is the typical situation.  The parameters used for open channel flow calculations with a triangular flume are as follows:

  • B ,the surface width of the liquid  in ft for U.S. & m for S.I. units
  • λ , the sloped length of the triangle side  inft for U.S. & m for S.I. units
  • y,  the liquid depth measured from the vertex of the triangle in ft for U.S. & m for S.I. units
  • z. the side slope, typically specified in the form:  horiz:vert = z:1.

The triangular area is given by A = By/2.  You can see from the figure, however, that B = 2yz, so the area can be simplified to:  A = y2z.  

The wetted perimeter is: P = 2λ , but as with the trapezoidal flume:  l2 = y2 + (yz)2.   

This simplifies to the equation: P = 2[y2(1 + z2)]1/2

The triangular hydraulic radius can thus be calculated from:  R = A/P = y2z/{2[y2(1 + z2)]1/2}

 Hydraulic Radius Calculation with Excel Spreadsheets

 The equations discussed in the last three sections allow you to calculate the hydraulic radius for a rectangular, triangular or trapezoidal flume if the depth of flow and the necessary channel size and shape parameters are known.  An Excel spreadsheet like the one shown in the screenshot below, however, can be used as a very convenient hydraulic radius calculator.  

For Excel spreadsheet, hydraulic radius calculators, for rectangular, triangular, trapezoidal flumes and partially full pipe flow, go to the DOWNLOAD PAGE. under the "Manning Equation/Open Channel Flow" category.