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Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/21/2017 5:14 PM

Hello, I am conducting a research project which analyzes flow rates of open channels. In one particular portion of the project, I am analyzing an open channel that is receiving inflow from a separate source. Let us say for the purpose of this question that the flow rate of the water coming out from the separate source and into the open channel is entering the open channel with a flow rate of "Q" meters^3/sec. If this is the case, then once the water enters the open channel, will it continue to travel with the same flow rate "Q" regardless of the composition of the channel (ie. if the channel is earthen or if it is cemented)? Or instead, will the flow rate change? And if the flow rate changes, how will it change in accordance to the composition of the open channel? Finally, whatever the answer to this question maybe, how may I prove that answer mathematically and what equation would I use for such a proof?

Please keep in mind that the water flow in the open channel is uniform, open channel flow.

I have attempted to analyze this problem with the Manning formula. However, because the Manning formula does not offer a function input for inflow flow rate, I was unable to obtain an answer.

Thank you very much and please excuse my poor terminology as I am not a civil engineer student.

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#1

Re: Equation for comparing flow rates of canal as a function of inflow flow rate?

04/21/2017 6:22 PM

Is this tidal flow? If so it flows thusly....and varies daily....

https://tidesandcurrents.noaa.gov/publications/TidalCurrentsEducationalPamphlet4.pdf

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#2

Re: Equation for comparing flow rates of canal as a function of inflow flow rate?

04/21/2017 6:43 PM

Unless there are leaks or extraneous sources, the flow rate is constant everywhere along the length of any channel, but the velocity might not be.

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#3

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/21/2017 9:49 PM

I am assuming your channel doesn't leak!

The average flow rate at any point in the channel will be the volume flow rate Q divided by the cross-sectional area, A. The flow rate at any point will be maximum in the center and reduce to zero at the walls.

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#4

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/21/2017 10:14 PM

Using the Conservation of Energy, and assuming there are no fluid leakages and entrance/exit losses:

Q1 = Q2

The area (A) and mean fluid velocity (V) are variable. Since Q=AV in both channels:

A1*V1 = A2*V2

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#5

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 12:30 AM

Ask yourself what would happen if the flow rate (not velocity, but volumetric flow rate) were not constant everywhere in the channel. Assuming no leakage, etc., what would be the physical effects of:

1. a greater flow rate upstream than downstream?

2. a lesser flow rate upstream than downstream?

Now, consider a river ending in a waterfall (where the waterfall ends is irrelevant). Is the river's flow rate the same as the waterfall's at any cross-section even though their physical confines are vastly different?

Don't just look at the equations. Those equations were developed by someone who first had practical understanding. The equations are just the formal description in the precise language of mathematics, but they are no substitute for actually understanding what's going on. As Tornado pointed-out, the volumetric flow rate is constant over the channel's length, regardless of its cross-section and texture and assuming no spurious effects such as leakage. Under steady-state conditions it has to be.

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#6
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 1:10 AM

As a minor addition, it will be the mass flow rate that remains constant, as in the present case of (most if not all) liquids. For compressible gases, the volume flow rate will vary according to pressure/density, but the mass flow will be constant.

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#7
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 1:16 AM

Yes. A good (and possibly more radical) example is a liquid-fuelled rocket engine.

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#8

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 2:58 AM

Here's a fun exercise: you've no doubt seen a faucet turned on low such that the water exiting the faucet tapers into a smooth, somewhat trumpet-shaped stream

Assuming a constant flow rate F and given a faucet exit diameter of D, what is the equation for the stream's diameter as a function of vertical distance y from the exit? (increasing y measured downward and assuming the stream remains continuous and does not break up into droplets)

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#9

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 4:21 AM

It depends on how the flow is input to the channel. If there is a hydraulic break ie it enters from a level above maximum channel level, the channel flow always = input flow.

If the input is submerged, and the channel level can affect the flow in the input system, flow will vary with channel characteristics. To find how much, you would need to know the flow/level relation of the input and the channel, and maybe need to iterate to get an answer.

The Manning formula is OK for the channel part, but if the depth varies significantly, due to friction loss or slope of channel bottom, you might need integration.

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#10

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/22/2017 6:11 PM

Thank you everyone for the very insightful responses.

To answer the above statement, the water will be entering the open channel system from above. This will be done by tapping into shafts dug into the water table (see below).

The main point of my research will be to examine the system below. I will be looking for an increase in water volume from the start of the channel to the end of the channel. I will do this be measuring the flow rate at the start and end of the channel. (It is hypothesized that due to the porous nature of the rocks, water will seep into the underground portion of the canal, thereby significantly increasing the water volume in the canal.) I will then determine how much the water volume has increased. Then I will compare these volume increases against a theoretical version of the canal system in which this canal system is cemented and not porous (thus the added water volume is virtually zero). From this point forward, the research will focus on economics and the economic efficiency of a cemented system versus an earthen system, accounting for the fact that the earthen system is much more expensive to maintain than a cemented system.

