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Ind. Eng. Chem. Fundam. 1981, 20, 318-323
Rayleigh-Taylor Instabtlltles in Fluidized Beds AnJanl K. Dldwanla and 0. M.
Homsy’
Department of Chemical Engineering, Stanford Universily, Stanfod, Califwnia 94305
The gavitatlonal instability of fluidized medii in which h m r voidage regions lie below those of lower voidage was examined both experimentally and theoretically. The experiments were conducted in two-dimensional liquid fluidized beds. Potentlally unstable situations were produced by a rapid increase In fluid veloci, and except In extreme cases, no instability was observed. A linear instability theory was developed using the two-fluid equations of fluidization, and this theory predicted a high degree of instability, in disagreement with the observations. An interpretatii of these results is given which provides strong evidence for the existence of a yield stress in fluidized solids below a certain void fraction.
Introduction The phenomenon of local voidage fluctuations is characteristic of fluidized beds over their entire range of operation, e.g., bubbles in gas fluidized beds, instability voidage waves in liquid fluidized beds, etc. These fluctuations often lead to rapid spatial variations in voidage which can be well approximated as sharp voidage fronts with a jump discontinuity in particle concentration. Of these voidage fronts, the case in which the high voidage regions (less dense) are below those of low voidage is potentially gravitationally unstable. The situation is akin to the classical problem of the Raleigh-Taylor instability for single-phase fluids (Chandrasekhar, 1961), but with added complexity and important differences. These fronts often display complex time-dependent propagation dynamics. The theoretical analysis of their stability is further complicated by our lack of knowledge about the proper form of the constitutive equation, the values of the material constants appearing in the two-fluid continuum modelliig equations, and the proper boundary conditions to be satisfied across such fronts. Furthermore, there are at present no experimental observations which can be used directly to ascertain the importance of the Rayleigh-Taylor mechanism on the mechanics of the fluidized state. In spite of these shortcomings, the propagation and stability characteristics of these potentially unstable fronts, as will be discussed below, are believed to play a critical role in a wide variety of contexts, e.g., bubble splitting, transitions between fluidization regimes, transient bed operations, and modulometric measurements. The importance of Rayleigh-Taylor instabilities attracted early attention in connection with bubble splitting and bubble formation in gas fluidized beds (Rice and Wilhelm, 1958). Experimental observations indicate that large bubbles or slugs in fluidized beds break up through a splitting mechanism in which large spikes of the dense phase penetrate the bubble roof (Clift and Grace, 1972; King and Harrison, 1980). This mechanism of bubble splitting by gravitational collapse of the bubble roof has been analyzed by essentially equivalent treatments of Rice and Wilhelm (1958),Upson and Pyle (1973)and Clift et al. (1974). It is, however, not possible to make a direct comparison between existing theory and experiments on bubble splitting, since the theory assumes a velocity in the base state which is steady and unidirectional, while the actual flow around a bubble is axisymmetric and often time-dependent. Potentially unstable voidage shocks can be produced experimentally in nonbubbling liquid fluidized beds by a sudden increase in the fluidization velocity. Since the bed expansion is generally an increasing function of fluidization 0 196-43 13/8 1/ 1020-03 18$01.25/0
velocity, a sudden change in the latter causes a sharp change in the voidage. Slis et al. (1959)reported observations of the overall bed expansion of systems fluidized by water, following a sudden increase in the fluidization velocity. They observed disagreement between a theory which assumed the discontinuity travelled in a stable manner and their experiments, which they attributed to a smearing of the front. Upson and Pyle (1974)repeated the same experiments in two-dimensional beds, thus allowing visual observation of the propagating front. The rapidly evolving slug, which they attribute to RayleighTaylor instability, is clearly the result of flow maldistribution and channeling at the base of the bed, as is evident from their visualizations. Flow uniformity at the base of the bed, as we will discuss later, is essential for any careful investigation of instability phenomena of this nature. The gravitational instability of voidage fronts is also thought to be responsible for flow transition between some of the fluidization regimes. Canada et al. (1978)have observed that in large gas fluidized beds at high velocities, there is a flow transition between an apparent slugging regime and one called “turbulent”, the latter being characterized by void regions which are no longer distinct bubbles, but which are rapidly destroyed and re-formed. This “turbulent” regime is almost certainly a result of the gravitational instability of gas