Ind. Eng. Chem. Res. 1987,26, 911-921 g = gravitational acceleration, cm/sz g, = conversion factor, g.cm/(dyn.s2)
n = number of runs NEo= Eotvos number gd,2Ap/u = (3/4)Nw& N M = Morton number, gp:Ap/p:u3 = 3C&w,3/4Nk4 N = O3p:/gp:Ap = (NM)-l = Reynolds number, dppcut/p, Nwe = Weber number, dPu,2pJu SAA = surface-active agent (i.e., surfactant) uo,ut = velocity of fluid through orifice or nozzle, terminal velocity, cm/s V,, Vex,, V,, = volume of droplet, experimental value, value calculated from eq 1using diminished u due to SAA, cm3
dk
Greek Symbols
,I
p, P d , p,,
= viscosity of continuous phase and of water, P Ap = density of dispersed phase, continuous phase, and
the difference between the two, g/cm3 u
= interfacial tension, dyn/cm
Literature Cited Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles, Academic: New York, 1978; pp 172-188. Davies, J. T. In Advances in Chemical Engineering; Drew, T. B., Hoopes, J. W., Jr., Vermeulen, T., Eds.; Academic: New York, 1963;Vol. 4,p 33. Davies, J. T. Turbulence Phenomena; Academic: New York, 1972; p 313. Davies, J. T.; Rideal, E. K. Znterfacial Phenomena, 1st ed.; Academic: New York, 1961;pp 318, 335-337.
911
Davies, J. T.; Vose, R. W. Roc. R. SOC.London, Ser. A 1965,A286, 218. Edge, R. M.; Grant, C. D. Chem. Eng. Sci. 1971,25,1001. Emmert, R. E.; Pigford, R. L. Chem. Eng. h o g . 1954,50,87. Farmer, W.S. Unclassified Report ORNL 635, 1950; Oak Ridge National Laboratory, Oak Ridge, TN. Garner, F. H.; Hale, A. R. Chem. Eng. Sci. 1953,2,157. Garner, F. H.; Skelland, A. H. P. Chem. Eng. Sci. 1955, 4, 149. Garner, F. H.;Skelland, A. H. P. Ind. Eng. Chem. 1956,48, 51. Gordon, K. F.; Sherwood, T. K. Paper presented at American Institute of Chemical Engineers Meeting in Toronto, April 1953 (Cited as their ref 10 in Garner and Hale, 1953). Grace, J. R.; Wairegi, T.; Nguyen, T. H. Trans. Znst. Chem. Eng. 1976,54, 167. Harkins, W. D.; Brown, F. E. J. Am. Chem. SOC.1919,41,499. Hu, S.; Kintner, R. C. AIChE J. 1955,I , 42. Johnson, A. I.; Braida, L. Can. J. Chem. Eng. 1957,35,165. Kafesjian, R.; Plank, H.; Gerard, E. R. AZChE J . 1961,7,463. Klee, A. J.; Treybal, R. E. AZChE J. 1956,2,444. Scheele, G. F.; Meister, B. J. AZChE J . 1968,14,9. Schroeder, R. R.;Kintner, R. C. AZChE J. 1965,11,5. Skelland, A. H. P.; Caenepeel, C. L. AZChE J. 1972,18,1154. Skelland, A. H. P.; Vasti, N. C. Can. J. Chem. Eng. 1985,63,390. Skelland, A. H. P.; Wellek, R. M. AZChE J . 1964,10,491. Tailby, S.R.; Portalski, S. Trans. Znst. Chem. Eng. 1961,39,328. Thorsen, G.; Stordalen, R. M.; Terjesen, S. G. Chem. Eng. Sci. 1968, 23,413. Treybal, R. E. Liquid Extraction, 2nd ed.; McGraw-Hik New York, 1963;pp 184-185.
Receiued for reuiew April 16,1984 Accepted October 8,1986
Evolution of Pore Surface Area during Noncatalytic Gas-Solid Reactions. 1. Model Development Girish Ballal and Kyriacos Zygourakis* Department of Chemical Engineering, Rice University, Houston, Texas 77251
Novel random pore models are developed to predict the evolution of internal pore surface area during noncatalytic gas-solid reactions. These models can treat porous solids having micro- and macropores of different shapes and exhibiting widely ranging pore-size distributions. Pores are visualized as overlapping geometrical entities randomly interspersed in the solid matrix. Probabilistic arguments are used t o correlate the changes in pore volumes and surface areas t o the conversion of the solid reactant in the kinetic control regime. Surface area losses due t o coalescence of adjacent pores are rigorously accounted for and the model parameters are obtained from measurable physical properties of the unreacted solid. Numerical computations show a strong effect of the size distribution of the micro- and macropores on the predicted surface area-vs.-conversion curves. The problem of noncatalytic reactions between fluids and porous solids is considered in this study. These reactions take place on the walls of the internal pores of the solids, and in the absence of intraparticle diffusional limitations and heterogeneities in the chemical structure of the solid, their rates may be assumed to be proportional to the internal surface area. Since the solid reactant is consumed, however, the surface area and hence the gasification rate will vary continuously with the extent of reaction. The gasification reactions of char particles formed by devolatilizing parent coals are prime examples of such systems. The char particles exhibit bimodal pore-size distributions with many large macropores and a micropore network that usually results in high values for the internal surface area. The pores are randomly oriented and may be of many different shapes. In the case of some highranked coals (anthracites), the pores resemble slits or cracks rather than cylinders. Thus, the modeling of the
* Author to whom
all correspondence should be addressed.
