I n d . Eng. Chem. Res. 1990,29, 1404-1412
1404
Smith, R. A. Vaporisers: Selection, Design and Operation; Longman: New York, 1986. Srinivasan, K.; Krisna Murthy, M. V. Determination of the BulkSaturated Liquid Condition for Maximum Heat Fluxes a t Boiling Crises. Int. J. Heat Mass Transfer 1986,29 (12), 1963. Tong, L. S. Boiling Heat Transfer and Two-Phase Flow; Wiley: New York, 1965. Volejnik, M. An Investigation of Industrial Reboilers. Int. Chem. Eng. 1979, 19 (4), 689. Wazzan, A. R.; Procaccia, H.; David, J.; Fromal, A.; Pitner, P. Thermal-Hydraulic Characteristics of Presswised Water Reactors during Commercial Operation. Nucl. Eng. Des. 1988, 105, 285. Westwater, J. W. Boiling of Liquids. Adv. Chem. Eng. 1956, 1, 2. Whalley, P. B.; Hutchinson, P.; James, P. W. The Calculation of Critical Heat Flux in Complex Situations using an Annular Flow Model. Proc. Inst. Heat Transfer Conf., 6th, 1978,5, 65. Yilmaz, S. B. Horizontal Shellside Thermosyphon Reboilers. AIChE Symp. Ser. 1987,83 (257), 40; Chem. Eng. Bog. 1987,83 (ll),64. Zinemanas, D.; Hasson, D.; Kehat, E. Simulation of Heat Exchangers with Change of Phase. Comput. Chem. Eng. 1984, 8 (6), 367.
Palen, J. W.; Shih, C. C.; Yarden, A.; Taborek, J. Performance Limitations in a Large Scale Thermosyphon Reboiler. Ind. Heat Transfer Conf., 5th, 1974, 204. Palen, J. W.; Shih, C. C.; Taborek, J. Mist Flow in Thermosyphon Reboilers. Chem. Eng. Prog. 1982, 78 (71, 59. Pope, B. J. An Investigation of Natural Circulation Tube Reboiler Capacity and Performance. Ph.D. Thesis, University of Washington, Pullman, 1959. PPDS Version 10 (1986), The Inst. Chem. Eng., 165/171 Railway Terrace, Rugby CV21 3HQ, England. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworth: New York, 1982. Sarma, N. V.; Reddy, P. J.; Murti, P. S. A Computer Design Method for Vertical Thermosyphon Reboilers. Ind. Eng. Chem. Process Des. Deu. 1973, 12 (3), 278. Sciance, C. T.; Clover, C. P.; Sliepcevich, C. M. Nucleate Pool Boiling and Burmout of Liquefied Hydrocarbon Gases. Chem. Eng. Prog. Symp. Ser. 1967, 109 (77), 63. Shah, G. C. Troubleshooting- Reboiler Systems. Chem. Enp. - Prog. 1979, 75 (71, 53. Shellene, K. R.; Sternling, C. N.; Church, D. M.; Snyder, N. H. ExDerimental Studvof a Vertical Thermoswhon Reboiler. Chem. Eng. Prog. Symp. ker. 1968,64 (82), 102' Smith, J. V. Improving the Performance of Vertical Thermosyphon Reboilers. Chem. Eng. Prog. 1974, 70 (71, 68.
Received for review May 17, 1989 Revised manuscript received October 24, 1989 Accepted November 14, 1989
GENERALRESEARCH Generalized Viscosity Behavior of Fluids over the Complete Gaseous and Liquid States Huen Lee' and George Thodos* Department of Chemical Engineering, Northwestern University, Euanston, Illinois 60208-3120
Viscosity measurements reported in the literature have been used for the development of a generalized expression capable of predicting this transport property for all state conditions. These include the dilute and dense gaseous states and the saturated and compressed liquid regions. The present development shows that the excess viscosity, p - p*, can be predicted in a generalized manner from the relationship, 105(p - p * ) y = [exp(bg" dgK)]- 1, where b, d , a , and K are universal constants. The complex nature of the parameter g includes, besides density and temperature, the influence of the expansion characteristics of the substance, in the course of freezing, a t the triple point. The viscosity parameter, y = u ~ ~ ~ / ~ / is M unique ~ / ~ toT a~substance ~ / ~ , and for its definition requires triple-point values. Comparison between experimental and predicted viscosities, covering all fluid-state conditions and including the saturated and compressed liquid regions, yields an average absolute deviation of 3.21% (1563 points) for 24 substances examined.
+
Our present knowledge for the satisfactory prediction of transport properties continues to be limited to the dilute and moderately dense gaseous states of substances. Attempts to extend our background to include gases at high pressures and liquids existing at temperatures below their normal boiling point and for conditions approaching their respective triple points have not yet been properly resolved to permit the formulation of a unified approach for the prediction of viscosity. This difficulty stems largely from the complex nature of liquids associated with this state of aggregation and particularly as temperatures in the proximity of the freezing curve are approached. The
* Author t o whom correspondence should be addressed. Present address: Korea Advanced Institute of Science and Technology, Seoul, Korea. 0888-5885/90/2629-1404$02.50/0
correlation of the viscosity in a generalized manner in this region has not yet been properly resolved, and consequently, the prediction of this transport property continues to prove inadequate over the complete fluid state. The difficulty for treating, in a generalized manner, the complete liquid region is not unexpected because of the complex nature associated with the aggregation of the liquid state existing below the normal boiling point and particularly as the freezing state is approached. While, on one hand, the kinetic theory of gases offers a direct approach for the estimation of properties of substances in the gaseous state, the lattice theory, on the other hand, provides the theoretical background needed for treating the solid state. The liquid state bridges these two extremes. So far, this bridging has proven to be quite formidable. Nonetheless, the proper interpretation of this
62 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1405 bridging could prove quite valuable in permitting the prediction of properties for this compromising state of aggregation. Because of the elusive nature of this state, liquids can be considered to represent states of aggregation of solids, and for their description, solidlike theories have been advanced, while other theories consider liquids as dense gases. Since 1873, when van der Waals proposed the existence of continuity between the gaseous and liquid states, no significant development on the theory of the liquid state was advanced until the 19309 when the similarity of liquids and crystalline solids was proposed. This concept led to the development of the solidlike theory. Using this concept, Hildebrand (1971) presents an interesting treatment of the liquid state and decides against the adoption of the "quasi-crystalline" treatment of liquids. On the other hand, Lee (1971) points out reasons underlying the phenomenological existence of the liquid state, not from the point of view of similarity between the liquid and solid states but rather from the dissimilarity existing between them. The difficulty encountered in the correlation of transport properties at these extreme state conditions is largely coupled with the complex nature of the highly dense gaseous and liquid states. Under these conditions, the state of aggregation of both polar and nonpolar substances becomes highly sensitive to the structural aspects associated with their molecular configuration. In this connection, Crawford et al. (1975) point out that such nonuniformities become further accentuated for liquids as they are made to approach their triple-point temperature.
