Submerged and Free Flow

Calculations, Equations, Graphs

Based on ISO 9826 methodology

Parshall Flume

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Parshall flume diagrams

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Links on this page:  Equations  Comments about graphs 
Comparison of ASTM D1941 and ISO 9826  Variables   Units   Error Messages    References

Our Parshall flume calculation is based on the ISO 9826 (1992) standard.  The standard is valid for submerged as well as free-flowing Parshall flumes. A free flowing flume can be identified by the drop in water depth at the flume throat.  In submerged flow, the downstream water backs up into the throat swallowing the drop making the drop difficult or impossible to identify.  Analysis of submerged flow requires two head measurements - one in the approach channel and one in the throat.  Whereas, free flow requires only the upstream head measurement.

Parshall flumes must be built with their dimensions in strict accordance with specifications in published documents such as the ISO 9826 and ASTM D1941 (1991) standards or USBR (1997).  Flumes (like weirs) are designed to force a transition from sub-critical to super-critical flow. In the case of Parshall flumes, the transition is caused by designing flumes to have a narrowing at the throat and a drop in the channel bottom.  Such a transition causes flow to pass through critical depth in the flume throat. At the critical depth, energy is minimized and there is a direct relationship between water depth and velocity (and flowrate). However, it is physically very difficult to measure critical depth in a flume because its exact location is difficult to determine and may vary with flowrate. Through mass conservation, the upstream depth is related to the critical depth. Therefore, flowrate can be determined by measuring the upstream depth, which is a highly reliable measurement.  Like all of LMNO Engineering's software, our Parshall flume calculation was written in double precision.

Equations and Methodology             Back to calculation
The methodology for our Parshall flume calculation follows that of ISO 9826 (1992).  ASTM D 1941 (1991) also addresses Parshall flumes but has pages of tabular data which are more difficult to implement into a computer program compared to the figures and equations of ISO 9826.

Variable definitions can be found in the Variables section.

LMNO Engineering calculation allows 0<=h<=3 m and 0.01<b<16 m, but the calculation is most accurate when used within the ISO 9826 recommendations of h<=2 m and 0.152 <= b <= 15.24 m.
For b<=0.152 m, C and n values are from Herschy (1995).  For b>0.152 m, C and n values are from ISO 9826 (1992), equations 10 and 11.

Free flow
Free flow occurs when a hydraulic jump is visible at the throat; that is, when the downstream head is significantly less than the upstream head.  Our calculation defines free flow as occurring when H/h<=0.6 for b<3.05 m or when H/h<=0.8 for b>=3.05 m.  These criteria (called "modular limits") are similar, but not identical, to the ISO 9826 criteria.

For free flow,
Q = C hn    where Q is in m3/s and h is in m

Parshall flume coefficients

Submerged flow
Submerged flow occurs when a hydraulic jump is not visible at the throat; that is, when the downstream head is sufficiently high that it "drowns out" or "swallows up" the hydraulic jump.  In our calculation, submerged flow occurs when H/h>0.6 for b<3.05 m and when H/h>0.8 for b>=3.05 m.

For submerged flow, ISO equation 12 is re-written as:
Q = C hn - Qe    where C and n are found from the above figure based on width (b).  Qe accounts for the effects of submergence.

For b<3.05 m, ISO 9826 equation 13 provides the following equation for Qe which is implemented in our calculation:

Qe equation

For b>=3.05 m, Qe=CsQ3
Cs =(0.3281)b.  ISO 9826 Figure 2 is a graph of Q3 as a function of  H/h and h.  LMNO Engineering fit equations to all of the lines in the ISO figure for use in our calculation.  A graph using our equations for selected H/h ratios is shown:

Q3 graph

Comments about the Graphs                                          Back to calculation
When the graph "Q vs. H/h (h constant)" is selected, you may notice a discontinuity in the graph at H/h=0.6 for b<3.05 m or H/h=0.8 for b>=3.05 m.   The discontinuity occurs because the flow is transitioning from free flow to submerged flow.  For some ranges of H/h, you may observe something similar to the following:

Graph of Q vs. H/h, for b=0.5 m and h=0.5 mQ vs. H/h

The discharge for H/h<0.6 does not appear to be shown.  However, our software scales the y-axis based on the maximum value on the curve.  In this case, the maximum value on the curve is 0.398 m3/s which blends in with the boxed outline of the graph.  The flow is 0.398 m3/s for all H/h <0.6.

Comparison of ISO and ASTM methods             Back to calculation
Both ISO 9826 and ASTM D1941 present methods for computing discharge through Parshall flumes.  Since a Parshall flume is a standard flume, the two methods should be similar.  Looking at the two standards, they at first appear somewhat different because ISO 9826 is in SI units while ASTM D1941 uses predominantly English units.  Further, the ISO standard presents information in tables, equations, and figures while ASTM D1941 uses tables almost exclusively.  Curious as to how well the two methods compare, we developed the following table.  We also have done other comparisons but, due to space considerations, have chosen not to show the results on this web page.  In general, there is good agreement for free flow, but in some cases the submerged flowrates are significantly different, such as for the b=0.152 case shown below.

