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Ind. Eng. Chem. Res. 2003, 42, 6258-6262
CORRELATIONS Group Vector Space Method for Estimating the Critical Properties of Organic Compounds Xu Wen* and Wei Wenying School of Chemical Engineering, Tianjin University, Tianjin 300072, China
The specific position of a group in the molecule has been considered and a group vector space method for estimating the critical properties of organic compounds developed. Expressions for estimating the critical properties Tc, Pc, and Vc of organic compounds have been proposed and the numerical values of relative group parameters presented. The average percent deviations of estimation of the above three properties are 0.69, 2.15, and 1.43, respectively, which show that the present method demonstrates significant improvements in accuracy and applicability, compared to the conventional group methods. Introduction Critical properties are very important data in the chemical and petroleum industries, and they are applied widely in domains as state calculation, process simulation, and product design. It is not always possible, however, to find reliable experimental values of these properties for the compounds of interest in the literature, nor is it practical to measure the properties as the need arises. So, estimation methods are profusely employed. For the estimation of the properties of pure compounds, simple group contribution (SGC) methods1-6 are the most widely used. These methods provide the important advantage of quick estimates without requiring substantial computational resources. However, many of these methods are of questionable accuracy and utility. To overcome the above limitation, composite group contribution methods7,8 have been reported in the literature. New problems emerged along with the increment of regression accuracy of these methods. Model parameters of the group method are obtained from fitting property data of a great many substances, which behave like a statistical feature. Only if the number of substances in linear data regression is much more than that of parameters in the model does the group method have a function of extrapolation predicting. Compared with the number of substances in regression, the higher the parameter number in the model, the poorer the predicting function of the model is. If the number of model parameters is more than that of the substances, the value of the model parameter solved will not be unique. For simple group methods, only a single functional group was taken as the independent molecular structural unit. There are not more than 40 groups for various organic compounds in these methods. An * To whom correspondence should be addressed. E-mail:
[email protected].
addition of two or more adjacent simple groups was taken as the independent molecular structure unit for the composite group methods. When only interaction between two adjacent simple groups is taken into account, the number of independent molecular structural units for all organic compounds is 223, and the parameter number in property correlation will be the same. The method proposed by Marrero-Morejo´n and Pardillo-Fontdevila8 was the group interaction contribution method. Among their estimation of critical properties, the number of critical volume data in regression was only 289, which was not much larger than 223. For the Constantinou and Gani method, the estimation was performed at two levels: the basic level only used the contribution from the first-level groups, while the second level increased the consideration of the second-order groups. Their 78 first-level groups were not sufficient to describe the molecule of some common compounds, and under certain circumstances, the same molecule may be described in different ways because of overcomplication of their method. In our early work,9 a group vector space method for estimating the boiling and melting points of organic compounds was proposed. This paper extends the approach to estimation of the critical properties. Another purpose of this study is to develop the new group contribution method to limit the number of model parameters with higher accuracy. Group Vector Space for Organic Compounds In this work, we select 40 simple groups to describe organic compounds. These groups are the same as those used by Joback and Reid.1 The molecule is considered to be in a given space, and every group in the molecule is only a point in the space. To write conveniently, graphs with different numbers of points are all expressed as graphs with five points. Thus, the organic molecules can be expressed as the following seven
10.1021/ie0304098 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/04/2003
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6259
topologic graphs:
The matrices show that the space position of point i in the graph can be represented by an m-dimensional vector (bi1, bi2, ..., bim). Then the module ai of the point i vector is m
The topologic structure of a molecule can be described by the distance matrix of the molecule. The distance matrix is a square symmetric matrix. It is required to input n × n elements for the n-order square matrix. The input quantity for the large molecule is too large, and, furthermore, the distance matrix is unable to describe the molecular structure characteristics such as chains, rings, and branches sufficiently. To overcome the above limitation, the following group vector space method has been proposed. Considering the chain graph first, the dimension number of the space is equal to the number of end points on the chain, and one end point has determined a dimension of the space. The coordinate of an end point in the dimension determined by it is zero, while the coordinate of another point in this dimension is the distance from that point to the end point. For a cyclic graph, one ring represents a dimension. In that dimension, the coordinate of the ring point equals the number of points on the ring, and the coordinate of the nonring point equals the sum of the distance from the point to the ring and the number of points on the ring. If the route from the ring point to the end point is nonunique, the shortest route should be selected. So, the dimension number m of the space for a graph is equal to the sum of the number ke of end points and the number kr of rings in the graph. Every point in the graph has m coordinates in the m-dimensional space. The graph may be described by a space matrix, the number of rows in the matrix equals the number of points in the graph, and the number of columns equals the dimension number of the space. The space matrices of the above seven topologic graphs are as follows:
Ri ) (
bij2)1/2 ∑ j)1
(i ) 1-5)
The average square root of the module of some point i in the graph is defined as the module index vi of this point vector. That is 5
νi ) Ri/(
Rj2)1/2 ∑ j)1
(i ) 1-5)
The quantity vi is used to describe the point i position in the space. On the analogy of this, the module index vi of group i in the molecule is taken to characterize the position of that group in the molecular space. Thus, every simple group, except halogen groups, has its own independent module index. For the four halogen groups, their module indexes were determined to be the same as those of the hydrocarbon groups with which they were connected. Correlation and Group Parameters The expression of critical property f for simple group method is
f)a+
∑i ni∆fi
(1)
where subscript i represents the group type, ∆fi is the contribution value of the i-type group, ni is the number of i-type groups in the molecule, and a is the correlation constant.