layers of large horizontal extent characteristic of the apparent slug flow regime. In liquid fluidized beds, planar instability voidage waves are observed to develop a transverse structure in wide beds (Didwania and Homsy, 1981)and high voidage bubblelike clusters in narrow beds (El-Kaissy and Homsy, 1976). Flow visualizations reported in these works suggest that a secondary instability of the wave train, similar to a Rayleigh-Taylor instability mechanism, may be responsible for such behavior. Finally, the experimental investigation of gravitational instabilities remains very promising from a modulometric standpoint, an approach taken in another context by Anderson and Jackson (1969)and Homsy et al. (1980). The growth characteristics of these instabilities are strongly sensitive to the constitutive equations for the stress and to the rheological parameters appearing therein. Thus the experimentally observed instability characteristics of these propagating fronts, e.g., the wave of maximum growth rate, can be used to deduce values for the material constants appearing in the modelling equations. The present study has as its objectives: (a) to experimentally investigate the propagation and stability characteristics of voidage fronts in a two-dimensional liquid fluidized bed, (b) to analyze the stability of such voidage fronts using two-fluid continuum equations and compare 0 1981 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, I981 519
Table I. Roperties of GI= Beads (p. = 3990 kg/m') set no. A
dominant
code"
diam, cm
304 Class C (Ballotini No. 5)
0.0589-0.0417
diam, em Urn*, cm/s H-6 cm 0.059 0.73 59.5 0.1 2.05 42.2
B -16 t 1 8 meshCIassV 0.1-0.119 a Microbeads Division, Cataphote Corporation, Jackson, MS 39208. these results with our experimental observations, and fid y , (c) to use our experimental observations to judge the validity of the constitutive equation. We have obtained visualizations of the voidage fronts propagated by a sudden change in the fluidization velmity with a distributor section carefully designed to eliminate all large-scale flow and voidage nonuniformities under both steady and time-dependent flow conditions. We report below our observations (in some cases) of a high degree of stability of these potentially gravitationally unstable voidage fronts. Based upon modelling considerations, this high degree of stability is new and unexpected. The Experimental Section gives a description of the details and the results of our experiments. The predictions for the propagation and stability characteristics of voidage fronts are presented in Section 3. In Section 4,we present a comparison between theory and experiments and deduce the values of material constants. Finally, a discussion of the main features of our results and their possible implications is covered in Section 5. Experimental Section A plexiglass column of 30 X 3.15 ern cross section and 180 cm height was used for the experiments. The fluidization system was glass beads in water. The thickness of the bed was kept large compared to the particle diameter, so that the continuum theory applies. A diagram of the experimental apparatus and related details are given by Didwania and Homsy (1981). Table I gives the size range and density of the beads investigated. The experimentally determined values of voidage, superficial fluidization velocity, and bed height at minimum fluidization conditions and the ratio of bed thickness to particular diameter are also included. Before entering the base of the bed,fluidizing water passed through a 30-cm high packed bed section filled with glass beads (0.059 cm diameter) and three to five layers of fine wire screens. This design of the distributor section maintained flow uniformity in both steady and timedependent flow over a range of fluidization in the wavy regime. The bed was backlit with 2500-W flood lamps with a diffuser screen between the bed and lamps. The two-dimensionality of the bed permitted visualization of the propagating voidage fronts. Flow visualizations were carried out using video tape recording and 35-mm still photography. Two types of experiments were performed. In the first type. performed with both bead sizes, an initially fluidzed bed was subjected to a sudden change in the fluidization velocity, leading to the upward propagation of a voidage shock from the base of the bed. This change was accomplished by manually turning the control knob on the rotameter. Relative to the time scale of the resulting phenomena, this change was essentially instantaneous, takimg fractions of a second to accomplish. The resulting shock, described in detail below, was observed to be a few particle diameters thick. The velocity of this shock was measured by stopwatch as well as using a photo-detection package, similar to that of El-Kaissy and Homsy (1976),with substantial agreement between the two types of measurements. The step change in velocity was either an increase or decrease. The measured front velocities are presented
emf 0.37 0.383
bed depth, Dld, 53 28
Table It. Compuison between Kinematic Wave Theory and Experiment for a Step Decrease m Fluidization Velocity fluidization ,C t. ,% ,, range in-tigated em cmps cm/s set e., 1 G U$ 4.40 4.40 0.507 0.399 A
