Fluid-Particle and lntraparticle Mass Transport Rates in Slurries Takehiko Furusawa' and J. M. Smith* University of California, Davis, Calif. 9661 6
Fluid-particle and intraparticle mass transport rates were measured at 25°C and 1 atm pressure for the adsorption of benzene in aqueous slurries of activated carbon. The effects of particle size (1 61-91 2 p ) and stirrer speed on the fluid-particle mass transfer coefficient, k,, were well correlated using a Reynolds number based upon the energy dissipation rate per unit mass of water in the slurry. These results agreed reasonably well with literature data obtained from dissolution experiments. The experimental results strongly indicated that intraparticle diffusion had a retarding effect on the adsorption rate. However, effective diffusivities for the liquid-filled pores were much larger than values predicted from the molecular diffusivity of benzene in water using a normal tortuosity factor. Assuming that surface migration explains this discrepancy, a surface diffusivity of about 0.6 X 1 0-8 cm2/sec was obtained.
A g i t a t e d vessels containing slurries of adsorbent, or catalyst, particles are in growing use as adsorbers for pollutant removal and as reactors. The capacity and performance of such vessels is especially dependent upon mass transport rates. Hence, measurement and correlation of rate coefficients is important for the several transport steps involved in the overall adsorption or reaction process. Mass transfer from gas bubble to bulk liquid, which may be a significant resistance when one of the feed streams is a gas, has been studied extensively and a recent review with additional data is available (Misic and Smith, 1971). Less is known about transport coefficients between bulk fluid and particle surface. I n fact, there appears to be no published information based upon adsorption or reaction systems except that for mass transfer to slurries of particles of ion-exchange resin (Harriot, 1962). The other data were obtained from rates of dissolution of solid particles. One of the objectives of the present study was to determine fluid-particle rate coefficients for an adsorption system and to compare the results with those obtained from dissolution data (Brian and Hales, 1969; Harriot, 1962) where particle shrinkage and transpiration may have an effect (Brian, et al., 1969). Aqueous solutions of benzene with activated carbon particles in agitated, baffled vessels were employed. This system also permitted studying intraparticle diffusion resistances. The retardation of rates of adsorption or of reaction, due to intraparticle transport, using slurry-size particles (100-1000 p ) , was the second objective of the investigation. A transient, batch method was employed in which the water initially contained dissolved adsorbate, benzene. At time zero, activated carbon particles were added, and the course of adsorption followed by analyzing, for benzene content, small samples of water withdrawn from the vessel. This method eliminated the usually significant mass transfer step from gas bubble to liquid. Then fluid-particle and intraparticle transport coefficients could be more accurately, and conveniently, extracted from the data. It should be noted that the fluid-particle coefficients, determined here in the On leave from University of Tokyo.
absence of gas bubbles, might be affected by a dispersed gas phase. Experimental Section
Data were taken in Pyrex and stainless steel vessels whose dimensions are given in Figure 1. The ratios of impeller dimensions to tank diameter were the same for both adsorbers. Each vessel contained eight fixed baffles spaced evenly around the circumference. The ratio of baffle dimensions to vessel diameter was also the same in each adsorber. Flat-bottom vessels were used. Different levels of agitation were obtained by driving the impellers with variable speed motors. It has been shown (Nagata, et al., 1960) that for a vessel with eight baffles and of the geometry given in Figure 1 the power number, related to the power consumption by eq 1, is 9.5. That is
This value of the power number can be used t o calculate the energy dissipation rate per unit mass of water. The effect of the energy consumption rate on fluid-particle mass transfer was studied using this relation and by varying the impeller speed, N . The water level was the same as the vessel diameter in runs with both adsorbers. To reduce benzene loss, air-tight covers, with a system for sample withdrawal and an opening for feeding carbon particles, were used. The water surface was free, but the space between the liquid surface and the cover was made as small as possible. Preliminary tests showed that the benzene loss was 1.5% or less with the Pyrex adsorber and 0.9% or less with the stainless steel adsorber. I n preparation for a run, a measured amount of benzene was added to the water in the vessel, and the concentration COwas determined by analysis. The run was started by adding a known mass of carbon particles. Filtrasorb 200, granular activated carbon from the Calgon Corporation, was used. The material as supplied was crushed and sieved to prepare particles of average diameters from 161 to 912 p . The average Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
197
8 Blade Impeller
1
fi
1.0 0.9
0.) 0.7 0.6 0.1 0
0.4
U
u
0.1 0
12
6
18
21
30
TINE (YlN.1 VESSEL
H
D
Z
B
b
1 Stainless I 7 1/2" I 7 1/4" I D I 3/40 D I 1/10 D I Figure 1 . Adsorption vessels
Table 1.
