1758
I n d . E n g . Chem. Res. 1987,26, 1758-1766
A Corresponding States Treatment for the Viscosity of Polar Fluids Ming-Jing Hwangt and Wallace B. Whiting* D e p a r t m e n t of Chemical Engineering, W e s t Virginia University, Morgantown, W e s t Virginia 26506-6101
T h e extended corresponding states viscosity model of Ely and Hanley has been modified to extend it t o polar fluids. In addition to the Pitzer’s acentric factor for nonpolar fluids, used by Ely and Hanley, an association parameter for hydrogen-bonding compounds and a viscosity acentric factor for polar compounds are employed. Extensive comparisons with experimental data for more than 30 substances, including highly branched alkanes, naphthenes, and polar and hydrogen-bonding compounds, show a n average absolute deviation between experimental values and calculated values of less than 6%. Comparisons with other recommended methods based on the same conditions show that this model is better in both accuracy and applicability; it applies over the entire fluid density range from dilute gas to dense liquid. The prediction of viscosities of gases and liquids has been widely studied. However, most of these studies deal only with asymptotic limits, Le., dilute gas or dense liquid, and usually they are limited to certain types of fluids and a narrow range of experimental conditions. No method has yet been devised for the prediction of viscosities of both nonpolar and polar fluids over the entire range of P-V-T states. Starling et al. (1984) presented a multiparameter correlation using Pitzer’s acentric factor, the dipole moment, and an association parameter for the prediction of viscosity and thermal conductivity of a wide class of fluids, including polar and hydrogen-bonding compounds. However, the technique Starling et al. used is inherently limited to dilute gases because his model is an application of kinetic gas theory, which assumes that molecules undergo two-body collisions and therefore cannot account for the many-body collisions in a dense fluid. While gas viscosities can be estimated based on sound theory, there is no comparable basis for estimating liquid viscosities, although results from nonequilibrium molecular dynamics are making inroads in this area. In general, empirical correlations and equations from simplified theories have been the principal methods used for practical calculations. More often, it is preferable to use experimental data if they are available. A review of these empirical correlations is given by Reid et al. (1977). On the estimation of liquid viscosities, these authors concluded that “There is a problem in selecting a technique to join low- and high-temperature estimated viscosities.” The difficulties faced in the theoretical treatment of viscosity or other transport properties (thermal conductivity and diffusivity) arise from the mathematical complexity in handling nonequilibrium distribution functions (now becoming available through nonequilibrium molecular dynamics simulations) and the uncertainty of experimental data. Although, on a macroscopic scale, both viscosity and equilibrium properties reflect the effect of interaction, shape, and orientation of molecules, viscosity differs from the equilibrium properties in a very important respect: viscosity characterizes the resistance to changes in shape, orientation, and internal energy during transport processes in polar fluids. Any method for the prediction of viscosity of polar fluids must take this effect into account to give accurate predictions.
* Author to whom correspondence should be addressed. + Present address: Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, P.4 15261.
08S8-58S5/87/2626-1758$01.50/0
The most advanced approach for the prediction of transport properties is a corresponding states method developed mainly by Hanley and his co-workers (e.g., Hanley, 1976, 1977; Evans and Hanley, 1979; Ely and Hanley, 1981a-c). This approach works very well for nonpolar fluids over the entire fluid density range but, again, is limited to straight nonpolar hydrocarbons and a few other simple compounds. The failure of Ely and Hanley’s model (Ely and Hanley, 1981a) for highly branched hydrocarbons and polar fluids is due to the inability of their model to represent the significant changes in internal energy, shape, and orientation during transport processes in these compounds. This is because the corresponding states technique assumes a scaling of the distribution functions of each species. In the case of a corresponding states treatment for transport properties, this scaling must apply to the nonequilibrium distribution functions. For more complex molecules, this scaling begins to break down. We modify Ely and Hanley’s model by employing a viscosity acentric factor and an association parameter to characterize the effects of molecular shape, orientation, and intermolecular forces. The results are excellent. The overall average absolute deviation found in the correlation of data is about 6% for more than 30 substances, including highly branched alkanes and naphthenes and for polar and hydrogen-bonding compounds, over the entire range of fluid states from the dilute gas to the dense liquid. Apart from the empirically derived association parameter for hydrogen-bonding compounds and the viscosity acentric factor for polar fluids and highly branched hydrocarbons, only critical properties (Pc,T c ,V c )and molecular weight (M) are needed. (Pitzer’s original acentric factor is used for simple nonpolar fluids.) A set of mixing rules is needed to extend this work to mixtures. Conventionally, a variation of the van der Waals mixing rules is used. However, to account for the nonrandomness in asymmetric mixtures, we are now studying the use of local-composition mixing rules, particularly the recent density-dependent local-composition model (Whiting and Prausnitz, 1982). The advantage of using this model rather than other local-composition models is that this one corrects for nonrandomness at all fluid densities and satisfies the low-density boundary condition of randomness. which other models do not. Corresponding States Treatments for Nonpolar Fluids An approach has been developed, based on the extended 8 198i American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1759 corresponding states principle and the conformal-solution concept, for the prediction of thermodynamic and transport properties of fluids. In this approach, it is assumed that the configurational properties of a pure fluid or mixture can be expressed in terms of properties of another pure fluid, usually termed the “reference” fluid, a t the appropriately corresponding pressure and temperature (or density and temperature). The appropriate hypothetical states are determined by solving conformal-solution equations. A correlation is then developed so that the reference temperature and reference density may be estimated from readily available data, such as critical constants. Such a correlation leads to a so-called corresponding states treatment (Rowlinson and Watson, 1969). An extended corresponding states treatment improves the correlations by introducing state-dependent shape factors (Leach et al., 1968). Recently, Chen et al. (1982) presented a new conformal-solution theory to predict equilibrium properties of polar and nonpolar mixtures; Massih and Mansoori (1983) derived the statistical mechanical basis for shape factors of polar fluids. Over the last decade, the extended corresponding states principle has been applied, mainly by Leland, Rowlinson, and their successors, to predict the thermodynamic properties of pure fluids and their mixtures. The theoretical foundation of the approach is well defined for equilibrium properties but is less understood for nonequilibrium properties, although the underlying analogy is plausible. The development of the approach for the prediction of transport properties of fluids and fluid mixtures was mainly by Hanley and his co-workers (e.g., Hanley, 1976, 1977; Evans and Hanley, 1979; Ely and Hanley, 1981a-c, 1983) and by Gubbins and his co-workers (e.g., Mo and Gubbins, 1974, 1976; Murad and Gubbins, 1977, 1981; Murad, 1981). Hanley and his co-workers retain the general form of the analytical expressions for shape factors given by Leach et al. (1968). These factors are obtained by simultaneous solution of two thermodynamic conformal-solution equations for compressibility and reduced free energy. Hanley’s method gives remarkable results-typically less than 10% uncertainty-for viscosities and thermal conductivities of nonpolar hydrocarbons, but it fails for polar species. Since shape factors may have different values if different properties are used in the conformal-solution equations (Murad and Gubbins, 1977), Gubbins and his co-workers replaced the reduced free energy with the transport properties (viscosity, thermal conductivity, or self-diffusivity). Furthermore, Christensen and Fredenslund (1980) used two transport properties (viscosity and thermal conductivity) in place of those used in the usual treatment for thermodynamic properties. However, these correlations vary from one type of fluid to another and are suitable only for nonpolar substances.
