No. 4, 1979 345
Ind. Eng. Chem. Fundam., Vol. 18,
Table XV. Comparison of Total Enthalpy Changes from Experiment with Predictions
the parameter values used are given below -P= - -
enthalpy change
N*
c, c, C, c; C,
compn, mol %
temp range,K
pressure, exptl, bar J g-'
100 100 100 100 100 100 100 7 1.4128.6 91.0/9.0 91.019.0 91.019.0 61.0139.0 53.3146.7 40.9/59.1 40.9159.1 10.8189.2 10.8189.2 see Table XI1 see Table XI11
100-270 140-210 110-250 110-250 110-250 110-270 120-270 110-270 110-27 0 110-1 9 6 202-270 110-270 110-275 110-270 110-265 110-270 110-270 105-270 105-250
50.7 32.0 50.0 32.0 50.7 50.7 1.o 50.7 30.4 50.7 50.7 50.7 50.7 30.4 50.7 30.4 50.7 50.7 50.7
333.1 542.5 727.7 353.0 350.1 333.6 284.6 733.0 77 3.6 294.5 363.2 368.5 494.1 498.0 460.0 377.3 367.6 762.5 714.3
RT
IMRK, %dev 2.3 1.2 0 .o -1.1 -1.1 0.7 -1.1 0.9 1.3 3.1 -1.3 0.3 3.3 0.5 0.3 0.2 0.1 0.1 1.8
to be much higher than the predictions. It is argued that this is caused by a liquid-liquid phase separation a t low temperatures. (b) Small amounts of helium increase the liquid-phase heat capacity of a multicomponent mixture, but not that of pure methane and pure nitrogen. Although we are not sure of the exact mechanism involved, a liquid-liquid phase separation a t low temperatures might explain the increased heat capacity.
1.0
a1 = 1.0
1
a
V-b
V(V+b)
+ (1.45 + 1.62.~).
+ kl(
1-
kl = -0.127 k2 = 0.015
-
);
+ k2[1 -
(q)
+ 5.45*10-4~~-'
0.355.~- 0.65.10-4*~-'
b = 0 . 0 8 6 P7C5 11.0 - O.ZO.u(
2
- 1.0))
P = pressure; T = temperature; Tc = critical temperature; w
= acentric factor; and V = molar volume.
Literature Cited Chacg, S.D., Lu, B.C.-Y., Chem. Eng. Frog. Symp. Ser., No. 81,63,16 (1963). Gonzales, M. H., Subrernaniam, T. K., Kao, R. L., Lee, A. L., Paper 21 in "Proceedings of the First International Conference on LiquefiedNatural Gas", Chicago, April 1968, Institute of Gas Technology, Chicago, IN., 1966. Lammers, J. N. J. J., van Kasteren, P. H. G., Kroon, G. F., Zeidenrust, H., in "Proceedings of the Fifty-Seventh Annual Convention of the Gas Processors Association", New Orleans, March 1978. Van Aken, A. B., Lammers, J. N. J. J., Simon, M. M., in "Chemical Engineering in a Changing World. Proceedings of the Plenary Sessions of the First World Conference on Chemical Engineering", Amsterdam, June-Juiy 1976, p 51 1, W. T. Koetsier, Ed., Eisevier, Amsterdam, 1976. Yu, P., Eishayal, I.M.,Lu, B. C.-Y., Can. J. Chem. Eng., 47, 495 (1969).
Appendix
A modified Redlich-Kwong equation of state has been used for the prediction of enthalpies. This equation and
Receiued for reuiew July 17, 1978 Accepted July 23, 1979
The Gravity Flow of Gases, Liquids, and Bulk Solids Frederick A. Zenz" and Fredrlc E. Zenz Manhattan College, Rlverdale, Bronx, New York 10471
I t is demonstrated that the gravity flow of bulk solids issuing from an opening in the bottom of a bin, the flow of liquid over a weir, and the flow of gas under the slots of a bubble cap or an inverted weir, can all be represented by a common equation. This equation has been tested by observing solids lighter and heavier than water flowing out of bins submeraed in a tank of water and by observing the flow of water over a weir submerged in a tank of oil.
