Znd. Eng. Chem. Fundam. 1984, 23, 490-493
490
Porosity Estimation for Random Packings of Spherical Particles Norlo Ouchlyama National Industrial Research Institute of Kyushu, Tosu, Saga-ken, Japan
Tatsuo Tanaka Department of Chemical Process Engineeriw, HokkaMo Unhersky, Sapporo, Japan
An estimate of the porosity of a bed of randomly placed sdM spherical particles having various sizes can be based on the assumption that each particle is in contact with a fixed number of surrounding particles, each of a diameter equal to the average diameter of all the particles present. On this assumption the average porosity is dependent on the porosl?, of a cokction of spheres of uniform size and on the distribution of sizes in the heterogeneous mixture of particles. The calculated pore volumes in mixtures of particles of three different sizes are found to agree approximately wlth measured values.
Introduction Packing porosity is one of the fundamental properties of a bed of particulate materials. Much work has been done already on paeking problems and it is a well-known fact that the fractional void volume of a bed of mixed solid particles varies with the size distribution of the materials involved. The authors have recently proposed a theoretical formula t~ estimate the overall average porosity of a randomly packed mixture consisting of various sized spheres (Ouchiyama and Tanaka, 1981). While several confirmations were made of the theory in the original presentation, the new theory should be examined carefully by experiments. In this presentation, we attempt to estimate the porosity of mixtures of solid spheres of three sizes by the proposed theory. The theoretical results obtained will be examined by using past experimental data. Theoretical Treatment A Simplified Packing Model. Our theoretical considerations assume a simplified packing model. See Figure 1. Every sphere in a packing is in direct contact wit, several neighbors, each having the average diameter, D , defined as
where f ( D ) is the number-frequency size distribution of particles. Introducing the surface porosity around a particle, eA, which means the void area fraction on a spherical surface of diameter D + D, the coordination number, C(D), around the specified particle of diameter D is then exactly expressed as
In this treatment, for simplicity, we assume that the surface porosity is independent of the size and hence of the size distribution of particles. The surface porosity can then be estimated, as is stated later, from the porosity value for a packing of uniformly sized spheres. Porosity Estimation Based on the Model. According to the above model, every hypothetical sphere of diameter D + D in a packing encloses a specified solid particle of diameter D at ita core. On the other hand, each hypothetical sphere has to share, in part, the space in common with the other hypothetical spheres. The space commonly 0196-4313/84/1023-0490$01.50/0
--
occupied is favorably restricted within the spherical shell having inner and outer diameters equal to ( D D ) and ( D D), respectively. Here, the abbreviation ( D D ) is defined as ( D D ) = 0 for D 5 D (34 = D - D for D > D (3b) and the volume of the spherical shell, V,(D), is given as
+
-
-
V,(D) = -(D + D), - ‘ ( D D)3 (4) 6 6 First we discuss an expression for the space allocated to a specified solid particle. Let the space in common occupied by the number n of the hypothetical spheres be AV,,. Then, the space per sphere, AV,,/n, should be allocated equally to each solid particle. That is, the total volume of the space allocated to a specified solid sphere of diameter D , V , ( D ) ,can be expressed as V,(D) = r ( D D)3/6 + A V , + AV,/2 + AV3/3 + ... = T ( D D)3/6 + V m ( D ) / f i(5) 7T
-
where
AVl
-
+ AV2 + AV, + ... = V,(D)
(6)
Where no macropores exist we can now describe the total bulk volume of the space in the packing, VBT,as vBT
=
(8)
where N is the total number of particles in the packing. Here, “macropore”means the void space which could exist apart from solid surface at a distance greater than half an average diameter of the particles. The overall average volume porosity of the packing, ?, should therefore be
0 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
491
where
(15)
D = ?Ofi i=l
W Figure 1. A simplified packing model. Hatching calls attention to a spherical shell considered in the theory.
Equation 9 is a general expression for the overall average porosity of a mixed packing without macropores. It appears that the size dependence of the average number, ti, can be evaluated based on geometrical or statistical considerations. When the size dependence is discarded, however, the value of ii can be evaluated approximately. Taking the volume balance of the total solid particles in a packing, we obtain
where t,(D) is the average porosity of a spherical shell and is given as eq 11. See Figure 1.
From eq 10 together with eq 2, 4, and 11 A =
(12) For the particular case of the packing of uniformly sized spheres, combining eq 9 with eq 12 gives the following relation for the packing porosity zo.