Corresponding States for Polar Fluids The conformal-solution viscosity model (Ely and Hanley, 1981a) assumes that the viscosity of a fluid can be evaluated via the corresponding states argument where Allhs is an Enskog correction to mixture viscosity for size and mass difference effects and F, is a dimensional factor defined as
erence fluid, and M is the molecular weight. (In TRAPP, M a and qo are modified to take into partial account that the equilibrium shape factors may differ from those for nonequilibrium states. In this work, we have used these modifications.) For the special case of two-parameter corresponding states, the ratios of equivalent to actual temperatures and for equivalent to actual densities are equated to the ratios of well-known critical constants: m
m r
(3) Po = -PoC pa
(4)
PaC
The range of applicability of corresponding states can be broadened considerably via a perturbation scheme, e.g., an extended corresponding states model which involves “shape factors” (Leach et al., 1968). Here the two-parameter corresponding states formalism is maintained, except that eq 3 and 4 become
TO _
Ta
TOC TaCfl,,0[Ta*, v,*,%I
(5)
where 0 and 4 are the so-called shape factors (Leach et al., 1968), which are functions of Pitzer’s acentric factor, w , of the reduced temperature, T,*, and of the reduced volume, V,*. In general, the shape factors can be determined exactly for any pure fluid with respect to a reference fluid by simultaneous solution of the conformal-solution equations (Rowlinson and Watson, 1969) ZJT,, Pa) = ZO(T0, Po) (7)
Aao(Ta, p a ) = AoO(To, PO) (8) where Z is the compressibility factor and Ao denotes the molar reduced residual Helmholtz free energy defined as A(T,P)- Aid(T,p) (9) RT where the superscript “id” denotes a fluid that obeys the ideal gas law. Leach (1967) solved eq 7 and 8 and correlated the shape factors for pure normal paraffins, C1-C5, using methane as the reference and using experimental compressibility and vapor-pressure data. Their results are generalized as fln,O(Ta*, Va*,0,) = 1 + (w, - wo)F(T,*, Va*) (10) and A0 =
where F(T,*, V,*) = a l + bl In Ta++ (cl
+ dl/T,*)(V,+ - 0.5)
(12)
and G(T,*, V,*) = az(Va++ b2) + c2(Vu++ d q ) In T,+ (13) where T,’ = min ( 2 , max (Tu*,0.05)( (14) and
where a refers to the fluid of interest, 0 refers to the ref-
V,’
= min ( 2 , max (V,*, 0.05))
(15)
1760 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
-0-
8
- A -
4)
- b -
--.-
" 14 6
%
1B
Q:
1.6
-
1.4
-
0 1.2
-
t U
+
I
I
r
v Y U
1.0
-
50.0
-
-0 -0
W
n 4
Ob
t
Od
1 1 a5
I
I
I
I
06 0.7 a8 0.9 REDUCED TEMPERATURE
I
I
111
- 0 -
e
- A -
0
- b -
I
I
a5
06
1
I
I
0.7 08 0.9 R E D U C E D TEMPERATURE
I
in
Figure 3. Shape factors of neopentane (saturated liquid).
-0-
20-
-a1.8
-
- b -
4)
%
QI
''I
16
1.4
1.4
U
I
1.0
Figure 1. Shape factors of ethylene (saturated liquid).
1B
t
0 12
Q5
I
I
1
I
I
C6
07
08
09
10
-3
I
TEMPERATURE
05
Q6
0.7
REDUCED
CB 0.9 TEMPERATURE
1.0
F i g u r e 2. Shape factors of n-pentane (saturated liquid).
Figure 4. Shape factors of dichlorodifluoromethane (saturated liquid).