eq 1has been based on the data for argon, the application of this relationship can be extended in a generalized manner to other gases. Viscosities calculated with eq 1 were compared with the corresponding experimental values of argon to produce an average deviation of 0.50% (100 points). In addition, this relationship has also been applied to calculate the viscosities a t normal pressures for neon, krypton, and xenon to give average deviations of 3.04% (39 points), 1.20% (9 points), and 0.63% (11points), respectively. With the exception of helium and hydrogen, eq 1 is also capable of predicting accurately the viscosities for diatomic gases, carbon dioxide, carbon tetrachloride, and paraffinic, olefinic, naphthenic, acetylenic, and aromatic hydrocarbons. Existing Models for the Dense Gaseous a n d Liquid States By use of theoretical arguments, the influence of pressure on the viscosity of gases is reported by Enskog (1922), who utilizes the Boltzmann equation and kinetic theory considerations applied to binary collisions of rigid spherical molecules to present relationships for the viscosity, thermal conductivity, and self-diffusivity of gases at elevated pressures. However, the Enskog approach, although interesting from a theoretical point of view, does not provide an accurate means for predicting the viscosity of dense gases as pointed out by Lennert and Thodos (1965). The complex nature associated with the liquid state has proven to be the limiting factor for the development of a generalized approach capable of predicting the viscosity for this state of aggregation. To overcome this limitation, Batschinski (1913) utilizes the free volume concept as a correlating parameter for the prediction of viscosity of liquids. Continued interest in the application of the free volume concept has been expressed by Cohen and Turnbull (1959), Macedo and Litovitz (1965),and Gubbins and Tham (1969). In this context, Dymond and Brawn (1977) utilize, for the viscosity behavior of the liquid state, the close packing volume of their molecules. Their expression has been generally accepted as capable of predicting reliable viscosities for the compressed liquid state by allowing this molar volume to depend on temperature; however, their relationship does not prove adequate for predicting viscosities at high densities in the proximity of the freezing curve as shown by Trappeniers et al. (1980).
Viscosity for the Dilute Gaseous State The theoretical aspects associated with the viscosity of dilute gases have been well documented by Maxwell (1868). Using kinetic theory arguments, Maxwell was the first to show that the viscosity of dilute gases was independent of pressure. Later, Hinchfelder et al. (1954),using pairwise interactions, apply these arguments to spherical molecules for the prediction of viscosity using molecular potential parameters. However, their approach is not capable of predicting adequately this transport property for complex molecules. The earliest attempt to extend the corresponding states principle to viscosity for dilute gases is credited to Kamerlingh Onnes (1881). In this connection, he was able to express the viscosity in reduced form as c ( ( T , / M ~ P , ~ ) ' / ~Dimensional Analysis Based on Triple-Point Properties and postulated that this normalized viscosity is the same Following the general treatment of viscosity for the for all substances a t a given reduced temperature. Licht dense fluid state, it follows that the excess viscosity, p and Stechert (1944) and later Flynn and Thodos (1961) p*, may be related to, besides the state variables of temand Stiel and Thodos (1961) utilize the viscosity modulus, perature and molar volume, the triple-point values, instead 6 = Tti6/M 1/2P:/3,to correlate with reduced temperature of the commonly accepted corresponding critical state the generalized viscosity behavior at normal pressures for values. By use of dimensions of mass, length, time, and a number of both polar and nonpolar gases. In their study, temperature, this dimensional analysis approach shows Stiel and Thodos (1961) utilize 52 gases, for which z, that the normalized excess viscosity, ( p - p * ) y , becomes ranged from 0.307 for neon to 0.250 for n-nonane, and conclude that the relationship between reduced viscosity, T" ( p - p * ) y = kR ' J 2 p * t , and T R is independent of 2,. In this context, Rorris (2) Wf (1979) utilizes dimensional analysis and applies this approach to the comprehensive correlation of available argon where y = u ~ ~ ~ / ~ / TM= ~TIT,, / ~ and T ~ ~= /p/pa. ~ , The data to present, for the normalized viscosity of the dilute viscosity parameter, y, is identical with that suggested by gaseous state, the relationship, Andrade (1934), who proposes that the normalized viscosity of the liquid state in the proximity of the freezing 35.50 point is lO5(p*0) = 0.576TR0.910 < T R < 15 exp(lO/ TR1i2) (1) (3) where p* is in poises and the viscosity parameter p = u , ' / ~ / M1/2Pc1/2. Despite the fact that the development of
Equation 3 assumes that the produce
p ( u p / M ll2Tt1i2)
1406 Ind. Eng. Chem. Res., Voi. 29, No. 7 , 1990 300~10-~
NEON 1(=0.2876 M=20.183
W
:I
Hayne~(1973)
r I A- 20
-
saturated liquid t .r=1285 0 7.1493
a 0
n r:l€67
c
'?1 -
Vermesse and Vidal (1973) 4~~3678
IO
/
1
----
Trappaniers.van der Gulik and van den Hooff (1980) .
