Ind. Eng. Chem. Res. 2008, 47, 2075-2085
2075
Estimation of the Heat Capacity of Organic Liquids as a Function of Temperature by a Three-Level Group Contribution Method Zdenka Kolska´ Department of Chemistry, Faculty of Science, J. E. Purkinje UniVersity, C ˇ eske´ Mla´ dezˇe 8, 400 96 U Ä stı´ nad Labem, Czech Republic
Jaromı´r Kukal Department of Computing and Control Engineering, Institute of Chemical Technology Prague, Technicka´ 5, 166 28 Prague 6, Czech Republic
Milan Za´ bransky´ and Vlastimil Ru˚ zˇ icˇ ka* Department of Physical Chemistry, Institute of Chemical Technology Prague, Technicka´ 5, 166 28 Prague 6, Czech Republic
A new group contribution method for estimating the heat capacity of pure organic liquids, as a function of temperature in the range from the melting temperature to the normal boiling temperature, was developed. A large set of critically assessed data for 549 compounds was used for group contributions calculation. Values obtained using the method developed here were compared with estimations that were determined using the Za´bransky´ and Ru˚zˇicˇka and Chickos group contribution methods. A statistical analysis of the regressed data was also performed, which indicated the confidence of the regressed parameters and other related information. The average relative errors for the new method are as follows: 1.2% for 549 compounds from the basic set covering data over the temperature range indicated above, 1.5% for 404 data points at 298.15 K for compounds from the basic dataset, and 2.5% for 149 compounds from an independent test set with data obtained at 298.15 K. Introduction Isobaric heat capacity of liquid (Clp) is an important thermodynamic quantity of a pure compound. Its value must be known for the calculation of an enthalpy difference, which is required for the evaluation of heating and cooling duties. Liquid heat capacity also serves as an input parameter, for example, in the calculation of the temperature dependence of the enthalpy of vaporization, for extrapolation of the vapor pressure and the related thermal data via their simultaneous correlation, etc. Heat capacity is one of the directly measured properties. Calorimetrically determined experimental data are known for some 2000 organic and inorganic compounds.1,2 A substantial portion of them were determined at only one temperature, mostly at 298.15 K, or in a narrow temperature range. A comparison of available experimental data on liquid heat capacities and of industrially important compounds clearly reveals a need for an estimation method to supplement the missing data. As in most engineering applications, the heat capacity of a liquid is required to calculate the enthalpy difference; an analytical function is desirable to represent the temperature dependence of the heat capacity. Several estimation methods were presented in the literature,3-13 some which are based on a group contribution approach.14-30 In this work, we applied the three-level group contribution method that has been reported by Marrero and Gani31 and extended it to the estimation of heat capacity of liquids as a function of temperature. This is a useful development insofar as the Marrero and Gani method allows the estimation of a pure
compound physical chemical property that is either temperatureindependent or required to be measured at a specified temperature, primarily 298.15 K.32-35 New Group Contribution Method Methodology Description. The new group contribution method for the estimation of heat capacity of organic liquids is based on the methodology developed by Constantinou and Gani32 and Marrero and Gani.31 The estimation of properties involves a three-level calculation procedure that covers groups of the first level for the estimation of simple monofunctional molecules, and groups of the second and third levels for improving the more-complex prediction of compounds.31,35 New Model. We extended the Gani et al. approach for estimating the heat capacity of liquids as a function of temperature, ClP(T). The temperature dependence of the group contributions was expressed through an empirical polynomial equation.1 The new model for the estimation of ClP(T) has the form of the following equations:
Clp(T) ) Clp0(T) +
l l (T) + w ∑ MjCp2-j (T) + ∑I NiCp1-i j l (T) z ∑ OkCp3-k k
(1)
with
C pl qthlevel-i ,j, or k(T) ) aq-i, j, or k + bq-i, j, or k
* To whom all correspondence should be addressed. Fax: +420 220445018. E-mail:
[email protected]. 10.1021/ie071228z CCC: $40.75 © 2008 American Chemical Society Published on Web 02/20/2008
(100T ) +
dq-i, j, or k
(100T )
2
(2)
2076
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008
l In eq 1, Cp1-i (T) is the contribution of the first-level group of type i, Cp2-j(T) is the contribution of the second-level group of l (T) is the contribution of the third-level group type j, and Cp3-k of type k. Ni, Mj, and Ok denote the number of occurrences of the individual groups (of type i, j, or k, respectively) in a l compound. Cp0 (T) (which could be considered as the contribution of the zero-level group) is an additional adjustable parameter. Variables w and z are weighting factors that are assigned to 0 or 1, depending on whether the second-level and third-level contributions, respectively, are used or not. In eq 2, aq-i, j, or k, bq-i, j, or k, and dq-i, j, or k are adjustable parameters for l l (T), Cp2-j (T), and the temperature dependence of Cp0(T), Cp1-i l Cp3-k(T). All compounds were described by the same set of groups as those defined in the paper by Kolska´ et al.35 Database. A large database of critically assessed data on liquid heat capacity1,2 (hereafter referenced as the basic set) was used for all group contribution parameter calculations. The pure organic compounds included in the database range in the molar mass from 41 g/mol up to 462 g/mol and cover many families, such as hydrocarbons (saturated, cyclic, unsaturated, aromatic), halogenated hydrocarbons, and compounds that contain O, N, and S atoms. A list of names of individual families is given in Appendix A. The basic set used in this work was identical to that used by Za´bransky´ and Ru˚zˇicˇka.29 Procedure for Estimation of Model Parameters. The parameters to be fitted were the following: aq-i, j, or k, bq-i, j, or k, and dq-i, j, or k. For the parameter estimation, we adopted two approaches. First, in the so-called hierarchic approach (denoted as new model-H) the parameter calculation was performed as described earlier31,35 in three consecutive steps. Second, in the nonhierarchic approach (denoted as new model-NH), all parameters were calculated in a single step. Mathematical Background. Determination of adjustable parameters of any group contribution method results in a solution of an overdetermined system of linear equations Tt ) u, where T ∈Rm×n, t ∈Rn, and u ∈Rm; m is the number of experimental input data and n is the total number of group contributions (n ) n1 + n2 + n3, where n1, n2, and n3 are the numbers of contributions in the first, second, and third levels). The traditional least-squares method is based on the minimization of the function SSQ(t) ) |Tt - u|2. The explicit optimum solution, t* ) (T′T)-1T′u, is not numerically stable for a large number n and for ill-posed tasks. The numerical difficulties can be decreased using the hierarchic approach, which will be demonstrated for three levels of optimization used in this work. The contribution vector can be decomposed as t ) (p,q,s), where p ∈Rn1, q ∈Rn2 and s ∈Rn3. The adequate decomposition of the system matrix comes to T ) (P|Q|S) and the original set of equations can be written as Pp + Qq + Ss ) u. The three-level hierarchic approach of the least-squares optimization is based on the step-by-step minimization of the function SSQ*(p,q,s) ) |Pp + Qq + Ss - u|2. Let t* be the optimum solution of the minimization task SSQ(t) ) min. Then, SSQ(t*) e SSQ*(p*,q*,s*); this inequality follows from the definition of the global optimum t* in the theoretical case (“theoretical case” means that the truncation error is absent). However, in a real case, the role of the truncation error and that of the ill-posed linear system can cause a turnover of the inequality SSQ(t*) e SSQ*(p*,q*,s*), when the hierarchic solution is numerically better than the global one. The global solution with a suppressed truncation error was preferred in this work and is represented by the nonhierarchic approach. Our procedure was based on the regularization of the least-
