Ind. Eng. Chem. Res. 1990,29, 488-494
488
Literature Cited Beddow, J. K. Description of Particulate Assemblies. In Particulate science and technology; Chemical Publishing: New York, 1980; p 725. Calmanovici, C. E.; Ishii, R. C. Solubilization tests with Brazilian phosphate from North and Northeast regions using sulphuric and nitric acids. Fertilizantes 1987, IO (21, 2-6. Carberry, J. J. Fluid-solid Noncatalytic reactions. Chemical and Cataiytic Reaction Engineering; McGraw-Hill: New York, 1976. Costa, M. L.; Si, J. H. S. Os fosfat,oslateriticos da AmazBnia oriental: geologia, mineralogia, geoquimica, e correlaqBo com as bauxitas da Amazhia. Proceedings, Congresso Brasiileiro de Geologia, 31, Sante Catarina; SBG: SBo Paulo, Brazil, 1980; Vol. 3, pp 1459-72. Damasceno, E. C.; Born, H.; Liberal, G. S.; Melo, M. T. V.; Beisegel, V. R. Recursos minerais de fosfato no Brasil. Proceedings, Encontro Nacional de Rocha FosfSitica, 4 Brasilia, June /July 1988; IBRAFOS: Brazilia, 1988, in press; p 13. Gremillion, R.; Lehr, J. R. Characterization of a phosphate rock sample from Olinda, Brazil: sample MR 639. IFDC: Muscle Shoals, .4L. 1979; p 11 (report not, published). Ishaque, M.; Ahmed, 1.; Nassem, N. A. Study of decomposition of Jordan and Lagarban phosphate rock with nitric acid, hydrocloridic acid and sulphuric acid. Fert. News 1983, 28 (2), 43-9. Janikowski. S. M.; Robinson, N.: Sheldrick, W. F. Insoluble phos-
phate losses in phosphoric acid manufacture by the wet process: theory and experimental techniques; The Fertiliser Society: London, 1964; Vol. 81, p 51. Lehr, J. R.; McClellan, G. H. Phosphate Rocks: Important Factors in Their Economic and Technical Evaluation. Proceedings of the CENT0 Symposium on Mining and Beneficiation of Fertilizer Minerals, Istanbul, Turkey, Nov 19-24, 1973; pp 192-242. Levenspiel, 0. Noncatalytic Fluid-solid Reactions. Chemical Reaction Engineering; John Wiley & Sons Inc.: New York, 1972. McClellan, G. H.; Klem, F. J. Phosphate rock characterization. Proceedings, Fertilizer Seminar: Trends in consumption & production, New Delhi, Dec 1-3,1977; FAI/IFDC: New Delhi, 1977; P 9. Mendonca, J. C. G. S.; Braga, A. P. ; Campos, M. Consideracoes sobre a mineralizacao fosforo-uranifera da jazida de Itataia - CE. Proceedings, Congresso Brasileiro de Geologia, 31, Santa Catarina, Brazil; 1980; Vol. 4. Siqueira, N. V. M.; Lima, W. N. Estudo geoquimico de alterap6es e distribuiqBo de elementos em perfil lateritic0 desenvolvido sob influencia do f6sforo-Pirocha (MA). Proceedings, Congresso Brasileiro de Geologica, 32, Salvador, 1982; SBG: SBo Paulo, 1982; Val. 5 , pp 1991-2002. Receiued for reuiew April 18, 1989 Reuised manuscript received October 6, 1989 Accepted November 3, 1989
Generalized Treatment of Thermal Conductivity Enhancement in the Critical Region H u e n Lee Department of Chemical Engineering, Korea Institute of Technology, 400, Kusong-dong, Yusung-gu, Taejon, C'hung-chong nam-do, Korea
,4generalized method has been developed to predict the anomalous thermal conductivity enhancement of pure fluids in the critical point region. The power-law hypothesis and National Bureau of Standards (NBS) equation of state were adopted to express thermodynamic behavior in the critical region. This predictive procedure requires only critical constants of pure fluids. Values calculated with this generalized model when compared with corresponding thermal conductivity enhancement data yield for 16 fluids an approximate overall average absolute deviation of 12% within the critical region bounded by 1 < T R < 1.03 and 0.7 < pR < 1.3. As early as 1934, Kardos (1934) noted in his hot-wire measurement of thermal conductivity for carbon dioxide the presence of an irregular pattern for this transport property near 31 O C . However, for temperatures above this value, this irregular behavior gradually diminished, and the resulting measurements conformed to a pattern consistent with normal behavior. Many different experiments of thermal conductivity for carbon dioxide have also showed this behavior since then (Sellschopp, 1934; Guildner, 1962; Simon