0888-5885/87/2626-0911$01.50/0
evolution of the internal surface area (or equivalently of the gasification rate) with conversion is a complex task for such systems. One of the earliest attempts to model such reactions was that by Petersen (1957). He solved the reaction-diffusion problem in a single pore and in an assembly of cylindrical pores of uniform size. However, he neglected the intersection of new reaction surfaces, which is the dominant factor after the initial stages of gasification. Szekely and Evans (1970)used Petersen’s formulations to obtain the reaction rate-vs.-conversion relationship. Recently, Ramachandran and Smith (1977)also considered a structural model, assuming cylindrical pores of uniform radius and neglecting intersection of new reaction surfaces during gasification. Another approach to this problem is by using population balance techniques. Hashimoto and Silveston (1973) adopted this approach to account for pore-size distribution and coalescence of pores as they grow due to reaction: However, their model contains numerous parameters which need to be fitted in order to obtain the reaction 0 1987 American Chemical Society
912 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
rate-vs.-conversion relationship. The population balance approach was also used recently by Simons and Finson (1979) and Simons (1979). A number of semiempirical relations were, however, needed to correlate the important physical properties of chars to pore statistics. Some of the more recent random pore models are those proposed by Bhatia and Perlmutter (1980) and by Gavalas (1980). Bhatia and Perlmutter considered the pore structure to consist of cylindrical pores parallel to each other. The overlapping of these parallel pores was taken into account by correlating them to the nonoverlapped pore volume, following the approach of Avrami (1940). The final form of the model describes the pore structure with two parameters that can be obtained from measurable properties of the solid. It is simple, very easy to apply, and predicts accurately the gasification rates of many chars. The model has been applied to solids with bimodal poresize distributions (Bhatia and Perlmutter, 1981). However, two parameters were again used and were obtained by integrating the distribution functions. Although the model can be applied to solids with arbitrary pore-size distributions, the use of only two “average” parameters to describe complex pore structures may be restrictive and may lead to inaccurate predictions in some cases. Despite the inaccuracies of pore characterization techniques, they can still provide detailed information about the micropore and macropore structures, allowing for separate treatment of the two distributions. Gavalas has also proposed a random capillary model considering randomly oriented, overlapping cylindrical pores in space. The pore-size distribution is, however, characterized only by two moments. Hence, a bimodaltype distribution cannot be accounted for satisfactorily, and the two parameters have to be adjusted in order to get a good fit. Moreover, the effect of the noncylindrical nature of the pores was not accounted for. The problem of bimodal pore-size distribution has been considered by Zygourakis et al. (1982). The pore structure is visualized to consist of two types of entities: spherical cavities representing the macropores and the cylindrical pores representing the micropore structure. The reaction rate-conversion relationship has been correlated to the structural properties of the initial char. Theoretical predictions have been shown to be in good agreement for a wide set of experimental data by Dutta et al. (1977) and by Dutta and Wen (1977). However, the coalescence of pores was accounted for by introducing two parameters, which had to be adjusted in order to obtain a satisfactory fit of the experimental data. Also, the effect of the nonsymmetric shape of the pores, as well as the effect of the ash content on the reaction rate, was neglected. In the present work, a different approach has been used to model the random pore structure. In the next section, a simple model is developed in which the pores are assumed to be generated from random overlapping of parallel uniform cylindrical cavities. The model parameters are shown to be directly related to experimentally measurable quantities such as surface area and porosity. The evolution of surface area with conversion can then be predicted from the experimentally measurable properties of the initial char. The simple model defines our approach to this problem and is extended to account for cylindrical pores with noncircular cross sections and bimodal pore-size distributions in subsequent sections. Finally, the effect of the ash content of chars is considered.
Models for Unimodal Pore-Size Distribution Cylindrical Parallel Pores (Model 1-A). Consider a cube with a side equal to h as shown in Figure 1. The
+
Side
AA
Figure 1. Representative cube for model 1-A.
cube is assumed to be a representative unit of the pore structure and contains N parallel cylindrical pores of radius r. Although the cylinders are parallel, they do overlap randomly. All the pores are of length h, so that any closed pore volume in the porous structure has been neglected. It is evident from the figure that, although we have considered cylinders of radius r only, the resultant pore structure after overlapping has a distribution of effective pore sizes. Of course, the resultant pore-size distribution is likely to be unimodal since we have considered the overlap of uniform-sized cylinders. Since the pores are parallel to each other, the pore volume is obtained by multiplying the total overlapped cross section of pores by the side of the cube. This total cross section of the pores, shown by the hatched area in Figure 1,will be calculated next. Consider a conceptual statistical experiment where N circles of radius r are thrown, independently of each other, on side AA of the cube in Figure 1. The centers of the circles are assumed to uniformly distributed throughout square AA. The shaded portion shown in Figure 1 is then the set of points overlapped by at least one circle. Henceforth, the shaded portion will be termed as coverage and be denoted by X. It is a Lebesque measurable subset of the Euclidean space, E2. Let the area of the shaded portion, which is the measure of X , be denoted by p ( X ) . If the aforementioned statistical experiment is repeated a large number of times, a distribution of p ( X ) will result. The quantity of interest here is the expected value of the resultant random variable, p ( X ) , and will be denoted by E ( p ( X ) ) . Robbins’ theorem (1944, 1945) will be used in order to calculate this quantity. The probability that a point, x,on face AA is covered by one circle of radius r is
w2 h2 + 4hr m 2 The probability of the point, x,not being covered by the circle is, of course, (1- Prl). N such circles are thrown at face AA, independently of each other. Hence, the probability that a point, x, is covered by at least one of the circles is Pr, =
+
Pr(x E X ) = 1-
4hr + h2 h2 + 4hr + rrr2
According to Robbins’ theorem, the expected value of the measure, p ( X ) , is given by the Lebesque integral of Pr(x E X ) over square AA. Thus
E(pCL(X))= where
p(X)
1 Pr(x E X ) d d x ) AA
is the Lebesque measure on the Euclidean
Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 913 space, E2. By definition, p ( X ) = Jx dp(x). Hence, in the present case, this expression simplifies to
h2
= h2[ I - (
+ 4hr + r r 2
h2+4hr h2 + 4hr + xr2
)"I
where x1 and x2 represent the two coordinates of point x. The pore volume, V , in the representative cube is then
v = h3[ 1 - (
h2+4hr
h2
+ 4hr + r r 2
)"I
(2)
Let S denote the surface area of the pores inside the cube. If the radius of N cylinders increases by dr, the corresponding increase in pore volume would be given by d V = S dr. Hence, dV/dr = S. By differentiating eq 2 with respect to r, we obtain the expression for the surface area, S
I
+
h2 2hr (h2 + 4hr)(h2 4hr
+
+ ar2)
The expressions for surface area and pore volume can be written in terms of the convenient dimensionless groups, t and 6, defined as
(5)