squares task with the adequate parameter λ > 0. This approach is called the Tikhonov regularization36 or the LevenbergMarquardt method,37 respectively. The corresponding formula is t+ ) (T′T + λIn)-1T′u. The objective of the regularization is to obtain SSQ(t+) e SSQ*(p+,q+,s+) < min (SSQ(t*),SSQ*(p*,q*,s*)). The recommended value of the parameter λ is 10-6. In the calculation of the contribution vector t ) (p,q,s) in this work, we used the relative error minimization, so the original system was modified to Trel ) T diag-1(u) and urel ) 1m in this case. The group contribution parameters are given in Table 1. In this table, we present parameters that have been determined by the nonhierarchic approach, as well as by the hierarchic approach. Although it turned out that the nonhierarchic approach is slightly superior (see the next part), for compatibility with commonly used software packages (e.g., the CAPEC program package38-41), we present both sets of parameters. The statistical significance of group-contribution values was evaluated using the average relative error (ARE), which is given by eq 3, and using the median of relative errors (MED):
ARE[Cpl ] )
1
n
∑
n i)1
l l |Cp,exp - Cp,est |i l Cp,exp,i
× 100
(3)
where n is the number of data in the database and the suffixes “exp” and “est” denote the experimental and estimated values, respectively. The median is a number that is obtained by separating the higher half of a sample from the lower half. The median of the relative error (RE) values is found by arranging all of them from the lowest value to the highest value and selecting the one in the middle. Table 2 gives the ARE and MED values for Cpl estimation over a temperature range. The correlation statistics are shown for the first level, the second level, and the third level for the hierarchic approach and for a single level for the nonhierarchic (global) approach. The results in Table 2 clearly demonstrate that the nonhierarchic approach is slightly superior, in terms of the ARE and MED values for the entire basic dataset. More detailed results are shown for individual molecular types in Appendix A. Figure 1 provides a visualization of the correlation of experimental data on Cpl at 298.15 K for compounds from the basic dataset determined by our model. Note that, except for a few data points, all of the data have been fitted to a high degree of accuracy over a wide range of Cpl values, from ∼80 J K-1 mol-1 up to ∼2000 J K-1 mol-1. Performance Analysis and Model Application The performance of the developed model for estimation of the liquid heat capacity was analyzed in terms of comparison with other estimation models, as well as in terms of extrapolation features, predictive capability, and consistency. Comparison with Other Models. Two methods, which were formulated by Za´bransky´ and Ru˚zˇicˇka29 and Chickos et al.,24,28 were selected for a comparison of the Cpl estimation results. The former method is representative of a classical Benson’s type second-order group-contribution method that is capable of estimating the liquid heat capacity over the temperature range from the melting temperature to the normal boiling temperature. The latter method is also based on the Benson’s group additivity approach of the first-order type, but it is capable of estimating Cpl at only one temperature: 298.15 K. The Chickos
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2077 Table 1. List of Group Contributions and Their Values for Liquid Heat Capacity (Cpl , Expressed in Terms of J K-1 mol-1), Determined by the Nonhierarchic (NH) and Hierarchic (H) Approachesa New Model, Nonhierarchic-NH No. 0 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
group name additional adjustable parameter CH3 CH2 CH C CH2dCH CHdCH CH2dC CHdC CdC CH2CH2dCdCH CH2CH2dCdC CHtC CtC aCH aC fused with an aromatic ring aC fused with a nonaromatic subring aC, except as above aN in an aromatic ring aC-CH3 aC-CH2 aC-CH aC-C aC-CHdCH2 aC-CHdCH aC-CdCH2 aC-CtCH OH aC-OH COOH aC-COOH CH3CO CH2CO CHCO CCO aC-CO CHO aC-CHO CH3COO CH2COO CHCOO HCOO aC-COO COO, except as above CH3O CH2O CH-O aC-O CH2NH2 CHNH2 CNH2 CH3NH CH2NH CH3N CH2N aC-NH2 aC-NH aC-N NH2, except as above CH2CN CCN aC-CN CN, except as above aC-NCO CH2NO2 aC-NO2 ONO2 CH2Cl CHCl CHCl2 CCl2 CCl3
a
b
New Model, Hierarchic-H No.a
a
b
d
7.24
0
89.53
-40.13
5.19
First-Level Group Contributions -10.75 17.70 -1.15 16.19 3.21 0.41 50.97 -19.12 3.71 53.24 -26.31 4.51 -14.86 47.46 -8.04 18.12 40.76 -9.36 -13.26 64.86 -13.50 -0.58 80.98 -21.01 -150.51 186.33 -37.28 34.08 9.23 1.39 50.34 3.06 1.38 25.94 -0.53 3.95 4.16 28.24 -4.93 -1.28 8.17 -0.43 384.23 -151.04 15.62 539.17 -291.04 41.64 474.70 -258.65 37.43 10.79 5.74 -0.98 20.23 9.63 -0.43 49.18 -6.09 1.15 96.82 -30.00 3.37 52.70 -15.10 1.17 -0.29 38.36 -5.08 -134.98 110.23 -17.46 -62.49 82.00 -13.56 58.32 -2.92 1.88 13.08 -17.97 9.02 178.51 -34.86 1.98 12.58 20.70 -0.21 -738.97 370.11 -39.06 127.88 -47.29 10.19 65.01 15.15 -3.53 67.08 10.77 -3.23 -23.93 32.22 -4.45 103.77 -17.30 1.21 31.10 11.01 -2.20 26.21 18.40 -2.50 58.31 -4.75 5.65 91.18 23.69 -9.10 7.84 20.47 -2.85 -40.23 40.10 -5.38 87.72 -6.65 0.79 -3.40 43.61 -7.28 7.24 25.20 -2.53 42.67 9.68 -1.36 54.36 3.90 -1.66 40.33 23.30 -5.76 48.21 27.05 -4.95 34.31 39.40 -7.34 66.17 11.97 -2.34 14.59 44.77 -8.21 221.83 -45.82 0.63 -46.47 75.91 -13.72 78.76 8.23 -5.69 133.67 -25.76 2.28 90.55 -16.34 2.06 41.79 8.14 -2.78 51.85 -3.48 0.44 36.93 7.45 1.85 47.87 11.05 -1.77 21.97 20.75 -3.50 9.56 -3.63 2.19 28.13 24.91 -3.35 88.87 -13.06 4.00 380.42 -171.77 23.67 -99.91 132.09 -23.00 10.00 21.75 -2.18 93.83 -23.16 3.38 36.39 18.82 -1.98 43.52 24.65 -4.95 57.68 16.51 -1.30
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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
1.04 19.64 12.20 25.40 15.13 57.81 35.28 46.90 22.45 37.77 43.17 27.11 -3.00 -2.09 -4.91 -3.73 -9.24 19.83 14.90 37.15 97.94 88.73 20.15 237.57 -30.89 78.77 -99.86 163.82 -2.21 -739.60 69.38 52.07 79.70 -40.63 112.43 32.92 46.66 34.25 74.76 -1.46 -52.52 72.73 12.98 21.08 48.82 66.98 116.47 83.30 61.60 220.92 84.91 209.44 -5.20 -391.63 124.73 156.25 38.67 41.12 21.25 24.68 36.89 5.54 44.07 93.49 380.97 -103.93 17.62 87.64 31.35 36.42 49.56
8.29 0.78 7.90 -2.25 9.02 -21.62 -2.15 0.39 33.10 7.91 10.62 0.04 35.78 8.58 8.60 10.46 12.84 -0.12 11.61 4.17 -37.93 -28.81 25.04 -66.05 44.81 -16.25 71.89 -29.49 30.79 363.53 -2.63 26.42 -2.77 68.57 -21.22 10.22 5.07 35.65 13.70 52.01 58.54 3.36 38.74 14.60 4.92 -9.64 -23.43 3.17 15.06 -94.89 -12.05 -69.77 42.02 253.86 -23.07 -58.04 13.64 6.56 25.29 36.07 10.70 19.92 14.42 -14.92 -174.45 135.83 17.23 -18.64 24.59 32.16 23.46
0.76 0.80 -1.03 0.16 0.99 6.38 2.17 -1.01 -10.93 1.49 -0.39 3.70 -6.70 -0.47 -0.96 -1.51 -1.59 0.05 -0.33 -0.61 5.66 2.60 -2.86 6.45 -5.20 4.10 -7.70 1.83 -1.98 -37.22 1.91 -5.81 -0.97 -11.07 1.53 -1.96 -0.28 -5.17 -2.99 -7.91 -6.92 -0.87 -7.68 -0.71 -0.40 0.60 1.32 -1.24 -2.83 15.48 3.11 8.74 -7.08 -37.13 2.43 8.37 -4.38 -1.16 -2.79 -7.46 -1.75 -2.59 -1.53 4.14 24.47 -23.90 -1.47 2.55 -3.30 -6.71 -2.67
105.94
-51.40
d
2078
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008
Table 1. (Continued) New Model, Nonhierarchic-NH No.