0
Vermesse ond Vidal (1975)
T'LW5
0 7=1215
x 1=3594
0 s=3857
4 I
05
S
I
06
8
I
07
S
I
08
t
1
d
I
09
P ID a=-
I
I
II
12
~
l
13
I
~
I
Herremon and Grevendonk (1974) saturated liqufd 0 7=1100 0 7.1467 0 ~ = 1 2 1 5 X 7.1597 + 7:1340 9 7.1707
~
14
Ptl
Figure 1. Dependence of ( p - p*jy versus w relationship on temperature for the dense gaseous and liquid states of argon.
is constant for all substances; however, this restriction is found to apply only for the simple monatomic molecules such as neon, argon, krypton, and xenon. For diatomic, triatomic, and polyatomic configurations, eq 3 deviates progressively, and for liquid propane, at its freezing point, this relationship produces the value of 2300 X 10". This represents a 45-fold variation over that encountered for simple fluids.
Comprehensive Treatment of Viscosity Measurements for the Dense Gaseous and Liquid States The dependence of excess viscosity on density advanced by Shimotake and Thodos (1958) has been shown to apply satisfactorily for dense gases and liquids existing at temperatures above their normal boiling points. Jossi et al. (1962) utilize this dependence and postulate a generalized excess viscosity relationship of the form, based on critical constants, ( p - p*)[ = f b R )
(4)
where the viscosity parameter ( = TC1l6 jM'/2P:/3 and p R = p / p , . Although eq 4 proves adequate in relating properly the viscosity behavior of the fluid state for a number of both polar and nonpolar substances, it does not satisfactorily account for the behavior of liquids existing below their normal boiling point and particularly in the vicinity of the triple-point region. To present a comprehensive treatment for the complete fluid state, this study was undertaken in an unrestricted manner with special emphasis placed in the highly compressed gaseous and liquid states. Since the freezing line represents a limiting state condition for the existence of the fluid state, the triple point has been adopted in this study as a frame of reference instead of the commonly used critical point. The extensive experimental measurements for viscosity available in the literature for the subcritical and supercritical states of neon and argon offer adequate information to describe the ( p - p * ) y versus w = p / p l t behavior of these substances for densities ranging from the moderately compressed gaseous state to liquids approaching their freezing line. Figure 1 presents the dependence of (@ - p*)y on w = P I P l t for argon based on the saturated and compressed liquid measurements of Haynes (19731, the supercritical temperature data of Vermesse and Vidal (1973), and of Trappeniers et a]. (1980). The data of Haynes cover the supercritical, saturated liquid, and compressed liquid-state regions. In particular, it was noted that bifurcations from the saturated liquid curve prevail
I
~
~
0.5
06
07
~
08
~
0.9
IO
~
~
1.1
~
12
13
a=P Plt
Figure 2. Dependence of (fi - p*jy versus w relationship on temperature for the dense gaseous and liquid states of neon.
for the three normalized temperatures, T = 1.285,1.493and 1.667,for which compressed liquid viscosities are reported by Haynes (1973). Similar bifurcations are also noted for the supercritical data of Vermesse and Vidal (1973) a t T = 3.678 and of Trappeniers et al. (1980) at 7 = 2.663,3.594, and 3.857. This type of dependence is also noted with neon in Figure 2 for which the compressed liquid data of Herreman and Grevendonk (1974) exhibit similar bifurcations for the six subcritical temperatures of T = 1.100, 1.215, 1.340, 1.467, 1.597, and 1.707 and the supercritical data of Vermesse and Vidal (1975) at T = 12.15. Similar patterns of behavior were also noted for methane, ethane, and propane using the extensive viscosity measurements of Diller (1981). This bifuractional behavior exhibited with all these substances and shown in Figure 1 for argon and in Figure 2 for neon shows that density alone cannot properly describe the dependence of excess viscosity, p p*, in a generalized manner in these fluid regions. However, the combined temperaturedensity parameter, Fluf, expressed by eq 2, although more complex, may prove to be more amenable to a generalized treatment. Consistent with eq 2, the combined influence of temperature and density has been expressed, using data for argon, neon, methane, ethane, and propane, into the single generalized variable, x=-
w 70.07w2.73
This density-temperature parameter has been applied to the viscosity measurements of argon and neon and has been shown to possess the ability of eliminating the presence of bifurcations by producing the individual functional relationships for each of these substances shown in Figure 3. A similar treatment was carried for 22 additional substances which range from the simple monatomic to the more structurally complex molecules such as the n-alkanes to n-octadecane, benzene, cyclohexane, and tetramethylsilane. All of the 24 substances included in this study and their sources of viscosity measurements are presented in Table I. These substances are also included in Table I1 along with the basic constants associated with each of them. The combined (@ - p*)y versus x relationships resulting for these 24 substances are presented in
l
~
Ind. Eng. Chem. Res., Vol. 29, No. 7,1990 1407
200r-5
Table I. Sources of Viscosity Measurements for Substances Included in This Study substances sources monatomic Trappeniers et al., 1964;Vermesse and neon Vidal, 1975;Forster, 1963;Herreman and Grevendonk, 1974;Slyusar e t al. ~~~
It& M Iv,,,cm?g-mole -816.21 0.2878
Neon 24.54 20.183 o Argon 83.79 39.948
1973 Haynes, 1973;Trappeniers e t al., 1980; Vermesse and Vidal, 1973 Ulybin et al., 1978 Ulybin and Makarushkin, 1977
argon krypton xenon diatomic nitrogen oxygen fluorine hydrocarbons methane ethylene ethane propane n-butane n-pentane n-hexane cyclohexane benzene n-heptane n-octane n-nonane n-decane n-octadecane miscellaneous carbon dioxide carbon tetrachloride tetramethylsilane
,B
P
28.21
0.1602
50 40 W
*?s-
Van Itterbeek et al., 1966;Vermesse, 1969 Haynes, 1977 Haynes, 1974
30 -
20 -
Y
IO -
Diller, 1981 Vargaftik, 1975 Diller, 1981 Diller, 1981 Vargaftik, 1975 Lee and Ellington, 1975 Vargaftik, 1975 Jonas et al., 1980 Parkhurst and Jonas, 1975 Vargaftik, 1975 Brazier and Freeman, 1969 Vargaftik, 1975 Vargaftik, 1975 Vargaftik, 1975
8-
-
6-
54-
3tk Figure 3. Relationship of ( p - p*)y versus I: representing the dense gaseous and liquid regions of neon and argon.
