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 Andrzej Matuszkiewicz September 26, 1999 13:39

Water-hammer

If a valve is abruptly closed in a pipe where liquid flows, it creates a series of pressure waves going back and forth, phenomenon known as a water-hammer. It seems to me that this problem can be difficult to deal with for both density and pressure based numerical methods. It is a zero Mach problem and certainly it exist because of pressure-density coupling.

I would appreciate any comments and/or references.

Andrzej

 John C. Chien September 26, 1999 14:21

Re: Water-hammer

(1). It is hard to get that experience. In our daily life, it takes time to shut down the faucets in the kitchen or bathroom. (2). In a power plant, it could happen, maybe under emergency conditions. So, I don't know whether it is a real problem or just a phenomenon (3). There are many commercial CFD codes around, I guess, it should be a simple case to simulate the flow in a straight pipe after one end is closed suddenly. Maybe, the commercial CFD code users would like to take this challenge....flow moving in a straight pipe at a constant speed initially, then with one end suddendly stopped (sealed), the subsequent flow development.

 D.M. Lipinski September 27, 1999 05:02

Re: Water-hammer

Water hammer phenomena are described by the hyperbolic PDEs. The method of characteristics is very well suited for the analysis of such phenomena. You can search any database for the "hydraulic transients" keyword and you'll get plenty of references. One dimensional analysis (of one or multi-phase systems) is the most common, e.g. for the pipe networks. For the start, you may have a look in the book by Chaudhry (Chaudhry, M.H., Applied Hydraulic Transients, 1979(?)).

regards

DML

 Heinz Wilkening September 27, 1999 11:44

Re: Water-hammer

Hi,

as John already pointed out, in a ideal world the fluid has to deaccelerate to zero in zero time, therefor infinite acceleration and also infinite pressure.

But nature knows it ways out of this dilema by the combination of different mechanisms, I just can think of three imidiately but there are more I guess.

1.) As John pinted out, a valve does not close in zero time.

2.) Even Water and other liquids are somewhat compressible.

3.) The pipesystem itself react not stiff but a little elastic.

As you may not now a priori under which conditions which point is more relevant, you have to take into accound all. In resent past I heard of acctive research projects doing that. If you are interested I can try to find them again.

Ciao Heinz

 andy September 27, 1999 12:26

Re: Water-hammer

I can see why you would view the phenomena as a problem for a traditional density based formulation but why is it a problem for a pressure based formulation? These have problems for Mach Numbers above 2 or so (where the physics strongly indicates that "density transport" is what is going on) but not at low speeds.

 Andrzej Matuszkiewicz September 27, 1999 23:23

Re: Water-hammer

Thanks for all answers. It is true that the closing time is always finite, but in an emergency situation it can be short enough to cause numerical problems. Water-hammer is of importance for nuclear reactor safety, pipeline systems, boilers overpressure, etc. All these systems have check-valves that go on if the pressure or temperature on one side exceed some limit value. That creates water-hammer with a maximum pressure that is a function of closing time, so parametric numerical studies would be helpful for design of safe closing systems. It seems to me that it may be a problem for pressure based methods as well because compressibility cannot be ignored since it is a main driving force for these phenomena. In case of liquid the pressure-density relationship is very steep, so the new time step pressure may not be accurate enough for a good evaluation of new time step density. That in turn may corrupt the stability of a pressure based numerical scheme. Anyway, the best way to check it out is to try it, so I will. Heinz, any references are very helpfull.

Thanks a lot.

Andrzej

 andy September 28, 1999 04:00

Re: Water-hammer

I was referring to compressible pressure based methods not incompressible ones. Broadly, these behave pretty much the same as density based methods in the transonic and the lower end of the supersonic range but have problems (of the serious kind) going to higher speeds. I assume they are implemented in some commercial programs. Can anyone confirm this?

 Sergei Chernyshenko September 28, 1999 12:37

Re: Water-hammer

In reality, this phenomenon involves the elasticity and inertia of the tube wall, which are more important in this case than the compressibility of the fluid. This phenomenon can be quite accurately described within 1D approximation, and analiticly. This is very old staff, famous american physicist Robert Wood used the theory to determine the location of the leakage in the city water supply system, and the theory was developed, if my memory does not fail be, by Joukowskii. May be about 100 years ago, and, what a suprise (:)) it still works. I do not remember an exact reference now, but, certainly, this can be found in textbooks on hydraulics.

 clifford bradford September 28, 1999 18:55

Re: Water-hammer

actually john it is a very sserious problem in complex piping systems such as power plants. my brother who worked in a power plant tells me that control systems have to be in place to avoid it as it can lead to pipe failiure

 Andrzej Matuszkiewicz September 28, 1999 21:23

Re: Water-hammer

Do you know any reference to compressible pressure based methods?