B
0.507 0.566 0.576
0.455 0.516 0.440
4.60 5.00 4.30
4.70 4.97 4.34
0.457 0.531 0.565 0.579
0.406 0.441 0.523 0.525
8.76 8.11 9.09 8.82
8.70 8.14 9.01 9.05
U d65
1G
Us/
U d 63
Figure
1. Still photograph of propagating potentidy unstable voidage honk Initial voidage = 0.38; final voidage = 0.40.
in Table II for a step decrease in the fluidization velocity. Also given in the table are theoretical values discussed below. The voidage front remained sharp during the time necessary to traverse the entire bed height. We show in Figure 1 a 35-mm still picture of the bed following a step increase in the fluidization velocity. The dark horizontal band is a lateral support. The sharp voidage front, visible below the support, was found?to traverse the entire bed height with no tendency to become unstable to gravitational instability modes of the Rayleigh-Taylor type. In some cases, as in Figure 1,the fronts were slightly wavy instead of planar, but this waviness is not an instability. Because of the difficulties in maintaining time-dependent flow uniformity for higher u./ud such behavior could be observed only over a limited range of fluidization. The experimentally measured front propagation velocities in the case of a step increase are presented in Table 111. At higher u./ud, following a step increase in the fluidization velocity, the effect of flow maldistribution and channeling dominated the time dependent flow. Instead of one-dimensional planar propagation, localized jets leading to high voidage slugs were observed. These slugs, as far as we could ascertain, were identical with those observed by Upson and Pyle (1974) and could be easily shown to be caused by the flow manifold below the distributor section.
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Table 111. Comparison between Kinetmatic Wave Theory and Experiment for a Step Increase in Fluidization Velocity fluidization Cex , Ctheory, range invesset E~ c m g cm/s tigated A
0.380 0.391 0.391
0.402 0.409 0.420
3.60 4.46 4.40
3.56 4.41 4.36
1 4 Us/ C J d < 2
B
0.398 0.406 0.413
0.424 0.449 0.449
7.23 7.50 7.41
8.03 7.81 7.93
1 G Us/
U d
Q
1.5
Because of our inability to observe gravitational instabilities in the first type of experiment, we performed a second, more drastic type of experiment with the smaller particles. An initially unsaturated bed was suddenly ex) . caused the posed to a high velocity fluid (u, = 7 ~ ~ This entire packed bed to lift off the distributor plate to a height of about 10 cm. As the saturation front propagated into the packed section, the lower interface between clear liquid and the particulate phase could be observed. Although initially planar, this interface rapidly developed a twodimensional waviness with a clearly defined wavelength. Particles fell along troughs in the wave, with regions of high voidage being trapped between troughs. The instability was observed to be very rapid, taking less than 1 s to develop to large amplitude. The dynamics of these instability phenomena were analyzed with the aid of a video tape recordingsystem. These visualizations indicate that the instability indeed has features very similar to that of Rayleigh-Taylor instability mechanisms for an interface separating two single-phase incompressible, viscous fluids. The value of the most dangerous wavelength as measured in several duplicate experiments was in the range 3.9-4.3 cm. This wavelength was found to be essentially independent of flow rate for this second type of experiment. Theory The experimental conditions following a sudden change in the fluidization velocity can be described by Figure 2. Two uniformly fluidized regions with differing voidages, eO2 and colt exist one above another. The interface separating the two regions propagates upward with velocity C and finally takes over the upper boundary of the second region, which also propagates upwards with velocity V,. A relationship between C and Vo2can be obtained from a total solids balance, as given by p&(h - z ) ( l - €02) + p&(l - €01) = pSAho(1 - €02) (1) Here A is the cross section of the bed and ho is the initial height of the bed before the step change. Differentiating eq 1with respect to time and identifying dhldt as V , and dz/dt as C, we obtain
