It has already been mentioned that strong pore diffusion effects were present and thus, attempts to infer actual mechanisms from these results should be discouraged. The occurrence of two simultaneous reactions (with common components) together with multicomponent surface and diffusion phenomena would be a significant undertaking worthy of future research. The fact that the reaction orders for hydrogen and steam were relatively high could be indications that only the outermost portions of the catalyst were being utilized. However, the problem is sufficiently complex to render this to speculation only. Whereas many of the questions which have been raised here must await further research which should be conducted in the absence of pore diffusion effects, the results do indicate the importance of the shift reaction. This should be fully appreciated before attempting to utilize methanation rate data for feed compositions with carbon oxide concentrations much different than those under which the original data were taken. Since it is likely that first generation methanators will be of the fixed bed variety and will employ recycle, the rate data collected over the course of this investigation were fit to a straightforward power law model. The following rate expression was obtained
with the methanation rate, rCH4, expressed in g-mol/g of catalyst-h and the partial pressures expressed in atmospheres. The reaction rate orders in this expression are slightly different than those shown in Figures 3-5 due to interactive effects among the various experiments which are accounted for in eq 3. In view of the discussion which precedes this expression, it is important to appreciate its limitations. It is valid for catalysts at least as large as 1.6 mm, for temperatures greater than or equal to 300 "C, and for methanator feeds of high CO:,
concentration. There is some question as to the maximum concentrations of steam which would be applicable because of the possible loss of catalyst activity if the concentrations were too high. The maximum concentration of steam which was employed in this study was 60%. L i t e r a t u r e Cited Akers, W. W., White, R. R., Chem. Eng. Prog., 44, 553 (1948). Betta. R. A.. Piken, A. G., Shelef, M., J. Catal., 35, 54 (1974). Binder, G. C., White, R. R.,Chem. Eng. Prog., 48, 563 (1950). Dent, F. J., Moignard, L. A., Eastwood. A. H.. Blackturn, W. H., Hebden, D., "An investigation into the Catalytic Synthesis of Methane for Town Gas Manufacture", Gas Res. Board Comm., No. 20/10 (1948). Fontaine, R. W., Harriot, P., 41st Chemical Engineering Symposium, i8EC Division, ACS, Pittsubrgh, Pa., Apr 1975. Lee, A. L., Feidkuchner, H. L., Tajbl, D. G., Prepr. Pap. Nat. Meet., Div. Fuel Chem., Am. Chem. SOC.,14, 126 (1970). McGill, R. N., Richardson, J. T., 41st Chemical Engineering Symposium, 18EC Division, ACS, Pittsburgh, Pa., Apr 1975. Mills, G. A,, Steffgen. F. W., Catal. Rev., 8, 159 (1973). Pursiey, J. A., White, R. R., Sliepcevich, C.. Chem. Eng. Symp. Ssr., 48, 51 (1952). Richardson, J. T., Friedrich, H., McGili, R. N., J. Catal., 37, l(1975). Schoubye, P., J. Catal., 14, 238 (1969). Schoubye, P., J. Catal., 18, 118 (1970). Smith, J. M., "Chemical Engineering Kinetics", 2d ed. pp 413-414, McGraw-Hili, New York, N.Y., 1970. Van Herwijnen. T., Van Doesburg, H.,DeJong, W. A,, J. Catal., 28, 391 (1973). Vannice, M. A,, J. Catal., 37, 462 (1975). Vlasenko, V. M., Yuzefovich, G. E., Russ. Chem. Rev., 38, 728 (1969). Wei, V. T., Chen, J., 4th Joint AIChE-CSChE Conference, Vancouver, B.C., Sept 1973. Wen, C. Y., Chen. P. W., Kato, K., Galli, A. F., Prepr. Pap. Nat. Meet., Div. Fuel Chem., Am. Chem. Soc., 12, 104 (1968). White, G., 41st Chemical Engineering Symposium, I&EC Division, ACS. Pittsburgh, Pa., Apr 1975.
Recei~ledfor reuieu: January 9, 1976 Accepted August 2, 1976 T h e financial support o f t h i s research on the p a r t o f U n i o n Carbide Corporation, T a r r y t o w n Technical Center, is gratefully acknowledged.
