Determine the Rate of Change of Pressure Across a Valve

[Purple_Dan]

Hello, I am Primarily a Software Engineer, but have some background knowledge from my A-Level and Uni days.
However, I have the following problem.

I'd like to find out the rate of change in pressure across a valve.
Here are the things I know about the system:

C_v at 100% open = 0.004 (I believe it's the US coefficient)
Pressure drop across valve can be made up as this is variable within the system, let's assume 500psi for this example.
Assuming it's pure water at 25°C with a specific gravity of 1 (because I don't want to get into all that mess).

Also, am I right in thinking that if the C_v is 0.004 at 100%, then the C_v at 50% will be 0.002?

Any help would be greatly appreciated.

Thanks,
Dan

 

[HuskyNamedNala]

Empirical data is your best option, especially for something transient like this. Why do you need the transient data?

 

[Purple_Dan]

Empirical data is your best option, especially for something transient like this. Why do you need the transient data?

I agree, empirical data would be a great commodity to have. Unfortunately, I'm writing the code to control the valve in an office in the West Midlands and the site is in Aberdeen.

I need to control the valve to maintain a constant psi/sec drop. I was going to do a simple, "if drop is too big, close valve a bit, else open valve a bit" logic. But I thought it would be nicer to try and do it properly.

 

[HuskyNamedNala]

Hmm. sounds like a cool problem. Maybe write a basic predictor-corrector method to estimate the derivative? If the derivative exceeds some value, then reduce flow to XX, re estimate derivative, repeat?

Wish I got cool projects like that.
[edit[
It also just occurred to me that you might be able to utilize optimization methods. For constant flow rate you want the derivative to equal 0, so you can set this up as a minimization problem.

 

[HuskyNamedNala]

Also, where is the flow rate being measured? This is important so you don't get erroneous readings due to turbulent fluctuations.

 

[montoyas7940]

Valve data is typically published by the manufacturer. Different types will vary greatly and most are far from linear.

Do have any other data for your valve?

 

[Purple_Dan]

Also, where is the flow rate being measured? This is important so you don't get erroneous readings due to turbulent fluctuations.

I'm not actually measuring the flow rate, I'm measuring the pressure. Thus I can determine the current psi/sec drop. I then need to alter the control valve to maintain a certain drop. I can determine the flow rate from the pressure drop across the valve using the flow coefficient, but I'm having trouble finding the rate of change in pressure from that.

From what I can gather, the flow coefficient is in units of "gallons per minute per psi". If I have the pressure drop across the valve in psi, can I find the pressure drop per minute?
I have a hunch that I'll need the volume of the system, in which case I'll ask them to fill it up and measure how much water comes out. But ideally, I want to make commissioning as quick as possible.

You can probably tell I'm not much for fluids and mechanics. I'm more electronically minded.

 

[Purple_Dan]

Valve data is typically published by the manufacturer. Different types will vary greatly and most are far from linear.

Do have any other data for your valve?

Aha! I've found a graph that tells me the flow coefficient at different open levels. It's a 30VRMM valve, if anyone wants a look.

So that answers one question, thanks!

 

[HuskyNamedNala]

How you are measuring the pressure will determine the correlation between pressure reading and flow rate. For example, if you use a Pitot static tube you can equate your reading to the dynamic pressure. If you have a sensor at the bottom of the tube, you can say the pressure reading will be the hydrostatic less dynamic. You will need the surface area of the inlet and outlet to determine the flow rate. As montoyas7940 stated, valve curves are usually published to customers.

 

[montoyas7940]

I'm not actually measuring the flow rate, I'm measuring the pressure. Thus I can determine the current psi/sec drop. I then need to alter the control valve to maintain a certain drop. I can determine the flow rate from the pressure drop across the valve using the flow coefficient, but I'm having trouble finding the rate of change in pressure from that.

From what I can gather, the flow coefficient is in units of "gallons per minute per psi". If I have the pressure drop across the valve in psi, can I find the pressure drop per minute?

Rate of flow is proportional to the Sqrt of Dp.

Psi/sec? What do you mean?
Dp across the valve is just Dp across the valve. Do you need to control the rate at which Dp changes as you open or close the valve?

 

[montoyas7940]

I need to control the valve to maintain a constant psi/sec drop. I was going to do a simple, "if drop is too big, close valve a bit, else open valve a bit" logic. But I thought it would be nicer to try and do it properly.

