- Email: [email protected]
Modelling of the simultaneous comminution and chemical reaction in a non-catalytic gas-solid batch reactor
Pergamon Plh
Chemical Engineeriny Science, Vol. 52, No. 16, pp. 2715 2728, 1997 ,~? 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain ~50@119-2:5111~97~11@@54-7 0009-2509/97 $17.00 + 0.00
Modelling of the simultaneous comminution and chemical reaction in a non-catalytic gas-solid batch reactor Stefan Bade and Ulrich Hoffmann* Institut fiir Chemische Verfahrenstechnik, TU Clausthal, Leibnizstr. 17, 38678 Clausthal-Zellerfeld, Germany (Received 29 November 1994; accepted in revised form 16 December 1996) Abstract--The shrinking-core model for reaction control was applied to particle size distributions, particularly to the logarithmic normal distribution. Furthermore, a simultaneous comminution of the solid was considered in the model; in the first case the process of the batchwise comminution was described with a model including two kinetic parameters, in the second case a more complex material balance model was introduced and this operates with rates of breakage and distribution parameters. A comparison is given between model predictions and experimental results for the reaction of gaseous hydrogenchloride with metallurgical grade (MG) silicon to give trichlorosilane. The experimental results were obtained in a newly developed apparatus, the so-called reaction-mill, which can simultaneously act as a vibration mill and as a chemical reactor. © 1997 Elsevier Science Ltd
1. I N T R O D U C T I O N
Among the several available models for non-catalytic gas-solid reactions, the shrinking-core model is the most widely used. But the modelling of non-catalytic gas-solid reactions is often restricted to a definite particle size. However, in practice, always particle size distributions of the solid reactants are used. Until now heterogeneous non-catalytic gas-solid reactions were not discussed under the point of view of a simultaneous comminution process during the chemical reaction, because there exists a lack of experimental apparatuses to run both the processes simultaneously. Although many experimental results of comminution and its influence on succeeding chemical reactions were obtained (Heinicke, 1984), the development of the so-called reaction mill is just in its beginning (Senna and Okamoto, 1989; Bade, 1994). In this connection, a vibration mill is the preferred apparatus as a basis for a reaction mill, because of its effective generation of fine particles and because of its possible mechanical activation of the solid. So combining the two process steps opens new and most promising possibilities to run a process more effectively, because effects of the comminution process, that exceed the change in the dispersity of the solid, can be used for the chemical reaction. But with the development of reaction mills a need for models which are able to
*Corresponding author.
describe the two simultaneously running processes arises. Until now no models exist that can be used in such cases. The present paper deals with the development of such a model. The basis for the combined kinetics of comminution and chemical reaction is the one-parametric shrinking-core model with its kinetic equations for the technically interesting case of reaction control. The batchwise comminution kinetic is developed with both a two-parametric model and a material balance model, that describes both the rates of breakage and the distribution of the fragments for every particle size fraction. 2. MODELLING Let us first consider a non-catalytic gas-solid reaction of a non-porous solid with a gas, where the chemical reaction exclusively proceeds at the external surface of the solid: viAl(g) + colSt(s) ~ VEA2(g) + co2S2(s).
(1)
Let the following assumptions be valid: (1) In the beginning and during the reaction the solid particles have a spherical geometry and are not porous, (2) the particles only shrink during the reaction and no ashlayer is formed, (3) the irreversible chemical reaction proceeds in an isothermal way and is first order with respect to the gaseous reactant, (4) the chemical reaction determines the kinetic of the whole process, (5) the concentration of the gaseous reactant is constant
2715
2716
S. Bade and U. Hoffmann
during the reaction and (6) the chemical reaction exclusively proceeds at a sharp interface at position rc. The the shrinking-core model for the gaseous and solid educts A~ and S~ (here and in the following simplified and named as A and S) can be written as:
dNs dt
(D14rcrZCA. Vl
(2)
The particle size distributions [eqs (5) and (6)] are characterized by two parameters: the standard deviation a and the median/~. In Fig. 1 the normal and the logarithmic normal distribution are illustrated for a dimensionless (R/p) = 0.6 and a = 0.2. The cumulative particle size distribution Q3 is obtained by integration of q3 and it is standardized with eq. (8):
The time dependence of the radius of the unreacted core r~ is given by
Q3 =
q3(R)dR =
1
(8)
0
dr~ ~t kCA dt vl Ps
(3)
And after integration the conversion X~ of the solid with the definite radius R~ is obtained as
(rc3
X,=I-\~//
=1-
(
1
--v, p-~o/ .
(4)
This equation is only valid with a definite radius R~ of the particle. But if there exists a particle size distribution, for example, the normal [eq. (5)] or logarithmic normal distribution [eq. (6)] the overall conversion of the solid X [eq. (7)] is obtained by integrating the product of the conversion function X~ and the most commonly used mass related particle size distribution q3 over the whole interval of particle sizes 1
q3(R)
exp[-1-(R-P~z]
(5)
1 1 [ I(In(R/P)~2~ q3(R)=a~-~Rexp-2\ a ] ] X=
(6)
X~(t, R)q3(R)dR.
(7)
0
In this work the conversions were calculated only with the logarithmic normal distribution, because the normal distribution is more suitable for very narrow distributions while the logarithmic normal distribution is more suitable for practically existing distributions, which have a broader particle size distribution. Combining eqs (4) and (6) with respect to eq. (7) results in the overall conversion X if a logarithmic normal distribution is considered: x-
~-2 ~ I ~ [ O"
1-
(i
~°' kCAt'3] jR --=,,:exp
0
~1
x [- ~(~)2]dR.
(9)
The integration of eq. (9) can be carried out using the integration method of Gaul3-Legendre including a calculation of the weight function; generally the numerical integration was carried out with 200 points. But carrying out the integration numerically, attention must be paid to the fact that the conversion function Xi [eq. (4)] can mathematically result in values greater than unity, which is physically meaningless. So if a particle with radius R~ has reached the
1.5 1.2
/ /
i ~/logarilhminormal c distribution \
/
%:0
.1
02
o.a
o.;"oi
~'~rmaldisltibution
"o.i"oli'"oi8
(R/p)/-
o.g
1.o
11
1;2"i.3
Fig. 1. Dimensionless normal and logarithmic normal distribution for (R/#) = 0.6 and a = 0.2.
Simultaneous comminution and chemical reaction
2717
o=0 0.9 0.8
0.7
0.5 X 0.5 0.¢ 0.3 0.2 0.i
0
l , , I t l t t [ , l i , , i i ' i [ t ' i t l l l I I l i l t l l l s ' i
I000
2000
3000
'lii'iillillllllll
4000
I
5000
BOO0
tls Fig. 2. Influenceof the logarithmic normal distribution on the overall conversion X of the solid [eq. (9)] in isothermal case; # = 465 x 10 - 6 m; (ol/v 1 = 1/3; k = 6 x 10-* m s- t; CA = 30 mol m- 3; Ps = 91,039 molm -3.
conversion unity, in the further computation the conversion stays constant at unity. In Fig. 2 the influence of the particle size distributions on the overall conversions is illustrated. It can be recognized that with a constant /~ the influence of the standard deviation on the conversion is relatively small; but with values of a greater than 0.1 the influence is distinctly observable: In the starting phase, the amount of finer particles accelerates the conversion rate, while at a high degree of conversion the larger particles effect a reduction of the conversion rate. A batchwise comminution process now effects a reduction of the median/~ and a change in the standard deviation or. To describe the kinetics of comminution the time-dependent functions of a and/~ have to be determined. Approaches to model the time dependences of # and a caused by comminution are given in eqs (10) and (11) #(t) = #0 e x p ( - k , t ) + / ~ ( 1 - exp(-kut) )
(10)
For relatively narrow particle size distributions the median ~ can be approximated with the mean particle size ~ [eq. (12)]:
.~= ~ xiAQ3,i = ~ xiq3,1mxi~ ~. i=1
In Fig. 4 the resulting time-dependent particle size distributions are given for various times. Combining eqs (9)-(11) gives the overall conversion function of the solid, when the particle size is not changed by the chemical reaction. But if we now discuss about the simultaneous comminution and chemical reaction according to the shrinking-core model without ashlayer formation, it is necessary to modify eq. (10), because the size of the particles is not only reduced by comminution but also by the shrinking process. Therefore, eq. (3) has to be integrated [eq. (13)]: rc = R
e~l kCat VI
or(t) = a0 e x p ( - k~t) + 6~(1 - e x p ( - k~t))
(11)
Fig. 3 gives an example of the comminution of metallurgical grade silicon in a vibration mill. The median p is shown in dependence of the duration of the comminution process. It can be seen that an exponential expression describes the time dependence of the median # quite well; analogously a similar exponential expression is chosen for the standard deviation, although it fits the experimental values quite well only for longer durations of the comminution process.
