Ind. Eng. Chem. Res. 1987, 26, 1362-1365
1362 c5 = variable defined in (10) c6 = variable defined in (16) c = coolant heat capacity = reactant stream heat capacity
8 $ = molar flow rate of key species A
k = reaction rate constant K = adsorption rate constant L = reactor length m = mass flow rate of coolant M = mass flow rate of reactant rA = reaction rate per unit volume of reactor R = reactor radius T = temperature of reactant gas t = temperature of coolant U = overall heat-transfer coefficient x = conversion of key species A z = dimensionless length 2 = axial coordinate of reactor Greek S y m b o l
-AH = heat of reaction
Subscripts 0 = at the reactor inlet z = 0 1 = at the reactor exit z = 1
A = key reactant species max = maximum value Literature Cited Adomaitis, R. A.; Cinar, A. The Bifurcation Behavior of a n Autothermal Packed Bed Tubular Reactor; Illinois Institute of Technology: Chicago, 1986, submitted for publication. Akella, L. M.; Lee, H. H. AIChE J. 1983, 29(1), 87. Barkelew, C. H. Chem. Eng. Sci. 1959,25, 37-46. Bilous, 0.;Amundson, N. R. AIChE J. 1956, 2(1), 117. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. Gray, P.; Lee, P. R. Combust. Flame 1965, 9, 201-203. Henning, T.; Perez, T. Chem. Eng. Sci. 1986, 41, 83-88. Hlavacek, V.; Marek, M.; John, T. M. Collect. Czech. Chem. Commun. 1969, 34, 3568-3880. Morbidelli, M.; Varma, A. Chem. Eng. Sci. 1985, 40, 2165. Oroskar, A.; Stern, S. A. AIChE J . 1979, 25(5), 903. Thomas, P. H. Proc. R. SOC.London, Ser. A 1961, A262,1922-1206. van Welsenaere, R. J.; Froment, G. F. Chem. Eng. Sci. 1970,25,1503. Received for review March 13, 1986 Revised manuscript received January 16, 1987 Accepted March 4, 1987
Superscripts * = at the hot spot location of reactor
A Group Contribution Method for Liquid Thermal Conductivity Manoj Nagvekar and Thomas E. D a u b e r t * Department of Chemical Engineering, The Pennsylvania S t a t e University, University Park, Pennsylvania 16802
An estimation method, based on the principle of second-order group additivity, has been developed for predicting the thermal conductivity of organic liquids. By use of experimental data, group contributions have been determined for two constants in the modified Reidel’s temperature correlation for liquid thermal conductivity. The proposed method is completely generalized and fairly easy to use as the only input parameter required; besides, the molecular structure is the critical temperature. In terms of accuracy, it compares favorably with the existing prediction methods. The average deviation for 226 compounds, for which experimental data are available, is 5.9%. Thermal conductivity is a very important transport property, especially for liquids, since it is required for most heat-transfer calculations, including forced convection. Unfortunately, it is an extremely difficult property to measure, primarily due to the development of convective currents and heat losses during the course of the experiments. Under these circumstances, it is necessary to turn to an empirical method to estimate the liquid thermal conductivity. In this paper, we propose a prediction method which is based on the group contribution approach. Group Contribution Methods When the contribution of an “element” of a molecule (atom, bond, group, etc.) to a particular property is constant regardless of the nature of the rest of the molecule, the property is termed an additive one. Many molecular properties have been shown to obey additivity rules, to varying degrees of accuracy. Knowing the structure of the molecule, an additive property may be easily and quickly estimated by summing up the contributions corresponding to the “elements” of the molecules. Benson and Buss (1958) have presented a very good discussion of additivity rules and classified them in a hi-