Ulybin and Makarushkin, 1976 McCool and Woolf, 1972 Parkhurst and Jonas, 1975
Figure 4 is best represented analytically by an expression of the form,
+
Figure 4. Each of these relationships passes through the origin, ( p - p * ) y and 0 and x = 0; however, those shown in Figure 4 present their behavior in the highly dense gaseous and liquid states. Of these, propane shows the highest and carbon dioxide the lowest viscosity dependence on x . The functional dependence of the relationships in
i05(p - p * ) y = [exp(axm px")] - 1
(6)
Equation 6 properly accounts for the limiting boundary condition at the origin. A nonlinear regression analysis, involving the use of a CDC-6600 computer, generated for each substance the coefficients a and 0and the exponents
Table 11. Basic Constants Associated with Substances of This Study critical constants
triple point Vlt,
UC,
monatomic neon argon krypton xenon diatomic nitrogen oxygen fluorine hydrocarbons methane ethylene ethane propane n-butane n-pentane n-hexane cyclohexane benzene n-heptane n-octane n-nonane n-decane n-octadecane miscellaneous carbon dioxide carbon tetrachloride tetramethylsilane
viscosity parameters B= 7" u;/61
u,,2/31
T,, K
P,, atm
cm3/mol
T,, K
cm3/mol
20.183 30.948 83.80 131.30
44.45 150.65 209.41 289.73
26.86 48.02 54.18 57.64
41.79 75.50 92.29 118.29
24.54 83.79 115.94 161.36
16.21 28.21 34.31 42.66
0.08001 0.04694 0.03155 0.02547
0.2878 0.1602 0.1071 0.0839
3.15381 0.48551 0.25406 1.45769
28.013 126.20 32.00 154.58 37.997 144.30
33.54 49.77 51.47
89.20 73.39 66.19
63.14 54.363 53.48
32.29 24.81 22.29
0.06896 0.05127 0.04548
0.2411 0.2040 0.1757
0.70825 3.53877 7.00469 1.05417 1.11968 3.57348 5.00948 0.893 49 1.08057 4.01593 9.59931 1.04883
190.555 282.40 305.5 369.99 425.18 469.6 504.7 554.15 562.09 540.3 568.8 594.6 617.6 745.0
45.39 49.70 48.50 42.10 37.47 33.30 29.30 40.40 48.34 27.00 24.50 22.80 20.80 11.9
100.00 129.0 141.72 195.70 254.91 304.0 370.0 311.70 258.66 431.94 492.0 548.0 608.0 1178.2
90.66 104.00 89.88 85.44 134.86 143.40 177.84 279.83 278.693 182.57 216.38 219.66 243.51 301.35
35.37 42.62 45.58 60.28 78.44 95.10 113.45 106.30 87.31 129.20 150.38 165.58 184.25 327.73
0.07984 0.06020 0.05979 0.05592 0.05396 0.05291 0.05332 0.04466 0.04108 0.05286 0.05311 0.05290 0.05350 0.05906
0.2825 0.2259 0.2454 0.2505 0.2070 0.2048 0.1893 0.1462 0.1334 0.1890 0.1799 0.1797 0.1740 0.1717
0.76741 1.541 73 1.84003 2.37851 4.39887 2.16846 1.52288 5.12360 3.58505 2.25619 1.70809 1.82314 1.24569 5.086 20
44.01 304.19 153.823 556.40
72.85 45.00
94.04 276.00
216.55 250.00
37.37 91.96
0.03766 0.03067
0.1145 0.1039
0.73743 2.56821 2.971 94 0.18875 5.07501 24.516 53
448.61
27.84
362.00
182.00
114.14
0.05387
0.1857
1.81048 3.83246
16.043 28.054 30.070 44.097 58.124 72.151 86.178 84.16 78.115 100.198 114.232 128.259 142.286 254.484
88.23
M1/*P>/* M1/2T,1/2
coeff and exponents: eq 6 B m n
M
a
0.68364 3.423 57 3.56094 2.55279
3.331 50 3.51689 4.03939 5.38675 1.60224 3.901 98 4.53043 0.07939 1.038 12 4.23447 3.91625 4.51129 4.781 40 1.45383
1.69658 7.90429 12.04691 4.072 48
7.54251 8.10601 6.93993 12.97611 3.66827 6.05866 10.605 85 2.94903 3.54578 7.94773 5.35946 9.78271 1.09696 5.12622
6.66740
-0.03577 1.02750 1.12762 0.59049
0.96793 0.50363 0.88972 1.28334 -0.72447
1408 Ind. Eng. Chem. Res., Vol. 29, No. 7 , 1990 3000x10~5,
, , , , , I
I
20001
,
I
, 1
Table 111. Triple-Point Molar Volumes for the Solid and Liauid States 0
“F
neon argon krypton xenon nitrogen oxygen fluorine methane ethylene carbon dioxide carbon tetrachloride cyclohexane benzene
800
600
200
‘“F
‘!I
24.54 83.79 115.94 161.36 63.14 54.36 53.48 90.66 104.00 216.55 250.00 279.83 278.69
13.98 24.66 29.66 37.10 29.31 23.77 21.60 30.94 39.06 29.09 87.1 101.1 77.0
16.21 28.21 34.31 42.66 32.29 24.81 22.29 35.37 42.62 37.37 91.96 106.30 87.31
1.160 1.144 1.157 1.150 1.102 1.044 1.032 1.143 1.091 1.285 1.056 1.051 1.134
‘“t”’
20
Zoo0
4
3
X
Figure 4. Relationship of
(p
- p * ) y versus x representing the highly
dense gaseous and liquid state regions for a number of simple and complex molecules.