(16)
Restrictions on the Porosity Minimum Attainable in a Mixed Packing. If macropores exist, the preceding theory would underestimate both the total bulk volume of the space in a packing and the packing porosity. In order to consider the probable effect of macropores we introduced an assumption that imposes restrictions on the minimum porosity value for the packed mixture. The assumption is that a multicomponent mixture never has a packed bulk volume smaller than the total volume of both the pore space and the solid space when there is one fewer component. That is VB(m) 2 VB(m-1) (17) where VB(m) = total packed volume of m-component mixture in question, and VB(m - 1) = packed volume of an (m - 1)-component mixture. Rewriting the inequality V,(d Vs(m - 1) I (18) 1-z 1-~(m-1) where V,(rn) = total solid volume of m-component mixture in question; V, (rn - 1) = solid volume of an (rn - 1)-component mixture; z = packing porosity of the mixture in question; and E(m - 1) = packing porosity of an (n1)-component mixture. As Vs(m - 1)/V,(m) is the fractional solid volume, u(m - l),of the (m - 1)-component mixture, it follows that F 1 1 - (1- ~ ( -ml))/u(m - 1) (19) This inequality places restrictions on the minimum porosity value attainable in a mixed packing. An existence of macropores is suggested when eq 14 indicates a porosity value smaller than the above minimum porosity. In such a case, the minimum porosity value should be adopted as the overall average porosity of the packing. Inequality 19 has to be satisfied by all of the mixture combinations, including (m - 1)components, whose number equals m. For a ternary mixture of components 1,2, and 3, therefore, 3 combinations come into effect. That is 1 - Eij Zl1--
Using this equation, we can evaluate the surface porosity E A from the porosity value obtainable in the packing of uniformly sized spheres. According to the above theory, the packing porosity without macropores can be estimated by giving the porosity value z0 and the size distribution function f ( D ) of particles. For a discrete size distribution of an m-component mixture, the theoretical iesults are summarized as follows: Di = particle size of ith component; f i = number fraction of ith component
FDi3fi
i=l
ui
+ uj for i.j = 1.2, 1.3, 2.3
(20)
The porosity value et, for a binary mixture can in turn be estimated by using both eq 14 and inequality 19 with m = 2. Calculations and Discussions The above theory is based entirely on the assemblage of spherical particles, whereas deviations in the shape from sphere are often encountered in practice. The precise discussion on the shape problems is beyond the scope of this paper. Excepting such an extreme shape as a plate or a needle, however, the effect of particles’ shapes appears to be reflected on the packing porosity z of the theory. In almost the same way of the mathematical treatments, we have already calculated the porosity values for a binary mixture which agreed well with experiments (Ouchiyama and Tanaka, 1981). In the following, we examine the
492
Ind. Eng. Chem. Fundam.. Vol. 23, No. 4, 1984 FINE
FINE
0.432
MEDIUM
COARSE
Figure 2. Isoporosity lines experimentally obtained by Standish and Borger (1979): coarse = 12.7 mm; medium = 9.6 mm; fine = 6 mm.
0.432
0.432
MEDIUV
COARSE
Figure 5. Isoporosity lines experimentally obtained by Kawamura et al. (1971): coarse = 0.8C-2.30 mm, medium = 0.30-0.80 mm; fine = 0.0884.30 mm.
FI\
1 41c
FIZF
0.43(
0.410
0.410
MEDIUM
COARSE
Figure 3. Isoporosity lines drawn based on the theoretical results of Figure 4.
0.430
0.430
VED I U Y
COARSE
Figure 6. Isoporosity lines drawn based on the theoretical results of Figure 7.
8 410 409
409
408 408 JOY
408 I O R
YEDIUN
409 409
410
CO.IKS1
Figure 4. Overall average porosity values computed by the theory: coarse = 12.7 mm; medium = 9.6 mm; fine = 6 mm; z,, = 0.410.
theory by using the measured values for mixtures of particles of three different sizes. Past experimental da@i on ternary mixtures have often been presented as isoporosity lines on triangular diagrams. Of particular interest are the mixing ratios at which the most dense packings occurred. Figures 2 and 5 show the two typical experimental results by Standish and Borger (1979) and Kawamura et al. (1971), respectively. An experimental result similar to Figure 2 can also be observed in the report of Ridgway and Tarbuck (1968) and the one shown by Figure 5 in the article by Cunningham (1963). According to our theory, on the other hand, the overall average porosity of ternary mixture ( m= 3) is a function of (1)the porosity value for a single component, zo, (2) the vi) of the three composizes, D,, and (3) the fractions nents. Assuming suitable values for zo and D,, we evaluated mixture porosities at the volume fractions u, of each com-
vi,
\II,)
11$1
< (
\R,'
Figure 7. Overall average porosity values computed by the theory: coarse = 1.55 mm; medium = 0.55 mm; fine = 0.194 mm; zo = 0.430.
ponent, varying each by 10%. The number fractions f, were calculated from
2 ui/D?