spectively. Integrating these continuity equations across the propagating voidage shock, we obtain (1 - eo2) (V02 - C) = (1 - eoJ(V01- C) (4a) ~ o z ( ~ o-20 = eol(lJo1 - 0 (4b) Noting that
and, using eq 2, eq 4a and 4b yield vo, = 0
The predicted propagation velocities, C , using the above expression, are presented in Tables I1 and I11 for comparison purposes. The stability of the potentially unstable front was analyzed earlier, using a simplified set of dynamic equations, by Didwania and Homsy (1980). The simplified analysis treated the fluidized suspension in each of the regions as an effective single-phase Newtonian fluid. The quasi-static approximation was used to account for the time dependence of the base state. (The time dependence is always present in a finite bed; the region of high voidage grows and that of low voidage shrinks as the shock propagates.) The analysis predicted unconditional instability of such fronts. Below we use the same two-fluid continuum modelling equations, as discussed earlier by Homsy et al. (1980), to investigate the effect of various material constants and the interparticle force on the stability characteristics of such fronts and to seek a better agreement between theory and experiments. The equations of motion, with a Newtonian closure for stress in both phases, are as follows. fluid phase:
solid phase:
(€01 - 602)C
(2) (1- €02) The equations of continuity for the two phases, satisfied in both regions, are fluid phase: v 0 2
=
at
(2 +
(Uj -
u;)
)]
axj
- p & l - €)g6jz
+
a€ a w
--+I= 0 (34 at ax, solid phase:
a(i -
a((i - +,)
+ 8x1 = O (3b) at where u, and u, are fluid and solid phase velocities, re-
The subscripts f and s refer to fluid and solid phases, respectively; as,a4,6, and 9 are drag, virtual mass, bulk, and shear viscosities, respectively; p is the pressure, and g is the accleration due to gravity. The notation is that of Homsy et al. (1980).
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 521
fluid phase:
Fluidizing water
Figure 2. Schematic of the bed mnditian following a step change in fluidieationvelocity.
The linearized boundary conditions satisfied a t the interface are as follows
The situation in Figure 2 can be represented by uniform voidage and velocities for both phases in each region. t = f, (84 ui = (O,O,UO) (8b) ui = (O,O,V,) (8c) can be evaluated from the initial and final bed heights, respectively. Vo1and V, can be obtained for eq 6a and 2. The initial and final fluidization velocities can be used to evaluate U,, and Uoz. The equations of continuity are thus satisfied and thme governing the motion yield for each region. fluid phase:
IVu.1, I = solid pbase: (since pd = p m for uniform fluidization) as - (Ui - U J b . = (1- fdb, - pr)g6, (9b) %
A m s the propagating interface, continuity of fluid phase pressure and conservation of mass for both phases are satisfied. We examine the stability of the interface in a frame of reference attached to it and moving upward with constant velocity C. Our analysis assumes that the propagating interface is far from both the top and the base of the bed. No voidage perturbations were permitted in any of the two regions. Two-dimensional small perturbations of the velocities for both phases, pressure ui', vi', p i , and the interface, v were permitted. We neglect the second- and higher-order terms in these small perturbations and nondimensionalize these perturbed variables using a proper combination of U,,, dp (particle size) and the fluid density, pf. We obtain, in each region, the following linearized equations and appropriate boundary conditions, after dropping the primes solid phase:
where [ 112= [ l1 - [ 12. All perturbations decay to zero far away from the interface in both regions. The dimensionless material constants and parameters appearing above are a3
=~ S ~ ~ / P ~ U O I a4
= a4/~1
Ref = pfUodp/wt Re. = P J J O I ~ / ~
Fr = UoI2/gdp = P/Pf R =P
u= ~
~ P .