Andrzej

 xiaofeng yang September 28, 1999 23:37

Re: Water-hammer

if the problem can be assumed to one dimensional flow, it could be solved by 1-D method of characteristics which has been well developed and found its application in numerical simulation of the high-pressure fuel injection system for the diesel engines. In the injection system, a valve (a needle valve inside a nozzle) is abruptly opened and closed with a small injection duration. I think there is a similar phenomenon with the water-hammer.

 John C. Chien September 29, 1999 10:17

Re: Water-hammer

(1). Your are right. (2). I think you can install some extra air chambers along the pipelines, or install the flexible joint in-line. (3). Or some smart valves with Fuzzy Control Logic chip in it to turn off the flow smoothly.

 clifford bradford September 29, 1999 11:29

Re: Water-hammer

you can get in contact with professor Narendra Simha at the university of Miami's Mechanical engineering department. they had a seminar presenter there last year who spoke about water hammer. Dr Simha can give you the person's contact information

 Andrzej Matuszkiewicz September 30, 1999 00:11

Re: Water-hammer

I have not dealt with the problem in the past but I did start by reading a paragraph in textbook

Mechanics of Fluids by M.C. Potter & D.C. Wiggert, Prentice Hall, 1977, pp. 563-567.

This is what I learned. They derived equation

Delta P = - (rho*Delta V)*a

where Delta P is pressure amplitude, rho is density, Delta V is velocity jump, and a is pressure wave speed. They refer to it as the Joukowsky equation. To my lay mind (rho*Delta V) represents an inertia. Wave speed a includes both compressibility of fluid and elasticity of pipe wall. Inspection of their equation for wave speed (11.5.16) indicates that it is zero (so is pressure amplitude) for incompressible fluid, however it is not zero for infinitely rigid pipes. It means that the phenomenon exists without pipe elasticity but do not exist without fluid compressibility.

Sometimes, it is very instructive to go through examples in textbooks. Their example 11.11 shows that for water flow in a steel pipe only 15% of pressure wave amplitude is due to the pipe elasticity but as much as 85% is due to the water compressibility. Even for a layman as I am it is not difficult to figure out why it happens. Generally, solids are more rigid than fluids.

Regards,

Andrzej

 Sergei Chernyshenko September 30, 1999 06:21

Re: Water-hammer

Hi, Andrzej.

The equation you cite is a standard relation accross a shock, and can be found in any textbooks on gas dynamics. lt can be derived from mass and momentum conservation. It is difficult to say which factor on the RHS is inertia, they all are to a certain extent. It should be applicable to a pipe with rigid walls, certainly. For elastic, it might be more complicated, since the cross-section can change across the shock, too, but I do not know.

>Inspection of their
>equation for wave speed (11.5.16) indicates that it is zero >(so is pressure
>amplitude) for incompressible fluid,

Wave speed in incompressible fluid is not zero, it is infinity, if the walls are rigid. So, is it not some inconsistency? Check what happens when the walls are rigid. Here can be a key to deeper understanding.

>It means that the phenomenon exists without pipe elasticity >but do not
>exist without fluid compressibility.

Kghm. Suppose, I have a pipe with very thin and very elastic walls, say, a rubber pipe, and I impose a sudden pressure drop at one end. Then, compressibility of the fluid will hardly be important. However, the cross-section will change, and this will propagate to the other end of the pipe, and not with an infinite speed, of course. So, the phenomenon exists in incompressible case. Whether it is described by a particular theory is another question.

>example 11.11 shows that for water flow in a steel pipe only 15% of pressure wave
>amplitude is due to the pipe elasticity but as much as 85% is due to the water
>compressibility. Even for a layman as I am it is not difficult to figure out why
>it happens. Generally, solids are more rigid than fluids.

Why, Andrzej, surely you understand that this depends also on the thickness of the walls, and quite strongly. It is compressibility of the fluid compared to flexibility of the walls what counts.

(This is not important for water-hammer, but are you sure about that rigidness? Fluids flow easily but they do not change their volume easily, compare the numbers, the compressibility. Or just the speed of sound. I do not think the difference is really that large.)

So, in your place at this point I would ask myself, am I sure that I understood the text correctly? I am also sorry to say that there are books with are just incorrect. I am in no way implying that the particular book is bad, I am only saying that if you wish to take something from a book without really understanding it, then a good practice is to consult several books.

As we say in Russia, I do not know an English equivalent, there is no king road in science (seems some teacher once told it to his royal pupil somewhere in Europe). So, trust your brains more than books and work hard to understand what really are in books.

And, finally, if 1D approximation is sufficient for you than it is just a wave equation quite easy to solve numerically, too. This, I believe, unswers your initial question.

