I 38
Ind. Eng. Chem. Process Des. Dev. 1984,23, 138-145
A Model Representing the Change of Pore Structure during the Activation of Carbonaceous Materials Kouichl Miura and KenJlHashimoto' Department of Chemical Engineering, Kyoto University. Kyoto, 606, Japan
This paper presents a new model for calculating the change in the pore structure during the activation of carbonaceous materials. The carbonaceous material is assumed to be divided into two parts, the nonorganized carbon and the stacks consisting of graphitic layers of carbon (Crystallite). When the carbonaceous material is activated
the more reactive nonorganized carbon is gasified, rapidly forming a reaction interface, and then the less reactive crystallite is gasified gradually, thus forming micropores within the crystallite. The number of pores created in the crystallite was calculated on the basis of the probability concept used by Wolff. Then the analytical relationships among the pore surface area, the pore volume, the solid conversion, and the reaction time were established. The validity of the model was examined by comparing the values calculated by the model with the char gasification data obtained by the authors.
Introduction The highly adsorptive capacities of activated carbons are mainly associated with their internal pore properties such as pore surface area, pore volume, and pore size distribution, which develop during the activation of chars with steam, carbon dioxide, and so on. Development of a model, by which the changes in such pore properties during the activation can be predicted, is very useful for estimating the adsorption capacities of the produced activated carbons. The activation process is a typical gas-solid reaction. Many models have been presented for gas-solid reactions including the shrinking-core model (Yagi and Kunii, 1955), the grain model (Szekely et al., 1970, 1971; Sohn and Szekely, 19721, and the crackling-core model (Park and Levenspiel, 1975). These models are, however, mainly concerned with the description of the transient change of the solid conversion and are incapable of describing the change in the pore structure. The first model taking the change in pore structure into consideration was presented by Petersen (1957). This model is simple enough for practical use, but it cannot be applied to the activation reaction because it was developed under the assumption of uniform pores. Several models based on the population balance method have been presented to describe the change in the structure of porous media (Hashimoto and Silveston, 1973; Miura et al., 1975; Bhatia and Perlmutter, 1979). These population balance models, however, contain many parameters which must be determined experimentally and are tedious for practical use because of its mathematical complexity. Recently, Bhatia and Perlmutter (1980) have improved their first model with the approximate treatment for the initiation of new pores developed by Avrami (1939) in order to obtain a simple relationship between pore surface area and solid conversion. The improved model was successfully applied to the correlation of two of the three data seta presented by the authors (Hashimoto et al., 1979), but this model does not reflect the structure of the char. Wolff (1959) assumed that the char is formed from small cubes which are stacks of graphitic crystalline planes (crystallite). Upon the activation of the char in such a structure, the activating agent randomly attacks the planes and removes the large segments of the individual planes to form micropores of only a few angstroms width. He adopted the probability concept to analyze the random 0196-4305/84/1123-0138$01.50/0
removal of the individual planes and developed an analytical relationship between the pore surface area and the density of the activated carbon. This model involves only one parameter, namely, the number of planes in a crystallite. To correlate experimental data by this model, however, the number of planes in a single crystallite must be more than forty, a number which is much greater than those presented by many investigators. As for the structure of the char, much research (Austin and Hedden, 1954) has been performed including the pioneer work of Franklin (1951). Based on X-ray diffraction measurements, Franklin classified the carbonized materials into two groups: graphitizing carbon and nongraphitizing carbon. The schematic representation of the structure of each carbon is shown in Figure 1. In the figure straight lines represent graphitic layers, the stacks of which are called crystallite, and the curved lines a t the periphery of the individual crystallites indicate the nonorganized carbon. Carbons prepared by pyrolysis of organic substances at lo00 OC are said to consist of groups of from two to four parallel graphitic layers less than 20 A in diameter and comprised of 15 to 45% of nonorganized carbon. The crystallite and the nonorganized carbon react with the active agents at different rates. The nonorganized carbon includes hydrogen, hydrocarbon radicals, or other functional groups, and is more reactive than the crystallite carbon. However, the models which take into account the difference in the reactivities between the crystallite and the nonorganized carbon have not been presented. In this paper a new model is presented for predicting the changes in pore surface area, pore volume, and solid conversion during the activation. When deriving the model, it was assumed that the more reactive nonorganized carbon is gasified rapidly forming a reaction interface within the grain, and then the less reactive crystallite comes in contact with the oxidizing gas and is gasified by the random removal of the crystallite layers (Wolff, 1959) from the outer surface to the interior of the grain. The validity of the proposed model was examined by comparing the pore properties calculated by the model with the authors' experimental data (Hashimoto et al., 1979, 1981). Development of the Model Basic Idea of the Model. Chars used as raw material for activated carbons are often pelletized with some binder. In such a case a single particle of the char is composed of grains having a diameter of 1-2 pm, as shown in Figure 1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 Crystallite,
139
Partially gasified crystallite
/Nonorganlzed carbon
a
(a) Real situation
(a) Graphitizing carbon
(b) Non-graphitizing carbon
Figure 1. Schematic representation of the crystalline structure of the carbon (Franklin, 1951).