and Eckert, 1963; Le Neindre et al., 1973; Becker and Grigull, 1977). This anomalous thermal conductivity enhancement in the critical region has also been noticed for a number of fluids ranging from helium-3 (Pittman et al., 1982),which possesses maximum quantum effects, to the simple monoatomic argon and xenon (Trappeniers, 1982) and projecting into the behavior of water (Le Neindre et al., 1973; Sirota et al., 1976; Tufeu and Le Neindre, 198713) and heavy water (Tufeu et al., 1986), which are associated with strongly polar and hydrogen-bonding contributions. It thus becomes evident that the thermal conductivity of fluids exhibits an anomalous enhancement in a large range of densities and temperatures around the critical point. In addition to thermal conductivity, many thermophysical properties such as the isothermal compressibility, thermal expansion coefficient,. heat capacity, and viscosity 08SS-5~85/90/2629-0488$02.50/0
all diverge as the critical point is approached. While the critical thermal conductivity enhancement of fluids has been noticed up to temperatures 20% above the critical temperature, the critical viscosity enhancement of fluids only appears at temperatures less than 3% from the critical temperature. However, the development of a generalized method for the prediction of thermal conductivity of fluids in the critical region has been strongly hindered because of the lack of reliable experimental measurements associated with this region. Therefore, it may prove expeditious in this treatment to examine the critical behavior of thermal conductivity for a variety of fluids including both the nonpolar and polar substances, though for only 16 fluids the experimental data appeared in the literature. Theoretical Remarks The thermal conductivity of fluids near the critical point is commonly separated into a normal thermal conductivity, A,, in the absence of critical fluctuations and a critical enhancement, Ah,, due to the critical fluctuations:
x
= h,
+ Ahc
(1)
The general behavior of thermal conductivity slightly above the critical temperature as a function of density is schematically shown in Figure 1 (Basu and Sengers, 1977). 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 489
p I.',/'
the critical region, Hanley et al. (1976) extended eq 6 by using an empirical asymptotic function as follows: Ax, = (KT/~v[)P(C, - c,)f(t,Ap*) (7a) where the function f(t,Ap*) satisfies the boundary conditions lim f ( t , A p * ) = 1 (7b) t,Ap'-r0
and lim
f(t,Ap*) =
0
(7c)
f,Ap*-m
I I
Y
1
I I
Pc
P Figure 1. Typical thermal conductivity-density relationships of pure fluids at three supercritical temperatures, T,< T , < T2< T3.
Outside the critical region, the thermal conductivity, A, is to be identified with the normal thermal conductivity, A,. Inside the critical region, A, is defined empirically by extrapolating the behavior of the normal thermal conductivity outside the critical region into the critical region. It has been theoretically indicated that some of the kinetic coefficients may be changed rapidly near the critical point due to nonlinear coupling between the hydrodynamic modes. Because one mode (such as heat mode or viscous mode) can break up into two or more other modes, the transport properties are enhanced and can in fact diverge at the critical point. The calculation of these coupling constants and the resulting divergences in transport properties are now commonly referred to as the mode coupling theory. Theoretical prediction of the mode coupling theory (Kawasaki, 1976) can be obtained from the simple argument that assumed that the mobility of clusters is purely a diffusional process. For the diffusional involvement of these clusters, the Einstein relation for the diffusion coefficient can be expressed as follows: D = KT/{ (2) where K is the Boltzmann constant, T the temperature, and 5 the friction coefficient. When this friction coefficient is identified with the hydrodynamic friction coefficient predicted by Stoke's law for a spherical droplet of radius 5, it becomes l =6 ~ 7 5 (3) where 7 is the shear viscosity. Substituting