FLUME GRAPHING CALCULATOR SOFTWARE Parshall, Trapezoidal, Rectangular, and U flume Plus: Equations, Formulas, References

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Links on this page:  Equations   Variables   Units   Error Messages    References

Introduction
Flumes are used to measure flowrate (discharge) in open channels.  They typically have widths from a few cm to 15 m or so.  The water depth in the approach section of flumes typically can be between a few cm and about 2 m.  Flumes, compared to weirs, have the advantage of less head loss through the device, yet are more complicated to construct and more difficult to analyze.

Head is measured in the flume upstream of the throat - in the so-called "approach channel".  For Parshall flumes, head is measured upstream from the throat a distance of 2/3 of the length of the approach channel (x=length of approach channel in the above diagram).  For the other three flumes, head is measured upstream from the throat a distance of 3 to 4 times the maximum expected head.  This location is somewhat arbitrary because the head does not vary too much with position, so the exact location of the head measurement is not as important as for a Parshall flume.  Since the rectangular, trapezoidal, and U flumes can have a raised throat (a hump), it is important to note that head is measured from the top of the hump rather than from the bottom of the approach channel.

This web page has calculations for four types of flumes - Parshall, rectangular, trapezoidal, and U shape.  Each has advantages and disadvantages related to construction, installation, head measurement, sedimentation, and analysis.  Parshall flumes are the most common.  They were studied extensively in the mid 1900s.  Their analysis is well documented in many texts.  Their analytical maturity is exemplified by having both ASTM and ISO standards written for them (ASTM, 1991; ISO, 1992).  Recently, Parshall flumes have gone out of favor due to their construction complexity and likelihood to trap sediment compared to newer flume designs.

Rectangular and trapezoidal flumes function by having a constriction at the throat and/or a raised invert (bottom) at the throat.  Either feature can cause critical flow at the throat in a properly operating flume.  These flumes are simpler to construct, can be more easily fit into an existing channel, and can trap less sediment than a Parshall flume.  However, the methodology relating discharge to measured head is more complex.

U-flumes, similar to Palmer-Bowles flumes but with a semi-circular throat, are ideal for use in culverts or pipes.  Critical flow is achieved by narrowing the throat or by raising the bottom of the flume at the throat.  Analysis of U flumes is similar to that of the trapezoidal flume.

All flumes must be built with their dimensions in strict accordance with specifications in published documents such as the ISO and ASTM standards.  Otherwise, discharge analysis must be conducted for the specific flume beginning with theory and proceeding to experimentation to modify the theory by physical observations.

Regarding analysis of flumes, flumes (like weirs) are designed to force a transition from sub-critical to super-critical flow. In the case of flumes, the transition is caused by designing flumes to have a narrowing at the throat, raising of the channel bottom, or both.  Such a transition causes flow to pass through critical depth at 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.

The LMNO Engineering flume calculations are based on ASTM and ISO standards for flumes. These standards were developed from theoretical relationships and modified by experimental observations conducted over several decades.  Our calculations are valid only for unsubmerged flows.  An unsubmerged flow 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.

Like all of LMNO Engineering's software, our flume calculation was written in double precision.

Equations and Methodology
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The methodology for the flume calculations follows that of ISO 9826 (1992) for the Parshall flume and ISO 4359 (1983, 1999) for the rectangular, trapezoidal, and U flumes.   There are other sources of equations and methodology for flumes such as ASTM D 1941 (1991) for Parshall flumes and USBR (1997) for other flumes.  Also, Herschy (1995) offers essentially a re-compendium of the ISO equations.  Many other textbooks on open channel flow discuss flumes, but not in the degree of detail of the ISO or ASTM standards.

LMNO Engineering decided to follow the ISO methodology for the four flume types.  ISO presents the methodology as a series of equations and graphs which have been agreed upon by an international panel after decades of research involving theory and experimentation.  The ISO standards explain the validity and accuracy of their methodology.  For Parshall flumes, the ASTM and ISO methods present similar, but not identical, methods.  We have selected to follow the ISO method because it presents equations that are valid for a wider range of flume sizes.

Variable definitions can be found in the Variables section.

Parshall Flume (ISO 9826, 1992)
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).