(12)
i=1
-/~(t)
(13)
PS
and to be combined with eq. (10); it results in eq. (14), which describes the time dependence of/~ according to simultaneous comminution and chemical reaction: /~(t) = (/~oexp(-- kut) +//oo(1 - e x p ( - kut)) )
to1 kCat h
Ps
(14)
By using eq. (14) attention must be paid to the fact that/~ of eq. (10) has to be in the radius form, so that the eqs (10) and (13) are compatible. Furthermore, the
2718
S. Bade and U. Hoffmann 1000
900' 800 700 E
O
600
"3 ~" 500
400 E
I
o
3O0 200 100 0
0
500
1000
1500
2000
2500
3000
3500
4000
time/s Fig. 3. Comminution of metallurgical grade silicon in a vibration mill; exponential decrease of the median /~ with the duration of comminution; ku = 1.12 x 10-3 s-1; ]-/O = 930/~m; #o0 = 5 pm.
0.030I
'~-~II EI0'020 t=3OOOs
o.olo
H ~ ~t/t=2000 s
0"0000
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
x/pm
Fig. 4. Logarithmic normal distributions for various times during a comminution process; ~0 = 930 #m; #~ = 5 Mm; tr0 = 0.12; tr~ = 0.55; k~, = 1.12 × 10-3 s- 1; k, = 8.3 x 10 -4 s-1.
use of eq. (14) implies t h a t b o t h the c o m m i n u t i o n process a n d the chemical reaction exclusively generate spherical particles, If we n o w c o m b i n e eqs (9), (11) a n d (14) the overall conversion of the solid can be calculated assuming that the s t a n d a r d deviation of the particle size distribution a is not influenced by chemical reaction but only by c o m m i n u t i o n . In Fig. 5 the overall conversions were calculated for three cases: conversion with a c o n s t a n t particle size distribution
for t = 0 a n d t ~ oo a n d the conversion with simultaneous c o m m i n u t i o n a n d chemical reaction with time d e p e n d e n t # a n d tr related to eqs (11) a n d (14). In Fig. 5 it is noticed t h a t with simultaneous comm i n u t i o n a n d chemical reaction a S-shaped conversion function is obtained, while the conversion functions with t i m e - i n d e p e n d e n t particle size distributions are not S-shaped. So if the c o m m i n u t i o n process is very slow, the conversion function a p p r o a c h e s the
Simultaneous comminution and chemical reaction
2719
1 0.9 0.8
p=2.5wm;o=0.55
0.? 0.6
X 0.5 0.4 0.3 0.2 0.i 0
~j~l,,,I,,,,,,,~,la,,,,,,,,I,,,,,,,,,l~,t,,,,,,I,,~,,,,,,I,,,,,,,,,I,,,,,,,ll
0
500
1000
1500
2000
2500
3000
3500
4000
t/s Fig. 5. Isothermal conversions of the solid with constant particle size distributions for t = 0 (# = 465 #m, cr = 0, 12) and t --,@ (/a = 2.5 #m, tr = 0.55) compared with the conversion for simultaneous comminution and chemical reaction (k, = 1.12 x 10- 3 s- 1; k, = 8.3 × 10-4 s- 1); k = 2 × 10-4 m s- ~; CA = 30 mol m- 3; Ps = 91,039 molm-3;) col/v1 = 1/3.
case for t = 0, while for an extreme fast comminution process the conversion function approaches the case for t ~ o c . The described first model to simulate the process of comminution is a two parametric one, which requires the determination of the median # and the standard deviation tr as functions of time. Here all the time the particle size distribution is characterized by the selected function of q3, e.g. normal or logarithmic normal distribution. A more detailed description of the batch comminution process is possible with the population balance model (Gardner and Austin, 1962; Reid, 1965; Sch6nert, 1971). The whole particle size interval is divided into as many as possible narrow particle size fractions, n. The first fraction (i = 1) contains the coarsest particles, while the last fraction (i = n) contains the finest particles. The condition for the particles to belong to the class i is given in eq. (15): xi+ ] < x ~< xi
(15)
and always is x~ < xj
for
i > j.
(16)
The mass proportion of the class i is called m~ and the sum of all masses m~ is unity during the whole comminution process (material balance):
~m i
=
1
(17)
i=1
Now two sets of coefficients are introduced to model the comminution process:
(1) rates of breakage w i that describe the proportion leaving the class i by comminution per unit of time; dimension: s-1. (2) distribution parameters bij that describe the distribution of the fragments; blj corresponds to the proportion that reaches the class i when particles of class j are comminuted. The alteration with time of the mass proportion in every fraction results from the per unit of time entering fragments of coarser particles subtracting the mass proportion leaving this fraction; so a material balance model can be used for describing the comminution kinetics. By that for n classes a system of ordinary differential equations of first order results: dmi --=--wlmi dt
i- 1 + 2 bijwjmj j=x
(18)
with the initial condition m~(t = O) = mio. The comminution of the finest particles in the fraction with i = n is (by definition) not considered in this model, so that an accumulation in this fraction occurs. This system of n differential equations requires the determination of ( n - 1 ) rates of breakage wi and (n(n - 1)/2) distribution parameters bit. With the simplification of a constant rate of breakage independent of particle size (w = const.) and of distribution parameters bis that are constant during the process of comminution, the calculation can be carried out for an experimentally determined or estimated rate of breakage w and for (n(n - 1)/2) estimated distribution parameters b~j. In fact, the rates of breakage wl depend on the particle size and they decrease with decreasing
2720
S. Bade and U. Hoffmann
particle size. Furthermore, the distribution parameters b~j depend on the dispersity of the solid which alters during a batch comminution process. In the case of dividing the whole particle size interval into n = 10 classes, 45 distribution parameters have to be estimated with the condition of eq. (19):
bij
=
1 for j = 1. . . . . (n - 1).
(19)
i=j+l
The values for the rates of breakage wzare obtained by the loss of mass in class i during comminution. In the experimental procedure each narrow fraction has to be comminuted and from the resulting exponential loss of mass with time the rates of breakage w~ are obtainable. Then the rates of breakage wi can be used to calculate an arithmetic mean value of w. The distribution parameters b~j can be obtained by determining the distribution of the fragments after comminution of each narrow fraction; doing this, it is important to comminute only for short times because the developing fragments can further be broken up; because of the short comminution intervals the analysis of the fragments is quite difficult. The number of b~ which have to be determined can be reduced by introducing similar distributions of the fragments. Further details of experimental determination of w~and b~ are described elsewhere (Austin and Bhatia, 1971/1972; Mular, 1970; Klimpel and Austin, 1970). In the following example of modelling a comminution process, the rate of breakage w is assumed to be constant and independent of particle size; it is obtained by calculating the arithmetic mean value of w~. In Table 1 the estimated values of the distribution parameters bij are shown for the comminution of metallurgical grade silicon in a vibration mill. The particle size interval is divided into 10 classes and the distribution parameters are assumed to be constant during the comminution process. The values for b~j are obtained in comminution experiments of short duration (15 s) with each narrow fraction for j = 1. . . . . 7. The distribution parameters b9, s and blo,8 are assumed to be 0.5, blo,9 has to be 1. The experimentally determined parameters b~j include a rela-
tively large experimental error because of the short period of comminution and the resulting analytical difficulties; so the determined values bij represent only rough estimates. The time-dependent courses of the mass proportions mi are shown in Fig. 6. The calculation of the system of 10 differential equations [eq. (18)] was carried out using a Runge-Kutta routine of fourth order. In Fig. 6 it can be seen that the simulation of the batchwise comminution process results for the mass proportion of the first class (i = 1) in an exponential decrease, while the mass proportion of the class with the finest particles (i = n) shows a s-shaped increase; here an accumulation takes place because the breakage of this class is not considered in the population balance model (w = 0 by definition). The mass proportions of the eight classes numbered 1 < i < 10 run all through a maximum that shifts with increasing fineness of the particles to later times. Now the chemical kinetics according to the shrinking-core model without ashlayer formation, additionally, can be introduced into the system of differential equations [eq. (18)]. The material balance of one class is schematically illustrated in Fig. 7. The flow into the class i occurs not only by comminution of the other (i - 1) fractions but additionally by shrinking caused by chemical reaction of the particles in the class with number (i - 1). On the other hand, the particles leave the class i due to breakage and they reach the other (n - i) fractions according to the distribution parameters; simultaneously, the particles leave the class i due to a shrinking of the particles caused by chemical reaction and they enter the class (i + 1). The time constants z~ represent the time necessary to reach the particle radius of the (i + l) class according to a shriking of the particle. These time constants can be calculated by using the shrinking-core model in the case of reaction control: eq. (3) can be integrated between the limits of Ri and Ri+ 1 and the time constants r~ are written as [eq. (20)]: Vl Ps rl = - - - (Ri -- Ri+ 1). ~01 k C a
(20)
Table 1. Estimated values of the distribution parameters blj for division of the whole particle size interval into 10 classes b/j
j = 1
2
3
4
5
6
7
8
9
i= 2 3 4 5 6 7 8 9 10
0.03 0.05 0.I0 0.20 0.20 0.15 0.10 0.09 0.08
0.06 0.08 0.12 0.15 0.20 0.20 0.11 0.08
0.08 0.12 0.15 0.20 0.25 0.12 0.08
0.I0 0.12 0.15 0.20 0.30 0.13
0.10 0.20 0.30 0.25 0.15
0.20 0.30 0.35 0.15
0.30 0.50 0.20
0.50 0.50
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
b~j i=j+I
Simultaneous comminution and chemical reaction
2721
0.20 0.18 0.16 0.14 0.12 "~-0.i0 0.08
0.06 0.04 0.02 0.00
0
200
600
400
800
I000
1200
1400
1600
1800
2000
tls Fig. 6. Mathematical simulation of a batchwise comminution process under consideration of l0 particle size fractions; 45 estimated distribution parameters b~ and a constant rate of breakage w = 0.005 s- 1.