O / ~ O l
v = VO/~Dl
(12)
where p = p.(l-to) + p,t@ Equtions l l a - c express continuity of normal and tangential component of the averaged velocity and of fluid phase pressure across the voidage shock. The boundary conditions (1ld-g) are obtained by integrating the mcmentum equations for both phases across the shock. We assume the small disturbance of the interface to be of the
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Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981
3!-
~
~ ~ __ _
IO
K y . WAVE
_
L
20
NUMBER,
30
-
40
cm-'
Figure 3. Growth constant (most unstable branch) vs. horizontal wavenumber;tw = 0.39; = 0.42; Fr = 0.038; (a) all = &42 = 0.0; Re,l = Red = 0.013;Ren. = 1.29; (b) b41 = a42 = 0.0;Re,l = Red = 0.012,Ren = Ree = 1.29;(c) b41 = &4z = 4.0;Resl = Red = 0.012;Ren = Rem = 1.29.
form e d + h . This yields a set of linear constant-coefficient oridinary differential equations satisfied by the perturbations in the vertical direction. These have solutions of the form CA,eS*, with the S , depending upon u and material parameters. Using the fact that
-d2Pf -
ax,ax,
-0
and applying the boundary conditions, one obtains a determinamental equation for the growth constant, u. This we write symbolically as
The positive values of the growth constant correspond to the instability of the interface. The above dispersion relation was numerically investigated for parameter values corresponding to our experimental conditions. The range of material constant values used for this purpose were deduced earlier from experimental measurements on instability voidage waves by Homsy et al(1980). Since the continuum theory remains valid for length scales much greater than the particule diameter, the range of horizontal wavenumber investigated was 0-10 cm-'. Corresponding to each wavenumber, for a given set of parameters, three positive values of u satisfied the above dispersion relation. Of these three modes, the one corresponding to the largest of value of a is of practical interest. Figure 3 shows a plot of u for this most unstable mode as a function of the wavenumber, for a set of experimental conditions described therein. This mode exhibita a maximum in the growth constant for a particular value of horizontal wavenumber. The behavior is similar to the Rayleigh-Taylor instability of viscous incompressible single phase fluids. We note that the effect of increasing effective viscosity of the soilds phase, represented through Res, is to decrease the value of maximum in growth constant and the corresponding wavenumber. Our results are insensitive to variations in the value of S3 and are slightly affected by changes in the value of S4 as shown in Figure 3. The drag function, &3, however, has a pronounced effect on the value of the distance (i.e., sl, $2) from the interface over which the perturbations die out. We also note that the values of maximum growth constant and the corresponding wavenumber are close to the values obtained by using the simplified theory described earlier. Thus, when no voidage variation is permitted, the interparticle
force does not have any significant effect on the instability characteristics. In our second type of experiment, we observed a rapid growth of the disturbances of the voidage front with a preferred horizontal wavelength. For single-phase fluids, there is a most dangerous wavelength that dominates the experimental observations and the value of it is known to be highly sensitive to the viscosities of the two fluids on both sides of the interface. In our second type of experiment, the bottom fluid is clear water and the top suspension has a known effective density but unknown viscosity. By assuming the top suspension to be an effectively Newtonian single phase fluid, we can deduce a value for its viscosity from our experimentally observed value of the most dangerous wavelength. Menikoff et al. (1977) have analyzed the unstable normal modes for Rayleigh-Taylor instability of two superposed incompressibleviscous fluids over a wide range of viscosities and densities. They report that the most dangerous wavenumber, with appropriate scaling, can be rendered remarkably insensitive to the fluid properties. Using their results, and our experimentally observed wavelength range 3.9-4.3 cm, we estimate the viscosity for the top suspension to be in the range 13-19 P.