I should read more carefully.
I would suggest a flow element for Dp instead of measuring Dp across the valve. Then use a flow control valve for flow control.
To maintain a constant flow, if Dp is high across the element then close the valve a little.

 

[Purple_Dan]

How you are measuring the pressure will determine the correlation between pressure reading and flow rate. For example, if you use a Pitot static tube you can equate your reading to the dynamic pressure. If you have a sensor at the bottom of the tube, you can say the pressure reading will be the hydrostatic less dynamic. You will need the surface area of the inlet and outlet to determine the flow rate. As montoyas7940 stated, valve curves are usually published to customers.

The pressure is measured using a 4-20mA sensor, it's a hydrostatic test, no air bubbles or anything. The transmitter located before the control valve, there is no transmitter after the control valve, so I'm assuming it to be 0psi (ignoring atmospheric pressure).

 

[Purple_Dan]

Rate of flow is proportional to the Sqrt of Dp.

Psi/sec? What do you mean?
Dp across the valve is just Dp across the valve. Do you need to control the rate at which Dp changes as you open or close the valve?

Yes, I know the change across the valve. I want to know, if I open the valve x%, what will the pressure be after X seconds. That way I can just open the valve to the required %, rather than guessing and inching it open. I'd rather not guess when there could be 23000psi in the system!

 

[HuskyNamedNala]

If the transmitter is only before the control valve, how will you regulate the flow? You need at least 1 upstream and downstream of the valve. It may not be feasible, but it would be best to have the downstream sensor 10 diameters from the outlet. Turbulent fluctuations will gives you oscillating readings which will require you to take a time average before any calculations are executed. The best you can do at this is point is estimate the incoming mass flow rate, then correlate that with your valve curve. Keep in mind that valve curve you have is only applicable to the ranges it was tested for. If you have an anomalous event you should not rely on it.

 

[HuskyNamedNala]

I am sorry, I keep thinking in terms of mass flow rates, you can equate mass flow to pressure as was discussed earlier. I am used to thinking in those terms, sorry.

 

[HuskyNamedNala]

Yes, I know the change across the valve. I want to know, if I open the valve x%, what will the pressure be after X seconds. That way I can just open the valve to the required %, rather than guessing and inching it open. I'd rather not guess when there could be 23000psi in the system!

You won't find that transient data. Maybe the vendor has a specific test that examined that, but almost always curves are for steady state. Nobody here can tell you how long it will take to equalize. Call the suppler and see if they can help.

 

[montoyas7940]

Ok, scratch most of that other stuff. You are measuring static pressure and then controlling the rate of pressure reduction/increase in a hydrostatic test vessel.

 

[Purple_Dan]

Ok, scratch most of that other stuff. You are measuring static pressure and then controlling the rate of pressure reduction/increase in a hydrostatic test vessel.

Yes, that probably makes things a lot simpler, that might have been worth mentioning before!
Sorry everyone!

Although, I will seek your advice should my client ever want to automate Gas testing!

 

[jim hardy]

I'd like to find out the rate of change in pressure across a valve.

rate of change in pressure vs time or pressure vs valve position?
if versus position ignore remainder of this post.

Pressure drop across valve can be made up as this is variable within the system, let's assume 500psi for this example.
Assuming it's pure water at 25°C with a specific gravity of 1 (because I don't want to get into all that mess).

Water being incompressible and dense and having inertia
must be accelerated or decelerated as your valve changes its Cv.
In a long straight pipe full of water , the inertia edit: mass of that long column of water has inertia that'll manifest itself as transient pressure opposing the change in flow - just like inductance for us electricals. In control systems it can wreak havoc.

So i think you might need to know something about the system in which your valve resides

Late entry : I see this is a manually adjusted needle valve - probably not an issue.

 

[montoyas7940]

I agree with Nala the Husky (my sister also has a dog named Nala).

You don't have enough info about the system. You might ask the test facility to provide empirical data based on their experience.

 

[Purple_Dan]

I agree with Nala the Husky (my sister also has a dog named Nala).

You don't have enough info about the system. You might ask the test facility to provide empirical data based on their experience.

I expected as much. Still, it was worth a try. The problem with getting data is that they're installing a new system, I'm just automating it. So I'll just have to run some tests during commissioning.