Scope of Measurements
1. Average particle diameter, dp: 161-912 fi 2. Volume of liquid in adsorber, V : Pyrex 800 cma Stainless steel 4700 cms 3. Slurry concentration, m,: 0.666-1.92 X 10-3 g/cma 4. External surface area, 8,: 0.337-0.0507 cm-1 5. Xgitation speed: Pyrex adsorber 300-980 rpm 165-780 rpin Stainless steel adsorber 6. Particle size distribution and average particle diameter Range of sieve openings, p
dp,P
147-175 175-2 10 210-246 246-295 295-351 35 1-4 17 351-495 417-495 495-589 689-701 701-833 833-991
161 192 228 270 323 384 423 456 542 645 767 912
Figure 2. Typical concentration decay curves (m, = 0.85 X g/cm3; N = 500 rpm; Co = 1.407 X 10-6 mole/ cm3). Dotted lines are curves through experimental results; solid lines were predicted from single-resistance model with a nonlinear isotherm
in adsorption data could be detected using the two procedures. For 423-p particles, a slight difference was observed, but this was less than the error in analyzing samples for benzene content. It was concluded, therefore, that the pores were filled with water during adsorption. Samples of water, normally 40 pl, were withdrawn by inserting microsyringes in the system. The samples were then introduced into a gas chromatograph equipped with a flameionization detector and a 4 ft long column of Porapak Q (80100 mesh particles). Reproducibility runs for small particle sizes showed deviations of less than 5%. For the largest particle sizes the deviations were larger, perhaps due to small deviations from a uniform particle concentration in the adsorber. Most runs lasted about 30 min except for the isotherm measurements when samples were taken after 3 hr. Typical data are shown by the experimental points in Figure 2, which illustrates runs for different particle sizes but for the same power consumption (stirrer speed). Equilibrium Results
Runs to steady-state concentrations of benzene were carried out for different initial concentrations CO.The concentration n of benzene adsorbed on the carbon was evaluated from the equation
co - c** diameter was taken as the arithmetic average of the two sieve openings defining the sample used (see Table I). The manufacturer supplied the following properties of the particles: solid (true) density, pt = 2.04 g/cm3; particle density, pp = 0.956 g/cma; surface area = 800-900 m2/g. These data indicate a particle porosity, cP = 0.53. Before use the particles were dried in air a t llO°C until constant weight was attained (more than 12 hr). I n some runs the particles were evacuated to a pressure of less than 1 mm and a small quantity of water was added under vacuum conditions. This mixture was then introduced into the adsorber. In duplicate runs the air-dried particles were added directly to the water in the adsorber. For 161-p particles no difference 198 Ind. Eng. Chern. Fundam., Vol. 12, No. 2, 1973
m, The isotherm so obtained a t 25°C is shown in Figure 3. The Langmuir isotherm fit the data well as shown by the plot of C / n vs. l/C in Figure 4. The line through the experimental points corresponds to the constants given in eq 3
n = - - KC 1
+ aC
12.5 x
-
1
1 0 3 ~
+ 4 . 7 1 X 10%'
(3)
The surface coverage ordinate in Figure 3 was obtained by using a surface area of 850 m2/g and by taking the surface occupied by one molecule of adsorbed benzene to be 33.7 X 10-16 cm2.
2A
0.5
2.0
0.4 0
I
P
1.6
. . B
0 0 Y
I
I
0.3
I
z
%
.