Discussion and Conclusions A comparison of our experiments and predictions in Tables I1 and I11 indicate that the one-dimensional shock model predicts the propagation velocities of the voidage fronts reasonably well. The agreement for a step decrease in fluidization velocity is slightly better than that for a step increase in the velocity. For a step increase, in the case of large size particles, the experimentally observed values of the propagation velocities are consistently smaller than those predicted. We also note that the potentially gravitationally unstable fronts exhibit a high degree of stability, contrary to our theoretical predictions. Our stability analysis predicts that any small disturbance of the interface should have grown by a fador of at least e7-e8 during its residence time in the bed. This predicted behavior, of course, assumes a range of values of the material constanb and depends rather sensitively on the assumed shear viscosity values for the two phases. Only a very large value of shear viscosity (- 200 P)can explain the high degree of stability that we observe for the potentially unstable fronts. Such a large value is far in excess of what might be considered reasonable. The shear viscosity value that we infer for the highly concentrated suspension from our second type of experiment agrees well with the ones obtained by other indirect methods (see Grace, 1970; Davidson et al. 1977). Our experimental studies have shown that a potentially unstable voidage shock propagates with a surprisingly high degree of stability in concentrated fluidized suspensions. The propagation velocities of these shocks can be predicted reasonably well by simple kinetmatic wave theory. In our second type of experiment, we observe the propagating interface between the top fluidized suspension and the underlying fluidizing water region to be highly unstable. We can obtain from instability analysis a reasonable estimate for the effective shear viscosity of the top fluidized suspension, treating it as an effective single-phase Newtonian fluid. A quasi-static theory, allowing for interaction between the solid and the fluid phases, assuming no mean voidage fluctuations and under Newtonian constitutive assumption for both phases, fails to account for the observed high degree of stability of these voidage shocks in the fist type of experiment. This implies that we need to examine the
Ind. Eng. Chem. Fundam. W81, 20,323-332
validity of the assumptions underlying the theory. The assumption of no mem voidage fluctuation is a reasonable one for analyzing the Rayleigh-Taylor instability modes, because allowance of mean voidage fluctuations only leads to appearance of additional dilation instability modes, e.g., instability voidage waves (see Pigford and Baron, 1965; Anderson and Jackson, 1968; and Homsy et al., 1980). The quasi-static assumption well describes the experimental conditions when the shock is far from both the top and the base of the bed and is a valid one, considering the long experimental residence times of our experiments. The assumption of Newtonian constitutive equation for the stress at very high particle concentrations, is therefore a questionable one. We interpret our first type of experiment as follows: over the voidage range covered, the particulate phase exhibits a yield stress; i.e., it will not flow under applied stress until a critical level is reached. In a carefully controlled experiment, the fluctuations which may drive an instability are presumably small, and therefore do not result in stresses above the yield value. This would account for the observed high degree of stability. The theory ignores this effect and predicts a high degree of instability for reasonable values of the material parameters. It is easy to O I show,&however, that any linear instability theory containing a finite yield stress will predict TU) instability, in agreement with our observations. The presence of a finite yield stress is also in accord with widely accepted behaviors of cohesionless granular materials; see Homsy (1979) for a partial discussion and references. It is clear that if this yield stress is due to particle kinematics alone and not due to forces of electrostatic, molecular, or capillary origin, it should vanish above a certain voidage. Due to the flow distribution described earlier, we were not able to examine the conjecture experimentally. It remains to account for our second type of experiment, in which gravitational instabilities were observed. We interpret these as follows: after the large increase in flow, sufficient pressure exists at the base of the unsaturated bed to lift it off the distributor. As the saturation front propagates into the packed section, uniform sedimentation
323
occurs until the point that the layer is dilated above the void fraction for which the yield stress vanishes. Rayleigh-Taylor instability then rapidly develops, leading to vigorous macroscopic mixing in the saturated region.