An Antoine Type Equation for Liquid Viscosity Dependency to Temperature Edward Goletz, Jr., and Dimitrios Tassios* New Jersey lnstitute of Technology, Newark, New Jersey 07102
An Antoine type equation, containing three adjustable parameters, is presented that successfully correlates liquid viscosity to temperature over the full range of boiling to freezing point. A maximum error of 5 % was observed for 60 hydrocarbons and 14 polar compounds. The only exception was propene with a maximum error of 12 %. A simplified two-parameter form of the equation correlated all compounds to within 5 YO of the reported viscosity, except for the polar compounds with hydrogen bonding. The two-parameter form performs better than the Andrade, Walther, and Doolittle equations in both the correlation and extrapolation of liquid viscosity data.
Introduction
The viscosities of fluids are required in most engineering calculations where fluid flow or mixing is an important factor, such as in the design of reactors, heat exchangers, fractionating towers, cooling towers, and distillation units. For Newtonian fluids, viscosity is defined as the measure of the internal fluid friction, which.is the constant of proportionality between the shear stress per unit area a t any point and the velocity gradient. Liquid viscosity is characterized by strong temperature
dependency especially close to the boiling and freezing points. While numerous correlations have been proposed, a t temperatures below the boiling point and above the freezing point, the viscosity-temperature data are reasonably well represented (Reid and Sherwood, 1966) by the Andrade equation (1930)
n = aeb/T
(1)
where n = viscosity of liquid, absolute, and T = absolute temperature. Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
75
Table I. Sources and Type of Viscosity Data Compound Hydrocarbons and water Nitriles
Type
I Ii
Reference
Kinematic Absolute
Acetone MEK and chloroform Alcohols
Absolute Absolute
Ethylene glycol Aniline
Absolute Absolute
180
American Petroleum Institute (1953) Timmermans (1950);Weast (1970);Wright (1961) Weast (1970) Lange (1967);Timmermans (1950);Weast (1970) Timmermans (1950); Weast (1970) Curme (1953) Lange (1967); Weast (1970)
Absolute
Table 11. Sums of Squares for Hydrocarbon Correlations. Z Varied in Equation c = 239 + Ztb
I
I
1
b
-
I
0
OCTADECANE
. I WADECANE
' DODECANE
160
CYCLOHEXANE
140
A
-
PROPEhE
OCTANE
100
80
60
Sums of squares x lo3 Values of Z: Compound
-0.57
-0.38
-0.19
Hexane Octane Decane Dodecane Tetradecane Hexadecane Octadecane Eicosane
4.53 34.27 80.43 187.02 430.90
1.26 2.18 7.12 14.79 41.84 97.01 188.00 252.37
10.99 6.94 4.96 9.95 6.61 9.78 11.04 11.95
28.26 27.94 29.34 46.73 68.96 122.98 177.90 213.56
Values of Z:
-0.37
-0.19
-0.01
Benzene Propene" Decene Cyclohexane
0.04 4.09 0.93 11.75
0.08 2.26 2.07 17.75
0.12 1.08 3.82 40.0
0
2580.19
40
0.00
-A
0 -0.60
-c.40 VALUES OF Z IN EQCATICN:
A, -0.00
-0.20
C
-
239
The General Expression for C An expression, similar to that recommended (Dreisbach, 1959) for C' in the Antoine equation, was considered
c = 239 + Ztb The Andrade equation is similar to the commonly used vapor pressure-temperature correlation
- B'/T
(2)
The Antoine equation, which is extensively used for more accurate vapor pressure-temperature calculations (Thomas, 1946), simply substitutes ( t C') for T in eq 2
+
log P = A'
- B'/(t
+ C')
(3)
On the basis of the excellent results obtained for correlating vapor pressure data with temperature using the Antoine equation, the performance of a modified Andrade equation log n = A
+ B / ( C + t ) , t "C
(4)
is examined in this paper. Two procedures were followed in establishing the values of the parameters A, B, and C. In the first, a regression of the data was employed in determining all three constants. In the second, following Dreisbach's (1959) suggestion that C' = 239 - 0.19tb ( t b : normal boiling point), a general expression for C was sought. Once such an expression was found and C was determined, the values of A and B were established by regressing all the viscosity-temperature data for each compound. The second approach, because it results in a two-adjustable parameter expression that is simpler to use, is given more attention in this paper. Data sources for the 60 hydrocarbons and the 14 polar compounds used in this study are presented in Table I. Kinematic viscosity data were reported for the hydrocarbons and water and were used without conversion to absolute viscosity. 76
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
Z tb
This decision was based on convenience as well as on the desire to establish the applicability of this empirical correlation to both types of viscosity data.