1.2
I
0.2
c
%
0
H 5
0.8
0.1
0.4
0
0.2
0
c n 0
0.2
0.4
c
0.6
0.8
0.8
0.6
0.1
1.0
106 , g mole/ 0 3
I
Figure 4. Langmuir isotherm plot
1.0
x 106,lmolslcm3
Figure 3. Equilibrium adsorption isotherm for benzene on Filtrasorb 200
1 .o
1.0
0.9 0.7
0.9 0.8 0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.8
2
I n the analysis of the data it is assumed that the adsorption of benzene a t a site within the carbon particle is rapid with respect to mass transfer. It is also assumed that n and C are related by the Langmuir isotherm, eq 3, which postulates reversibility. While the rapidity of the adsorption does not appear questionable, it was desirable to be certain that the adsorption is reversible. To check reversibility, the water was separated from the carbon particles after an equilibrium adsorption run. Then the particles were added to the adsorber containing distilled water. The desorption process was followed by analysis of water samples taken a t subsequent times. The adsorption and desorption results, shown in Figure 5 , indicate relatively rapid desorption. Note that the driving force for desorption is much less than that for adsorption. Scope of Data
A11 data were taken a t 25OC and 1 atm total pressure. The initial concentration used for the rate measurements was 1.407 x 10” mole/cm3, a value less than the solubility of benzene in water. The chief variables were particle size and impeller speed, as shown in Table I. Method of Analysis
The experimental data are interpreted by supposing a three-step model: (1) mass transfer of benzene from bulk liquid to particle surface; ( 2 ) intraparticle diffusion; and (3) adsorption a t an interior site. It is assumed that step 3 is rapid with respect to the first two steps. I n a well-agitated slurry adsorber mixing in the liquid phase is rapid. Hence the concentration C of adsorbate and concentration ?n, of carbon particles in the liquid are nearly uniform throughout the vessel. Then the change in C with respect to time is related to the fluid-particle mass transfer coefficient by the equations
dc = - k,S,(C dt
- C,)
(4)
2 LT P
4
. 0 ”
,”
-
n 0
0.3
0.3
0.2
0.2
2
0
0.1
0,l
25
0
100
75
50
TIME , (MIN.1
Figure 5.Adsorption and desorption runs (d, = 323 p ; N = 430 rpm, m, = 0.8443 X mo1e/cm3) COfor adsorption run = 1.407 X l o + mole/cm3; steady-state concentration at end of desorption run, ( C B R ) d = 0.201 X 1 O + moIe/cm3
The differential mass balance of benzene within the particles is, assuming a constant total, effective intraparticle diffusivity Dt *
D , ( %dr2
an, +2Kr) r 3r - p p - at
dC*
-
-
‘P ’
at
(6)
Boundary and initial conditions are
C,
(t
0
=
=
0) for 0
5 r 5 d,/2
(9)
Since equilibrium is assumed for step 3, n, and C, are related by the isotherm, eq 3, or 5
r =
at
2( ac,
1
KCr
) at
+ UC,
g
r
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1 9 7 3
(10) 199
IC
I
8
8-
I
I
I
600
800
I
0 Pynr Adsorbar, dp = 323 Microns S.S. Adrorbr, dp = 230 Mlcronr
-
A 6
-
r
I
. Y
H, H
2
1
1w
2W
400
600
800
I000
100
200
400
1000
IMPELLER SPEED (RPM)
PARTICLE SIZE, d p , MICRONS
Figure 6. Effect of particle diameter on mass transfer coefficient (impeller speed = 500 rpm, Pyrex adsorber)
Equations 4-10 can be solved numerically to relate the observed C us. t curves (cf. Figure 10) to rate coefficients, k , and Dt. This procedure was used to evaluate Dt and is discussed later. X simpler procedure can be employed to obtain k,. Since C, approaches zero and C approaches Co, as t 0, eq 4 shows that
-
Figure 7. Effect of impeller speed on mass transfer coefficient
can be obtained from slopes of the curves obtained when the data are plotted as C/CO us. t , according to eq 11. However, the uncertainty in evaluating the slope of a curve a t t = 0 can be avoided by using eq 14. This expression shows that a plot of { [(C/Co) - (1/1 m , K ) ] ] us. t on semilogarithmic paper will give a straight line a t t = 0. Equation 14 is applicable a t t = 0 since the influence of intraparticle diffusion does not yet affect the results and since the isotherm becomes linear as time approaches zero (see Figure 3). From the initial slope, - [(l m , K ) / ( m , K ) ] k,S,, of the straight line, k , can be evaluated. The outer surface area of the particles was obtained from m, by assuming spherical particles of diameter dp;that is
