## Rate of water vapour transfer vs. barrier thickness?

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### Rate of water vapour transfer vs. barrier thickness?

Hello

I am new. I am working on a moisture diffusion problem. I am trying to discover how important barrier tickness is to the rate that water vapour can get across a barrier.

Suppose I have a 1m x 1m barrier of say PE (polythene) sheet and it lets in X grams of water per hour through water diffusing through the plastic itself. Now suppose I make the barrier of polythene four times thicker. Does the rate of water transfer go down to 1/4 of what it was, or a 1/16?

i.e. Is the rate of water vapour transfer:
A. Inversely proportional to the just thickness of the barrier OR
B. Inversely proportional to the SQUARE of the thickness of the barrier?

Also does it matter if the water is transferring under pressure or by simple passive diffusion?
My thinking is that when under pressure it would flow much like electrical resistance - in which case to get the total resistance you simply add up the resistance of each bit of stuff in the way.

HOWEVER it seem to me that passive diffusion is different because the water molecules at any point are just as likely to do their random walks in the wrong direction - in which case the square of the distance seems more intuitive!

Either way we need to find a way to explain how ice not that far down in the Antarctic is exactly the composition that is was 100s of thousands (millions?) of years ago.

Many thanks

J
ship69

### Re: Rate of water vapour transfer vs. barrier thickness?

ship69,

Welcome to the forums!

I can't remember this stuff and apparently I'm too tired to derive it correctly. Below are my quick notes, but they're highly likely to contain errors.

I think that you're asking about a fairly straightforward application of Fick's first law, so
ship69 » November 19th, 2015, 2:37 pm wrote:i.e. Is the rate of water vapour transfer:
A. Inversely proportional to the just thickness of the barrier OR
B. Inversely proportional to the SQUARE of the thickness of the barrier?
would be $\left(\text{A}\right)$ in the passive case. This is, the flux $\left(\left[{\equiv}\right]\frac{\text{mol}}{{\text{m}}^{2}{\times}\text{s}}\right)$ is proportional to the concentration.

In the more pressure-driven case, it should be more like osmosis. In this case the profile of the driving profile should be changing over the barrier, so then the concentration profile changes more sharply than in the passive case.

Stuff like Fick's law is a first-order approximation thing, and you'll probably want to improve upon it for actual industrial processes. For example, when we're making computer chips, we diffuse ions from a vapor into the silicon. To bring this process up to the point where we can reliably produce computer components, the Deals-Grove model was constructed. I'd note that Deals-Grove is concerned with the case where the diffusing component reacts with the solid phase, which likely isn't appreciably an issue in your use case.

For more examples, I'd suggest using Google images to search for 'diffusion gradient profile'.
Natural ChemE
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### Re: Rate of water vapour transfer vs. barrier thickness?

No I don't get this.

First of all, let's simplify concentration. Consider a 100% water tight building but with one wall consisting of plastic (PE) film through which some water can diffuse. In this situation imagine that we have a steady, high concentration of water vapour on the outside of the building and a steady lower concentration of water inside the building (due to a dehumidifier running).

So the concentrations of water in the air both sides are now constant.

Yes, it is intuitively obvious that the Flux (i.e. the amount of substance[in this case water] that gets through per unit area per unit time) will be directly proportional to the difference in concentrations [of water].

It is pretty obvious that pressure driven and non-pressure driven are two completely different situations, and completely different mathematics must apply. In the pressure-driven case the walk of individual molecules is absolutely not random whereas in passive diffusion they are.

PRESSURE-DRIVEN
By analogy, suppose you had a setup consisting of long hosepipe (say 10 meters long) and you had filled it full of sand (to create resistance) and you had then turned on a tap to fill the sand with water.

Now suppose we attach the hosepipe to a different water tank and this time all the water in the tank contains blue ink or dye. If attached with a steady pressure it is easy to imagine that after a while at least some of the dye arriving at the other end of the hosepipe. This feels like conventional electricity flow Ohm's law V=IR

i.e. I (the flow of electrons) equals the V (the pressure of electrons) divided by R (the resistance).