 

[Purple_Dan]

What would I need to know about the system? I know the pressure and the flow coefficient of the valve. I also have a temperature sensor in there (we're not really using it for anything at the moment). I know it's just water, which we can make assumptions about. I don't know the volume of the system, but everything else I can think of is accounted for.

Thanks for all your input so far guys!

 

[montoyas7940]

The only extra info you need is how much volume changes as pressure changes in the vessel. The volume goes up due to the elastic nature of the vessel. It is like a balloon that does not get much bigger as you inflate it because it is really strong. But it does get a little bigger at high pressures. How much bigger (and at what pressures) is what you need to know. And then it becomes a simple problem. The valve at X position and P' pressure passes V volume/time which changes pressure by P/time. Rinse and repeat as pressure changes.

There are off the shelf controllers available.

 

[montoyas7940]

What do you test at 23000 psi ?

 

[Purple_Dan]

What do you test at 23000 psi ?

"Christmas Tree Valves" that go into the sea bed to extract oil. I don't want to be stuck up there too long because they can't afford to keep me there with the falling oil prices!

 

[HuskyNamedNala]

Running tests will take a life time! Call the manufacturer of the valve, ask if they can procure transient valve curves. I have had to do this for a similar project. Trust me, it is hopeless trying to predict and run tests on these unless you have a lot of time.

 

[Chestermiller]

You are determining the pressure drop/flow rate relationship for the valve from the information the manufacturer provided on the valve coefficient. This is step 1. But, to get the time dependence of the pressure drop, you need to consider what is happening in the part of the system upstream of the valve. What does the system look like upstream of the valve?

Chet

 

[Purple_Dan]

You are determining the pressure drop/flow rate relationship for the valve from the information the manufacturer provided on the valve coefficient. This is step 1. But, to get the time dependence of the pressure drop, you need to consider what is happening in the part of the system upstream of the valve. What does the system look like upstream of the valve?

Chet

Unfortunately, I haven't been on site yet. But I do have a P&ID. I'll upload the part of the P&ID that we're interested in as the test pressure is isolated from the source.
Also, I'm not sure how much of the P&ID I can show before I break the confidentiality agreement!

The pressure to the left of the isolation valve is 0psi at this point. The pressure after the control valve is going to drain, so that's 0psi also. I don't know what the HP outlet looks like, but that goes to the valve that we're testing.
The test transmitter is the pressure sensor.
It's all filled with water as it's a hydrostatic test. Tests can be anywhere between 50-23000psi.

 

[Chestermiller]

Sorry. I'm not too good at reading upside down.

Chet

 

[Purple_Dan]

Sorry. I'm not too good at reading upside down.

Chet

Apologies! Here you go!

 

[russ_watters]

You can't find the pressure drop across the valve unless you know the flow rate. So if you are changing the valve position, which changes both the pressure drop and flow rate, you need to know the characteristics of the entire system to find what the new pressure will be at the next increment of valve position.

 

[Purple_Dan]

You can't find the pressure drop across the valve unless you know the flow rate. So if you are changing the valve position, which changes both the pressure drop and flow rate, you need to know the characteristics of the entire system to find what the new pressure will be at the next increment of valve position.

I already know the pressure drop and the flow coefficient, from that I can determine the flow rate, but these are just instantaneous values. I want to know what the pressure drop will be after 1 second while I'm changing the valve position, which changes the flow coefficient. If I can determine that from the flow rate, how would I calculate what the flow rate will be in 1 second? I'm assuming I'll need the volume of the system...

It doesn't even have to be too exact, just a clue as to what a good position to put the valve in would be good. Then I can implement a control system based off that if I need to.
At the moment I am essentially completely blind as to how to implement this.

 

[HuskyNamedNala]

Like I told you, you won't find any analytical solution. You need to implement a predictor corrector algorithm, where based on the change in flow rate in the recent past, you estimate the derivative, then adjust the valve, re-measure the flow rate and correct. Repeat until you achieve steady state.
If you haven't taken a numerical methods class, you might be in trouble...

But from your posts, I gather what you are trying to do is use existing data to pre-calculate what the valve will be at steady state, rather than do a real time computation. Simply put, you won't be able to do this. Yes, theoretically there might be an exact solution to your problem, but practically the effort expended in looking for this solution will not yield a good return on your investment... unless you happen to solve the N-S equations, in which case you'll become a millionaire and world famous and won't have to work ever again.