more accurate. For the simple case described above, this intuitive argument can be put in the form of more rigorous mathematical formulation, to obtain quantitative estimates for the appropriate values of h. This will be done in two stages. Firstly, the variance of the distribution for the measure of coverage, p ( X ) ,will be computed to get a lower limit for the proper values of h. Secondly, the surface area-vs.-conversion relationship will be computed for increasing values of h to show that values of h larger than this lower limit result in surface area-vs.-conversion curves that converge to an asymptotic one. As discussed earlier, eq 1 gives the expected value of the measure of coverage. It represents the average value of the area covered by at least one circle, when the statistical experiment is repeated a number of times. However, higher moments are needed to specify the probabilistic distribution completely. The variance of the distribution is of particular importance. Smaller values of the variance imply a sharper distribution, and hence, the values of the measure of coverage obtained from the statistical experiments fall in a narrow range around the expected value. The variance of the coverage distribution will be computed next. Consider two points, x = ( x l , x z ) and y = (y1,y2),on face AA of the representative cube shown in Figure 1. As explained earlier, N circles of radius r are thrown on the face of the cube. Let X represent the set of points overlapped by at least one circle and let Y represent the set of points not overlapped by any of the circles. Let p ( X ) and p ( Y) denote the respective measures. When the statistical experiment described above is repeated a large number of times, probabilistic distributions for p ( X ) and p ( Y ) will be generated. According to Robbin's theorem, the second moment of the probabilistic distribution for p ( X ) is given by
E(w2(Y))=
where
FAG)=
1
1 + 45 + 45 +
5=;
r
and (7)
1WX,Y) E'
ddx,y)
(8)
where p ( x , y ) represents the Lebesque measure on E4and P r ( x , y ) represents the probability that x E Y and y E Y. The probability, P r ( x , y ) ,depends on the coordinates of points x and y . Hence, the second moment of X distribution is given by
EG2(Y))= Let todenote the value of 5 for the unreacted solid. As the solid reacts, the pores grow in size and [ increases with conversion. However, a t any conversion, x,, 5 will be directly related to its initial value, Eo, by the relation
where the subscripts, 1 and 2, refer to the coordinates of points x and y . The points x and y belong to Y when none of the centers of the circles fall within distance r from them. When the distance s between x and y is larger than 2r, the function Pr is a constant and is given by pr(xl,Xz,Yl,Y2) =
For a known value of h, eq 4 and 5 can be applied for the unreacted solid to obtain the model parameters N a n d Q from the experimentally measurable properties, e, p ,and S,. The model parameters can then be used to ottain surface areas at various convertions by using the previously given relationship between [, Eo, and conversion. Although the parameter h cannot be measured experimentally, an appropriate value can be chosen as follows. It is obvious that the probabilistic approach outlined before should become more accurate for larger values of N and smaller values of [. This indicates that large values of h should be used in order to make the statistical experiment
h2
+ 4hr + r r 2
when s
> 2r (10)
+ (yl- x,)2)'/2. However, when the distance between x and y is smaller than 2r, the function Pr depends on the distance, s, and is given by the expression if s = ((yz - x J 2
914 Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 I
a,
\ In-
\
\
0 .&
-a U
m
\\\
'\
20
0
40
60
80
1
I .
- 5 . 0 -4.5
100
Conversion, ( X ) Figure 2. Effect of change in h (or 8,)on surface area-conversion relationship for model 1-A. pp = 0.18 g/cm3, e = 0.00576. (1) S = 18 m2/g, h = 0.1 cm or S, = 180 m2/g, h = 0.01 cm. (2) S, = 18 "$g, h = 0.01 cm. (3) S,= 18 m2/g, h = 0.001 cm or S, = 1.8 m2/g, h = 0.01 cm. (4)S,= 18 m2/g, h = O.ooO1 cm or S, = 0.18 m2/g,h = 0.01 cm.
The quadruple integral in eq 9 can be reduced to a double integral by making use of the transformation i = 1, 2 u; = y i - x i Equation 9 is then reduced to
The double integral in eq 12 can be evaluated by using eq 10 and 11. After some mathematical manipulations, the following expression can be obtained
27rr2 - 2r2 arccos ( t ) + 2r2t(1 - t2)'/' h2
1".
+ 4hr + ?rr2 (?rth2+ 4r2t3- 8rt2h)dt
(13)
By use of eq 13 for the second moment of the distribution for p(V, the variance can be obtained as U2(X) -- E ( p 2 ( X ) -) E2(Y(X))= E(P'(V) - E2(P(Y))=
h4
(I
2a52 - 1 + 45
+
*52
)"(
1- 4
~ - 5854 ~+
The dimensionless variance of the coverage distribution can thus be computed from the known values of N and 5 at any conversion. A small value of variance implies that the values of coverage obtained from different experiments fall in a narrow range around the expected value. Equation 14 suggests a range of values of h to be used in the model
-4.0 -3,s Log (hl
-i.o
-2
.o
Figure 3. Effect of change in h on dimensionless variance of the probabilistic distribution for model 1-A. pp = 0.18 g/cm3, e = 0.00576, S,= 18 m2/g.
since the variance decreases with increasing h. Thus, for a certain minimum acceptable value of the variance, eq 14 gives the appropriate value of h. Any larger value of h is acceptable to make the measure of coverage obtained from eq 1 accurate enough. Figure 2 shows the surface area-vs.-conversion relationship for a char obtained from the Hydrane process. Only the micropore structure of this char has been considered to obtain the model parameters, since the aforementioned model is appropriate for unimodal pore-size distribution. The curves shown on Figure 2 are for a fixed value of microporosity and for increasing values of h. It is evident from this figure that the surface area-vs.-conversion relationship is identical for any value of h larger than 0.01 and is given by a limiting curve. The dimensionless variance for this case is shown in Figure 3 as a function of h on a logarithmic scale. The variance clearly decreases very sharply with increasing h. It is evident that the variance is indeed extremely small, for any value of h larger than 0.01; hence, the model is statistically meaningful and accurate. This observation supports the intuitive feeling that sufficiently large values of h will result in large values of N and small values of 5, implying sharper probabilistic distribution. Hence, only the first moment should be necessary to specify the coverage distribution accurately, and any further increase in the parameter h should not have any significant effect on the predicted surface area-conversion relationship. The surface area-vs.-conversion relationship as shown in Figure 2 can be explained as follows. Initially, the pores enlarge due to the reaction occurring on the pore surface, thus increasing the area. However, as the pores enlarge, they coalesce with the neighboring pores, resulting in the decrease of the pore surface area. As a result of these competing phenomena, the surface area increases a t low conversions. It passes through a maximum and finally decreases as the effect of pore coalescence becomes more important at higher conversions. Figure 4 shows the effect of microporosity of the unreacted char on the model predictions. With a decrease in char microporosity,the pores become smaller in size and are located further apart from each other. Hence, the pores can grow to higher conversions before the coalescence becomes predominant. Hence, the maximum in the surface area-vs.-conversion curve becomes larger and is shifted toward higher conversions. It seems that the limiting position of conversion occurs at about 40% conversion. This is in good agreement with the
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 915
I
I
h2 + 4hkb + 4k2b2- 4kb2 h2 + 4hkb + 4k2b2
Pr(x E X)= 1 -
Robbin's theorem can again be used to obtain the total cross-sectional area of the pores. The cross-sectional area is then multiplied by h to obtain the pore volume in the representative cube, yielding
h2 + 4hkb + 4k2b2- 4kb2 h2 + 4hkb + 4k2b2
(16)
As the reaction proceeds, the pore walls are consumed and the reaction surface recedes from its original position. Let the reaction surface recede by de, thus increasing the pore volume by dV. It can be easily shown that the surface area is related to the change in volume by the expression
o !
