frost. Since water frosts generally have very low densities when deposited under these conditions, a rather small percentage of water ice in the frost was presumably responsible for the substantial reduction i n density shown in Figure 1. Nomenclature
G I
= =
Io
= =
z
dimensionless geometrical correction factor number of yradiation counts; a measure of the intensity of radiation through a layer of frost number of counts without frost deposit frost layer thickness, em
GREEKLETTERS pm = mass attenuation factor, g/cm2 fif = average frost density, g/cm3 literature Cited
P. L. T., Reid, R. C., Brazinsky I., “Cryogenic Frost Properties,” Cryog. Technol. 5 (5), 205 (1969)). Ogunbameru, A. N., Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1971. Shah, Y . T., Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1968. (Partially reprinted in Brian, P. L. T., Reid, R. C., Shah, Y . T., IND.ENG. CHEM.,FUNDAM. 9,375 (1970)).
A. N I Y I OGUNBAMERU1 P. L. THIBAUT BRIAN2 ROBERT C. R E I D * Department of Chemical Engineering Massachusetts Institute of Technology Cambridge, Mass. 02139 1 Present address, Department of Chemical Engineering, University of Ife, Ile-Ife, Nigeria. 2 Present address, Air Products and Chemicals, Inc., Allentown, Pa.
RECEIVED for review, May 26, 1972 ACCEPTED,May 7, 1973
Brazinsky, I., Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1967. (Partially reprinted in Brian,
Activation Energy: Not Involved in Transport Processes in Liquids Mean free paths of molecules in simple liquids are very much shorter than their diameters; diffusion occurs by a succession of small displacements, not by leaps through barriers requiring energy of activation. Changes of viscosity and diffusivity with temperature can be accurately and more simply expressed in nonexponential formulas than by plotting their logarithms against reciprocal temperatures.
T h e method most commonly used for representing the temperature dependence of viscosity and diffusivity of liquids is to plot their logarithms against l/T, by analogy with the Arrhenius equation for chemical rate constants, and interpreting the slopes of the lines thus obtained as “activation energy.” This designation implies the presence of barriers against freedom of flow, imagined as consisting of some sort of quasi-lattice structure. We wish to call attention to evidence that no activation is involved in these processes, and t h a t their variations with temperature are quite accurately and more simply represented by nonexponential equations. Much of the evidence is summarized in Chapter 3 of “Regular and Related So1utions,IJ b y Hildebrand, et al. (1970). We invite special attention to a paper b y Dymond and Alder (1966), who showed that “calculated values for transport coefficients of the rare gases a t temperatures and densities greater than the respective critical ones agree within about 10% with the experimental results in both absolute values and temperature dependence, without involving a n activation barrier. I n this a priori theory, the cross section, that is, the square of the hard-sphere diameter, is determined as a function of temperature from the available equilibrium data, within the framework of the van der Waals theory.’’ Years of study and experiment with simple liquids have led us to concepts that we describe as follows. (1) All molecules participate equally in thermal agitation that produces maximum disorder. It is represented by the distribution functions deduced from X-ray scattering, which yields diffuse rings, quite unlike the spots obtained from
crystals. All characteristics that distinguish crystals from liquids disappear upon melting; vacancies that may exist in a crystal become randomly distributed in its liquid as intermolecular space. Consequently, it makes no physical sense to extend to liquids a theory of diffusion in solids that postulates the presence of holes of molecular size. It is not necessary to assume even for diffusion in solids a mechanism that includes activation energy, since realistic values of coefficients of diff usion can be calculated for hard-sphere systems that are without fluctuations in potential energy (Bennett and Alder, 1971). (2) No directive force, such as an electric field causing ions to migrate, or gravity causing sedimentation of suspended particles, acts upon the molecules of a nonpolar liquid. They diffuse simply because thermal motions keep them ever on the move. Their mean displacement with time depends (a) upon temperature and (b) upon the ratio of intermolecular volume, V , t o the volume, Vo, at which the molecules become too closely crowded to permit either diffusion or bulk flow. (3) R e have shown (Hildebrand, 1971; Hildebrand and Lamoreaux, 1972) with scores of examples that fluidity, 6, the reciprocal of viscosity, over ranges of liquid molal volume from Vo nearly to the critical molal volume, conforms closely to the equation
6
=
B(V - Vo)/Vo
(1)
Values of B depend inversely upon the capacity of molecules to absorb the externally imposed momentum of viscous flow by reason of their mass, flexibility, softness, or inertia of rotation. Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