The values of the constants for eq 12 and 13 are given elsewhere (Ely and Hanley, 1981a). This correlation has been successfully applied for the prediction of thermodynamic properties (e.g., Leach et al., 1968; Teja, 1975) and transport properties (Murad, 1981; Ely and Hanley, 1981a, 1983). However, as pointed out earlier, even though eq 8 ensures that the free-energy surface will be the same in reduced form for the two substances, it does not ensure that the transport property surfaces will be the same. One can observe this by simply changing the conformal equation of the thermodynamic property (eq 8) to a transport property, e.g., by taking viscosity as an example of transport properties, solving eq 7 and 1 instead of eq 7 and 8 to obtain the viscosity shape factors, 8, and $,, and comparing these with the thermo-
dynamic shape factors, O and 4. Here the subscripts, a and 0, correspoinding to actual and reference substances, respectively, are omitted for convenience. We solved the nonlinear systems (eq 1 and 7 and eq 7 and 8) by using the subroutine ZSPOW from IMSL (International Mathematical and Statistical Library), which is based on Powell's algorithm. (Further details are given elsewhere (Hwang, 1984).) Figures 1-6 show the deviation between the thermodynamic shape factors, 0 and 4, and the viscosity shape factors, O,, and +,,, for six systems representing straight nonpolar hydrocarbons, branched nonpolar hydrocarbons, and polar and hydrogen-bonding compounds. The deviation is significant except for ethylene and n-pentane, which are nonpolar hydrocarbons. It is also observed that the effect of the density shape
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1761
- 0 -
2.0
- 6 -
effects of molecular shape, orientation, internal degrees of freedom, and intermolecular forces besides the hydrogen bond. For the cases of straight nonpolar hydrocarbons and a few simple compounds, w,, reduces to Pitzer’s acentric factor, which has been used successfully in Ely and Hanley’s work. The parameters, w,, and k, are determined by optimizing the objective function
e
4
-.- 0 -
1.8
a,
4?,
1.6
OBJ = C[(q,CalCd/q,exptl)- 11’
(17)
1
1.4
Here qCalcdis the vector of calculated viscosities and qYpt1 is an experimental datum. An optimization subroutine ( z x x s ~based ) on a finite-difference Levenberg-Marquardt method from IMSL was used to perform the optimization of eq 17. A commercially available computer program (TRAPP)written by Ely and Hanley (1981b) at NBS for the evaluation of qCalCdvia the corresponding states scheme, outlined at the beginning of this section, was modified for this purpose. In principle, the choice of a reference fluid is not restricted, but in practice, it is limited to the very few substances for which sufficient reliable data are available over a wide enough range of experimental conditions for the equation of state (E.O.S.) and for the viscosity. Methane is the obvious choice. Here, we retain the same methane E.O.S. and viscosity correlation used in Ely and Hanley’s work.
U
0 1.2 +
u Y 4
1.0
w
n U
50.8
Q6
I
I
a5
I
0.6 0.7 REDUCED
1
I
OB
0.9
I
1.0
TEMPERATURE
Figure 5. Shape factors of methanol (saturated liquid).
Treatment of Data - 0 -
c=j
- 0 -
r+
For systems for which relatively accurate E.O.S. and viscosity correlations cannot be found in the literature, Ao is calculated from experimental enthalpy and entropy data via the thermodynamic relation
-.-+,e, - 0 -
Most of the viscosity data were taken from the monographs of Touloukian et al. (1975), Stephan and Lucas (1979), and Vargaftik (1975). One-dimensional and two-dimensional cubic-spline interpolation subroutines (IQHSCU,BCCU) from IMSL were used to interpolate thermodynamic and viscosity data at specific temperatures and pressures.
E 1.2 1 L
o.6
t 1
1
0.5
I
I
0.6
0.7
REDUCED
I
I
0.8
0.9
1
I
1.0
TEMP E RAT URE
Figure 6. Shape factors of water (saturated liquid).
factor (4) on viscosity is not as significant as the effect of the temperature (energy) shape factor (e). On the basis of these observations, we retain the density shape factor function (eq 111, but rewrite the energy shape factor function (eq 10) as
e,,(T,*, V,*,
k) = 1 + ( w , , ~- wo)J’(T,*, V,*)
u,,,,
+ k/T,*
(16)
where k is an “association parameter” that is zero for compounds without hydrogen bonding. The effect of association is approximated by a linear term in reciprocal reduced temperature. This form is used because of its simplicity and its ability to meet the theoretical limit that at high temperatures the hydrogen bond loses it dominance. The viscosity acentric factor, w”, characterizes the
Results The methane equation of state and viscosity correlations as well as the shape-factor correlations were extended into a pseudoliquid region (Ely and Hanley, 1981a) to cover the range of states which are likely to be encountered for fluids and mixtures of interest but are not covered by the actual methane fluid surface. By use of the reference equation of state and the reference viscosity correlation (see Ely and Hanley (1981a)), the viscosity shape factors (e, and @,,) were obtained from simultaneous solution of eq 7 and 1; the thermodynamic shape factors (0 and 6)were obtained from simultaneous solution of eq 7 and 8. Shape factors for the saturated liquid state are plotted as a function of reduced temperature in Figures 1-6 for six representative systems: ethylene, n-pentane, neopentane, dichlorodifluoromethane (Freon 12), methanol, and water. The deviations of viscosity shape factors from thermodynamic shape factors are insignificant for straight-chain nonpolar hydrocarbons (see Figures 1 and 2 for ethylene and n-pentane) but are significant for other compounds. For very dilute gases, the two conformal equations (e.g., eq 7 and 8) become identical and the solution is indeterminant. In this case, however,
1762 Ind. Eng. Chem. Res., Vol. 26, No. 9, 198’7 Table I. Acentric Factors and Association Parameters for Pure Fluids“ compd no. of data pts temp range, K Nonpolar Fluids 1. n-octane 106 320-670 2. 1-hexene 43 280-375 3. carbon tetrachloride 70 236-800 4. neopentane 66 311-444 5. cyclopentane 5 273-313 6. cyclohexane 66 290-773 7. methylcyclohexane 52 290-530 8. ethylcyclohexane 52 290-530 103 270-650 9. benzene 10. ethylbenzene 62 300-500 11. m-xylene 14 273-403 12. o-xylene 31 268-418 13. sulfur dioxide 14. ethyl ether 15. chlorobenzene 16. acetone 17. bromotrifluoromethane 18. methyl chloride 19. chloroform 20. trichlorofluoromethane (R-11) 21. dichlorodifluoromethane (R-12) 22. chlorotrifluoromethane (R-13) 23. dichlorofluoromethane (R-21) 24. chlorodifluoromethane (R-22) 25. trichlorotrifluoroethane (R-113) 26. dichlorotetrafluoroethane (R-114) 27. ethyl acetate 28. n-propyl acetate 29. n-butyl acetate 30. acetic acid 31. ammonia 32. water 33. methanol 34. ethanol 35. 1-propanol 36. 2-propanol 37. 1-butanol 38. aniline
w
w.