0
)"I
(
v = h3[ 1 -
20
40
60
so
s =-av= av
100
Conversion Figure 4. Effect of change in microporosity on the surface areaconversion relationship for model l-A. pp = 0.18 g/cm3, S, = 18 m2/g, h = 0.01 cm.
models by Gavalas and by Bhatia and Perlmutter for unimodal pore-size distribution, predicting a maximum a t 39.3% conversion for limiting cases. Figure 2 also shows the effect of the increase in initial surface area (S ) on the surface area-conversion relationship for a particufar value of h. It can be seen that the effect of initial surface area becomes smaller for larger values of surface area until a limiting relationship is obtained. The variance of the coverage distribution could be computed relatively easily for the simple case described above. However, the computations become much more difficult for more complex cases, accounting for different pore shapes and for bimodal pore-size distributions. Hence, for these cases, the range of h obtained for the simple model described earlier will be used as a guide, and the value of h giving the limiting surface area-conversion relationship will be chosen to obtain statistically accurate behavior.
Overlapping Parallel Pores with Rectangular Cross Section (Model 1-B) As described earlier, the shape of the pores is not necessarily cylindrical. Microscopic investigations on the porous structure of some chars indicate that the pores in those structures resemble cracks or slits more closely than cylindrical cavities (Mahajan and Walker, 1978). In these cases it seems more reasonable to represent the pore cross section as a result of overlapping of rectangles or ellipses rather than circles. Hence, the statistical experiment described earlier will be revised to account for the effect of pore shape. Let us throw N rectangles with the smaller side equal to 2b on face AA of the cube in Figure 1. Let the aspect ratio, defined as the ratio of the larger to the smaller side of the rectangle, be denoted by k . The quantity of interest remains the fractional area of face AA covered by a t least one rectangle. Since the rectangles are thrown a t random, the probability that a point, x,on face AA is covered by one rectangle is 4kb2 h >> kb > b Prl = h2 + 4hkb + 4k2b2 Since the rectangles are thrown independent of each other, the probability that a point x is covered by a t least one of the N rectangles is then
de
ab
k - 1 av b ak
The partial derivatives can be evaluated from eq 16 and substituted in eq 17 to get S = h3N
+ 4hkb + 4kb2(k - 1) h2 + 4hkb + 4k2b2 4(k + l)h2b + 16k2hb2+ 16k2(k - l ) b 3 (h2+ 4hkb + 4k2b2)2 h2
]
(18)
Equations 16 and 18 can be written in dimensionless form to obtain modeling equations for surface area and porosity. It is found that the porosity and surface area can still be given by the modified versions of eq 4 and 5, given below: 6
= 1 - FBN(i)
(44
P = NFB"-YS: )GB(S:)
(54
if 1
+ 4k5 + 4(k - l ) k t 2
b
E= + 4k5 + 4 k 2 t 2 4(k + 1) + 16k2t2+ 16k2(k - l ) t 3 G B ( ~= ) ( 1 + 4k5 + 4k252)2
FB(5) =
1
(19)
(20)
For a known value of the aspect ratio, k , eq 4a and 5a
can be solved together with eq 19 and 20 for the initial char to obtain the model parameters, and N. These model parameters can then be used at various conversions to obtain the surface area-vs.-conversion relationship. The aspect ratio, k , in these equations accounts for the noncylindrical nature of the pores. An approximate estimate for it could be obtained from microscopic investigations of the initial char. In the absence of such information, it should be treated as a model parameter and could be obtained by fitting the experimental data for the particular char. The effect of the aspect ratio, k , on the surface areavs.-conversion relationship is shown on Figure 5. When k equals unity, the cross section of the pores is square, and they can grow up to higher conversions before coalescing with neighboring pores. Hence, the maximum in surface area-vs.-conversion curves is most pronounced and occurs a t the highest possible conversion. As k increases, the pores start coalescing at lower conversions, and hence, the maximum exhibited by the surface area-vs.-conversion curve becomes smaller. Of course, any value of k smaller
916
Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 h2 + 4hr1
Pr(x E X ) = 1 -
h2 + 4hr1 + r r 1 2
However, the expression also includes the probability that the point may be covered by a large circle in addition to a smaller one. The probability of interest is that of a point is covered by at least one small circle but not by any of the large circles or by a large circle overlapping a smaller circle. A large circle can overlap a smaller circle if the distance between their respective centers is less than re = rl + r2 Hence, in order to account for this effect, Nl small circles of radii r1 and N2 large circles with extended radii re are thrown on side AA. The required probability can then be shown to be given by 01
0
20
40
60
so
1 100
Conversion
[ (
)”I(
P,, =
h2+4hrl
1-
h2
Figure 5. Effect of change in aspect ratio k on surface area-conversion relationship for model 1-B. pp = 0.18 g/cm3, e = 0.005 76, S, = 18 m2/g, h = 0.01 cm.
h2+4hre
+ 4hr1 + rr12
+ 4hre + we2
h2
+ r2
re = rl
(21)
The probability that a point is covered by any of the large or small circles is
>”(
h2 + 4hr1
P,=l-
h2
+ 4hrl + r r 1 2
h2 + 4hr2 h2 + 4hr2 + r r Z 2 (22)
The probability that a point is covered by a large circle can be calculated from the difference of these equations:
c than unity will give the same surface area-conversion relationship as its inverse, since both of these cases represent identical shapes of pores. The curve of k equal to unity is almost identical with that obtained by model 1-A.
