Channel flow problem

[rs1600]

Hi,

im posting this here as you guys have been helpfull in the past and im hoping you can give me some pointers with the following senario?

I have a 50m long smooth concrete channel, the cross section is 0.55m deep x 0.175m wide. water flows or weirs into this channel all along one side (50m) at a rate of 330m3/hr. (its a drain channel down one side of a pool)

one end of the channel blanked off, the other end is open and discharges via gravity into a large tank below.

What i need to deturmine is the maximum running level in the channel, i am assuming the maximum depth will be at the far side (blanked off side) gradually decreasing to virtually zero at the discharge point (this would give me a gradient?). i have looked at using the mannings equation and various on-line calculators however i am unsure if the way the water enters the channel will affect this? Im generally a little unsure anyway to be honest!

If anybody can help me calculate the above and explain how i could calulate this in the future for similar senarios i would be greatfull.

Kind regards,
R.

 

[BigInch]

How the water enters the channel will affect the depth of flow. If water enters uniformly all along, of course the water towards the blanked off end will add to all the water entering towards the drained end, so in fact the depth would be increasing towards the drain point, if the water was not being drained out. So the depth at any point along the channel is a function of the water entering through the upstream end of the channel plus that water spilling into the channel at that point, minus the water leaving downstream in the channel. If you know the water spilling into the channel in each 1 meter length of channel, you could easily calculate the beginning depth from water upstream, add the water spilling in, calculate the new depth and then calculate the flowrate going into the next 1 meter length. Some of the water spilling into each 1 meter length would tend to backflow into the upstream 1 meter segment, if the channel slope would not let the total amount of water flow out to the downstream segment. In that case you could assume that the difference becomes part of the upstream segment's volume recalculate, and so on.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

 

[rs1600]

Thanks BigInch for the reply, I kind of understand what you are saying but guess I'm not so sure how to actually go about solving this?

The problem I have is I need to ensure the channel is deep enough to accept the flow of water without flooding at any point, it is complicated by the need for excess volume in the channel to cater for any surge if water in the pool is displaced, I can calculate the surge volume but what I need to be satisfied with is there is capacity left over in the channel?

To do this I need to determine what the running level will look like under normal flow conditions I.e. 330m3 evenly displacing over the 50m length, with drainage from one open end of the channel, I then have an idea what is left over for any surge, or indeed if i need to increase the depth / volume of the actual channel itself!

If you could add anything to the above to help me with the figures that would be greatly appreciated?

Thanks,
R.

 

[BigInch]

OK, stepping back from the details of the problem of channel flow, the essential process becomes more clear. Breaking up the channel into 20 lengths of 2.5 meters and looking at the flow into and out of each segment over time steps of 1 second ... 300 m3/hr / 50 meters / 3600 sec/hr = 0.005 m3 spilling into each segment.

It becomes clear that, in a 1 second time step, all fluid entering a segment will be removed to downstream segments during that 1 second, if the channel velocity is 2.5 m/s. So the problem becomes finding out what minimum velocity you need, given a maximum channel depth of 0.55 meters, such that you don't get overflow.

From looking at a spreadsheet I made to get a rough idea of what increased depths do to channel flow, it seems that the changes in velocities in each segment are not very sensitive to the small changes in depths that might be expected to occur from 0.005 m3/second spilling into each segment every second, once the model reaches stable channel flow. The primary controlling variable is average velocity in the channel. Only small deviations from that velocity appears to be able to create a rather large difference in depth, sometimes doubling the channel depth. I did not actually evaluate that mathematically with a proper channel model, just looked at sensitivity to what might be a change in flowrate to change in depth descriptive function. So since the maximum channel velocity you could possibly need can be found as I have already done above, all you need to do is design the channel slope to get that velocity. With that velocity, any volume entering a segment is promptly removed during any time step you'd like to consider.

It ammounts to designing the channel flow capacity to drain all the volume entering in 1 second, so 330 m3/h = 0.005 m3/sec - segment, divide that by your channel width to get the depth and make sure your channel slope is steep enough to yield a velocity = channel segment length/1 second velocity. If you half that velocity, then to be safe, maybe you'd also want to double the height of the channel to be safe.

Try looking at it from that perspective, just a volume into - volume out of each segment.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

 

[rs1600]

Thank you, i have read you reply and while i appreciate your input to describe what is happening, unfortunatly im struggling to be able to crunch the numbers.