+
+
Hence, the slope a t t = 0 of a plot of C/CO vs. t is equal to -k,S,. From such slopes k , can be extracted. When intraparticle-diffusion resistance is negligible (the single-resistance case) and when the isotherm is linear (n = KC), a n analytical solution for C us. t is possible. Since C, = C, and n, is also uniform through the particle, eq 6 and 7 may be replaced by dn mB- = k,S,(C dt
- C,)
Also eq 10 becomes dn dt
dCs dt
- -- K -
(13)
Equations 4, 12, and 13, with initial conditions (5) and n = 0 = 0, can be solved analytically to give
at t
C Co 1
*
1 + m,K + 1 + m,K
exp[
-
___ + msK k,S,t]
m,K
(14)
Equation 14 is not helpful for deciding if intraparticle diffusion affects the adsorption in our case, since the isotherm is not linear. The correct method is to compare the general solution (solution of eq 4 to 10) for C = f(t) with the solution of the single-resistance case using the nonlinear isotherm. The latter requires a numerical solution of eq 4, 10, and 12 with the appropriate initial conditions. Such comparisons, which show that intraparticle mass transport effects are significant, are discussed later. Fluid-Particle Mass Transfer
Concentration vs. time curves were first measured for various particle sizes in the Pyrex adsorber. I n principle k, 200 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
s, =
6ms d,Pt(l -
(15) ep)
The range of surface areas is shown in Table I. The results are plotted as ( S h ) / ( s ~ ) ' / us. ~ d, in Figure 6. The Sherwood and Schmidt numbers were evaluated taking the molecular diffusivity of benzene in water as 1.13 X cm*/sec (Calderbank, 1966). Xass transfer coefficients were evaluated, in a similar way, from data a t various agitation speeds taken in both reactors. The results, plotted in Figure 7, show a n effect of energy dissipation on mass transfer rate. The basis for correlating k , as a function of particle size and agitation speed is Kolmogoroff's theory (Levins and Glastonbury, 1972, Shinnar and Church, 1960) of isotropic turbulance in a n agitated slurry. This theory states that the Reynolds number should be defined as
and (17) where the eddy size, 7, is a function only of the energy dissipation rate per unit mass of particle-free slurry and the kinematic viscosity Y. Thus
180
10
5;
.
E I
m
I 16
( R v~ --I ~(01' I
Id 2 11
107
1pi
'
"3
Figure 8. Correlation of mass transfer coefficients
The Reynolds numbers are based upon the particle diameter and the mean square relative velocity between two points a t a distance d, apart. Such velocity can be derived by Kolmogoroff's theory. The energy dissipation rate is related to the power number by the expression
Values of e were obtained from eq 19 using N , = 9.5 for our adsorbers. Then 7 was calculated from eq 18. The values of 7 were 34 fi at the minimum impeller speed and 23 fi a t 500 rpm for the Pyrex adsorber and 39 fi at the minimum rpm in the stainless steel vessel. Since these eddy sizes are much less than d,, the Reynolds number was evaluated by eq 17. All our data (for various N and d,) are plotted in Figure 8 as Sh/(Sc)'I3 us. (Re)13. Brian and Hales (1969) proposed this method of plotting earlier, and correlations suggested by these authors for their own data, and the measurements of Harriott (1962), are shown as lines in Figure 8. The results of our measurements lie generally between the adsorption (ion exchange) and dissolution correlations based upon Harriott's data and verify the use of E as a correlating parameter. Recently (Levins and Glastonbury, 1972) weaknesses in the application of Kolmogoroff's theory to slurries have been pointed out. Improvements were noted when the ratio of impeller to vessel diameters, in addition to E , is used for scaleup. Both vessels in our work had the same value of this ratio, so that the worth of this ratio as a correlating parameter could not be tested. Also, our data did not provide justification for using (Scj' '. This was employed in Figure 8 solely for the purpose of comparing our data with those of others. Figure 9 shows our data and those of other investigators plotted us. the Reynolds number. This plot verifies that the Sherwood number is approximately proportional to (R~)I','' and indicates the range of Schmidt numbers covered by the several studies.