erarchical manner. The primary assumption in constructing additivity rules is that the influence of neighboring elements decreases rapidly with distance in a molecule. This assumption determines the definition of what constitutes a “group”, since the distance at which groups cease to interact must be assumed before the groups can be specified. In this study, a second-order contribution scheme, which is based on nearest-neighbor interactions, was used. In the second-order scheme, a group is defined as a polyvalent atom together with its ligands, a t least one of which must also be polyvalent. Thus, this scheme is restricted to compounds having at least two polyvalent atoms. The nomenclature followed in this study is to first identify the polyvalent atom and then its ligands. Thus, C-(C)(H), represents a C atom attached to another C atom and three H atoms, i.e., a primary methyl group. Cd represents a double bonded carbon atom, C , a triple bonded carbon atom, and Cb a carbon atom in the benzene ring. For multiple-bonded carbon atoms, the other multiplebonded carbon atom is implied since these atoms must always occur in pairs. This scheme of classification for group additivity can only distinguish among isomers when
0888-5885/87/2626-1362$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1363
AA,W/(m K) 0.1542 X -0.2599 X 0.2470 X 0.6935 X 0.3082 X -0.1569 X 0.2470 X 0.2751 X -0.1475 X 0.2020 x
-0.1191 x 10-1 lo-' -0.7994 x 10-1 lo-' -0.5950 X
lo-'
lo-'
lo-' lo-'
lo-' lo-' 10-1
0.5190 X lo-' 0.3544 X lo-' 0.4205 X lo-'
0.3227 x lo-' loT2 -0.7712 X lo-' -0.1064 0.3874 X 0.2465 X lo-' -0.3677 X lo-' -0.1034 -0.6957 X -0.2081 0.1447 X lo-' -0.3710 0.3906 0.2838 0.1138
-0.1818
X X X
X
10"
0.2156 X lo-' 0.1696 X lo-' 0.1802 X lo-' 0.8114 X -0.1317 X lo-'
C-(Cl)(F)z(C) C-(Cl)z(F)(C) perfluorocycloalkane ring correction
-0.2172
-0.8871 X 0.3529 X lo-' 0.9558 X lo4 0.2097 X 0.1209
-0.1032 0.7060 X -0.1435 X 0.8535 X -0.1752 -0.3575 x -0.1064 -0.1223 0.5130 X 0.5895 X
0.4455 X 0.1773 X 0.2971 X 0.4010 X 0.2734 X 0.2267 -0.4417 X 0.1968 X
10" lo-' lo-'
lo-' lo-'
lo-' lo-' lo-' 10-2
lo-' lo-'
0.1927
the isomerism arises from differences in the groups present in the molecule as in the case of n-butane and isobutane. It cannot distinguish between isomers which differ only in the geometrical relationship between identical groups, e.g., 2-methylhexane and 3-methylhexane. In the case of the cis-trans and the ortho-meta-para isomers, a correction is introduced to account for steric effects.
Development of the Method The approach chosen here was to use an equation form to represent thermal conductivity as a function of temperature and then determine group contributions for the constants involved in the temperature-dependent form. The thermal conductivity of most liquids, with the exception of water and the polyols, decreases with temperature. Below or near the normal boiling point, the decrease
X
0.3774 X lo-' -0.2919 X lo-' -0.1531 -0.5950 X 0.8430 X lo-' 0.2748 -0.3900 X 0.7520 -0.6290 -0.3780
X X X
lo-'
lo-'
lo-'
0.7959 X 0.1562 -0.1054 X lo-' 0.1429 0.1592 X lo-' 0.2127
lo-'
0.9477 X 0.1162 0.6516 X -0.1484 X lo-' 0.5263 X lo-' -0.2 200 -0.8033 X lo-' 0.3689 X lo-' -0.6457 X lo-' -0.1844 X lo-'
0.2585 X lo-' -0.9010 x 10-1 0.3340 X lo-' 0.6335 X lo-' -0.4000 X 0.6904 X 0.6042 X 0.3618 X 0.7245 X 0.7245 X 0.3683 X
0.8119 X
amine ring correction
0.2165 X lo-'
0.1774 X lo-' 0.1498 X lo-' 0.1355 X lo-' 0.1389 X lo-' 0.1389 X lo-' 0.4504 X
0.2579 X lo-' 0.8392 X 0.5310 X lo-' 0.9368 X 0.5661 X lo-'
N-(Co)(C)
-0.1520 X lo-' 0.6609 X lo-' 0.1395 X lo-' 0.7459 X lo-' -0.3040 X lo-'
co-iNjicj N-(Co)(H)z
AB, W/(m K)
o-(cO)z O-(CO)(Cd) O-(H)(Cd o-(c)(cb) cyclohexanol ring correction 1,2-propylene oxide ring correction 1,4-dioxane ring correction