The steady state valve curves will give you an approximate starting point. You will have to compute the derivative of the pressure drop in real time and estimate. This is the best solution I can think of.

 

[HuskyNamedNala]

Or... what you could do is open the valve, wait 10-20 seconds for steady state to be achieved, then readjust, repeat, etc. This method is not very elegant and is very slow, but it will work too.

 

[Chestermiller]

Are you trying to control the pressure drop at constant flow rate, the flow rate at constant pressure drop, or a combination of the two? If you're trying to control a combination of the two, then you need to quantify the interaction of the valve with the process equipment upstream of the valve (in terms of the fluid mechanics interactions). So far, aside from the control equipment, you haven't told us anything about the process equipment upstream.

Chet

 

[Purple_Dan]

Are you trying to control the pressure drop at constant flow rate, the flow rate at constant pressure drop, or a combination of the two? If you're trying to control a combination of the two, then you need to quantify the interaction of the valve with the process equipment upstream of the valve (in terms of the fluid mechanics interactions). So far, aside from the control equipment, you haven't told us anything about the process equipment upstream.

Chet

Other than that P&ID, I know relatively little about the equipment, I don't know lengths of pipes, diameters, volume or anything. I know that there's 0psi to the left of the isolation valve, 0psi below the control valve and the space inbetween can be between 50 and 23000psi.
If we take out all the unnecessary gumpf like gravity and pressure loss due to friction, what equation that will give me a prediction of the pressure? What variables will I need to know?

Thanks, I know it must be annoying not having all the data, but I don't have it either!

 

[russ_watters]

I already know the pressure drop and the flow coefficient, from that I can determine the flow rate, but these are just instantaneous values. I want to know what the pressure drop will be after 1 second while I'm changing the valve position, which changes the flow coefficient.

That doesn't make sense: If you already knew the pressure drop at each valve position, all you would need to do is divide the difference by how long it takes the valve to get to its next position.

 

[russ_watters]

If we take out all the unnecessary gumpf like gravity and pressure loss due to friction...

Pressure drop due to friction is the core issue you need to address!

 

[HuskyNamedNala]

You don't seem to understand, you're not going to get this analytically. Neglecting viscosity will get you Euler equations:
http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics )

You can't even simplify the equations for 1-D because to conserve mass you would need to include 2 velocity components; presuming the velocity along the length of the pipe will vary.

Why are you so adverse to implementing the method I told you about?

 

[Purple_Dan]

Pressure drop due to friction is the core issue you need to address!

Yes, but I only care about the friction at the valve, not in the pipes.

 

[Purple_Dan]

Why are you so adverse to implementing the method I told you about?

Because, while I agree with you and would love for it to be that simple, I don't think the customer will allow me to wait a few minutes until it's got it right. Also, I know there's a better way to do it, and I'm a bit of a perfectionist. Sorry.

Also, I don't see what's wrong with Euler equations, with enough iterations, you get pretty close to the right answer. At least, I did in Uni...

 

[Purple_Dan]

That doesn't make sense: If you already knew the pressure drop at each valve position, all you would need to do is divide the difference by how long it takes the valve to get to its next position.

I know the psi at each position, what I don't know is what the psi will be after 1 second if the valve is open at 50%, I don't even have a clue what it would be. Just a nice guess would be good.

 

[HuskyNamedNala]

Because, while I agree with you and would love for it to be that simple, I don't think the customer will allow me to wait a few minutes until it's got it right. Also, I know there's a better way to do it, and I'm a bit of a perfectionist. Sorry.

Also, I don't see what's wrong with Euler equations, with enough iterations, you get pretty close to the right answer. At least, I did in Uni...

It is ironic that you believe you know a better way to solve this problem, yet with more information than any of us on this forum you are asking around for solutions. Good luck with whatever method you chose. The analytical route won't be fruitful.

 

[montoyas7940]

I know the psi at each position, what I don't know is what the psi will be after 1 second if the valve is open at 50%, I don't even have a clue what it would be. Just a nice guess would be good.

Because you only know the valve flow vs vessel pressure, you can't know how much static pressure will change in the vessel over time without first discovering how much volume you have to remove to achieve the change.

 

[jim hardy]

Scientific Wild Guess here, a SWIG:Seems to me that with 23,000 psi and a needle valve this must be some sort of leakage test rig.