I say all this in order to ask this question: Can I assume that the flow rate of a theoretical cemented canal will be the same as the flow rate of an earthen canal at the start of the canal?

Also, can I find the water volume increase in the earthen canal from the inflow point to the discharge point simply by using the following equation:

(Final Q - Initial Q)(Final time - Initial time)=Volume added to earthen canal

Again, thank you all for your help and I am sorry for how basic these questions might seem. I am only a high school student with no engineering courses!

Also, here is a picture of how these canals actually look. As you can see, they are open channel systems. (Keep in mind that the underground canals are much larger than this one though.)

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#33
In reply to #10

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/28/2017 12:35 PM

At any section of the canal that is above the groundwater level, where the water flow is laminar, through a chanel with porous rock surfaces, the flow-quantity will tend to have a slight net-decrease, not a net-increase...

( Qfirst-downstream-point ) - ( Qlost-to-porousness, etc. ) = ( Qnext-downstream-point )

If the canal were very veeeeerry long, the water would ultimately all be lost through canal porousness, etc...

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#11

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 11:17 AM

We may not answer you query in the affirmative, that is, if we are being forthright and honest. Earthern stuctures may well break down, and splinter into many channels, and water can easily soak into such earthen flow channels, and this is completely variable in nature, can only be estimated with experience in surmising these matters with upstream and downstream measurements.

If you wish to know Q(t) (flow rate) at a point in the flow, then that is easily possible if the full velocity and geometric profile is known. Inflow flow rate is redundant double speak. Just say flow input, and flow output. Or upstream flow, and downstream flow.

Obviously, all tributaries matter, as well as all branches. Uptakes by organisms growing at the water's edge or in the water, vertebrates, and even evaporation may be brought into question and consideration mathematically.

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#12
In reply to #11

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 11:45 AM

Thank you for your response.

The earthen structures I am examining are highly maintained so as to avoid their break down or splintering off into channels. My question is simply this. How can I determine the amount of water added to the earthen channel from inflow point to discharge point? Can I simply say (ΔQ)(Δtime)= added water volume.

If this equation is invalid, how can I determine the water volume added/lost from inflow point to discharge point given that I will measure the Q at inflow and Q at discharge? (I could also measure Q at any other point along the way as I have access to the shafts and tunnels).

Finally, you mentioned above that given that I knew the full velocity and geometric proportions, I could determine Q. (I am assuming this equation is Q=AV, yes?) Is there a way to measure the velocity of the water at a certain point in a channel experimentally without a flow meter? If so, how might I do this?

Again, thank you for your help.

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#13
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 12:36 PM

If the canal is open to the atmoshere, and wide enough, and long enough, there will be significant losses due to evaporation, such as the various canals of the California Water Projects (i.e.: the so-called All-American Canal from the Colorado River towards San Diego, etc.)...

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#18
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/25/2017 7:55 PM

Depending on environmental conditions, etc., evaporation can consume less than .25 %, and more than 1.00%, of total water volume...

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#14
In reply to #12

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 1:31 PM

The old fashioned way: scatter some rose petals on the surface and time them from A to B, assumption that the channel has an average size and shape from A to B. Another assumption is that no water is added or lost between A and B.

Problem: how in the heck does anyone know what the velocity is on bottom of channel from the rose petals? Answer: they don't.

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#16
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 2:57 PM

James Stewart,

Is there a way to use the rose pedal method to determine the average velocity considering that the velocity will approach 0 as the water approaches the sides of the canal? Or is the decrease in velocity close the canal walls so small that it is negligible?

Thanks,

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#17
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/24/2017 3:34 PM

In the range of approximation of a large channel, the boundary layer is nearly infinitesimal, but the problem is with the boundary itself presently what is normally a tortuous path for water, that results in large eddy currents, whirlpools, etc. This makes flow estimation a bit more complicated.

I suppose if one had a drone that could monitor what was initially a line deposit of rose petals, the dispersion of these in linear distance could reveal something of the complexity of flow in said channel. Obviously, the residence time of rose petals in eddies would of some interest.

End result, if all the rose petals are eventually seen crossing the line B from line A, then all of computed statistics of a "random" class might be computed and studied.

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#19
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 12:55 AM

Below I have attached a picture of a canal I will analyze. As you can see, it is fairly straight at certain parts and has what appears to be a fairly smooth flow. If I found sections such as this along the canal and used the rose pedal method, would that give me a reasonably accurate measure since the flow is smooth and the canal is straight with constant geometric proportions?

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#20
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 2:04 AM

That's partly right, but the velocity near the bottom and sides of the canal is somewhat less, so you might need to apply a correction (say 0.8?) to the rose-petal measurement. You can refine the correction if you have an annubar to sample velocities across the canal and at various depths.