Models for Bimodal Pore-Size Distributions Overlapping Parallel Pores with Circular Cross Section (Model 2-A). So far, the statistical experiment under consideration involved throwing a large number of uniform-sized circles or rectangles on a square. Due to the random overlapping of these entities, the resultant pore structure shows a distribution in effective pore sizes. However, since uniform-sized geometrical figures were used in the experiment, the resultant pore structure can only represent a pore-size distribution that is unimodal. In order to accurately represent a bimodal pore-size distribution, the statistical experiment must be modified. Let us consider the following two populations of geometrical entities: N1circles with radius r1 and N 2 circles with much larger radius r2. These circles are thrown at random on face AA of the representative cube as before (see Figure 6). The hatched area on Figure 6 thus represents the total pore cross section. An area overlapped only by small circles represents the cross section of micropores. The remaining area is the cross section of macropores. The objective here is to obtain separate expressions for the cross-sectional area of micro- and macropores. The probability that a point is covered by at least one of the small circles is
h2
+ 4hre + ar,2
h2 + 4hr1
Side A A
Figure 6. Representative cube for model 2-A.
)”’+ )”[ ( + + + (
h2 + 4hre
P,=1-
h2
h2 + 4hrI + r r 1 2
h2
4hre
4hre
we2
h2 ~ ~ ~ 7 ~ r 2 2( 2)3 )N ’ ]
Using Robbin’s theorem and evaluating the required probabilities from eq 21 and 23, we can obtain the crosssectional areas for the micropores and macropores. The cross-sectional areas should be multiplied by h to obtain the micro- and macropore volumes as
h2 + 4hre h2 + 4hre + we2
v, = h3[ 1 -
-
h2 h2
(
+ +
h2 4hre h2 + 4hre rr:
+ 4hr1
)”’[
+ 4hr1 + r r 1 2
(
)+ h2 + 4hre
h2 h2
+ 4hre + rr: 4hr2
)”] ]
(25)
h2 + 4hr2 + ~r~~
The surface areas of the micro- and macropores can be computed by taking appropriate derivatives with respect to rl and r2. The expressions for micro- and macropore volumes and surface areas can then be put in convenient
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 917 dimensionless forms to obtain the expressions
' --1
' I
I.
Equations 26-29 can be solved simultaneously for the model parameters, &, N,, and N2,since the micro- and macroporosities of the unreacted solid are usually experimentally measured. Equations 26-29 can again be solved a t various conversions, using the known values of model parameters, in order to obtain the change in micro- and macroporosities and surface areas with conversion. The surface area-conversion relationship for this case is then a function of total porosity, surface area, and the fractions of these quantities attributed to micropores. It was found that the model is most sensitive to the value of the microporosity. The effect of this parameter on the total surface area-vs.-conversion relationship is shown in Figure 7 . These results are as expected, since small values for the microporosity imply that the micropores are smaller and are more widely spaced. Hence, they can grow larger before coalescence becomes important, and the maxima in the surface area-vs.-conversion curves are more pronounced. Overlapping Parallel Pores with Circular and Rectangular Cross Sections (Model 2-B). As described earlier, the shape of micropores is far from being cylindrical in certain coals. This effect can be accounted for by representing the micropores by overlapping rectangles instead of circles, as was done in the case of unimodal distribution. The statistical experiment described earlier will be modified to include two kinds of geometrical entities: N1 rectangles with side 2b and aspect ratio k and N 2 circles with radius rz, which is much larger than b or kb. These entities are again uniformly distributed on face AA of the representative cube. The area covered only by small rectangles represents the cross-sectional area of micropores, and the total area covered by at least one of the big circles of rectangles represents the total cross-sectional area of micro- and macropores combined. The objective again is to obtain expressions for the volume of the micro- and macropores. The probability that a point is covered by at least one of the small rectangles is given by eq 15. This expression also includes the probability that the point be covered by any of the circles in addition to small rectangles. After this effect is accounted for, the probability that the point be covered by a t least one of the small rectangles, but not by any of the circlks, is given by
sl0,
h2 + 4hkb
P,=[l-(
+ 4k2b2- 4kb2 h2 + 4hkb + 4k2b2
[
)"'I IN'
+ 4hre h2 + 4hre + ar,2 h2
if b re = r2 + - ( k 2
+ 1)
0
20
60
40
Conversion,
80
100
Figure 7. Effect of change in microporosity on predictions of model 2-A. pp = 0.18 g/cm3, c, = 0.3902, Sgr = 16 m2/g, S,, = 2 m2/g, h = 0.01 cm. 0
1
v
Conversion,
(%I
Figure 8. Effect of change in aspect ratio k on predictions of model 2-B. p = 0.18 g/cm3, e = 0.3902, em = 0.00576, Sgr = 16 m2/g, S , = 2 mB/g, h = 0.01 cm.
Similarly, the probability that a point be covered by at least one of the rectangles or circles is P T = l -
(
h2
+ 4hkb + 4k2b2- 4kb2 h2 + 4hkb + 4k2b2
r(
h2 :4ir:::r22
The probability of coverage by a large circle only then can be computed by taking the difference and is given by
P,=l-
h2
(
+ 4hkb + 4k2b2- 4kb2 h2 + 4hkb + 4k2b2 h2 + 4hre
h2 + 4hkb + 4k2b2- 4kb2 h2 4hkb + 4k2b2
+
>" [ -
h2 + 4hre + rr,2
(30)
(%I
)"'I(
-
h2 T4ir:?;rz2
>"'
(31)
These probabilities can be used with Robbin's theorem to evaluate cross-sectional areas corresponding to micro-
918 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
and macropores. The cross-sectional areas can then be multiplied by h to obtain micro- and macropore volumes. The surface areas corresponding to micro- and macropores can be evaluated by appropriate combinations of partial derivatives with respect to r , b, and k . The expressions are more complicated than before, due to the added parameter, k , and are given below in dimensionless form:
t Side A A
Figure 9. Representative cube for the 3D cylindrical pore model.