(1
Id
1
Figure 9. Correlation of mass transfer coefficient with Reynolds number
therm. The value for Dt which gave predicted decay curves in agreement with the data was about ten times greater than the molecular diffusivity of benzene in water. Yet for diffusion within the volume of the pores the effective diffusivity should be less than the molecular value because of tortuosity and void fractions less than unity. A possible explanation for the high Dt value is that surface migration on the pore walls contributed importantly to intraparticle mass transport. This possibility is strengthened by the work of Kawazoe, et al. (1965), who studied the recovery of benzene in a packed column of activated carbon particles and found that surface diffusion was a significant contribution. Also surface migration has been observed (Schneider and Smith, 1968) in the adsorption and transport of light hydrocarbons (ethane, propane, and n-butane) on silica gel, and these authors summarize what is known about surface diffusion. To evaluate intraparticle diffusion quantitatively, eq G was rewritten in terms of the separate contributions of surface (Os)and pore volume ( D e )
De($+:$)+DS('%L%
2 dC, dr2
+
Ep
sp
=
9)
dC,
-
dt
, + acr -~ dt
sp
E,
(20)
I n this formulation D, is defined in terms of the flux S,,moles per unit time per unit cross-sectional area of the particle. T h a t is
Intraparticle Diffusion
The solid curves in Figure 2 were predicted assuming a single-resistance model (that is, solution of eq 4, 5, 10, and 12) using the measured k , values and eq 3 for the nonlinear isotherm. The results, which are typical for all the runs, show increasing deviation from the data as the run proceeds. The slower decay curves suggest that a n additional resistance affects the rate of adsorption. Hence, the two-resistance model (eq 4-10) was solved numerically with the nonlinear iso-
!There C,,,, is expressed as the moles adsorbed per unit of pore volume. Since adsorption a t a site on the carbon has been supposed to be fast iyith respect to mass transfer, C, and Cr,spare related by the adsorptioii isotherm. If the nonlinear isotherm is used, the resulting relatioriship is complex. Then \\hen eq 20 is expressed in terms of Cr alone, it is very difficult to extract the surface diffusivity. This problem can be avoided by noting Ind. Eng. Chem. Fundam., Vol. 1 2 , No. 2, 1973
201
0
10
5
dure gives an approximate value of D, equal to 0.6 X 10-8 cm2/sec. Other surface diffusivities for liquid-filled pores were not found in the literature. Our result is a n order of magnitude or more lower than D , values reported for gas-filled Pores (Schneider and Smith, 1968). However, the density of molecules near the surface for the liquid-filled pore is much greater than for gas-filled pores. Thus the large concentration of obstructing water molecules could retard surface migration of the adsorbed benzene. It should be emphasized that the quantitative interpretation of surface diffusion given here is approximate. Constant D, has been assumed in writing eq 20, yet the surface coverage varies with radius. Also the assumption of a linear isotherm, even for the initial period of a run, may not be valid throughout the particle. With intraparticle diffusion resistance, n, a t large radii could exceed the linear range while n, near the center of the particle mas still very low.
15
TIME (MI#.)