/
//
\
wimi N
,-1
,~b~j.jr.j
mi(t)
(
mi/x i
Class i
mi. 1 Iti. 1
Fig. 7. Schematic representation of the material balance of one particle size fraction with simultaneous comminution and chemical reaction according to the isothermal shrinking-core model.
In eq. (20) Ri is the mean particle radius of class i that can be calculated with eq. (21): R i = (X i q'-
Xi+ 1)/4.
wi +
mi + ~ bqwjmj + j=l TI-I
xA#m)
1
1000
2 3 4 5 6 7 8 9 10
800 630 400 250 160 100 63 25 10
Ri(pm) 450 357.5 257.5 162.5 102.5 65 40.75 22 8.75 2.5
r(s)
%(s)
8194 6509 4689 2959 1866 1183 742 401 159 45.5
1685 1820 1730 1093 683 441 341 242 113.5 45,5
(21)
Therefore, in the case of simultaneous comminution and chemical reaction the differential equation [-eq. (18)] can now be written as [eq. (22)]:
dt
Table 2. Calculation of the time constants r and z~ with the isothermal shrinking-core model in the case of chemical reaction control; k = 5 × 10- 4 m s- 1; Ps = 91039 tool m- 3; 091/vl = 1/3; Ca = 30 molm -3
(22)
In Table 2 an example is given to show (1) how to divide the particle size interval into 10 classes, (2) how to calculate the mean radius Ri and the time constants z for complete conversion and % for shrinking in the next class. In Fig. 8 a mathematical simulation of a batchwise comminution process with simultaneous chemical reaction according to the shrinking-core model (Table 2) is illustrated. The difference of Fig. 8 compared to Fig. 6 can be seen. While by using eq. (18) in the class with the finest particles (i = 10) an accumula-
tion occurs because the rate of breakage is zero, by using eq. (22) the accumulation in this class is avoided because now the particles shrink by chemical reaction. Therefore the mass proportion of this class (i = 10) runs through a maximum too. At this time, the flow into this class is equal to the consumption by chemical reaction. Only if the chemical reaction is infinitely slow an accumulation in this class occur. Furthermore, it is noticed that the mass proportions m~ reach the value zero faster and that the maxima show a reduced height compared to Fig. 6. All these effects are caused by the chemical reaction, Because the chemical reaction consumes the solid reactant to give gaseous reaction products, the condition given in eq. (17) is no longer valid. In the case of chemical reaction, eq. (17) now has to be formulated
2722
S. Bade and U. Hoffmann
0.10 O.Og
/
8
g
1
0.08
10
0.07
0.06 -0.05 0.04 0.03 0.02
0.01 0.00
500
1000
1500
2000
2500
3000
t/s Fig. 8. Mathematical simulation ofa batchwise comminution process with simultaneous chemical reaction according to the isothermal shrinking-core model; consideration of 10 particle size fractions (Table 2); 45 estimated distribution parameters b u (Table 1); w = 5 x 10-3 s-1; k = 5 x 10 -4 m s-1; CA = 30 mol m-3; ps = 91,039 molm-3; col~v1 = 1/3.
1.00 0.00 0.80 0.70
with slmiJItaneouscomml
O.BO "L 0.50 X
/
/
shrinking-core model
0.40 0.30 0.20 0.10 0.000
=
,
i
i
,
i
i
=
J
i
600
i
i
i
i
i
i
i
i
|
i
i
i
i
i
1200
i
i
,
i
I
,
1800
i
i
i
i
i
i
i
=
I
i
2400
i
i
=
i
i
J
i
i
3000
t/s Fig. 9. Comparison of the isothermal conversions of the solid reactant without comminution [eq. (4)] and with simultaneous comminution: 10 particle size fractions (Table 2); 45 estimated distribution parameters b u (Table 1); w = 5 x I0- 3 s- 1; k = 5 x 10 -4 m s- i; Ca = 30 mol m - 3; Ps = 91,039 mol m - 3; oJ1/vl = 1/3.
as [eq. (23)]: ~mi
~< 1.
(23)
/=1
The total conversion of the solid can be calculated using eq. (24): n
X = 1-
~, mi. i=1
(24)
In Fig. 9 the conversions of the solid are calculated b o t h with the simple shrinking-core model Feq. (4)] a n d with a simultaneous c o m m i n u t i o n process according to the system of differential equations [eq. (20)]. It is noticed that the conversion rate in the case of simultaneous c o m m i n u t i o n is higher t h a n in the case of pure chemical reaction, where the complete conversion of the solid is reached after T = 8194 s
2723
Simultaneous comminution and chemical reaction (Ri = 450 #m, see Table 2), while with simultaneous comminution the complete conversion of the solid is already reached after 2000 s. Furthermore, in the case of simultaneous comminution a s-shaped course of conversion results. The maximum conversion rate is obtained when the particle size fraction with the finest particles (i = n) runs through its maximum. The second model to simulate the process of comminution described requires the determination of a large amount of parameters dependent on the number of classes. In the ideal case the classes have to be very narrow and for every class the experimental determination of the rate of breakage and the distribution parameters is necessary. So this model needs a great effort and it is to be discussed in which cases this effort is worthwhile. Certainly, this model has to be used if there exists an unusual particle size distribution, which cannot be described with the normal or logarithmic normal distribution, e.g. the development of a bimodal distribution function during comminution can be described with the second model. Furthermore, this comminution model has to be used if one is interested in the exact particle size distribution, which can be extracted for all times by calculating the mass proportions in dependence on time. 3. EXPERIMENTALRESULTS An apparatus where the simultaneous comminution and its chemical reaction with a gas takes place is the so-called reaction mill. It is a vibration mill with variable forms of vibration (rotatory, vertical), that is heatable up to 450°C, pressure resistant up to 2 MPa
and has a gas inlet and outlet. The cylindrical reaction chamber is made of stainless steel and has a volume of 393 cm 3 (d = 100 mm, h = 50 mm). It is constructed to exceed very high accelerations of maximum 65 g with high relative amplitudes adjustable from 0.02 to 0.14, because the input of mechanical energy is considered to have a distinct influence on the course of the chemical reaction. This apparatus is described elsewhere (Bade, 1994; Bock, 1994). The heterogeneous non-catalytic reaction that was investigated in the reaction mill was the exothermic reaction of metallurgical grade (MG) silicon with gaseous hydrogenchloride to give trichlorosilane as the main product [eq. (25)]; a thermodynamic analysis showed that this reaction is essentially irreversible. Si(s) + 3 HCI(g) ~ SiHCls(g) + H2(g ), AH ° = -223.2 kJ mol- 1
(25)
The reason to prefer a vibration mill as a chemical reactor for this reaction is that the MG silicon contains about 10 wt% impurities, above all iron. The iron is hydrochlorinated to the solid by-product FeC12, which is regarded as reaction inhibiting. Indeed, it was shown (Bade, 1994; Schr6ter, 1992) that the chemical reaction carried out singularly without comminution does not reach a complete conversion of the solid reactant silicon in the lower temperature region up to 600 K. It is expected that the inhibiting effect of FeC12 is cancelled, when the process runs under the influence of mechanical energy in a vibration mill. Furthermore, the chemical reaction [eq. (25)] thermally runs only approximately above 573 K, while in the reaction mill it was shown that the
0.020 0.018
with comminution
without comminution
0.016 0.014 0.012 >~0.010 0.008 experiment
•
O.OOfi
•
model
0.004 0.002
0.000= 0
/ , ,e,, ~, . . . . . . . 500
il . . . . . . . . . 1000
I ......... 1500 t/s
i~,,L 2000
.....