Sums of squares not multiplied by lo3.
log P = A'
+
Figure 1. Best value of 2 for hydrocarbons.
(5)
To establish the best value for 2, the viscosity-temperature data for 12 randomly chosen hydrocarbons were fitted into eq 4 for several values of C obtained by varying 2 in eq 5. The resulting quality of fit, defined by the sum of squares N
s = c ( n u b s - ncal)i2
(6)
1=1
where N is the number of data points, is presented in Table 11. Typical results are shown in Figure 1. While, for most alkanes, a minimum around 2 = -0.19 is realized, minima a t other values of 2 are observed. For example, for n-octane, the minimum value occurs around 2 = -0.40. However, with this value, compounds such as octadecane and hexadecane would have viscosity errors greater than 10% (Goletz, 1973). Overall, considering the sharp minima observed around Z = -0.19 and the shallow ones for the other cases, a value of 2 = -0.19, similar to that for vapor pressure data, represents a good choice. This conclusion is further substantiated from the quality of the results obtained. Results Table 111 presents the accuracy of correlating liquid viscosity to temperature for 12 n-alkanes by using the two- and three-parameter forms of the proposed equation. Twenty data points per compound were used where available. The average errors were 0.68% and 1.03% and the maximum errors were 3.3% and 4.7% for the two- and three-parameter forms, respectively.
Table 111. Correlations for Viscosities of Liquid N-Alkanes. Modified Andrade Equation-3 Parameters vs. 2 Parameters Range of errors
Av % error
No. of carbon atoms
Temp range of data, "C
No. of data points
2 3 4 5 6 7 8 9 10
-175 to -90 -185 to -45 -90 to -5 -125 to 35 -95 to 65 -90 to 95 -55 to 125 -50 to 150 -25 to 170 -25 to 195 -5 to 215 -5 to 235
17
11
12 13
3-Parameter
20 13 20 20 20 20 20 20
20 20 20
Average % error for 230 data points
2-Parameter
0.8 0.8 0.7 0.6 0.5 0.7 0.7 0.6 0.5 0.6 0.7 1.0
0.9 1.1 1.1 1.4 1.0 1.7 0.9 1.0 0.6 0.9 0.7 1.0
0.68
1.03 Range of errors
%Parameter
2-Parameter
0.0 to 1.3 0.1 to 2.4 0.1 to 1.8 0.0 to 1.3 0.0 to 1.5 0.0 to 1.7 0.0 to 3.3 0.0 to 1.3 0.0 to 1.1 0.0 to 1.2 0.0 to 2.0 0.0 to 2.9
0.0 to 2.9 0.0 to 2.5 0.1 to 2.9 0.0 to 4.7
0.0 to 3.3
0.0 to 4.7
0.1 to 2.8 0.0 to 4.1 0.0 to 3.8 0.0 to 2.8 0.0 to 1.8 0.0 to 2.0 0.0 to 2.6 0.0 to 3.5
Table IV. Correlations for Viscosities of Polar Compounds Constants in eq 4 Compound Hydrogen cyanide (HCN) Acetonitrile Propionitrile Caprinitrile Acetone Methyl ethyl ketone Chloroform
Temp range of data, "C
No. of data points
Av % error
Range of errors
A
B
C"
-13.3 to 25
10
0.2
0.0 to 1.0
-4.05224
610.39
234.1
-20 0 20 -80
6 8 9 12 4
0.8 0.7 0.9 1.0 0.1
0.3 to 1.8 0.2 t o 1.6 0.0 to 2.5 0.0 to 2.1 0.0 to 0.3
-3.28828 -3.63853 -3.50479 -3.34732 -3.61610
550.96 672.91 743.33 554.95 675.49
223.5 220.5 192.7 228.4 223.8
9
0.9
0.0 to 2.4
-3.15506
638.44
227.3
to 40 to80 to 140 to 41
0 to80
-13
to 60
Following the successful performance of both the two- and three-parameter forms, attention was focused on the twoparameter form for two reasons: (i) it is simpler to use with the values of A and B obtainable through a plot of log n vs. l l ( t C). (ii) as shown in Table 111,the two-parameter correlation gives errors that are comparable to those resulting from the three-parameter form. The API data on 17 n-alkylcyclohexanes, 5 n-alkylbenzenes, and 19 n-monoolefins were tested. For the 60 hydrocarbons, including a total of 762 data points, the maximum error was 4.8% with an average absolute error of 0.6%. The only exception was propene with a maximum error of 12% (near the freezing point) and an average absolute error of 4%. Detailed results are given by Goletz (1973).
