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gas model: simplify and improve our Z factor calculations
Lubomir found better compressibility data for the pure gases that we need for scuba, making the air table superfluous: we get good values from just regular linear mixing of the Oxygen, Nitrogen and Helium calculations. Also, rather than using a quintic polynomial, a cubic one does sufficiently well, making for smaller code and fewer coefficients. And judging by the reactions from people on G+ (as well as just looking at how good the fit is with the air data), this is all the right way to do this, and this thus removes the Redlich-Kwong equation. All-credit-goes-to: Lubomir I. Ivanov <neolit123@gmail.com> Signed-off-by: Linus Torvalds <torvalds@linux-foundation.org> Signed-off-by: Dirk Hohndel <dirk@hohndel.org>
This commit is contained in:
Linus Torvalds
Dirk Hohndel
parent
12f36c92e1
commit
a173a3ce79
1 changed files with 37 additions and 126 deletions
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@ -5,159 +5,70 @@
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#include "dive.h"
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/*
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* This gives an interative solution of hte Redlich-Kwong equation for the compressibility factor
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* according to https://en.wikipedia.org/wiki/Redlich–Kwong_equation_of_state
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* in terms of the reduced temperature T/T_crit and pressure p/p_crit.
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*
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* Iterate this three times for good results in our pressur range.
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*
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* Generic cubic polynomial
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*/
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static double redlich_kwong_equation(double t_red, double p_red, double z_init)
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{
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return (1.0/(1.0 - 0.08664*p_red/(t_red * z_init)) -
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0.42748/(sqrt(t_red * t_red * t_red) * ((t_red*z_init/p_red + 0.08664))));
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}
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/*
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* At high pressures air becomes less compressible, and
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* does not follow the ideal gas law any more.
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*/
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#define STANDARD_TEMPERATURE 293.0
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static double redlich_kwong_compressibility_factor(struct gasmix *gas, double bar)
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{
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/* Critical points according to https://en.wikipedia.org/wiki/Critical_point_(thermodynamics) */
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double tcn2 = 126.2;
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double tco2 = 154.6;
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double tche = 5.19;
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double pcn2 = 33.9;
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double pco2 = 50.5;
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double pche = 2.27;
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double tc, pc;
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tc = (tco2 * get_o2(gas) + tche * get_he(gas) + tcn2 * (1000 - get_o2(gas) - get_he(gas))) / 1000.0;
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pc = (pco2 * get_o2(gas) + pche * get_he(gas) + pcn2 * (1000 - get_o2(gas) - get_he(gas))) / 1000.0;
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return (redlich_kwong_equation(STANDARD_TEMPERATURE/tc, bar/pc,
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redlich_kwong_equation(STANDARD_TEMPERATURE/tc, bar/pc,
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redlich_kwong_equation(STANDARD_TEMPERATURE/tc, bar/pc,1.0))));
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}
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/*
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* Generic quintic polynomial
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*/
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static double quintic(double bar, const double coefficient[])
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static double cubic(double bar, const double coefficient[])
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{
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double x0 = 1.0,
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x1 = bar,
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x2 = x1*x1,
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x3 = x2*x1,
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x4 = x2*x2,
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x5 = x2*x3;
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x3 = x2*x1;
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return x0 * coefficient[0] +
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x1 * coefficient[1] +
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x2 * coefficient[2] +
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x3 * coefficient[3] +
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x4 * coefficient[4] +
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x5 * coefficient[5];
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x3 * coefficient[3];
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}
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/*
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* These are the quintic coefficients by Lubomir I. Ivanov that have
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* been optimized for the least-square error to the air
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* compressibility factor table (at 300K) taken from Wikipedia:
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* Cubic least-square coefficients for O2/N2/He based on data from
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*
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* bar z_factor
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* --- ------
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* 1: 0.9999
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* 5: 0.9987
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* 10: 0.9974
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* 20: 0.9950
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* 40: 0.9917
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* 60: 0.9901
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* 80: 0.9903
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* 100: 0.9930
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* 150: 1.0074
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* 200: 1.0326
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* 250: 1.0669
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* 300: 1.1089
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* 400: 1.2073
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* 500: 1.3163
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* PERRY’S CHEMICAL ENGINEERS’ HANDBOOK SEVENTH EDITION
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*
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* with the lookup and curve fitting by Lubomir.
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*
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* NOTE! Helium coefficients are a linear mix operation between the
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* 323K and one for 273K isotherms, to make everything be at 300K.
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*/
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static const double air_coefficients[6] = {
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+1.0002556612420115,
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-0.0003115084635183305,
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+0.00000227808965401253,
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+1.91596422989e-9,
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-8.78421542e-12,
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+6.77746e-15
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static const double o2_coefficients[4] = {
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+1.00117935180448264158,
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-0.00074149079841471884,
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+0.00000291901111247509,
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-0.00000000162092217461
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};
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static const double n2_coefficients[4] = {
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+1.00030344355797817778,
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-0.00022528077251905598,
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+0.00000295430303276288,
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-0.00000000210649996337
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};
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static const double he_coefficients[4] = {
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+1.00000137322788319261,
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+0.000488393024887620665,
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-0.000000054210166760015,
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+0.000000000010908069275
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};
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static double o2_compressibility_factor(double bar) { return cubic(bar, o2_coefficients); }
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static double n2_compressibility_factor(double bar) { return cubic(bar, n2_coefficients); }
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static double he_compressibility_factor(double bar) { return cubic(bar, he_coefficients); }
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/*
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* Quintic least-square coefficients for O2/N2/He based on tables at
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*
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* http://ww.baue.org/library/zfactor_table.php
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*
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* converted to bar and also done by Lubomir.
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*/
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static const double o2_coefficients[6] = {
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+1.0002231211532653,
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-0.0007471497056767194,
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+0.00000200444854807816,
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+2.91501995188e-9,
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-4.48294663e-12,
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-6.11529e-15
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};
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static const double n2_coefficients[6] = {
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+1.0001898816185364,
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-0.00030793319362077315,
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+0.00000327557417347714,
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-1.93872574476e-9,
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-2.7732353e-12,
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-2.8921e-16
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};
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static const double he_coefficients[6] = {
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+0.9998700261301693,
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+0.0005452130351730479,
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-2.7853712233619e-7,
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+5.5935404211e-10,
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-1.35114572e-12,
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+2.00476e-15
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};
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static double air_compressibility_factor(double bar) { return quintic(bar, air_coefficients); }
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static double o2_compressibility_factor(double bar) { return quintic(bar, o2_coefficients); }
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static double n2_compressibility_factor(double bar) { return quintic(bar, n2_coefficients); }
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static double he_compressibility_factor(double bar) { return quintic(bar, he_coefficients); }
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/*
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* We end up using specialized functions for known gases, because
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* we have special tables for them.
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*
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* For air we use our known-good table. For other mixes we use a
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* linear interpolation of the Z factors of the individual gases.
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* We end up using a simple linear mix of the gas-specific functions.
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*/
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double gas_compressibility_factor(struct gasmix *gas, double bar)
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{
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double o2, n2, he; // Z factors
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double of, nf, hf; // gas fractions ("partial pressures")
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if (gasmix_is_air(gas))
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return air_compressibility_factor(bar);
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o2 = o2_compressibility_factor(bar);
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n2 = n2_compressibility_factor(bar);
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he = he_compressibility_factor(bar);
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of = gas->o2.permille / 1000.0;
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hf = gas->he.permille / 1000.0;
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of = get_o2(gas) / 1000.0;
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hf = get_he(gas) / 1000.0;
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nf = 1.0 - of - nf;
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return o2*of + n2*nf + he*hf;
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