pz Assumed situation
2 layers were
b
gas if ied (XJ
= 0.4)
-----
Crystallite
----(no pores) (a)
i I
I
! 0
RO
0 1~~ r,
nL= s
----I.=: -----
9 '
( 2 pores)
Figure 2. Schematic representation of the model.
(b)
2. The interspace of the grains forms macropores of activated carbons (Walker and Rusinko, 1955). Each grain consists of stacks of 3 to 5 graphitic layers (crystallite) and nonorganized carbon. Since the nonorganized carbon is much more reactive than the crystallite, it is gasified, forming a reaction interface within a grain. Then the crystallite located outside the reaction interface can get in contact with the oxidizing gas and is gasified gradually from the outer surface to the interior of the grain. Therefore the concentration distribution of the oxidizing gas is represented as shown in Figure 2 if the chemical reaction is controlling and the original grain is impervious to oxidizing gas. Thus it is reasonable to apply the shrinking-core model, in which the surface reaction is controlling, to the gasification of the nonorganized carbon. Concerning the gasification of the crystallite, three views have been presented in the literature. Wolff (1959) assumed the random removal of the individual planes as stated earlier. Arne11 and Barss (1948), Dubinin et al. (1964)) and Kalback et al. (1970) considered that not individual planes in the crystallite but the whole crystallite might be gasified a t the same time. Walker et al. (1954) hypothesized that the crystallites are not uniform in size and that the smaller crystallites are gasified faster than the larger ones. Since no definite conclusion as to which view is correct can be drawn presently, Wolff's assumption was used here to facilitate the calculation of the pore surface area. The following assumptions were made to develop the model. (1) The resistances of both interphase and intraparticle diffusions are negligibly small. (2) The gasification reactions of the nonorganized carbon and the crystallite are f i t order with respect to the concentration of oxidizing gas and zero order with respect to the solid concentration. (3) The grains comprising the particle are of spherical shape and uniform in size. (4) The crystallites are uniform in size and can be represented by cubes of size d.
Figure 3. Formation of micropores in the crystallite.
The Relation between Solid Conversion and Reaction Time. 1. Gasification of Nonorganized Carbon. Since all grains undergo uniform changes in the case of chemical reaction controlling, one may pay attention to the change of only a single grain. When the reaction interface of the nonorganized carbon arrives a t the radial position rc within the grain as shown in Figure 2, the mass balance equation of the carbon at this stage is written as
Integrating eq 1with the initial condition rc = rg at t = 0 yields kacAdMc
rc = rpg- Pa
Equation 2 is rewritten in dimensionless forms as Ec=1-u ( u I 1 )
E, = 0
(34
( u > 1)
(3b) where E, = rJrg and u is a dimensionless time defined by (4)
The parameter, t*, is the time at which the reaction of the nonorganized carbon is completed. It is obtained by letting r, 0 in eq 2. 2. Gasification of the Crystallite. Each crystallite is supposed to react as shown in situation (a) of Figure 3a. However, it is known that the gasification of the crystallite proceeds from the side faces of the crystallite, since the periphery of the planes, which is composed of some functional groups, is very reactive. Therefore it is rea-
-
140
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
sonable to expect that once the reaction begins at the periphery of a certain plane the plane is gasified rapidly and completely to form a micropore. Then situation (a) in Figure 3a is replaced by situation (b). Based on this consideration, the conversion xI = 0.4 for a single crystallite composed of 5 layers, for example, corresponds to the situation in which two layers are gasified. Therefore it is reasonable tQ represent the reaction interface by the surface area of the side faces of the cube, 4d2 (1- xl). when a discrete number of layers are gasified, for example, when x1is equal to 0, 0.2, 0.4, 0.6,0.8, or 1.0 for the crystallite composed of 5 layers. By using the approximation that the reaction interface is represented by 4d2 (1- xl) for all x1values (0 < x1< l), the gasification rate of the crystallite a t the radial position r within the grain can be represented as