Note that all the partial derivatives can be obtained analytically. Equations 32-35 can be used to obtain the surface area-conversion relationship from the known pore structure characteristics of unreacted char, as shown in Figure 8. The aspect ratio, k , however, will have to be an adjustable parameter if sufficient microscopic data are not available to obtain a good approximation to its actual value. As expected, an increase in the shape factor results in a smaller maximum in the surface area-vs.-conversion curve. Again, it should be mentioned that the curve obtained by setting k equal to unity is almost identical with that obtained by using model 2-A. Models with Three-Dimensional (3D) Pore Orientation. The previous models have considered cylindrical pores with different cross-sectional shapes and have accounted for bimodal pore-size distributions. However, the pores were assumed to be parallel to each other. In an actual porous solid, however, the pores are randomly oriented with respect to each other and intersect a t various angles. In order to account for this, the statistical experiment described earlier should be further modified. Let us consider the case of cylindrical pores with two different sizes, as in model 2-A. In addition to face AA, circles of two different sizes are thrown on two other faces of the cube which are perpendicular to face AA (Figure 9). The area overlapped by a t least one circle on each face represents the cross-sectional area of pores perpendicular to that face. Equations 21 and 23 are still valid to obtain the cross-sectional area of pores in the x , y, and z directions. However, the total pore volume will be less than before, due to the intersection of perpendicular pores. This is the simplest approach since the intersection of pores is limited to the perpendicular direction. The results obtained from this simple three-dimensional case can be compared with the case of parallel pores to estimate the effect of intersection of pores with various orientations. More complex statistical experiments can be designed to account for random pore orientation if necessary.
Let P, denote the probability that a point inside the cube is covered by a micropore parallel to the x , y, or z axis. Let P, denote the probability that it be covered by a macropore. Obviously, P,, and P, are given by eq 21 and 23, respectively. In the three-dimensional case, however, extra care is needed so that the points covered by two or more pores perpendicular to each other are not counted more than once. Initially, the probability that a point is covered by three pores, all perpendicular to each other, is considered. The probability that a point is covered by two micropores and one macropore is P,"P,, and the probability of it being covered by two macropores and one micropore is P,Pm2. Similarly, the probability that a point is covered by three macropores or three micropores is PP3and Pm3,respectively. The probability that a point be overlapped by two pores perpendicular to each other is considered next. The probability that a point is covered by two micropores should be P,". However, this expression also includes the three-pore intersections involving the two micropores under consideration. After accounting for this factor, the probability that a point is covered by two micropores oriented in the x and y directions, but not by any pore oriented in the z direction, will be P,2(1- P, - P,). All other probabilities involving two pore intersections can be evaluated in a similar fashion. Following the aforementioned procedure, one may calculate the probability that a point is covered by only a micro- or macropore but not by any pore oriented in the other two directions. The total pore volume can be obtained by adding one-, two-, or three-pore coverage probabilities. Similarly, the summation of all probabilities of points not covered by any of the macropores gives the micropore volume. The final expressions for micro- and macropore volumes are tp
= w 3 - 3w2 E,
+ 3w2g+ 3~ - 6 ~ +g 3wg2 = g3 + 3g - 3g2
(36) (37)
where
where the functions F A and G A are given by eq 6 and 7 , respectively. The micro- and macropore surface areas can be obtained by combining appropriate partial derivatives of micro- and macropore volumes with respect to rl and r2.
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 919 In
p, = (3g2 + 3 - ~ ~ ) [ ~ N ~ F A ~ ~ ' ( C ; , -) G A ( C ; ~ )
I
GA(te) FAN'([e)NIFAN'-' (51) GA (f 1) + F A ~ ~ ( ~ ~ ) N ~ F A " - ' ( ~ ~+) ~AN1(~l)~2~ANz-'(f2)GA(f2)l GA(E~) 2FA"(f1)N8AN'-'
(le)
(40)
p,, = (3w2 - 6~ + 6 ~ +g 3 - 6g + 3g2) X [-(I - FAN'(E1))N2FANT1(5e)GA(Se) + F ~ ~ ' ( f e ) N l F A ~ ' - ' ( f l ) G A (+f 1(3W2 ) ] - 6w + 6wg) [N2FANT1(te)GA(te) - FAN'(~1)N2FANz--'(~e)G*(fe) FANz([e)NIFAN1-l (61) GA(fi) + FAN2(f2)N1FAN'-' (f 1) GA(S1) I (41)
A s described earlier, eq 36-41 can be solved for the unreacted solid to obtain the model parameters, f l 0 , f z 0 , N,, and N2. The same set of equations can then be solved a t various conversions to obtain the relationship between surface area and conversion. The method of allowing the pores in three mutually perpendicular directions has been shown in detail for bimodal cylindrical pores (i.e,, model 2-A). However, the same method can also be used with minor modifications for other models (i.e., models 1-A, 1-B, and 2-B). The equations for the three-dimensional version of the unimodal cylindrical case (model 1-A) are t
= 3v - 3v2
. c (
U
m
P
3-0 B i m o d a l
--
m
1-0 B i m o d a l
al
>
m-
al U
m
: a
OJ-
m
-1
04
0
u I
20
60
40
Conversion.
(%I
80
100
Figure 10. Comparison for model predictions for models 1-A and 2-A and the three-dimensional cylindrical pore model. pp = 0.18 g/cm3, h = 0.01 cm. (1)For model 1 - A e = 0.00576, S, = 18 m2/g. (2) For model 2-A and 3D model: e,, = 0.005 76, ,e = 0.3902, S , = 16 m2/g, S , = 2 m2/g. 0
+ v3
p = (3 - 6~ + 3V2)NF,4N-1(f)G~(f) if
u = 1 - FAN([)
C; = r / h
Figure 10 shows the comparison of the predictions from the one-dimensional unimodal and bimodal models for a Hydrane 150 char. The macropore structure has been neglected in obtaining the model parameters for the unimodal model. Most of the internal surface area for this char is due to micropores. However, as the reaction proceeds, the macropores enlarge and engulf a large number of micropores. A substantial amount of reaction surface is lost in this process. Hence, the maximum in surface area-conversion curve becomes smaller and is slightly shifted toward lower conversions, when the effect of macropore enlargement is taken into consideration. The same figure also shows the comparison between one- and three-dimensional bimodal distribution models. In the former case, the entire length of a pore is involved as it coalesces with an adjacent pore, as opposed to the threedimensional model where the effect of coalescence is limited to a neighborhood of the point of contact. Hence, the pore coalescence effect is felt a t a smaller conversion in the case of model 2-A, resulting in a smaller maximum in the surface area-vs.-conversion relationship. It is obvious that model 2-A provides a lower bound for the surface area-vs.-conversion curves. The results of Figure 10 seem also to inducate that the consideration of bimodal pore-size distribution has a much more pronounced effect than that of three-dimensional pore orientation. It should, therefore, be always included in a detailed model.