Figure 10. Comparison between experimental results and predictions based upon two-resistance equations with D, = 0.6 X 10-8 cmz/sec (N = 500 rpm; Co = 1.407 X 1 0 -6 moIe/cma)
from Figure 3 that the isotherm is approximately linear up mole/g of carbon. This to n values of the order to 0.8 X adsorbed concentration is not reached until several minutes of each run have elapsed. Hence, if the comparison with the data is made during this initial period, the linear isotherm may be used. Then, from the definitions of n,, C,, and C,,,p
Conclusions
Fluid-particle mass transport coefficients measured in slurry adsorbers were found to agree with those obtained by dissolution of nonporous particles in agitated vessels. However, the most important conclusion of this study is the evidence for significant intraparticle diffusion resistance in slurries with particles as small as 161 p , While no other information n as found for diffusivities in liquid-filled pores, the experimental data indicate values much higher than expected. If surface diffusion is the explanation, the required surface diffusivity is about 0.6 X 10-8 cm2/sec. Nomenclature
a
c c,
and
c,,,
= PpK
c,
(23)
CP
Substituting these expressions into eq 20 to eliminate n, and c r , s p gives
where Dt = De f 9 K D ,
(25)
EP
Equation 24 is the same as eq 6 when the latter is modified by substituting the linear isotherm expression for &/at. Equations 4, 5, 24, and 7-9 were solved numerically to give predicted decay curves. Then, using the experimental results fork,, the value of Dt was determined which gave curves which agreed best with the initial part of the experimental decay curves. Figure 10 shows typical results and indicates that Dt is about 1.5 X 10-4 cm2/sec, or 10-fold larger than the niolecular diffusivity. This result iz, about two orders of magnitude greater than the effective diffusivity, De, in the pore volume, as estimated from the equation
using a reasonable value [a = 4 (Satterfield, 197O)I for the tortuosity factor. Hence, the pore volume contribution to Dt can be neglected in evaluating D, from eq 25. This proce202
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
= constant in Langmuir isotherm, cm3/mole
= concentration of
benzene in bulk liquid, moles/cma concentration in liquid at outer surface of particle Cr concentration in liquid-filled pore a t radius r C r , s p = concentration of adsorbed benzene a t radius r, moles/(cma of pore volume) co = concentration in bulk liquid a t start of adsorption run c,, = Concentration in bulk liquid a t steady state (equilibrium) dI = impeller diameter, cm a, = average particle diameter, cm D = molecular diffusivity of benzene in water, cm2/sec De = effective diffusivity of benzene in liquid-filled pores of carbon particles] cm2/sec Dt = total, effective diffusivity in particles, cm*/sec D, = surface diffusivity, defined by eq 21, cm2/sec k, = mass transfer coefficient between bulk liquid and outer surface of particles, cm/sec K = adsorption equilibrium constant, cm3/g of carbon m, = mass of carbon particles per unit volume of particlefree slurry, g/cma n, = concentration of adsorbed benzene at radius r of particle, moles/g of carbon N = rotation speed of impeller, rps N, = povier number, defined by eq 1 Ai, = surface diffusion flux, moles/(sec) (cross-sectional area of particle) P = power consumption of the slurry due t o agitation, ergs/sec - radial coordinate in carbon particle (assumed r spherical), cm (Re)I = Reynolds number for q > d,, defined by eq 16 sc = Schmidt number, u/D Sh = Sherwood number, k,d,/D s, = outer surface of carbon particles per unit volume of particle-free slurry, c m - 1 t = time, sec volume of particle-free liquid in adsorber, cm3 TB = mass of particle-free liquid in adsorber, g
v
= =
=
GREEKLETTERS c
= energy dissipation rate in slurry per unit mass of
ip
= = = = = =
V
q p~ pp Pt
particle-free liquid, ergs/(g sec) porosity of carbon particles tortuosity factor, defined by eq 26 kinematic viscosity, cm*/sec eddy size, defined by eq 18 density of particle-free liquid, g/cma density of carbon particles, g/cma = density of solid Phase, true density, of carbon Particles, g/cma
Harriot, P., A.1.Ch.E. J . 8, 93 (1962). Kawazoe, K., Fukuda, Y., Kagaku Kogaku (Abr. Ed. Engl.) 3, 250 (1965). Levins, D. M., Glastonbury, J. R., Chem. Eng. Sci. 27,537 (1972). Misic, D. M., Smith, J. M., IND.ENG.CHEM., FUNDAM. 10, 380 (1971). Nagata, S., Yamamoto, K., Hashimoto, K., Naruse, Y., Kagaku Kogaku 24, 99 (1960). Satterfield, C. N., “Mass Transfer in Heterogeneous Catalysis,” M.I.T. Press, Cambridge, Mass., 1970. Schneider, P., Smith, J. M., A.I.Ch.E. J . 14, 886 (1968). Shinnar, R., Church, J. M., Ind. Eng. Chem. 52, 253 (1960). RECEIVED for review May 12, 1972 ACCEPTEDJanuary 12, 1973
literature Cited
Brian, P. L. T., Hales, H. B., A.Z.Ch.E. J . 15, 419 (1969). Brian, P. L. T.. Hales, H. B.. Sherwood. T. K.. A.Z.Ch.E. J . 15. 727’ (1969). ’ Calderbank, P. H., “Mixing, Theory and Practice,” Vol. 1, Chapter 6, Academic Press, New York, N. Y., 1966.