0.3995 0.2850 0.1940 0.1970 0.1960 0.2120 0.2360 0.2430 0.2120 0.3020 0.3210 0.3250
0.3413 0.1114 0.4707 0.7844 0.4249 0.7985 0.5934 0.5080 0.3646 0.2940 0.1980 0.2807
Polar Fluids (without Hydrogen Bonding) 86 200- 1250 0.2510 58 409-522 0.2810 20 273-463 0.2490 41 250-650 0.3090 145 290-435 48 253-660 0.1560 41 250-650 0.2160 18 233-373 0.1880 112 250-575 0.1760 40 170-500 0.1800 38 209-373 0.2020 64 250-500 0.2150 91 230-480 0.2520 60 209-500 0.2550
-0.1945 0.0000 0.2165 2.0163 0.2469 0.3547 0.3150 0.3720 0.2830 0.2821 0.1766 0.2802 0.7269 0.6405
Hydrogen-Bonding Fluids 21 273-873 46 290-500 46 290-500 16 303-673 85 310-800 85 290-773 66 290-650 105 270-600 52 320-550 40 430-530 31 223-673 13 273-393
0.3630 0.3290 0.4170 0.4540 0.2500 0.3440 0.5590 0.6350 0.6240 0.5900 0.3080
0.5638 0.9586 0.6147 1.1500 0.2286 0.8905 1.6497 1.5725 1.7465 1.2637 2.0221 6.9552
assoc. Darameter k
-0.0635 -0.1876 -0.0743 -0.2146 -0.0269 -0.3115 -0.3394 -0.1964 -0.2103 -0.0521 -0.241 3 -2.1888
Viscosity data are for various pressures from 1 bar to several hundred bar.
the solution can be obtained in a simpler manner (Chen et al., 1982) because at very low densities any proper equation of state approaches the viral equation of state. The equivalent temperature can be found by solving
where B, and Bo represent the second virial coefficients of a fluid, a,and the reference fluid, 0, respectively; C, and Co are the third virial coefficients of these fluids. The second virial coefficient and the third virial coefficient can be approximated by [ ( z - l)/p] and [2(2 - 1 - Ao)/p2], respectively, at any low density. From the equivalent temperature, To,the equivalent density, po, is given by
In this work, however, we found that the simpler method offers no advantage over the Powell’s method used in subroutine ZSPOW, not to mention that this method results in a discontinuity in computed shape factors at the arbitrarily low density at which the simpler method is used. For gases at low densities, fluid properties are functions of temperature only, as are the thermodynamic and the viscosity shape factors. It is therefore possible to find a
cut-off density to eliminate the numerical problem encountered in this region. Leach (1967), in his study on thermodynamic shape factors, found that, at reduced densities less than 0.5, the shape factors become essentially independent of density; he also found that, at reduced densities greater than 2.0, the factors again become independent of density in the liquid and dense-fluid regions. Leach’s shape factors were correlated for the normal paraffins, C1-CI5, and have been observed to exhibit deviations for polar fluids for the reasons we have already discussed. However, we have retained the generalized formulations and coefficients of Leach’s shape factors (eq 10-15) but allowed the acentric factor in the temperature shape factor equation (eq 10 for 0) to vary for polar fluids with an additional term (K/T,*) to characterize the association in hydrogen-bonding fluids (see eq 16). Two reasons support these modifications. First, the density shape factor equation (eq 11 for 4) remains unchanged since 0 (temperature or energy) seems to have a stronger effect than 4 (density) on differences between equilibrium to nonequilibrium states. Second, the function, F(T,*, Vn*), in eq 10 satisfies an essential temperature dependence for viscosity, according to which the viscosity of a gas increases with increasing temperature whereas the viscosity of a liquid decreases. This function, F , is plotted vs. reduced temperature in Figure 7 at the two boundaries (reduced density = 0.5 and 2.0). The boundaries of reduced temperature and reduced density (eq 14 and 15) were set by
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1763 Table 11. Summary of Calculated Resultsn this work comod 1. n-octane 2. 1-hexene 3. carbon tetrachloride 4. neopentane 5. cyclopentane 6. cyclohexane 7. methylcyclohexane 8. ethylcyclohexane 9. benzene 10. ethylbenzene 11. m-xylene 12. o-xylene av
N
Hanley and Ely
BIAS MAX Nonpolar Fluids 19.36 106 5.49 2.09 7.76 2.71 -0.61 43 70 (17) 7.67 (5.91) 5.59 (-0.32) 19.39 (-16.75) 25.48 6.08 -1.24 66 -0.05 0.73 5 0.50 -2.07 -30.96 66 7.05 -4.09 -21.38 52 7.74 -1.68 -15.18 52 3.62 103 (63) 4.18 (5.20) -0.32 (1.92) 13.75 (13.75) -0.49 -8.24 62 1.48 2.39 0.44 -5.75 14 31 3.92 0.79 -11.25 4.46 -0.14
13. sulfur dioxide 14. ethyl ether 15. chlorobenzene 16. acetone 17. bromotrifluoromethane 18. methyl chloride 19. chloroform 20. trichlorofluoromethane (R-11) 21. dichlorodifluoromethane (R-12) 22. chlorotrifluoromethane (R-13) 23. dichlorofluoromethane (R-21) 24. chlorodifluoromethane (R-22) 25. trichlorotrifluoroethane (R-113) 26. dichlorotetrafluoroethane (R-114) av
86 58 20 41 145 48 41 18 112 40 38 64 91 60 (32)
