or an immersed cooling coil were obtained for the aerated tower as follows
(h/pC p ) (p 2 / A p pg )
( C pp / k ) 2/3 Vis -0.05 = 0.054Ug1l4
when U l I 54 m/h and U , I1000 m/h
( h / p C p) (p 2 / A p w g ) ( C pp / k )2/3 Vis -O, O5 = 0.3 when U I I54 m/h and U , I1000 m/h, and
(h/pC p ) (p 2 / Ap pg ) ' I 3 ( C pp / k ) ' I 3 Vis -O. O5 = 0.054U,'/4(U154)'/~
when 54 m/h. Heat transfer coefficients for an aerated tower with a n o n Newtonian liquid can also be correlated by equations for Newtonian liquids by using the apparent viscosity calculated from the flow curve and the local average shear rate which is obtained as follows in this study. jlav = 5 0 . 0 ~ ~
when u y I4.0 cm/s ( U , 2 144 m/h), and = 100u,0.5; yavj= 0.195u2.0
when u gI4.0 cm/s.
r = latent heat of steam condensation, kcal/kg ti = temperature at inlet, "C t o = temperature at outlet, "C u = velocity, cm/s U = velocity, m/h uh = overall heat transfer coefficient, kcal/m2 h "c w = mass flowrate, kg/h y = shearrate,s-' qav = average shear rate, s-l r = rate of condensation, kg/h m I.( = viscosity at bulk temperature, P wa = apparent viscosity, P ww = viscosity a t wall temperature, P p = density, kg/m3 pf = density of condensate, kg/m3 Ap = density deference between gas and liquid, kg/m3 r = shear stress, dyn/cm2 jt, ( = (h/Cpp)(p2/Appg)1/3(Cpp/k)2/3) = j factor Vis ( = p w / p ) = viscosity correction term
Subscripts c = cooling coilside g = gasside j = jacketside 1 = liquidside s = steamside w = cooling water side
Literature Cited
Nomenclature C p = specific heat, kcal/kg "C d, = inside diameter of coil tube, m d, = nozzle diameter, m D = inside diameter of aerated tower, m D, = diameter of coil helix, m D, = rate of steam condensation, m3/h g = gravitational acceleration, m/h2 h heat transfer coefficient, kcal/m2 h " C H = liquid height in aerated tower, m H , = lowest coil level, m H , = lowest jacket level, m k = thermal conductivity, kcal/m h "C K = fluid consistency index, dyn sm/cm2 L = overall height of heat transfer device, m m = exponent in power law rheological equation Q = amount of heat transferred, kcal/h
Akita, K., Ph.D. Thesis, Kyoto University, 1972. Fair, J. R.. Lambright, A. I.,Andersen, J. W., Id.Eng. Chem., Process Des. Dev., I,33 (1962). Hart, W. F., lnd. Eng. Chem., Process Des. Dev., 15, 109 (1976). Inoue, I., Unno, H., Kagaku Kogaku, 36, 65 (1972). Jescke, D., Zh. Ver. Deut. lng., 23, 69 (1926). Kast, W., lnt. J. Heat Mass Transfer, 5, 329 (1962). Kato, Y , Repr. Fac. Eng. Yamanashi Univ. Japan, 8, 31 (1957). Kato. Y., Kagaku Kogaku, 26, 1068 (1962). Kolbel, H., Borchers, E., Muller, K.. Chem. lng. Tech., 30, 729 (1958). Kolbel, H., Borchers, E., Martins, J., Chem. hg. Tech., 32, 84 (1960). Konsetov, V. V.. lnt. J. Heat Mass Transfer, 9, 1103 (1966). Nagata. S., Nishikawa. M.. Itaya, M., Ashiwake, K., KagakuKogaku, Ronbunshu, 1, 5 (1975). Nusselt, W., Zh. Ver. Deut. lng., 60, 541. 569 (1916). Sieder, R. A., Tate, G. E., ind. Eng. Chem., 28, 1429 (1936). Yoshitome, H., Makihara, M.. Tsuchiya, Y., Kagaku Kogaku, 28, 228 (1964).
Received f o r reuieu May 11, 1976 Accepted September 8,1976
Chebychev Polynomial Correlation Equations of Composition and Bubble/Dew Point Temperature of Binary Mixtures Containing Ethane with Propane, Butane, and Pentane Bruce R. Corn, James H. Weber, and Luh C. Tao' Department of Chemical Engineering, University of Nebraska, Lincoln, Nebraska 68588
Correlation equations based on Chebychev polynomials are presented for the bubble and dew temperatures of ethane-propane, ethane-butane, and ethane-pentane binaries. They cover the entire composition range and the pressure range of 100 to 600 psia.