6260 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 Table 1. Values of Group Parameters no. of groups
groupa
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
-CH3 >CH2 >CH>C< dCH2 dCHdC< dCd tCH tC(>CH2)R (>CH-)R (>CCdO)R OdCH-COOH -COOdCdO -NH2 >NH (>NH)R >N-Nd (-Nd)R -CN -NO2 -SH -S(-S-)R
a
critical temperature (K) ∆TcPi × 102 ∆TcIi × 102 ∆Tc0i × 102 0.316 3.197 5.335 10.064 -3.270 0.448 5.407 -2.000 -13.81 4.188 -1.813 2.250 12.628 -1.576 2.427 16.230 0.032 -3.894 -3.687 5.686 -2.853 3.477 3.520 0.012 -0.019 -0.567 -9.417 -0.003 -1.705 -0.277 9.588 -8.637 0.002 4.239 4.147 0.000 -2.122 -1.817 -2.976 1.655
1.249 0.375 -0.609 -2.204 2.496 0.589 -0.441 1.399 11.767 -1.892 1.296 -0.140 -2.782 1.126 -0.098 -2.461 1.675 2.826 2.195 -0.936 2.257 0.237 4.427 4.224 1.830 -0.187 4.036 0.437 2.956 2.141 -3.464 4.010 1.462 -1.067 1.700 3.374 1.440 1.472 0.957 -0.730
-0.099 -0.310 0.125 0.091 0.139 0.457 -0.639 -0.030 -1.805 0.000 1.042 0.207 -0.732 0.634 0.450 -0.898 0.405 0.024 0.258 0.907 1.789 0.001 -0.002 -1.022 0.000 0.000 0.362 -0.009 -0.199 0.001 0.730 0.000 0.000 0.049 0.000 0.000 0.516 -0.262 0.724 0.000
critical pressure (MPa) ∆PcPi × 102 ∆PcIi × 102 ∆Pc0i × 102 5.217 3.669 17.498 32.628 -4.921 8.191 26.685 0.970 -1.233 -2.566 -4.994 -0.974 36.201 -0.816 9.065 10.466 -5.351 -7.831 -19.38 3.329 -21.68 7.489 5.386 -3.497 0.004 -18.23 -10.58 -0.075 -6.955 -8.925 -6.612 6.511 0.002 12.477 10.625 0.000 -6.060 -8.681 -17.08 15.282
2.506 2.229 -3.314 -7.638 7.298 -1.135 -6.463 0.684 4.497 1.428 3.510 1.050 -8.441 2.268 -1.886 -6.255 -0.984 5.046 6.338 2.081 4.341 -1.806 0.860 4.443 2.713 11.070 3.649 2.035 4.007 2.497 1.591 -1.648 0.804 -4.550 -0.733 1.302 4.349 4.614 5.483 -5.623
-0.139 0.016 0.495 -0.062 -0.494 1.023 -0.751 0.000 0.008 -0.293 6.391 -0.007 -2.968 -0.765 1.236 -0.077 0.001 -1.772 1.498 -1.895 7.442 -0.001 0.000 0.013 0.000 0.000 0.000 0.000 -0.982 -0.001 0.000 0.000 0.000 -0.712 -0.001 0.000 1.016 0.732 2.699 0.000
critical volume (cm3/mol) ∆VcPi ∆VcIi ∆Vc0i 1.152 3.600 11.532 19.359 -1.913 2.681 4.865 1.501 -388.7 1131.1 2.751 -2.312 15.172 1.314 3.296 -1.882 -109.6 1.217 4.734 -10.99 0.006 -7.041 0.861 7.080 0.008 -4.091 -4.270 -0.001 1.932 13.068 -12.33 31.016 0.000 -3.421 -0.346 -0.017 -3.000 -5.304 -7.890 0.001
4.159 2.249 -1.766 -4.743 5.114 1.578 0.340 1.515 333.2 -475.3 2.444 2.739 -3.698 2.280 0.567 3.684 44.684 -0.921 -0.432 7.909 3.801 9.436 5.680 2.466 4.858 6.814 5.205 2.138 1.639 -2.204 7.260 -8.095 0.000 2.213 6.257 5.930 3.169 5.721 7.001 6.642
-0.111 0.057 0.499 -0.110 -0.055 0.734 -0.082 0.000 -110.9 -0.656 -1.494 1.215 0.540 0.523 -0.091 -0.610 0.000 1.829 -0.349 0.002 0.389 0.000 0.005 0.024 0.000 0.001 0.000 0.000 0.149 0.000 0.138 0.000 0.086 1.459 -0.001 0.012 -0.949 0.205 1.718 0.000
R represents the group on the ring. AC represents the connection to the aromatic ring.