At those kinds of pressures we must think differently about steel pipes and valves. Instead of rigid bodies they become elastic like balloons, and compressibility of water becomes noticeable.
A steel cylinder expands when you pressurize it. So it holds more water. As you let water out through your valve pressure decreases as the walls relax.
30,000 psi in steel gives ~ 1 part in a thousand elongation.
If you know the ratio of wetted area to wall cross section of your pressure vessel you can estimate its pressure versus volume curve, .

Without knowing the elasticity of that Christmas Tree Valve, which is a function of its dimensions, ... it's a Un-SWIG.

I could be way off base, frequently am...
Do you know for sure what parameter it is of that "Christmas Tree" they're wanting to measure?

here's a lead for compressibility of water.

http://ipims.com/data/fe31/E3131.asp?UserID=&Code=3298

i hope I'm on track here. feel free to correct.

 

[Purple_Dan]

Scientific Wild Guess here, a SWIG:Seems to me that with 23,000 psi and a needle valve this must be some sort of leakage test rig.

At those kinds of pressures we must think differently about steel pipes and valves. Instead of rigid bodies they become elastic like balloons, and compressibility of water becomes noticeable.
A steel cylinder expands when you pressurize it. So it holds more water. As you let water out through your valve pressure decreases as the walls relax.
30,000 psi in steel gives ~ 1 part in a thousand elongation.
If you know the ratio of wetted area to wall cross section of your pressure vessel you can estimate its pressure versus volume curve, .

Without knowing the elasticity of that Christmas Tree Valve, which is a function of its dimensions, ... it's a Un-SWIG.

I could be way off base, frequently am...
Do you know for sure what parameter it is of that "Christmas Tree" they're wanting to measure?

here's a lead for compressibility of water.

http://ipims.com/data/fe31/E3131.asp?UserID=&Code=3298

i hope I'm on track here. feel free to correct.

Well colour me impressed, I hadn't taken that into consideration. Unfortunately, like many of the questions I have been unable to answer, they test all different sizes of Christmas Tree Valves!
In your expert opinion, would the water compress enough to drastically affect my calculations? And would the volume change much at 23000 psi? I suppose that would depend on the specifics of the pipes and valves, seeing as hot water bottles are harder to inflate than balloons...

I hadn't thought that the water might compress, as my A-Level Physics knowledge didn't cover that!

 

[jim hardy]

I hadn't taken that into consideration.

i think that's what the guys have been trying to say when they ask for system details and what volume you need to move.
If i hit on the right words, well, it was just serendipity...

In your expert make that very amateurish, jh opinion, would the water compress enough to drastically affect my calculations? And would the volume change much at 23000 psi? I suppose that would depend on the specifics of the pipes and valves, seeing as hot water bottles are harder to inflate than balloons...

Well, keep in mind I'm just an old electronics instrument guy from a power plant . I have watched over the shoulders of genuine mechanicals and hope some will chime in here.

hopefully the system shape and size are among those things you know but can't divulge.

Let's just oversimplify and SWIG that the steel is loaded to 30,000 psi in the steel not in the water.
Ordinary steel can take that stress without yielding, exotics several times more.
So the designers would set wall and fastener thicknesses to limit stress to a fraction (maybe 70%?) of yield at design pressure.
so every dimension would increase by 0.1%, (look up Young's Modulus)
hence volume at design pressure would go up by ~ 1.001^3 = 1.003 .

I used to take care of an acoustic system attached to our reactor vessel, made from 4 inch and more thick steel..
It emits creaks and groans and snaps as you pressurize it and the metal stretches.Compressibility of water is usually ignored.
Per that link it's in the range of 2 to 4 X 10-6 per psi and gets smaller as pressure increases.
If we just use 3E-6
then at 20,000 psi it might compress say 3E-6 X 2E4 = 6E-2 = 6%.
That's probably a high number because as you see from that graph its compressibility gets smaller as pressure increases.
And dissolved gas in the water will have an effect.
But for the first few thousand psi water might compress 3E-6 X 1E3= 3E-3 , 0.3% per thousand psi.

At least that's what it looks like to this old instrument guy,
>>>and i'd appreciate corrections by a genuine mechanical engineer.