group name
72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
CH2F CHF2 CF2 CF3 CCl2F HCClF CClF2 aC-Cl aC-F aC-I aC-Br -I, except as above -Br, except as above -F, except as above -Cl, except as above OCH2CH2OH CH2SH CHSH CSH aC-SH -SH, except as above CH3S CH2S CHS aC-SCO3 C2H3O CH2 (cyclic) CH (cyclic) C (cyclic) CHdCH (cyclic) CHdC (cyclic) CH2dC (cyclic) NH (cyclic) O (cyclic) CO (cyclic) S (cyclic) -O-, except as above -S-, except as above
111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
(CH3)2CH (CH3)3C CH(CH3)CH(CH3) CH(CH3)C(CH3)2 C(CH3)2C(CH3)2 CHndCHm-CHpdCHk (k, m, n, p ) 0, ..., 2) CH3-CHmdCHn (m, n ) 0, ..., 2) CH2-CHmdCHn (m, n ) 0, ..., 2) CHp-CHmdCHn (m, n ) 0, ..., 2; p ) 0, ,.., 1) CHCHO or CCHO CH3COCH2 CH3COCH or CH3COC CHCOOH or CCOOH CH3COOCH or CH3COOC CHOH COH CHm(OH)CHn(OH) (m, n ) 0, ..., 2) CHm(OH)CHn(NHp) (m, n, p ) 0, ..., 2) CHm(NH2)CHn(NH2) (m, n ) 0, ..., 2) NC-CHn-CHm-CN (n, m ) 1, ..., 2) COO-CHn-CHm-OOC (n, m ) 1, ..., 2) OOC-CHm-CHm-COO (n, m ) 1, ..., 2) CHm-O-CHndCHp (m, n, p ) 0, ..., 3) CHmdCHn-F (m, n ) 0, ..., 2) CHmdCHn-Cl (m, n ) 0, ..., 2) CHmdCHn-CN (m, n ) 0, ..., 2) CHndCHm-COO-CHp (m, n, p ) 0, ..., 3) CHmdCHn-CHO (m, n ) 0, ..., 2) CHmdCHn-COOH (m, n ) 0, ..., 2) aC-CHn-X (n ) 1, ..., 2), where X is a halogen aC-CHn-O- (n ) 1, ..., 2) aC-CHn-OH (n ) 1, ..., 2) aC-CH(CH3)2 aC-C(CH3)3
New Model, Hierarchic-H
a
b
d
No.a
a
b
d
6.30 8.74 12.93 -29.59 35.62 14.09 -10.02 18.32 8.20 15.62 6.08 -95.22 -21.39 -11.58 39.99 -5.75 31.55 54.58 90.39 40.68 -14.27 45.30 53.18 54.06 99.88 130.69 49.63 -5.86 104.05 18.20 18.76 49.82 43.14 -209.01 123.35 33.03 -2.80 -30.04 -48.86
12.33 25.00 19.38 55.77 22.90 22.86 45.46 11.79 14.34 25.52 22.28 85.97 32.75 12.82 -5.52 59.46 20.87 10.27 -18.41 15.13 21.96 7.02 8.70 22.96 -18.58 -29.16 -16.59 11.43 -54.34 6.76 0.52 -27.08 -15.85 172.27 -82.77 8.24 24.33 24.29 56.79
1.27 -2.39 -2.34 -6.90 -2.58 -1.30 -5.93 -1.69 -2.09 -5.08 -3.60 -14.69 -5.29 -1.08 -0.14 -4.83 -3.14 -2.48 2.69 -2.91 -3.68 -0.08 -2.35 -6.19 1.85 4.45 4.07 -0.54 8.72 -2.66 2.28 5.26 5.44 -28.22 16.64 -1.69 -3.96 -4.13 -9.06
72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
48.23 33.50 35.62 -23.33 41.45 31.20 32.33 36.94 12.20 36.07 29.22 -87.17 -10.87 -12.79 23.19 -36.22 38.20 51.69 57.67 61.12 43.47 51.47 34.28 66.68 120.51 126.26 40.12 0.60 22.63 83.25 19.00 56.86 40.16 47.34 -2.31 9.62 33.15 -130.06 -26.06
-18.42 3.27 -0.02 47.21 15.75 12.21 7.80 -2.11 8.78 12.20 4.63 82.44 24.80 12.63 -15.79 80.39 16.82 11.05 9.27 1.81 -4.34 3.49 23.43 9.42 -27.86 -22.66 5.99 7.28 -0.97 -38.77 -0.72 -21.05 -14.67 12.58 15.12 20.64 -4.87 108.77 34.45
6.94 2.36 1.68 -4.69 -0.77 0.32 2.20 0.92 -0.54 -2.86 -0.26 -14.38 -3.73 -0.83 6.02 -8.35 -2.50 -2.62 -2.89 -0.69 0.96 0.41 -4.96 -3.94 2.36 2.78 1.53 0.12 -0.13 5.10 2.63 4.49 5.51 -3.89 -2.72 -3.18 1.14 -18.70 -4.55
Second-Level Group Contributions 2.30 -6.45 1.55 -4.29 8.08 -2.02 -35.67 24.12 -3.81 -70.53 50.14 -9.22 -54.47 39.79 -6.90 42.52 -64.77 16.23 22.93 -33.28 8.05 22.37 -33.23 7.89 -0.34 -15.08 5.06 -364.47 252.44 -42.14 -71.15 54.13 -9.95 -84.87 63.96 -12.01 -127.32 62.69 -7.40 -338.60 142.49 -16.75 -393.77 284.02 -47.66 -356.37 249.59 -37.58 -169.45 150.35 -31.31 -194.13 102.90 -13.55 36.85 -43.85 9.13 -46.52 39.70 -9.47 167.72 -57.46 4.29 -178.24 125.41 -20.24 45.65 -52.96 11.86 43.05 -39.85 7.54 -32.11 -6.64 5.20 9.56 -3.63 2.19 63.70 -55.53 10.43 163.18 -105.60 17.22 850.61 -562.94 92.16 57.09 -45.85 9.65 -60.08 48.57 -8.26 -233.61 171.75 -28.90 5.51 -12.78 3.67 43.76 -23.35 4.39
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
19.48 -23.06 -13.18 -44.90 -37.80 -8.21 1.13 -3.40 -16.74 -339.11 -9.11 -10.48 -107.39 -114.97 -124.66 -216.13 94.71 -72.71 -136.35 -12.58 91.78 -108.97 7.25 6.94 -18.07 0.00 2.36 148.10 948.72 106.37 -1.05 -88.19 -11.49 4.95
-16.15 20.55 12.25 32.11 26.83 7.32 -0.82 2.39 15.08 233.99 7.88 8.76 47.19 -44.39 78.06 138.02 -54.40 35.13 73.44 0.87 -2.29 78.24 -6.77 -8.08 16.79 0.00 -0.38 -83.06 -610.36 -59.16 0.67 58.31 9.08 -3.79
3.01 -4.46 -2.46 -6.66 -5.56 -1.29 0.12 -0.53 -2.98 -39.34 -1.43 -2.21 -4.55 20.91 -10.03 -17.23 6.36 -4.05 -9.81 0.67 -5.77 -12.39 1.24 2.28 -3.81 0.00 -0.23 11.52 97.75 7.48 -0.11 -8.20 -1.55 1.12
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2079 Table 1. (Continued) New Model, Nonhierarchic-NH No.