+
P o l a r Compounds As expected, viscosity-temperature correlation for polar liquids is more difficult than for nonpolar ones (Doolittle, 1951b). The polar compounds used in this study were separated into two groups as follows: (1)polar compounds where hydrogen bonding is not significant; (2) polar compounds where hydrogen bonding is significant, such as the alcohols, water, and ethylene glycol. In the first group, liquid viscosity-temperature data of four nitrile compounds, acetone, methyl ethyl ketone, and chloroform were investigated. Unlike the hydrocarbons, the viscosities of these compounds were reported in the literature in centipoises and, therefore, absolute viscosity was correlated with temperature. The results of these correlations are shown in Table IV. The viscosity-temperature correlation for all these compounds was excellent, having a maximum error of 2.5% and an average error of only 0.7% for the 58 data points investigated. With the exception of HCN, however, no ex-
perimental data close to the freezing and boiling points were available. In the second group, compounds where hydrogen bonding is present, the value of 2 = -0.19 no longer gives satisfactory correlations. As shown in Table V, however, successful results were obtained by using different values for 2. Notice that a single value of 2 = 0.4 was sufficient for all four alcohols. Discussion a n d Conclusions Table VI presents a comparison of the two-parameter expression with some of the often used equations for determining viscosity, especially for hydrocarbons: the Andrade, the Doolittle (1951a,b), and the Walther (1933) equations. The last one, a double exponential type, is well known as the basis of the ASTM paper for determining the viscosity-temperature profile of lubricating oils (Doolittle, 1951b). As shown, the Doolittle equation is better than the Walther and Andrade equations, but not as good as the two-parameter equation, specifically at temperatures near the freezing point of nheptadecane. In addition, the two-parameter equation is easier to use than the Doolittle and Walther equations. The absolute viscosity data reported by Doolittle for n-heptadecane were used in this comparison. Extrapolation from a limited number of experimental data points is often desirable in practical applications. For this purpose, four points were randomly selected-mid-range between the freezing and boiling points-and viscosities outside this range, especially close to the two limits, were predicted for five points using the Andrade and the two-parameter expression. The results, summarized in Table VII, clearly demonstrate the superiority of the two-parameter Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
77
Table V. Correlations for Viscosities of Polar Compounds with Hydrogen Bonding Temp range of data, "C
Compound Alcohols Methyl Ethyl n-Propyl n-Butyl Others" Water Aniline Ethylene glycol (I
C = 239
-84.2 -98.1 0 -50.9 2 5 4
No. of
data point
Constants in eq 4
Av % error
Range of errors
A
B
C
to 50 to 70 to 100 to 100
16 12 12
1.5 2.4 1.4 0.7
0.7 to 2.6 0.2 to 5.0 0.4 to 3.3 0.0 to 1.4
-4.70439 -5.12473 -6.81202 -7.16916
1190.83 1534.63 2269.71 2528.01
264.8 270.3 277.8 286.2
to95 to 100 to177
20 15 19
0.3 1.5 0.7
0.0 to 0.8 0.0 to 4.6 0.0 to 3.5
-3.63148 -2.88534 -3.66017
542.05 545.45 1009.01
129.0 104.7 130.5
+ Zti,, where 2 equals f 0 . 4 , -1.1, -0.55,
11
-0.73 for the alcohols, water, ethylene glycol, and aniline, respectively.
