Unstable Solutions to Thermodynamic Differential Algebraic Equations From Equation Solvers.

[NNadir]

All of humanity's efforts to address climate change have failed, and thus it will fall to the up coming generations to reverse it, this with reduced access to rapidly depleting resources including not only endangered chemical elements, but of course, access to things which very much depend on the environment, specifically food and water.

This is a far more challenging engineering task than it would have been to eliminate the use of dangerous fossil fuels but I've comforted myself about the way we've screwed future generations by noting that they will have access to some tools we lacked, in particularly computational capabilities of which my generation could only dream.

For a number of years now, I've been interested in the alternative fuel DME, dimethyl ether, not just its synthesis and use as a fuel, but also in some properties connected with its transport. DME is an easily liquified gas, and as such, I think - though I haven't seen it widely discussed - may have utilization in certain heat transfer applications, including long distance heat transport and heat storage (utilizing recent advances in materials science.) The use as a working fluid in heat transfer, either as a heat pump fluid or a refrigerant - each the reverse of the other - involves phase changes, as does transport in pipelines.

Although I have very little experience with computer modeling of thermodynamic equations of state, I decided to poke around in some of the usual journals I read to see what people are saying about this capability and I came across some interesting papers that are relevant to my daydreams about transporting and storing heat in a DME media.

For example there is this one about pipelines - CO₂ pipelines - involved in a scheme to address climate change that will not work, carbon capture and sequestration: Solution of the Span-Wagner Equation of State Using a Density Energy State Function for Fluid-Dynamic Simulation of Carbon Dioxide (Ind. Eng. Chem. Res., 2012, 51 (2), pp 1006–1014) which contains the following fun text:

"

Carbon capture and storage (CCS) has been proposed as a strategy to reduce carbon emissions to the atmosphere. An important part of the CCS chain is transportation, either in pipelines or in tanks (on boats, vehicles or trains), from a capture point to a storage site. To ensure efficiency and safety in these operations, it is important to have accurate simulation tools to control the processes, and also to perform risk analysis. For instance, an issue with transportation of the supercritical substance (CO₂ is transported at high pressure, in order to minimize pipeline dimensions) is crack propagation. If a pipeline ruptures, the crack may propagate along the pipe, depending on the speed of the crack, relative to the speed of the expansion wave inside the pipe.1, 2 To simulate this process, an accurate model for the flow inside the pipe is needed. Another example is the global trade of CO₂ quotas, which has also been suggested as a means to reduce CO₂ emissions.3 From an economic perspective, it is important to describe the CO₂ properties accurately in order to determine the amount of CO₂ stored and transported.

.

"

The authors refer the reader to the original Span-Wagner equation utilized "to describe the CO₂ properties accurately" and make this note:

We see that the equation contains a total of 51 terms, many of which include logarithms and exponentials. This means that this is an expensive function to evaluate, and efficient numerical methods are critical to obtain satisfactory run times for dynamic simulations.

But, one needs to be very careful about how one performs these calculations, as I learned reading another related paper from the same scientific organization, this one:

Time Efficient Solution of Phase Equilibria in Dynamic and Distributed Systems with Differential Algebraic Equation Solvers (Ind. Eng. Chem. Res., 2013, 52 (5), pp 2130–2140)

This paper contains this mildly disturbing text:

"

The most common approach to solve systems with phase equilibria is to perform the calculations in inner loops, separated from the higher-level modeling. This method will be referred to as the traditional methodology in the rest of this work. The advantage is that algorithms tailored for robust flash calculations can be applied. A disadvantage is that the approach leads to nested iterations loops. Numerical noise is when the fluctuations of internal variables due to limited accuracy are comparable to the predefined solution tolerance, which can give an unstable solution.

"

The authors explore the use of the commercial MATLAB program and Fortran to reduce the calculation time.

The issue here is that one can get kind of disconnected to the nuts and bolts of equation solvers, and in fact, face "butterfly effects" in which the calculations go awry.

I've given my son, starting an engineering program this fall, the task of programming the solutions to the Soave-Reddich-Kwong equation and the Peng Robinson equations in both Mathematica and Matlab, the former because I bought it for him and the latter because the engineering universities all seem to use it.

I need to make him aware of these kinds of risks when he goes to a deeper level than pure play.

Esoteric, but interesting.

[lostinhere]

1. Interesting

Are these articles behind paywalls or easily accessible?

[NNadir]

3. They may well be. I generally access papers at Universities I visit during lunch and on weekends.

It's a wonderful life.

At many universities one can access papers for free on line.

I pay for library access at one University I use, and another gives me two hours a day guest privileges.

[gibraltar72]

2. Huh?

[eppur_se_muova]

4. "It's very hard to make predictions, especially about the future."

And that's *before* chaos enters the picture.