-
T h e Gravity F l o w of Bulk Solids It has been demonstrated (Zenz, 1975) that the efflux
of freely flowing bulk solids from a bin bearing a round hole or a slot in its bottom and situated in dry ambient air can be represented by the dimensionally consistant relationship
W , = g1J2pBh,'Jz/(tan a)lJ2 (1) where W , = solids efflux rate in lb/s X ft2 of hole area, g = acceleration field, ft/s2, pB = solids bulk density, lb/ft3, a = bulk solids angle of internal friction, deg, and h, = narrowest dimension of the opening through which the bulk solids are flowing, ft. In instances where the solid particles are large relative to the opening, an area cor0019-7874/79/1018-0345$01 .OO/O
rection which consists of reducing the length and width, or the diameter, of the opening by 1.5 times the particle diameter (Zenz, 1962) must be applied. Equation 1is also only valid for openings whose narrowest corrected dimension is a t least 16 times the particle diameter. For reasons which might not be obvious at the moment, consider the addition of a term to the right-hand side of eq 1 which at least would not be in conflict with the data on which it was founded
where pf = density of the surrounding medium, lb/ft3. To
0 1979 American
Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
a first approximation the added term is intuitively reasonable since efflux rate must approach zero when the density of the flowing material approaches that of the medium into which it is tended to flow. Equation 2 can then be rearranged to the form A S 2
where the term Ap represents the difference in density between the flowing bulk solids and the surrounding medium. In this notation eq 3 always leads to a positive value for W , whether the direction of flow is downward, as in the usual cases where the solids are heavier than the surrounding medium, or upward, as would be conceivable if a bin full of ping pong balls were submerged in a sunken ship. The angle a represents the inclination of the shear plane between the flowing and the stationary solids measured from the horizontal. It represents the shallowest direction vector of the particles flowing toward the hole and must be experimentally determined for any given material. It is a characteristic of each material to the same extent as angle of repose. Those few reported measurements of a which can be found in the literature suggest it generally lies in the neighborhood of 70". The Gravity Flow of Bulk Liquid The flow of water over a weir, as given by the Francis equation, can also be regarded as a manifestation of gravity induced efflux of matter through a flow area. As conventionally employed in the hydraulic design of distillation trays or in the metering of liquids in open channel flow, the Francis equation is expressed as h ",,, = 0.48(GPM/L 'rw)2'3 (4) Equation 4 has been shown (Zenz, 1962) to be transposable into the form
w,= P L ( h ' 2 ) ' / 2
(5) where pL = liquid (water) density, lb/ft3, h'h, = liquid crest over the weir, in., WL = lb of liquid/s X ft2of flow area above the weir, where this flow area is (h'a X Lrr,)/144, and L '& = weir length, in. Equation 5 is not dimensionally consistent. However, it could be made consistent by adding a term
,.
17
CFM
-
- ---
h " z So"
Figure 2. Bubbling flow of gas under inverted weir.
Obviously this can be considered a contrivance to rationalize 70", which is the angle necessary in eq 7 to have it yield correct answers for water flowing over weirs in an ambient air environment. It raises the question of whether the 70" represents a specific value of a which might vary for other liquids differing substantially from water and whether tan a would be a valid substitution for what appears as a constant in eq 6. The right-hand side of eq 7 can again be multiplied by an additional factor which would not alter its agreement with the data on which it was based, yielding
PL
-WL2 =pL
where h, in eq 6 is now in units of feet and g is the surrounding acceleration field in units of ft/s2. The quantity 32.2/12 in the denominator of eq 6 equals the tangent of 69.6", which by analogy to eq 1 tempts substitution of tan a in place of 32.2/12, suggesting that liquids also shear along a preferred internal plane of very nearly the same slope as that exhibited by bulk solids. Though it is difficult to conceive any fluid as having such an angular property, closer examination of the behavior of liquid flow over a weir indicates that liquid level does exhibit an angular decay approaching the weir (Perry and Chilton, 1973) as shown somewhat exaggerated in Figure 1. The liquid level sloughs off toward the weir from some distance A upstream, where A is reportedly about 2 to 3 times the crest head. If A were 2.75 the angle a in Figure 1 would correspond to 70" and suggest that eq 6 could be written "112
6
(8)
This can be rearranged to give
\1/2
w,= (tan 70") l/*PLhw'~z
!h
Figure 1. Weir flow of water in ambient air,
(7)
&,AP
tan 70"
(9)
where Ap = the density difference between the flowing liquid and the surrounding medium, lb/ft3. Gravity Flow of Gases and Vapors The operating slot opening of a bubble cap submerged in tray liquid has been shown (Zenz, 1952, 1953) for a rectangular slot to be given by CFM = 2.43 where CFM = gas flow through the slot, ft3/min, w = slot width, in., and So = height to which slot is blown open, in. If viewed inverted as illustrated in Figure 2, this represents the flow of gas over a weir in a liquid medium (the inverse of the phases in the previous Francis weir equation) SO that So becomes equivalent to h,. Algebraic rearrangement results in altering (10) to the form
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
347
Figure 3. Arrangement of test apparatus.