eq 3 into eq 2, the diffusion coefficient becomes D =KT/~T~( (4) Kawasaki (1976) assumed a direct analogy between this diffusion coefficient and thermal diffusivity to couple these two phenomena into the relationship AA,/pAC, = ~ T / 6 n 7 ( (5) where AC = C, - C,. In the derivation of eq 5, it is assumed tiat the shear viscosity, 7, does not exhibit an anomalous behavior near the critical point. This assumption is not strictly justified, but the anomalous behavior of shear viscosity turns out to be sufficiently small so that it can be neglected for most practical purposes. Equation 5 can be rearranged to present the enhancement of thermal conductivity as follows: Ax, = ( K T / ~ ~ v [ ) P ( C - ,c,) (6) In order to make eq 6 vanish for conditions removed from
where E = T R - 1 and Ap* = pR - 1. Utilizing the measurements for the thermal conductivity enhancement of carbon dioxide, Hanely et al. (1976) proposed the form and developed the coefficients of this function to be f ( t , A p * ) = exp(-l8.66(tl2 - 4.25(Ap*I4) (8) By use of the thermodynamic relation, (9) C, - C, = (T/p2)(aP/aT),z(ap/aP)~ where l/p(dp/dP)T is the isothermal compressibility, and substituting eq 9 into eq 7a, the following relationship results: Ax, = [ ( ~ T ~ / 6 7(1 ~/4P))(aP/dT ),2(ap/ap)TI / exp(18.661~1~ + 4.251Ap*I4) (10)
where 4 is a long-range correlation length that represents the average radius of the critical state clusters. This long-range correlation length can be determined experimentally from light-scattering or X-ray-scattering data. Assuming that the correlation length, 5, can be approximated by the Ornstein and Zernike formula, Sengers (1972) obtained the following relationship: [ = R(nKTKT)1/2 (11) where n is the number density, R a proportionality constant, usually referred to as the short-range correlation length, and KT the isothermal compressibility. The expression given by eq 10 possesses semiempirical arguments for its development and satisfactorily represents the thermal conductivity enhancement in the critical region for a few substances for which experimental information is available. However, it should be recognized that the application of this relationship requires extensive information for a substance in the critical region. More often than not, such information is not readily accessible, and therefore, the application of eq 10 becomes limited to substances whose physical properties are well defined in the critical region.
Generalized Behavior along the Critical Isochore The theories and experimental data that exist at the present time do not allow unambiguous resolution Qf the abnormal behavior of the thermal conductivity near the critical point of a pure fluid. Even the modern theories include a number of adjustable parameters or they possess physical properties a t the critical point, which are very difficult or impossible at the present level to measure or predict. This lack of information makes the current theories meaningless for the estimation of transport properties in the critical region. Also, no unified approach in a generalized manner has been attempted for the prediction of the critical thermal conductivity behavior. As a first step of treating the anomalous thermal conductivity enhancement in the critical region, the generalized thermal conductivity behavior along the critical isochore is qualitatively investigated. The thermal conductivity enhancement of pure fluids along the critical isochore is
490. Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990
-. .
t
A
a t C
E
. +. I
OOOIi 00001
I
1
( 1 ( 1 ( / 1
000l
1
1
I l l l l l
001
I
I
~~~~t~~
I
01
~
~
1
1
1
1
1
u
1
10 .
I0
E