where p1 is the density of the crystallite and kl is the surface reaction rate constant for the Crystallite. Rearranging eq 5 gives
This equation is valid after the reaction interface of the nonorganized carbon passes the radial position r and the crystallite comes in contact with the oxidizing gas. The time at which the reaction interface passes the radial position r is obtained from the following equation, because the shrinking rate of the reaction interface, -dr,/dt, is constant in this case tinit = (rg- r)/(-drc/dt)
weight of mineral material free char (mineral-materialfree-basis, abbreviated to m.m.f.b.) can be represented by = +
v, v,
(13) where Vm is the initial value of V,, to is the porosity of char, and ngrepresents the number of grains contained in a unit weight of char (m.m.f.b.) and is calculated as
Substituting eq 9 and 14 into eq 13 and rearranging the equation gives the following analytical relationship between V, and u
K2
- e-Ku(l
+
+ -$))],/(pa(l
- 6)
+ p16) (15)
Calculation of Pore Surface Area Sp The pore surface area based on a unit weight of char (m.m.f.b.1 was represented by S,, and its initial value was specified by So As the reaction interface of the nonorganized carbon shrinks, the pore surface area involved in the unit weight of char decreases from Sm to Sfo12.On the other hand,. micropores are created in the crystallite located in the radial position between rg and rc in the grain. Then the pore surface area Sf at a certain instant is given by
(7)
Calculating drc/dt using eq 2 and rearranging the resulting equation gives tinit = (1 - E)t*
(8)
where 4 = r/rr Integrating eq 6 with the initial condition of xl = 0 a t tinitresults in x1 = 1 - exp(-K(u - 1 + 4)) (9) where K is a dimensionless parameter defined by
K =~(~I/~J(P (rg/d) ~/PJ
(10)
3. Equation for Calculating the Solid Conversion x. The conversion of a single grain, x , is represented by
- &)pa+ 6 p l ) + 1 r g 4 r r 2 ( 1rc
where 6 is the volume fraction of the crystallite in the char excluding void space and mineral material. Now the case of chemical reaction controlling is considered, where x is equal to the overall conversion of the pellet. Substituting eq 9 into eq 11and integrating, the resulting equation gives the following relationship between x and u
where s is the surface area of a single crystallite located at the radial position, r, and is calculated based on the assumption made by Wolff (1959). Imagine that a crystallite composed of five graphitic layers has been gasified by 40% (x, = 0.4); namely, two of the five layers are gasified. Assuming that the individual layers are gasified in random fashion; case (a) and case (b) in Figure 3b occur in the same probability. However, the number of pores (fissures) created in the crystallite is zero for case (a) but two for case (b). There are many other cases besides (a) and (b) as to which layers are removed. Considering all the probabilities, the average number of pores created in the nL layer of crystallite by random removal of nx layers was formulated as (Wolff, 1959)
The number of pores with width of only one layer, N,, was also obtained as N, =
nxbL - nx)(nL - nx - 1) ~ L ( I ~-L1)
(18)
By utilizing Nt and N 1 ,s was formulated as s = 2d2 + 4d2(1- X I )
Derivation of Equation for Calculating Pore Volume Vp The pore volume (void space). V ,based on a unit
+ (2Nt - NJd2
(19)
where the first term on the right-hand side is the surface area of the top and bottom faces of the cube, the second term is the surface area of the side faces, and the last term is the surface area of the micropores created within the cube.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
To carry out the integration in eq 16, the surface area of a single crystallite, s, must be represented by a continuous function of xl,because x1 is related to r through eq 9. Although both Nt and Nl are originally defined in terms of discrete values of nL and nx, as shown in eq 17 and 18, they are assumed to be applicable for any values of the conversion by replacing nx/nL for xl. Thus eq 17 to 19 can be represented by the following continuous functions of x1. Nt = -nLxt (nL- l)xl (20)
141
1
+
c-I
I
s = 2(3 - 2x1)
- l)(nL+ nLq - 2) + xl(nL- nLxlnL -1
Figure 4. Typical relationships between x and u.
