On the Linearization of Antoine's Law - Industrial & Engineering

2734

Ind. Eng. Chem. Res. 2009, 48, 2734–2737

CORRELATIONS On the Linearization of Antoine’s Law Pascal Floquet UniVersite´ de Toulouse, Laboratoire de Ge´nie Chimique UMR 5503 CNRS/INPT/UPS, ENSIACET/INPT 118 route de Narbonne, 31078 Toulouse Cedex 4, France hexahydrophtalate (DIBE),4 and 2,6,10,14-tetramethylpentadecane (Pristane).5

Introduction Historically, many different formulas and graphical representations have been suggested for the vapor pressure-temperature relationship for a pure component.1 Clapeyron’s formula, which was proposed in 1834 and thermodynamically stated by Clausius in 1864, gave birth to the well-known Clausius-Clapeyron differential relationship, d ln P° -∆Hv ) d(1 ⁄ T) R∆z

B T

∑

nexp

(log Pexp° - log Pcalc°)2 )

∑

(3)

The three coefficients A, B, and C must be determined classically, thanks to experimental data. From a numerical point of view, eq 2 is clearly linear, with respect to the coefficients A and B, in terms of the variables log P° and 1/T; thus, if a substance behaves according to the Clausius-Clapeyron model, a plot of log P° against 1/T should yield a straight line with slope sB. In contrast, Antoine’s equation (eq 3) is a nonlinear equation with respect to the parameters A, B, and C and for the same variables log P° and 1/T. This equation is a well-known nonlinear model in the field of chemical engineering education and is often used as an example for the purpose of identifying a nonlinear model.2 In this work, we propose to promote a change of variables, leading to a linearized Antoine’s equation. It is easy, with this model, to compute the previously mentioned coefficients A, B, and C via a classical multivariate linear regression method. Thus, these coefficient values are good candidates for initial points of a nonlinear regression of a complete nonlinear Antoine’s law. To prove the adequacy of a such an initial guess, three examples have been chosen with experimental values of temperature (Texp) and vapor pressure (Pexp): carbon tetrachloride (CCl4) (from the contribution of Hildebrand and McDonald in ref 3), di-isobutyl

[

nexp i)1

(

log Pexp° - A -

Logarithmic relative criterion (LRC) : log Pexp° - log Pcalc° nexp log Pexp°

∑

(

(2)

This assumption, or a stronger one of constant heat of vaporization, was soon determined to be one of the main reasons why the Clausius-Clapeyron equation would only fit the experimental data in a limited temperature range. A common way to improve the fitting ability is to use Antoine’s equation for the vapor pressure of a saturated liquid, which introduces a third parameter (C) in the approximated Clausius-Clapeyron equation (eq 2): B log P° ) A T+C

The nonlinear Antoine’s model (eq 3) can be fitted with experimental data via the minimization of distinct criteria. Classical criteria include the following: Logarithmic absolute criterion (LAC) :

(1)

Here, ∆Hv is the latent heat of vaporization and z is the compressibility factor (z ) P°V/(RT), where P° is the vapor pressure, V the corresponding volume, R the gas constant, and T the temperature). Assuming that the term ∆Hv/(∆z) is constant leads to log P° ) A -

Several Nonlinear Regressions on Antoine’s Law

∑

nexp i)1

[

)

2

B Texp + C

)]

)

(

B Texp + C log Pexp°

log Pexp° - A -

)

]

2

2

Absolute criterion (AC) :

∑

nexp

(Pexp° - Pcalc°)2 )

∑ Relative criterion (RC) :

∑

nexp

(

nexp i)1

[

Pexp° - Pcalc° Pexp°

∑

nexp i)1

[

(

B Texp + C

(

B Texp + C

Pexp° - exp A -

)

2

)]

)

Pexp° - exp A Pexp°

)

]

2

2

Figure 1. Values of AC criterion versus coefficients B and C (A ) 15) for the CCl4 example.

