Formula for flow through weir

[kachofool]

Hey

I'm trying to find a formula that will let me calculate the required flow in a weir for a specified head and length. For example, if I had a weir that was 1 foot long and had a head of 0.2", and I wanted to find the flow.

I have a few problems... first, there seem to be many variations on two main formulas that might be used for this situation (Bazin and Francis). I tried to do a crude experiment using a plastic box, cutting out a weir and trying to use either formulas to get the correct flow, which I measured before hand. Neither formula is close to what I measured and they both yield very different results from each other. I will show the formulas I have used at the end of this post. It is probably a good idea to note that the experiment I did was not accurate in any sense, but I expected to receive a somewhat similar flow number from the equations. Instead, I was off by orders of magnitude!

For my application, the velocity of approach can be negated. There will also be a clinging nappe; the water won't freefall over the weir, instead it will cling to the weir surface and flow down. In Bazin's experiments, he determined that the required flow rate relative to a normal free fall (no clinging nappe) is 1.279.

My ultimate purpose is to determine the minimum flow rate required for a given head and weir width, where a clinging nappe is present.

The formulas I used were
Francis: Q = 3.33*L*(H^1.5)
Bazin: Q = 3.29 * (0.405 + 0.00984/H)* L * sqrt(2g) * H^1.5

I would appreciate any sort of input. I don't exactly have the resources to perform a proper experiment, but would feel better if I had some sort of confirmation that the results the formula were yielding were realistic (I was concerned because the results from the two formulas themselves were so far apart).

Thanks,
Preet

 

[pleckner]

Do you have or can you get your hands on Cameron?

Page 2-10 and 2-11 have formulas and a table for weir flow.

First off, Cameron reports the Francis formula as:

Q = 3.3 x (L-0.2H)x H^1.5

Q = ft3/sec
L = length of weir opening in ft, should be 4 to 8 times H
H = head on weir in ft; to be measured at least 6 ft back of weir opening.

The width of the full flow path before the weir should be L + 6H. If this were a river, L + 6H would the width of the river, bank-to-bank.

Also, this formula appears to be good for Rectangular Weirs with End Contractions.

IF this weir is of a triangular notch configuration, the equaitons are different.

 

[kachofool]

Thanks for the quick reply! I'm using the same Francis formula, but I have a supressed weir, where the width of the weir is the width of channel, so there are no contractions.

The (L-0.2H) you see is for two end contractions, and (L-0.1H) is for one contraction. With zero contractions, the term just becomes L.

In the specific tests I did, the width of the weir was 0.7 ft, and the head had a height of 0.0164 ft, which fits in with the conditions you mentioned for L and the width of the full flow path.

Is Cameron a journal/publication? Where can I find this? Most tables I've come across start reporting heads at about 50mm (0.164 feet), however for my application, the head needs to be about a tenth of that height.

Just to mention, in my experiment I got a flow rate of 0.00374 cfs, and the francis formula gave 0.0049 cfs. Considering that a clinging nappe has an increased flow of about 28% (this was both mentioned by Bazin and verified here: (

http://herkules.oulu.fi/isbn9514259777/html/chapter7.html))

, the corrected flow rate from the francis formula yields 0.00628 cfs.

In any case, tables with smaller heads around my range would really help me out since I'm writing a report and need my information to be verified.

Thanks,
Preet

 

[rmw]

I have seen the same formula that Cameron has in Hydraulics text books too.

Do an advanced search on this site for Cameron Hydraulics as it has been discussed several times.

rmw

 

[JStephen]

Those formulas don't look that different. If you're getting results a lot different, I'd look at the units very closely. Make sure you're not using inches instead of feet, or leaving out a g-c factor where you need it, confusing lbm with lbf or something similar. Those equations are not dimensionally consistent, meaning that the constants have to have units which are not indicated.

I would anticipate that at a very small scale (0.2"), results could vary considerably due to surface tension effects, which are not included in the normal weir-flow equations. But that wouldn't affect how the equations agreed with each other.

 

[kachofool]

While I try to obtain Cameron Hydraulic Data, could anyone tell me whether or not the tables offered are from experimental results or just values from using an equation?

JStephen, the formulas both work properly at high heads, but differ quite a bit at lower heads.

I'm trying to find tables with proper experimental results, and will go with the formula that most often yields the closest result. In my application, the flow calculated doesn't need to be exact, but I was hoping to find something that would give me an accuracy of +- 20%.

 

[pleckner]

The table in Cameron is based on the equations and not actual tests. However, the equation is based on test data.

