Flow rate over a circular weir 3

[imatasb]

Hi everyone,

I have a tank with a baffle at the beginning. I am collecting some lights just before the baffle. I will like to drain these lights because otherwise they end entrained with the liquid. I want to install a pipe that finishes in a 4 or 6" funnel.

Which formula can I use to calculate the overflow into the pipe, based on the head of the liquid.

I have an actuated valve on the outlet pipe so I was thinking of opening at a regular frequency to drain those ligths but I would to know which the flowrate that I can expect from that circular weir.

Thank you.

 

[katmar]

Basically, you are looking at a modified version of the Francis Weir Formula.

The standard Francis formula for a straight weir is

h = k * (Q/L)^2/3

where
h is crest height (inch)
k = 5.4 for a straight weir with zero approach velocity
(k decreases as the approach velocity increases)
Q is liquid flowrate (cubic feet per second)
L is weir length (feet)

For a circular weir with a crest of less than 20% of the pipe diameter Robinson and Gilliland (Elements of Fractional Distillation, McGraw Hill, 1950) recommend a value of 3.9 increasing to 4.3 for the value of k as the pipe diameter varies from 0.87 to 2.07 inch.

For your pipe you are looking at a k value of between 4.3 and 5.4, which is probably accurate enough for you to set the initial valve frequency because you would have to fine tune it on the plant anyway. The angle of the funnel will also affect the k value, but I have never seen a correlation that takes this into account.

Obviously you must substitute the pipe circumference for the weir length.

regards
Harvey



Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[imatasb]

Thank very much for your answer harvey.

 

[imatasb]

Thank you very much for your answer harvey

 

[IFRs]

How does it make sense that the crest height decreases as the approach velocity increases? Somehow this seems counterintuitive. Can you explain?

 

[katmar]

If the approach velocity increases, then the velocity over the weir will be higher as well. If you have the same volumetric quantity flowing at a faster rate it takes less cross sectional area. The width (weir length) is the same, so the crest height drops.

By analogy with pipe flow - if you have 100 gpm flowing at 1 ft/sec you need a 6.4" pipe, but if your flow velocity is 6 ft/sec you only need a 2.6" pipe. In weir flow the numbers are not this extreme, but I think the principle holds.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[IFRs]

Why would the flow remain constant? If the approach velocity increases, won't the crest height and flow both increase? It "feels" like there should be more "pressure" behind the flow from the increased velocity that would translate into an increased crest height. Surely if a river flows faster it will pour more water through a weir? I must be missing something....

 

[katmar]

Hi IFRs, sorry - I misunderstood your question. What I wrote was correct in that for a given flowrate the weir crest will decrease as the approach velocity increases.

But the question you are posing (as I now understand it) is that if for a given physical set-up the approach velocity increases what will happen. I agree with you that the flowrate will increase (all that water has to go somewhere!), and maybe even the crest, but to get that same higher flowrate with zero approach velocity you would need an even greater weir crest.

So in terms of the equation I originally posted, the "k" value which gives the relationship between flowrate and weir crest will decrease as the approach velocity increases.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[bimr]

The flow rate over a weir is a function of the head on the weir.

Francis formula:

Q = 3.33 * ( L- 0.2H ) * H (1.5)

http://www.cheresources.com/weir.pdf

IFRs, you are correct in your thinking. As the head decreases, the flow also decreases

since Q = VA

the velocity also decreases.

Katmar is confusing Bernoulli's theorem and the francis formula (flow across a weir). The H in the Francis formula is the elevation head (total head available to the fluid minus the velocity head) so it does not include a component for velocity head (V2/2g).

 

[bimr]

More resources:

http://www.engineeringtoolbox.com/weirs-flow-rate-d_592.html

 

[IFRs]

I think you are both right. I presume that the Francis formula as shown by katmar when solved for Q is the same as the Francis formula shown by bimr. Marks handbook also has correction factors for crest length vs channel width and crest heignt vs depth below crest.

 

[katmar]

No, I am not confusing Bernoulli and Francis. I have observed personally that the Francis k-value varies with approach velocity. I was working with a stilling box that split an incoming stream into several separate outgoing streams using overflow weirs. The intention was for each of the outlet streams to be the same, but because the box was not symmetrical the approach velocities varied from weir to weir, and the outlet flows varied too.

IFRs is correct in that there are several correction factors to the Francis formula for particular geometries of weirs. An enormous amount of work has been done on weirs (particularly with water) because they are used so much in the water treatment field.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[bimr]

The Francis formula has been found to be acceptably accurate for weirs with sharp, square edges, a smooth vertical upstream face, a deep upstream pool behind the weir, and negligible effects of approach velocity.

Velocity distribution has also been found to influence weir discharge as Katmar noted.

If a standard weir application where discharge equations are to be used, consideration needs to be given to measurement of the head on the weir but also to velocities, weir sharpness, and general conditions of the weir plate.

One would think that the effects of velocity should be discarded in this application since imatasb would want a low velocity to avoid skimming too much water with his skimmed oil. Therefore, the weir should be oversized to get the necessary low velocity.

However, if Katmar wants to add the effects of velocity, the method of changing the coefficient that he suggested is incorrect. The coefficient in the francis formula are used for weir geometry, not the effects of velocity.

The Francis Formula is actually:
Q= 1.84 ( L-.1nh)((H+ha)3/2 -(ha)3/2)

where n is the number of end contractions and where ha is the Velocity Head. The effect of velocity is equal to (V2/2g).

A chart showing the simplified Francis formula can be found in this old mining report and may be useful for the rough estimate that imatasb actually needs.

http://www.em.gov.bc.ca/DL/GSBPubs/Bulletin/Bull15/Bulletin_015-Hydraulic_Mining_Methods.pdf

 

[IFRs]

imatasb -
Are you installing a pipe whose opening is horizontal, at the elevation of the baffle? If so, it's more like a drain than a weir and may not be accuratly quantified by a weir analogy.

 

[katmar]

IFRs - you bring up a very important point by calling it a drain, which it very much is. In my original post I mentioned that the weir crest should be limited to 20% of the pipe diameter to ensure that the weir model can be used. If the crest is more than 20% the pipe will go into a different flow regime and the weir model cannot be used.

But in fact a proper design should also include an analysis of the pipe to ensure that it is self venting or it will flood. Especially if the bottom diameter of the funnel is much smaller than the top diameter of the funnel.

A few years ago I saw an overflow from a sand filter that was exactly as imatasb described, i.e. using a funnel. It was being overloaded and the flow was going through a cycle of about 30 seconds. The overflow into the funnel was more than the drain pipe could take away in self venting mode and it gradually backed up until the whole funnel was flooded. This gave sufficient head to start a syphon and the drain suddenly sucked itself dry, and then started filling up again and going through the cycle again.

This type of weir-cum-drain was also very popular for distillation column downcomers until about 30 years ago - which is why it was described in the Robinson & Gilliland book I mentioned earlier.

What this whole long winded story is leading up to is that it is fine to consider a pipe with a horizontal opening as a weir, provided that the limits of applicability are observed.

bimr - if you modify the weir crest with a velocity head factor (as in your latest formula) or you tweak the k-value as I suggested - it all comes back to the same thing. You are making empirical corrections to a basic formula and in the end they are all approximations. The OP was asking whether the principle of using a funnel as a weir was reasonable. We have got side-tracked in detail that doesn't affect the conclusion that what he wanted to do was perfectly fine.

Harvey

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[bimr]

Maybe a floating skimmer would be more practical:

http://www.megator.com/oil_skimmers.htm