m and n presented in Table 11. A review of the relationships presented in Figure 4 indicates that, although the viscosity behavior of each substance is a unique function of the density-temperature variable x , these relationships do not, as yet, conform to a generalized pattern. An additional parameter associated with the molecular orientation a t the liquid-solid phase transition would prove of value in combining these relationships into a single unifying behavior capable of predicting viscosity in a generalized manner. Generalized Viscosity Behavior for the Dense Gaseous and Liquid States Using the statistical mechanic arguments of Zandler et al. (1968) to account for the dissimilar behavior of the bulk thermodynamic and mechanical properties of structurally similar molecules, it is possible to show that the fractional volume expansion at the triple point defines a key parameter that should exert considerable influence in the near solid region of a liquid. Therefore, the volume expansion factor at the triple point defined as = Ult/ULlt (7) has been adopted and applied to as many substances for which experimental information is available. Information of this type is rather limited, particularly for the procurement of uStvalues, the solid molar volume at the triple point. Table I11 presents molar volumes for 13 substances for which adequate information exists to permit the calculation oft. These values range from t = 1.032 for fluorine to c = 1.285 for carbon dioxide, and consequently, in the extreme case, it may be stated that 1.000 < t 1.285, where t 1.000 represents a limiting state condition. Later, it was shown that for propane, t = 1.022, the lowest value encountered with the substances included in this study. In order to treat viscosity in a generalized manner, the density-temperature variable, x, has been related to t ,
-
2
0.3
I
l
0.4
l
l
0.5
l
l
l
0.6
0.7
l
l
0,8
l
~
09
l
10
B Figure 5. Generalized relationship for the viscosity behavior of fluids in the dense gaseous, saturated liquid, and compressed liquid states.
through a dependence similar to x , to define the preliminary comprehensive parameter, g = x / @ t ) * l X r , The introduction of this density-temperature-volume expansion factor parameter enabled the separate relationships of Figure 4 to collapse into the single dependence of ( p - p*)y versus g as shown in Figure 5. This final generalized relationship suggests that the functional dependence of ( p - p*)y on g is of the same form as eq 6, but instead involving g, the generalized modulus which includes collectively the influences of density, temperature, and the volume expansion factor. Preliminary estimation for the values p , s, and r needed to define the comprehensive variable, g, has been based on information relating to argon because of the reliable triple-point density values available for the solid and liquid states of this substance (Wilsak, 1982). By use of a nonlinear regression analysis, involving experimental viscosity measurements and corresponding values of density-temperature values expressed as x , the following best values result for argon: p = 0.976, s = 2.3566, and r = 0.6673. These constants have been adopted to define the generalized variable, X
g =
(0.g76t)2.3566/~0,‘M3
(8)
and therefore, the final generalized excess viscosity relationship becomes 105(p- p*)y = [ e ~ p ( 2 . 9 3 2 8 g 8+. ~4.5424g0.9228 ~~~ )I - 1 (9)
/
i
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1409 Table IV. Volume Expansion Factor, c, Solid Molar Volume, v , ~ ,Modified van der Waals Covolume Parameter, b , and Deviations for Viscosity Resulting from Equation 9 viscosity t = vlt/v,, uB,cm3/mol 2, n b, cm3/mol Pk % dev monatomic neon 1.14704 14.13 0.308 1.7905 11.84 81 4.32 argon 1.14396 24.66 20.51 134 3.36 0.293 1.7450 krypton 1.152 51 29.77 0.291 1.7390 24.90 19 0.85 xenon 37.47 1.13843 31.52 56 3.34 0.287 1.7270 diatomic 1.12160 28.79 0.289 1.7330 23.92 111 4.74 nitrogen 1.088 55 22.79 19.62 1.61 oxygen 0.288 1.7300 99 fluorine 1.085 31 20.54 17.69 2.23 71 0.288 1.7300 hydrocarbons 1.13654 31.12 0.288 1.7300 26.45 92 6.27 methane 1.082 40 ethylene 0.276 1.6942 39.38 33.24 14 1.26 1.054 13 43.24 36.29 1.29 ethane 0.274 1.6883 125 1.022 04 propane 58.98 49.62 68 6.38 0.271 1.6794 1.044 68 75.09 n-butane 0.274 1.6883 65.22 0.65 7 1.040 17 91.43 1.42 n-pentane 0.263 1.6559 75.09 243 1.052 31 n-hexane 107.81 90.45 1.76 32 0.260 1.6471 1.076 35 cyc1ohexane 98.76 0.277 1.6972 80.57 3.10 25 1.10083 benzene 65.59 3.35 35 79.31 0.271 1.6794 1.04132 n-heptane 124.07 5.71 36 0.263 1.6559 106.69 1.055 60 n-octane 142.46 119.45 0.259 1.6442 4.47 27 1.045 57 n-nonane 17 158.36 133.97 0.260 1.6471 2.58 1.050 95 175.32 n-decane 3.29 136 142.50 0.248 1.6123 1.040 46 n-octadecane 314.99 256.97 0.229 1.5579 2.99 8 miscellaneous 1.206 27 carbon dioxide 30.98 0.274 1.6883 24.10 57 1.39 carbon tetrachloride 1.073 89 85.63 0.272 1.6824 70.66 27 1.99 1.063 79 107.30 tetramethylsilane 92.43 0.273 1.6853 43 8.74 total 3.21 1563
where the excess viscosity, p - p*, is expressed in poises. Equation 9 represents in a generalized manner the excess viscosity behavior of nonpolar substances and therefore should be amenable to the calculation of t, the volume expansion factor, from information relating viscosity with density and temperature. This approach was adopted to establish e parameters for all the substances presented in Table IV, including argon. Whenever possible, these calculated values of c, presented in Table IV, can be compared with the corresponding values given in Table 111, which resulted directly from experimental measurements.