group name
145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
aC-CF3 (CHndC)(cyc)-CH3 (n ) 0, ..., 2) CH(cyc)-CH3 CH(cyc)-CH2 CH(cyc)-CH CH(cyc)-C CH(cyc)-CHdCHn (n ) 1, ..., 2) CH(cyc)-CdCHn (n ) 1, ..., 2) CH(cyc)-Cl CH(cyc)-OH CH(cyc)-NH2 CH(cyc)-SH CH(cyc)-SC(cyc)-CH3 AROMRINGs1s2 AROMRINGs1s3 AROMRINGs1s4 AROMRINGs1s2s3 AROMRINGs1s2s4 AROMRINGs1s3s5 AROMRINGs1s2s3s4 AROMRINGs1s2s3s5 AROMRINGs1s2s4s5 PYRIDINEs2 PYRIDINEs3 PYRIDINEs4 PYRIDINEs2s3 PYRIDINEs2s4 PYRIDINEs2s5 PYRIDINEs2s6 PYRIDINEs3s4 PYRIDINEs3s5 AROMRINGs1s2s3s4s5 CH2OCHO CH2COOCH2 CCOOCH2 CHCOOCH2 CH3COOCH2 OCH2CH2O CH2COOCH3 OCH2O CH2SSCH2 (CH2OCH2)(cyc) (CH2OCH)(cyc) (CHdCHOCHdCH)(cyc) (COOCH2)(cyc) (CH2NHCH2)(cyc) aCaNaC (CH2SCH2)(cyc) (dCHSCHd)(cyc) (dCHSCHd)(cyc) (3 F) (5 F) (perFlouro)
199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218
a
b
8.94 8.25 49.82 -27.08 -60.66 37.50 -50.85 31.05 -151.74 87.41 -35.95 20.41 1.49 -22.73 -33.33 7.74 -78.46 40.50 -173.40 96.39 -198.45 149.07 -14.27 21.96 -45.91 27.08 24.35 -20.88 -5.82 1.12 -49.93 25.88 -32.01 14.84 22.01 -14.75 -15.05 7.75 -60.51 29.24 -70.85 47.97 -32.41 20.35 -76.59 53.81 -4.29 0.74 -11.90 6.64 -21.96 12.13 -9.50 6.43 -12.26 4.65 -15.61 5.79 -95.43 62.60 -21.29 15.56 -16.32 7.33 73.70 -55.26 -40.23 40.10 16.41 -37.05 -23.93 32.22 7.84 20.47 -21.27 34.02 23.69 -11.32 21.12 -35.22 19.75 -20.97 11.00 -15.56 -126.93 100.97 49.63 -16.59 -151.24 110.58 -174.57 123.71 271.87 -166.98 61.37 -47.23 47.26 -37.50 -21.48 12.32 -78.10 55.00 58.27 -51.18 20.27 -29.74 35.87 -45.37 Third-Level Group Contributions HOOC-(CHn)m-COOH (m > 2, n ) 0, ..., 2) 47.68 35.65 OH-(CHn)m-OH (m > 2, n ) 0, ..., 2) -776.13 474.81 NC-(CHn)m-CN (m > 2) -31.39 29.87 aC-(CHndCHm)cyc (fused rings) (n, m ) 0, ..., 1) -361.00 234.70 aCsaC (different rings) -949.70 542.18 aC-CHn(cyc) (fused rings) (n ) 0, ..., 1) -352.63 223.53 aC-(CHn)m-aC (different rings) (m > 1; n ) 0, ..., 2) -162.91 94.57 CH(cyc)-CH(cyc) (different rings) -16.53 32.62 CH multiring -82.22 50.32 aC-CHm-aC (different rings) (m ) 0, ..., 2) -328.63 177.13 aC-(CHmdCHn)-aC (different rings) (m, n ) 0, ..., 2) -134.98 110.23 aC-S-aC (different rings) -509.98 283.05 aC-O-aC (different rings) -476.00 250.62 aC-CHn-O-CHm-aC (different rings) (n, m ) 0, ..., 2) -30.04 24.29 AROM.FUSED[2] -392.10 164.35 AROM.FUSED[2]s1 -392.02 161.30 AROM.FUSED[2]s2 -426.19 177.11 AROM.FUSED[2]s2s3 -363.01 152.00 AROM.FUSED[2]s1s4 -344.79 133.58 AROM.FUSED[2]s1s3 -402.80 165.45
New Model, Hierarchic-H
d
No.a
a
b
d
-3.23 5.26 -5.87 -4.97 -13.13 -3.20 6.84 0.06 -4.38 -11.09 -26.01 -3.68 -4.31 4.24 0.26 -2.94 -1.30 2.98 -0.47 -3.03 -7.66 -2.69 -8.73 0.25 -0.83 -1.52 -0.45 -0.06 -0.11 -10.84 -1.96 -0.54 10.65 -5.38 11.15 -4.45 -2.85 -9.40 0.99 10.86 4.28 4.70 -20.41 4.07 -20.98 -22.84 24.72 9.64 6.61 -2.63 -10.33 10.74 8.20 12.02
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
27.80 -1.77 -4.22 11.24 -84.48 -69.43 31.37 39.30 85.90 8.85 -116.79 0.00 -2.87 4.78 8.43 -15.93 -12.07 32.27 -18.74 -26.33 -19.78 18.66 -25.54 11.65 4.04 -6.03 10.95 8.19 4.84 -99.60 -0.83 4.13 -11.43 -21.70 18.74 23.03 -19.92 -3.46 37.76 30.37 5.03 42.25 -1.41 -8.92 -9.66 -24.78 18.78 85.98 3.29 -47.55 -65.12 9.25 -9.12 0.99
-21.80 1.60 2.53 -9.71 40.25 24.49 -22.71 -25.73 -63.30 -39.69 91.89 0.00 1.00 -4.24 -4.57 7.90 4.91 -17.71 15.33 11.40 13.51 -14.12 19.35 -8.29 -2.39 3.10 -4.16 -5.94 -4.80 63.93 4.97 -3.26 8.84 17.54 -14.29 -20.53 16.43 -0.73 -20.87 -17.66 -8.63 -38.27 2.28 7.78 3.91 12.17 -10.30 -59.16 -3.60 35.65 48.87 -6.59 5.13 -1.12
3.92 -0.27 -0.37 1.87 -5.04 -1.83 4.13 3.69 11.85 13.58 -16.34 0.00 -0.06 0.83 0.54 -0.89 -0.38 2.50 -2.80 -1.31 -2.12 2.85 -3.19 1.30 0.22 -0.47 0.46 0.86 0.81 -10.67 -1.04 0.38 -1.32 -3.23 2.41 4.46 -3.06 0.36 2.79 2.99 2.29 8.39 -0.66 -1.56 -0.27 -1.54 1.12 10.39 0.95 -6.83 -9.35 1.10 -0.71 0.39d
-11.76 -74.60 -8.09 -37.84 -78.94 -34.99 -13.04 -7.48 -8.17 -24.66 -17.46 -41.17 -34.67 -4.13 -17.68 -16.51 -18.53 -15.61 -12.83 -17.16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
54.95 -578.38 12.93 7.34 16.65 -5.73 -114.37 108.59 -14.06 19.44 0.54 -22.17 -43.71 -0.01 -8.67 30.93 -20.53 33.70 17.88 -16.41
25.28 311.30 -14.66 2.93 -4.81 4.19 58.67 -48.77 5.01 -12.71 -0.21 5.46 10.48 0.00 4.97 -16.04 8.67 -10.53 -10.67 6.87
-9.02 -43.20 2.86 -1.73 0.32 -0.64 -7.13 6.03 -0.48 1.93 0.02 -0.28 -0.34 0.00 -0.69 2.22 -0.94 0.72 1.15 -1.02
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Table 1. (continued) New Model, Nonhierarchic-NH
New Model, Hierarchic-H
No.