i=l
Figures 4 and 7 summarize these computer estimates of the overall average porosities for the corresponding two ternary mixtures, respectively. Each Arabic number on both figures is a thousand times the porosity value and the one bracketted with parentheses is the minimum porosity value determined by the inequalities (20). Referring to the numbers, we drew the isoporosity lines on the triangular
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Ind. Eng. Chem. Fundem. 1984, 23, 493-499
diagrams which are shown separately in Figures 3 and 6. Comparing them with the experimental figures, the most dense packing appears on the coarse-fine axis in Figures 2 and 3, respectively, and inside the triangular diagrams in the Figures 5 and 6. Including the configurations of isoporosity lines, we can see a general agreement between the theory and the experiments. These results suggest wide application of the proposed theory. Nomenclature
C(D)= coordination number, dimensionless Q,Di= diameter of particle, m
D = average diameter of particles from eq 1, m f ( D ) = number frequency size distribution of particles, fi = fractional number of ith component, dimensionless rn = number of components composed of particles of different sizes, dimensionless N = total number of particles, dimensionless n = number of hypothetical spheres, dimensionless ii = average value of n, dimensionless VB (rn) = total packed volume of rn-component mixture in question, m3 VB (rn - 1)= packed volume of an (rn - 1)-componentmixture, m3 VBT = total packed volume of mixture in question, m3 V, (0)= total volume of space allocated to a specified particle, m3
V,,,(D) = volume of spherical shell, m3 Vu(rn - 1)= solid volume of an (rn - 1)-componentmixture, m3 Vu (rn) = total solid volume of rn-component mixture in question, m3 ui = fractional solid volume of ith component, dimensionless u (rn - 1) = fractional solid volume of an (rn - 1)component mixture, dimensionless Greek Letters z = overall average porosity of mixed packing, dimensionless zo = overall average porosity of packing of uniformly sized spheres, dimensionless t A = surface porosity around a particle, dimensionless tij = overall average porosity of packing of components i and j , dimensionless t (rn - 1) = overall average porosity of an (rn - 1)-component mixture, dimensionless t,(D) = average porosity of a spherical shell, dimensionless Literature Cited Cunningham, G. W. React. M t e r . 1983, 6, 1. Kawamura, J.; Aokl, E.; Okusawa, K. Kagaku K q k u 1971, 35, 777. Ouchiyama, N.; Tanaka. T. I&. Eng. Chem. Fundem. 1981, 20, 66. Standlsh, N.; Borger, D. E. Powder Techno/. 1979, 22, 121. RMgway, K.; Tarbuck, K. J. Chem. Recess Eng. 1988, 49, 103.
Received for review August 31, 1983 Accepted April 11,1984
High-pressure Phase Behavior of Binary Mixtures of Octacosane and Carbon Dioxide Mark A. McHugh,' Andrew J. Seckner, and Thomas J. Yogad Department of Chemical Engineering, Universlty of Notre &me, Notre Dame, Indkna 46556
The high-pressure fluid phase behavior of binary mixtures of octacosane and COPis experimentally investigated. Solubilities of octacosane in supercritical CO, and mixture molar volumes are determined for isotherms of 34.7, 45.4, 50.2, and 52.0 OC over a range of pressures from 80 to 325 atm. The solubility data are obtained by two different experimental techniques. The pressure-temperature projection of the two branches of the three-phase solid-liquid-gas freezing point depression curve is also determined. The octacosane-CO, LCEP is determined as 32.2 OC and 72.6 atm. The UCEP, which is at a pressure greater than 650 atm, could not be determined due to the pressure limitation of the experimental apparatus. Phase diagram constructions are used qualltatively to explain the observed phase behavior and to provide Information on the expected phase behavior of the octacosane-CO, system at pressures higher than those experimentally investigated.
Introduction
Supercritical solvent extraction (i.e., extraction with a solvent which is a t a temperature above its critical temperature and a pressure above its critical pressure) is currently being considered as an alternative to conventional separation techniques, such as liquid extraction and distillation. Although supercritical solvents have been used for a variety of separation problems (Schneider et al., 1980; Paulaitis et al., 1983; McHugh, 1984), there is still a need for fundamental research with supercritical solvents to fully understand and capitalize on the unique processing capabilities of these types of solvents. The high-pressure fluid phase behavior of mixtures is one area of research t U.S.Steel
Chemical Research, Monroeville, PA. 0196-4313/84/1023-0493$01.50/0
with supercritical solvents which needs to be expanded. Of interest in this study is the phase behavior of one class of binary mixtures which consists of a heavy nonvolatile solid and a light supercritical fluid. In this instance the melting temperature of the heavy solid, T,,, is greater than the critical temperature of the light component, T,,, and the molecular size, shape, structure, and critical conditions of the two components differ substantially. The phase behavior for this type of binary mixture is depicted in the P-T diagram shown in Figure 1 (Rowlinson and Richardson, 1959). CD and MH are the pure component vapor pressure curves, MN the heavy component melting curve, and EM the pure heavy component sublimation curve. Points D and H represent pure component critical points. For this type of system, the critical mixture curve which represents the critical conditions for mixtures of 0 1984 American Chemical Society
