by
Trent Guidry
1. January 2012 09:49
The Antoine correlation is a simple equation that can be used to model the saturated liquid vapor pressure of a pure chemical substance.
The analytical derivatives of the Antoine correlation along with a Gauss Newton algorithm can be used for the nonlinear least square regression of the correlation parameters.
Antoine Vapor Pressure Correlation
Gauss Newton for nonlinear least squares Update Formula
Gauss Newton Jacobian
Antoine Correlation Gradient
Antoine Correlation Least Squares Regression Jacobian
Gauss Newton Error
Antoine Least Squares Error
Example
Vapor pressure of water data
| 0 |
4.6 |
| 20 |
17.5 |
| 40 |
55.3 |
| 60 |
149.4 |
| 80 |
355.1 |
| 100 |
760 |
Iteration 0
Error: 130.064672
Parameter Values:
| A |
20.58 |
| B |
5200.00 |
| C |
273.00 |
Jacobian
| 4.6292 |
-0.0170 |
0.3230 |
| 16.9889 |
-0.0580 |
1.0290 |
| 52.8037 |
-0.1687 |
2.8027 |
| 143.2204 |
-0.4301 |
6.7161 |
| 346.9296 |
-0.9828 |
14.4776 |
| 764.3159 |
-2.0491 |
28.5666 |
Error
| -0.0292 |
| 0.5111 |
| 2.4963 |
| 6.1796 |
| 8.1704 |
| -4.3159 |
JTJ
| 728149.3364 |
-1978.6961 |
27985.4611 |
| -1978.6961 |
5.3818 |
-76.1911 |
| 27985.4611 |
-76.1911 |
1079.7754 |
JTE
Iteration 1
Error: 127959.979830
Parameter Values:
| A |
18.20 |
| B |
3658.97 |
| C |
225.97 |
Jacobian
| 7.4502 |
-0.0330 |
0.5339 |
| 27.7962 |
-0.1130 |
1.6811 |
| 85.0749 |
-0.3199 |
4.4006 |
| 222.6711 |
-0.7787 |
9.9631 |
| 513.9223 |
-1.6797 |
20.0868 |
| 1070.4289 |
-3.2839 |
36.8614 |
Error
| -2.8502 |
| -10.2962 |
| -29.7749 |
| -73.2711 |
| -158.8223 |
| -310.4289 |
JTJ
| 1467582.5198 |
-4582.3569 |
52424.1835 |
| -4582.3569 |
14.3276 |
-164.1605 |
| 52424.1835 |
-164.1605 |
1883.9846 |
JTE
| -433070.3148 |
| 1354.0131 |
| -15512.9533 |
Iteration 2
Error: 2979.670602
Parameter Values:
| A |
18.38 |
| B |
3867.50 |
| C |
230.86 |
Jacobian
| 5.1012 |
-0.0221 |
0.3702 |
| 19.3961 |
-0.0773 |
1.1920 |
| 60.5480 |
-0.2235 |
3.1918 |
| 161.6204 |
-0.5557 |
7.3885 |
| 380.2124 |
-1.2231 |
15.2169 |
| 806.5652 |
-2.4378 |
28.4957 |
Error
| -0.5012 |
| -1.8961 |
| -5.2480 |
| -12.2204 |
| -25.1124 |
| -46.5652 |
JTJ
| 825298.2362 |
-2536.2161 |
30181.6888 |
| -2536.2161 |
7.8039 |
-92.9972 |
| 30181.6888 |
-92.9972 |
1109.8942 |
JTE
| -49438.0184 |
| 152.3515 |
| -1818.5255 |
Iteration 3
Error: 2.988742
Parameter Values:
| A |
18.46 |
| B |
3916.80 |
| C |
231.12 |
Jacobian
| 4.5589 |
-0.0197 |
0.3343 |
| 17.5806 |
-0.0700 |
1.0919 |
| 55.5551 |
-0.2049 |
2.9603 |
| 149.8850 |
-0.5149 |
6.9270 |
| 355.9384 |
-1.1441 |
14.4029 |
| 761.4062 |
-2.2995 |
27.2005 |
Error
| 0.0411 |
| -0.0806 |
| -0.2551 |
| -0.4850 |
| -0.8384 |
| -1.4062 |
JTJ
| 732313.2542 |
-2247.9292 |
27060.5736 |
| -2247.9292 |
6.9088 |
-83.2808 |
| 27060.5736 |
-83.2808 |
1005.3589 |
JTE
| -1457.1865 |
| 4.4995 |
| -54.5131 |
Iteration 4
Error: 0.006883
Parameter Values:
| A |
18.46 |
| B |
3913.32 |
| C |
230.86 |
Jacobian
| 4.5265 |
-0.0196 |
0.3324 |
| 17.4860 |
-0.0697 |
1.0874 |
| 55.3274 |
-0.2043 |
2.9512 |
| 149.4151 |
-0.5137 |
6.9115 |
| 355.0836 |
-1.1423 |
14.3797 |
| 760.0062 |
-2.2971 |
27.1692 |
Error
| 0.0735 |
| 0.0140 |
| -0.0274 |
| -0.0151 |
| 0.0164 |
| -0.0062 |
JTJ
| 729406.0251 |
-2240.7533 |
26971.2510 |
| -2240.7533 |
6.8922 |
-83.0707 |
| 26971.2510 |
-83.0707 |
1002.7135 |
JTE
Final
Parameter Values:
| A |
18.46 |
| B |
3913.25 |
| C |
230.86 |
Error: 0.006876
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