Good morning, On 3 March 2016 at 08:24, Robert Helling <[email protected]> wrote:
> Good evening, > > > On 02.03.2016, at 18:44, Linus Torvalds <[email protected]> > wrote: > > > > On Wed, Mar 2, 2016 at 9:25 AM, Dirk Hohndel <[email protected]> wrote: > >> > >> So since Robert's formula /should/ be the right way to calculate the > >> compensation factors, let's figure out what about it is broken and use > >> "matches the wikipedia data" as a measuring stick for that. > > I think (unless my implementation is wrong),this is as good as it gets > with a two parameter model that is also not particularly tuned to our use > case of air-like gases at 300K with 1–300bar pressure. > > > > > Well, the thing is, Robert's formula isn't actually physical, it's > > fundamentally an approximation too. > > > > In fact, it's arguably much less physical than the van der Waals > > equation, that at least tries to model the physical behavior, while > > afaik the Redlich–Kwong equation is _purely_ an empirical > > approximation. > > > > The first paragraph in the wikipedia page really does sum it up: > > > > https://en.wikipedia.org/wiki/Redlich%E2%80%93Kwong_equation_of_state > > > > so to some degree, Lubomirs least-square polynomial would actually be > > superior: it is the same kind of approximation, but it's an > > approximation that has been specialized for one particular gas and > > pressure/temperature range we happen to care about. > > > > The Redlich–Kwong equation is intended to be much more general, but > > exactly because of that it's much less accurate at least for one case > > we care about. > > > > Side note: according to Wikipedia there are various newer refined > > versions with more complexity (and some with more per-gas constants). > > So it's possible that we could still get it all - both the "multiple > > gases" _and_ "sufficient accuracy“. > > I really don’t know what is the correct approach. > > First of all, all this compressibility business is really a small effect > and we are talking about small differences here, so all of this is somewhat > academic, in particular as Linus has pointed out that we always ignore the > effect of gas temperature which can be as much as 15%.. > > The other thing to say is, after rereading the wikipedia page I realized > that I was getting the calculation for mixtures wrong. I had assumed that > it works like for van der Waals that you are supposed to take weighted > averages of the critical data, but you don’t. But the difference this makes > for air is much smaller than the difference to the empirical numbers. With > the corrected procedure for mixtures I also calculated the curve for some > typical trimix to see how important the effect of the gas composition is. > Turns out, it is 3-4 times the difference between the table data and the > computed Z. Or put differently: This is more significant than the > difference between measured and modeled Z’s. Or: Taking the tabulated > values for air and pretend they are the same for trimix gives an error 3-4 > times the error from using the model for air. > > As Linus said correctly, this model is semi-empirical, it uses some > physical intuition about the general form of the correction but then plugs > in measured values (and it is supposed to hold also in a regime where the > gas is close to liquid). But there are only two per gas but this might be > an aesthetic point. > > The problem is that beyond the air table from wikipedia, we (or at least > I) don’t really have empirical data. We don’t know how to extrapolate to > other gases or we don’t know what to match or which values to take for > models with more parameters. > > I think what we need here is an executive decision from our beloved > maintainer: How do you want to proceed, there are essentially three > options: Linus’ table interpolation, Lubomir’s quadratic fit to that table > which both cannot handle other gases than air or this semi-empirical model > with its intrinsic error (in which case I would provide a new patch to get > the mixing right, the above mentioned calculation is in mathematica). All > have advantages and disadvantages. > > I think we can fit the data set better. Firstly, the experimental Z values from Wikipedia fit very well with a quadratic equation (less expensive than a cubic). Secondly, we can adjust the coefficients of the quadratic equation according to temperature with linear interpolation between the values at 250 K, 300 K and 350 K. K = Ax^2 + Bx + C where A = a1.T + a2 B = b1.T + b2 C = 0.999421 T = temp in Kelvin a1 = -3.03810E-08 a2 = 1.12395E-05 b1 = 1.033437E-05 b2 = -3.367652E-03 This equation is fitted to the data for air at 300 K, and allows adjustment according to temperature. See the attached graph. We could factor according to gas mix, with a simplified factor correlated from Redlich–Kwong equation values at say 300 K and 100 bar. I haven't got my head around that equation yet, and have far too much work to do right now. Cheers, Rick
Z correlation.pdf
Description: Adobe PDF document
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