(22)
When inserting eq 22 into eq 16 and integrating the equation, one has the relationship between Sf and u as follows
Equations 12,15, and 23 are the final results obtained by the proposed method. The parameters involved in the equations are divided into two groups. The parameters nL, d, pa, pl, and 6 represent the properties of the char, and K and t* are dimensionless parameters which contain the reaction rate constants. Some parameters are interrelated. The size of the crystallite d in the unit of angstrom is related to nL by d 3.4nL (24) The following equation describes the volume fraction of the crystallite 6 and the densities ~c
= (1 - 6)pa +
PI
(25)
where p c is the true density of the char (m.m.f.b.). Since the density of the crystallite p1 is almost equal to the density of the graphite (2.26 g/cm3), the density of the nonorganized carbon pa can be calculated from eq 25. The Procedure To Predict the Change in the Pore Surface Area. The prediction of the change in the pore surface area, Sf,during the activation is the main concern of the presented model. If the quantities of the char, nL, 6, Sm, p c , etc., are exactly known, only the parameters K and t* are necessary for the prediction. To determine K and t* only the x vs. t relationship has to be measured. The experimentally obtained x vs. t relationship is com-
pared with the relationships calculated by eq 12 through a curve-fitting method for determining K and t*. Then the relationships between Sfvs. t and between V ,vs. t can be predicted by using eq 23 and 15, respectively. Typical relationships between x and u (= t / t * ) are shown in Figure 4 for nL = 5 and 6 = 0.5. Since the effect of 6 on the relationships was found to be slight from the examination of the parameter sensitivity and the value of nL = 5 is expected to be reasonable, this figure can be used to estimate roughly K and t* by use of the measured x vs. t relationship. In many cases, however, the exact value of nL and 6 are not known. In such cases the Sfvs. x relationship must be measured in addition to the x vs. t relationship under certain experimental conditions. The experimental data are utilized to determine the parameters nL, 6, K, and t* through eq 12 and 15 with the aid of a simulation method. The determined parameters serve to predict the change in Sf under other experimental conditions.
Comparison of Model Simulation and Experimental Data Experimental Data. The authors (Hashimoto et al., 1979) activated three kinds of chars made from Miike coal (abbreviated to MC), Victoria coal (VC), and a coconut shell (CS) using a batchwise fluidized bed reactor to produce activated carbons, and they investigated the changes in pore structure during the activation in detail. In this work the pore surface areas of the chars and the produced activated carbons were measured by the nitrogen adsorption method at 77.5 K, and so the pore surface areas of the chars could not be determined experimentally. In general, the measurement of the pore surface area of the chars is difficult by the nitrogen adsorption method (Marsh, 1964). Recently, the authors (Hashimoto et al., 1981) also activated a char made from waste phenolformaldehyde resin (PF) to produce activated carbons. In this recent work the pore surface areas were measured by the COz adsorption method a t 195 K. Therefore the pore surface area of the char could be determined experimentally. In both works the pore characteristics were represented on the basis of unit weight of char including mineral material. These experimental results were converted to the values based on unit weight of mineral material free char (m.m.f.b.), because the analytical relationships were obtained on the mineral material free basis. In the authors’ laboratory Murphree (1977) measured the x vs. t relationships for VC and MC under the chemical reaction controlling regime. The authors (Hashimoto et al., 1981) also measured the x: vs. t relationship for PF. Here the data of Sfvs. x , V , vs. x, and x vs. t for VC, MC, and PF were used to examine the validity of the presented model. The pore surface area of the activated carbon made
142
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table I. Values of Quantities Employed for t h e Simulation ___ --_ _ ~
vi ,
Sfo,
,i
sample
cm’,g of s.m.
g/cm3
E”
nL
m’ig of s.m.
VC MC PF
0.311 0.293 0.248
1.56 1.83 1.82
0.212 0.242 0.310
5 5 5
391
p
C’
u x
r
-This
model
Bhatia s model Petersen s model
Table 11. Parameters Determined through t h e Simulationa ~~
K
vc
5.4 5.1 2.7
____
MC PF
6
Sfo, Pa, m’/g of s.m. g/cm3
Time.
_____0.31 0.39 0.48
579 118
-
*
+
+ = 4L0/SfO2Pc
1000
[min]
Comparison between the Calculated Curves and the Experimental Data. Relationships between x and t Figure 5 shows the comparisons for VC between x vs. t relationships obtained experimentally and those calculated by the models. The values of the parameters employed for the calculation are given in the legend for Petersen’s and Bhatia and Perlmutter’s models. For the three temperatures of 750,800, and 850 “C, good agreements were obtained between the experimental data and the curves calculated by the presented model (solid lines). The broken lines (Bhatia and Perlmutter’s model) and the chain lines (Petersen’s model) almost coincide for x values less than 0.7. For x larger than 0.7, the model of Bhatia and Perlmutter seems to fit the experimental data better. Compared with the presented model, however, these two models fit the experimental data poorly. Relationships between Sfand x. Figure 6a shows the comparisons for VC between the Sf vs. x relationships obtained experimentally and those calculated by the models. The curve calculated by the presented model (solid line) fits the experimental data well over the whole range of x. The curve calculated by Bhatia and Perlmutter’s model fits the experimental data fairly well. Bhatia and Perlmutter (1980) estimated the parameters tc. and Smas 6.9 and 457 m2/g, respectively, for correlating roughly the experimental data for VC. In this work the parameters were determined by the Simplex method as stated earlier. Therefore the parameters were a little different ($ = 5.6, Sm= 487 m2/g) from those determined by Bhatia and Perlmutter, and the fit between the experimental data and the calculated curve was improved by employing these values. The curves calculated by Wolffs (chain line) and Petersen’s model (dotted line) fit the experimental data poorly. Figure 6b shows similar comparisons for MC. The Sf vs. x relationship calculated by the presented model reproduces the experimental data well. The other models, however, fit the data poorly. It is noted that the nL value in the Wolff model were found to be extraordinarily large as 9.8 X lo5. Similar comparisons for P F are shown in Figure 6c. In this case the curves calculated by the presented model and Bhatia and Perlmutter’s model fit the experimental data very well. The model of Wolff also correlates the experimental data fairly well. This is because the sample P F was composed of pure components and was expected to be more highly graphitized than VC and MC which were made from coal. In fact, the 6 value for P F was larger than that for VC or MC as given in Table 11. As the 6 value increases, the assumption employed by Wolff (6 = 1) becomes valid. Petersen’s model, however, cannot fit the
.