Put another way every time a molecule moves (i.e. 'wiggles') it is more likely to move in a certain direction. The progress is likely to be steady over distance.

And if we hold the two concentrations steady, and the physical pressure steady and allow time to run until we get a steady state, if it takes 1 hour for 1 gram of dye to get past the 10 meters mark it would take 2 hours for 1 gram to get to past the 20 meters mark.

==> Ficks 1st law fits.

NOT PRESSURE DRIVEN
However now consider the situation without pressure. i.e. You simply on the tap but there is no pressure driving the dye-containing water. In this case the only thing driving the dye forwards is a random walk. i.e. It's just as likely for the molecules to go forwards as backwards. Although the dye will quickly cover the first 1mm, it's easy to imagine that no measurable quantities of dye will arrive after a whole year - maybe not even after 100 years.

And if we hold the two concentrations steady, (and the physical pressure remaining the same on both sides) and allow time to run until we get a steady state, the amount of bulk flow [i.e. flux] over a longer distance (such as 10 meters) will be very close to zero.

Taking an individual molecule it's progress, over a very short distance will be extremely rapid. However the further and further it needs to go, the greater and greater are the chances that it's overall walk will not happen to have gone in that direction. i.e. The further done the pope you go, the less and less likely it becomes than any given molecule will happen to have randomly walked that far!

==> Surely in this situation rate of bulk flow is not directly proportional to distance!

With thanks

J
ship69

Bump?
ship69

### Re: Rate of water vapour transfer vs. barrier thickness?

Simply put are we saying that diffusion and flow under pressure follow the same equations?
ship69

### Re: Rate of water vapour transfer vs. barrier thickness?

ship69 » November 24th, 2015, 6:03 am wrote:Simply put are we saying that diffusion and flow under pressure follow the same equations?

In general we can say that both pressure-driven flow and diffusion are effected by a driving force. Driving force is a gradient of some physical quantity:
• Flow under pressure:
• Material flows from higher pressure to lower pressure.
• Resistance described as viscosity.
• Background medium is called aether.
• Viscosity is not known to be a function of aether.
• So, aether is considered extraneous.
• Flow calculations are complicated by other reactions to pressure, primarily compression.
• Diffusion:
• Material flows from higher concentration to lower concentration.
• Resistance described as diffusion coefficients.
• Background medium is called the bulk phase.
• Diffusion coefficients are known to be a function of the bulk phase.
• So, bulk phase is an important factor.
• Flow calculations are complicated by other reactions to concentration, primarily chemical reactivity.
Natural ChemE
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### Re: Rate of water vapour transfer vs. barrier thickness?

> Resistance described as viscosity.
The viscosity is surely a property of the fluid itself. e.g. treacle would have a much higher viscosity than water and would therefore flow down a pipe slower.

> Background medium is called aether.
e.g. A pipe full of jammed solid with sand would allow water to pass through faster than a pipe jammed solid with clay.

> Viscosity is not known to be a function of aether.
Confused: Are you saying the medium (e.g. sand or clay) is irrelevant?

> So, aether is considered extraneous.
What do you mean by "extraneous"? Do you mean it doesn't affect the speed of flow? Or that it's too hard to calculate? Or by convention is never part of the calculations or what?

> Background medium is called the bulk phase.
This sounds incorrect. Surely "bulk phase" means the majority of the liquid - so as to distinguish with "surface phase" where interactions like "surface tension" and chemical reactions happen.

Either way you are failing to answer the question I just asked: The jargon may be different but are the equations the same?

P.S. Given that the water that goes in remains unchanged when it comes out, for reasons of simplicity, I am assuming chemical reactions to be irrelevant.
ship69

### Re: Rate of water vapour transfer vs. barrier thickness?

ship69,

This PDF kinda explains some of the topic, with the concentration profile over the film thickness profile in Slide #3.