0.6317 X 0.7611 X 0.2206 X 0.2501 X -0.8033 X -0.6457 X 0.2303 0.2465 X 0.3685 X 0.5005 X -0.1094 X -0.1168 X 0.5053 X -0.4417 X 0.3300 X -0.2030 X -0.3495 X 0.2577 X 0.5632 X
0.4468 X lo-' -0.5349 X lo-' -0.2054 -0.1487 -0.3521 -0.4529 0.3035 X lo-' -0.2152 X lo-'
lo-'
lo-' lo-' lo-'
lo-' lo-'
lo-'
0.1070 0.2207 X 0.1112 -0.3871 X 0.3689 X -0.1844 X -0.2768 0.1209 X
lo-' lo-' lo-' lo-'
lo-'
0.4715 X lo-' 0.9075 X lo-' 0.2967 X lo-' 0.6865 X lo-' -0.1236 0.3035 X lo-' -0.1484 X lo-' -0.4157 X lo-' 0.2551 lo-' 0.2871 lo-' 0.2444
lo-' lo-'
lo-' lo-'
lo-' lo-' lo-' lo-'
lo-' lo-' lo-'
is nearly linear. Over wider temperature ranges, the following correlation due to Reidel (1951a-c) has been commonly used:
1
where k = liquid thermal conductivity, T, = reduced temperature, T/T,,and T,= critical temperature. In this work, a modified version of Reidel's form k = A + B(l - T,)2/3 (2) was used. The constants A and B were determined for each compound in an extensive data base containing virtually all of the available liquid thermal conductivity data. A summary of the data base including the data references is available (Nagvekar et al., 1984).
1364 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Table I1 EI'OUD
C-(C)(W, C-(C)dH) C-(H),(O)(C)
O-(CO)(C) CO-(C)(O) C-(H),(CO) total
no. 2 1 1 1 1 1
AA,W/(m K) 2(0.8730 X 0.9418 X lo-' 0.1773 X lo-' -0.1094 X lo-' 0.7611 X 0.9477 X lo-' A = 0.2819 X lo-'
AB, W/(m K) 2(0.1113) -0.1483 -0.5349 x 10-1 0.2967 X 0.2207 X 10-1 0.1162 E = 0.1620
The next step in the development of the method involved the determination of the contribution of an individual group to the constants A and B in eq 2. The definition of what constitutes the best value of a group contribution is not very rigid. If the property being investigated is in perfect agreement with the principle of group additivity, the group contribution value could be determined from the difference in property values of two compounds which differed only by the unknown group. Ideally the contributions for each group could be determined by performing a simultaneous regression over the entire data base. This was not feasible, however, since more than 200 compounds are included in the data base involving about 80 different groups. Furthermore, the use of a limited data base provides a better test of the predictive ability of the method. The determination of the groups was carried out using the Levenberg-Marquardt function minimization technique (More et al., 1980). The method was applied to the various families of compounds, beginning with the n-alkanes which require only two groups to construct any straight-chain saturated hydrocarbon and two additional groups to represent the branched chain compounds. Seven groups are needed to represent all the noncyclic monoolefins. Due to lack of data, however, only five groups could be determined. For the same reason, no contributions could be determined for the groups in the allenes and conjugated dienes and alkynes. By use of the group contributions determined for the hydrocarbons, the procedure was extended to non-hydrocarbons. Each family of compounds usually involved the determination of about four or five new groups. A t each stage, the contributions for the groups determined before were kept constant. A few of the group contributions were unique determinations for cases where only one compound could be located to rep-
resent a group. The values of the group contributions are listed in Table I. The group contribution values have the units of W/(m K). Use of t h e Method The proposed method may be used to find the liquid thermal conductivity of any compound for which the necessary groups are available, within the reduced temperature range, 0.3 < TI < 0.8. As an example, consider isobutyl acetate for which the molecular structure is CH3
I
OBaroncini et a1.k (1981) method.