27. ethyl acetate 28. n-propyl acetate 29. n-butyl acetate 30. acetic acid 31. ammonia 32. water 33. methanol 34. ethanol 35. 1-propanol 36. 2-propanol 37. 1-butanol 38. aniline av for all fluids
21 46 46 16 85 85 66 105 52 40 31 13 2138
AAD
Polar Fluids 4.11 9.69 2.29 2.81 5.53 3.68 4.10 3.66 3.92 7.69 6.12 1.73 4.54 4.40 (2.71) 4.28 3.50 1.75 2.40 7.00 8.18 11.03 6.19 6.05 7.06 5.16 12.26 11.84 6.87 5.27
(without Hydrogen Bonding) -1.32 -18.94 8.13 28.90 0.63 -6.87 0.01 6.41 16.15 4.65 -1.36 17.33 0.68 8.00 -9.42 -1.94 1.12 13.30 5.29 13.08 0.24 -15.14 0.24 6.26 -0.41 13.64 1.00 (-1.08) 12.16 (-9.42) 1.21
Hydrogen-Bonding Fluids -0.17 -9.79 0.12 -6.50 0.07 -8.84 -3.17 -12.51 -7.24 -16.26 -4.24 -43.28 6.53 34.37 1.13 27.26 0.41 -21.59 -2.19 -12.42 5.21 28.83 -1.15 -37.01 -0.64 0.20
AAD
BIAS
MAX
6.46 20.00 12.70 (29.05) 26.71 27.52 45.73 31.47 25.69 10.29 (13.65) 1.72 22.42 6.13 19.74
4.09 20.00 -3.04 (-29.05) -26.34 -27.52 -45.73 -31.47 -25.69 -9.50 (-12.42) 0.02 22.42 5.52 -9.77
19.50 35.63 -51.43 (-51.43) -41.94 -29.82 -62.21 -39.38 -37.16 -31.72 (-31.72) -7.92 26.51 8.88
7.67 14.18 5.52 12.83
2.73 13.97 5.52 -12.83
-13.52 27.59 7.48 -24.99
5.25 4.14 15.21 7.14 9.55 6.90 3.92 31.33 21.27 (35.76) 11.18
-5.05 -0.06 -14.96 -2.50 2.04 3.24 -2.51 -30.36 -18.76 (-35.76) 11.35
-31.02 -8.10 -28.48 -12.28 -16.60 11.31 -6.27 -56.45 -47.25 (-47.25)
13.49 10.98 5.90 15.72 10.41 52.42 17.34 20.35 21.68
8.69 10.14 5.50 -15.72 -3.90 30.10 -8.55 -12.07 -9.44
77.59 38.79 11.34 -27.66 -17.64 102.91 -37.02 -46.80 -51.96
46.96 36.04 22.84 17.60
-46.38 -36.04 -7.06 -7.07
-89.29 -76.76
"AAD = average absolute percent deviation, BIAS = percent bias, MAX = maximum percent deviation; values in parentheses are for high densities only.
Leach in 1967 and are retained in many studies in which his correlations are applied. Also, these boundaries were found to be satisfactory in this study for viscosity. Over 2000 data points were correlated. The error in the data quoted is estimated to be 5-15% and is probably much worse for very viscous species. The optimized values of wq and k for 38 compounds are given in Table I. Shown in Table I1 are the average absolute percentage deviation (AAD), the average percentage error (BIAS), and the maximum percentage error (MAX), estimated from this study and also from Ely and Hanley's method. The new method shows significant improvement over that of Ely and Hanley. This is to be expected because their model was intended only for nonpolar fluids; Ely and Hanley make no claim that their method can work for polar species. We include the comparison in Table I1 because there is a great temptation to use the TRAPP program for calculation of polar-fluid viscosities, and many in industry have done so because there was no alternative. Although this work shows better results than Ely and Hanley's method for both gases and liquids, the results are better for liquids than for gases, as can be seen from the
4
b
\
4\ . a\
I
I
1
parenthesized values in Table 11. The phenomenon of dilute gas viscosity is primarily due to individual collisions between randomly moving molecules, while in liquids it is mainly the short-range interacting forces that account
1764
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
Table 111. Comparison of Calculated and Experimental Low-Pressure Gas Viscosities Thodos et this work Starling et al. aLn AAD MAX AAD MAX AAD MAX compd N T,K q , WP benzene 3 301-473 73-117 2.99 -4.53 0.56 -1.00 1.96 3.30 2.41 3.99 1.59 -2.40 3.13 5.20 carbon tetrachloride 3 398-573 133-190 5 308-573 72-129 2.95 7.64 1.46 2.30 1.54 3.50 cyclohexane 8.41 -13.49 1.39 -3.00 1.60 -3.10 sulfur dioxide 7 283-1173 120-432 4 373-598 93-153 2.57 5.53 1.22 1.80 4.77 7.10 acetone 15.96 -17.14 5.78 11.30 2.00 3.70 3 273-673 90-251 ammonia chloroform 5 293-623 100-208 3.59 6.93 1.87 3.40 1.10 1.80 4 383-573 111-165 5.54 -7.77 0.29 0.50 1.50 3.00 ethanol 4 373-598 93-153 3.54 -6.22 0.50 1.00 2.80 4.00 ethyl acetate 7.42 -11.12 0.20 0.30 6.22 8.60 4 393-573 103-150 2-propanol 6.93 12.30 0.28 0.70 2.46 -5.20 6 308-593 101-195 methanol 4.11 -5.46 4.28 5.00 1.12 2.10 methyl chloride 4 293-403 106-147 3 298-548 104-144 1.19 2.93 0.81 1.10 2.63 3.60 1-propanol overall 5.20 1.56 2.53 "~s
Golubev" AAD MAX 4.66 6.90 5.23 6.10 4.08 5.40 24.68 93.00 4.87 -6.20 10.26 -13.00 7.44 8.00 10.75 -11.00 1.55 -2.20 5.90 -6.70 22.16 -22.00 1.60 -2.60 6.26 -7.10 8.42
Reichenberg" AAD MAX 2.66 4.40 2.03 -2.60 3.20 -4.20
Van Velzen" AAD MAX 26.75 -34 22.20 -52 21.05 -42 6.1 -6.1 6.8 15 34.25 -51 32.0 -38 14.4 20 44.67 69 14.58 34 4.1 -4.7 4.65 8.8 3.27 -3.6 4.1 2.50 11.2 -2 1 5.38 9.9 165 128.7 21.44
Morris" AAD MAX 4.23 6.1 21.78 -53 11.72 -22 16.53 -26 9.75 12 22.0 -28 6.63 15 28.0 -32 4.43 -6.1 24 8.1 5.58 13 10.4 22 48.67 -57 24.33 -33 21.25 -29 9.2 9.4 73.0 94 18.90
4.55
2.50
0.70 0.25 2.57 3.02 1.86 2.72 1.00 1.96
-1.40 0.70 -4.10 4.20 2.80 4.10 -1.60
reported in Reid et al. (1977).