I ......... 2500
I ......... 3000
3500
Fig. 10. Modelling of the solid conversion in the reaction mill at 543 K with the shrinking-core model [eq. (9)]; k, = 3.5 x 10-4 s-l; k~ = 3.0x 10-4 s-l; k = 3.8x 10 6ms-l; CHCl= 23 molm-3; Ps = 91,039 molm-3; ~1/vl = 1/3,
2724
S. Bade and U. Hoffmann
0.020 with commlnutlon
0,018
without comrninution
O.OlO
0.014 0,012
~
0.010
0,008 0.008
1
~
experiment
0.004 !
0.002 i
0.000,~
; ;
.........
500
!= .........
1000
, .........
1500
= .........
2000
, .........
2500
, .........
3000
i
3500
t/s Fig. l 1. Modelling of the solid conversion in the reaction mill at 553 K with the shrinking-core model [eq. (9)]; k,=3.5xl0-'*s-1; k =3.0xl0-'*s-1; k=5.4xl0-6ms-1; C.cl=22.6molm-3; Ps = 91,039 molm-3; COl/V1 = 1/3.
reaction can take place at temperatures down to 373 K (Bade, 1994). Results of investigations concerning the comminution process have shown that the brittle non-porous solid M G silicon (feed size: 800-1000/zm) can be efficiently reduced to a particle size of about 5/~m in the vibration mill and that the particle size distributions can be described quite well with the logarithmic normal distribution [eq. (6)]. So here the first model, described above, was used to model the simultaneous comminution and chemical reaction. In Figs 10 and 11 experimentally observed conversions of the solid in the reaction mill at 543 and at 553 K, respectively, are shown. The courses of conversions are divided into two parts: (1) in the first part the chemical reaction runs simultaneously with comminution, (2) in the second part only the chemical reaction proceeds. In the first part the modelling was carried out using eq. (9) combined with eqs (11) and (14), in the second part the calculation was carried out with the particle size distribution parameters at the time when the comminution process was interrupted. The parameters of k, and k, were previously determined. They are functions of the amplitude, the numbers of revolution, the acceleration and the fillings of grinding balls and charging material of the vibration mill. It is seen that the experimental conversions can be described very well with the model that considers the comminution process with two kinetic parameters ku and k,. In the temperature region 543-553 K the influence of temperature is relatively high, while the influence of mechanical energy is not so important, so
that the chemical reaction is mainly dominated by the surface enlargement during the comminution process. In this connection, the fresh surfaces produced are so active that the chemical reaction starts at temperatures below 573 K, while with pure thermal excitation it only starts above 573 K (Schr6ter, 1992; Bade, 1994). With decreasing reaction temperature, the model used here predicts increasing deviations from the experimental conversions, because the influence of mechanical energy in the reaction mill becomes more and more dominating.
4. DISCUSSION The described models for calculating the conversions of a solid reactant during the influence of mechanical energy have shown that they can represent the experimentally observed conversions, in particular in the higher temperature region, where the dispersity of the solid educt plays a dominating role. To validate the models, higher conversions of the solid have to be determined experimentally. The discussion of the limiting cases (very slow chemical reaction; very slow comminution) leads to the conclusion that a reaction mill is only interesting in those cases where the chemical reaction and the comminution have comparable rates. Otherwise the two processes can be separately carried out without getting into any disadvantage. The two models for the comminution process show some differences: the first model is a more empirical model and describes the macroscopic effect of comminution, while the second model is a more theoretical one and is based on the physical properties of the
2725
Simultaneous comminution and chemical reaction solid in dependence on the particle size. So the first model only needs two kinetic parameters ku and k,, while the second model needs (n - 1) rates of breakage and (n(n - 1)/2) distribution parameters. In the described example above the relationship between the kinetic parameters for comminution and chemical reaction in the first model is 2 : 1, in the second model it is 46 : 1. So it is necessary to consider the comminution problem: if it is easy to handle (use of conventional particle size distributions) the first model is a good and simple approach, if it is unusual (e.g. bimodal or other distribution that cannot be described with conventional functions) one has to determine as many rates of breakage and distribution parameters as possible to use in the second model. This determination requires a very high experimental effort because very narrow particle size fractions have to be comminuted for very short times and the analysis of the particle size distributions has to be very exact. In general, the first comminution model is preferable because of its ease of application and the minor effort needed to determine the two desired kinetic parameters. The limit of both comminution models is that the influence of the surrounding gaseous medium cannot be considered; but the gaseous medium influences the comminution characteristics as well as the temperature. So in combining the two kinetics of comminution and reaction the influence of the surrounding medium on particle breakage is assumed to be negligible. In the above represented modelling of the shrinking-core model with particle size distributions, only
the technically most important case of reaction control was considered (Assumption 4). Let us now discuss the other two cases of ash diffusion and film diffusion control. The expression of the conversion for film diffusion control can be written as (Szekely et al. 1976): Xi
3koCAt vl psR
(2) 1
(26)
The conversion expression of ash diffusion control is represented by (Szekely et al. 1976) 1-3(1-Xi)
2/3+2(1-Xi)=
t
-
(ol 6DeCAt Vl psR 2 (27)
which can be rearranged as X'=l-[sin(~sin-l(
1
~°l2DeCa~t\'~sR2 vl ) ) +~]3 (28)
These conversion expressions can be used to calculate the overall conversion with eq. (7), if a particle size distribution of the solid educt exists. Again the restrictions that were made for evaluating the integral of eq. (7) must be paid attention. In Fig. 12 the conversions for the three limiting cases and for the same time of complete conversion of the mean particle size are represented. If eqs (26) and (28) are used with a simultaneously running comminution process, some differences appear compared to the case of reaction control. In the case of film diffusion control, it is possible that
O.g O.B 0.7 0.6
ash dif
X 0.5 0.4 0.3 0.2
film diffusion
0.1 0
"t"~llllll[ll,l,llll]llltllll,]tl~l,t,,lltllllllllllll,
100
200
H t,lltllll=lll]ll,,l=ll
300
400
500
600
I
700
800
t/s Fig. 12. Comparison of the isothermal conversions calculated with the shrinking-core model incl. a LND for chemical reaction, ash and film diffusion control [eq. (7)]; # = 500#m; ~r = 1; CA = 200molm 3; o;l/v~ = 1/3; k = 5.58 x 10 -4 m s- 1; D~ = 4.65 x 10-8 m 2 s- 1; kg = 1.86 × 10 -4 m s- 1; z(~) = 1224 s.