F i g u r e 2. Relationships between A i , and t for a number of fluids: (D) helium-3; ( 0 )argon; (A)xenon; (A)oxygen; (0) carbon dioxide; ( 4 ) sulfur hexafluoride; ( 0 )methane; (4) ethane; (+) propane; (M) n-butane; (e)isobutane; (v)ammonia; (0, Sirota e t al., 1976) and (b,Tufeu and Le Neindre, 1987b) water; (v)heavy water. Table I. Critical Constants and v Values for 16 Fluids Used in This Study fluid M Tc,K Pc, MPa po, kg/ms Z, Y helium-3 3.016 3.3099 0.11678 41.45 0.310 1274.7 argon 39.948 150.725 4.865 535 0.290 771.5 xenon 131.300 289.734 5.8400 1110 0.287 1393.4 oxygen 31.999 154.580 5.043 436.2 0.288 680.9 carbon 44.010 304.127 7.3753 467.8 0.274 722.1 dioxide sulfur 146.050 318.687 3.7605 730 0.284 2018.8 hexafluoride methane 16.043 190.555 4.595 162.7 0.286 534.1 ethane 30.070 305.33 4.8718 206.5 0.279 775.5 propane 44.097 369.82 4.247 221 0.276 1073.0 n-butane 58.124 425.16 3.797 228.0 0.274 1368.4 iaobutane 58.124 407.85 3.631 227 0.276 1392.0 chlorotrifluoro- 104.459 302.0 3.87 579 0.278 1689.2 methane chloropenta154.467 353.2 3.23 613.5 0.277 2386.6 fluoroethane ammonia 17.031 405.4 11.303 235 0.243 392.3 water 18.015 647.13 22.06 322.2 0.229 293.2 heavy water 20.031 643.89 21.66 357 0.226 316.0
commonly described by a simple power law (Sengers, 1972) expressed as Ah, = AXc(t,pc) = At+ where $ is the critical exponent having a universal constant for all fluids irrespective of molecular details and A the critical amplitude to be specifically determined from the experimental measurements associated with each fluid. The dependence of AX, on t for a number of fluids was plotted in Figure 2. In a previous work (Lee, 1987), a dimensional analysis approach was adopted to give the following relationship: AX,v/a = t+ (13) where v = M 1/2T,lP,3/2V2/6 is the thermal conductivity parameter and a the characteristic constant specific to each fluid. The critical constants taken from Sengers et al. (1981) and u values for a number of fluids are presented in Table I. Comparing eq 12 with eq 13, A is identical with a/u. Figure 3 indicates that AXcv/a has a linear dependence on t in log-log coordinates with a slope of approximately -0.63 for t < 0.1, which coincides with the critical exponent of correlation length predicted from two modern theories, namely, renormalization group (Siggia e t al., 1976) and mode coupling (Kawasaki, 1976). However, the nonlinear relationship between AX,v/a and t appears for e > 0.1, which means that the critical en-
E
Figure 3. Relationships of A Q / a and t for a number of fluids. The symbols are the same as in Figure 2. Table 11. Basic Parameters i n Equation 22 and Literature Sources of Experimental Thermal Conductivity Data in the Critical Region fluid U zn E, lit. sources helium-3 0.381 66 0.489 2.96 Pittman et al., 1982 argon 1.23330 0.183 2.27 Trappeniers, 1982 xenon 1.80526 0.183 2.27 Trappeniers, 1982 oxygen 1.48823 0.183 2.21 Weber, 1982 carbon dioxide 2.11394 0.141 2.01 Michela et al., 1962 sulfur hexafluoride 1.83702 0.172 2.86 Letaief et al., 1986 methane 1.39147 0.164 2.03 Roder, 1985 ethane 1.83193 0.147 2.03 Desmhest and Tufeu, 1987 propane 2.531 32 0.137 1.83 Tufeu and Le Neindre, 1987a n-butane 2.836 66 0.13P 1.88b Nieto de Caatro et al., 1983 isobutane 2.731 10 0.140 1.93 Nieuwoudt et al., 1987 chlorotrifluoromethane 2.302 17 0.15CP 1.97b Venart et al., 1976 chloropentafluoroethane 2.668 61 0.146" 1.95* Hahne et al., 1989 ammonia 5.690 05 0.109 1.37 Tufeu et al., 1984 water 8.978 24 0.100 1.20 Sirota et al., 1976 10.52910 Tufeu and Le Neindre, 1987b heavy water 8.70006 0.100 1.20 Tufeu et al., 1986 azovalue calculated from eq 24. bE, value calculated from eq 25.
hancement of thermal conductivity decays with a critical exponent more negative than -0.63 and further that the critical exponent is a function of t. A linear regression method was used to obtain the best values of a for 15 fluids. These values are presented in Table 11. It is of great interest that the values of a are closely related to the corresponding critical compressibility factors as shown in Figure 4. The resulting functional relationship between a and 2, simply determined by the least-squares analysis can be expressed as i=O
where Bo = 60.7304, B, = -348.1854, and B2= 495.0016. I t becomes quite evident from Figure 3 that the thermal conductivity enhancement of pure fluids along the critical isochore can be predicted in a generalized manner by using the most important two parameters, v and a which can be directly calculated from the critical constants of fluids.