For a narrow pressure interval, vapor pressures of pure substances may often be correlated by a simple equation such as the Antoine equation. To cover a wide pressure range u p
to the vicinity of the critical regions, Gibson (1967) used the orthogonal Chebychev polynomial to obtain the vapor pressure equations of eight chemical species. Utilizing this concept Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
137
Table I. Correlations Coefficients for Ethane-Propane Binary Variable
ff0
ff1
18.56941 1487.057 419.0169 515.3298 665.4519
1373.086 2645.249 278.8503 -197.1009 -226.1788
46.77276 1557.814 427.2622 515.9714 665.3705
-501.0846 -1606.201 -384.9973 -65.33519 -149.7127
a3
ff2
Bubblepoint Polynomial Coefficients -2878.202 8.806047 -5176.690 -1733.538 251.9714 -2210.289 151.7148 -56.64799 513.0362 -734.0510 Dewpoint Polynomial Coefficients 7185.195 -15059.81 18267.70 -37130.59 3234.741 -5835.617 49.20888 -84.43983 262.6774 -429.1453
ff4
a5
afi
1585.702 2536.125 1672.048 0.0 330.5902
0.0 2760.878
-1062.015
0.0 0.0 0.0
0.0 0.0 0.
8361.492 19455.83 1300.780 0.0 199.4271
65.84666 895.5156 2733.800
0.0 0.0
ff4
ff5
016
-21920.30 -50771.31 -13443.06 203.2571
8238.181 19200.19 8028.273
0.0 0.0
-1732.809
0.0 0.0
0.0 0.0
21722.99 10304.23 57409.02 0.0
-663.8071 14440.28 -18836.18
0.0 0.0
0.0
-1075.114 0.0 0.0
Table 11. Correlations Coefficients for Ethane-Butane Binary Variable
ff0
103.1059 1897.114 477.7834 605.7247 305.000 85.90536 1856.434 471.8114 605.5416 305.000
ffl
3
a2
ff
Bubblepoint Polynomial Coefficients -7515.900 20532.28 -16481.86 47050.55 -3893.222 10517.43 802.2174 -624.9096 43.25835 46.48775 Dewpoint Polynomial Coefficients -8757.961 32602.14 2487.421 -20017.74 65573.49 5699.678 -5623.130 29907.68 785.3729 -5.458635 -549.1062 1105.384 -159.3259 -97.90316 256.5624 600.1707 327.7622 367.2690 -573.0894 -326.9851
0.0
-47465.75 -76696.23 -63803.04 -742.8732 -236.4942
0.0
0.0 0.0
Table 111. Correlations Coefficients for Ethane-Pentane Binary Variable A ao a1
Tmm Tmax
A a0
a1
Tmm Tmax
a0
a1
98.62325 2021.508 461.1762 683.0391 836.2667
4843.429 10408.88 2391.425 -932.8692 -286.7945
110.8145 2049.743 459.3917 683.4777 835.2000
-15414.57 -36884.88 -4412.087 15.91909 1.589248
ff2
a5
ff4
Bubblepoint Polynomial Coefficients -36246.65 88991.14 -91612.41 -83924.25 201473.7 -196384.9 -14181.65 31768.56 -32649.00 1219.167 -557.2309 0.0 -300.0419 287.8022 0.0 Dewpoint Polvnomial Coefficients 183264.3 -723901.2 1398740. 444605.5 -1785755. 3489078. -178664.5 316008.2 50471.15 2259.391 - 1540.291 -1004.274 608.1731 -497.0491 216.6302
for mixtures, Corn e t al. (1974,1975) demonstrated the feasibility of obtaining two correlating equations for mixtures of methane and ethane, one for bubble point and the other for dew point as functions of composition, over the range of 100 to 600 psia. Such equations are useful in the design of partial and total condensers as well as for thermodynamic calculations of partial derivatives along the bubble point or dew point surfaces (e.g., Houser and Weber, 1961). This approach of direct correlation of bubble/dew point data thus provides a simple and convenient alternative to the use of equations of state. In this study, 159 data points (Matschke and Thodos, 1962), 99 points (Kay, 1960)and 115 points (Reamer et al., 1960) of ethane-propane, ethane-n-butane, and ethane-n-pentane, respectively, were used to obtain these correlations in terms of the following equation by utilizing the Chebychev polynomial technique described previously (Corn et al., 1974, 1975). 138
a3
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
33779.05 60493.54 12574.15 0.0
217.0140 7237.304
0.0
0.0 0.0 0.0
-1299320. -3260875. -279684.5
456594.4 1149124. 96183.86
0.0
0.0 0.0
-626.6026
( A + T)log p = ao + u ~ X
+
ff6
(1)
where X = ( 2 T - [T,,, Tmin])/(Tmax - Tmi,).These coefficients T,,,, T,i,, A , ao, and a 1 are defined as polynomials of mole fractions of ethane as Zna,zR. For the bubble point equation z = x and for the dew point equation z = y . Tables I, 11, and I11 summarize these polynomial coefficients for ethane-propane, ethane-butane, and ethane-pentane, respectively. Table IV shows percent errors relative to literature values of data points within their respective pressure ranges. Detail comparisons, between calculated and data values for intervals of 0.10 mol fraction have been tabulated and are available (Corn, 1975). These results indicate that use of the Chebychev polynomial can result in good correlations to represent the bubble or dew points and mixture composition by a single equation for binary mixtures. Once obtained, these equations provide a coherent relationship for the entire composition range and would
