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Parshall Flume Design Calculator

Flow rate equals 4 times throat width times upstream depth raised to 1.522 times throat width raised to 0.026

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Parshall Flume Flow Equation

A Parshall flume uses a converging-diverging channel shape to create a predictable relationship between upstream water depth and discharge. The standardized geometry means no individual calibration is needed. This calculator applies the general equation for throat widths between 1 and 8 feet under free-flow conditions.

Q = 4B × h₁¹˙⁵²² × B⁰˙⁰²⁶

How It Works

A Parshall flume is a fixed hydraulic structure with a converging inlet section, a narrow throat, and a diverging outlet. Water accelerates through the constriction, creating a critical-flow condition that produces a unique, predictable relationship between upstream water depth and volumetric flow rate. The general equation Q = 4 × B^1.026 × h₁^1.522 applies to standard flumes with throat widths between 1 and 8 feet under free-flow (non-submerged) conditions.

Example Problem

A Parshall flume with a 2 ft throat width measures an upstream depth of 1.5 ft. What is the flow rate?

  1. Identify the known values: throat width B = 2 ft and upstream depth h₁ = 1.5 ft.
  2. Determine what we are solving for: the volumetric flow rate Q in ft³/s.
  3. Write the Parshall flume equation: Q = 4 × B^1.026 × h₁^1.522.
  4. Compute the throat width term: B^1.026 = 2^1.026 = 2.036.
  5. Compute the depth term: h₁^1.522 = 1.5^1.522 = 1.855.
  6. Multiply all terms: Q = 4 × 2.036 × 1.855 ≈ 15.1 ft³/s. This is approximately 6,770 gallons per minute.

When to Use Each Variable

  • Solve for Flow Rate — when you know the throat width and upstream depth measurement.
  • Solve for Throat Width — when designing a new flume to handle a target flow rate with a given depth.
  • Solve for Upstream Depth — when you need to predict the water level for a given flow and flume size.

Key Concepts

A Parshall flume is a fixed hydraulic structure with a converging inlet, a narrow throat, and a diverging outlet. The geometry creates a critical-flow condition in the throat, producing a unique relationship between upstream depth (h₁) and discharge. Under free-flow conditions, only the upstream depth measurement is needed; under submerged conditions, both upstream and downstream depths are required with a correction factor.

Applications

  • Wastewater treatment: measuring influent and effluent flows for regulatory compliance reporting
  • Irrigation: monitoring water deliveries from canals and ditches to individual farms
  • Industrial discharge: measuring and documenting process water outflows per EPA permit requirements
  • Hydrology: gauging stream and river flows at permanent monitoring stations
  • Stormwater management: monitoring outfall flows during storm events for NPDES permit compliance

Common Mistakes

  • Using the calculator under submerged-flow conditions without applying the submergence correction — submergence reduces flow for a given upstream depth, and ignoring it overestimates discharge
  • Measuring upstream depth at the wrong location — the measurement must be taken at the specified gauge point (typically 2/3 of the converging section length upstream from the throat)
  • Applying the general equation outside its valid throat-width range — the exponent and coefficient change for very small (< 1 ft) or very large (> 8 ft) flumes

Frequently Asked Questions

Why are Parshall flumes preferred for measuring open-channel flow?

Parshall flumes require no moving parts, are self-cleaning (the converging shape flushes sediment), need no individual calibration (standardized geometry), and maintain accuracy of ±3–5% under free-flow conditions. They also create less head loss than sharp-crested weirs, making them suitable for flat-gradient channels where backwater is a concern.

How do you size a Parshall flume for your channel?

Start with the maximum expected flow rate and an acceptable upstream depth (typically 30–70% of the converging section depth). Use the equation Q = 4 × B^1.026 × h^1.522 to find the throat width B that accommodates both the maximum and minimum expected flows. Standard sizes (1, 2, 3, 4, 6, 8 ft) are available — choose the closest standard width.

What is the difference between free-flow and submerged flow?

Free-flow means the downstream water level does not affect the upstream reading. Submerged flow occurs when tailwater backs up into the throat, requiring a correction factor. Most flumes are designed for free-flow conditions. Submergence begins when the downstream-to-upstream depth ratio exceeds about 0.7.

How accurate is a Parshall flume?

Under free-flow conditions with proper installation, Parshall flumes achieve accuracy of ±3–5%. Errors increase with submergence, improper approach conditions, or sediment buildup. Careful attention to the gauge location and level installation are critical to maintaining accuracy.

What is the formula for Parshall flume flow rate?

The general free-flow equation is Q = 4 × B^1.026 × h₁^1.522, where Q is flow rate in ft³/s, B is throat width in feet, and h₁ is upstream depth in feet. This equation applies to standard flumes with throat widths between 1 and 8 feet. Smaller and larger flumes use different coefficients and exponents.

Can a Parshall flume measure flow in both directions?

No, Parshall flumes are designed for unidirectional flow. The converging-diverging geometry only creates the critical-flow condition when water flows in the correct direction (from the wide converging inlet through the narrow throat). Reverse flow produces unreliable readings.

How do you install a Parshall flume?

The flume must be installed level across its width (no lateral tilt) with the correct longitudinal slope. The approach channel should be straight for at least 10 channel widths upstream, with uniform flow and no turbulence. The gauge (stilling well or ultrasonic sensor) is placed at the standard upstream measurement point, typically 2/3 of the converging length from the throat.

Parshall Flume Flow Formula

The general free-flow equation for Parshall flumes with throat widths between 1 and 8 feet:

Q = 4 × B1.026 × h11.522

Where:

  • Q — volumetric flow rate, in cubic feet per second (ft³/s)
  • B — throat width of the flume, in feet (ft)
  • h1 — upstream water depth (head), in feet (ft), measured at the standard gauge location

The exponents (1.026 and 1.522) are empirical, derived from extensive laboratory and field calibration. This equation applies only under free-flow (non-submerged) conditions.

Worked Examples

Wastewater Treatment

What is the influent flow rate at a treatment plant with a 3 ft flume?

A municipal wastewater treatment plant uses a 3 ft wide Parshall flume. The upstream depth gauge reads 2.0 ft.

  • Q = 4 × 31.026 × 2.01.522
  • Q = 4 × 3.087 × 2.879
  • Q ≈ 35.6 ft³/s

That is about 16,000 gallons per minute — a typical daily peak flow for a medium-sized plant. The operator records this reading hourly for compliance reporting.

Irrigation

What throat width do I need for a 20 ft³/s irrigation canal?

An irrigation district needs to measure up to 20 ft³/s in a canal. The maximum expected upstream depth is 1.8 ft. What throat width is required?

  • B1.026 = Q / (4 × h1.522)
  • B1.026 = 20 / (4 × 1.81.522) = 20 / (4 × 2.459)
  • B1.026 = 2.034
  • B ≈ 1.98 ft

A standard 2 ft Parshall flume fits this application. The closest standard size is always chosen since the empirical coefficients are calibrated to standard widths.

Stormwater

What upstream depth will a 4 ft flume show during a 50 ft³/s storm event?

A stormwater outfall is monitored with a 4 ft Parshall flume. During peak flow of 50 ft³/s, what upstream depth should the gauge read?

  • h1 = (Q / (4 × B1.026))1/1.522
  • h1 = (50 / (4 × 4.072))0.6571
  • h1 = (3.069)0.6571
  • h1 ≈ 2.14 ft

The operator can set a high-water alarm at 2.1 ft to flag peak storm events. If h exceeds about 70% of the converging section depth, check for submergence effects.

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