Table VI. Comparison of Observed Viscosities of n-Heptadecane with Calculated values % Error
Temp, K
Viscosity observed, P x 10'
Andraden
293.2 323.2 373.2 423.2 473.2 523.2 573.2
43.09 21.70 10.78 5.984 3.938 2.785 2.030
-6.2 +3.4 +5.2 f1.7 -1.1 -2.4 -0.6
Doolittle (1951b). I n n = -2.94718
Andrade equation
Two-parameter equation
Compound
Avabs error
Avabs error
n-Decane n-Tetradecane n-Octadecane n-Butylcyclohexane Acetone 1-Decene
5.56 8.04 7.48 7.42 2.74 2.56
7.6 16.7 15.2 11.2 8.7 3.8 _ _ _ -
0.86 2.50 2.88 4.32 1.78 1.06
1.6 4.7 4.5 7.3 3.7 2.1
5.63
10.53
2.23
3.98
_-
Av
Doolittlea
Eq 4h
-4.3 -0.2 +0.9 -0.1 -0.6 -0.8 $0.8
-2.3 +1.1 f0.9 -0.5 -0.9 -0.7 -1.1
+2.1 0.0 -2.7 -3.8 -2.4 +0.6 +6.4
+ 929.1/(181.6 + t ) ,
Table VII. Comparison of Extrapolated Viscosities
Max error
Walther
Max error
expression for the six compounds tested. For 1 4 0 f the 30 points, the Andrade equation gave errors over 5% with a maximum error of 16.7%. For the two-parameter expression, the maximum error was 7.3% with only two points exceeding 5% error (Goletz, 1973).
In conclusion, use of eq 4 with 2 = -0.19 results in a twoparameter equation that correlates viscosity-temperature data for liquid hydrocarbons and polar compounds with no or weak hydrogen bonding to within 5% of the reported values. For polar compounds, with significant hydrogen bonding the three-parameter form is needed for correlation within 5%. The two-parameter form is clearly better than the Andrade equation. Even for the hydrogen-bonded compounds, the two-parameter expression (2 = 0.19) gave results that are equivalent t o the Andrade equation as seen in Table VIII. Use of the two-parameter equation is recommended hence, for its accuracy and easy use. The three-parameter equation yields, of course, better accuracy, but caution should be exercised in obtaining the parameter values by regression. Poor choice of starting valuejn the regression subroutine can lead to errors larger than those obtained from the two-parameter one. For example, for n-eicosane, maximum errors of 4.@?/0and 2.7% were obtained by the three and two parameter forms, respectively (Goletz, 1973). In closing, it should be mentioned that the correlations are applicable to both absolute and kinematic viscosity data.
Table VIII. Polar Compounds with Hydrogen Bonding Andrade equation Compound Methyl alcohol Ethyl alcohol n-Propyl alcohol n-Butyl alcohol Water Ethylene glycol Ani1ine
Sums of squares 0.520 0.80 0.015 0.16 33.10 110.0 4.90
C = 239 - 0.19t1,. 78
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1 , 1977
Antoine type equation" Max error 3.0 5.0 3.3 2.0 6.6 18.4 17.6
Sums of squares 0.018 40.78 0.008 14.55 20.57 35.2 2.83
Max error 4.5 17.3 2.9 10.4 4.8 11.0 14.6
Nomenclature A', B', C' = constants in eq 3 A , B, C = constants in eq 4 a, b = constants in eq 1 N = number of data points n = liquid viscosity, absolute or kinematic (see Table I) s = sum of squares, eq 6 T = absolute temperature, K t = temperature, "C t h = normal boiling point, "C Z = constant,eq5
Subscripts obs = observedvalue cal = calculated value
Press, Pittsburgh, Pa., 1953. Andrade, E. N. da C.. Nature (Londoh),125, 309 (1930). Curine, G. O., "Glycols", Reinhold, New York, N.Y., 1953. Doolittle, A. K.,J. Appl. Pbys.,22, 1473 (1951a). Doolittle, A. K., J. Appl. Pbys.,22, 1032 (1951b). Dreisbach, R. R., Adv. Chem. Ser., No. 22, 4 (1959). Goletz, E.. M.S. Thesis, Newark College of Engineering, 1973. Lange, N. A., "Handbook of Chemistry", revised 10th ed. McGraw-Hill, New York, N.Y., 1967. Reid, R. C., Sherwood, T. K., "The Properties of Gases and Liquids", 2d ed, Chapter 9, McGraw-Hill. New York, N.Y., 1966. Timmermans, J., "Physico-Chemical Constants of Pure Organic Compounds", VoI. I, Elsevier, New York, N.Y., 1950. Thomas, G. W., Cbem. Rev., 38, l(1946). Walther, C., Proc. WorldPet. Congr., London, 2, 419 (1933). Weast, R. C.. "Handbook of Chemistry and Physics", 51st ed, Chemical Rubber Co., Cleveland, Ohio, 1970-1971 Wright, F. J., J. Cbem. Eng. Data, 6, 454 (1961).