10.1021/ie800892v CCC: $40.75  2009 American Chemical Society Published on Web 02/04/2009

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2735 Table 1. Comparison of Initialization Techniques on Antoine Regression Problem with Several Criteria: CCl4 Coefficient initialization

A

B

MLAE

15.8502

2794.9

Zero Clapeyron Antoine Linear

15.872 17.567 15.850

2807.1 3824.4 2794.9


13.393 17.346 15.850

1606.8 3678.2 2794.9


16.020 16.020 16.020

2890.5 2890.5 2890.5


16.000 16.000 16.000

2879.2 2879.2 2879.2


15.883 15.883 15.883

2813.3 2813.3 2813.3

C -46.64 LAC Criterion -46.041 -0.026 -46.638 LRC Criterion -112.460 -6.436 -46.638 AC Criterion -41.925 -41.925 -41.925 RC Criterion -42.488 -42.488 -42.488 BC Criterion -45.739 -45.739 -45.739

norm

number of iterations

number of function evaluations

6.06 × 10-6 2.50 × 10-4 6.12 × 10-6

39 2 2

298 12 12

4.14 × 10-5 6.94 × 10-6 1.83 × 10-7

85 9 2

664 65 12

0.939 0.939 0.939

234 108 24

1770 816 197

0.024 0.024 0.024

220 95 25

1681 730 202

0.472 0.472 0.472

224 57 11

1699 441 80

Table 2. Comparison of Initialization Techniques on Antoine Regression Problem with Several Criteria: DIBE Coefficient initialization

A

B

MLAE

11.4945

2658.0


13.394 13.394 13.394

3817.6 3817.6 3817.6


9.606 9.606 9.606

1768.1 1768.1 1768.1


13.254 13.254 13.254

3681.1 3681.1 3681.1


14.024 14.024 14.024

4251.5 4251.5 4251.5


12.869 12.901 12.878

3475.2 3495.4 3481.4

C -209.45 LAC Criterion -154.95 -154.95 -154.95 LRC Criterion -256.00 -256.00 -256.00 AC Criterion -163.13 -163.13 -163.13 RC Criterion -136.51 -136.51 -136.51 BC Criterion -170.10 -169.19 -169.82

Boublik et al.3 criterion (BC) :

∑

nexp

norm

[(log Pexp° - A) × (Texp + C) + B]2

All of these criteria examine least-squares minimization problems that are not exactly equivalent, in terms of parameter solutions or final criterion value. In this work, all of the leastsquares fitting are performed with the LSQNONLIN program, which is obtained from the Matlab Optimization Toolbox and is based on the Levenberg-Marquardt method with line search for a medium-scale problem.6,7 For comparison purposes, the tolerances associated with these problems (the termination tolerance on the function value (εF) and on variable change (εX)) are set to 10-7. [Note: Three termination criteria have been applied: nonevolution on variables (evolution less than εX), nonevolution of the least-squares function (evolution less than εF), and first-order optimality conditions.]



1.09 × 10-2 1.09 × 10-2 1.09 × 10-2

88 122 49

684 943 377

1.32 × 10-2 1.32 × 10-2 1.32 × 10-2

88 176 49

671 1348 371

0.236 0.236 0.236

188 248 70

1433 1897 522

0.109 0.109 0.109

134 131 59

1038 1004 446

31.550 31.551 31.550

7949 20878 16641

31800 83512 66568

minima of the optimization problem. Figure 1, for example, shows the carbon tetrachloride (CCl4)3 example for a fixed value of coefficient A (A ) 15), where the response surface (criterion AC) is plotted versus various values of coefficients B and C. It can be observed than the “minimum valley” is quite narrow and may explain the numerical difficulties that are encountered. A good initial point of the regression procedure provides us a less-difficult path tracing to the solution and a clear advantage in terms of global minimum localization.