2726
S. Bade and U. Hoffmann
reduction of the particle size by the eomminution process results in a change in the controlling mechanism: The larger particles can react according to film diffusion control, but in the reaction of the finer particles the resistance of the external gas film diminishes so that the chemical reaction controls the conversion. Here it is possible that one has to fix a temperaturedependent limiting particle size diameter and this separates the film diffusion and reaction control. So one has to operate with two equations for integration to evaluate the total conversion. In the case of ash diffusion control Assumption 2 is no longer valid. Assumption 2 has now to be formulated so that the particles react with constant radius because an ashlayer that consists of solid product $2 is formed. Reduction of particle size is only caused by comminution, so that eq. (7) has to be combined with eqs (10) and (11) to model the simultaneous comminution and chemical reaction. If the ash of product $2 has the same comminution characteristics as the solid educt S~, the kinetic parameters ku and k, do not have to be determined under reaction conditions; but if there is a difference in the behavior, which is more probable, the kinetic parameters of comminution have to be determined for the case of simultaneous chemical reaction. The second model of comminution can also be discussed with the two other limiting cases. The time constants zi for film diffusion control can be derived from eq. (26) with Xi = t/% (Szekely et al. 1976). It can be written analogous to eq. (20): Zi
vx Ps (Ri -- Ri+ 1). cox 3k~Ca
(29)
So the combination of eq. (29) with eq. (22) yields the system of differential equations that can be used to calculate the courses of the mass proportions and the conversion of the solid. If there exists a change in the controlling mechanism with decreasing particle size (from film diffusion to chemical reaction control) a combined calculation with eqs (29) and (20) is necessary. In the case of ash diffusion control, time constants cannot be defined because the particles do not shrink with time according to the proceeding chemical reaction. So the material balance (Fig. 7) is simplified because the size reduction occurs only by comminution. But now an accumulation takes place in the fraction with i = n and conversion of unity is given when all particles in this fraction are converted to the solid product $2. Here eq. (24) for determining the solid conversion is no longer valid. Use of eq. (7) is universal. It can be applied to all particle size distributions, e.g. logarithmic normal distribution, normal distribution, Gates-Gaudin-Schuhmann distribution, RRSB (Rosin, Rammler, Sperling and Bennet) distribution and all other particle size distributions, that can be used in special cases. Furthermore, other conversion expressions Xi can be used in eq. (7), which result from other models, e.g.
grainy-pellet model or crackling-core model. But when using these expressions with a simultaneous comminution process attention must be paid to the fact that the assumed structure that develops in the solid particles during chemical reaction, strongly is destroyed by particle breakage. So it seems that these complicated gas-solid models are not so suitable to calculate conversions of the solid, because the effect of destroying structures in the solid dominates over the effect of developing structures. Naturally, eq. (7) can be used with other particle geometries, e.g. cylinder or plate, but here the geometries that result from chemical reaction and comminution, have to match.
5. SCOPE The problem discussed here to model the simultaneous running processes of comminution and chemical reaction in a reaction mill is a new field of modelling heterogeneous systems especially for gas-solid reactions. But other heterogeneous noncatalytic systems are conceivable, too: liquid-solid, solid-solid and gas-liquid-solid systems are the potential for reaction mills. Here the use of other models for the chemical reaction is necessary. Particularly the reaction between two solids requires a more complicated model for the chemical reaction, because the reaction is difficult to investigate without the influence of comminution and yet a model does not exist that describes the reaction between two solids. The case of gas-solid reactions in a reaction mill is of technical interest for reaction control only. Film diffusion control is not desirable and ash diffusion can be excluded because the ash layer is removed by the influence of mechanical energy. But the input of mechanical energy influences not only the dispersity of the solid but also the reactivity of the solid: it is increased because of a mechanical activation. So the chemical reaction proceeds at lower temperatures with increased reaction rates. In this connection it is necessary to modify the chemical kinetics: formation, relaxation and chemical reaction of active centres that are generated in the solid reactant have to be considered. The reaction rate is now proportional to the number of active centres (Heinicke, 1984; Bade, 1994). This approach is particularly useful in the lower temperature region, while in the higher temperature region the approach with the surface-related rate can be used. The selection of a system of substances for use in a reaction mill demands a brittle solid, which is even more brittle at elevated reaction temperatures and in a mill for fine grinding. This is because a mechanical activation of a solid can only be observed if it is comminuted to a particle size of several pm (Heegn, 1989). Vibration mills have a very high effectiveness with regard to formation of fine particles. But other mills, e.g., ball mill and impact mill are also suitable as reaction mills.
Simultaneous comminution and chemical reaction Vibration mills have the advantage to have a very good transport of heat because of the multiple contacts between particles and reactor wall, grinding balls as well as other particles. So exothermic reactions can be carried out almost isothermally. The behavior of the solid particles in the reaction mill can be described similar to an aerosol state in a fluidized bed reactor (Bade, 1994; Bock, 1994) and it is thinkable that the modelling of the simultaneous comminution and chemical reaction can also be carried out using fluidized bed models (Bade, 1994). A further development is the construction of continuous operating reaction mills. The advantage for modelling of the two processes is that in the steady state the particle size distribution is time independent but dependent on the reactor length. So the simple eq. (7) can be used and the determination of the kinetic parameters of the comminution process is not necessary. Finally, it can be mentioned that the understanding of a comminution process and, in particular, its effects on a proceeding chemical reaction is just in its beginning and the modelling of a simultaneous chemical reaction and comminution mainly suffers from a lack of comprehension of the processes that are caused by the comminution in the microstructure of the solid.
Acknowledgements
The authors very much appreciate the support of this project within the scope of the Sonderforschungsbereich 180 by the Deutsche Forschungsgemeinschaft.
n q3 rc t w x xi
2727
number of particle size fractions, dimensionless particle size distribution (related to mass), m - 1 radius of unreacted core, m time, s rate of breakage, s- 1 diameter of particle, m limiting diameter of class i, m mean diameter of the particle size distribution, m
Greek letters median of a particle size distribution, m vi stoichiometric coefficients of gaseous reactants, dimensionless Ps molar density of solid, mol m - 3 a standard deviation of a particle size distribution, dimensionless z time constant in the shrinking-core model for total conversion, s z~ time constant in the shrinking-core model to reach the next fraction, s @ stoichiometric coefficients of solid reactants, dimensionless Subscripts 1, 2 0 oc i, j
refer to components refers to t = 0 refers to t = mark the number of particle size fraction REFERENCES
A,S
CA De
AH o
Ns Q3 R
Ri Xi X bij d g h k ko
k. k~ mi
NOTATION gaseous and solid reactants, respectively concentration of gaseous component A, molto -3 effective diffusion coefficient, m 2 s standard enthalpy of reaction, J m o l - 1 moles of solid component S, mol cumulative particle size distribution (related to mass), dimensionless radius of particle, m mean radius of particle in class i, m conversion of solid particle with radius Ri, dimensionless over all conversion of solid, dimensionless distribution parameter, dimensionless diameter of reaction chamber, m acceleration due to gravity = 9.81 m s -2 height of reaction chamber, m rate constant of the chemical reaction, ms-I mass transfer coefficient, m s rate constant related to the median #, s rate constant related to the standard deviation a, s-1 mass proportion of particle size fraction i, dimensionless
Austin, L. G. and Bhatia, V. K. (1971/1972) Experimental methods of grinding studies in laboratory mills. Powder Technol. 5, 261-266. Austin, L. G. and Luckie, P. T. (1971/1972) The estimation of non-normalized breakage distribution parameters from batch grinding tests. Powder Technol. 5, 267-271. Bade, S. (1994) Einsatz einer Reaktionsschwingmiihle zur simultanen Zerkleinerung und chemischen Reaktion yon Ferrosilicium mit Chlorwasserstoff. Ph.D. thesis, Technical University Clausthal. Bock, U. (1994) Leistungsaufnahme und Kinetik der Fiillung in einer Schwingmiihle bei grofAen Beschleunigungen und Amplituden. P h . D . thesis, Technical University Clausthal. Gardner, R. P. and Austin, L. G. (1962) A chemical engineering treatment of batch grinding. Symp. Zerkleinern, pp. 217-248. VDI, Diisseldorf. Heegn, H. (1989) Uber den Zusammenhang von Feinstzerkleinerung und mechanischer Aktivierung. Aujbereitungstechnik 30, 635-642. Heinicke, G. (1984) Tribochemistry. Akademie, Berlin. Klimpel, R. R. and Austin, L. G. (1970) Determination of selection-for-breakage functions in the batch grinding equation by nonlinear optimization. Ind. Engng Chem. Fundam. 9, 230-237. Mular, A. L. (1970) The determination of selection elements and lumped parameters for grinding mills. Bull. Canad. Mining & Metall. 821 826.
2728
S. Bade and U. Hoffmann
Reid, K. J. (1965) A solution to the batch grinding equation. Chem. Engng Sci. 20, 953-963. Sch6nert, K. (1971) Mathematische Simulation von Zerkleinerungsprozessen. Teil 1: Das allgemeine Modell und der station/ire Sonderfall. Chem. Ing. Technik. 43, 361-367. Schr6ter, M. (1992) Reaktions- und zerkleinerungstechnische Untersuchungen zur Trichlorsilansyn-
these fiir die Entwicklung einer Reaktionsmfihle. Ph.D. thesis, Technical University Clausthal. Senna, M. and Okamoto, K. (1989) Rapid synthesis of Ti- and Zr-Nitrides under tribo-chemical condition. Solid State lonics 32/33, 453-460. Szekely, J., Evans, J. W. and Sohn, H. Y. (1976) Gas-Solid Reactions. Academic Press, New York.