Table IV. Comparison of Calculated and Experimental Liquid Viscosities
compd acetic acid aniline benzene carbon tetrachloride chlorobenzene cyclohexane cyclopentane neopentane ethanol ethyl acetate ethyl benzene ethyl ether 2-propanol 1-propanol o-xylene m-xylene methanol overall
N 4 4 6 4 4 4 3 3 3 4 4 4 3 3 4 4 3
T,K 283-383 268-393 278-463 273-373 273-393 278-353 253-323 258-283 273-348 293-463 253-413 273-373 280-323 283-373 273-413 273-413 278-333
7,cP 1.5-0.4 13.4-0.7 0.8-0.1 1.4-0.4 1.1-0.3 1.3-0.4 0.7-0.3 0.44.3 1.8-0.5 0.5-0.1 1.2-0.2 0.3-0.1 3.3-1.1 2.9-0.4 1.1-0.3 0.8-0.2 0.8-0.4
this AAD 21.05 21.27 10.43 20.29 2.08 7.96 1.50 17.34 3.90 3.62 6.72 22.22 48.8 17.69 4.78 3.10 6.13 13.05
work MAX -22.4 -46.9 21.2 22.1 -6.3 12.66 2.48 20.07 7.07 -6.32 8.94 -27.63 -60.55 -28.98 -9.21 6.68 8.50
Thomas" AAD MAX 43.25 -51 19.47 0.98 5.5
-41 1.9 15
31.33 67.33 6.85 2.15 9.9 83.67 73.3 9.15 7.53 49.0 32.01
-37 -79 19 -5.6 16 -90 -86 -16 16 -56
Orrick and Erbar" AAD MAX 12.95 -22 27.37 20.25 2.0 42.0 32.0 1.83 22.47 9.75 1.75 3.25 16.33 8.47 3.85 1.05 34.37 14.18
-46 22 -5.1 -5 1 -36 -3.5 27 27 -2.9 11 -24 -9.8 5.0 1.4 59
"As reported in Reid et al. (1977).
for the viscosity. It is concluded that the parameters, w,, and k , can account for the effects of the strong interacting forces on liquid viscosity for nonnormal paraffins while Pitzer's acentric factor, w , cannot. The 38 compounds that we have studied are classified into three categories. I. Nonpolar Fluids. Except for n-octane, ethylbenzene, and o-xylene, whose transport properties are predicted well by Ely and Hanley, there is a significant difference between w,, and w for these compounds, and consequently, significant improvement is observed in our calculated values. These compounds include cyclic hydrocarbons such as cyclopentane and cyclohexane, spherical compounds such as carbon tetrachloride, and double-bonding species such as 1-hexene. The effects of internal degrees of freedom, molecular size, shape, and deformation seem to be accounted for by the optimized viscosity acentric factor, w,,. 11. Polar Fluids without Hydrogen Bonding. Most of the compounds in this category are halogen-substituted hydrocarbons (Freons) simply because complete viscosity data for these compounds are available from the refrigerant industry. Very little difference is observed when w is replaced by w,, for Freons of low molecular weight. The mass (size) effect on viscosity, however, is reflected in w?. Figure 8 shows an increasing difference between a,,and w with increasing molecular weight for Freons. Only gas viscosity
data are correlated to obtain w,, for sulfur dioxide and acetone, but still the prediction is better, particularly for acetone. 111. Hydrogen-Bonding Fluids. The hydrogenbonding fluids studied in this work range from acetates and acids to alcohols and include ammonia, water, and aniline. The calculations are greatly improved when the viscosity acentric factor, a,,, and association parameter, k , are employed. The effect of association vanishes at high temperatures as k/T,* approaches zero. Aniline, with viscosities 4 times greater than that of water at room temperature, has very large values of w,, and k , while w,, and k for 2-propanol are lower than for other alcohols, possibly because the hydroxyl group is hindered by the neighboring methyls. Since Pitzer's acentric factors for bromotrifluoromethane and for 2-propanol are not found in the literature, estimated values for w (0.2 for CBrF, and 0.6 for 2-propanol) were used in the calculations. A comparison between this work and previously recommended methods (Reid et al., 1977) and the multiparameter correlation of Starling et al. (1984) for some nonpolar, polar, and hydrogen-bonding fluids is shown in Table I11 for gas viscosities and in Table IV for liquid viscosities. The experimental values used for comparison in Tables I11 and IV are taken directly from Tables 9-4 and 9-12 in Chapter 9 of Reid's book. These data are from a source
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1765 1.2
1.0
1c
I While
1
: w
black : q, 0 0 --tRW 0:. -CF2&?