Several Initial Points for the Nonlinear Regression All the nonlinear regressions are performed with the following initial points:

Interaction between Coefficients C and D

• “Null” initial point: all the initial coefficient values A, B, and C are set to zero;

Difficulties with the nonlinear regression procedures (LAC, LRC, AC, RC, BC or other criteria) arise from the correlation between the coefficients B and C and from the number of local

• “Clapeyron” initial point: initial coefficient values A and B are set to the solution of the linear Clausius-Clapeyron equation (eq 2); the initial value of C is set equal to zero;

2736 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 Table 3. Comparison of Initialization Techniques on Antoine Regression Problem with Several Criteria: Pristane Coefficient initialization

a

A

B

MLAE

20.4388

4274.8


20.479 20.479 20.438

4296.8 4296.8 4274.8


20.360 20.360 20.360

4242.1 4242.1 4242.1


26.276 23.998

8647.3 6726.6


21.603 21.123

4938.5 4657.6


20.459 20.460 20.458

4285.5 4286.3 4285.5

C


norm

-116.09 LAC Criterion -115.42 -115.42 -116.07 LRC Criterion -116.79 -116.79 -116.79 AC Criterion


1.12 × 10-3 1.12 × 10-3 1.14 × 10-3

57 58 3

440 445 19

1.80 × 10-4 1.80 × 10-4 1.80 × 10-4

578 418 21

4304 3147 154

1524 2466

6100 9868

25000 16534

100004a 66140

7295 17647 393

29184 70592 1576

NONCONVERGENCE 13.91 28.424 -38.82 28.820 RC Criterion NONCONVERGENCE -96.33 5.198 -104.59 4.912 BC Criterion -115.76 9.807 -115.74 9.807 -115.76 9.807

Maximal number of function calls.

(

)

If we denote X as the following (nexp × 3) matrix, 1 T1exp T1exp log P °1exp 1 T2exp T2exp log P °2exp X) l l l exp exp exp Tnexp log P °nexp 1 Tnexp

( )

and Y as the following array,

log Pexp°1

Y)

log Pexp°2 l log Pexp°nexp

then the multilinear least-squares regression gives the coefficient array B C

A C -1 C via the solution of the following linear system: β)

• “Linearized Antoine” initial point: all the initial coefficient values A, B, and C are set to the solution of the linearized Antoine equation (given below).

() A-

Figure 2. Values of the response log P° with T and T log P° for the Pristane example.5

This new linear initial point that we proposed is obtained via the following transformation of Antoine’s equation (eq

(XTX)β ) XTY Then,

3): AT + AC - B f C log P° ) AT - T log P° + (AC - B) log P° ) T+C V A B T log P° log P° ) A - + TC C C (4)

(

) ()

Equation 4is linear with the coefficients A/C, -1/C, and (A s B/C), in terms of transformed variables log P°, T, and T log P°. It is then possible to identify these coefficients linearly (and, thus, the original ones A, B, and C) by fitting log Pexp° with Texp and Texp log Pexp°.

A)B)

β2 β3

β2 β23

C)-

+

β1 β3

1 β3

The adequation of the linearized model can be seen in Figure 2, which gives the response log P° with T and T log P° for the Pristane example.5 The scatter points are experimental points and the surface corresponds to the multilinear model.

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2737

simple, noniterative manner. This equation is easy to compute and allows one to obtain quickly good results to the nonlinear regression of vapor pressure versus temperature of pure components. Examples relevant from thermodynamical literature are used to show the quality of our MLAE parameters to converge to a nonlinear regression solution of Antoine’s model and to compare our approach with the classical ones. Notations

Figure 3. Convergence versus initialization for the DIBE example and the RC criterion. Table 4. Literature Results (Norm of the Criterion) on the Antoine Regression Problem with Several Criteriaa Value parameter Antoine’s coefficient A B C criterion LAC LRC AC RC BC

CCl4

DIBE

15.938 2844.1 -44.213

12.866 3471.6 -170.35

20.489 4301.7 -115.26

2.58 × 10-3 4.79 × 10-4 0.9974 0.1596 0.7093

1.06 × 10-1 1.80 × 10-1 0.69629 0.37515 31.56

3.37 × 10-2 2.97 × 10-1 41.129 5.5148 9.8759

Pristane

a Values shown in bold and italic font have been chosen by the authors, if known.