Generalized Behavior in the Critical Region The scaling laws are based on the phenomenological consequence of the physical intuition that the anomalous
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 491 sibility along the critical isochore, which can be related to the constants in the NBS equation by
r
= x 0Y E1-1E 2 (1-?)/20
(19)
Equation 18 implies that the scaled isothermal compressibility is only a function of the scaling variable, x. In analogy to eq 18, Sengers (1972) defined a scaled thermal conductivity as
Figure 4. Dependence of
CY
upon 2,.
critical behavior depends on one length only, namely, the correlation length which measures the size of the critical fluctuations. A t the present time, there are no experimental grounds for doubting that the scaling hypothesis is a true reflection of the most essential features of the critical phemomena. In this connection, the scaling law analysis for thermal conductivity enhancement in the critical region proposed originally by Sengers (1972) was adopted in this study as a first approximation. This approach assumes that the chemical potential is a homogeneous function of two independent thermodynamic variables, temperature and density. By applying a Taylor series expansion of the chemical potential of a classical equation in powers of Ap* and 6 around the critical point, the following expression can be straightforwardly obtained: AP* A p * IA p * 16-1
= h(x)
where h(x) is a function of the scaling variable x defined as
In eqs 15 and 16, A@* is a normalized chemical potential difference A@* = { g ( p , T ) - p ( p c , T ) \ p c / P c , 6 the critical exponent of the critical isotherm, and p the critical exponent of the coexistence curve. The function h(x)should satisfy a number of conditions formulated by Griffiths (1967). An empirical equation for h(x) that satisfies most but not all of these conditions is given by the following expression (Sengers and Levelt Sengers, 1978):
where El and E2are empirically determined constants, xo is the value of x along the coexistence curve, and y = p(S - 1) is the critical exponent of the isothermal compressibility. Equation 17 is often referred to as the National Bureau of Standards (NBS) equation. Since p 2 K ~= ( d p / d ~ )the ~ , normalized isothermal compressibility, K * T = P A T , can be expressed by
where l? is the critical amplitude of isothermal compres-
where A is a function of temperature and density that corrects an asymmetry of the peaks for AX, along isotherms with respect to p = p c and becomes unity if Ax, is symmetric with respect to p = p c . The scaling function f&) in eq 20 can be approximated by the square root of the scaling function fKT(x) in eq 18. It should be also noted from eq 20 that the scaled thermal conductivity enhancement in the vicinity of the critical point is, to a good approximation, only a function of the scaling variable, x. For a complete analysis for the anomalous thermal conductivity enhancement of a fluid near the critical point, it becomes therefore neccessary to formulate the generalized model based on the scaling hypothesis and power law assumption described in the previous section. By use of eqs 17-20, the functional dependence of f J x ) on the scaling variable x can be written in the form of
r
where x
Y = -
+ xg XO
In addition to the NBS equation used in this study, the parametric scaled equations have been also applied for the universal representation of the thermodynamic properties of fluids in the critical region (Sengers and Levelt Sengers, 1978, 1984, 1986). Recently, the practical representation for the critical thermal conductivity enhancement has been developed by incorporating a finite cutoff into the asymptotic mode-coupling integrals for the diffusivity associated with the critical fluctuations (Olchowy and Sengers, 1988, 1989). It has been verified from the hypothesis of universality of critical behavior that the critical exponents p, 8, and y and the constant E2should be the same for all fluids. From a comprehensive analysis of experimental equation of state data, Levelt Sengers et al. (1976) obtained the following values: p = 0.355, 6 = 4.352, y = 1.190, and E2 = 0.287. With the values thus adopted for the critical exponents p, 8, and y and the constant Ez, the thermal conductivity enhancement of pure fluids in the critical region can be then completely determined by two substance-dependent parameters, xo and El. The values of these parammeters for a number of fluids taken from Sengers et al. (1981) are presented in Table 11. From the analysis so far derived, the generalized thermal conductivity behavior of pure fluids near the critical point can be then expressed as log ( A X , u x o ~ ~ z / ~ E 1 3 ~ z ) A p * I r = i z gf*X,( ) y)
(22)
492 Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 1
0
..
O5
.
.< ,
$ 0
1
0 4
iI
c c
r
.
/
c-:
-.
I
P
0 022
Figure 5. Deviations between the experimental thermal conductivity data and the values calculated by using eq 22: (0) ethane; (A) propane; (0) isobutane; ( 0 )ammonia; (A)water, (W) heavy water.
