125
Ind. Eng. Chem. Fundam. 1986,25, 125-129
Subscripts
Matljevic, E.; Couch, J. P.; Kerker, M. J . Phys. Chem. 1962, 66, 111. Matljevlc, E.; Mathal, K. G.; Otterwill, R. H.; Kerker, M. J . Phys. Chem. 1961, 65, 826. Matljevlc, E.; Tezak, B. J . Phys. Chem. 1953, 57,951. Napper, D. H. Trans. Faraday Soc. 1968, 6 4 , 1701. Napper, D. H. J . Colloid Interface Sci. 197Oa, 32, 106. Napper, D. H. J . Colloid Interface Sci. 197Ob, 33,384. Napper, D. H.; Netschey, A. J . Colloid Interface Sci. 1971, 37,528. Rles, H. E.; Meyers, B. C. Science 1968, 760, 1449. Sarkar, N.; Teot, A. S. J . Colloid Interface Sci. 1973, 43,370. Saunders, F. L.; Saunders, J. W. J . Colloid Sci. 1956, 77, 260. Senju, R. "Koroldo Tekitei-ho"; Nankodo: Tokyo, 1969. Sommerauer, A.; Sussmann, D. L.; Stumm, W. Kolloid-Z. 1968, 225, 147. Stumm, W.; O'Melia, C. R. J . AWWA 1968, 60,514. Suk, V.; Malat, M. Chemist-Analyst 1958, 4 5 , 30. Suzuki, A,; Kashiki, I . Kagaku Kogaku Ronbunshu 1982, 6 , 728 (in Japanese). Vincent, B. Adv. Colloid Interface Sci. 1974, 4 , 193-277.
a = anions taking part in colloid formation c = cations taking part in colloid formation Registry No. PVSK, 26837-42-3; MGCh, 88650-88-8; AgI, 7783-96-2; Zn(OH)2,20427-58-1; A1(N0J3, 13473-90-0; Hi-floc (SRU), 32037-29-9; Hi-floc (co-poly), 82013-86-3. Literature Cited Bratby, J. "Coagulation and Flocculation"; Uplands Press: Croydon, U.K., 1980a; p 142. Bratby, J. "Coagulatlon and Flocculation"; Uplands Press: Croydon, U.K., 1980b; p 84. Flory, P. J. "Principles of Polymer Chemistry"; Cornel1 University Press: New York, 1953; Chapter 13. Kashikl, I.; Suzuki, A. Kagaku Kogaku Ronbunshu 1982a, 6 , 722 (in Japanese). Kashiki, I.; Sauzuki, A.; Gotoh, K. Kagaku Kogaku Ronbunshu 1982b, 8 , 7 3 (in Japanese). La Mer, V. K. J . Colloid Sci. 1964, 19, 291. Matljevlc, E.; Abramson. M. B.; Schulz, K. F.; Kerker, M. J . Phys. Chem. 1960, 6 4 , 1157.
Received f o r review May 31, 1984 Accepted April 29, 1985
Porosity Estimation from Particle Size Distribution Norlo Ouchlyama Natlonal Industrial Research Institute of Kyushu, Tosu, Saga-ken, Japan
Tatsuo Tanaka" Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
The packing porosity of a bed of randomly placed spherical particles is generally described as a function of the size distribution of particles. This treatment takes into account the probable effect of macropores on the mixture porosity. For mixtures having the Gaudin-Schuhmann size distribution of particles, computed estimates of the average porosity are found to agree in general with known data.