Note that this profile assumes that there's an "other side", i.e. the right-hand side in that figure, that receives the diffusing material. By contrast, consider the case in which we drop a chunk of polymer into a pool of water; then we're more interested in saturation time, at which the maximum amount of water ${M}_{\infty}$ is dissolved. Slide #9 shows the evolution toward saturation in terms of $\frac{{M}_{t}}{{M}_{\infty}}{\rightarrow}1$ as $t{\rightarrow}{\infty}$. Unlike in the case in Slide #3 which has a sloping profile at steady state, the saturation case in Slide #9 ultimately comes to a more flat concentration throughout the entire material as maximum saturation is achieved.
Note:
Since in theory it takes infinitely long to actually reach full saturation, saturation time ${t}_{\text{sat}}$ is often defined as $t$ such that $\frac{{M}_{t}}{{M}_{\infty}}=0.99$, or some other similar approach to the theoretical maximum. In general it's not meaningful to stress about the "real equilibrium" since other effects will take precedence at infinity, e.g. protons evaporating, or less dramatically the plastic itself dissolving.

There're a lot of complexities we could consider, or we could just stick to an idealized Fick's-first-law perspective. I'm having some trouble judging exactly how technical to get.

Could you explicitly state your question in a more complete way? This would help me provide a better answer in-line with what you actually want to know.
Natural ChemE
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### Re: Rate of water vapour transfer vs. barrier thickness?

ship69 » November 20th, 2015, 7:00 am wrote:Taking an individual molecule it's progress, over a very short distance will be extremely rapid. However the further and further it needs to go, the greater and greater are the chances that it's overall walk will not happen to have gone in that direction. i.e. The further done the pope you go, the less and less likely it becomes than any given molecule will happen to have randomly walked that far!

This situation is called dynamic equilibrium. Sure stuff's moving all around, but the apparent flows that we're concerned with are actually the net flows.

For example, say that water's diffusing through plastic. Sure water molecules may be randomly walking at the midpoint in the plastic, but since there's more just before the midpoint than just after it, the net movement is away from the source and toward the sink. This is, if the midpoint is at $x=\frac{1}{2}$, then the net flow is positive because $\large{{\left\frac{{\text{d}}C}{{\text{d}}x}\right|}_{x={\frac{1}{2}}}}{\gt}0$.

As you can reason, the net movement in a random walk situation like this should be proportional to the concentration gradient. This is why we say that the concentration gradient is the driving force.

As you would expect, when the plastic is fully saturated with water due to having been immersed in water for a long period of time, water still diffuses through the plastic. However, we say that the net flow of water through the plastic is essentially zero because randomly walking water won't prefer one direction over another when the plastic has an even concentration of water throughout, as at equilibrium.
Natural ChemE
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### Re: Rate of water vapour transfer vs. barrier thickness?

OK, here's the thing. Although mathematics may be the most concise way to describe certain things, it is also an extremely painful way to describe other things. Moreover in my case as soon as you resort to mathematics then, despite having a degree in zoology, you lose me.

Either way something strange is going on when the gas bubbles trapped in ice only 50 - 100 meters down in ice are said to accurately represent what was happening 100s of thousands of years ago. SURELY something here is NON-linear.

As far as I am concerned if one can't explain something in simple terms then one probably doesn't understand it properly oneself.

I find it enjoyable to think about things in intuitive, non-mathematical terms, but it that doesn't work for you then I should probably bow out of this discussion gracefully now.
ship69

### Re: Rate of water vapour transfer vs. barrier thickness?

ship69,

Sorry for the slow response; it's been busy lately.

Just to clarify, are you asking me to:
1. use fewer symbols, e.g. "concentration" instead of "$C$"; or
2. avoid concepts like fluxes, equilibrium, concentrations, etc.?
Basically I'm not sure what "math" above may've been a source of confusion.
Natural ChemE
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