The following polynomial form was adopted to represent the functional dependence of f*x,on log y :
However, since both the power law, eq 12, and the NBS equation, eq 17, are in general universally valid in the critical region of It1 < 0.03 and IAp*l < 0.25 (Levelt Sengers et al., 1976), the thermal conductivity data of helium-3, argon, xenon, oxygen, carbon dioxide, ethane, propane, isobutane, ammonia, water, and heavy water located within these temperature and density ranges are selectively used to determine the coefficients of eq 23. The sources of data for these fluids are listed in Table 11: A, = -0.829 45, A l = -0.12628, A2 = -0.21651, A, = 0.04224, and A, = -0.00288. In order to illustrate how accurately eq 22 predicts the critical thermal conductivity behavior, the deviations between experimental thermal conductivity data for the various types of fluids and values calculated by using eq 22 are plotted in Figure 5. Equation 22 indicates that four parameters, u, cy, xo, and E,, should be precisely determined for the better prediction of critical thermal conductivity enhancement. The parameter u can be directly calculated from the critical constants of pure fluids, while for the determination of the parameters cy, xo, and El both thermal conductivity data along the critical isochore and equation of state data in the critical region are required. Unfortunately, these critical region data are only available for a very few substances in the literature. It becomes therefore necessary to develop the estimation method of the parameters cy, xo, and El. In the previous section, the dependence of cy upon 2, has been already found in the form of eq 14, and therefore, the parameter a could be obtained from the critical compressibility factor of pure fluids. In a similar way, values of the parameters xo and El for a number of fluids were plotted against the corresponding critical compressibility factor as shown in Figures 6 and 7, respectively. The functional dependence of xo and El on 2,determined by the least-squares analysis can be expressed as ' XO(Z,) = zciz,' (24) i=O
where Co = 2.7811, C1 = -22.1621, and C2 = 45.6701 and (25) where Do = 8.3551, D, = -69.5364, and D, = 167.5192. For
0 2 60
024
032 3
030
028
ZC
Figure 6. Dependence of xo upon 2,.
SF,e
3 0 r
t-
E, 2
0
HE-4,0
i
lD20 t
/
02"
l
O
022
[
"
024
'
~
I
'
026
028
'
I
030
I
J
032
ZC
Figure 7 . Dependence of El upon Z,.
. I'
Figure 8. Deviations between the experimental thermal conductivity data and the values calculated by using eqs 22,24, and 25: ( 0 ) n-butane; (A)chloropentafluoroethane.
n-butane and chloropentafluoroethane, the values of xo and El were not reported in the literature, and therefore, eqs 24 and 25 were directly used to determine these values. With the calculated values of xo and El, the critical thermal conductivity enhancements, Ax,, for these two fluids were directly calculated from eq 22 and compared with the corresponding experimental data (Figure 8). The overall AAD % between the values of critical thermal conductivity enhancement estimated from eq 22 and those from experimental data for fluids used in this study was about 12%. This result is considered to be highly acceptable in
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 493 the critical region. In summary, the overall estimation procedure developed so far can be stated as follows: (1) calculate v from critical constants of a fluid; (2) calculate a, xo, and El from eqs 14,24, and 25, respectively; and (3) calculate Ahc from eqs 22 and 23.
Conclusions This study is considered to be the first attempt for the generalized prediction of anomalous thermal conductivity behavior of pure fluids in the critical point region. The power-law hypothesis and NBS scaled equation of state representing the thermodynamic behavior near the critical point were adopted to analyze the experimental thermal conductivity data for 15 fluids used in this work. The basic parameters, considerably affecting thermal conductivity enhancement in the critical region, were found to be thermal conductivity modulus (v), critical amplitude ( a ) , value of scaling variable x along the coexistence curve ( x o ) , and critical region parameter in the NBS equation of state (El). Three parameters, a , xo, and E l , could be well correlated with the critical compresibility factor. This predictive method developed in a generalized manner thus requires only the critical constants of pure fluids. However, the validity of this method remains to be tested further when additional experimental information regarding thermal conductivity enhancement in the critical region becomes available.
Acknowledgment This work was supported by the Korea Science and Engineering Foundation and the University Awards Program of Korea Institute of Technology.