from MC had a maximum a t about 50% of the solid conversion, whereas that from VC decreased, and that from P F increased, almost monotonously with the solid conversion. Therefore these three sets of data were considered to be suitable to test the flexibility of the model. Determination of Parameters. Parameters in the Presented Model. To calculate x, Sf,and Vf by the model reasonable values must be assigned to the parameters, but the exact values of nL and 6 are not known for the three chars. Referring to Franklin’s work (1951),it was supposed that nL = 3-5, 6 = 0.3-0.6 for these chars. Based on this work nL was fixed at 5 in the succeeding calculation to facilitate the determination of the other parameters. The three parameters 6, K , and Smfor VC and MC and the two parameters 6 and K for PF were determined by the so-called Simplex method so that the simulated curves might fit the experimental data sets of Sfvs. x and Vf vs. x. By using the determined 6 and K , x vs. u relationships were calculated and were compared with the experimental data sets of x vs. t to determine t*. The physical properties of the chars employed for the simulation are given in Table I, and the parameters determined are listed in Table 11. The values of 6 for VC, MC, and P F were 0.31,0.39, and 0.48, respectively. These values are within the range expected from Franklin’s work. Parameters in the Other Models. The models presented by Petersen (1957), Wolff (1959), and Bhatia and Perlmutter (1980) were also employed to calculate x, Sf, and Vf for comparison. The parameters involved in these models, if they are adjustable, were also determined by the Simplex method so that the simulated curves might fit the experimental data sets of Sfvs. x and V, vs. x. The determined parameters are as follows: Petersen’s model, Sm for VC and MC, no adjustable parameter for PF; Wolff s model, nL for VC and MC, no adjustable parameter for PF; Bhatia and Perlmutter’s model, Smand $ for VC and Mc, for PF. The parameter involved in the last model is defined by
+
t
Figure 5. Comparison between the calculated curves and the experimental data of x vs. t for sample VC. Parameters employed: Petersen’s model (tp* = 43.6 min at 800 “C, 23.2 min at 850 “C), Bhatia-Perlmutter’s model (I) = 5.6, tB* = 98 min at 800 “C, 51.5 min at 850 “ C ) .
1.54 1.56 1.40
These values were determined by use of the data This was calculated by inserting obtained a t 850 “ C . p c and 6 into eq 25. a
100
10
b
sample
(26)
where Lo is the total pore length per unit weight of char (m.m.f.b.). The relationships between x and t can also be calculated through Petersen’s and Bhatia and Perlmutter’s models. To calculate the relationships the rate parameters tp* and tB*(see Nomenclature section about their definitions) must be estimated. They were determined by use of the method similar to that employed for determining t*.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 143 I
I
I
l
l
l
l
l
l
l
l
7001 This model
i T
E
yi 0.5
:: < L"
Calculated by this modei
0
-I
/ 0.5 1 .o
OO
Conversion, x
1
loot
i,,ll0.5 ililll.o OO 1
Conversion, x
Table 111. Reaction Rate Constants Determined
C-I
600
vc
Woiff
TLi-
MC PF
E
i1
21
5
Bhatia and Perlmutter
Conversion, x I
I
I
I
I
C- 1 l
l
1
1000
-\
&... -..--.---.._.______ 'Petersen
0.5
[-I
Figure 7. Comparison between the calculated curve and the experimental data of pore volume for sample MC.