Effect of Deposition of Ash/Inerts In the discussion so far, the solid is assumed to react completely during the pore enlargement process. In gassolid reactions like char gasification, however, the small amount of ash present in the char remains unreacted. Bhatia and Perlmutter (1980) have considered a similar case where either an inert material or the solid product of reaction forms a fine layer on the pore walls during the
"18 3
m? c1
0
1
0
20
40
60
Conversion,
80
(XI
I
100
Figure 11. Effect of ash/inerta deposition on the model predictions of three-dimensional cylindrical pore model. f a = 0.092, p = 0.18 g/cm3, h = 0.01 cm, e,, = 0.00576, e, = 0.3902, Sgp= 16 m B /g, ,S = 2 m2/g.
reaction. The effect of this layer is then taken into account in the form of a diffusional resistance which increases gradually as the layer builds up. In the case of chars, however, microscopic investigations (Renton, 1982) indicate that the ash may be present in the form of particles larger than the smaller pore radii. In such cases, the pore surface area could be divided into the free carbon surface area and the surface area that is covered by ash and is thus not available for reaction. As a simple approximation, the fraction of surface area not available for reaction, fa, will be assumed to be proportional to the two-third power of the volume fraction of ash a t any conversion. It can be easily shown that the fraction, fa, is related to the conversion by
r f*=l-
1+ 1
1
Ro(1 - x,)
i213
1
where Ro represents the ratio of volume fraction of carbon to that of ash in the unreacted char. Equation 42 can be used along with the appropriate expressions for surface areas in earlier models to account
920 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 Table I. Random Pore Models and Their Essential
Features
AI
~~~
pore-size shape of pore cross model distribution section 1-A unimodal circular l-B unimodal rectangular 2-A bimodal circular for both microand macropores 2-B bimodal rectangular (micropores), circular (macropores) 3D model bimodal circular for both microand macropores
01
I'
/
7-0 00
20
40
60
80
cn
100
Conversion, ( X I Figure 12. Evolution of initial porosities and the average micropore radius with conversion. pp = 0.18 g/cma, h = 0.01 cm, e,, = 0.00576, e, = 0.3902, S , = 16 m2/g, ,S = 2 m2/g.
for the effect of ash deposition. The fractional surface area available for reaction decreases gradually with increasing conversion during the early stages of the reaction process. This decrease is much sharper Ft higher conversions,where the weight fraction of ash becomes much more significant. Figure 11shows the effect of ash content on the available surface area-vs.-conversion relationship for the Hydrane 150 char using the three-dimensional cylindrical pore model. The maximum in the curve becomes smaller and is shifted toward lower conversions, as expected. In the case of chars with smaller ash content, the curve would be relatively unaffected at lower conversions and will go down sharply, when most of the carbon in the char has reacted.
Prediction of Other Structural Properties In the previous sections, the evolution of surface area with reaction was predicted and discussed for various models. However, it can also be used to predict evolution of other structural characteristics, like average pore radii of micro- and macropores and the evolution of pore volumes that were initially characterized as micro- and macropore volumes for the unreacted char. Figure 12 shows the evolution of these quantities for the Hydrane 150 char. Most of the reaction occurs in the pores characterized as micropores for the unreacted char. Hence, the volume associated with these pores increases almost linearly, while the volume associated with the larger pores remains nearly constant. However, these volumes should not be considered as actual micro- and macropore volumes a t higher conversion, since the size of initial micropores grows considerably during the gasification process (as shown in Figure 12) and most of them would be considered more appropriately as macropores rather than micropores at higher conversions. The total porosity increases linearly with conversion and reaches unity at complete conversion. Thus, all of our models satisfy the pore volume balance at any stage of gasification, although the assumption of constant particle size is not realistic a t higher conversions. Summary Several models have been developed for predicting changes in the porous structure of a solid during reaction in the kinetic control regime. Table I summarizes the main features of these models. The model parameters are directly related to the structural properties of the unreacted
~~
pore orientation parallel parallel parallel
parallel 3 mutually perpendicular directions
solid and are experimentally measurable. The bimodal nature of the pore-size distribution found in certain solids is taken into account by allowing two populations of pores with significantly different radii to overlap randomly. The noncylindrical nature of pores found in certain porous structures is accounted for by considering one of these populations to consist of rectangular slits. The aspect ratio of such pores could be estimated experimentally from detailed microscopic investigations. In the absence of such data, however, it should be considered as an adjustable parameter. The presence of pores with various orientations is partly taken into consideration by allowing pore intersections at three mutually perpendicular directions. More elaborate models could be easily developed to account for random three-dimensional orientation of pores. However, our results seem to indicate that this might not be essential. On the other hand, the effects of bimodal pore-size distributions and of pores with noncylindrical cross sections are very pronounced and should be included in a detailed model. The simplest model (l-A) gives predictions that are quite similar to those obtained by the models proposed by Gavalas (1980) and Bhatia and Perlmutter (1980). However, our approach can be easily extended to more complicated pore geometries and, in this sense, appears to be of great interest for applications to many reacting systems. Experimental investigations have been carried out to verify the model predictions, and the results will be reported in a companion paper along with the verification of model predictions with literature data. Extensions of the present models to include noncylindrical pores (e.g., spherical cavities) are straightforward and will be considered in a future communication.