L i t e r a t u r e Cited
Receiued for reuiew January 15, 1976 Accepted September 16,1976
American Petroleum Institute, "Selected Values of Physical and Thermodynamic Properties of Hydrocarbonsand Related Compounds", Project 44, Carnegie
Calculation of Sieve and Valve Tray Efficiencies in Column Scale-up Gavin R. Garrett,'
Robert H. Anderson, and Matthew Van Winkle
University of Texas, Department of Chemical Engineering, Austin, Texas 787 12
This article presents the results of an experimental fractionation efficiency study comparing the results obtained from a I-in. Oldershaw and an 18-in. column. The two binary systems used were benzene-1-propanol and l-propanol-toluene. The efficiency of the large fractionation column was calculated from the measured efficiency of a small sieve column, using a method similar to that of the "AIChE Bubble Tray Design Manual". Calculated efficiencies for the large sieve tray column were in close agreement with experimental results. The calculated efficiencies for the large valve tray column were not in as close agreement with the experimentally determined efficiencies.
Distillation is such a widely used industrial process for separating chemical components that no small effort has been expended in trying to improve the performance of distillation equipment. Many types of equilibrium devices have been considered for separations use in attempting to attain equilibrium of vapor-liquid phases. However, even the best device is less than perfectly efficient. Also, any fractionation design calculation method, no matter how sophisticated or complex, must still ultimately utilize some value for efficiency for any given column. Although various efficiency data and efficiency equations have been reported in the literature, there has not been reported a generalized method for efficiency prediction of a plant-sized column which utilizes, as a basis, the efficiency from a small laboratory column. The purpose of this effort is to examine a way to calculate the efficiency of a small plantsized column, first given a measured efficiency on a similar laboratory-sized column. Scope of Work The basic purpose of this study was to determine fractionation efficiency, related to several of the parameters affecting efficiency, while using a large and small column; tray I T o whom correspondence should be addressed a t P.O. B i g Spring, Texas 79720.
Box 1311,
hydraulics as affected by weir height, reflux ratio (LA') and percent flood, interrelated within one binary system. Two binary systems were studied: benzene-1-propanol and 1propanol-toluene. To minimize the number of experiments while maximizing the resultant data, it was decided to focus the prime attention on the most interesting case for each binary system. Then the parameters were varied in an increasing series of multidimensional concentric rings around the prime case. This is the Box-Behnken response-surface experimental design method (Box, 1960). Tables I and I1 (supplementary material) show the coded Box-Behnken design for these variables and the numerical values for the design levels. In addition to the coded runs, extra data were taken a t various percents of flood, from 15 to 97.5%of flood. Experimental A p p a r a t u s Two columns were used for this experiment. The large column was 18 in. in diameter; two circular glass sections of conical shape permitted visual observation of column operation. These sections were 24 in. in length and had a maximum inside diameter of 18%~in. a t the ends with a minimum inside diameter of 17l7k2inches in the middle. Their thickness varied from 1 in. at the ends to in. a t the middle. Brass flanges and rubber inserts were used to match the glass sections to the rest of the column. The basic arrangement for this hardware was first used by Todd (1971). The tray decks themselves were Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
79
