What is the Parshall flume equation?

For free flow, Q = C·Ha^n, where Q is discharge in ft³/s, Ha is the upstream measuring head in feet, and C and n are empirical constants for each standard throat width. For example a 2-ft flume uses C = 8.00 and n = 1.55, and a 6-inch flume uses C = 2.06 and n = 1.58 (USBR Water Measurement Manual, Table 8-6).

Where is the Parshall flume head measured?

The upstream measuring head Ha is taken in the converging section, about one-third of the way in from the entrance — not in the approach channel. It is referenced to the flume crest (floor of the level converging section). The water surface there is drawn down from the approach channel, so Ha is not the same as upstream channel depth.

When is a Parshall flume submerged?

Discharge at a given Ha is unaffected until the submergence ratio Hb/Ha exceeds 50% for 1-, 2- and 3-inch flumes, 60% for 6- and 9-inch flumes, 70% for 1- to 8-ft flumes, and 80% for 10- to 50-ft flumes (USBR). Above those limits the free-flow equation over-predicts flow and a submergence correction is needed; corrections above 90% submergence are not reliable.

How accurate is a Parshall flume?

A correctly built, correctly set, free-flowing standard Parshall flume measures within about ±3 to ±5 percent, plus head measurement error. Construction tolerance matters more as size decreases, because flume sizes are not geometrically proportional and each is a separately calibrated device.

Can I scale Parshall flume dimensions for a non-standard throat width?

No. Standard Parshall flumes are not geometrically similar — a 12-ft flume is not three times a 4-ft flume in every dimension — so the C and n values apply only to flumes built to the standard dimensions for that throat width. Truncated (Montana) flumes can use the free-flow equation but cannot be corrected for submergence.

Parshall Flume Calculator — Flow from Head

Free-flow discharge through a standard Parshall flume from the upstream measuring head, for every throat width from 1 inch to 50 feet, with an optional downstream head to check whether the flume is still in free flow.

Parshall flume coefficients by throat width

Each standard size was calibrated separately (some of the largest by scale model), so C and n are not a smooth function of W and flumes must be built to the standard dimensions for their size. Coefficients below are USBR Table 8-6; submergence limits are from the same section.

Free-flow rating Q = C·H_aⁿ (ft³/s, ft) and submergence limit
Throat width W	C	n	Q at H_a = 1.0 ft (cfs)	Free-flow limit H_b/H_a
1 in	0.338	1.55	0.338	50%
2 in	0.676	1.55	0.676	50%
3 in	0.992	1.55	0.992	50%
6 in	2.06	1.58	2.06	60%
9 in	3.07	1.53	3.07	60%
1 ft	3.95	1.55	3.95	70%
2 ft	8.00	1.55	8.00	70%
3 ft	12.00	1.57	12.00	70%
4 ft	16.00	1.58	16.00	70%
5 ft	20.00	1.59	20.00	70%
6 ft	24.00	1.59	24.00	70%
7 ft	28.00	1.60	28.00	70%
8 ft	32.00	1.61	32.00	70%
10 ft	39.38	1.60	39.38	80%
12 ft	46.75	1.60	46.75	80%
15 ft	57.81	1.60	57.81	80%
20 ft	76.25	1.60	76.25	80%
25 ft	94.69	1.60	94.69	80%
30 ft	113.13	1.60	113.13	80%
40 ft	150.00	1.60	150.00	80%
50 ft	186.88	1.60	186.88	80%

Source: USBR Water Measurement Manual, 3rd ed., Chapter 8 §10, Table 8-6 and submergence limits. USBR lists the 8-ft flume at the boundary of both the 70% and 80% groups; this calculator uses the more conservative 70%. The 1-in flume passes 0.03 ft³/s at the minimum recommended head of 0.2 ft; below 0.2 ft, head-reading precision and surface tension degrade accuracy for flumes up to 1 ft wide.

Worked examples

Example 1 — 2-ft flume in an irrigation lateral

Given: W = 2 ft, H_a = 1.50 ft, H_b = 0.90 ft.

Q = 8.00 × 1.50^1.55 = 15.00 ft³/s

= 15.00 × 448.83 = 6,732 gpm = 9.69 MGD

S = 0.90/1.50 = 60% < 70% limit

Free flow — Q = 15.0 cfs is valid without correction

Example 2 — Checking the USBR manual's own values

The Water Measurement Manual's worked examples, reproduced from the equation:

3-in flume, H_a = 0.20 ft: Q = 0.992 × 0.20^1.55 = 0.0819 ft³/s (manual: 0.082)

3-ft flume, H_a = 2.10 ft: Q = 12.00 × 2.10^1.57 = 38.5 ft³/s (manual table: 38.4)

20-ft flume, H_a = 3.25 ft: Q = 76.25 × 3.25^1.60 = 502.6 ft³/s (manual table: about 503)

Equation and published tables agree to within 0.3%

Example 3 — When the flume is submerged

Given: W = 2 ft, H_a = 1.50 ft, H_b = 1.20 ft.

S = 1.20/1.50 = 80% > 70% limit

Submerged — the free-flow value of 15.0 cfs overstates the actual flow; apply the USBR submergence correction (Figure 8-16 for 1- to 8-ft flumes) or lower the tailwater

Parshall flume or long-throated flume?

USBR no longer generally recommends Parshall flumes for new installations. Long-throated (ramp or Replogle) flumes are cheaper to build, tolerate submergence up to about 90% with a single head reading, and can be rated by computer from as-built dimensions. Parshall flumes remain common because thousands are already in service and some state water laws and compacts require them. Designing a Parshall flume to operate submerged is poor practice: head-reading error adds 4–20% to the discharge uncertainty on top of the 3–5% free-flow accuracy.

References: U.S. Bureau of Reclamation (2001). Water Measurement Manual, 3rd ed., rev. reprint, Chapter 8, §10 Parshall Flumes. Parshall, R.L. (1950). Measuring Water in Irrigation Channels with Parshall Flumes and Small Weirs. USDA SCS Circular 843. ASTM D1941, Standard Test Method for Open Channel Flow Measurement of Water with the Parshall Flume.