Nomenclature b = half length of smaller side of rectangular pore cross section fa = mass fraction of ash in char FA, GA = functions defined by eq 6 and 7 FB, G? = functions defined by eq 19 and 20 h = side of the representative cube k = aspect ratio of rectangular pore cross section Pr, = probability of a point being covered by a specific geometrical entity Pr(x E X ) = probability of a point being covered by at least one geometrical entity P,,, P,,, = probability that a point is covered by at least one micropore or macropore, respectively (models l-A, l-B, 2-A, and 2-B) R, = ratio of volume fraction of carbon to ash in unreacted char r = radius of cylindrical pore cross section S = pore surface area in the representative cube S, = surface area per unit weight of the solid x , = fractional conversion based on initial reactive solid V = pore volume in the representative cube
.Ind. Eng. Chem. R e s . 1987,26,921-927
921
Greek S y m b o l s
Dutta, S.; Wen, C. Y.; Belt, R. J. Ind. Eng. Chem. Process Des. Dev.
p = dimensionless quantity, ppS& e = volume fraction of pores m representative cube f = dimensionless radius of circular pore cross section (or dimensionlesshalf length of smaller side of rectangular pore
1977, 16, 20. Gavalas, G:R. AIChE J. 1980,26, 577. Hashimoto, K.; Silveston, P. L. AIChE J. 1973,19, 259. Mahajan, 0. P.; Walker, Pp. L. In Analytical Methods for Coal and Coal Products; Karr, C., Ed.; Academic: New York, 1978;Vol. 1, p 125. Petersen, E. E. AIChE J. 1957, 3, 442. Ramachandran, A. A,; Smith, J. M. AIChE J . 1977, 23, 353. Renton, J. J. In Coal Structure; Meyers, R. A., Jr., Ed.; Academic: New York, 1982; p 283. Robbins, H. E. Ann. Math. Stat. 1944, 15, 70. Robbins, H. E. Ann. Math. Stat. 1945, 16, 342. Simons, G. A.; Finson, M. L. Combust. Sci. Technol. 1979,19, 217. Simons, G. A. Combust. Sci. Technol. 1979, 19, 227. Szekely, J.; Evans, J. W.Chem. Eng. Sci. 1970, 25, 1091. Zygourakis, K.; Arri, L.; Amundson, N.R. Ind. Eng. Chem. Fundam. 1982, 21, 1.
cross section), r / h (or b / h ) particle density
pp =
Subscripts
= micropore property m = macropore property 0 = unreacted solid property 1, 2 = populations of different entities for bimodal pore-size distribution models p
Literature Cited Avrami, M. J. Chem. Phys. 1940,8,212. Bhatia, S . K.; Perlmutter, D. D. AIChE J. 1980, 26, 379. Bhatia, S. K.; Perlmutter, D. D. AIChE J. 1981, 27, 226. Dutta, S.; Wen, C. Y. Ind. Eng. Chem. Process Des. Deu. 1977, 16, 31.
Receiued f o r review May 14, 1984 Revised manuscript receiued November 1, 1985 Accepted October 15, 1986
A Lumped Kinetic Model for Hydroprocessing Coal Extract James M. Chen* and Harvey D. Schindlert Lummus Crest Znc., Bloomfield, N e w Jersey 07003
A lumped kinetic model for hydrotreating 850+ O F coal extract is proposed in this study. This model distinguishes hydrogenation and cracking reactions that occur in the catalytic hydrotreating of coal extract. T h e reaction rates of aged Shell 324M Ni/Mo catalyst used t o hydrotreat Illinois No. 6 coal extract were determined. By use of these results, the effects of temperature and space rate on the 850+ O F conversion, as well as donor hydrogen content, were investigated. In addition, the effects of feed properties and catalyst aging on the hydrotreating performance were illustrated. Separation of liquefaction from hydrotreating reactions in direct coal liquefaction affords optimization of individual reaction stages. Integration of these two stages by recycle of the hydrotreated bottoms to the liquefaction solvent enables production of an all-distillate yield. Recent development of this integrated process, the Integrated Two-Stage Liquefaction process (ITSL), has shown a greater yield of distillate products and a higher hydrogen utilization efficiency than preceding processes, as reported by Schiffer et al. (1982), Chen et al. (1983), Schindler et al. (1984a), and Rao e t al. (1982). Much of the work in coal liquefaction, such as Curran et al. (1967) and Neavel (1975), has shown that a good hydrogen-donor solvent is essential to achieve maximum coal conversion by stabilizing coal fragments and preventing repolymerization reactions. Winschel and Burke (1983) have shown that the solvent quality can be empirically correlated with the IH NMR analysis and that the hydrogen required for liquefaction comes from the dehydrogenation of cyclic compounds in the solvent. In the ITSL process, the hydrogen-donating capability of the solvent is provided and maintained by hydrogenation in the separate hydrotreater. Most of the hydrotreated solvent in the ITSL process has a boiling point greater than 650 O F and normally
* Current address: Engelhard Corporation, Menlo Park, NJ 08817. t Current address: Science Applications International Corporation, Paramus, NJ 07652. 0888-5885/ 87 / 2626-0921$01.50/ 0
contains about 25-4570 850+ O F material. Use of this heavy solvent in ITSL contrasts with preceding processes, where only a lighter distillate solvent was used. With this heavy solvent, very little information is known about the effect of hydrotreating conditions on the solvent quality. Recently, Stephens and Chapman (1983) reported the hydrogenation kinetics of pyrene in the presence of ground Shell 324M Ni/Mo catalyst. However, pyrene is still a relatively light compound. To produce an all-distillate product yield, a balanced amount of the 850+ O F nondistillate material also needs to be converted in the hydrotreater. Although the reactions for extract conversion have been investigated by Rao et al. (1982) and Nalitham et al. (1983) to determine the yield structure, these kinetics give very little insight as to the parametric effects on the recycle solvent properties. Coal extract normally contains a substantial amount of polynuclear aromatics. Under high hydrogen pressure and in the presence of a catalyst, these molecules may proceed to smaller molecules via hydrogenation and cracking, followed by the rehydrogenation reactions. With the equilibrium limitation and difference in temperature dependence between the hydrogenation and cracking rates, it has been known that lowering reaction temperatures increases the selectivity for hydrogenation reactions, resulting in more solvent hydrogenation for a given extract conversion. Lowering temperature, however, reduces the reaction rates and increases the required reactor size to achieve a certain 850+ OF conversion. In addition, it may overhydrogenate the solvent, which could render it a poor 0 1987 American Chemical Society