387
The primary effect of temperature, together with pressure, is to determine values of V.Above the critical volume i t also alters values of B , as found by Hildebrand and Lamoreaux. As 1’ expands through the critical volume and beyond and paths between collisions become longer, permitting molecular velocities to approach those in free space where momenta are proportional to T’’, fluidity a t a specified volume is proportional to T-’/’. A third although minor effect of temperature in this region is a small decrease in effective molecular diameters. Values of Voand B for liquids below their boiling points can easily be obtained by plotting 9 against V;straight lines are always obtained for simple liquids. The intercept a t @ = 0 gives Vo and the slope gives B/l/’o. Sonspherical molecular shapes may reduce freedom of motion when J’ - V Ois very small, resulting in some bending away from the straight line near the bottom. This model of the motions of molecules in liquids applies equally well to diffusion, as outlined by Hildebrand (1971). Dymoncl (1972) has since shown by molecular dynamics that eyact self-diffusion coefficients for hard-sphere systems can be calculated from the ratio V/VO, nhere Vo is the volume of close-packed spherrs. When we published our first paper on diffusion (Haycock, et al., 1953), m-e too plotted log D against 1/T, but during the ensuing years we gradually learned more about the liquid state
until we now “have a better idea” that we gladly share with others in this paper. B. J. ALDER Lawrence Livermore Laboratory Cniversity of California Livermore, Calif. 94450
J. H. HILDEBRAND* Department of Chemistry University of California Berkeley, Calif. 9472’0
RECEIVED for review March 21, 1973 ACCEPTEDApril 3, 1973 Acknowledgment is made to the U. S. Atomic Energy Commission and to the donors of The Petroleum Research Fund, administered by the American Chemical Society, each for partial support of this research. literature Cited
Bennett, C., Alder, B. J., J . Chem. Phys. 54, 4796 (1971). Dymond, J. H., Trans. Faraday Soc. 68, 1789 (1972). Dymond, J. H., Alder, B. J., J. Chem. Phys. 45, 2061 (1966). Haycock, E. W., A41der,B. J., Hildebrand, J. H., J. Chem. Phys. 21, 1601 (1953). Hildebrand, J. H., Science 174, 490 (1971). Hildebrand, J. H., Lamoreaux, R. H., Proc. S a t . Acad. Sci. U . S. 69. 3428 (19721. HildAbrand,‘J. H’., Prausnitz, J. AI., Scott, R. L., “Regular and Related Solutions,’’ Van Nostrand-Reinhold, New York, N. Y., 1970.
Dynamics of Packed-Bed Adsorbers Using the Cell Model Analysis of response curves to a pulse input provides a method of studying rate parameters in packed beds. The cell model o f Deans and Lapidus is used here to derive relations between the moments of the response curves and the equilibrium and transporl rate coefficients for a packed-bed adsorber. The relations, which are restricted to linear adsorption rates, are obtained b y adapting moment equations for a well-mixed slurry to each cell of the packed bed. When the Peclet number representing axial mixing in the void spaces is 2, the moment equations based upon the axial dispersion and cell models become identical.
M e a s u r e m e n t of response curves (to pulse inputs) from packed beds of adsorbent particles has been widely used for evaluating transport and adsorption rate parameters (for example, Eberly, 1969; Kucera, 1965; Suzuki and Smith, 1971). Analysis of such curves has been based upon the axial dispersion model of packed-bed behavior. Recently (Furusawa and Smith, 1973), it has been shown that response curves can be used t o evaluate rate parameters in slurry adsorbers using a stirred-tank model. This procedure for a single-stage wellstirred tank can be combined with the cell model for packed beds (Deans and Lapidus, 1960) to provide a n alternate method for evaluating rate parameters. I n this communication the cell model is used to derive relationships between the moments of the response curve and the mass transport and adsorption rate constants in the bed. These relations become identical Jritli those derivable from the axial dispersion model when the axial Peclet number is 2, a condition that exists a t high flow rates where the cell model is most likely to be valid. The Deans and Lapidus cell model supposes that the void volumes between particles in a packed bed constitute a twodimensional array of well-stirred tanks. These small volumes have a dimension of the order of the particle diameter, d,. 388
Ind. Eng. Chem. Fundom., Vol. 12, No. 3,1973
For a n isothermal application, temperature and composition are assumed to be constant in the radial direction. Then the bed may be regarded as a series of stirred tanks in the axial direction. The moments of the response curve in the effluent will be the sum of the moments from each stirred tank or cell. For example, if the first absolute moment is defined
la
Ct dt
!J1
=
(1)
7
s,
Cdt
n (p1)be-d
(!Jl)l,cell = n(!Jl)cell
=
(2)
2=1
where n is the number of cells in series. Similarly, if the second central moment is defined as
1-
C(t -
I*, =
~
1
dt)
~
lm Cdt
(3)