Introduction Packing problems of solid particles are frequently encountered in a wide field of science and technology. Both the packing porosity and the number of contacts between neighbor particles have an essential relation to the material and the process properties of solid particles. We have already proposed a simple expression for the number of contacts (Ouchiyama and Tanaka, 1980). Many experimental studies have been done of the fractional void volume of a bed of solid particles, and it is a well-known empirical fact that the packing porosity varies with the size distribution of the materials involved. From a theoretical point of view, on the other hand, much work has been devoted to the regular packings of solid spheres, and a few investigators such as Furnas (1931), Westman and Hugill (1930), Tokumitsu (1964), and Kawamura et al. (1971) have examined the mixture porosity of a bed of solid particles of different sizes. Until now, however, nobody has succeeded in generally describing the mixture porosity as a function of the size distribution of particles. Recently we have proposed a theoretical formula to estimate the packing porosity of a mixed bed from the size distribution function of the solid particles (Ouchiyama and Tanaka, 1981, 1984). We have tested the theory with mixtures of the binary, ternary, Gaudin-Schuhmann, Gaussian, and log normal particle size distributions. 0196-4313/86/1025-0125$01.50/0
General agreement has been observed between the theory and experiments. For the continuous size distribution of particles such BS that of Gaudin and Schuhmann, however, some remarkable deviations of computed values were recognized from experiments. It was inferred that the existence of macropores would have caused the discrepancy. As to the discrete size distribution of particles, on the other hand, the effect of macropores has satisfactorily been taken into the theory. This paper presents a modified relation regarding the effect of macropores on the packing porosity. This treatment enables one to estimate more precisely than before the packing porosity of a mixture having an arbitrary particle size distribution. The theory will be applied to mixtures having the Gaudin-Schuhmann size distribution. Theoretical Treatment Porosity Estimation Based on a Simplified Model. Our theoretical consideration starts with a simplified model for the coordination number. See Figure 1. A central solid sphere of diameter D is in direct contact with a fixed number of surrounding spheres, each having the average diameter D. Introducing the surface porosity, eA, which means the void area fraction on a spherical surface of diameter D + D and which is assumed to be independent of the sizes of the particles involved, the coordination 0
1986 American Chemical Society
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Figure 1. A simplified model for the coordination number. Hatching calls attention to a spherical shell considered in the theory.
number, C(D),around the central particle of diameter D is exactly expressed as
where
D = $”bf(D) dD
porosity of a mixed packing without macropores. Restrictions on t h e Minimum B u l k Volume Attainable in a Mixed Packing. If macropores exist, the above theory would underestimate the total bulk volume of the space in a packing. In order to consider the probable effect of macropores, we introduced an assumption which imposed restrictions on the minimum bulk volume attainable in a mixed packing. In our previous paper (Ouchiyama and Tanaka, 1984),we imagined that some constituent particles of a mixture can be embedded in the packing interstices between the other residual particles. Thinking the matter over, however, we realize that the larger particles cannot be embedded in the interstices of the isolated packing of smaller particles although the reverse seems possible. We now assume that a multicomponent mixture never has a packed bulk volume smaller than the total volume of both the pore space and the solid space, VB,when smaller particles are taken away from the original mixture in question. On the other hand, the total volume above, VB,cannot be smaller than the bulk volume computed from the theory for a bed without macropores. Therefore, the following inequality should stand for an arbitrary size of D’.
Here, f ( D ) is the number-frequency size distribution of particles and D, and D1 are the smallest and the largest sizes of particles, respectively. According to the above model, each hypothetical sphere of diameter D D in a packing has to share, in part, the space in common with the other hypothetical spheres. The space commonly occupied lies 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 d D (34 =D-D forDID (3b) and the volume of the spherical shell, V,(D), is given as
vBT
-
+
-
D)3 - T ( D D)3 (4) 6 6 If we let the average number of hypothetical spheres taking part in the common occupancy be ii, then the total volume of the space allocated to a specified solid sphere of diameter D , V,(D), can be expressed as V m ( D )= -(D
7T
V,(D) = s ( D
- D)3 + Vm(D)/17i
(5)
Where no macropores exist, we can represent the total bulk volume of the space in the packing, V B T , as
VBT=
L, V , ( D ) N f ( D )dD Dl
(6)
where N is the total number of particles in the packing. Here, “macropore”means the void space which could exist away from solid surfaces a t a distance greater than half an average diameter of the particles. The overall average volume porosity of the packing, Z, should therefore be estimated from VBT -1 - (7)