Estimation of e For the calculation of e, the molar volumes of the saturated liquid and solid states at the triple point must be available. Values for the saturated liquid molar volume are readily accessible from the literature. However, if unavailable, this value can be estimated from correlative relationships such as that given by Rackett (1970)
400
3001 200
vrt= 1.206 3 20 01/
NI
IO IO
At the triple point, this relationship becomes for saturated liquids ult = ugc(1-Tt/T3°'m
(11)
If experimental information is not available for the saturated liquid state, eq 11 can be applied with confidence to obtain the liquid molar volumes at the triple point. On the other hand, information relating to the solid molar volumes at the tiple point is available only for the few substances presented in Table 111. To expand this limited list of substances, the values of E presented in Table IV were used with experimental saturated molar liquid volumes to obtain corresponding solid molar volumes at the triple point for all the 24 substances included in this study. These values have been found to relate directly with the modified van der Waals covolume parameter b, as shown
I
20
I
I
I
30 40 50
I I I l l 70 100
I
200
1
m
I
1
500
b, cmyg-mole
Figure 6. Dependence of vSt upon the modified van der Waals covolume parameter, b.
in Figure 6. This parameter is defined through the modified van der Waals equation of state, (P+%)(u-b)=RT
+
where b = [(n - l)/(n + l)]u, and n = 22, + (42: l)1/2. Therefore, for the calculation of b, all three critical constants must be available. The linear dependence of uat upon b, shown in Figure 6, passes through the origin and can be expressed through the simple relationship, uSt = 1.20b (13)
1410 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
to account for molecular structures ranging from the simple spherical geometry of neon up to and through the long chainlike orientation of n-octadecane. The relationship of Figure 6 properly accommodates other configurations such as those associated with benzene and cyclohexane. Equation 13 proves to be more dependable for predicting solid molar volumes at the triple point over that suggested recently by Sharma (1980) and should be preferred over experimental measurements which are very difficult to obtain. On the other hand, experimental liquid molar volumes at the triple point are ordinarily more dependable, and as such they should be preferred over values predicted by eq 11. Example. Calculate the viscosity of p-xylene in the liquid state at 50 "C (323.15 K) and atmospheric pressure for which the density is reported to be p = 0.835 g/cm3. This substance is classified as very weakly polar with a dipole moment of 0.1 D. The following basic constants are associated with p-xylene ( M = 106.17): T, = 616.2 K, P, = 34.65 atm, p c = 0.280 g/cm3; Tt = 286.423 K, u, = 379.18 cm3/mol, z, = 0.260. (a) Volume Expansion Factor at the Triple Point, e = vlt/vst: The Rackett equation (eq 11) produces the following liquid molar volume a t the triple point, ult
= ~,g,[l - Tt/Tc]o.286 =
(379.18)(0.260)(1-28~423~616~2)0'~ = 122.91 cm3/mol to yield for the liquid density at the triple point the value plt = M/ult = 106.17/122.91 = 0.8638 g/cm3. Extrapolation of experimental p-xylene liquid density measurements to the triple point produces a molar volume of ult = 122.71 cm3/mol (Campbell, 1983). The corresponding solid molar volume is calculated by the use of eqs 1 2 and 13, where = n = 2z, + (42; 2(0.260) + [4(0.260)2+ 1I1l2= 1.647
+
and n-1 b=- 1.647 - 1379.18 = 92.65 cm3/mol n + 1'' - 1.647 1 to yield for the solid molar volume at the triple point of p-xylene, the value ust = 1.20(92.65) = 111.18 cm3/mol. No experimental value for ust is available for p-xylene. Therefore, the volume expansion factor at the triple point for this substance becomes e = 122.91/111.18 = 1.1055. (b) Determination of p*, the Viscosity for the Dilute Gaseous State at 50 "C. By use of eq 1, p*p, the normalized viscosity for the dilute gaseous state of p-xylene at 50 "C (TR= 323.15/616.2 = 0.5244), becomes - 35.50/e~p(10/0.5244~/~) = 105p*8 = 0.576(0.5244)0.91 0.320 X lo5 where p = uC1l6/M1/2Pc1/2 = 379.181/6/106.171/234.65112 = 0.04436 to yield for the dilute gaseous state of p-xylene a t 50 "C, a viscosity value of p* = 0.320 X 10-5/0.04436 = 7.217 X P
+
(c) Determination of p, the Liquid Viscosity at 50 "C. For this fluid state, the triple point becomes the reference state to produce the following normalized temperature and density: r = T / T t = 323.15/286.423 = 1.1282 w = p / p l t = 0.835/0.8638 = 0.9667
Thus, the density-temperature variable becomes
x=-=
w
0*9667 = 0,9593 1.12820.07(0.9667)273
T0.07u273
and therefore the generalized density-temperature-volume expansion factor parameter becomes X 0.9593 g = (0.g76t)2.3566/~0.83(0,976 x 1.1055)2.3566/0.9593~.~'~ 0.7980 By using eq 9, the generalized excess viscosity, ( p - p * ) y , then becomes
+ 105(p- p*)y = e~p[2.9328(0.7980)~.~~~* 4.5424(0.7980)0.9228] - 1 = 61.588 X lo5 where U,2J3
--
= M 1/2T,'/2
122.912/3 = 0.1418 106.171/2286.4231/2
to yield for the viscosity of liquid p-xylene at 50 "C 61*588 + 7.217 X = 411.55 X P 0.1418 Vargaftik (1975) reports for liquid p-xylene at 50 "C and atmospheric pressure the experimental value of p = 456 X P, to yield a deviation of 3.2%. P =
General Remarks The anomaly of the viscosity behavior encountered in the immediate critical point region has not been included in this study since such an undertaking deserves a separate comprehensive investigation. However, the present study overcomes the limitations associated with the prediction of viscosity of the liquid state of substances existing below their normal boiling point and extends in the vicinity of temperatures approaching their freezing state. This extension requires, besides the involvement of the critical constants, information relating to the triple point. The final expression given by eq 9 accommodates not only the dense gaseous state but the saturated and compressed liquid states as well. The introduction of t, the volume expansion factor at the triple point, has been found to possess a unifying influence which permits the extension of the corresponding states principle for the behavior of viscosity into the saturated and compressed liquid states. The generalized parameter, g, requires no adjustable parameters and is found to apply for nonpolar substances through the relationship given by eq 9. This relationship defines the excess viscosity, ( p - p * ) y , for the entire fluid region, including the saturated and compressed liquid states existing for all temperatures and particularly for temperatures existing below the normal boiling point and approaching the freezing curve. The results of the present study overcome limitations associated with the method of Ely and Hanley (1981) that utilizes the critical constants, the Pitzer acentric factor, and a shape factor to predict viscosities for the entire fluid region. These investigators present an overall average deviation of 8.42% (1869 points) for 35 hydrocarbons and carbon dioxide, however, they state that "these deviations become somewhat worse, and negative as the freezing point of the fluid is approached." Table IV presents average deviations for each of the 24 substances included in this study. These deviations range from 0.65% (7 points) for n-butane to 8.74% (43 points) for tetramethylsilane. The overall average deviation for the 24 substances is 3.21% (1563 points). This deviation falls within the accuracy of experimental error.