IR-11) IR-121
O,* -CE)cL (R-131 a,. -CCHFZL IR-221 L O C 2 F a IR 1131 9,. -CZWU (R-1141
-
-
.
e
o
e
Y
0.4
Acknowledgment ;
4
0.2
O
plicability, provided some viscosity data are available for determining the two new parameters. T o modify the NBS computer program to incorporate our model, very few statements need to be added. Apart from the parameters, k and a,,, the method requires only the common characterization parameters of the pure components as input: TC, P c , V c ,w, and M. The extension of this work to correlate the viscosities of asymmetric fluid mixtures is under investigation.
*
B 0
el0
l&
o 0
1;o
MOLECULAR
0
l&
WEIGHT,
1& M I 9 / 9 moll
l&
zbo
210
2ko
’
Figure 8. Acentric factors for Freons.
different from that presenting the data correlated in our work to obtain a,,and k. Unfortunately, the high degree of inconsistency between different data sources causes large deviations between calculated and experimental values for some species, as is shown in Tables I1 and IV for acetic acid. In addition, the deviations also become somewhat worse a t the extrapolated experimental conditions of this work (see, e.g., neopentane). Still, it can be seen from Table IV, which is for liquid viscosities, that this work is superior to other methods, in accuracy. For gas viscosities, this work is just as good as other methods as seen from Table 111, because this work and all the other methods calculate values within the limits of experimental error. It should be pointed out that our method can be applied to a wide class of compounds and over the entire range from dilute gas to dense liquid; the other can be used only for specific types of fluids and none can be applied to both gases and liquids. The density of a fluid is also predicted via this corresponding states scheme. The density is predicted to better than 1% for nonpolar hydrocarbons and a few other simple fluids (Ely and Hanley, 1981a). However, since no effort was made to modify the density shape factor equation, larger deviations for polar fluids are expected. In general, 5-15% error in predicted density for polar fluids is observed. Like most other methods, this method is not recommended for use near the critical region; the predicted values in the critical region can have an error as great as 20%. For compounds which are not investigated in this work, the parameters, wv and k, can be estimated from values for similar compounds studied in this work. For example, wv of halogen-substituted hydrocarbons can be estimated from Figure 8, which shows that w, is a simple function of molecular weight.
Conclusions We have extended Ely and Hanley’s conformal-solution viscosity model for the prediction of fluid viscosities from 60 nonpolar substances to 90 substances. Thirty new compounds are added, including polar and hydrogenbonding species. The basis of this work is the extended corresponding states approach with modifications to the temperature shape-factor equation. Besides retaining the remarkable predictions given by Ely and Hanley for nonpolar fluids, which are accurate to within 8% overall, this work is also able to correlate the viscosity data for nonnormal paraffins within an average absolute deviation of about 6%. Comparisons between the results of this work and those from other methods show that this work is better than others in accuracy and ap-
We thank the U.S. National Science Foundation for financial assistance.
Nomenclature A = Helmholtz free energy AAD = average absolute percent deviation = 1 / N - )7yptl)/vLexptjloo% a,b,c, etc. = coefficients B = second virial coefficient BIAS = percentage bias = 1/N q(Jexptl1 100%
((vidcd
[(vLcalcd- v r e x p t 9 /
C = third virial coefficient F = dimensional scaling for viscosity H = molar enthalpy k = association parameter M = molecular weight MAX = maximum percent deviation = max[(vLCaiCd v~exptl)/vLex~tl]lOO%, for all i P = absolute pressure R = universal gas constant S = molar entropy T = absolute temperature V = molar volume 2 = compressibility factor, PV/RT
Greek Symbols = viscosity 0 = temperature (energy) shape factor p = density 4 = density shape factor o = acentric factor q
Subscripts 7 = viscosity CY = fluid of interest 0 = reference fluid Superscripts c = critical value calcd = calculated value exptl = experimental value id = ideal gas * = reduced by value at critical point
0 = reduced residual value
Literature Cited Chen, Y. P.; Angelo, P.; Naumann, K.-H.; Leland, T. W. GRI Project Report, GRI Contract 5014-363-0180, 1982; Rice University, Houston. Christensen, P. L.; Fredenslund, Aa. Chem. Eng. Sci. 1980,35,871. Ely, J. F.; Hanley, H. J. M. Ind. Eng. Chem. Fundam. 1981a, 20,323. Ely, J. F.; Hanley, H. J. M. Natl. Bur. Stand. Note 1981b, 1039. Ely, J. F.; Hanley, H. J. M. J . Res. Natl. Bur. Stand. 1981c, 86, 597. Ely, J. F.; Hanley, H. J. M. Ind. Eng. Chem. Fundam. 1983,22, 90. Evans, D. J.; Hanley, H. J. M. Phys. Reu. 1979, A20, 1648. Golubev, I. F. “Viscosity of Gases and Gas Mixtures: A Handbook”, Natl. Tech. Inf. Serv. Report TT7050022, 1959. Hanley, H. J. M. J . Res. Natl. Bur. Stand. 1977, 82, 181. Hanley, H. J. M. Cryogenics 1976, 16, 643. Hwang, M.-J. M.S.Ch.E. Thesis, West Virginia University, Morgantown, 1984. Leach, J. W. Ph.D. Thesis, Rice University, Houston, 1967. Leach, J. W.; Chappelear, P. S.; Leland, T. W. AIChE J. 1968, 14, 568.
