# Parshall Flume Equations for Open Channel Flow Rate Calculations

## Introduction

Parshall flumes are widely used for open channel flow measurement, especially for flows containing suspended solids, as for example the flow in wastewater treatment plants. A Parshall flume has a converging section, throat, and diverging section, somewhat like a venturi flume. It also has prescribed variations in the channel bottom slope as shown in the diagram in the next section. Parshall flume equations are available for flow rate calculations based on a measured head, for a Parshall flume constructed with prescribed dimensions as discussed in the next section.

## Parshall Flume Configuration and Dimensions

The size of a Parshall flume is designated by its throat width. Dimensions are available for Parshall flume throat

widths from 1 inch to 50 ft. The 1 inch flume will carry a flow of 0.03 cfs at 0.2 ft of head, while a 50 ft Parshall flume will carry 3,000 cfs at a head of 5.7 ft. The diagram at the left shows the overall configuration of a Parshall flume. The picture at the right shows a small Parshall flume in operation for open channel flow measurement.

Standard dimensions for Parshall flumes with throat widths from 1 ft to 8 ft are shown in the table below the figure at the left.

Image Credit: Montezuma Valley Irrigation Company

For the range of throat widths in the table, the other dimensions in the diagram are constant at the following values:

E = 3'-0", F = 2'-0", G = 3'-0", K = 3 inches, N = 9 inches, X = 2 inches, Y = 3 inches.

## Free Flow and Submerged flow in a Parshall Flume

The flow through a Parshall flume is said to be “free flow” when the flow rate through the throat of the flume is not affected by the downstream flow. When this is the case, a hydraulic jump should be visible in the Parshall flume throat. If the downstream level is high enough so that flow is backed up into the throat, the hydraulic jump isn’t visible, and the flow is said to be submerged flow. A quantitative criterion to differentiate between free flow and submerged flow uses head measurements at two locations, Ha and Hb as shown in the diagram at the left. That criterion for several ranges of throat width is summarized here:

For 1” < W < 3” : free flow for Hb/Ha < 0.5; submerged flow for Hb/Ha > 0.5

For 6” < W < 9” : free flow for Hb/Ha < 0.6; submerged flow for Hb/Ha > 0.6

For 1’ < W < 8’ : free flow for Hb/Ha < 0.7; submerged flow for Hb/Ha > 0.7

For 8’ < W < 50’ : free flow for Hb/Ha < 0.8; submerged flow for Hb/Ha > 0.8

## The Parshall Flume Free Flow Equation

The Parshall flume free flow equation is Q = CHan, where Q is the flow rate through

the flume in cfs, Ha is the head over the flume in ft (measured at the point shown for Ha in the diagram above), and C and n are constants for a Parshall Flume of given throat width. Values for C and n are given in the table at the right for selected throat widths ranging from 1 in to 50 ft.

Graphs and tables are available for determination of the flow rate through a Parshall flume under submerged flow conditions.

Also for Excel spreadsheet templates that can be downloaded to make flow rate calculations for Parshall flumes for both free flow and submerged conditions, with either U.S. or S.I. units, see the article, “Excel Formulas for Calculations with Parshall Flume Equations.”

## Example Calculation

**Problem Statement:** Consider a 6 ft Parshall flume operating under free flow condition with a head, Ha = 3.5 ft. What is the flow rate through this Parshall flume?

**Solution:** For a Parshall flume with 6 ft throat width, C = 24.00 and n = 1.59, from the table above. The Parshall flume free flow equation can then be used:

Q = CHan = (24)(3.51.59) = **175.9 cfs**

## References

References for further information:

1. U.S. EPA, *Recommended Practice for the Use of Parshall Flume and Palmer Bowlus Flumes in Wastewater Treatment plants,* EPA600/2-84-180, 1984.

2. Wahl, Tony L., *Equations for Computing Submerged Flow in Parshall Flumes,* Bureau of Reclamation, Denver, Colorado, USA.

3. U.S. Dept. of the Interior, Bureau of Reclamation, *Water Measurement Manual*, 2001 revised, 1997 third edition.

## This post is part of the series: Open Channel Flow Measurement

Open channel flow rate measurement is usually done by measuring a change in water depth. It can be done with a weir or flume. Common types are the sharp crested weir (including V-notch weir, rectangular weir, and cipolletti weir), the broad crested weir, the Parshall flume and venturi flume.