10
[-I Figure 6. Comparisons between the calculated curves and the experimental data of pore surface area. (a) Sample VC. Parameters employed Petersen's model (S, = 474 m2/g of a.m.); Wolff s model (nL = 27.7); Bhatia-Perlmutter's model (S, = 487 m2/of s.m.), $ = 5.6). (b) Sample MC. Parameters employed: Petersen's model (S, = 337 m2/g of s.m.); Wolffs model (nL = 9.8 X lo6); Bhatia-Perlmutter's model ( S , = 103 m2/g of s.m.), $ = 92). (c) Sample PF. Parameters employed Petersen's and Wolff s models (no adjustable parameter); Bhatia-Perlmutter's model ($ = 15.1). Conversion, x
experimental data of PF either. The above discussions indicate that the presented model is evidently better than the previous models for calculating the Sfvs. x relationship. Indeed, the number of the adjustable parameters involved in the presented model (two or three: 6 and K , or 6, K , and Sm) is larger than that in the other models, but this model is expected to reflect well the physical structure of the char. Each parameter also has a physical meaning, and, for example, the value of 6
7 50 800 850 850 850
404 146 78.0 81.0 135
0.44 1.3 2.5 2.5 1.3
1.5 4.3 8.3 7.8 2.4
determined by the simulation lay within the range expected from Franklin's work. Bhatia and Perlmutter's model also fitted the experimental data fairly well, though the model does not reflect the physical structure of the char. Therefore this model may be applied for calculating the Sf vs. x relationship. Wolff s model may be valid when the char is highly graphitized, but cannot be applied to the activation of the chars made from coal. Petersen's model cannot be utilized for calculating the Sfvs. x relationship associated with the activation of the char, because the assumption of uniform pores is not acceptable in the activation of the char. Relationships between V , and x. Figure 7 shows the comparison between the Vf vs. x relationship calculated by the presented model and that obtained experimentally. The curves calculated by the other models are not shown because they almost coincide with the curve calculated by the presented model. The slight deviation of the calculated curve from the data points a t lower x values is due to the closed pores involved in the char of MC (Hashimoto et al., 1979). Good agreement was also obtained for the other samples between the curves calculated by the models and the experimental data. Therefore either model can be applied for calculating the change in V,. The Validity of the Assumption of Chemical Reaction controlling. The values of the reaction rate constants, kaj r g and kl, were estimated by using the predetermined parameters, K and t*, and other quantities listed in Tables I and 11. Table I11 gives the determined values of ka/rgand kl for VC at three temperatures (750, 800, and 850 "C) and the values for MC and PF at 850 OC. At 850 OC the k,jr and kl values for VC are almost equal to those for MC. $he values for PF are smaller than those for VC or MC. This may be because the size of grain and the functional groups involved in the chars are different among the samples. The value of rg is around 0.5 Km = 5 X cm. Therefore k, is greater than kl by the order of 2 to 3 for each sample, indicating that the nonorganized carbon is much more reactive than the crystallite. This finding supports the basic idea employed for developing the model. The Arrhenius plots of k, j r g and kl are shown in Figure 8. The activation energies were obtained as 43.4kcal/mol from the figure for both rate constants. This value seems
144
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table IV. Effect of nL o n the Other Parameters sample
6
K
m’/g of s.m.
3 5 7 3 5 7 3 5 7
0.22 0.31 0.38 0.28 0.39 0.46 0.34 0.48 0.59
5.8 5.4 5.2 5.6 5.1 4.7 2.9 2.7 2.6
559 579 599 117 118 120
vc MC \
I
9.0
9.5
fi x lo4
a
i
PF
to be greater than that obtained under the diffusion controlling regime, indicating that the assumption of chemical reaction controlling is supposedly valid. To examine quantitatively whether the assumption is valid, the value of the modulus, which is a kind of Thiele modulus presented by Szekely et al. (1976),was calculated by use of the data obtained at 850 “C. The modulus is defined by
where Ro is the particle radius, ea is the porosity of the macropore portion, and De is the effective diffusivity of oxidative gas (steam in this case). When Ho is less than 0.3, the reaction of the nonorganized carbon proceeds under chemical reaction controlling. Then the reaction of the crystallite is also in the regime of chemical reaction controlling, because the reaction rate of the crystallite is much slower than that of the nonorganized carbon. The following values were employed for the calculation X
cm; ea = 0.177
-
Conclusion The carbonaceous materials from which activated carbons are produced are considered to be composed of the nonorganized carbon and graphitic layers of carbon (crystallite). The model was developed for calculating the change in the pore structure of such carbonaceous materials during the activation. In the development of the model the reactions of the nonorganized carbon and the crystallite were treated separately. The micropores were expected to be created within the crystallite by the random removal of the indiTidual planes. By counting the number of pores created within the crystallite based on the probability concept, the change in the pore surface area could be calculated. Then the analytical relationships were obtained for representing the transient changes in the solid conversion, the pore surface area, and the pore volume. The validity of the model was elucidated from the fact that the curves calculated by the model fitted the char gasification data of the authors well. Acknowledgment The authors are grateful to the Ministry of Education, Science and Culture, Japan for the award of Grant-in-Aid for Scientific Research, Special Project Research.
k J r g = 2.52 s-l; De = 0.05-0.09 cm2 s-l where De was estimated by the random pore model (Wakao and Smith, 1962). Inserting the above values into eq 27 gives
Ho = 0.10-0.13
-
from the real structure of the crystallite. Therefore the selection of nL = 5 may be acceptable. A fixed value of nL greatly facilitates the application of the presented model.
