Partial thermodynamic equilibrium and Gibbs energy minimisation

[lalbatros]

Believe it or not, but working in a cement plant led me to an interresting question in thermodynamics.

The original problem is related to lumps formation in a silo containing hot cement.
It involves gypsum dehydration and the reverse reaction, hemihydrate (or plaster) hydration:

CaSO4*2H2O = CaSO4*0.5H2O + 1.5 H2O(g)​

in words:

gypsum = hemihydrate + 1.5 H2O(g)​

The cement fed to the silo contains typically 2%(mass) of gypsum, 2%(mass) of hemihydrate, 96%(mass) of ground clinker, and 50%(volume) of air.
Since the cement is hotter in the center, more water vapor is produced there by dehydration.
This vapor migrate to the colder side where it eventually produces the reverse reaction: hemihydrate hydration.
This produces lumps near the walls of the silo.

So far, so good. I had also a good time evaluating the moisture content in air mixed with cement when the dehydration reaction saturates. It increases exponentially with cement temperature. And for sure the rate of this process must be linked to this amount of water vapor.

But then came the bad idea: in the end, the water vapor plays only a "kinetic" role: it permits the displacement of water from the center to the edge of the silo. If the amount of water vapor is negligible compared to the tons of gypsum/hemihydrates, it should play no role in the calculation of the thermodynamic equilibrium.

Therefore I considered a new model:

- I assumed that the temperature profile in the silo changes much more slowly than the chemical reactions proceed
- I assumed Gibbs energy minimisation can then be applied in this situation to calculate the (partial) equilibrium

In other words, I considered the following reaction:

a (hot gypsum) + b (hot hemihydrate) + c (cold gypsum) + d (cold hemihydrate)
=
a' (hot gypsum) + b' (hot hemihydrate) + c' (cold gypsum) + d' (cold hemihydrate)​

and I calculated the equilibrium of this reaction by Gibbs energy minimisation.

And now, the problem is: the equilibrium calculated from this last reaction tends to assign the gypsum to the hot region at the expense of the gypsum in the cold region. The contrary of the expected result.

And this is where you can help me:

- have I made a mistake in the reasoning
- have I overlooked something
- is there more to read about such kind of "partial equilibrium"
- ...

your toughts and suggestion are welcome.

Michel

 

[Bystander]

You might want to refresh your memory as far as enthalpies of reaction --- or, do a little dinosaur hunting (mixing plaster for jacketing bones) --- you are examining a system that exhibits counter-intuitive behavior.

 

[lalbatros]

Bystander,

I appreciate your grass-root comment.

But, please, note that it is not the system that exhibits a strange behaviour: the hot gypsum (90°C) dehydrates in the center of the silos and the water vapor recombines with the hemihydrate (or plaster) in the cold region of the silos (there are no dinausor artifact there, but that doesn't matter). This is well known since decades by all cement manufacturers and this is why they cool cement below 60°C before storage. Note also that gypsum in housing is a fire-retardant precisely because of the heat necessary for this dehydration that occurs mainly above 80°C.

I am doing process modeling on a daily basis for a living and I like this job quite a lot.
This time I wanted to have some fun over the weekend by considering this elementary question of gypsum dehydration-hydration in a unusual way. And this led me to the observation:

Gibbs energy minimisation cannot be applied as such to this inhomogeneous system​

Over night I think I understood the reason for that: the process of evaporating crystaline water from gypsum and rehydrating it at a lower temperature could produce some work due to the (saturation) pressure difference between the hot and the cold side. This work is lost in the actual process. This potential loss of free energy is not taken into account in my balance and this is probably why I get this wrong result.

However, my understanding is not yet satisfactory. Specially because I still want to know if the actual equilibrium (with a stationary temperature gradient) can be obtained from a minimisation principle. And I would like mainly to understand the physics.

Hope you enjoy the question,

Michel

 

[lalbatros]

the question exposed more clearly

The attached picture shows clearly the process I am taking as example.

https://www.physicsforums.com/attachments/6822

Let's consider first the equilibrium when the vessels are not connected:

Gibbs energy minimisation gives the saturation pressure of the dehydration reaction in the two vessels.
The result is the well known equation

$$\Delta G = 0$$​

Applied to gypsum dehydration it gives:

$$\Delta G = G_{hemigydrate} + 1.5 G_{H2O(v)} - G_{gypsum} + 1.5 RT ln p_{H2O} = 0$$​

The calculations then give the final equilbrium (un-connected vessels):

for the vessel at 90°C a saturation pressure of 0.7 atm
for the vessel at 60°C a saturation pressure of 0.2 atm​

When the vessels are connected, vapor should flow from the hot vessel to the cold vessel:

It is clear for the above discussion that when the vessels are connected the flow of water vapor will persit until:

- either all the gyspum is depleted on the hot side
- or all the hemihydrate is rehydrated on the cold side​

There is no need in this simple example to derive the final result from any minimisation principle, since the end result is easily obtained.

However, it is strinking that:

• when the vessels are not connected, a minimization of the Gibbs energy provides the equilibrium pressure in both vessels
• while when the vessels are connected, the Gibbs energy minimization does not apply (I tried, naïvely, and it gives the wrong opposite result)

In addition, in more complicated examples, a minimisation principle might be quite useful, if one exists. This would be the case when more that two minerals are involved as well as more than two temperatures.

Any suggestion or discussion would be welcome.
Am I wrong somewhere?
Is something else than G to be used in this (partial equilibrium) situation?

Thanks

Michel

ps:
It is well known that a fixed temperature gradient can lead to an unstationnary behaviour, even for an infinite time. Therefore my question may be very stupid, since a partial equilibrium may not exist at all, generally speaking. However, in my example, it is clear that such a situation will not occur, and a stationary equilibrium will be reached. This situation looks very much like a solid-solid reaction mediated by the water vapor. Therefore, it is maybe not totally wrong to expect a minimum principle to apply in this situation.

 

[Bystander]

'Tain't clear whether you're more interested in "irreversible thermo" along temperature gradients or the lumpy cement: in the first case, Ch. 28 in Lewis and Randall is a good place to start, not going to get you across the finish line; second case, you'll need to sort industry "folklore" from fact, hunt up real data on the anhydrite-hemihydrate-dihydrate-water equilibria as functions of temperature, water activity over calcium aluminate and silicates, strength of dihydrate "cementation" of mineral grains (zero), and make the judgment call whether 4% of the mix "wags" the rest of the dog.

 

[lalbatros]

Bystander,

My interrest is clearly in thermodynamics more than in lumps.

I use these thermodynamic data for much of my daily work since a long time.
The first source of data I use is https://www.amazon.com/gp/product/3527287450/?tag=pfamazon01-20.
Other specialised sources are available for pyroprocessing and for hydration chemistry.
Therefore, the data are not the problems, but the principles are my motivation.

You are right to mention silicates and aluminates for the real world discussion of my example.
These can react faster than hemihydrate, and they can even contribute to the dehydration of gypsum, at high temperatures. And alkaly-sulfates play also a big role here.

But my first interrest, right now, is not real world.
All the details of the real-world chemistry in my chosen example can be dismissed.
Say that whe have only sulfates and inert minerals in the vessels.
I am interrested to find out if and how partial equilibrium with a temperature difference can be calculated by an extension of the minimum free energy principle.

This explains why I first posted in the "physics" area, since thermodynamics is the topic.
But my post suffered a quantum leap to the "chemistry" area because of a short chemical formula!
People are becoming more and more "specialised" today!

Thanks

Michel