1766
Ind. Eng. C h e m . Res. 1987, 26, 1766-1773
Massih, A. R.; Mansoori, G. A. Fluid Phase Equilib. 1983, 10,57. Mo, K. C.; Gubbins, K. E. Chem. Eng. Commun. 1974, 2, 281. Mo, K. C.; Gubbins, K. E. Molec. Phys. 1976, 31, 825. Morris, P . S. M.S. Thesis, Polytechnic Institute of Brooklyn, New York, 1964. Murad, S.; Gubbins, K. E. Chem. Eng. Sci. 1977, 32, 499. Murad, S.; Gubbins, K. E. AZChE J . 1981. 27, 864. Murad, S. Chem. Eng. Sci. 1981, 36, 1867. Orrick, C.; Erbar, J. H., unpublished results, Oklahoma State University, Stillwater, 1974 (as reported in Reid et al. (1977)). Reichenberg, D. AIChE J . 1975, 21, 181. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed. McGraw-Hill: New York, 1977. Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969, 24, 1565. Starling, K. E.: Chung, T . H.; Lee, L. L. Ind. Eng. Chem. Fundam. 1984, 23. 8.
Stephan, K.; Lucas, K. Viscosity of Dense Fluid; Plenum: New York, 1979. Teja, A. S. AZChE J . 1975, 21, 618. Thodos, G.; Yoon, P. AIChE J . 1970, 16, 300. Thomas, L. H. J . Chem. Sot. 1946, 573. Touloukian, T. S.; Liley, P . E.; Saxena, S. C. Thermophysical Propertics of Matter, 11; Plenum: New York, 1975. Van Velzen, D.; Cardozo, R. L.; Langenkamp, H. Ind. Eng. Chem. Fundam. 1972, f l , 20. Vargaftik, N. B.; Tables on the Thermophysical Properties of Liquids and Gases, 2nd ed. Wiley: New York, 197.5. Whiting, W. B.; Prausnitz, J. M. Fluid Phase Equilib. 1982, 9, 119.
Received for review J u n e 6 , 1985 Revised manuscript received J u n e 1, 1987 Accepted J u n e 22, 1987
Study of Char Gasification in a Reaction/Adsorption Apparatus Stratis V. Sotirchos* and John A. Crowley D e p a r t m e n t o f Chemical Engineering, University of Rochester, Rochester, N e w York 14627
T h e reaction of an activated carbon (coconut char) with COz was studied in a reaction/adsorption apparatus which allows successive reactivity and physical adsorption measurements to be made on the same solid sample. Reaction and surface area evolution data were obtained in the temperature range from 800 to 900 "C. All reaction rate trajectories obtained in this study showed a maximum in the reaction rate, 2-3 times higher than the initial rate, at about 85% conversion. However, there was no correlation between these results and the evolution of the internal surface area although the reaction appeared t o take place initially in the kinetically controlled regime. 1. Introduction A large number of physical and chemical rate processes is involved in the reaction between porous solid particles and gaseous reactants, but the experimental and theoretical analysis of such reactive systems is mostly complicated by the progressive evolution of the solid pore structure in the course of the reaction. The overall gassolid reaction process includes diffusion of the gaseous reactants and products in the surrounding gas phase, diffusion and reaction in the porous network, and heat transport both in the interior of the particle and in the surrounding gas phase. There are two general limiting cases for noncatalytic gas-solid reactions, namely, the regime where the reaction rate is controlled by external diffusion only and that where it is solely determined by intrinsic kinetics. Depending on the relative rates of reaction and mass transport in the surrounding gas phase, either of the two limiting cases may prevail or the system may react in the general transition regime where both kinetics and intraparticle and external diffusional limitations are important. If the intraparticle diffusional limitations are much stronger than the external ones, one more limiting case may arise in which the process is solely controlled by intraparticle diffusion. Because of the pore structure evolution of the solid, the rates of intraparticle diffusion and reaction in a noncatalytic gas-solid system change continuously, leading in turn to a change in the overall reaction mechanism. For instance, a system may start reacting under external diffusion control but move into the kinetically controlled regime after some conversion level. * Author t o
whom correspondence should be addressed.
OSSS-5SS5/87/2626-1766$01.50/0
Initial reactivity or pore structure experimental data, therefore, may be useless for noncatalytic gassolid systems unless there is some way, a theoretical model perhaps, that will allow us to predict the reactivity and the pore structure properties of the solid at conversions different from zero. Similarly, reactivity vs. conversion data alone present interpretation problems because of diffusional effects and dependence of the overall reactivity on the pore structure properties, in particular, on the available internal surface area. Consequently, simultaneous reactivity and pore structure studies are needed for data analysis and model testing. Although the pore structure usually provides some basis for the estimation of the intraparticle diffusivity, experimental studies for intraparticle mass transport would also be useful. The literature for intrinsic or global kinetic parameters for carbon gasification is somewhat disperse. Although there is a difference of opinion about the true activation energy of the reaction, most studies place it in the range 50-60 kcal/mol (Dutta et al., 1977; Laurendeau, 1978). The order of reaction with respect to CO, has also been investigated in many of the previous kinetic studies. A t atmospheric pressure, the order of reaction approaches unity (Gadsby et al., 1948; Dutta et al., 1977), while i t becomes zero at higher pressures (Fuchs and Yavorsky, 1975). The reaction temperature is of utmost importance in studying reaction rates of the char-C02 system since it determines whether the behavior of the system is influenced by diffusional limitations, external or intraparticle. For carbon or char gasification, the transition regime where both kinetics and diffusional limitations are important begins in the range 90G1200 "C (von Fredersdorff, 1955; Walker et al., 1959; Donnelly et al., 1970). Most studies have been conducted in the above temperature
0 1987 American Chemical Society