10.0
CK-’]
Figure 8. Arrhenius plots of k , and k 8 / r gfor sample VC.
Ro = 5
Sf 0 ,
nL
(28)
This value is less than 0.3. Therefore the assumption of chemical reaction controlling was found to be valid. Effect of n L Value on the Behavior of the Presented Model. In the previous sections, nL was fixed a t 5 by referring to Franklin’s work. Here it was examined whether the selection of nL affects the value of the other parameters and the fit of the presented model. Table IV summarizes the parameters determined by the simulation method for each of nL = 3, 5, and 7 . It was found that the parameter K and the pore surface area of the char Smwere not affected greatly by the selection of nL for each sample and that the value of 6 tended to increase slightly with the increase of n> For each combination of the parameters in Table IV the fit of the simulated curve and the experimental data was found to be good. These facts indicate that some other reliable information such as the data of X-ray diffraction analysis is required to obtain the correct nL and 6 values. The difference in the 6 value, however, is not so large as far as nL lies between 3 and 7, and the value of nL = 5 is not apart
Nomenclature CAO= concentration of reactant gas, mol/cm3 De = effective diffusivity within particle, cm2/s d = size of crystallite assumed by cube, A or cm Ho = particle Thiele modulus, dimensionless K = dimensionless parameter (= 4 (kl/k,) ( p a l p l ) ( r g / d ) ) k , = reaction rate constant of nonorganized carbon, cm/s k , = reaction rate constant of crystallite, cm/s k , = reaction rate constant of reactant solid which appears in tB* and tP*, cm/s Lo = total pore length per unit weight of char (mineral-material-free-basis, m.m.f.b.) MB = molecular weight of reactant solid which appears in tB* and tp*, g/mol M , = molecular weight of carbon, g/mol Nt = total number of pores created in a crystallite N , = number of pores of one layer width in a crystallite ng = number of grains per unit weight of char (m.m.f.b.) nL = number of graphitic layers in a crystallite nx = number of graphitic layers removed during gasification Ro = radius of particle, cm r = radial position within grain, cm rc = radial position of reaction interface of nonorganized carbon, cm rg = radius of g h n , cm r, = mean radius of pore of the starting material which appears in tp*,A or cm
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 145-150
145
Registry No. Carbon, 7440-44-0.
Sf = surface area based on unit weight of starting material (m.m.f.b.), m2/g of s.m. s = surface area of a crystallite, cm2 or m2 t = time, s or min t* = time of completion of reaction of nonorganized carbon, s or min tB* = l/(k,CA@BS,), s or min tp* = rOpc/(ksCA&B), s or Fin tfit = time at which the reaction of crystallite begins, s or min u = dimensionless time (= k,CA&t/r,p,) V , = pore volume based on unit weight of starting material (m.m.f.b.), cm3/g of s.m. x = overall conversion (m.m.f.b.), dimensionless x , = conversion of a crystallite at r within grain, dimensionless
Literature Cited Arnell, J. C.; Barss, W. M. Can. J . Res., Sect. A 1048, 26, 236. Austin, A. E.; Hedden, W. A. Ind. Eng. Chem. 1054, 46, 1520. Avraml, M. J. Chem. Phys. 1030, 7 , 1103. Bhatia. S . K.; Perlmutter, R. D. AIChE J . 1070, 25, 295. Bhatia, S.K.; Perlmutter, R. D. AIChE J . 1080, 26, 379. Dubinln, M. M.; Piavnik, G. M.; Zaverlna, E. D. Carbon 1964, 2 , 261. Franklin, R. E. Proc. R . SOC.London, Ser. A 1051, 209, 196. Hashimoto, K.; Sliveston, P. L. AIChE J . 1973, 79, 259, 265. Hashimoto, K.; Miura, K.; Yoshikawa, F.; Imal, I.Ind. Eng. Chem. Process Des. Dev. 1070, 78, 72. Hashimoto, K.; Mlura, K.; Miyoshi, Y.; Masuda, T.; Yamanishi, T. J . Chem. SOC.Jpn., Chem. Ind. Chem. 1081, 1815. Kalback, W. M.; Brown, L. F.; West, R. E. Carbon 1970, 8, 117. Marsh, H.; WynneJones, W. F. K. Carbon 1064, 1 , 269. Mlura, S.;Silveston, P. L.; Hashimoto, K. Carbon 1075, 73,391. Murphree, B. E., M.S. Thesis, The University of Tennessee, Knoxville, TN, 1977. Park, J. Y.; Levenspiei, 0. Chem. Eng. Sci. 1075, 30, 1207. Petersen, E. E. AIChE J . 