Without going into detail, we want to point out that the value of ii in eq 5 can be evaluated approximately, as summarized later, when its size dependence is discarded. Equation 7 is a general expression for the overall average
(8)
D, d D‘ 5 D1
--
+
2
where
+
7T
L,V,(D)NY(D)d~ D1
D,
~ Y D=) MD)/s:~(D) D
(94
d~
(9c)
Note that when V,(D) of inequality 8 is evaluated from eq 5, the values of ti and D should be based on the above size distribution function, f’(D),instead of f ( D ) in the related equations. Inequality 8 places restrictions on the minimum bulk volume attainable in a mixed packing. The existence of macropores is suggested when eq 6 gives a value smaller than the above minimum volume. In such a case, the minimum value should be adopted as the total bulk volume of the packing. Summary of t h e Theoretical Relations. Equation 6 for no macropores is included in inequality 8, and all the size variables of the theory can be expressed in a dimensionless form by defining x = D/D,; k = D,/D,; x ’ = D‘/DI (10) Denoting the number-frequency size distribution of particles as a function of the dimensionless size x by g ( x ) , the theoretical relations are summarized as follows:
- 11-Z
1
for arbitrary x’; k 5 x ’ 5 1 (11)
where
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 127 I '
I
1
I
l
l
1
1
I
Equation 12a is a rearranged version of eq 16a of our previous paper (Ouchiyama and Tanaka, 19811, where zo is the porosity value for a packing of uniformly sized spheres. For the discrete size distribution of particles, the above equations can be rewritten. By numbering the size components of particles from the largest to the smallest in a mixture, we obtain
- 1- 1-S
0.2
m
i=l
for arbitrary p ; 1 I p Im (13) where
D
0,4
0,6 0.2 0,4 0.6 0,2 0.4 0,6 Distribution Constant, 2' Figure 2. Effects of the distribution constant, q, on the overall average porosity, z (replotted from experiments of Kawamura et al. (1973)). A-2, B-2, and C-2 mean the three different powders used. Maximum sizes were 2.98, 4.17, 6.33, and 8.50 mm. Minimum sizes were fixed (-0.08 mm) but not specified.
1u,
D
2"
.-w
o
m
e Here, m is the total number of size components of particles in a mixture and gi is the fractional number of particles of ith component.
4
a
?
5
0
6
0.000683 0.001367 0.002737
Discussion First, the new restrictions represented in this paper will be discussed in connection with binary and ternary mixtures of particles which were already examined in our previous papers (Ouchiyama and Tanaka, 1981,1984). For a binary mixture (m = 2), eq 13 is reduced to the relations corresponding to p = 1 and 2. The relation for p = 2 is the one for the packing without macropores. The relation for p = 1 is rewritten as
0
0,2
0.4 0,6 0.8 Distribution Constant, a
1.0
Figure 3. Effects of the distribution constant, q, on the overall average porosity, z (replotted from the packing data reported by Dinger et al. (1982)).
Next, let us apply the theory to mixtures having the Gaudin-Schuhmann size distribution of particles. The number-frequency size distribution function of the dimensionless size x is 1
g(x) = x q - ' / L (15) where u1 is the fractional solid volume of the larger particles. This restriction completely agrees with eq 18 of our previous treatment (Ouchiyama and Tanaka, 1981),which was satisfactorily used to account for the effect of macropores in a binary mixture. For a ternary mixture ( m = 3), eq 13 presents two relations for p = 1and 2 in addition to the one for no macropores (p = 3). According to our preceding theory (Ouchiyama and Tanaka, 1984), on the other hand, the mixtures of six combinations of the size components had to be additionally examined in the ternary mixture, three for m = 3 and three for m = 2. Using eq 13 together with eq 14a and 14b, we estimated packing porosities for the mixtures corresponding to Figures 4 and 7 of our previous paper (Ouchiyama and Tanaka, 1984). AU the computed values coincided with those already reported. This suggests that the previous restrictions included some redundant conditions with regard to the effect of macropores.
xq-*
dx
(16)
We tried in our first paper (Ouchiyama and Tanaka, 1981) with this size distribution of particles to estimate packing porosities by discarding the effect of macropores. In the higher range of q values, combined with the smaller size ratio of k,however, we recognized an opposite dependence between the theory and experiments. In the meantime, further experimental data have become available on the Gaudin-Schuhmann size distribution of particles. Figure 2 shows the experiments of Kawamura et al. (1973), who measured packing porosities varying q values a t several different size intervals. Their experiments clearly reveal that the average porosity decreased with increasing q values in the lower range of q values, and the reverse was true in the higher range. The latter dependence was also observed in the experiments of Andreasen (1930). The dependences similar to Figure 2 were also found in the computer packing which was reported by Dinger et al. (1982) as Figure 3. Returning to our present theory, we can now estimate the packing porosity in consideration of the effect of ma-
128
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 I. I
.