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1411
Nomenclature a = van der Waals pressure parameter, eq 12 b, d = universal constants, b = 2.9328 and d = 4.5424, eq 9 b = van der Waals covolume parameter, cm3/mol, eq 12 f = exponent, eq 2 g = density-temperature-volume expansion factor parameter, eq 8 k = proportionality constant, eq 2 m,n = exponents, eq 6 M = molecular weight n = exponent for modified van der Waals equation, eq 12 P = pressure, atm P, = critical pressure, atm R = gas constant T = temperature, K T , = critical temperature, K TR = reduced temperature, T I T , Tt = triple-point temperature, K u = molar volume, cm3/mol u, = critical volume, cm3/mol ult = liquid molar volume at triple point, cm3/mol OR = reduced volume, u/u, vet = solid molar volume at triple point, cm3/mol x = density-temperature variable, w/ro~07w*"9 , eq 5 z, = compressibility factor, Pcu,/RTc Greek Letters a = exponent, eq 2; also a universal constant, a = 8.3264, eq 9
a, 0 =
coefficients, eq 6
/3 = viscosity parameter, ~ , l l 6 / M ~ / ~ P , 'eq / 2 ,1 y = viscosity parameter, ~ ~ t 2 / ~ / M ~eq/ 2~ T ~ l / ~ ,
= volume expansion factor at triple point, ult/uSt, eq 7 = universal constant, 0.9228, eq 9 p = viscosity of fluid P p* = viscosity of dilute gas, P f = viscosity modulus, Tcl/6/Ml/zP2/3 p = density, g/cm3 pc = critical density, g/cm3 pR = reduced density, p / p c plt = triple-point liquid density, g/cm3 7 = normalized temperature, T I T , w = normalized density, p / p l t t
K
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Forster, S. Viscosity Measurements of Liquid Neon, Argon and Nitrogen. Cryogenics 1963,3,176-177. Gubbins, K.E.; Tham, M. J. Free Volume Theory for Viscosity of Simple Nonpolar Liquids. AZChE J . 1969, 15, 264-271. Haynes, W. M. Viscosity of Gaseous and Liquid Argon. Physica 1973,67,440-470. Haynes, W. M. Measurement of the Viscosity of Compressed Gaseous and Liquid Fluorine. Physica 1974, 76, 1-20. Haynes, W. M. Measurements of the Viscosity of Compressed Gaseous and Liquid Oxygen. Physica 1977,89A,569-582. Herreman, W.; Grevendonk, W. An experimental study of the shear viscosity of liquid neon. Cryogenics 1974,14,395-398. Hildebrand, J. H. Motions of Molecules in Liquids: Viscosity and Diffusivity. Science 1971, 174,490-493. Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. Jonas, J.; Hasha, D.; Huang, S. G. Density Effects on Transport Properties of Liquid Cyclohexane. J. Phys. Chem. 1980, 84, 109-112. Jossi, J. A.; Stiel, L. I.; Thodos, G. The Viscosity of Pure Substances in the Dense Gaseous and Liquid Phases. AIChE J. 1962, 8, 59-63. Kamerlingh Onnes, H. Allgeine Theorie der Fliissigkeiten. Verhand, Akad. Amsterdam 1881,21, 8-53. Lee, L. L. S. The Percus-Yevick Theory of Radial Distribution Functions, Thermodynamic Properties and Vapor-Liquid Equilibria of Binary Mixtures of Molecules Interacting with the Kihara Potential and Applications to the Argon-Nitrogen System. Ph.D. Dissertation, Northwestern University, Evanston, IL, 1971. Lee, A. L.;Ellington, R. T. Viscosity of n-Pentane. J. Chem. Eng. Data 1965,10,101-104. Lennert, D. A.; Thodos, G. Application of the Enskog Relationships for Prediction of the Transport Properties of Simple Substances. Ind. Eng. Chem. Fundam. 1965,4,139-141. Licht, W., Jr.; Stechert, D. G. The Variation of Viscosity of Gases and Vapors with Temperature. J. Phys. Chem. 1944,48,23-48. Macedo, P. B.; Litovitz, T. A. On the Relative Roles of Free Volume and Activation Energy in the Viscosity of Liquids. J. Chem. Phys. 1965,42,245-256. Maxwell, J. C. On the Dynamical Theory of Gases. Phil. Mag. J. Sci. 1868,35 (Series 4), 129-145, 185-217. McCool, M. A.; Woolf,L. A. Pressure and Temperature Dependence of the Self-diffusion of Carbon Tetrachloride. J. Chem. SOC., Faraday Trans. Z1972,168, 1971-1981. Parkhurst, H. J., Jr.; Jonas, J. Dense Liquids 11. The effect of density and temperature and viscosity of tetramethylsilane and benzene. J. Chem. Phys. 1975,63,2705-2709. Rackett, H. G. Equation of State for Saturated Liquids. J. Chem. Eng. Data 1970,15,514-517. Rorris, E. Generalized Viscosity Behavior for the Inert Gases in their Dilute, Dense Gaseous, Saturated Liquid and Compressed Liquid States. M.S. Thesis, Northwestern University, Evanston, IL, 1979. Sharma, B. K. Corresponding states correlations and triple point solid volume of simple liquids. Phys. Stat. Sol. 1980, 99 (2), K121-Kl26. Shimotake, H.; Thodos, G. Viscosity: Reduced-State Correlation for the Inert Gases. AZChE J. 1958,4,257-262. Slyusar, V. P.; Rudenko, N. S.; Tret'yakov, V. M. Viscosity of elementary substances along the saturation line and under pressure. 111. Neon. Ukr. Fiz. Zh. 1973, 18, 190-194. Stiel, L.I.; Thodos, G. The Viscosity of Nonpolar Gases a t Normal Pressures. AIChE J. 1961, 7,611-615. Trappeniers, N. J.; Botzen, A.; van den Berg, H. R.; van Oosten, J. The Viscosity of Neon Between 25 "C and 75 OC at Pressures up to 1800 Atmospheres. Corresponding States for the Viscosity of the Noble Gases up to High Densities. Physica 1964,30,985-996. Trappeniers, N. J.; van der Gulik, P. S.; van den Hooff, H. The Viscosity of Argon at Very High Pressure, up to the Melting Line. Chem. Phys. Lett. 1980, 70,438-443. Ulybin, S. A,; Makarushkin, V. I. Viscosity of carbon dioxide at 220-1300 "K and up to 300 megapascals. Teploenergetika 1976, 6, 65-69. Ulybin, S. A.; Makarushkin, V. I. Viscosity of Xenon a t Temperatures of 170-1300 "K and Pressures up to lo00 bar. High Temp. 1977,15, 430-434. Ulybin, S. A.; Makarushkin, V. I.; Skorodumov, S. V. Viscosity of Krypton at Temperatures 120-1300 OK and Pressures up to IO8 dynes/cm3. High Temp. 1978, 16, 233-238.
