8/13/2019 soln-l28

1/5

Non-linear regression analysis to determine Antoine equation constants W.R. Wilco

Use Solver to find the best values for A, B and C in the Antoine equation (ln(p) = A - B/(T + C)),i.e., those giving the maximum value of R

2. Use the template at the tab below.

As initial guesses start with A=10, C=-10, B=100 and with the negative of those values.

How do the new values of A, B, C and R2compare with those from the previous exercise?

Using 10, -10 and 10, almost the same, assuming you did previous exercise correctly.

Do the values of A, B, C and R2found by Solver depend on the initial guesses?

Sometimes.

Why is it better to base the calculations on ln(p) rather than p?

ln(p) because p varies over orders of magnitude and one would not expect the

magnitude of the error to be the same for all of these. The percent error,

on the other hand, is more likely to be about the same (although surely not

identical). Consequently, the error in ln(p) will be less dependent on p.

In using least squares on ln(p) we are assuming the percent error is

independent of p. While probably not true, it's much better than assuming

the absolute error is independent of p.

8/13/2019 soln-l28

2/5

x, Clarkson University, spring 2004

8/13/2019 soln-l28

3/5

Solution to Laboratory Assignment 28

Analysis of data on vapor pressure of carbon monoxide from Perry's Chemical Engineers Handbook

Fit to Antoine equation (ln(p) = A - B/(T + C)). Where T is absolute temperature and A, B, C are constants to be determined.

Solution: Use solver to maximize R2starting with A = 19.09475378 19.088814A=10, C=-10, B=100 and with the negative of those values C = -16.07954872 -16.13833

Click on cells to see formulas used. B = 498.1702521 497.26078

from solver Values from previous exercise

Correlated Deviation squared

p (torr) T (C) p (Pa) T (K) ln(p) (ln(p)-ln(pavg))2

ln(p) (ln(pexp)-ln(pcor))2

1 -222.0 133 51.2 4.89 29.44 4.89 0.000008

5 -217.2 667 56.0 6.50 14.57 6.60 0.009569

10 -215.0 1333 58.2 7.20 9.75 7.25 0.003372

20 -212.8 2666 60.4 7.89 5.91 7.84 0.002175

40 -210.0 5333 63.2 8.58 3.02 8.51 0.00495660 -208.1 7999 65.1 8.99 1.77 8.92 0.004256

100 -205.7 13332 67.5 9.50 0.67 9.40 0.010159

200 -201.3 26664 71.9 10.19 0.02 10.16 0.000832

400 -196.3 53329 76.9 10.88 0.32 10.90 0.000168

760 -191.3 101325 81.9 11.53 1.46 11.52 0.000033

202650 89.7 12.22 3.61 12.32 0.010854

T (C) p (atm) 506625 102.5 13.14 7.93 13.33 0.036632

-191.3 1 1013250 112.2 13.83 12.32 13.91 0.006498

-183.5 2 2026500 123.5 14.52 17.67 14.46 0.004462

-170.7 5 3039750 131.3 14.93 21.24 14.77 0.024975

-161.0 10 10.32 129.70 0.1189-149.7 20 0.999082878 R

2using solver to maximize by varying A, B, C

-141.9 30 0.999082699 R2using A, B, C from previous exercise

The solution (values of A, B, C) depends on the initial values. Some combinations may not give a solution.

It is better here to do regression on ln(p) because p varies over 3 orders of magnitude.

Using ln(p) gives equal emphasis to minimizing the percent error for both large and small p.

If p is used, regression strongly emphasizes large values, so the result will be a poor fit

for small values of p.

Original Data Modified units Experimental

8/13/2019 soln-l28

4/5

8/13/2019 soln-l28

5/5

4.00

8.00

12.00

16.00

50 60 70 80 90 100 110 120 130 140

Vaporpressur

e,

Pa

Temperature, K

Antoine correlation for CO

Experimental

Correlation