With a hope to help,

Yours Sergei

 Andrzej Matuszkiewicz September 30, 1999 21:54

Re: Water-hammer

It was my slip. Of course the speed of pressure wave in incompressible fluid flowing in an inelastic pipe is infinite but in no way it changes the general conclusion of my message. The expression for the speed of pressure wave I used accounts for pipe wall thickness, bulk modulus of elasticity of fluid (B), and elastic modulus of pipe wall(E). It looks as a reasonably good approximation. For water and steel

B/E ~ 0.01

Thus, for any combination of fluids and solids that are of practical importance the influence of fluid compressibility is on the order of 85% and that of pipe elasticity is on the order of 15% as in the cited example. Even though the pipe elasticity is not negligible, it is much less important than fluid compressibility.

Andrzej

 Sergei Chernyshenko October 1, 1999 07:09

Re: Water-hammer

Hi, Andrzej,

>Of course the speed of pressure wave in incompressible fluid
>flowing in an inelastic pipe is infinite but in no way it changes the general
>conclusion of my message.

Of course it does not. The absence of the influence of the wall thickness on the wave speed does. By the way, all this is not that simple. Really, the amplutude of the pressure wave is determined by the amplitude of initial disturbance (Hit more hard, the wave is more strong). The corresponding wave speed is then determined by the pressure drop. There is a difference between a speed of a shock (that was in that formula you cite) and speed of the propagation of small disturbances. This is a long story, just check you understand all this.

But mutual importance of the two effects can be estimated from their contribution to the wave speed anyway, so we can go on.

>The expression for the speed of pressure wave I used
>accounts for pipe wall thickness, bulk modulus of >elasticity of fluid (B), and
>elastic modulus of pipe wall(E).

OK. So what happens with that ratio of contributions from compressibility and wall elasticity as the wall thickness tends to zero?

>Thus, for any combination of fluids and solids that are of >practical importance
>the influence of fluid compressibility is on the order of >85% and that of pipe
>elasticity is on the order of 15% as in the cited example.

Independently of the wall thickness? This is quite interesting indeed, but I would like to be sure. Could you check carefully, and confirm? I have no books handy now. I would be grateful. You see, the idea that fluid compressibility is more important even if the walls are very very thin (and therefore very very elastic even if made of steel) does not agree with my physical intuition. So, I am worried. It should depend on the wall thickness, not only the wall material.

Sure, there are pipes where the wall compressibility is not important at all. But it seems that when I was taught that stuff (kghm, it was more than 20 years ago, I could forget) they told us that taking into account the wall elasticity was the key element of the theory, at least as far as the practice was concerned, so I expected another ratio for practical applications indeed.

Yours Sergei

 Heinz Wilkening October 2, 1999 09:25

Re: Water-hammer

Hi Sergei,

> Sure, there are pipes where the wall compressibility
> is not important at all. But it seems that when I was
> taught that stuff (kghm, it was more than 20 years ago,
> I could forget) they told us that taking into account
> the wall elasticity was the key element of the theory,
> at least as far as the practice was concerned, so I
> expected another ratio for practical applications indeed.

I agree with your physical intuiation, but for solids you should not mix the expressions compressibility and elasticity. In the event of a waterhammer you mean elasticity, do you? This elasticity will cause the crossetion of a pipe to increase while a pressure peak in a fluid is passing by. Depending on the compressibility of the fluid this will cause the pressurepeak to be lower.

But how can we say one efect is 85% or 15% from what? If the pipe is not elastic and the fluid is total incompressible the pressurewave would run at an infinite speed which an infinite pressure amplitue I guess!?

Ciao Heinz

 Sergei Chernyshenko October 2, 1999 10:09

Re: Water-hammer

Hi, Heinz,

>for solids you should not mix the
>expressions compressibility and elasticity.

This (not mixing them) is exactly what I did.

>But how can we say one efect is 85% or 15% from what?

Sure, this is somewhat uncertain. What I usually mean in such case is: let us use a theory neglecting wall elasticity and compare with experiment, the error may be, say, 15%. Then, we neglect compressibility instead. The error is, say, 85% (they do not necessarily add up to 100% but this is not so important). Generally, we can compare not with an experiment but with a theory which takes into account both effects, if experimantal data are unavailable. This, I believe, is what Andrzej did.

>If the pipe is not elastic
>and the fluid is total incompressible the pressurewave >would run at an infinite
>speed

Yes.

>which <misprint? I read it as with> an infinite pressure amplitue I guess!?

For a suddenly closed valve, yes. But if you close the valve not suddenly, the wave is formed anyway, it moves with an infinite speed, but its amplitude is finite. I could comment further on this, but, anyway, what is your point here?

Yours Sergei

 Rainer Kurz October 4, 1999 18:32

Re: Water-hammer

Yes, this is a real world problem. However, I think there might be even an analytical slution if secondary effects (such as friction nad densityi cahnges) are negleceted.

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