group name
a
b
d
No.a
219 220 221 222 223
AROM.FUSED[3] AROM.FUSED[4a] AROM.FUSED[4p] PYRIDINE.FUSED[2] PYRIDINE.FUSED[2-iso]
-304.97 -738.46 -762.03 -394.59 -375.09
122.21 301.95 309.79 160.60 150.09
-12.63 -31.23 -31.93 -16.45 -15.03
21 22 23 24 25
a
a
b
-27.44 37.17 24.07 13.29 32.72
9.10 -16.16 -15.28 -7.59 -18.06
-0.68 1.74 2.12 0.96 2.38
For the hierarchic approach, serial numbers were assigned for the first-level, second-level, and third-level group contributions independently.
Table 2. Results for Estimation of Liquid Heat Capacity as a Function of Temperature Obtained by the Present Model (eq 1) for the Basic Dataset (549 Compounds) ARE (%)
MED (%)
NGRa
New Model, Hierarchic-H 1.9 1.2 1.6b 1.0 1.5c 0.9
110 + 1d 88 25
New Model, Nonhierarchic-NH first + second + third levels 1.2 0.8
223 + 1d
first level second level third level
a NGR is the number of group contributions determined at individual estimation levels for the hierarchic approach (new model (H)) or the number of all group contributions calculated for the nonhierarchic approach (new model (NH)). b Includes all compounds represented by the first-level and the second-level group contributions. c Includes all compounds represented by the first-level, second-level, and third-level group contributions. d The digit “1” denotes the additional adjustable parameter.
Table 3. Results for Comparison of Cpl at 298.15 K Obtained by the New Models (Nonhierarchic Approach (New Model-NH) and Hierarchic Approach (New Model-H) and by the Methods of Za´ bransky´ and Ru˚ zˇ icˇ ka and Chickos et al. for the Basic Dataset and the Test Dataset of Compounds method
ARE (%)
MED (%)
Basic Dataset new model-H 404 new model-NH 404 Za´bransky´ and Ru˚zˇicˇka (ZR) 404 Chickos et al. (CH) 399
1.6 1.5 1.8 3.9
1.1 1.0 1.1 3.0
Test Dataset 149 149 149 147
2.7 2.5 3.0 4.2
2.1 2.0 2.0 3.1
new model-H new model-NH Za´bransky´ and Ru˚zˇicˇka (ZR) Chickos et al. (CH)
NC
Figure 2. Comparison of the experimental, estimated, and predicted heat capacities at 298.15 K for n-alkanes. For an explanation of terms “estimated” and “predicted”, see the section entitled “Extrapolation Features”.
Figure 1. Comparison of the experimental data and the estimated and predicted values for liquid heat capacity at 298.15 K. For an explanation of terms “estimated” and “predicted”, see the section entitled “Extrapolation Features”.
method cannot be used for some more-complex compounds, in contrast to the Za´bransky´ and Ru˚zˇicˇka method. The average relative error of the new model for the basic dataset of 549 compounds over the temperature range was 1.2%; the error by the Za´bransky´ and Ru˚zˇicˇka method for the same dataset was also 1.2%. For a comparison of heat capacity estimation at 298.15 K, we used both methods (Za´bransky´ and Ru˚zˇicˇka and Chickos) and evaluated them for 404 compounds from the basic set and also for 149 additional compounds that were not used in the parameter calculation. The latter set of 149 compounds is hereafter called the test set. Because of the unavailability of
some groups in the Chickos et al. method,24,28 the number of heat capacity values estimated by this method is lower. Results of this comparison are presented in Table 3 and in Appendix A. It turned out (see Tables 2 and 3) that the new model is superior, in terms of the ARE and MED values for the entire basic set, as well as for the test set. Furthermore, the use of the nonhierarchic approach in the new model yields slightly better results. Extrapolation Features. Figures 2-5 show the extrapolation capability of the developed model, in terms of the Cpl prediction at 298.15 K for n-alkanes C4-C30, 1-alkanols C2-C19, 1-iodoalkanes C2-C7, and methyl esters of n-alkanoic acids C3C16. Note that our datasets contained the following data: for n-alkanes, C5-C20 (the basic dataset) and C4-C30 (the test dataset); for 1-alkanols, C2-C14 (the basic dataset) and C11C19 (the test dataset); for 1-iodoalkanes, C2-C3 (the basic dataset) and C4-C7 (the test dataset); and for methyl esters of
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2081
Figure 6. Comparison of the results for liquid heat capacity estimation at 298.15 K. Legend: New model-NH, this work, nonhierarchic approach; New model-H, this work, hierarchic approach; ZR, the Za´bransky´ and Ru˚zˇicˇka method; and CH, the Chickos et al. method. Figure 3. Comparison of the experimental, estimated, and predicted heat capacities at 298.15 K for 1-alkanols.
Figure 7. Comparison of relative errors of liquid heat capacity estimation at 298.15 K. Legend: New model-NH, this work, nonhierarchic approach; New model-H, this work, hierarchic approach; ZR, the Za´bransky´ and Ru˚zˇicˇka method; and CH, the Chickos et al. method.
Figure 4. Comparison of experimental, estimated, and predicted heat capacities at 298.15 K for 1-iodoalkanes.