Acknowledgment Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. The financial support of the Solids and Particulates Processing Program of the NSF Division of Engineering is also gratefully acknowledged. Literature Cited Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1968, 7 , 12. Anderson. T. 8.; Jackson, R. Ind. fng. Chem. Fundam. 1969, 8, 137. Canada, G. S.; Maeughnn, M. H.; Staub,F. W. AICM Symp. Ser. 1978, 78, No. 176, 14. Chandrasekhar, S. “klydrodynamlc and Hydromagnetlc Stablllty”; Oxford Unhwsity: Oxford, 1961; Chapter X. Cllft, R.; Grace, J. R. Chem. Eng. Scl. 1972, 27, 2309. CUft, R.; Grace, J. R.; Weber, M. E. Ind. Eng, Chem. Fundam. 1974, 13, 45. Davidson, J. F.; Harrlson, D.; ouedes de Carvalho, J. R. F. Ann. Rev. Fbkl Mech. 1977, 9 , 55. Didwania, A. K.; Homsy, G. M. “The Stabillty of the Propagation of Sharp Voldage Fronts In Uquld Fluldired Beds”; Proceedings of ThM Enginwrlng Foundatlon Conference on Fluidlzation; Plenum: New York and London, 1980. an r r 109-116. -DldwanG, A. K.; ! G. M. Int. J. Mu/t@hse Fbw, 1981. To appear. El-Kaissy, M. M.; Homsy, 0. M. Int. J. Muff@haseFbw 1976, 2 , 379. Grace. J. R. Can. J . Chem. Eng 1970, 48, 30. Homsy, G. M. “A Mscusskn of Some Unsdved Problems in the Mechanics of Fluldized Beds”; Proceeding of NSF Workshop on FluMlzatlon and FluidParHcle Systems, Troy, NY, 1979. Homsy, G. M.; ECKalssy, M. M.; Didwania, A. K. Int. J. Mukbhase Fbw 1980, 8, 305. KIng, D. F.;Herrlson, D. “The Bubble Phase In Hl@+Pressure Fluidired Beds”, Proceedings of Third Englneerlng Foundation Conference on Fluidlzation; Plenum: New York and London, 1980; pp 101-108. Menkoff, R.; MJoIsness,R. C.; Sharp, D. H.; Zemach, C. phys. Fluids 1977, 20, 2000. Plgford, R. L.; Baron, T. Ind. Eng. Chem. Fundem. 1985, 1, 81. Rice, W. J.; Wllhelm, R. H. AICM J. 1858, 4 , 423. Slls, P. L.; Willemse, Th. W.; Kramers, H. Appl. Scl. Res. 1959, AB, 209. Upson, P. C.; Pyle, D. L. “The Stablllty of Bubbles In Fluldized Beds”; Pre ceedlngs, Internatlone1 Symposium on Fluidization and Its Appllcatlons; Toulouse, France, 1973 pp 207-222. Upson, P. C.; Pyle, D. L. Chem. Eng. Scl. 1974, 29, 71.
Receiued for review September 29, 1980 Accepted June 16,1981
Prediction of Transport Properties. 1 Viscosity of Fluids and Mixtures James F. Ely’ and H. J. M. Hanley Thermphysical Ropertles Division, Natlonal Engineering Labofatory, National Bureau of Standattls, Boulder, Colorado 80303
A model for the prediction of the viscosity of nonpolar fluid mixtures over the entire range of PvTstates is presented. The model is an extension of an earlier version (Haniey, 1976) to molecular weights which roughly correspond to that of Cm. The proposed model Is based on an extended corresponding states principle and requires onty critical constants and Pitzer’s acentric factor for each component as input. Extensive comparisons with experimental data for pure fluids and binary mixtures are presented. The average deviation between experiment and prediction is 8 % for pure species and 7 % for mixtures.
Introduction The purpose of this article is to present a reliable selfconsistent method for predicting the viscosity of nonpolar fluids and their mixtures over a wide range of thermodynamic states from the dilute gas to the dense liquid. We
stress that the method is predictive and, in principle, the number of mixture components is unrestricted. In general, engineering calculations of transport properties are based on empirical correlations, limited to narrow ranges of temperature and pressure and often to pure
This article not subject to U.S. Copyright. Published 1981 by the American Chemical Society
