[PATCH 2/2] gas model: replace Redlich-Kwong with least-square quintic

Wed Mar 2 16:07:16 PST 2016

On 3 March 2016 at 01:28, Lubomir I. Ivanov <neolit123 at gmail.com> wrote:
> On 3 March 2016 at 00:55, Lubomir I. Ivanov <neolit123 at gmail.com> wrote:
>> On 3 March 2016 at 00:28, Linus Torvalds <torvalds at linux-foundation.org> wrote:
>>> Lubomir, what tool did you use for the least-square fitting?
>>>
>>
>> those equations in particular came from this website:
>> http://polynomialregression.drque.net/online.php
>>
>>> Because it would be interesting to see if we can do this for the table at
>>>
>>>   http://www.baue.org/library/zfactor_table.php
>>
>> i can take a look a these a bit later - e.g. in 30 minutes.
>>
>
> are those pressure values in PSI?
> we want them in bars i assume, based on the previous table?
> http://lists.subsurface-divelog.org/pipermail/subsurface/2016-February/024642.html
>

i've used a small C program to convert the pressures here
http://www.baue.org/library/zfactor_table.php

from psi to bar with the 0.0689475728 coefficient and rounded the bar
value to the nearest integer.

i've also added the AIR calculation, just in case.
one important thing to note here, since we are aren't using something
like spline interpolation, but instead, using polynomial regression,
deviations *can* happen near the data set edges.

so for instance, the tables start at 7 bars and end at 276 bars, but
if the input is 1 bar or something like 500 bars, the value might not
follow the fitted curve to the data set and deviate. this is usually
worse with something like Lagrangian interpolation, tolerable to quite
bad with polynomial regression and pretty good with splines (but
splines are expensive and i don't have a ready algorithm / website for
them).

--------------------------------------------------------------
oxygen:
---------
bar / z
7 0.995
14 0.990
21 0.986
28 0.981
34 0.977
41 0.973
48 0.969
55 0.966
62 0.962
69 0.959
76 0.956
83 0.954
90 0.951
97 0.949
103 0.947
110 0.945
117 0.944
124 0.943
131 0.942
138 0.941
145 0.940
152 0.940
159 0.940
165 0.941
172 0.941
179 0.942
186 0.943
193 0.944
200 0.945
207 0.947
214 0.948
221 0.950
228 0.953
234 0.955
241 0.957
248 0.960
255 0.963
262 0.966
269 0.969
276 0.972

biquad:
y = 1.000166641097901 - 0.0007420889598606622*x +
0.00000188569152312097*x^2 + 4.01140790998e-9*x^3 - 8.81067154e-12*x^4

quint:
y = 1.0002231211532653 - 0.0007471497056767194*x +
0.00000200444854807816*x^2 + 2.91501995188e-9*x^3 - 4.48294663e-12*x^4
- 6.11529e-15*x^5

--------------------------------------------------------------

nitrogen:
---------
bar / z:
7 0.998
14 0.997
21 0.995
28 0.994
34 0.993
41 0.993
48 0.993
55 0.993
62 0.993
69 0.994
76 0.995
83 0.996
90 0.997
97 0.999
103 1.001
110 1.003
117 1.005
124 1.008
131 1.011
138 1.014
145 1.017
152 1.021
159 1.024
165 1.028
172 1.032
179 1.036
186 1.041
193 1.045
200 1.050
207 1.054
214 1.059
221 1.064
228 1.069
234 1.074
241 1.080
248 1.085
255 1.091
262 1.096
269 1.102
276 1.107

biquad:
y = 1.0001872105114884 - 0.00030769385613513414*x +
0.00000326995780587478*x^2 - 1.88687434484e-9*x^3 - 2.9779061e-12*x^4

quint:
y = 1.0001898816185364 - 0.00030793319362077315*x +
0.00000327557417347714*x^2 - 1.93872574476e-9*x^3 - 2.7732353e-12*x^4
- 2.8921e-16*x^5

--------------------------------------------------------------

helium:
bar / z:
---------
7 1.004
14 1.007
21 1.011
28 1.015
34 1.018
41 1.022
48 1.026
55 1.029
62 1.033
69 1.036
76 1.040
83 1.043
90 1.047
97 1.050
103 1.054
110 1.057
117 1.061
124 1.064
131 1.067
138 1.071
145 1.074
152 1.078
159 1.081
165 1.084
172 1.088
179 1.091
186 1.094
193 1.097
200 1.101
207 1.104
214 1.107
221 1.110
228 1.114
234 1.117
241 1.120
248 1.123
255 1.127
262 1.130
269 1.133
276 1.136

biquad:
y = 0.9998885418429937 + 0.0005435539837400252*x -
2.3960530861159e-7*x^2 + 1.999279624e-10*x^3 + 6.760135e-14*x^4

quint:
y = 0.9998700261301693 + 0.0005452130351730479*x -
2.7853712233619e-7*x^2 + 5.5935404211e-10*x^3 - 1.35114572e-12*x^4 +
2.00476e-15*x^5

--------------------------------------------------------------

air:
bar / z
7 0.997
14 0.995
21 0.993
28 0.991
34 0.990
41 0.989
48 0.988
55 0.987
62 0.987
69 0.986
76 0.987
83 0.987
90 0.988
97 0.988
103 0.990
110 0.991
117 0.992
124 0.994
131 0.996
138 0.999
145 1.001
152 1.004
159 1.007
165 1.010
172 1.013
179 1.016
186 1.020
193 1.024
200 1.028
207 1.032
214 1.036
221 1.040
228 1.045
234 1.049
241 1.054
248 1.059
255 1.064
262 1.069
269 1.074
276 1.079

biquad:
y = 0.9997260053715754 - 0.0003827060866716186*x +
0.00000279540706068042*x^2 + 2.0260744986e-10*x^3 - 5.54727693e-12*x^4

quint:
y =  0.9994355774568318 - 0.00035668306234655186*x +
0.00000218474273138185*x^2 + 5.8403793405e-9*x^3 - 2.780101081e-11*x^4
+ 3.144563e-14*x^5

--------------------------------------------------------------

lubomir
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