Figure 5. Comparison of experimental, estimated, and predicted heat capacities at 298.15 K for methyl esters of n-alkanoic acids (MEAA).
n-alkanoic acids, C4 (the basic data set) and C3-C16 (the test dataset). It is shown that the model predicts this property with reasonable accuracy along the carbon number scale. Experimental data and the estimated and predicted values for some
members of the homologous series are also plotted. In these figures, as well as throughout the entire paper, the term “estimated” is used when the heat capacity estimation is performed for a compound that has a carbon number within the range of compounds from the basic dataset, whereas the term “predicted” is used when the property estimation is performed beyond the carbon number range of the basic dataset compounds. Thus, the predicted values characterize the extrapolation capability of the developed model. Analysis of Results of Regression: Statistical Approach. For an exact analysis of all obtained results, we applied some statistical tools that have been described earlier in detail.35 First, we tested that obtained data comes from a normal distribution. After rejecting this hypothesis, we continued with the comparison by nonparametric methods of mathematical statistics. We compared the experimental data and the estimated values obtained for Cpl at 298.15 K using the following methods: our new model, the nonhierarchic approach (new model-NH); our new model, the hierarchic approach (new model-H); the method by Za´bransky´ and Ru˚zˇicˇka29 (ZR); and the method by Chickos24,28 (CH). All of the procedures and tests used have been already explained.35 The results are presented in so-called “box-andwhiskers” plots and are shown in Figures 6 and 7. An explanation of the box-and-whiskers plot is given in Figure 8. This diagram creates a plot for each dataset (sample), which is divided into four equal areas of frequency (quartiles). The plot shows the most extreme values in the data (maximum and minimum values), the lower and upper quartiles, and the median. A box encloses the middle 50% values. The median of values is drawn as a vertical line inside the box. Horizontal lines, known as whiskers, extend from each end of the box. The left (or lower)
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Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008
Figure 8. Explanation of the box-and-whisker plot.
whisker is drawn from the lower quartile to the smallest point within 1.5 interquartile ranges from the lower quartile. The other whisker is drawn from the upper quartile to the largest point within 1.5 interquartile ranges from the upper quartile. The rectangular portion of the box extends from the lower quartile to the upper quartile, covering the center half of each sample. The centerlines within each box show the location of the sample medians. The cross symbols (+) in the box near the median represent the location of the sample means. The whiskers extend from the box to the minimum and maximum values in each sample, except for any outside or far-outside points (distant and extreme values in Figure 8), which are plotted separately. Outside points (distant value) lie more than 1.5 times the interquartile range above or below the box; they are shown as small squares. Far-outside points (extreme value) lie more than 3.0 times the interquartile range above or below the box; they are shown as small squares with crosses through them. For comparison of the models, this figure shows the following relation: the more similar the box-and-whisker plots of compared sets, the better the agreement between compared data. Especially, the median and mean values, and the lower and upper quartile, should be located at the same position. The statistical tests for all applied models confirmed whether the models are suitable or unsuitable for estimation of the properties being examined, because there is (or is not) a significant difference between the medians obtained by the individual models. The plots in Figure 6 clearly show how the individual models are able to describe the experimental data. It is evident that both of our new models, as well as the model by Za´bransky´ and Ru˚zˇicˇka, show good agreement with experimental data. Plots in Figure 7 show the results for a comparison of RE values obtained by the individual methods. The median and means of the relative errors obtained by our new model are smaller than those obtained by other models. The new models that have been presented in this paper provide the lowest estimation error. The method reported by Za´bransky´ and Ru˚zˇicˇka yields estimation errors that are similar to the errors obtained by our models. The use of the Chickos method results in significantly diverse relative errors, in comparison with other methods. Conclusions A new group-contribution method for estimating the liquid heat capacity as a function of temperature over the range from the melting temperature up to the normal boiling temperature has been developed. The method is based on a novel threelevel group-contribution model that contains a larger set of group contributions, compared to a classical Benson-type method, and, therefore, is capable of being applied for prediction of the heat capacity of more-complex molecular structures, including polyfunctional and multiring compounds. Group-contribution pa-
rameters were calculated via two approaches: a hierarchical one that has been used exclusively so far and entails evaluation of parameters in three consecutive steps, and a nonhierarchical one in which parameters were evaluated in a single step. The new method was compared with two other groupcontribution methods, for predictions both over a temperature range and at a single temperature (298.15 K). The new method was determined to provide liquid heat capacity values that were in good agreement with experimental data and, therefore, should be suitable for applications in engineering calculations. The nonhierarchical approach was proven to be superior to the hierarchical approach, in terms of the relative deviation from experimental data. The group-contribution parameters that have been developed by the hierarchical approach will be applied in existing computer-aided systems for the design, simulation, and analysis of chemical processes. Appendix A: Error Estimation Methods Table A1 gives a detailed summary of the estimation methods for a variety of families of compounds. NC denotes the number of compounds in the individual family, ARE is the average relative error, and MED is the median of relative errors. Appendix B: Example Application of the New Model Table B1 gives an example of using our model to estimate the heat capacity of liquid 1,3,5-trimethylbenzene and its comparison with experimental data and with values obtained by other methods. Appendix C: New Model, First Version In our first attempt, we adopted the methodology developed by Constantinou and Gani32 and by Marrero and Gani.31 We extended their approach for predicting the heat capacity of liquids and modified the method to estimate heat capacity at some selected temperatures (298.15 K and several reduced temperatures (Tr ) 0.3, 0.4, 0.5, 0.6, where Tr ) T/Tc and Tc is the critical temperature). The results are presented in Table C1. The model for predicting the heat capacity at an individual temperature was given as follows: n
l Cpl ) Cp0 +
m
0
NiCi + ω ∑ MjDj + z ∑ OkEk ∑ i)1 j)1 k)1
(C1)
where Cpl represents the liquid heat capacity at an individual temperature (298.15 K or a specified reduced temperature l (Tr ) 0.3, 0.4, 0.5, or 0.6,), Cp0 is the adjustable parameter, Ci is the first-level group contribution of type i, Dj is the secondlevel group contribution of type j, and Ek is the third-level group contributions of the type k. Ni, Mj, and Ok denote the number of occurrences of individual group contributions. Parameters ω and z are assigned values of unity and zero, depending on the usage of the individual level group contributions.31 Although the results obtained for heat capacity at individual temperatures were satisfactory, the analysis of the obtained values for individual groups revealed that group values did not follow the expected monotonic increase with temperature. Therefore, we developed a new version of the model defined by eqs 1 and 2.