Q = C hn    where Q is in m3/s and h is in m

Rectangular Flume
LMNO Engineering calculation allows 0<=h<=3 m, 0.01<b<16 m, b<B<104 m, 0<=P<3 m, 0<L<104 m, and F<1.  Note that if P=0, then B must be > b.  Likewise, if B=b, then P must be > 0.  The calculation is most accurate when used within the ISO 4359 recommendations of  h<=2 m, 0.1m<=b<=B, F<=0.5, h/b<=3, (bh)/[B(P+h)]<=0.7, h/L<=0.5, and h>=0.05 or h>=0.05 L (whichever is greater).

Order of computations (ISO 4359, 1983):

Cv can only be computed if  hbCd/A<0.93.

Trapezoidal Flume
LMNO Engineering calculation allows 0<=h<=3 m, 0.01<b<16 m, b<B<104 m, 0<=P<3 m, 0<L<104 m, F<1, and 0<m<100, and 0<M<100.  The calculation is most accurate within the ISO 4359 recommendations of h<=2 m, 0.1m<=b<B, F<=0.5, h/L<=0.5, and h>=0.05 or h>=0.05 L (whichever is greater).

Order of computations (ISO 4359, 1983):

Let H=h and obtain Cs from the graph below.  Note that the graph is only valid for 0.02 < mH/b < 5.

Then, Cv from numerical solution of:

Cv can only be computed if  hbCs/A<0.93.

Since Cs and Cv are functions of both H and h, re-compute H=h Cv2/3, Cs, Cv, and Q.  ISO 4359 suggests re-computing Q three times, but we re-compute Q until there are at least four significant digits of accuracy.  Then, V and F are computed from the final Q.

U-Flume
LMNO Engineering calculation allows 0<=h<=3 m, 0.01<b<16 m, d<D<104 m, 0<=P<3 m, 0<L<104 m, and F<1.  The calculation is most accurate within the ISO 4359 recommendations of h<=2 m, 0.1m<=d<D, F<=0.5, h/L<=0.5, and h>=0.05 or h>=0.05 L (whichever is greater).

Order of computations (ISO 4359, 1983):

Let H=h and obtain Cu from the graph below.  Note that the graph is only valid for  0.1< H/d < 3.

Then, Cv from numerical solution of:

Cv can only be computed if  hdCu/A<0.93.

Since Cu and Cv are functions of both H and h, re-compute H=h Cv2/3, Cu, Cv, and Q.  ISO 4359 suggests re-computing Q three times, but we re-compute Q until there are at least four significant digits of accuracy.  Then, V and F are computed from the final Q.

Variables
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ISO 4359 and ISO 9826 specify the indicated units for the equations shown above.  Our calculation allows you to specify a variety of units.
m=meters, s=seconds
A=Cross-sectional area of approach channel [m2].
b=Bottom width of flume throat [m].
B=Bottom width of approach channel [m].
C=Parshall flume constant [empirical units].
Cd=Coefficient of discharge for rectangular, trapezoidal, and U flumes [unit-less].
Cs=Shape coefficient for trapezoidal flume [unit-less].
Cu=Shape coefficient for U flume [unit-less].
Cv=Coefficient of approach velocity for rectangular, trapezoidal, and U flumes [unit-less].
d=Diameter of throat of U flume [m].
D=Diameter of approach channel of U flume [m].
F=Froude number of flow in approach channel [unit-less].  F<1 is slow or sub-critical flow.  F>1 is fast or super-critical flow.
g=Acceleration due to gravity, 9.8066 m/s2.
h=Measured head [m].  If there is a hump, then it is the vertical distance between the top of the hump and the water surface.
H=Total head [m].  Measured head plus velocity head.  H=h Cv2/3
k=Constant used in trapezoidal flume computation [unit-less].
L=Length of flume throat [m].
m=Side slope of trapezoidal flume throat.  Horizontal to vertical (H:V).
M=Side slope of trapezoidal flume approach channel.  Horizontal to vertical (H:V).
n=Parshall flume power constant [unit-less].
P=Hump height [m].
Q=Flowrate through flume [m3/s].
T=Top width of approach channel [m].
V=Velocity in approach channel [m/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 appear, the calculation and graphing is halted.
"Must have h >= 0".  Negative head was entered.
"h must be < 3 m".  Head cannot exceed 3 m.
"Must have D, d > 0".  For the U flume, upstream diameter or throat diameter was entered as <= 0.
"Must have D > d".  For the U flume, upstream diameter must be greater than throat diameter.
"Must have b > 0".  Parshall, rectangular, and trapezoidal flumes must have throat width > 0.
"Must have B > 0".  Rectangular and trapezoidal flumes must have approach width > 0.
"Must have B > b".  Trapezoidal flume must have approach width greater than throat width.
"Must have B >= b".  Rectangular flume must have approach width at least equal to throat width.
"Must have 0.01<b<16 m".  Rectangular and trapezoidal flume throat width must be in this range.
"Must have B < 10000 m".  Rectangular and trapezoidal flume approach width must be in this range.
"Must have 0.01<d<16 m".  U flume throat diameter must be in this range.
"Must have D < 10000 m".  U flume approach diameter must be in this range.
"P must be >0 if B=b".  Rectangular flume must have P>0 if B=b or B>b if P=0.  Of course, it can have both P>0 and B>b.
"Must have P >= 0".  Hump height cannot be negative (P does not apply to Parshall flumes).
"Must have P < 3 m".  Hump height cannot exceed 3 m (P does not apply to Parshall flumes).
"Must have L > 0".  Throat length must be > 0 (L does not apply to Parshall flumes).
"Must have L < 10000 m".  This is an LMNO Engineering upper limit.  The ISO standards specify throat lengths for each flume type as a function of expected flowrate.  Rarely will a flume have a throat length longer than several meters.
"Must have 0< M,m < 100".  For the trapezoidal flume approach channel side slopes (M) and throat side slopes (m) must both be between 0 and 100, measured as horizontal to vertical slope.