where WG = gas flow through open or blowing slot area, lb/s X ft2. Equation 11 is again not dimensionally consistent however, if h'a were expressed in units of feet and the gravitational term included then it could be written as .-
v
I.
PG
or (13) Again the substitution of tan 70" for what might otherwise be a numerical constant is open to question. By analogy to the previous derivation it could be regarded as specific for the air-water interface and therefore more generally as tan a. Subtracting a negligible pG from the p L term r
I
(14)
so that WG2
-=pG
&WAP
tan 70"
(15)
Tests of the Equivalent Equation The equivalence in gravity flow of gases, liquids, and solids becomes obvious in comparing eq 3, 9, and 15. Ws2= gpBhwAp/tan a
(3)
WL2= gpLh,Ap/tan 70"
(9)
WG' = gpGhwdp/tan70"
(15)
This common relationship raises several questions such as: (1)Is the gas-liquid interface related to some internal
liquid shear angle? (2) Is tan 70" simply a constant or is it related to a property of the liquid phase such as viscosity? (3) Would a be altered in a medium other than ambient air? (4) Is the Ap term applicable or simply fortuitous in the foregoing derivations of (3) and (9)? (5) Would eq 3 apply if Ap were reduced by submerging a bin in a tank of liquid? The resolution of questions 1, 2, and 3 would involve a long program of extremely careful investigations for which specific analytical tools might conceivable also need to be developed. However, questions 4 and 5 should conceivably be resolved relatively easily with a few simple observations of both the efflux of solids from a bin and the flow of water over a weir within a surrounding medium much denser than air. Experimental Apparatus and Results Figures 3a, b, and c illustrate the arrangements of exploratory test apparatuses intended to resolve questions 4 and 5. In Figure 3a water was delivered through a tube to a weir box submerged in a tank of fuel oil. As opposed to observations of the flow of water over a dam in ambient air, where the density difference is 62.4 - 0.0765 i= 62.32 lb/ft3, the submerged weir operated under a density difference of 62.4 - 52.1 = 10.3 lb/ft3. As illustrated by the typical results in Figure 3a, eq 8 correctly represented the observations. Though the apparatus was not arranged to permit determination of such flow angles as illustrated in Figure 1, it appears from the data that the surrounding medium very likely had a negligible effect, if any. Such flow angles would be expected to depend principally on the viscosity or shear characteristics of only the fluid actually flowing and not on its surrounding medium. The data in Figures 3b and c suggest that the angles of internal friction in these large-particle systems was still controlled by the solids and essentially unaffected by the surrounding fluid. The proper effluxing density is that of the liquid-plus-particlemixture; the interstitial fluid travels along with the solids. This is effectively another demonstration that the same situation must exist during bulk solids efflux in particle-gas systems. This is the cause of the vacuum, or lower pressure, observed in the region
348
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
above the orifice during efflux from bins in ambient air surroundings. The density-difference term in eq 3 appears to be substantiated for cases in which the particles are lighter, as well as heavier, than their surrounding fluid medium. Significance and Implications
The fact that the gravity flow of gases, liquids, and solids can all be represented by a common dimensionless equation is obviously of considerable interest. It also illustrates the inadequacy of the simple Francis weir equation in application to liquids other than water and to systems wherein the surrounding medium is of comparable density. Of perhaps greater significance is the implication that the viscosity of liquids and hence also of bulk solids is manifested in the angular term in eq 3, 9, and 15. The experiments reported in Figures 3b and c were carried out with rather large particles, in which case the viscosity of the mass was essentially that of only one phase. The same experiments carried out with very fine powders yielded in some instances complete blockage of the orifices and hence no flow. It could be argued that those examples were not cases of high viscosity but perhaps cases of high surface