2
0‘
C
1
1
0,6
1
-
l
1
I
0.2 0.4 0.6 0.8 Distribution Constant, q
I
1
2
I 1.0
Figure 4. Theoretical estimates of the overall average porosity (z0 = 0.4). 0 5
1
0
0,4
0.4 0,6 0,8 Distribution Constant, 4
0,2
1,0
Figure 6. Theoretical estimates of the overall average porosity (so = 0.6).
0.3
lu, >
-
Y
-0,2
0,7
0 L
a a
-
0,l
U
0
0,6
?,2
0.4
0,E
0.8 Distribution Constant, c
1.0
Figure 5. Theoretical estimates of the overall average porosity (zo = 0.5).
0,5
-
0.4
-
lW >I
Table I. Values of the Size Ratios k in Figures 4-6 and 7 integral Darameter k integral Darameter k 1 0.500 8 0.003 91 2 0.250 9 0.001 95 3 0.125 10 0.OOO 977 4 0.0625 11 0.000 488 12 0.000 244 5 0.0313 6 0.0156 13 0.000 122 7 0.00781
cropores. The overall average porosity of mixtures having the Gaudin-Schuhmann size distribution of particles is a function of (1)the porosity value for a single component, e,, (2) the size ratio, k , and (3) the distribution constant, q. Assuming suitable values and varying the lower integral limit x’of eq 11 stepwise between k and 1, we calculated the porosity values from eq 11to 12a. Several theoretical results are illustrated in Figures 4,5, and 6, where values of e, equal to 0.4, 0.5, and 0.6, respectively, are assumed. In each figure, the integral parameters are the assumed k values which are summarized in Table I. In contrast with our previous calculations (Ouchiyama and Tanaka, 1981),Figures 4-6 point out that, for the smaller range of k , there exists a certain critical value of q, qc, which gives the most dense packing for the Gaudin-Schuhmann size distribution having a fixed size interval. This new theoretical trend fully explains the opposite dependence on the q value in Figures 2 and 3. An interesting theoretical indication in Figure 7 can be obtained by replotting the lowest porosity against qc with constant parameters of k and q,. Figure 7 shows that the q value giving the densest packing a t a fixed size interval should depend on the porosity value of the single component, eo. Unfortunately, no exact data were given for values of k and zo regarding
+-
‘Z0.3 0
-
CL 0
0.2
-
0.1
0‘ 0
I
I
0,2 0,4 0.6 0,8 Distribution Constont, 4,
Figure 7. Theoretical relationship between most dense packing.
Z,
1.0
qe, k , and z,, in the
Figure 2, and details were not reported of the packing procedures as to Figure 3. For a quantitatively precise discussion, further investigations are needed. However, general agreement can be observed between the theory and the reported data. These results suggest wide application of the proposed theory.
Nomenclature
C(D)= coordination number, dimensionless D,D‘= diameter of particle, m D = average diameter of particles, m D,= diameter of smallest particles, m D1 = diameter of largest particles, m f(D)= number-frequency size distribution of particles, m-l f’(D) = defined by eq 9c, m-l g(x) = number-frequency size distribution of particles, dimensionless gi = fractional number of ith component, dimensionless i, p = integer, dimensionless k = D, f D1, dimensionless
129
Ind. Eng. Chem. Fundam. 1986,25, 129-135
m = number of components of particles of different sizes, dimensionless N = total number of particles, dimensionless N’ = defined by eq 9b, dimensionless r i = average number of hypothetical spheres, dimensionless q = distribution constant, dimensionless V,, = total bulk volume of mixture, m3 V,(D) = total volume of space allocated to a specified sphere, m3 V,(D) = volume of spherical shell, m3 vi = fractional solid volume of ith component, dimensionless x , x’y DID,, D’lD,, dimensionless f = DID,, dimensionless
average porosity of packing of uniform sized spheres, dimensionless
Zo =
Literature Cited Andreasen, A. H. M. Koiiok9-2. 1930, 5 0 , 217. Dinger, D. R.; Funk, J. E., Jr.; Funk, J. E., Sr. Proc. 4th Int. Symp. Coal Slurry Combust. 1982, 4 , 1. Furnas, C. C. Ind. Eng. Chem. 1931, 2 3 , 1052. Kawamura, J.; Aoki, E.; Okusawa, K . Kagaku Kogaku 1971, 3 5 , 777. Kawamura, J.; Hayami, H.; Ariyoshi, K . ; Hosoi, E. Yogyo Kyokaishi 1973, 81, 7. Ouchiyama, N.; Tanaka, T. Ind. Eng. Chem. Fundam. 1980, 19, 338. Ouchiyama, N.; Tanaka, T . Ind. Eng. Chem. fundam. 1981, 2 0 , 66. Ouchiyama, N.; Tanaka, T . Ind. Eng. Chem. Fundam. 1984, 2 3 , 490. Tokumitsu, 2 . Zairyo 1984, 13, 752. Westman, A. E. R.; Hugill, H . R. J . Am. Ceram. SOC. 1930, 13, 767.