Chemical Engineeriny Science, Vol. 52, No. 16, pp. 2715 2728, 1997 ,~? 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain ~50@119-2:5111~97~11@@54-7 0009-2509/97 $17.00 + 0.00
Modelling of the simultaneous comminution and chemical reaction in a non-catalytic gas-solid batch reactor Stefan Bade and Ulrich Hoffmann* Institut fiir Chemische Verfahrenstechnik, TU Clausthal, Leibnizstr. 17, 38678 Clausthal-Zellerfeld, Germany (Received 29 November 1994; accepted in revised form 16 December 1996) Abstract--The shrinking-core model for reaction control was applied to particle size distributions, particularly to the logarithmic normal distribution. Furthermore, a simultaneous comminution of the solid was considered in the model; in the first case the process of the batchwise comminution was described with a model including two kinetic parameters, in the second case a more complex material balance model was introduced and this operates with rates of breakage and distribution parameters. A comparison is given between model predictions and experimental results for the reaction of gaseous hydrogenchloride with metallurgical grade (MG) silicon to give trichlorosilane. The experimental results were obtained in a newly developed apparatus, the so-called reaction-mill, which can simultaneously act as a vibration mill and as a chemical reactor. © 1997 Elsevier Science Ltd
1. I N T R O D U C T I O N
Among the several available models for non-catalytic gas-solid reactions, the shrinking-core model is the most widely used. But the modelling of non-catalytic gas-solid reactions is often restricted to a definite particle size. However, in practice, always particle size distributions of the solid reactants are used. Until now heterogeneous non-catalytic gas-solid reactions were not discussed under the point of view of a simultaneous comminution process during the chemical reaction, because there exists a lack of experimental apparatuses to run both the processes simultaneously. Although many experimental results of comminution and its influence on succeeding chemical reactions were obtained (Heinicke, 1984), the development of the so-called reaction mill is just in its beginning (Senna and Okamoto, 1989; Bade, 1994). In this connection, a vibration mill is the preferred apparatus as a basis for a reaction mill, because of its effective generation of fine particles and because of its possible mechanical activation of the solid. So combining the two process steps opens new and most promising possibilities to run a process more effectively, because effects of the comminution process, that exceed the change in the dispersity of the solid, can be used for the chemical reaction. But with the development of reaction mills a need for models which are able to
*Corresponding author.
describe the two simultaneously running processes arises. Until now no models exist that can be used in such cases. The present paper deals with the development of such a model. The basis for the combined kinetics of comminution and chemical reaction is the one-parametric shrinking-core model with its kinetic equations for the technically interesting case of reaction control. The batchwise comminution kinetic is developed with both a two-parametric model and a material balance model, that describes both the rates of breakage and the distribution of the fragments for every particle size fraction. 2. MODELLING Let us first consider a non-catalytic gas-solid reaction of a non-porous solid with a gas, where the chemical reaction exclusively proceeds at the external surface of the solid: viAl(g) + colSt(s) ~ VEA2(g) + co2S2(s).
(1)
Let the following assumptions be valid: (1) In the beginning and during the reaction the solid particles have a spherical geometry and are not porous, (2) the particles only shrink during the reaction and no ashlayer is formed, (3) the irreversible chemical reaction proceeds in an isothermal way and is first order with respect to the gaseous reactant, (4) the chemical reaction determines the kinetic of the whole process, (5) the concentration of the gaseous reactant is constant
2715
2716
S. Bade and U. Hoffmann
during the reaction and (6) the chemical reaction exclusively proceeds at a sharp interface at position rc. The the shrinking-core model for the gaseous and solid educts A~ and S~ (here and in the following simplified and named as A and S) can be written as:
dNs dt
(D14rcrZCA. Vl
(2)
The particle size distributions [eqs (5) and (6)] are characterized by two parameters: the standard deviation a and the median/~. In Fig. 1 the normal and the logarithmic normal distribution are illustrated for a dimensionless (R/p) = 0.6 and a = 0.2. The cumulative particle size distribution Q3 is obtained by integration of q3 and it is standardized with eq. (8):
The time dependence of the radius of the unreacted core r~ is given by
Q3 =
q3(R)dR =
1
(8)
0
dr~ ~t kCA dt vl Ps
(3)
And after integration the conversion X~ of the solid with the definite radius R~ is obtained as
(rc3
X,=I-\~//
=1-
(
1
--v, p-~o/ .
(4)
This equation is only valid with a definite radius R~ of the particle. But if there exists a particle size distribution, for example, the normal [eq. (5)] or logarithmic normal distribution [eq. (6)] the overall conversion of the solid X [eq. (7)] is obtained by integrating the product of the conversion function X~ and the most commonly used mass related particle size distribution q3 over the whole interval of particle sizes 1
q3(R)
exp[-1-(R-P~z]
(5)
1 1 [ I(In(R/P)~2~ q3(R)=a~-~Rexp-2\ a ] ] X=
(6)
X~(t, R)q3(R)dR.
(7)
0
In this work the conversions were calculated only with the logarithmic normal distribution, because the normal distribution is more suitable for very narrow distributions while the logarithmic normal distribution is more suitable for practically existing distributions, which have a broader particle size distribution. Combining eqs (4) and (6) with respect to eq. (7) results in the overall conversion X if a logarithmic normal distribution is considered: x-
~-2 ~ I ~ [ O"
1-
(i
~°' kCAt'3] jR --=,,:exp
0
~1
x [- ~(~)2]dR.
(9)
The integration of eq. (9) can be carried out using the integration method of Gaul3-Legendre including a calculation of the weight function; generally the numerical integration was carried out with 200 points. But carrying out the integration numerically, attention must be paid to the fact that the conversion function Xi [eq. (4)] can mathematically result in values greater than unity, which is physically meaningless. So if a particle with radius R~ has reached the
1.5 1.2
/ /
i ~/logarilhminormal c distribution \
/
%:0
.1
02
o.a
o.;"oi
~'~rmaldisltibution
"o.i"oli'"oi8
(R/p)/-
o.g
1.o
11
1;2"i.3
Fig. 1. Dimensionless normal and logarithmic normal distribution for (R/#) = 0.6 and a = 0.2.
Simultaneous comminution and chemical reaction
2717
o=0 0.9 0.8
0.7
0.5 X 0.5 0.¢ 0.3 0.2 0.i
0
l , , I t l t t [ , l i , , i i ' i [ t ' i t l l l I I l i l t l l l s ' i
I000
2000
3000
'lii'iillillllllll
4000
I
5000
BOO0
tls Fig. 2. Influenceof the logarithmic normal distribution on the overall conversion X of the solid [eq. (9)] in isothermal case; # = 465 x 10 - 6 m; (ol/v 1 = 1/3; k = 6 x 10-* m s- t; CA = 30 mol m- 3; Ps = 91,039 molm -3.
conversion unity, in the further computation the conversion stays constant at unity. In Fig. 2 the influence of the particle size distributions on the overall conversions is illustrated. It can be recognized that with a constant /~ the influence of the standard deviation on the conversion is relatively small; but with values of a greater than 0.1 the influence is distinctly observable: In the starting phase, the amount of finer particles accelerates the conversion rate, while at a high degree of conversion the larger particles effect a reduction of the conversion rate. A batchwise comminution process now effects a reduction of the median/~ and a change in the standard deviation or. To describe the kinetics of comminution the time-dependent functions of a and/~ have to be determined. Approaches to model the time dependences of # and a caused by comminution are given in eqs (10) and (11) #(t) = #0 e x p ( - k , t ) + / ~ ( 1 - exp(-kut) )
(10)
For relatively narrow particle size distributions the median ~ can be approximated with the mean particle size ~ [eq. (12)]:
.~= ~ xiAQ3,i = ~ xiq3,1mxi~ ~. i=1
In Fig. 4 the resulting time-dependent particle size distributions are given for various times. Combining eqs (9)-(11) gives the overall conversion function of the solid, when the particle size is not changed by the chemical reaction. But if we now discuss about the simultaneous comminution and chemical reaction according to the shrinking-core model without ashlayer formation, it is necessary to modify eq. (10), because the size of the particles is not only reduced by comminution but also by the shrinking process. Therefore, eq. (3) has to be integrated [eq. (13)]: rc = R
e~l kCat VI
or(t) = a0 e x p ( - k~t) + 6~(1 - e x p ( - k~t))
(11)
Fig. 3 gives an example of the comminution of metallurgical grade silicon in a vibration mill. The median p is shown in dependence of the duration of the comminution process. It can be seen that an exponential expression describes the time dependence of the median # quite well; analogously a similar exponential expression is chosen for the standard deviation, although it fits the experimental values quite well only for longer durations of the comminution process.