Comparison of ASTM D1941 and ISO 9826
Width, b Upstream Head, h Flow Regime ASTM Flow, Q ISO Flow, Q
0.152 m
(6 inch)
0.24 m
(0.7874 ft)
Free Flow 0.0400 m3/s(1) 0.0408 m3/s(2)
Submerged, H/h=0.84 0.0309 m3/s(3) 0.0237 m3/s(4)
7.62 m
(25 ft)
0.6096 m
(2.0 ft)
Free Flow 8.13 m3/s(1) 8.13 m3/s(2)
Submerged, H/h=0.9 7.34 m3/s(5) 7.36 m3/s(6)
(1)  based on Table 2 in ASTM D1941.  (2)  based on Equation 10 in ISO 9826.
(3)  based on Tables 2 and 6B in ASTM D1941.
(4)  based on Equations 10, 12, and 13 in ISO 9826.
(5)  based on Tables 2, 9A, and 9C in ASTM D1941.
(6)  based on Equations 11, 12, and 14 and Figure 2 in ISO 9826.

            Back to calculation
ISO 9826 specifies the indicated units for the equations shown above.  Our calculation allows you to specify a variety of units.
m=meters, s=seconds
b=Throat width [m].
C=Parshall flume constant [empirical units].
Cs=Submergence coefficient [unit-less].  This is not the submergence ratio.
h=Measured upstream head [m].
H=Measured downstream head [m].  Only needed if the flume is submerged, or to determine mathematically if the flume is submerged.  Usually, one can visually see if there is a hydraulic jump, but determining the ratio H/h is a quantitative method.
H/h=Submergence ratio.  Flume is submerged if H/h>0.6 for b<3.05m or if H/h>0.8 for b>=3.05 m.
n=Parshall flume power constant [unit-less].
Q=Flowrate (discharge) through flume [m3/s].
Qe=Reduction in flowrate due to submergence [m3/s].
Q3=Flow factor for determining Qe if b>=3.05 m [m3/s].

Units             Back to calculation
Most units are self-explanatory.  MGD is millions of gallons (US) per day.

Error Messages given by calculation               Back to calculation
Input checks in top half of calculation.  If one of these messages appears, the calculation and graphing is halted.
"Must have h >= 0".  Negative head was entered.  If h=0, calculation automatically sets Q=0.
"h must be < 3 m".  Head cannot exceed 3 m.
"Must have H > 0".  For submerged flow, downstream head must be positive.
"Must have H/h > 0", "Must have H/h<0.99".  If submerged flow is selected, H/h must be in this range.  However, if b<3.05 m and H/h<=0.6, flow is considered to be free flow and will be computed using Q=C hn.   Likewise, if b>=3.05 m and H/h<=0.8, flow is considered to be free flow.
"Must have b > 0".  Throat width must be positive.
"Must have 0.01<b<16 m".  Flume throat width must be in this range.

Input checks for graph.  If one of these messages appears, the graph will not proceed.  Note that if any value is out of range in the upper portion of the calculation, a graph will not be shown.
"Min h must be >=0", "Min H/h must be >=0".  Minimum head or H/h ratio entered for graph cannot be < 0.
"Max h must be <= 3 m".  Graph has maximum limit of h=3 m.
"Must have 0.6<Max H/h<=1".  Message will only appear if b<3.05 m.  Q is independent of H/h until H/h becomes greater than 0.6.  So to avoid a "boring" graph showing a constant value for Q, the calculation requires that max H/h be in the range where submergence affects Q.
"Must have 0.8<Max H/h<=1".  Message will only appear if b>=3.05 m.  Q is independent of H/h until H/h becomes greater than 0.8.  So to avoid a "boring" graph showing a constant value for Q, the calculation requires that max H/h be in the range where submergence affects Q.
"Min/Max must be < 0.99".  Minimum value entered for graph must be less than 0.99 times maximum value entered for graph. Otherwise, the minimum and maximum are too close together to have good axis labels for the graph.
"Min must be < Max".  Minimum value entered for graph must be less than maximum value entered for graph.

Run-time errors.  The following message may be generated by the graphing portion of the calculation:
"No flow: H/h too low".  Will occur if all of the flowrates for the graph are 0.0.  For example, if b=0.5 m, H/h=0.9, and a head range of 0 to 0.05 m is entered for the graph, no water will flow through the flume since there is insufficient head difference.  All of the Q's will be 0.0, so the message is shown.  For these b and H/h values, h must be greater than 0.118 m before there will be flow.

References               Back to calculation
American Society for Testing and Materials (ASTM D1941-91).  1991.  Standard test method for open channel flow measurement of water with the Parshall flume.   Available at

Herschy, Reginald W.  1995.  Streamflow Measurement.  E & FN Spon (an imprint of Chapman and Hall).  2ed.

International Organization of Standards (ISO 9826).  1992.  Measurement of liquid flow in open channels - Parshall and SANIIRI flumes.  Reference number: ISO 9826:1992(E).  ISO documents can be downloaded as .PDF files for a fee at

USBR.  1997.  U.S. Department of the Interior, Bureau of Reclamation.  Water Measurement Manual.  1997.  3ed.  Available from .

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LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd.   Athens, Ohio  45701   USA   (740) 592-1890

(March 17, 2010: thickened graph line for better visibility)