In this study, the group contribution was divided into two parts: the position contribution of the group and the independent contribution of the group. The critical property f is expressed as follows: ni
f)a+
νj∆fPi + ni∆fIi + ∆f0i) ∑i ∑ j)1 (
where Tc, Pc, and Vc are the critical temperature, critical pressure, and critical volume, respectively, and the units employed are Kelvin, MPa, and cm3/mol. A great deal of experimental data in the literature10-12 have been used to optimize the values of group parameters in eqs 3-5 and are shown in Table 1.
(2) Estimation Results and Method Comparison
where ∆fPi is the position contribution of the i-type group, ∆fIi is the independent contribution of the i-type group, ∆f0i is the constant of the i-type group, and ni νj is the module index sum of the i-type groups. ∑j)1 To improve the property estimation accuracy, the power index “1” of property f in eq 2 is substituted by a variable, the optimum value of which can be obtained by trial computation. On doing this, the expressions of Tc, Pc, and Vc may be written as follows: ni
Tc/Tb ) 0.589 +
νj∆TcPi + ni∆TcIi + ∆Tc0i) ∑i ∑ j)1
Pc-0.5 ) 0.332 +
νj∆PcPi + ni∆PcIi + ∆Pc0i) ∑i ∑ j)1
(
(3)
ni
(
(4)
ni
Vc
0.6
) 11.237 +
νj∆VcPi + ni∆VcIi + ∆Vc0i) ∑i (∑ j)1
(5)
Critical temperatures of 585 compounds, critical pressures of 501 compounds, and critical volumes of 374 compounds are divided into six groups; the average deviations between the calculated value by this method and experimental data are listed in Table 2. The experimental data used came from the literature.10-12 The critical property predictions by the new method are compared with that by the SGC methods1,3-5 and by the composite group contribution methods7,8 in Table 3. The present method proposed gives, as shown in Table 3, more accurate predictions for critical pressures and volumes than the compared methods. The accuracy of critical temperature prediction by this method is slightly lower than that by the Marrero-Morejo´n and PardilloFontdevila8 method. In the Marrero-Morejo´n and Pardillo-Fontdevila method, a three-term expression with a second power of the group contribution was used for estimating the critical temperature. This may be be-
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6261 Table 2. Correlation Accuracy and the Error Distribution of the Critical Properties for Six Groups of Organic Compounds critical temperature (K)
critical volume (cm3/mol)
critical pressure (MPa)
compound
no. of data
absolute deviation
percent deviation (%)
no. of data
absolute deviation
percent deviation (%)
no. of data
absolute deviation
percent deviation (%)
aliphatic hydrocarbon naphthenic hydrocarbon aromatic hydrocarbon O,S compounds N compounds halogenated compounds total
116 45 51 216 59 98 585
3.05 3.55 4.11 3.98 4.03 4.69 3.91
0.55 0.59 0.73 0.68 0.92 0.79 0.69
117 59 35 160 50 80 501
0.05 0.08 0.12 0.095 0.13 0.11 0.09
1.58 1.81 2.53 2.12 2.48 2.95 2.15
98 19 29 126 30 72 374
4.87 5.21 7.65 6.02 6.12 5.09 5.63
1.21 1.49 1.61 1.42 1.52 1.62 1.43
Table 3. Comparison of the Accuracy between Widely Used Methods and the One Proposed Tc
Pc
Vc
method
AAEa (K)
APEb (%)
AAE (MPa)
APE (%)
AAE (cm3/mol)
APE (%)
Ambrose3,4 Klincewicz and Reid5 Joback and Reid1 Constantinou and Gani7 Marrero-Morejo´n and Pardillo-Fontdevila8 proposed one
4.3 7.5 4.8 4.85 2.79 3.91
0.7 1.3 0.8 0.85 0.48 0.69
0.18 0.30 0.21 0.113 0.106 0.009
4.6 7.8 5.2 2.89 2.92 2.15
8.5 8.9 7.5 6.00 4.56 5.63
2.8 2.9 2.3 1.79 1.45 1.43
a
AAE ) average absolute error. b APE ) average percent error.
Table 4. Results of the Critical Property Estimation for Octane group
ni
ni ∑j)1 νj
∆TcPi × 102
∆TcIi × 102
∆Tc0i × 102
∆PcPi × 102
∆PcIi × 102
∆Pc0i × 102
∆VcPi
∆VcIi
∆Vc0i
-CH3 >CH2
2 6
0.837 1.968
0.316 3.197
1.249 0.375
-0.099 -0.310
5.217 3.669
2.506 2.229
-0.139 0.016
1.152 3.600
4.159 2.249
-0.111 0.057
Table 5. Results of the Critical Property Estimation for 1,3,5-Triethylbenzene group
ni
ni ∑j)1 νj
∆TcPi × 102
∆TcIi × 102
∆Tc0i × 102
∆PcPi × 102
∆PcIi × 102
∆Pc0i × 102
∆VcPi
∆VcIi
∆Vc0i
-CH3 >CH2 (dCH-)R (dC
