Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 297

Table I Chen Churchill Wood

e s ( 7 ) e q ( 6 ) eq (5 ) abs % dev from 0.1000% 0.5771% 6.0503%

max % dev + 0.4651% + 0.6041% + 36.6541% Colebrook equation

min % dev -0.2286% -2.6096% -5.7461%

can be used for calculations of fluid flow problems with Reynolds number greater than 4000 up to lo7. Despite the fact that this equation can be handled more conveniently in a computer than the values from the Moody chart, it is implicitly expressed in terms of f D (i.e., both the right- and left-hand terms in eq 4 contain f D ) so that trial and error is required if the pressure gradient is not specified. In order to overcome this drawback, an explicit equation for the friction factor is needed for a considerable number of analytical calculations. During the past years since 1939, the most promising explicit equations have appeared as follows.

1. Wood Equation (1966). I t is valid for Re > 10000 and C €10 < 0.04.

( 5 ) where a = 0.094(t D)0.225 + 0.53(c/D), b = 8 8 . 0 ( ~ / 0 ) ~ , ~ , and

2. Churchill Equation (1977). The author claims that

(6)

f D = a + bRe-c

c = 1.62(t/D)O.l3 i . his equation holds for all Re and € I D .

f D = 8[(8/Re)12 + 1/(A + B)3/2]1/12

where

A = 2.457 In [ B =

1 (7/Re)0,9 + 0.27c/D

(37530/Re)16

The present article proposes another explicit equation as

- = -2.0 log ~ - 1 dz [ 3.7665D

which is also good for all values of Re and €10. The accuracy of the proposed eq 7 has been compared with the Churchill equation (6) and the Wood equation (5) against the Colebrook equation (4) for the range of Re from 4000 to 4(10)8 and c / D from 0.05 to 5(10)-7. For 36 comparison points, the results are given in Table I, where it is seen that the proposed eq 7 is almost identical with the Colebrook equation. The former equation is more convenient than the latter because it expresses f D explicitly. For this reason, eq 7 is recommended for use.

Literature Cited Churchill, S. W., Chem. Eng., 91 (Nov 7, 1977). Colebrook, C. F., J . Inst. Civil Eng., 133 (1939). Moody, L. F., Trans. ASME, 66, 641 (1944). Nikuradse, J., Forsch. Arb. Ing. Wes., No. 356 (1932). Von Karman, T., Nach. Ges. Wiss. Gotfingen, Fachgruppe I, 5 , 58-76 (1930). Wood, D. J., Civilfng., 60-61 (Dec 1966).

Department of Chemical Engineering University of Lowell Lowell, Massachusetts 01854

Ning Hsing Chen

Received for review February I , 1979 Accepted April 23, 1979

Acentric Factor and the Heats of Vaporization for Unassociated Polar and Nonpolar Organic Liquids

An empirical equation relates the heat of vaporization, AHv, the reduced temperature, T,, the acentric factor, w , and the temperature, Tfor T, ranging from 0.5 to 0.7. The equation accurately predicts AHv for numerous polar and nonpolar organic liquids.

Nath et al. (1976) have recently obtained an empirical equation relating the reduced vapor pressure, P,, the reduced temperature, T,, and the acentric factor (Pitzer et al., 1955) in the region of the coexistence of liquid and vapor phases. They found that this equation predicts the vapor pressures of polar organic liquids with the same order of accuracy as for normal liquids. It has been pointed out that w takes into account the changing magnitude of the acentric force field associated with the varying polar character of the molecules as accurately as dispersion forces between nonpolar molecules. More evidence is, however, needed to give support to the above viewpoint, and one would be interested to know if other equilibrium properties such as AHv for polar organic liquids can also be predicted with considerable accuracy by using this approach. Since the values of AHv/RT for normal liquids are described

(Hildebrand et al., 1970) to be a function of T , and w , the values of AH,/RT for 45 liquid hydrocarbons have been calculated at fixed values of T , equal to 0.50, 0.55, 0.60, 0.65, and 0.70 from the available vapor pressure data (Jordan, 1954; Meyer and Wagner, 1966; Fried and Schneier, 1968; Meyer et al., 1971) by assuming the vapor phase to be perfect and by neglecting the liquid-phase volume in the Clapeyron equation as was done earlier (Nath and Yadava, 1971). The values of AH,/RT in the T, range 0.5 to 0.7 have been fitted by the method of least squares to

(1)

where A = 31.4589 - 58.0378Tr + 33.86T:, B = 68.0973 - 147.6182Tr + 91.3371T:, and C = 43.5305 - 127.3007Tr + 91.8686T:. The critical-constant data used to calculate

AH,/RT = A + Bw + Cw2

00 19-7874/79/ 10 18-0297$0 1 .OO/O 6 1979 American Chemical Society

298 Ind. Eng. Chern. Fundarn., Vol. 18, No. 3, 1979

Table I. Experimental and Estimated Values of A H J R T for Various Polar and Nonpolar Organic Substances

AH, /RT substance w 106(~4P,z/T,4) T, eq 1 eq 2 (exptllb

A H,IR T (estd)