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2083 Table A1. Overview of Average Relative Error (ARE) and Median of Relative Errors (MED) Values of the Individual Estimation Methods for Various Families of Organic Compounds for the Estimation of Cpl at 298.15 K and over a Temperature Rangea New Model-NH family of organic compounds
NC
ARE (%)
New Model-H
MED (%)
NC
ARE (%)
MED (%)
Za´bransky´ and Ru˚zˇicˇka, ZR NC
ARE (%)
MED (%)
Chickos et al., CH NC
ARE (%)
MED (%)
Hydrocarbons aliphatic saturated hydrocarbons at 298.15 K over a temperature range aliphatic unsaturated hydrocarbons at 298.15 K over a temperature range cyclic saturated hydrocarbons at 298.15 K over a temperature range cyclic unsaturated and aromatic hydrocarbons at 298.15 K over a temperature range total at 298.15 K over a temperature range
21 24
1.5 1.5
1.3 1.5
21 24
2.0 2.0
1.3 2.0
21 24
1.7 1.0
1.3 1.0
21
1.9
1.9
26 25
1.5 0.9
1.3 1.0
26 25
1.7 1.0
1.3 1.0
26 25
2.0 0.5
0.6 0.5
26
4.1
3.2
35 43
1.5 1.0
0.8 0.8
35 35
1.4 1.1
0.6 1.0
35 43
1.8 1.0
0.9 0.8
35
4.2
3.1
41 65
1.1 0.6
0.7 0.5
41 65
1.2 0.9
0.8 0.7
41 65
1.3 0.8
0.9 0.6
41
4.0
3.0
123 157
1.4 0.9
1.0 0.7
123 157
1.5 1.1
1.1 1.0
123 157
1.7 0.8
0.9 0.7
123
3.7
2.7
Oxygen-Containing Compounds ethers at 298.15 K over a temperature range aldehydes, ketones at 298.15 K over a temperature range alcohols, phenols at 298.15 K over a temperature range acids at 298.15 K over a temperature range esters at 298.15 K over a temperature range heterocyclic O-compounds at 298.15 K over a temperature range miscellaneous O-compounds at 298.15 K over a temperature range total at 298.15 K over a temperature range
16 22
1.1 0.9
0.8 1.0
16 22
1.1 1.0
0.8 0.8
16 22
2.5 1.4
1.2 1.2
14
2.3
2.2
20 30
1.3 1.3
0.9 0.9
20 30
1.2 1.4
1.0 1.0
20 30
2.1 1.6
1.1 1.0
20
2.7
2.6
27 41
3.5 3.7
3.0 3.3
27 41
3.3 4.4
2.4 4.4
27 41
4.0 2.8
3.9 2.5
27
6.3
3.7
9 21
1.3 1.2
1.1 1.0
9 21
1.2 1.6
0.7 1.1
9 21
1.5 1.6
0.6 1.3
9
4.0
4.7
27 48
1.5 1.0
1.2 0.8
27 48
1.5 1.0
1.2 0.9
27 48
2.8 1.5
2.0 1.4
25
3.8
3.3
8 9
2.6 1.6
2.3 1.0
8 8
2.7 1.7
2.3 1.2
8 9
2.0 1.5
1.6 1.6
8
6.0
4.7
6 11
1.3 1.2
1.3 1.0
6 6
1.8 1.3
1.8 1.3
6 11
2.8 1.8
2.4 1.9
6
4.8
4.3
113 182
1.9 1.7
1.3 1.1
113 113
1.9 2.0
1.3 1.2
113 182
2.8 1.8
1.9 1.4
109
4.3
3.2
Halogenated Compounds chlorinated hydrocarbons at 298.15 K over a temperature range brominated hydrocarbons at 298.15 K over a temperature range iodinated hydrocarbons at 298.15 K over a temperature range fluorinated hydrocarbons at 298.15 K over a temperature range miscellaneous halogenated compounds at 298.15 K over a temperature range total at 298.15 K over a temperature range amines at 298.15 K over a temperature range nitriles at 298.15 K over a temperature range heterocyclic N-compounds at 298.15 K over a temperature range miscellaneous N-compounds at 298.15 K over a temperature range
20 27
0.8 0.7
0.5 0.7
20 27
1.8 1.1
1.3 0.9
20 27
2.1 1.1
1.7 1.0
20
4.0
4.0
7 7
1.3 1.6
0.7 1.2
7 7
1.6 2.0
0.8 1.4
7 7
1.1 1.1
0.6 0.7
7
2.9
3.1
4 4
0.7 0.6
0.7 0.8
4 4
0.5 0.7
0.5 0.9
4 4
0.5 0.5
0.3 0.7
4
2.0
1.8
14 21
0.5 0.8
0.4 0.7
14 21
0.9 1.1
0.7 1.0
14 21
1.4 1.4
0.8 1.1
14
3.2
2.7
12 19
1.4 1.0
1.5 1.1
12 19
1.3 1.2
1.1 1.2
12 19
1.6 1.1
1.2 1.2
12
4.5
4.8
57 78
0.9 0.9
0.6 57 1.4 0.8 0.7 78 1.2 1.1 Nitrogen-Containing Compounds
57 78
1.6 1.2
1.0 1.0
57
3.6
3.0
20 23
1.8 1.6
0.7 1.1
20 23
2.0 1.9
1.6 1.7
20 23
1.9 1.2
1.5 1.1
20
4.1
2.8
7 8
1.0 0.8
0.2 0.5
7 8
1.1 0.9
0.2 0.7
7 8
2.3 0.6
0.3 0.2
6
4.8
4.3
20 20
0.6 0.6
0.1 0.4
20 20
0.7 0.7
0.1 0.5
20 20
0.9 1.0
0.8 0.8
20
2.3
2.1
5 6
3.0 2.6
2.1 1.7
5 6
3.2 3.5
2.1 3.0
5 6
1.6 0.7
0.5 0.5
5
5.8
7.7
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Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008
Table A1. (continued) New Model-NH family of organic compounds
New Model-H
Za´bransky´ and Ru˚zˇicˇka, ZR
NC
ARE (%)
MED (%)
NC
ARE (%)
MED (%)
NC
ARE (%)
MED (%)
52 57
1.4 1.3
0.6 0.8
52 57
1.5 1.5
0.8 1.0
52 57
1.6 1.0
total at 298.15 K over a temperature range
Chickos et al., CH NC
ARE (%)
MED (%)
1.0 0.9
51
3.6
2.9
Sulfur-Containing Compounds sulfides at 298.15 K over a temperature range thiols at 298.15 K over a temperature range heterocyclic S-compounds at 298.15 K over a temperature range total at 298.15 K over a temperature range
17 20
0.9 0.8
0.7 0.6
17 20
0.9 0.9
0.8 0.6
17 20
0.8 0.7
0.6 0.5
17
3.9
3.4
17 17
1.1 1.0
1.0 1.0
17 17
0.9 0.9
0.6 0.7
17 17
0.8 0.8
0.6 0.8
17
3.0
2.8
6 9
1.3 1.2
0.8 1.1
6 9
1.6 1.1
1.5 0.4
6 9
0.5 0.6
0.4 0.4
6
2.9
2.4
40 46
1.0 0.9
0.8 0.8
40 46
1.0 0.9
0.7 0.6
40 46
0.8 0.7
0.5 0.6
40
3.4
3.1
2.3 1.8
5
9.7
9.2
1.6 1.6
13
4.7
5.6
2.7 1.5
1
15.2
15.2
1.1 0.8
399
3.9
3.0
Miscellaneous Compounds with a Mixture of Oxygen, Nitrogen, or Halogen compounds with halogen and oxygen at 298.15 K 5 1.9 1.6 5 1.5 0.9 5 1.7 over a temperature range 6 1.4 1.5 6 1.7 1.7 6 1.6 compounds with nitrogen and oxygen at 298.15 K 13 1.6 1.4 13 2.5 2.3 13 2.2 over a temperature range 22 1.5 1.4 22 2.0 1.6 22 1.9 compounds with halogen, oxygen, and nitrogen at 298.15 K 1 4.0 4.0 1 3.9 2.9 1 2.7 over a temperature range 1 2.2 2.2 1 2.1 2.1 1 1.5 overall total for T ) 298.15 K overall total for a temperature range
404 549
1.5 1.2
1.0 0.8
404 549
1.6 1.5
1.1 1.1
404 549
1.9 1.2
a For each family of compounds the first line gives values of the number of compounds (NC), the average relative error (ARE), and the median of relative errors (MED) of the individual estimation methods for estimation at 298.15 K and the second line gives the same values estimated over a temperature range.