(12)
i=1
-/~(t)
(13)
PS
and to be combined with eq. (10); it results in eq. (14), which describes the time dependence of/~ according to simultaneous comminution and chemical reaction: /~(t) = (/~oexp(-- kut) +//oo(1 - e x p ( - kut)) )
to1 kCat h
Ps
(14)
By using eq. (14) attention must be paid to the fact that/~ of eq. (10) has to be in the radius form, so that the eqs (10) and (13) are compatible. Furthermore, the
2718
S. Bade and U. Hoffmann 1000
900' 800 700 E
O
600
"3 ~" 500
400 E
I
o
3O0 200 100 0
0
500
1000
1500
2000
2500
3000
3500
4000
time/s Fig. 3. Comminution of metallurgical grade silicon in a vibration mill; exponential decrease of the median /~ with the duration of comminution; ku = 1.12 x 10-3 s-1; ]-/O = 930/~m; #o0 = 5 pm.
0.030I
'~-~II EI0'020 t=3OOOs
o.olo
H ~ ~t/t=2000 s
0"0000
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
x/pm
Fig. 4. Logarithmic normal distributions for various times during a comminution process; ~0 = 930 #m; #~ = 5 Mm; tr0 = 0.12; tr~ = 0.55; k~, = 1.12 × 10-3 s- 1; k, = 8.3 x 10 -4 s-1.
use of eq. (14) implies t h a t b o t h the c o m m i n u t i o n process a n d the chemical reaction exclusively generate spherical particles, If we n o w c o m b i n e eqs (9), (11) a n d (14) the overall conversion of the solid can be calculated assuming that the s t a n d a r d deviation of the particle size distribution a is not influenced by chemical reaction but only by c o m m i n u t i o n . In Fig. 5 the overall conversions were calculated for three cases: conversion with a c o n s t a n t particle size distribution
for t = 0 a n d t ~ oo a n d the conversion with simultaneous c o m m i n u t i o n a n d chemical reaction with time d e p e n d e n t # a n d tr related to eqs (11) a n d (14). In Fig. 5 it is noticed t h a t with simultaneous comm i n u t i o n a n d chemical reaction a S-shaped conversion function is obtained, while the conversion functions with t i m e - i n d e p e n d e n t particle size distributions are not S-shaped. So if the c o m m i n u t i o n process is very slow, the conversion function a p p r o a c h e s the
Simultaneous comminution and chemical reaction
2719
1 0.9 0.8
p=2.5wm;o=0.55
0.? 0.6
X 0.5 0.4 0.3 0.2 0.i 0
~j~l,,,I,,,,,,,~,la,,,,,,,,I,,,,,,,,,l~,t,,,,,,I,,~,,,,,,I,,,,,,,,,I,,,,,,,ll
0
500
1000
1500
2000
2500
3000
3500
4000
t/s Fig. 5. Isothermal conversions of the solid with constant particle size distributions for t = 0 (# = 465 #m, cr = 0, 12) and t --,@ (/a = 2.5 #m, tr = 0.55) compared with the conversion for simultaneous comminution and chemical reaction (k, = 1.12 x 10- 3 s- 1; k, = 8.3 × 10-4 s- 1); k = 2 × 10-4 m s- ~; CA = 30 mol m- 3; Ps = 91,039 molm-3;) col/v1 = 1/3.
case for t = 0, while for an extreme fast comminution process the conversion function approaches the case for t ~ o c . The described first model to simulate the process of comminution is a two parametric one, which requires the determination of the median # and the standard deviation tr as functions of time. Here all the time the particle size distribution is characterized by the selected function of q3, e.g. normal or logarithmic normal distribution. A more detailed description of the batch comminution process is possible with the population balance model (Gardner and Austin, 1962; Reid, 1965; Sch6nert, 1971). The whole particle size interval is divided into as many as possible narrow particle size fractions, n. The first fraction (i = 1) contains the coarsest particles, while the last fraction (i = n) contains the finest particles. The condition for the particles to belong to the class i is given in eq. (15): xi+ ] < x ~< xi
(15)
and always is x~ < xj
for
i > j.
(16)
The mass proportion of the class i is called m~ and the sum of all masses m~ is unity during the whole comminution process (material balance):
~m i
=
1
(17)
i=1
Now two sets of coefficients are introduced to model the comminution process:
(1) rates of breakage w i that describe the proportion leaving the class i by comminution per unit of time; dimension: s-1. (2) distribution parameters bij that describe the distribution of the fragments; blj corresponds to the proportion that reaches the class i when particles of class j are comminuted. The alteration with time of the mass proportion in every fraction results from the per unit of time entering fragments of coarser particles subtracting the mass proportion leaving this fraction; so a material balance model can be used for describing the comminution kinetics. By that for n classes a system of ordinary differential equations of first order results: dmi --=--wlmi dt
i- 1 + 2 bijwjmj j=x
(18)
with the initial condition m~(t = O) = mio. The comminution of the finest particles in the fraction with i = n is (by definition) not considered in this model, so that an accumulation in this fraction occurs. This system of n differential equations requires the determination of ( n - 1 ) rates of breakage wi and (n(n - 1)/2) distribution parameters bit. With the simplification of a constant rate of breakage independent of particle size (w = const.) and of distribution parameters bis that are constant during the process of comminution, the calculation can be carried out for an experimentally determined or estimated rate of breakage w and for (n(n - 1)/2) estimated distribution parameters b~j. In fact, the rates of breakage wl depend on the particle size and they decrease with decreasing
2720
S. Bade and U. Hoffmann
particle size. Furthermore, the distribution parameters b~j depend on the dispersity of the solid which alters during a batch comminution process. In the case of dividing the whole particle size interval into n = 10 classes, 45 distribution parameters have to be estimated with the condition of eq. (19):
bij
=
1 for j = 1. . . . . (n - 1).
(19)
i=j+l
The values for the rates of breakage wzare obtained by the loss of mass in class i during comminution. In the experimental procedure each narrow fraction has to be comminuted and from the resulting exponential loss of mass with time the rates of breakage w~ are obtainable. Then the rates of breakage wi can be used to calculate an arithmetic mean value of w. The distribution parameters b~j can be obtained by determining the distribution of the fragments after comminution of each narrow fraction; doing this, it is important to comminute only for short times because the developing fragments can further be broken up; because of the short comminution intervals the analysis of the fragments is quite difficult. The number of b~ which have to be determined can be reduced by introducing similar distributions of the fragments. Further details of experimental determination of w~and b~ are described elsewhere (Austin and Bhatia, 1971/1972; Mular, 1970; Klimpel and Austin, 1970). In the following example of modelling a comminution process, the rate of breakage w is assumed to be constant and independent of particle size; it is obtained by calculating the arithmetic mean value of w~. In Table 1 the estimated values of the distribution parameters bij are shown for the comminution of metallurgical grade silicon in a vibration mill. The particle size interval is divided into 10 classes and the distribution parameters are assumed to be constant during the comminution process. The values for b~j are obtained in comminution experiments of short duration (15 s) with each narrow fraction for j = 1. . . . . 7. The distribution parameters b9, s and blo,8 are assumed to be 0.5, blo,9 has to be 1. The experimentally determined parameters b~j include a rela-
tively large experimental error because of the short period of comminution and the resulting analytical difficulties; so the determined values bij represent only rough estimates. The time-dependent courses of the mass proportions mi are shown in Fig. 6. The calculation of the system of 10 differential equations [eq. (18)] was carried out using a Runge-Kutta routine of fourth order. In Fig. 6 it can be seen that the simulation of the batchwise comminution process results for the mass proportion of the first class (i = 1) in an exponential decrease, while the mass proportion of the class with the finest particles (i = n) shows a s-shaped increase; here an accumulation takes place because the breakage of this class is not considered in the population balance model (w = 0 by definition). The mass proportions of the eight classes numbered 1 < i < 10 run all through a maximum that shifts with increasing fineness of the particles to later times. Now the chemical kinetics according to the shrinking-core model without ashlayer formation, additionally, can be introduced into the system of differential equations [eq. (18)]. The material balance of one class is schematically illustrated in Fig. 7. The flow into the class i occurs not only by comminution of the other (i - 1) fractions but additionally by shrinking caused by chemical reaction of the particles in the class with number (i - 1). On the other hand, the particles leave the class i due to breakage and they reach the other (n - i) fractions according to the distribution parameters; simultaneously, the particles leave the class i due to a shrinking of the particles caused by chemical reaction and they enter the class (i + 1). The time constants z~ represent the time necessary to reach the particle radius of the (i + l) class according to a shriking of the particle. These time constants can be calculated by using the shrinking-core model in the case of reaction control: eq. (3) can be integrated between the limits of Ri and Ri+ 1 and the time constants r~ are written as [eq. (20)]: Vl Ps rl = - - - (Ri -- Ri+ 1). ~01 k C a
(20)
Table 1. Estimated values of the distribution parameters blj for division of the whole particle size interval into 10 classes b/j
j = 1
2
3
4
5
6
7
8
9
i= 2 3 4 5 6 7 8 9 10
0.03 0.05 0.I0 0.20 0.20 0.15 0.10 0.09 0.08
0.06 0.08 0.12 0.15 0.20 0.20 0.11 0.08
0.08 0.12 0.15 0.20 0.25 0.12 0.08
0.I0 0.12 0.15 0.20 0.30 0.13
0.10 0.20 0.30 0.25 0.15
0.20 0.30 0.35 0.15
0.30 0.50 0.20
0.50 0.50
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
b~j i=j+I
Simultaneous comminution and chemical reaction
2721
0.20 0.18 0.16 0.14 0.12 "~-0.i0 0.08
0.06 0.04 0.02 0.00
0
200
600
400
800
I000
1200
1400
1600
1800
2000
tls Fig. 6. Mathematical simulation of a batchwise comminution process under consideration of l0 particle size fractions; 45 estimated distribution parameters b~ and a constant rate of breakage w = 0.005 s- 1.