I layed out the limits for the derivation of the equation in my post above so I have to assume if your system configuration deviates, so will your results deviate from those given by the equation. Perhaps this is why your results are so different.

 

[kachofool]

You're correct... I don't have any doubt that my test data is wrong. However, I thought I'd get at least recognizable results. In any case, I think I will use this journal entry:

"Simple Discharge Relations for Sharp-crested Rectangular Weir and Right-angled Triangular Weir at Low Heads"

http://www.ieindia.org/publish/mc/0404/apr04mc4.pdf

which cannot be more related to what I need. It is my understanding that the plot shown in the document I have linked is from experimental data since the experiment 'device' is fully described.

Thanks a lot for all the help!

Preet

 

[JStephen]

Looking back at it again, that one equation has H in a denominator, which is going to make it blow up at some point. I would say it is probably adequate for a specific range and you're just outside that range. You obviously don't have infinite flow at zero H.

 

[BarryEng]

Why don't you use a V notch?
This method is often used in the centre of a rectangular weir so that very accurate results can be calculated for a range of flows from very low flows (V notch), up to large flows depending on the size of the rectangular weir. In gauging weirs on streams, this type of composite weir gives a much greater range while still maintaining accuracy.

Remember that for a V notch, flow is dependent on H^2.5 whereas for a rectangular weir, flow is dependent on H^1.5.
Hence for a low flow, the head is much larger on a V notch compared with a rectangular notch.

 

[kachofool]

The reason for finding out the flow does not have anything to do with measuring flow of an open channel of water, which is why I'm not using a v-notch.

My application has to do with building water features and I wanted to calculate the minimum req'd flow to create a cascading flow of water (clinging nappe) over a weir.

Thanks,
KF

 

[blueoak]

One problem may be that an ever decreasing head will decrease the weir coefficient. A clinging nappe has an a specific range where it increases flow. See US Army Corps:
EM1110-2-1603 Hydraulic Design of Spillways.
For clinging nappe weirs your coefficient is highly dependent on the upstream head. The referenced document gives some good guidance.

For clinging nappes you typically design an ogee or curve to extend this rate. I am not sure if the change from a 3.3 to a 3.08 will affect your results since your head term will increase.

It also looks like you are getting into that limbo area of laminar flow.

 

[kachofool]

Hello blueoak...

Could I ask what you mean by "A clinging nappe has a specific range where it increases flow"? I could not find this range in the document. Also, what do you mean when you say "that limbo area of laminar flow"?

Using the paper I mentioned in my earlier post, I calculated the flow rate using a weir width of 1m and a head ranging from 1mm - 10cm.

All values for the flow before 7mm were very odd in that the required flow decreased as the head increased. After 7mm, there was a non-linear increase in the relation, but the correlation was that flow rate increased as head increased, which makes more sense. I have enclosed my calculations and a couple of graphs for easy viewing. All calculations were done following the research paper I mentioned in my earlier post.

The paper in question does not mention what range of values are acceptable and instead is dependant on the ratio between the head and height of water from the channel bed. I wanted to know if anyone thinks the values from 1mm-7mm are correct? I was considering using a head of 4-5mm, which is why I am asking.

------------------------------

Weir Length is 1m in all cases

[from 1mm to 2cm]

head (m) Q (cub. m/s)

0.001 0.005522563
0.002 0.003974572
0.003 0.003339842
0.004 0.003007107
0.005 0.002821562
0.006 0.002722913
0.007 0.002681977
0.008 0.002682585
0.009 0.0027149
0.01 0.002772513
0.011 0.002851021
0.012 0.002947267
0.013 0.003058908
0.014 0.003184156
0.015 0.003321612
0.016 0.003470156
0.017 0.00362888
0.018 0.003797032
0.019 0.00397398
0.02 0.004159192
-------------------------

[from 1cm to 10cm]
-------------------------
head (m) Q (cub. m/s)

0.01 0.002772513
0.02 0.004159192
0.03 0.006388056
0.04 0.009159936
0.05 0.01236469
0.06 0.01594197
0.07 0.01985223
0.08 0.02406688
0.09 0.02856392
0.1 0.03332569

 

[kachofool]

The graphs for the data can be found here. Notice the curve behaviour in the range from 1mm-7mm.

[1mm-2cm]

http://img341.imageshack.us/img341/4051/gra1zq0.gif

[1cm-10cm]

http://img341.imageshack.us/img341/6942/gra2eu3.gif

Thanks,
Preet