Nomenclature
Table B1. Example of Using the New Model for 1,3,5-Trimethylbenzene
group
number of occurrences
additional adjustable parameter cp°(T) aCH aC-CH3 AROMRINGs1s3s5
1 3 3 1
a
b
d
105.94 -51.40 7.24 -1.28 8.17 -0.43 20.23 9.63 -0.43 -60.51 29.24 -3.03
Clp (298.15 K)exp ) 208.37 J K-1 mol-1 (value taken from Za´bransky´ et al.1,2) l Cp (298.15 K)New_model_NH ) 209.8 J K-1 mol-1 l l l RE ) 0.7%; RE ) (|Cp,exp - Cp,est |)/(Cp,exp ) × 100 l -1 -1 Cp(298.15 K)ZR ) 211.3 J K mol RE ) 1.4% Clp(298.15 K)CH ) 216.0 J K-1 mol-1 RE ) 3.7% Table C1. Results for Estimation of Heat Capacity (eq C1) at Selected Temperatures
Clp
number of compounds, NC
number of group contributions, NGR
Clp(298.15 K) Clp(Tr ) 0.3) Clp(Tr ) 0.4) Clp(Tr ) 0.5) Clp(Tr ) 0.6)
678 66 283 384 269
122 + 1c + 65d 21 + 1c + 10d 87 + 1c + 37d 89 + 1c + 48d + 14e 83 + 1c + 38d + 13e
average average absolute relative error, AAEa error, (J K-1 mol-1) AREb (%) 4.2 2.2 6.2 6.8 6.3
1.7 1.5 2.7 2.6 2.5
l a AAE ) (1/n)∑n |C l b c i)1 p,exp - Cp,est|i. See eq 3. Number of group contributions of the first level plus 1 adjustable additional parameter. d Number of group contributions of the second level. e Number of group contributions of the third level.
λ ) regularization parameter AAE ) average absolute error (J K-1 mol-1) ARE ) average relative error (%) Cpl ) isobaric heat capacity of pure liquid (J K-1 mol-1) Cpl (298.15 K) ) heat capacity at 298.15 K (J K-1 mol-1) Cpl (Tr ) X) ) heat capacity at reduced temperature Tr ) X, where X ) 0.3, 0.4, 0.5, 0.6 (J K-1 mol-1) m ) total number of experimental data points MED ) median of relative errors (%) n ) total number of group contributions n1, n2, n3 ) number of the first-level, second-level, and thirdlevel group contributions NC ) number of compounds NGR ) number of group contributions SSQ ) the sum of squares function SSQ* ) decomposed form of SSQ T ) temperature (K) Tc ) critical temperature (K) Tr ) reduced temperature; Tr ) T/Tc 1m ) unit vector of size m diag ) function for creating a diagonal matrix In ) identity matrix of size n p, q, s ) contribution vectors of the first-level, the secondlevel, and the third-level group contributions P, Q, S ) structural matrices of the first-level, the second-level, and the third-level group contributions t*, p*, q*, s* ) least-squares estimates t+, p+, q+, s+ ) regularized estimates t ) total contribution vector T ) total structural matrix Trel ) relative structural matrix u ) experimental data vector (individual Cpl values)
Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2085
urel ) relative experimental data vector (estimated Cpl values divided by the experimental ones) Subscripts i, j, k ) first-, second-, third-level group contribution of type i, j, k Acknowledgment This work was conducted within the IUPAC (Project No. 2004-010-3-100), the Institutional Research Plans (Nos. MSM 6046137307 and MSM 6046137306), and the Grant Agency of the Academy of Sciences of the Czech Republic (under Grant No. IAA 400720710). The authors acknowledge the use the ICAS (Version 9.0) software provided by R. Gani, especially its ProPred tool. Literature Cited (1) Za´bransky´, M.; Ru˚zˇicˇka, V.; Majer, V.; Domalski, E. S. Heat Capacity of Liquids. Critical ReView and Recommended Values; Journal of Physical Chemistry Reference Data Monograph No. 6; American Chemical Society: Washington, DC, 1996. (2) Za´bransky´, M.; Ru˚zˇicˇka, V.; Domalski, E. S. Heat Capacity of Liquids: Critical Review and Recommended Values. Supplement I. J. Phys. Chem. Ref. Data 2001, 30, 1199-1689. (3) Rowlinson, J. S. Liquids and Liquid Mixtures; Butterworths: London, 1969. (4) Yuan, T.-F.; Stiel, L. I. Heat Capacity of Saturated Nonpolar and Polar Liquids. Ind. Eng. Chem. Fundam. 1970, 9, 393-400. (5) Lyman, T. J.; Danner, R. P. Correlation of Liquid Heat Capacities with a Four-Parameter Corresponding States Method. AIChE J. 1976, 22, 759-765. (6) Tarakad, R. R.; Danner, R. P. An Improved Corresponding States Method for Liquid Heat Capacities. AIChE J. 1977, 23, 944-946. (7) Coniglio, L.; Rauzy, E.; Berro, C. Representation and prediction of thermophysical properties of heavy hydrocarbons. Fluid Phase Equilib. 1993, 87, 53-88. (8) Prasad, T. E. V.; Rajiah, A.; Prasad, D. H. L. On the Dependence of Liquid Heat-Capacity on Temperature and Molecular-Structure. Chem. Eng. J. Biochem. Eng. J. 1993, 52, 31-35. (9) Conesa, A.; Shen, S.; Coronas, A. Liquid densities, kinematic viscosities, and heat capacities of some ethylene glycol dimethyl ethers at temperatures from 283.15 to 423.15 K. Int. J. Thermophys. 1998, 19, 13431358. (10) Coniglio, L.; Trassy, L.; Rauzy, E. Estimation of Thermophysical Properties of Heavy Hydrocarbons through a Group Contribution Based Equation of State. Ind. Eng. Chem. Res. 2000, 39, 5037-5048. (11) Morad, N. A.; Kamal, A. A. M.; Panau, F.; Yew, T. W. Liquid specific heat capacity estimation for fatty acids, triacylglycerols, and vegetable oils based on their fatty acid composition. J. Am. Oil Chem. Soc. 2000, 77, 1001-1005. (12) Coniglio, L.; Rauzy, E.; Peneloux, A.; Neau, E. Use of heat capacities for the estimation of cubic equation-of-state parameterss application to the prediction of very low vapor pressures of heavy hydrocarbons. Fluid Phase Equilib. 2002, 200, 375-398. (13) Diedrichs, A.; Rarey, J.; Gmehling, J. Prediction of liquid heat capacities by the group contribution equation of state VTPR. Fluid Phase Equilib. 2006, 248, 56-69. (14) Luria, M.; Benson, S. W. Heat Capacities of Liquid Hydrocarbons. Estimation of Heat Capacities at Constant Pressure as a Temperature Function, Using Additivity Rules. J. Chem. Eng. Data 1977, 22, 90-100. (15) Vasil’ev, I. A.; Treibsho, E. I.; Korkhov, A. D.; Petrov, V. M.; Orlova, N. G.; Balakina, M. M. Inzh.-Fiz. Zh. 1980, 39, 1054-1060. (16) Ogiwara, K.; Arai, Y.; Saito, S. Correlation of the constant-volume heat capacity of liquid organic compounds by the group contribution method. J. Chem. Eng. Jpn. 1981, 14, 156-158. (17) Guthrie, J. P.; Taylor, K. F. Additivity methods for estimating heat capacities of vaporization for organic compounds. Can. J. Chem. 1984, 62, 363-372. (18) Dvorkin, P. L.; Ryzhova, G. L.; Lebedev, Y. A. Parameters for Calculation of the Heat-Capacity of Liquid Organic-Compounds at Different Temperatures. Bull. Acad. Sci. USSR, DiV. Chem. Sci. 1984, 33, 982-987.
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ReceiVed for reView September 11, 2007 ReVised manuscript receiVed December 14, 2007 Accepted December 17, 2007 IE071228Z