Missenard's (1965) method.
II
CHCH~OCCH~
I CH3
The various groups and their contributions to the constants A and B are given in Table 11. Thus, for isobutyl acetate,
k = 0.2819 X 10-1 + 0.1620(1 - T1)2/3 W/(m K) Additional examples and comparisons with experimental data are given by Nagvekar (1984).
Comparison with Existing Methods An extensive evaluation of the existing methods indicated that the Baroncini et al. (1981) method is the most accurate, except for compounds containing nitrogen for which it has not been developed. The Missenard (1965) method was found to have the least errors for the nitrogen-containing compounds. Overall results comparing the proposed method with the existing methods are shown in Table 111. Average and bias percent errors are shown for individual families of compounds. The comparison is not biased in favor of the proposed method, as most of the compounds have been included in the development of the Baroncini et al. (1981) method. The differences in the number of compounds evaluated are due to the unavailability of input parameters. All input parameters required for the evaluations were obtained from experimental sources, and no attempt was made to predict these parameters to avoid any additional source of error. The comparisonsindicate that except for the ethers and the esters, the proposed method has lower average errors
Table 111. Evaluation Results and Comparison with Existing Methods proposed group contribution method no. of 9'0 errors compds no. of points evaluated evaluated av bias hydrocarbons 54 374 3.7 -0.2 287 6.5 -0.7 20 acids -0.9 634 6.3 23 alcohols -4.8 43 8.3 9 aldehydes 30 1.2 3 0.0 amides 81 4.2 -2.1 11 amines 0.0 1 4 0.1 anhydrides 45 2.1 -0.3 bromides 8 -2.2 22 82 6.6 chlorides 1.2 243 9.7 29 esters 75 7.3 1.1 10 ethers 11 3.3 -1.7 fluorides 6 i 24 3.8 0.6 iodides -5.0 9 68 6.3 ketones 11 1.2 0.0 3 nitriles 0.6 4 37 1.6 nitro compds -2.7 4 27 3.7 phenols 3 14 3.5 2.4 other halides -0.7 overall 226 2089 5.9 m
0
existing methods no. of compds evaluated 31" 14a 19" 56 l b
2b 1b
9" 210 22" 7" 40 8" 9Q 2b I*
40 6' 166
no. of points evaluated 236 208 592 44 26 49 4 46 176 197 60 7 25 72 6 32 27
% errors
av 4.1 7.3 7.7 6.4 4.9 5.3 5.0 4.2 7.4 5.7 3.2 11.0
7.4 8.3 3.1 1.6 5.0
118
5.7
1925
6.3
bias 1.0 -3.4 -6.5 3.3 -4.9 0.3 -5.0 -2.6 -6.3 1.5 -0.1 -11.0 1.8
-1.7 1.6 -0.9 4.0 0.1 -2.7
I n d . E n g . Chem. R e s . 1987, 26, 1365-1372
than the presently available prediction methods and is applicable to a significantly larger number of compounds. It also offers the advantage of a single method applicable to all families of compounds. The proposed method cannot be used, however, for compounds whose molecules contain a single polyvalent atom. This should not be considered as a drawback of the method, as such compounds are usually common compounds for which extensive data compilations are available. Within a particular family or homologous series, the errors for the lowest members are usually higher than the average errors for the rest of the members. For these compounds also there are usually ample data available, and it is not necessary to use any prediction method. Conclusions A second-order group contribution method has been proposed for the prediction of liquid thermal conductivity of organic compounds. The groups utilized are consistent with previously determined methods for ideal gas heat capacity, ideal gas entropy, heat of formation and combustion (Benson, 19761, second virial coefficient (McCann and Danner, 1984), and critical temperature and pressure (Jalowka and Daubert, 1986). The method compares favorably with the most accurate prediction method available (Baroncini et al., 1981), which is not applicable to all families of compounds. It is considerably more accurate than the best available method applicable to all families (Missenard, 1965). The application of the method is fairly simple, depending only upon the availability of group contribution values. The only other input parameter required is the critical temperature, which if unknown can be estimated to a fair degree of accuracy (Danner and Daubert, 1983). Acknowledgment