At any given point, the net flow is the sum of flows of all upstream inputs (wells, surface water, rain, etc.) minus the sum of all upstream diversions (leaks, evaporation, withdrawals to serve users, etc.)

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#21
In reply to #20

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 4:50 AM

I don't suppose there is a way to measure, or at least take an educated guess at the correction? I will supposedly have access to an annubar in the future but it is uncertain when I will get funding for it. Is there a way to do without it?

Is there a way to figure out the correction by inputting the dimensions of the canal and some sort of roughness constant that could be determined by observation (like with the manning constant)?

Thanks

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#22
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 5:03 AM

Can you measure the depth at 2 points some distance apart? Using that and any change in bottom elevation (might need a theodolite or something) to find headloss, you can estimate flow from Manning. Headloss likely to be only a few mm so maybe not too accurate, but it's a start.

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#23
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 5:12 AM

Yes I think I can do that. It might be a bit difficult to figure out the best way to measure the slope but that might already be documented. I will look into obtaining a theodolite. (Is that the best instrument for measuring slope of the channel?) By head loss, are you referring to water loss due to evaporation? If so, I think it should be to a minimum because the channel, while it is open channel flow, is completely cut of from sunlight, or any light for that matter.

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#24
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 6:15 AM

I don't know much about surveying or theodolite use, but if you put marked poles in at each point (resting on the bottom of course) I think you can find the difference in bottom level.

By headloss I mean difference in top water level between the 2 points, to use in Manning. That's the difference relative to a horizontal line, forgetting about channel bottom (for the time being). You need the bottom level change as it affects the hydraulic radius in Manning, but it might not be too significant. Ideal situation is if the headloss is same as bottom level difference, so the depth (and hydraulic radius) is constant.

Evaporation will be quite negligible between the 2 points.

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#25
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 6:52 AM

So head loss would be graphed on x and y below while bottom level difference would be graphed on a and b?

But my question then becomes, does it really matter if I measure the flow rate with the manning equation at multiple points? As mentioned above, flow rate should remain constant unless water is lossed or gained from the system. So if the head loss, slope, or hydronic radius is changing, the water velocity might change but the flow rate would not, correct? Also, if I used manning, wouldn't the same go for water currents created by the sides of the channel or friction produced by the channel walls? The manning equation accounts for all this, correct? So, if this is the case, can I not simply measure flow rate with manning at just one point and then apply that everywhere, assuming that no change in total water volume exists? (Of course, I would still have to use two points to determine slope, as shown with a and b in the above picture).

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#26
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/26/2017 9:24 AM

To save me trying to get my head round all the posts, am I right thinking you just want to estimate the flow in the channel? The questions about rose petals suggests that. Pooh-sticks?

The Manning equation

uses Ih, the slope of the water surface, at any point. If you could measure the slope at a single point, that would be all you need, but in practice you can't, so you measure top water level between 2 points, preferably as far apart as possible, and divide level change by distance. I think that's what you say in last sentence. Of course, if no water is added or taken away the flow is constant.

The formula allows for roughness of the channel walls via the factor Ks, likely to be 80 - 90 in this case. If the depth and hence Rh varies I would just use the average over the length. Due to the uncertainty it's probably not worth going to a lot of trouble, eg integration etc.

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#28
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/27/2017 9:18 AM

Codemaster,

Sorry I'm a bit confused. Why would I need to average Rh? Flow rate is constant all throughout the canal so wouldn't that indicate that measuring flow rate (and thus Rh) at one instantaneous point would give me flow rate throughout the entire channel? Wouldn't I get a more accurate result by measuring Rh at exactly 1 point? Basically, because Q is always constant, instantaneous Q equals average Q, so wouldn't I either use ALL instantaneous variables to measure Q or ALL average variables to measure Q?

(Wouldn't this also be the same for slope? However, I recognize that slope would be difficult to measure experimentally at one point.)

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#29
In reply to #28

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/27/2017 11:49 AM

Only if you want a more accurate read on it than, holding up a spit on thumb in the breeze and calling it "good".

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#30
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/27/2017 4:20 PM

Hydraulic radius Rh is ratio of cross-section area to wetted perimeter, so for a rectangular channel breadth B, water depth H, Rh = B*H/(B + 2*H). If H varies, so does Rh. If you could find the slope at a point, you would only need the Rh at that point, but in practice you're likely to have average slope over a certain length of channel (as I said in #26). But unless you have high friction loss, and/or the channel bottom slopes significantly, taking the average Rh over the length is likely to be near enough, specially as there'll be some error in measuring the average slope. The only case when Rh is constant is if the friction gradient = bottom slope (as I said in #24).

If you're into maths and preferably have Mathcad you can work out a more general case.