Tables 1-3 show the complete nonlinear regression results for all the criteria and examples, in terms of coefficient initialization. It is easy to see that our multivariate linearized Antoine’s equation (MLAE) initialization procedure gives, in all cases, better results, compared to other initialization methods. The convergence of the nonlinear regression to optimal solution with the different initializations is shown, for the DIBE example4 and the RC criterion, in Figure 3. As foreseen, the proposed MLAE procedure is quite good for quick convergence to the optimal solution. Table 4 shows the literature results and the criterion chosen. In each case, the initialization proposed and the nonlinear regression end with a solution that is the same or better. Conclusion and Significance A multivariate linearized Antoine’s equation (MLAE) is presented in this short note. To our knowledge, this is the first time that this multilinear equation has been proposed in the literature. Compared with initial null parameters or “Clapeyron”type initial parameters, which do not take into account the highly nonlinear correlation between parameters B and C, our procedure gives a very good initial guess for the nonlinear regression of vapor pressure versus temperature. Fortunately, the multilinear expression of the linearized Antoine’s law is obtained in a

P° ) vapor pressure A ) first Antoine’s coefficient B ) second Antoine’s coefficient C ) third Antoine’s coefficient R ) gas constant; R ) 8.314 J K-1 mol-1 T ) temperature (K) V ) volume X ) experiment matrix (nexp × 3) Y ) response array (nexp × 1) z ) compressibility factor Acronyms AC ) absolute criterion BC ) Boublik’s criterion MLAE ) multivariate linearized Antoine’s equation DIBE ) di-isobutyl hexahydrophtalate LAC ) logarithmic absolute criterion LRC ) logarithmic relative criterion RC ) relative criterion Greek Letters β ) parameter array (3 × 1) ∆H ) heat of vaporization εX ) termination tolerance on the variable change (least-squares procedure) εF ) termination tolerance on the function value (least-squares procedure) Subscripts exp ) experimental calc ) calculated

Literature Cited (1) Wisniak, J. Historical Development of the Vapor Pressure Equation from Dalton to Antoine. J. Phase Equilib. 2001, 22 (6), 622–630. (2) Corriou, J. P. Process Control: Theory and Application; Springer: London, 2004. (3) Boublik, T.; Fried, V.; Ha´la, E. The Vapour Pressures of Pure Substances: Selected Values of teh Temperature Dependence of the Vapour Pressures of Some Pure Substances in the Normal and Low Pressure Region; Elsevier Scientific Publishers: Amsterdam, 1973. (4) Liu, Z.; Gao, Z.; Liu, R. Isobaric Vapor-Liquid Equilibria of the Binary System Maleic Anhydride and Di-isobutyl Hexahydrophtalate at 2.67, 5.33 and 8.00 kPa. Fluid Phase Equilib. 2005, 233, 23–27. (5) Bourasseau, E.; Sawaya, T.; Mokbel, I.; Jose, J.; Ungerer, P. Measurement and Prediction of Vapour Pressures of 2,6,10,14-tetramethylpentadecane (pristane). Fluid Phase Equilib. 2004, 225, 49–57. (6) Levenberg, L. A Method for the Solution of Certain Problems in Least-Squares. Q. Appl. Math. 1944, 2, 164–168. (7) Marquardt, F. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 1963, 11, 431–441.

ReceiVed for reView June 6, 2008 ReVised manuscript receiVed December 15, 2008 Accepted January 2, 2009 IE800892V

On the Linearization of Antoine's Law - Industrial & Engineering

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