Discharge Coefficient 3

[SMIAH]

I am curious to know which method/technique/equation or what degree of complexity everyone use to determine the coefficient of discharge C in the relation Q = CLH^3/2. This for, let's say, a broad-crested weir.

Keeping in mind that most of the larger dams have a complex geometry, and therefore physical models have been used extensively.

 

[cvg]

so, you are talking in reference to small dams? emergency spillways? or service spillways?

For design, we usually balance conservative assumptions (low C value and thus wider and deeper spillway) with a desire to accurately estimate the actual flow depth and rate (perhaps higher c values and more actual capacity in the spillway).

For a service spillway, you may want to go with a physical model. For an emergency spillway, go with conservative design. For a large dam, always go with a physical model if you can convince your client to spend the money.

 

[psmart]

The coefficient will actually varying with head, depending on the shape of the spillway. So you may need a C-vs-head curve, rather than just a single value. The variation can be up to 20% or more. Whether this matters depends on the goals of your analysis, but you need to keep in mind that "C" may not actually be a constant.

Peter Smart
HydroCAD Software

http://www.hydrocad.net

 

[cvg]

weir equation is not always the best approach either, even if the control section appears to be a weir. We have used Manning's equation for open channel type spillways where the weir is drowned and two dimensional hydraulic analysis to determine flow over spillways

 

[SMIAH]

Ok let's be more specific about this.

Suppose you have a broad-crested rectangular weir with no end contractions and no submergence effects (not a creager/ogee spape spillway).

The discharge coefficient over a broad-crested weir shouldn't be higher than 1.7 in Metric or 3.08 in English.

Some suggest that C should be around 1.1 (2.0 in English), when some others suggest a value of 1.45 (2.63 in English).

What equations do you use?

 

[gbam]

Brater,King, Lindell & Wei Handbook of hydraulics has the C vs head relationship that Peter mentioned. The 7th edition is metric, but I have googled and found pdf versions for english and metric.

 

[Drew08]

If I may take a bit different point of view:

The weir is nothing more than a combination of the energy equation and Froude equation. If you were to derive it, you'd find that C is a function of only the Froude number and gravity. Hence, the weir equation represents the energy of rectangular flow Q with some Froude F. Assuming critical flow fully develops (F = 1) then C = 3.1 (US). For some silly reason long ago engineers started using the C as an empirical catch all variable. When in fact the H is the real variable. Understanding this, you can avoid using C vs. H tables, which are nothing more than conversions between using H as the variable and using C as the variable.

The tricky part is determining the head losses between the point of reference to where critical flow develops. This varies greatly on the type of spillway and entrance conditions.

 

[cvg]

2.0 sounds extremely low, is this supported by any technical references to explain such a low weir coefficient is recommended?

2.63 is approximately equal to values that we have used for design of trapezoidally shaped emergency spillways. This is a generally very conservative value assuming there is free flow in the spillway channel. Again, this is a conservative number useful for design, maybe not so useful for estimating actual flow depths. Actual value will vary with depth of flow, approach conditions and geometry. See Brater and King, chapter 5 Q=CLH^3/2 and table 5-3 lists values for C ranging from 2.63 to 3.32. 2.63 is the lowest value listed in the table.

 

[SMIAH]

I have this book explaining the experiments of Blackwell, Bazin, the U.S. Deep Waterways Board and the U.S. Geological Survey. With the note "The table should give values of C within the limits of accuracy of the original experiments"
(3rd edition, 1939).

 

[psmart]

As cvg has suggested, the "right" answer really depends on your exact application - which hasn't been mentioned.

If you're trying to estimate flow rates based on observed head, you would want a more accurate value. But some calculations (such as a hydrograph routing through a pond) can be relatively insensitive to the exact weir coefficient. This is easy to verify by trying a couple of values.

Peter Smart
HydroCAD Software

http://www.hydrocad.net

 

[Drew08]

If a vegetated trapezoidal spillway is being considered say with a level section of >20', C values can be particularly low for low flow depths, with a major influence on dam performance. At low depths Manning's and energy losses are highest.

I've back calculated C values from spillway results using standard step methods (NRCS Sites, and HEC-RAS) and found C can be as low as 0.5 (US) for the first foot or so of head, increasing to and leveling off at about 2.6 (US) for 5 feet of head.

But I digress to my first comment that at the point where critical flow develops, C = 3.1 and H = [reservoir head] - [energy losses in the channel].

 

[SMIAH]

@Drew: Indeed, one can start from Energy in 1 = Energy in 2 to find a C of 3.1.

I have it here and I must say it looks pretty brilliant.

Indeed, it might be too much to check for the exact weir coefficient, but I don't like using C=1.45 blindly as HEC RAS sometimes makes me do.

 

[beej67]

Pink star for Drew08. I hadn't thought to go back to the derivation.

Hydrology, Drainage Analysis, Flood Studies, and Complex Stormwater Litigation for Atlanta and the South East -

http://www.campbellcivil.com

 

[SMIAH]

if a vegetated trapezoidal spillway is being considered say with a level section of >20', C values can be particularly low for low flow depths, with a major influence on dam performance. At low depths Manning's and energy losses are highest.

I've back calculated C values from spillway results using standard step methods (NRCS Sites, and HEC-RAS) and found C can be as low as 0.5 (US) for the first foot or so of head, increasing to and leveling off at about 2.6 (US) for 5 feet of head.

This is very interesting as I usually design with a C that is higher than 0.5 for the first foot or two. I guess that I should specify a lower C for such conditions to be more conservative.

 

[SMIAH]

Document attached for the derivation of C.

 

[Drew08]

If I can offer an alternate derivation maintaining a more general form that can be used for all flow regimes, not just critical flow. This definition of C can be used in situations were fully critical flow does not develop such as at a submerged weir.

Going back to my original example of a vegetated spillway. If the point of reference is the downstream end of the level section, then: C = 3.09 and H = [Stage] - [Losses]. If the reference point is the upstream end, then C = f(F,g) and H = Stage.

Determining the upstream Froude (F) in a vegetated spillway is a topic for another post.

 

[SMIAH]

Wow.

 

[SMIAH]

You do have to know the Froude Number, the width and the depth...