Nomenclature A = damping constant C, = specific heat at constant pressure C, = specific heat at constant volume D = diffusion coefficient El, E 2 = constants, eq 17 KT = isothermal compressibility n = number density P = pressure, MPa P, = critical pressure, MPa T = absolute temperature, K T , = critical temperature, K x = scaling variable, eq 16 xo = value of x along the coexistence curve y = variable, eq 21b 2, = critical compressibility factor Greek Symbols a = constant, eq 13
fl = critical exponent of coexistence curve r = critical amplitude of isothermal compressibility y = critical exponent of isothermal compressibility 6 = critical exponent along the critical isotherm c = normalized temperature, T R - 1 { = friction coefficient 7 = shear viscosity K = Boltzmann constant h = critical amplitude of thermal conductivity along the critical isochore X = thermal conductivity, W/(m K) A, = normal thermal conductivity, W/(m K) AX, = critical thermal conductivity enhancement, W/ (m K) p = chemical potential v = thermal conductivity modulus, M lI2Tc/P2/2V,5/6 E = correlation length
p = density, kg/m3 pR = reduced density Ap* = normalized density, pR - 1 \k = critical exponent of thermal conductivity along the critical
isochore
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Effects of Surface-Active Agents on Drop Size in Liquid-Liquid Systems A. H. P. Skelland* and Elizabeth A. Slaymaker School of Chemical Engineering, The Georgia Institute of Technology, Atlanta, Georgia 30332
Extensive new experimental data were obtained on the effects of various surface-active agents (SAA) on the size of drops formed at a nozzle tip in liquid-liquid systems. Drop formation times were 0.5-4 s. Of the many relationships for predicting the size of such drops under SAA-free conditions, only a few may be described as even semitheoretical. Of these, the Scheele and Meister equation, representative of the so-called two-stage model, was selected to test the proposition that use of the diminished interfacial tension due to SAA would be sufficient to cause the equation to predict drop size in SAA-contaminated systems. (This simple approach fails with certain other droplet phenomena). When so applied, the equation predicted 772 drop volumes, measured in the presence of SAA, with a mean deviation of only 1.8%,thus running almost through the middle of the data points. Various minor trends are identified. Rational design of liquid-liquid extraction columns requires knowledge of drop size. Perforated plate columns, for example, feature drops of disperse phase forming at the perforations on each plate, followed by detachment and rise (or fall) of the drops through the continuous phase and their subsequent coalescence beneath the plate above (or below). Mass-transfer rates are dependent upon conditions on each side of the interface during drop formation, rise, and coalescence and upon the interfacial area available for transfer throughout. An almost bewildering variety of procedures and expressions have been presented in the literature for the prediction of drop size from nozzles in liquid-liquid systems. Confining attention to the nonjetting region, some of these studies are as follows, in chronological order. For drops formed very slowly, such that kinetic and drag forces can be neglected, Harkins and Brown (1919) developed an expression for predicting the so-called "static drop volume". For drops formed more rapidly, kinetic and drag forces become significant, and analyses or correlations under such conditions have been presented by Hayworth and Treybal (1950), Siemes (1956), Ueyama (1957), Null and Johnson (19581, Rao et al. (19661, Scheele and Meister (1968), Narayanan et al. (1970), Kumar (1971), Heertjes et al. (19711, de Chazal and Ryan (1971), Izard (19721, Grigar et al. (19721, Skelland and Raval (1972, for power-law non-Newtonian fluids), Kagan et al. (1973), Kumar and Hartland (1982, 1984), Skelland and Vasti (1985), and Dalingaros et al. (1986).
If the object of the present study were to produce yet another correlation for predicting drop size, one would probably first apply most of the above developments in turn to determine which provided the best fit to our new data, regardless of the mechanism (or lack of it) upon which a given relationship is based. (Indeed, a prepublication reviewer asked, in effect, for essentially this.) However, this particular study has a different goal in mind, namely, to discover whether surface active contamination, commonly encountered industrially, can be accommodated for purposes of drop size prediction merely by using the diminished interfacial tension a t equilibrium (a) due to the anonymous surfactant, in some relation previously established for contaminant-free systems. The need for such an investigation stems from the fact that the effects of surface-active agents (SAA) are not always confined to changing the interfacial tension. Additional effects are such that a correlation obtained for some given droplet phenomenon in surfactant-free systems may not apply to the same systems containing trace amounts of surfactant, even when the diminished u due to the surfactant is used in the expression. Thus, Mekasut et al. (1979) showed that this simple procedure was inadequate for predicting either droplet eccentricity or oscillation frequency in surfactant-contaminated liquid-liquid systems. In the same vein, Skelland et al. (1987) found that criteria for the onset of droplet oscillation do not always retain their values for uncontaminated systems when used with the diminished u due to added surfactants.
0888-5885190/2629-0494$02.X/O C 1990 American Chemical Society