Input checks for graph.  If one of these messages appear, the graph will not proceed.  Note that if h is out of range in the upper portion of the calculation, a graph will not be shown.
"Min h must be >=0".  Minimum head entered for graph cannot be < 0.
"Min must be < Max".  Minimum head entered for graph must be less than maximum head entered for graph.
"Min/Max must be < 0.99".  Minimum head entered for graph must be less than 0.99 times maximum head entered for graph. Otherwise, the minimum and maximum heads are too close together to have good axis labels for the graph.
"Max h must be <= 3 m".  Graph has maximum limit of h=3 m.

Run-time errors.  The following messages may be generated during execution and will halt execution:
The following messages are triggered by the head entered in the upper half of the calculation.  Its value, in conjunction with the other values entered, may cause Cd, Cs, or Cu to be out of range.  However, since the graph is plotted for a range of heads, the graph may still be shown with the un-computable flowrates plotted at Q=0.

Run-time errors in upper half of calculation:
"Cd will be <0", "Must have (Cd)hb/A <0.93", "Must have 0.02< mH/b <5", "Must have (Cs)hb/A <0.93", "Must have 0.1< H/d <3", "Must have (Cu)hd/A <0.93".  These variables or ratios are out of the acceptable ranges (see Equations and Methodology above for further explanation).
"Infeasible input."  May occur in numerical routine for computing Cv if a combination of variables approaches machine precision.  This message very rarely, if ever, will occur since extreme values are screened out with the input checks.

Run-time errors for graph:
"No Q's computed".  All of the heads in the range entered for the graph result in parameters (such as Cd, Cs, or Cu) that are invalid, so no  flowrates could be computed.  Try lowering the minimum head and/or increasing the maximum head.
"Some Q's cannot be graphed".  Some heads in the range entered for the graph result in parameters (such as Cd, Cs, or Cu) that are invalid, so those could not be computed and were plotted at Q=0.

References               Back to calculation
American Society for Testing and Materials (ASTM D 1941-91).  1991.  Standard test method for open channel flow measurement of water with the Parshall flume.   Available at http://www.astm.org.

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

International Organization of Standards (ISO 4359).  1983.  Liquid flow measurement in open channels - Rectangular, trapezoidal, and U-shaped flumes.  Reference number: ISO 4359-1983(E).

International Organization of Standards (ISO 4359).  1999.  Technical Corrigendum 1 for:  Liquid flow measurement in open channels - Rectangular, trapezoidal, and U-shaped flumes.  Reference number: ISO 4359:1983/Cor.1:1999(E).

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 http://webstore.ansi.org.

USBR.  1997.  U.S. Department of the Interior, Bureau of Reclamation.  Water Measurement Manual.  1997.  3ed.  Available from http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/index.htm .