2,2-dimethylpropane" 0.197 0.0 0.60 11.28 11.05 11.35 2,3-dirneth~lpentane~ 0.297 0.0 0.52 15.39 15.56 15.31 2-methylheptane" 0.379 0.0 0.52 16.82 16.97 16.72 2,3-dimethylhexane" 0.341 0.0 0.52 16.26 16.43 16.21 3,4-dimethylhexanea 0.339 0.0 0.52 16.12 16.29 16.06 3-ethylhexane" 0.361 0.0 0.52 16.51 16.70 16.49 2,2,3-trimeth~lpentane~ 0.298 0.0 0.52 15.41 15.60 15.39 2,2,4-trimeth~lpentane~ 0.304 0.0 0.52 15.51 15.68 15.42 benzene 0.210 0.0 0.60 11.44 11.25 11.59 carbon tetrachloride" 0.191 0.0 0.52 13.58 13.85 13.71 methyl chloride" 0.152 1.77 0.55 12.02 11.99 11.88 ethyl chloride" 0.189 1.06 0.52 13.54 13.77 13.52 1-chloropropane" 0.223 0.62 0.52 14.12 14.44 14.30 2-propanone" 0.299 2.13 0.65 11.21 10.85 11.30 2-butanone" 0.321 1.09 0.65 11.44 11.07 11.47 2-pentanone" 0.346 0.77 0.65 11.71 11.32 11.74 pyridine" 0.246 0.48 0.65 10.66 10.17 10.62

" The values of AH, /RT for these liquids have not beewused to construct eq 1. The experimental values of AH,IRT refer to those obtained from the vapor pressure data. by assuming the vapor phase to be perfect and neglecting the liquid- - - , - phase volume in the Clapeyron equation.

T, and w were those reported by Kudchadker et al. (1968). For 47 organic liquids (not included in Table I) for which eq 1 was tested, the maximum percent deviation in AH, f R T obtained from eq 1 from the experimental values was 0.9; the average percent deviation was 0.3.

Calculated values of AH,/RT for some polar and nonpolar organic liquids are compared with experimental values in Table I. For a polar liquid to conform to the acentric factor theory (Pitzer et al., 1955), the magnitude of the expression p4PC2 f T,4 should be about 0.5 X lo4, when the units of the dipole moment, p, the critical pressure, P,, and the critical temperature, T,, are taken as esu, atm, and K, respectively. Table I, however, shows that eq 1 accurately predicts AH,/RT also for substances having higher values of p4PC2/T$, the highest being 2.13 X lo4 (see Table I) for 2-propanone.

Various methods for the prediction of AHv for liquids have been summarized by Reid and Sherwood (1966) and Reid et al. (1977). Accurate methods based on the law of corresponding states for prediction of AHv as pointed out by Reid and Sherwood (1966) are the Pitzer acentric factor correlation and the Chen Modification to the Pitzer acentric factor correlation. The latter correlation is AH, = T(7.90Tr - 7.82 - 7.11 log P1)/(1.07 - TI) (2) The values of AHJRT obtained from eq 2 have also been included for the sake of comparison in Table I which shows

that the values of AH, f R T obtained from eq 1 are quite accurate. I t should, however, be noted that the values of AH, f R T obtained from eq 1 will very accurately corre- spond to those obtained from the vapor pressure data by assuming the vapor phase to be perfect and by neglecting the liquid-phase volume in the Clapeyron equation.

Literature Cited Fried, V., Schneier, G. B., J . Phys. Chem., 72, 4688 (1968). HIMebrand, J. H., Prausnltz, J. M., Scott, R. L., "Regular Solutions", p 209, Van

Jordan, T. E., "Vapor Pressure of Organic Compounds", Interscience, New York,

Kudchadker, A. P., Alani, G. H., Zwolinski, B. J., Chern. Rev., 66, 659 (1968). Meyer, E. F., Renner, T. A., Stec, K. S., J . Phys. Chem., 75, 642 (1971). Meyer, E. F., Wagner, R. E., J. Phys. Chem., 70, 3162 (1966). Nath, J., Das, S. S., Yadava, M. L., I d . Eng. Chem. Fundam., 15, 223 (1976). Nath, J., Yadava, M. L., Indian J . Chem., 9, 1359 (1971). Pitzer, K. S.. Lippmann, D. Z., Curl, R. F., Jr., Huggins, C. M., Petersen, D. E.,

Reid, R. C., Prausnitz, J. M., Sherwocd, T. K., "The Properties of Gases and

Reid, R. C., Sherwocd, T. K., "The Properties of Gases and Liquids", 2nd ed,

Nostrand-Reinhold, New York, N.Y., 1970.

N.Y., 1954.

J . Am. Chem. SOC., 77, 3433 (1955).

Liquids", 3rd ed, McGraw-Hill, New York, N.Y., 1977.

McGraw-Hill, New York, N.Y., 1966.

Department of Chemistry University of Gorakhpur Gorakhpur 273001, India

Jagan Nath

Received for reuieu April 18, 1978 Accepted April 12, 1979