Integrating gives

and trying various flows till you get the length between the 2 points. The preliminary flow calculation gives a clue where to start. That might seem a roundabout way of doing it but it works OK. Alternatively could use a Find function to get Q from length, depths etc directly.

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#31
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

04/28/2017 4:06 AM

I forgot to add - I'll be away till Tiesday, but if you want to post some figures I'll work out the flow for you.

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#34
In reply to #30

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/04/2017 5:29 PM

Codemaster,

Thanks for the help. That equation that u posted looks very useful in my situation. However, I didn't fully understand it because I was unsure what all the variables were representing. Would u mind explaining the equation in a little more detail. I really appreciate your help.

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#35
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/04/2017 5:40 PM

OK I'm just off to bed, but I'll look at it tomorrow.

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#36
In reply to #34

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/05/2017 9:52 AM

Doing it the basic way, ie assuming the water depth is >> the surface level difference between the 2 points, you would use the equation in #26 to find velocity V. Multiply V by area B*H (see #30) to get flow Q.

In #30, first ignore

(which didn't all come out)

I inadvertently copied that from a specific case I'd been on!

Further down

is a rearrangement of #26, but I've called the hydraulic gradient s instead of Ih, as s was in the Mathcad, sorry if that confused.

is the formula relating depth H to length L along the channel (for a rectangular channel). It can be integrated to give L for a known s(H) as I said in #30, but it would be hard without Mathcad or some other numerical integration tool. Derivation is given in Brater & King Handbook of Hydraulics below. Note that uses D for depth whereas I've used H

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#37
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/05/2017 10:03 AM

I posted #36 quick as the computer was slowing down and I didn't want to risk losing it.

To continue - if you can't read the King & Brater bit, you can download it free.

Having said all that, it's probably not worth going to a lot of mathematical trouble, as you won't be able to measure the hydraulic gradient too accurately, and the result depends on an assumed value for the wall roughness factor Ks.

As I said before, (and see #22) if you post some figures I'll estimate the flow.

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#38
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/06/2017 3:36 AM

Ok thanks for the help. I think I have a better understanding of that equation now. But I'm still not sure, what exactly is the advantage of using the equation u posted rather than using the manning equation?

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#39
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/06/2017 4:18 AM

There's no difference. The Manning equation gives the slope of the water surface at any point. If the depth does not change significantly over the channel length considered, that's all you need.

The other equation applies if there is significant change, and incorporates the Bernouilli effect, but still uses Manning for the s term.

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#40
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/07/2017 5:04 AM

Thanks for clearing that up. However, I was also wondering what you meant by H prime in this equation and how can I find hydraulic gradient and velocity as a function of H prime? (Also, am I correct in assuming that g is the acceleration due to gravity?)

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#41
In reply to #40

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/07/2017 10:37 AM

It doesn't matter what H' is, which is why in integration it's known as a dummy variable. Just as eg

I called it H' as it's a measure of channel depth at any point, but it has to be distinguished from H for Mathcad.

You don't find hydraulic gradient and velocity as a function of H prime, they're linked by the Manning formula. In your case, doing it the basic way which I've suggested (at least for starters) you would measure the average hydraulic gradient, and calculate velocity from Manning.

You're right g is the acceleration due to gravity.

What you're doing here, measuring the level change over a known length (if you ever get round to doing it!) and using that to estimate flow is somewhat unusual. It's quite straightforward the basic way, but doing it the hard way and trying various flows till you get the length used (see #30) needs Mathcad or something similar. More often, when designing a system, the flow, channel length and downstream depth are known, and the upstream depth is calculated. Also needs integration if the depth is not >> the headloss, as I said earlier.

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#42
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/08/2017 4:44 AM

Don't know whether anybody noticed, but in

the 1 is wrong. Checking it in my head last night (it's quite simple) I realised the right answer is 333. Doing a check, it seems Mathcad is saying

is a true statement (0 = false, 1 = true)

Should be (letting Mathcad do the work)

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#43
In reply to #42

Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/08/2017 10:13 AM

I just now was reading that and going WTF?

Thanks for the Boolean clarification.

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#44
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/08/2017 10:41 AM

Good that somebody spotted it!

I often Ctrl= giving = in bold font if I want to copy something from Mathcad, as it then doesn't (usually) want to start evaluating something, or prompting for an input. But clearly it's possible to come unstuck. In the bottom equation the = is in text, so it doesn't do anything, but it's a tad more trouble to enter.

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#45
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Re: Equation For Comparing Flow Rates of Canal as a Function of Inflow Flow Rate?

05/08/2017 10:56 AM

I have had to watch the Matlab tutorial on converting date time (text) to datenum type data to allow plotting data logs as a function of elapsed time. I still have not seen how to plot against string values (as a single date-time vector). I need that one?

Any of you wizards out there have the spell for that, or do I need a new wand?

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