Acknowledgment is made to the Donors of The Petroleum Research Fun>, administered by the American Chemical Society, for the support of this work. Also, the gift of Filtrasorb by the Calgon Corporation is acknowledged.
Thermodynamics of Mixtures Containing Polar Liquids Jack Winnick University of Missouri, Columbia, M o . 66601
The partition function for polar liquids i s extended to mixtures using Scott’s two-fluid theory. The resulting expressions for the excess functions are tested against data for ten polar-polar and ten polar-nonpolar systems. One empirical mixture parameter i s required for each system. Its value i s set using an excess enthalpy datum. Good representation of the entire excess enthalpy and volume curves i s found for most systems, including those which are “cubic” in appearance. Excess entropy predictions are nearly always too high due to the degree of disorder assumed in the model.
R e c e n t l y a partition function was developed from a simple model of a pure nonpolar liquid (Winnick and Prausnitz, 1971a). Using a three-parameter theory of corresponding states, the partition function was generalized into a form which accurately correlated the volume and energy properties of nonpolar liquids ranging in complexity from argon t o high molecular weight paraffins and a giant siloxane. Two-fluid theory (Scott, 1956) was then utilized t o apply the partition function t o the description of nonpolar liquid mixtures (Winnick and Prausnitz, 1971b). Excess free energies, enthalpies, and volumes were calculated for a large number of mixtures of widely varying differences in component size and shape. Mixing rules required one empirical parameter, kI2,t h e deviation from the geometric-mean interaction energy. More recently, the pure-liquid theory was modified t o enable i t to describe polar liquids of simple structure (Winnick, 1972). The modification consisted of adding a reduced dipolar energy to the potential energy of each molecule in the liquid. For the eight species considered, no additional empiricism over t h a t required for nonpolar species was needed. It was found that the quantitative contribution of t h e polar energy of each was directly calculable and t h a t this contribution compared well with the theoretical value for cells formed of spherical molecules with point dipoles. The success of t h e above approach has prompted the present attempt a t describing excess properties of liquid mixtures containing polar components.
Theory
The configurational partition function for a pure polar liquid is
where
J ( p i ) = exp( -28.025exp(-4.84/Pf)
- 2.320/pi
+{}
(2)
and
Et
=
- [ L r i * / 8 i + mipi4/(Vi*)’RT]
(3)
For all pure components, mi = 1.0 x lo7 with units compatible with p , t h e standard dipole moment, in debyes, V* in cm3/mole, and RT in cal/mole. Here 8 is the molar volume reduced by a n empirically determined characteristic volume, V*. Similarly, p is the temperature reduced by a characteristic temperature, T*. The parameter, c, reflects the external degrees of freedom per molecule. The integration constant, {, is found from one vapor pressure datum but is not required in the treatment of mixtures. The characteristic energy, C*, is directly related t o the characteristic temperature Gi* = 9.008ciR2’i* Ind. Eng. Chern. Fundam., Vol. 12, No. 2, 1973
(4) 203
