# I Thermal expansion of liquid into gas void

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1. Mar 22, 2017

### Karlos

I have a question about thermal expansion of liquid into a gas void.

Imagine a closed upright cylinder filled mostly with water – say 99%, and the remaining 1% is gas.

Now imagine that you heat the cylinder and its contents.

The water will expand by ΔV owing to thermal expansion. The gas, being also heated, will also attempt to expand. But as gas is much more compressible than water, the water will expand into the gas void space.

The Gas pressure will then increase owing to the fact that it has now been compressed.

But what happens if the thermal expansion of the water is equal to the void space? i.e. what happens if the water is heated to an extent that ΔV = 1% of the cylinder volume (gas void)?

Surely the gas cannot be infinitely compressed.

So my question is:

At which point does water expansion halt? Whats the minimum possible compressed gas volume? Does it eventually dissolve into the water?

2. Mar 22, 2017

### BvU

Hi,

As a chemistry graduate you know about[/PLAIN] [Broken] Henry's law and you also know about vapor-liquid equilibrium.
So you can answer your own question when armed with some appropriate physical property data.

In your 99% scenario the liquid water will fill the entire volume rather rapidly (table A-4: when going from 20 to 50 $^\circ$C, $v$ goes up by 1% ). Further increase in temperature (table A-7) will let the pressure increase enormously due to the very low compressibility of water. At such pressures gases usually dissolve.

And the container will explode.

Last edited by a moderator: May 8, 2017
3. Mar 22, 2017

### Staff: Mentor

I think the OP was thinking of pure water being present in the container (although I'm not sure about this), so Henry's law wouldn't apply.

Of course, there is no guarantee that the container will explode, if the container is made beefy enough. But, thermal expansion of the container (and the deformation of the container from the internal pressure) might have to be included (considering the tiny volume difference at the start).

Last edited by a moderator: May 8, 2017
4. Mar 24, 2017

### Karlos

Perhaps I ought to put the problem in context.

I work in the oil industry - in a typical oil well, you have a pipe through which the oil flows from the rock deep in the ground, up to surface. Then you have several concentric pipes going outward from this pipe (called ‘casing’), and you have an annulus between each casing. These outer casings are cemented in place up to a certain depth below surface.

The ‘cylinder’ in this case is actually a closed annulus, the ‘water’ is brine or mud/hydrocarbons, and the gas is most likely methane.

When the well starts producing oil, these annuli will be heated above ambient temperature at surface because of the fact that the oil is coming from a deep reservoir at around 150degC (it is around 60degC by the time it reaches surface).

I am trying to estimate a realistic pressure increase in the annuli that can be expected to be caused by the thermal expansion of the liquid and gas in the annuli as the produced warm oil heats up the surroundings.

I took a simple approach (combined pV = znRT for the gas and ΔV =βVΔT for the liquid, and did a bit of algebra) but I found (not surprisingly) that the gas pressure became extreme (un realistic) once the liquid expansion approached the initial gas cap volume (hence my question). If there is zero gas to begin with, then the problem is much more simple! (but that isn’t what im asking)

There will of course be some ‘ballooning’ of the pipe to soak up some of the increase.

I imagine that as the pressure increases, more and more gas will dissolve into the brine/mud and the pressure will become less extreme. The problem now is that, how fast does this process occur? Is it possible for the liquid to fill the container and just eat up all the gas before the high gas pressure can be registered?

I should point out that my days as a chemist are a very very long time ago now, so my memory of diffusion dynamics are sketchy at best!

I can put the question to a 'petroleum engineering' forum, but I thought it would be good to get some ideas that just address the raw science behind it.

5. Mar 24, 2017

### Staff: Mentor

Isn't the annulus fluid circulated?

You made a poor choice for the equation of state of the liquid. A more appropriate choice would be:$$\rho=\rho_0e^{-\beta(T-T_0)+\alpha(p-p_0)}$$where $\rho$ is the water density at temperature T and pressure p and $\rho_0$ is the density at temperature $T_0$ and pressure $p_0$; $\alpha$ is the coefficient of volume thermal expansion and $\alpha$ is related to the bulk modulus. Basically, you omitted the compressibility of the liquid. You also neglected the possibility that some or all of the gas could dissolve in the liquid.

6. Mar 24, 2017

### BvU

You're welcome.
Definitely. In fact you don't have a problem in the usual PF sense yet: there is some concern and you want to get an insight on the ramifications. The mission should be to formulate a reasonable and sensible problem that can be useful in this matter. You tried in post #1 but I think we can agree that that is not a problem that can help with this.
A wolf is sheep's clothing ? I work in the chemical industry and they pay experts lot of money to work on issues like yours, so that they can get an operating permit. There's a good chance your issue has been dealt with extensively just to obtain such a permit -- unless your activities are in the darker parts of the world. Ask around !

Are you seriously involved in this or just interested ? In the first case hire an engineering consultant with the appropriate expertise and legal/insurance cover: PF can't bear the legal responsibilities.

- - -

In the second case (this is PF and if I wasn't interested and intrinsically helpful I wouldn't be here):
Find the official drawings, have them verified and make a sketch that takes the physics essentials into account. What's the weakest point ?, When is the risk at a maximum, etc. Gather all relevant physical properties. Chet already indicated that your 'vessel' may be elastic. I ensure you your pipe is. What kind of cement ? Zero porosity is a fiction. The mud and water is sealed in in the annulus -- how well ?

Post #2: you can assume all the gas goes into the liquid if the pressure is high enough. From then on what you have is an expanding liquid. And therefore an expanding pipe. Your volume is not fixed.

the gas is a detail unless there's enough of it. At some point you are back to your simple problem.

You bet. And if there's nothing else then that's all you have to consider.

yes. It is a kind of respite until your simple problem goes into existence.

You're doing fine.

7. Mar 27, 2017

### Karlos

Thanks to both once again,

You've given me some good pointers, Ill go away and re-think the problem and explore the industry experience a little.

Cheers!