tension and small interparticle dimensions. The significance of viscosity or surface tension is an obvious subject for further investigation, using such pure fluids as methyl alcohol and mercury as well as their mixtures with a variety of powders. Identification of the controlling property is of importance not only in resolving the relationships for the flow of slurries but in understanding the problems associated with the efflux of non-free-flowing powders. In any event the experiments reported here support the presented derivations of the common equations. Acknowledgment
Acknowledgment is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. Literature Cited Perry, R . H., ChiRon, C. H., Ed., "Chemical Engineers' Handbook", 5th ed.,pp 5-17, McGraw-Hill, New York, N.Y., 1973. Zenz, F. A,, Hydrocarbon Process., 54(5), 125-128 (1975). Zenz, F. A,, Hydrocarbon Process. Pet. Refiner, 41(2), 159-168 (1962). Zenz, F. A., Pet. Refiner, 32(1), 150-154 (1953). Zenz, F. A,. Chem. Eng., 169-173 (April 1952).
Receiued f o r review September 28, 1978 Accepted June 6, 1979
Vapor Pressures of Heavy Liquid Hydrocarbons by a Group-Contribution Method A. B. Macknick and J. M. Prausnitz' Chemical Engineering Department, University of California, Berkeley, California 94720
A group-contribution method gives parameters for a vapor-pressure equation based on a kinetic theory of fluids. All parameters are obtained from molecular structure only. Good representation is obtained for vapor-pressure data for 67 hydrocarbon liquids in the region 10-1500 mmHg. This group-contribution method is useful for estimation of vapor-pressures and enthalpies of vaporization for those heavy hydrocarbons where no experimental data are available.
An abundance of experimental vapor-pressure data is in the literature for low-molecular-weight hydrocarbons. However, with the exception of normal paraffins, reliable vapor pressures are scarce for compounds which have normal boiling points over 200 "C. For rational process design, especially for alternative energy processes, it is important to have good estimates for thermodynamic properties of heavy hydrocarbons. Since it is rarely feasible economically to determine experimentally all necessary data, correlations are often used to extend limited experimental information. However, for many high-molecular-weight hydrocarbons, no experimental data at all are available. Therefore it is desirable, when possible, to estimate physical properties from theoretically based correlations. This work presents a group-contribution method for determining parameters in a vapor-pressure equation. Many vapor-pressure equations have been proposed. Most are either empirical or integrations of the Clapeyron equation coupled with simplifying assumptions. In almost all cases these equations take a form where the pressure is an exponential function of the temperature and of the 0019-7874/79/1018-0348$01 .OO/O
adjustable parameters; this exponential dependence requires accurate prediction of parameter values. Since common vapor-pressure equations contain at least three, and often more, adjustable parameters with no clear physical significance, it is not possible to determine unambiguous values of these parameters by group contribution. Many equations use critical data to develop a corresponding-states relation. Such relations are suitable for most light compounds, but not for heavy hydrocarbons which are thermally unstable in the critical region. Critical data for heavy hydrocarbons are usually not known. Several authors have correlated vapor pressures for homologous series, but such correlations are limited to specific compounds. While such correlations are sometimes more accurate, a group-contribution method, based on carbon types, is more general because it can be used for a wide variety of hydrocarbons. Not only can one estimate the vapor pressure of a homologous series, but in addition, it is possible to estimate the effect of substituting various branches and side chains onto an unbranched parent molecule. 0 1979 American Chemical Society