The base of the channel is flat, i gather the gradient needed to add velocity and carry the water down the channel would therefore need to be come from the water gradient (maximum depth at furtherst point, relative to minimum depth at drain point) in the channel? Is this correct?

I think i now understand how i can calculate the actual water depth (for a given velocity) in a single 'section' of channel i.e in your example the depth in each 2.5m section would be 0.028m (0.005m3 flowing in / 0.175m channel width) assuming a velocity of 2.5m/s.

So lets assume if i had a gradient of 1:125 (eqivalent to 400mm water deepth at furthest point), then how would i calculate the flow velocity? and therefore know if the channel was deep enough?

The information i will have available will be - Flow rate into the channel & Channel length. From which i want to deturmine the channal dimensions. I gather the depth is the key to getting enough velocity, while the width will be more important to deal with the volume.

Appologies if this is a bit like 'pulling teeth' for you ha! But i am keen to understand this so your input is valued. Is there a formula i should be using? or any worked examples you can offer? The numbers are largely irrelevant as i just want to understand the working out.

Many thanks,
R.

 

[BigInch]

This one might help for the open channel flow part.

http://gilley.tamu.edu/BAEN 340 Fluid Mechanics/Lectures/Examples of uniform flow.pdf

Technically the channel should have a slope to make those equations work, but at the small slopes we're generally talking about, I don't think it will really make much difference if you calculate the slope of the channel you need and then just "turn it upside down" and say it is the channel that is flat with the water surface at that sloped. It's probably close, but I'm not betting the ranch on it just yet.

OK, so then divide the length of channel into 20 sections of 2.5 meters. Consider a 1 second time step and let all the water spill into each segment. Determine the depth and velocity in each segment. In the next 1 second time step, do for all segments:
1) calculate the velocity of flow volume of water entering each segment from the upstream segment,
2) the volume spilling into each segment, and
3) the volume leaving.
I don't think the potential for backflow upstream from one segment to the next is so great, as long as each segment has the same spill-in volume during each time step, so maybe we can ignore it.

I'll attach the spreadsheet I was playing with this morning. See what you think. It's kinda' kool, because of the slider you can operate to see the change in variables as you run through the time steps. I don't know how "real" it is, but it looks good. Maybe you can improve it.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

 

[katmar]

To do this calculation rigorously would be a major undertaking, but also a bit pointless because there would be too many uncertainties. I think the way to do it is to make a few conservative, simplifying assumptions to ensure you have a safe solution.

However you do it, you must start at the drain end because that is where most information is known. You need to find out what the depth of the flow will be as it flows out of your channel into the underground tank. Taking the simplifying assumption of treating the discharge as a weir with a width of 0.175 m it looks like the head (depth) at the point of discharge would be something like 0.4 to 0.5 meter.

Now you have the depth at one end and can start working backwards towards the blanked end. I agree with BigInch's approach of breaking it up into segments and treating the overflow into the channel as a point source at the start of the segment. In terms of calculating the gradient required this will give a more conservative value (i.e larger gradient) than if you did it rigorously with a continuous overflow.

The last segment (i.e. closest to the drain) would have the full flow of 330 m3/h. If you take this as a sloped open channel under uniform flow conditions with a depth of about 0.4 m and a width of 0.175 m then the slope required will be around 0.01 m/m or 10 mm per m of length. As you move away from the drain end the flow rate decreases and the depth increases, so the required gradient will decrease with each segment.

The first segment (at the blanked end, and using BigInches 20 segments) will have a flow of 330/20 = 16.5 m3/h. To be conservative guess the depth as only 0.5 m, and then the required slope is less than 1 mm per meter. You could put together a spreadsheet and calculate the gradient for each segment, or you could guess an average of say 7 mm per meter and say the depth at the blanked end is the 0.5 m at the drain end plus 50 times 7 mm = 500 + 50x7 = 850 mm.

One of the assumptions I made was steady uniform flow. In fact the water has to accelerate as it moves towards the discharge. The velocity at the outlet would be around 1.3 m/s, giving a velocity head of 86 mm. We should add this to the 850 mm to give 936 mm. Although the true depth would be less than this because of the conservative assumptions, it is sufficiently larger than your 0.55 m depth to make me worried. It certainly deserves to at least have a spreadsheet put together to determine more accurately the gradient per segment than my guessed average 0f 7 mm.

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