Heat exchanger and the second principle

[maajdl]

Hello,

I am developping a Python (/Pyomo) package for simulation and optimization of some industrial processes.
I want to create global (simplified) models for heat exhangers (and more) and avoid differential equations.
(to decrease the number of variables of the problem)
Most often the exchangers are counter-current.
To avoid nasty difficulties I avoid using the NTU method and prefer LMTD - like methods.

Doing so, I stumbled on a small question.

If we cut into small elements a counter-current heat exchanger, the second principle tells us that the output temperatures of the small elements must always remains in between the input temperatures. By induction, we see that this remains so for a "large" heat exchanger.

However, applying the second principle (DS>=0) to the full large exchanger does not garantee these limits on the output temperatures.
I did not anticipate that, but I observed this by solving the equations numerically.
This was however not a numerical problem, but a physical problem.
It occurs specially for very efficient exchangers with unbalanced flows.

The nasty consequence is that -apparently- I might be forced to cut the problem into small elements where the second principle will garantee the temperature limits in the elements and globally.

My question is: would it be possible to avoid cutting the model into small elements and keep the temperature ordering.

However note this:
For the simple heat exchanger model, I could simply write additional constraints on the temperatures (Tout1 within Tin1 and Tin2).
However, I want to go further than simple heat exchangers.
I want to model heat and mass exhcngers as well.
In that can the temperture ordering does not necessarily apply.
However the problem will remain:

How could I ensure the 2nd principle on the global scale as well as on the element scale ...
without cutting the model into elements?

Is there more theory about that?
Something like an additional constraint on the final states? Even with some additionaml assumptions ...

Thanks for your suggestions,

Maa

 

[JBA]

Your calculated values should inherently fall within the 2nd principle limits, if the calculated end temperature is outside those limits then that indicates there is a problem with your calculation method.

 

[maajdl]

You are right:

A global model -that does not decompose into elements- can find a solution respecting the 1st and 2d principle.
However, it does not necessarily provide the correct solution.
When this occurs, there is no way to get the same result with elements in series without allowing some elements to decrease entropy.

When adding a constraint on the output temperatures, the problem disappears totally.
This means that the model is correct but incomplete if the constraint on temperature is removed.

I would like to remove this temperature constraint.
I would like to replace it by something more general, that could be applied when there is combined heat and mass exchange.

For example, in a drying process, the wet and cold material might get out of the exhcanger even colder because of the evaporation.
In that case, the constraints on output temperatures are not correct anymore.
The constraint on entropy production remains correct.
However, it is not garanteed that a global constraint on entropy production will imply a local constraint on entropy production (for each element).

Therefore, a model based on differential equations might be compulsory.
If it was possible, I would like to avoid it, because of the big impact on the optimization.

Thanks for your comment

Maa