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Received f o r review January 18, 1990 Accepted February 15, 1990
Adsorption Equilibrium of Toluene from Supercritical Carbon Dioxide on Activated Carbon Chung-Sung Tan* and Din-Chung Liou Department of Chemical Engineering, National Tsing H u a University, Hsinchu, Taiwan 30043, Republic of China
The equilibrium loadings of toluene from supercritical carbon dioxide on activated carbon are reported in this study. The experimental data were obtained by measuring the outlet concentration of toluene from a column packed with activated carbon until it reached the inlet concentration. When the densities were fixed a t 0.32, 0.45, and 0.69 g/cm3, the Langmuir isotherm expression was found to correlate the experimental data satisfactorily for the temperatures 308, 318, and 328 K. As the concentration of toluene in supercritical carbon dioxide was kept constant, the crossover of the equilibrium loadings at different temperatures was observed at relatively high pressures. The pressure a t which the crossover occurred increased with increasing concentration.
Introduction Because supercritical carbon dioxide possesses several special characteristics and physicochemical properties, e.g., nonflammable, nontoxic, relatively inexpensive, higher mass-transfer rate, and adjustable extraction power for organic compounds depending on the density, it has proved to be an effective solvent for regenerating activated carbon loaded with organic compounds (Model1 et al., 1979; deFilippi et al., 1980; Tan and Liou, 1988, 1989a,b). The regeneration efficiency, defined as the fraction of the loaded amount to be desorbed at a fixed time, is in general dependent on the regeneration temperature and pressure. Tan and Liou (1988, 1989a,b) observed that an optimal temperature exists when the pressure is larger than a certain value. Below this value, the regeneration efficiency decreases with increasing temperature. But when the density was used as the operation variable, instead of the pressure, the regeneration efficiency was found to increase with temperature for a fixed density, and the abovementioned reversal of the temperature dependence was no longer observed no matter what density was used. Due to the lack of fundamental information, such as the adsorption equilibrium isotherm and the effective diffusion coefficient of organic compound in activated carbon, a lumped resistance model was proposed to interpret the regeneration data (Tan and Liou, 1988). Though this model could fit the experimental data well, it could not describe the reversal of the temperature dependence when the pressure was used as the operation variable. A more rigorous model involving adsorption equilibrium and intraparticle mass transfer, as suggested by Recasens et al. (1989), could also interpret the same data well, but the
* To whom correspondence should be addressed. 0888-5885/90/2629-1412$02.50/0
adsorption isotherm used in this model was lacking in experimental verification. In order to understand more about mass transfer during the regeneration, it is essential to have adsorption equilibrium data at supercritical conditions. Such data are quite scarce in the literature. As far as we know, only adsorption data for phenol on activated carbon from supercritical carbon dioxide have been reported (Kander and Paulaities, 1983). For this system, a general form of the adsorption isotherm covering a large range of the operating temperature and pressure was not obtainable. To predict the adsorption equilibria at supercritical conditions, one also needs adsorption equilibria data from aqueous solutions. In this study, measurements of the adsorption equilibrium of toluene from supercritical carbon dioxide on activated carbon were conducted. This system was selected because of the regeneration data for this system have been well documented (Tan and Liou, 1989a).
Experimental Section The experimental apparatus used for the adsorption measurements at elevated pressures is illustrated in Figure 1. The activated carbon (Degussa, WSIV) was first screened to obtain a 18-20-mesh fraction (the average particle size was 0.1 cm). This fraction was boiled in deionized water to remove fines and then was dried in an oven at 393 K. After drying, about 6.5 g of the prepared activated carbon was loaded in a 2.12-cm4.d. stainless steel 316 tube (adsorber). Glass beads of 0.1-cm diameter were also packed above and below the activated carbon packing, both with height of about 3 cm. With these pre- and postpacking sections, a uniform flow distribution in the adsorber may be achieved according to the observation of Tan and Wu (1988). 0 1990 American Chemical Society