Greek Letters z = overall average porosity of mixed packing, dimensionless t~ = surface porosity around particle, dimensionless
Received for review June 6 , 1984 Accepted April 24, 1985
Thermal Instability of Coal-Derived Naphtha Laurlne G. Galya*+ and Donald C. Cronauerr Gulf Research & Development Company, Pittsburg, Pennsylvania 15230
Paul C. Painter Department of Material Sciences, Pennsylvania State University, University Park, Pennsylvania 16057
Norman C. Ll Chemistry Department, Duquesne University, Pittsburgh, Pennsylvania 152 19
Formation of solid deposits from SRC-I1 in heat exchangers causes plugging and results in pilot plant shutdowns. A deposit obtained from a hydrogenation reactor was examined, and the deposit precursors were concentrated through distillation and ion-exchange chromatography. The results indicate that phenols and aikylated nitrogenous bases are responsible for deposit formation. Phenols react through coupling reactions to give furans and water. Alkylated nitrogenous bases react by pyrolysis to produce free radicals which cross-link to produce carbonaceous deposits. These reactions have been followed by thermogravimetric analysis, Fourier transform infrared spectroscopy, and gas chromatography in the temperature range 250-400 O C .
Introduction Upgrading of SRC-I1 naphthas requires nitrogen and oxygen removal by hydrogenation using a catalytic system. Black deposita form in both the vaporizer and inlet regions of the catalyst bed. This often causes plugging and reactor instability. The overall purpose of our project is to avoid these problems by understanding the fundamentals of the reactions leading to deposit formation. Two basic types of reactions have been implicated in fuel deposit formation. Oxidation reactions lead to insoluble sludges and acids during fuel storage. These can then deposit in reactor lines and subsequently can form coke. Pyrolytic reactions lead to carbonaceous deposits in hot reactor zones. Most of the previous work done in this area has been done on petroleum- and shale-derived materials; relatively little has been done on coal-derived fuels. Current address: Mobil Research and Development, Billingsport Rd, Paulsboro, NJ 08066. $Current address: Amoco Research Center, P.O. Box 400, Naperville, IL 60566. 0196-4313/86/1025-0129$01.50/0
This paper initially discusses previous work done in the area of fuel deposits. A brief discussion of the chemical nature of the deposit is then given, followed by a description of the methodology used and conclusions. Oxidative Deposits. Hydrocarbons are known to produce acids and sludges through autoxidative reactions in which free radicals are formed. The relative reactivity is determined by the ease of radical formation. This means that benzylic, allylic, and tertiary carbons are all likely centers for oxidation reactions to occur. Nitrogen compounds have been shown to contribute to oxidative instability in both petroleum and shale oils (Offenhauer et al., 1957; Thompson et al., 1951; Ward and Schwartz, 1962). The most deleterious forms of nitrogen compounds appear to be alkylated nitrogen heterocycles (Frankenfeld and Taylor, 1978, 1982). Included in these are 2,5-dimethylpyrrole, 2-methylindole, 2-methylquinoline, and 5-ethyl-2-methylpyridine(Nowack et al., 1980). Pyrroles have been shown to react via an autoxidative mechanism. Nonalkylated pyrrole oxidizes in n-pentane to form a deposit even though it is reported not to cause fuel instability (Frankenfeld and Taylor, 1982). Elemental oxygen con0 1986 American Chemical Society