1365
Physical Property Data of the American Institute of Chemical Engineers. Nomenclature A , B = constants in eq 2, W/(m K) AA, AB = contribution of a group to the constants A and B, w/(m K) k = liquid thermal conductivity, W/(m K) T , = critical temperature T, = reduced temperature Literature Cited Baroncini, C.; Di Fillipo, P.; Latini, G.; Pacetti, M. Int. J . Thermophys. 1981,2,21. Benson, S.W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976. Benson, S. W.; Buss, J. H. J. Chem. Phys. 1958,29,546. Danner, R. P.; Daubert, T. E. Manual for Predicting Chemical Process Design Data: Data Prediction Manual; Design Institute for Physical Property Data, American Institute of Chemcial Engineers: New York, 1983; Procedures 2A-2C. Jalowka, J. W.: Daubert, T. E. Ind. Enp. - Chem. Process Des. Deu. 1986,25,139. McCann, D. W.; Danner, R. P. Ind. Eng. Chem. Process Des. Deu. 1984,23,529. Missenard, F.A. Conductivite Thermique des Solides, Liquides, gaz et de Leurs Melanges; Editions Eyrolles: Paris, 1965. Missenard, F. A. C. R. Acad. Sci., Ser. 3 1965,260,5521. More, J. J.; Garbow, B. S.; Hillstrom, K. E. User Guide for Y I N PACK-1; Argonne National Laboratory: Argonne, IL 1980. Nagvekar, M., “A Group Contribution Method for Liquid Thermal Conductivity”, M.S. Thesis, The Pennsylvania State University, University Park, 1984. Nagvekar, M.; Daubert, T. E.; Danner, R. P. Documentation of the Basis for Selection of the Contents of Chapter 9 Thermal Conductivity i n Manual for Predicting Chemical Process Design Data; Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York.’ 1984: Table 7. Reidel, L. Chem. Ing. Tech. 1951a,23,59. Reidel, L. Chem. Ing. Tech. 1951b,23,321. Reidel, L. Chem. Ing. Tech. 1951c,23,465.
During the course of this work, M. Nagvekar was supported by funds provided by the Design Institute for
Received f o r review March 25, 1986 Accepted April 14, 1987
Newtonian and Inelastic Non-Newtonian Flow across Tube Banks Om Prakash, S.N. Gupta,* and P. Mishrat D e p a r t m e n t of Mechanical Engineering, I n s t i t u t e o f Technology, Banaras H i n d u University, Varanasi 221 005, I n d i a
Newtonian and inelastic non-Newtonian fluid flows across tube banks are analyzed by using two parallel-plate channel models. A simple parallel-plate channel model correlates the present and previous literature data on non-Newtonian fluids successfully, giving a single-valued correlation independent of tube spacings and the fluid rheology. Separate correlations are developed for staggered and inline arrangements. A comparison of a second model, the periodically converging-diverging channel flow, with experimental data shows that the flow does not expand from a minimum clearance to the maximum possible available space between tubes due to interaction of flow between them. It is further observed that the flow behavior index, n,has a marked effect on expansion ratio, D,/D,; the higher the pseudo-plasticity of the fluid, the higher is the expansion. 1. Introduction Most of the earlier attempts to correlate the friction factor and Reynolds number of Newtonian fluids flowing across tube banks are based on the conventional model
* Author t o whom correspondence should be addressed. Department of Chemical Engineering, Institute of Technology, Banaras Hindu University.
employing an equivalent diameter and the maximum velocity at the minimum cross section. Chilton and Genereaux (1933) presented pressure drop vs. Reynolds number data for inline and staggered tube arrangements and correlated the transition and turbulent region data using minimum clearange between two adjacent tubes as the characteristic length. In the turbulent region, the exponent on Reynolds number was found to
oasa-~ss~/s~/~~~~-0 i ~1987 ~ ~American $ o 1 . ~Chemical o / o Society