Table IV. Percent Errors of Correlation Equations System
Pressure range, psi
CS-C~
100-190
CZ-C~
c2-C~
100-550
100-600
Percent Errors (AveIMax) Type Bubble Point Dew Point Mole fraction
1.43110.8
2.33125.2
T
0.102/.687 0.666/4.83 0,47212.16
0.111/0.744 0.833/6.04 0.55315.86
0.063/0.289 0.367h.37 0.89513.54
0.05110.297 0.434/2.60 1.48h2.9
0.13810.516 0.75313.34
0.14010.630 1.2815.96
P Mole fraction T P Mole fraction T P
equations for the system of ethane and propane (Corn, 1975). K values of ethane varied from 1.05 to 1.38. Referring to literature values (Matschke and Thodos, 1962), 28 calculated points have errors less than 1%,30 points between 1 and 5% errors, and 11points more than 5%errors. Similarly, K values of propane ranging from 0.25 to 0.99 have 43 points with errors less than 1%, 19 points between 1 and 5%, and 8 points with errors above 5%. Nomenclature ao, a1 = Chebychev polynomial coefficient A = coefficient in eq 1 p = pressureinpsia T = temperature in O R X = normalized temperature z = mol fraction of ethane a = coefficients of polynomials for A, ao,
a1
and T,,,,
Tmin
facilitate computations in the two phase region due to their simple algebraic structure. T o illustrate the use of such equations in design calculations, Corn (1975) programmed on an IBM 360 computer the isobaric (300 psia) operation of a partial condenser for a feed stream of 20 mol % ethane and 80% propane at various condensate-to-feed ratios. The program used 7.3K storage space and 9.39 s of WATFIV batch processor execution time to compute 102 temperature points with a composition accuracy of 0.00005 mol fraction. A similar program for an isothermal operation of the same feed required the same storage space and the computation time was 9.48 s for 102 pressure values and their respective compositions of vapor and liquid phase. Seventy sets of K values were also computed from these
L i t e r a t u r e Cited Corn, B. R., Young, M. D.. Weber, J. H.. Tao, L. C., lnd. fng. Chem. frocess Des. Dev., 13, 95 (1974). Corn, B. R., Weber, J. H., Tao, L. C., lnd. Eng. Chem., Process Des. Dev., 14, 96 (1975). Corn, B. R., M.S. Thesis, Department of Chemical Engineering, University of Nebraska, 1975. Gibson, M. R., Bruges, E. A,, Mech. Eng. Sci., 9, 24 (1967). Houser, C.G., Weber, J. H., J. Chem. Eng. Data, 6, 510 (1961). Kay, W.B., lnd. Eng. Chem., 32,355 (1940). Matschke, D. E., Thodos, G., J. Chem. Eng. Data, 7, 232 (1962). Reamer, H. H.. Sage, 6.H., Lacey, W. N., J. Chem. Eng. Data, 5,44 (1960).
Received for review April 26, 1976 Accepted September 2, 1976 The authors acknowledge the financial aid from Continental Oil Company to Bruce Corn as well as the support of The Engineering Research Center.
Accurate Measurement of Activity Coefficients at Infinite Dilution by Inert Gas Stripping and Gas Chromatography Jean-Claude Lerol and Jean-Claude Masson Societe Rhbne-Poulenc Industries, Centre de Recherches de Decines, 69 150 Decines, France
Henri Renon, * Jean-Franqois Fabrles, and Henri Sannler Groupe “Reacteurs et Processus ”, €cole Nationale Superieure de Techniques A vancees-€cole 75272 Paris Cedex 06.France
Nationale Superieure des Mines de Paris
A method is presented for fast and accurate determination of the limiting activity coefficient of a solute dissolved in a liquid mixture. It is based on the study of the solute elution with time; the solute is stripped from the solution by a constant flow of inert gas. The variation of solute concentration in the gaseous phase is measured by gasliquid chromatography. Experimental results for several systems are in good agreement with data from retention time measurementsand extrapolation of vapor-liquid equilibria.
Introduction A rapid method of determination of vapor-liquid equilibria is a permanent need in industry for screening solvents used for separations by extractive distillation or liquid extraction and for the design of processes. Gas-liquid chromatography
(GLC) is very often used as an analytical method in connection with vapor-liquid equilibrium device (HBla et al., 1967). Methods using more specifically chromatography are of three types: (1)measurement of retention time for the elution of the solute by an inert carrier gas flowing in a column on a stationary phase of the solvent; (2) chromatographic analysis of Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
139