Maybe you want to ask the client some questions related to these effects. At least he'll know you're thinking about them..
You might get laughed off the stage,
or he might exclaim "At Last, somebody who asks the right questions".Good luck and keep us posted. I look forward to learning.

old jim

 

[russ_watters]

Yes, but I only care about the friction at the valve, not in the pipes.

You can't know one without the other.

I know the psi at each position, what I don't know is what the psi will be after 1 second if the valve is open at 50%...

Maybe I'm just being thick, but it seems to me that the second part contradicts the first part.

Or it could be that since we don't know some critical details about the system, we are speculating based on guesses.

In the beginning of the thread, it seemed like you are saying you are trying to calculate how pressure drop varies with valve position, as the valve position varies with time. But now it seems like you are asking about how pressure drop varies with flow, at a fixed valve position, as the system changes. So which is it? And if it is the system that is changing, how is it changing?

My first guess about the system was based on systems I see every day (trying to fit it into everyday experience): Take a heating hot water system, close the isolation valve at the discharge of the pump, and turn the pump on. Then open the valve slowly and see how flow rate and pressure drop across the valve and pump change with valve position.

Or is Jim right that this is some sort of leak test? Are you opening the valve and seeing how long it takes for the system to drain and pressure to drop to zero? Also, is this valve really holding back 23,000 psi on its own, giving it an initial pressure drop of 23,000 psi when you open it? There are additional problems with that, because the water flow will be supersonic, some will flash to steam as it goes through the valve due to the pressure drop alone (actually, you might get steam and snow at the same time), and it will heat-up tremendously due to the extraordinarily high friction of the extraodinarily high flow velocity. Of course, if 23,000 psi is just the static gravitational head and the entire system is a loop and the pressure drop is really only 10 or 20 or 50 psi across the valve, that issue won't exist.

 

[Purple_Dan]

You can't know one without the other.

Maybe I'm just being thick, but it seems to me that the second part contradicts the first part.

Or it could be that since we don't know some critical details about the system, we are speculating based on guesses.

In the beginning of the thread, it seemed like you are saying you are trying to calculate how pressure drop varies with valve position, as the valve position varies with time. But now it seems like you are asking about how pressure drop varies with flow, at a fixed valve position, as the system changes. So which is it? And if it is the system that is changing, how is it changing?

My first guess about the system was based on systems I see every day (trying to fit it into everyday experience): Take a heating hot water system, close the isolation valve at the discharge of the pump, and turn the pump on. Then open the valve slowly and see how flow rate and pressure drop across the valve and pump change with valve position.

Or is Jim right that this is some sort of leak test? Are you opening the valve and seeing how long it takes for the system to drain and pressure to drop to zero? Also, is this valve really holding back 23,000 psi on its own, giving it an initial pressure drop of 23,000 psi when you open it? There are additional problems with that, because the water flow will be supersonic, some will flash to steam as it goes through the valve due to the pressure drop alone (actually, you might get steam and snow at the same time), and it will heat-up tremendously due to the extraordinarily high friction of the extraodinarily high flow velocity. Of course, if 23,000 psi is just the static gravitational head and the entire system is a loop and the pressure drop is really only 10 or 20 or 50 psi across the valve, that issue won't exist.

Okay, so what happens is this:

• I build the pressure in the system to the desired target pressure via a pump
• I isolate the test area from the pump pressure
• Drop the pump pressure to zero (not via a control valve)
• The test runs and the test valve has to pass certain criteria
• I open the hold valve after the control valve
• I start opening the control valve to maintain a desired psi/sec drop

That last bit is what we care about.

I am only changing the valve
position, but from the specification documents of the valve I know the flow coefficient for the % that it is open.
Using this, I know the flow rate (or thereabouts).

So if a gallon of water has just burst out in 1 second, how do I calculate the psi drop.

If we ignore the compressibility of water and the change in volume of the system, which appears to be negligible for the accuracy I require.

And there really could be 23000psi across that control valve. I look forward to seeing snow steam, it sounds awesome...

And I'd like to apologise to everyone for being insufferable thus far!

 

[montoyas7940]

So if a gallon of water has just burst out in 1 second, how do I calculate the psi drop.

If we ignore the compressibility of water and the change in volume of the system, which appears to be negligible for the accuracy I require.

Except for the change in volume and to a much smaller extent the compressibility of the water there would be no flow as you open the valve. The change in volume is providing the flow.