1057, 3 , 443. Sohn, H. Y.; Szekely, J. Chem. Eng. Sci. 1072, 2 7 , 763. Szekely, J.; Evans, J. W. Chem. Eng. Sci. 1070, 25, 1087. Szekely, J.; Evans, J. W. Chem. Eng. Sci. 1071, 26, 1901. Szekely, J.; Evans, J. W.; Sohn, H. Y. "Gas Solid Reactions", Academic Press: New York, 1976; p 135. Wakao, N.; Smith, J. M. Chem. Eng. Sci. 1082, 17, 825. Walker, P. L.; Mckinstry, H. A.; Pustinger, J. V. Ind. Eng. Chem. 1054, 46, 1851. Walker, P. L.; Rusinko, F. J . Phys. Chem. 1055, 5 9 , 245. Wolff, W. F. J . Phys. Chem. 1050, 63, 653. Yagi, S . ; Kunli, D. "Proceedings, 5th International Symposium on Combustion"; Reinhold: New York, 1955; p 231.
Greek Letters 6 = volume fraction of crystallite within carbon phase, dimensionless ea = macroporosity, dimensionless to = microporosity, dimensionless [ = dimensionless radial position within grain (= r/rJ = dimensionless radial position of reaction interface of nonorganized carbon pa = density of nonorganized carbon, g/cm3 pc = density of carbon phase (= p,(l - 6) + p16), g/cm3 pI = density of crystallite, g/cm3 $ = dimensionless parameter defined by eq 26 Subscripts and Abbreviation 0 = value of the starting material m.m.f,b. = mineral-material-free-basis s.m. = starting material
Received for review March 31, 1981 Revised manuscript received May 25, 1982 Accepted May 24, 1983
Mass Transfer from Fine Particles in a Stirred Vessel. Effect of Specific Surface Area of Particles HMeharu Yagi, Toshifumi Motouchl, and Haruo Hlklta Department of Chemical Engineering, University of Osaka Prefecture, 4-804 Mom-Umemachi, Sakai City, Osaka, 59 1, Japan
A new method was devised to measure the mass transfer coefficient for fine particles in high concentration. In the concentrated suspension, the mass transfer coefficient and the effect on it of agitation decreased with increasing specific surface area of the particles, although variations in the physical properties of the suspension were negligible. These tendencies could be explained by a theoretical model in which the mass transfer from part of the particle surface was impeded by solute which had diffused from adjacent particles.
Introduction The mass transfer between particles and liquid in a stirred vessel is an important process in the chemical industry. Numerous experimental data on the mass transfer coefficient have been reported, together with equations to correlate them (e.g., Barker and Treybal, 1960; Calderbank and Moo-Young, 1961; Harriott, 1962; Brian et al., 1969; Nienow, 1969; Levins and Glastonbury, 1972; Nagata and Nishikawa, 1972; Kuboi et al., 1974; Sano et al., 1974; Boon-Long et al., 1978; Herndl and Mersmann, 1981; Conti and Sicardi, 1982). Theoretical bases of relations depend on various viewpoints of the relative motion of particle to liquid: Kolmogoroff s theory of local isotropic turbulence, the slip-velocity model, and the penetration model. Most works, except those of Harriott (1962) and Nagata and Nishikawa (1972), have focused on the mass transfer for 0196-430518411123-0145$01.50/0
large particles of above several hundred micrometers. Today, fine particles of micrometer order are in common use in industry. Catalyst particles in the slurry reactor, lime or limestone particles in the gas scrubber, and microorganisms in the fermenter are of micrometer order in size. In crystallization, information on mass transfer to the crystal of micrometer or smaller size is important not only to assess the performance of the crystallizer but also to examine the kinetics of nucleation. The mass transfer coefficient measured by Harriott (1962) for particles below 100 pm increased with decreasing particle size as predicted by the correlation Sh = 2, which can theoretically derived for mass transfer from a sphere to infinite stagnant fluid. But the mass transfer coefficients measured by Nagata and Nishikawa (1972) for particles below 10 pm gave almost constant values independently of size. Thus, in spite of 0
1983
American Chemical Society