/
//
\
wimi N
,-1
,~b~j.jr.j
mi(t)
(
mi/x i
Class i
mi. 1 Iti. 1
Fig. 7. Schematic representation of the material balance of one particle size fraction with simultaneous comminution and chemical reaction according to the isothermal shrinking-core model.
In eq. (20) Ri is the mean particle radius of class i that can be calculated with eq. (21): R i = (X i q'-
Xi+ 1)/4.
wi +
mi + ~ bqwjmj + j=l TI-I
xA#m)
1
1000
2 3 4 5 6 7 8 9 10
800 630 400 250 160 100 63 25 10
Ri(pm) 450 357.5 257.5 162.5 102.5 65 40.75 22 8.75 2.5
r(s)
%(s)
8194 6509 4689 2959 1866 1183 742 401 159 45.5
1685 1820 1730 1093 683 441 341 242 113.5 45,5
(21)
Therefore, in the case of simultaneous comminution and chemical reaction the differential equation [-eq. (18)] can now be written as [eq. (22)]:
dt
Table 2. Calculation of the time constants r and z~ with the isothermal shrinking-core model in the case of chemical reaction control; k = 5 × 10- 4 m s- 1; Ps = 91039 tool m- 3; 091/vl = 1/3; Ca = 30 molm -3
(22)
In Table 2 an example is given to show (1) how to divide the particle size interval into 10 classes, (2) how to calculate the mean radius Ri and the time constants z for complete conversion and % for shrinking in the next class. In Fig. 8 a mathematical simulation of a batchwise comminution process with simultaneous chemical reaction according to the shrinking-core model (Table 2) is illustrated. The difference of Fig. 8 compared to Fig. 6 can be seen. While by using eq. (18) in the class with the finest particles (i = 10) an accumula-
tion occurs because the rate of breakage is zero, by using eq. (22) the accumulation in this class is avoided because now the particles shrink by chemical reaction. Therefore the mass proportion of this class (i = 10) runs through a maximum too. At this time, the flow into this class is equal to the consumption by chemical reaction. Only if the chemical reaction is infinitely slow an accumulation in this class occur. Furthermore, it is noticed that the mass proportions m~ reach the value zero faster and that the maxima show a reduced height compared to Fig. 6. All these effects are caused by the chemical reaction, Because the chemical reaction consumes the solid reactant to give gaseous reaction products, the condition given in eq. (17) is no longer valid. In the case of chemical reaction, eq. (17) now has to be formulated
2722
S. Bade and U. Hoffmann
0.10 O.Og
/
8
g
1
0.08
10
0.07
0.06 -0.05 0.04 0.03 0.02
0.01 0.00
500
1000
1500
2000
2500
3000
t/s Fig. 8. Mathematical simulation ofa batchwise comminution process with simultaneous chemical reaction according to the isothermal shrinking-core model; consideration of 10 particle size fractions (Table 2); 45 estimated distribution parameters b u (Table 1); w = 5 x 10-3 s-1; k = 5 x 10 -4 m s-1; CA = 30 mol m-3; ps = 91,039 molm-3; col~v1 = 1/3.
1.00 0.00 0.80 0.70
with slmiJItaneouscomml
O.BO "L 0.50 X
/
/
shrinking-core model
0.40 0.30 0.20 0.10 0.000
=
,
i
i
,
i
i
=
J
i
600
i
i
i
i
i
i
i
i
|
i
i
i
i
i
1200
i
i
,
i
I
,
1800
i
i
i
i
i
i
i
=
I
i
2400
i
i
=
i
i
J
i
i
3000
t/s Fig. 9. Comparison of the isothermal conversions of the solid reactant without comminution [eq. (4)] and with simultaneous comminution: 10 particle size fractions (Table 2); 45 estimated distribution parameters b u (Table 1); w = 5 x I0- 3 s- 1; k = 5 x 10 -4 m s- i; Ca = 30 mol m - 3; Ps = 91,039 mol m - 3; oJ1/vl = 1/3.
as [eq. (23)]: ~mi
~< 1.
(23)
/=1
The total conversion of the solid can be calculated using eq. (24): n
X = 1-
~, mi. i=1
(24)
In Fig. 9 the conversions of the solid are calculated b o t h with the simple shrinking-core model Feq. (4)] a n d with a simultaneous c o m m i n u t i o n process according to the system of differential equations [eq. (20)]. It is noticed that the conversion rate in the case of simultaneous c o m m i n u t i o n is higher t h a n in the case of pure chemical reaction, where the complete conversion of the solid is reached after T = 8194 s
2723
Simultaneous comminution and chemical reaction (Ri = 450 #m, see Table 2), while with simultaneous comminution the complete conversion of the solid is already reached after 2000 s. Furthermore, in the case of simultaneous comminution a s-shaped course of conversion results. The maximum conversion rate is obtained when the particle size fraction with the finest particles (i = n) runs through its maximum. The second model to simulate the process of comminution described requires the determination of a large amount of parameters dependent on the number of classes. In the ideal case the classes have to be very narrow and for every class the experimental determination of the rate of breakage and the distribution parameters is necessary. So this model needs a great effort and it is to be discussed in which cases this effort is worthwhile. Certainly, this model has to be used if there exists an unusual particle size distribution, which cannot be described with the normal or logarithmic normal distribution, e.g. the development of a bimodal distribution function during comminution can be described with the second model. Furthermore, this comminution model has to be used if one is interested in the exact particle size distribution, which can be extracted for all times by calculating the mass proportions in dependence on time. 3. EXPERIMENTALRESULTS An apparatus where the simultaneous comminution and its chemical reaction with a gas takes place is the so-called reaction mill. It is a vibration mill with variable forms of vibration (rotatory, vertical), that is heatable up to 450°C, pressure resistant up to 2 MPa
and has a gas inlet and outlet. The cylindrical reaction chamber is made of stainless steel and has a volume of 393 cm 3 (d = 100 mm, h = 50 mm). It is constructed to exceed very high accelerations of maximum 65 g with high relative amplitudes adjustable from 0.02 to 0.14, because the input of mechanical energy is considered to have a distinct influence on the course of the chemical reaction. This apparatus is described elsewhere (Bade, 1994; Bock, 1994). The heterogeneous non-catalytic reaction that was investigated in the reaction mill was the exothermic reaction of metallurgical grade (MG) silicon with gaseous hydrogenchloride to give trichlorosilane as the main product [eq. (25)]; a thermodynamic analysis showed that this reaction is essentially irreversible. Si(s) + 3 HCI(g) ~ SiHCls(g) + H2(g ), AH ° = -223.2 kJ mol- 1
(25)
The reason to prefer a vibration mill as a chemical reactor for this reaction is that the MG silicon contains about 10 wt% impurities, above all iron. The iron is hydrochlorinated to the solid by-product FeC12, which is regarded as reaction inhibiting. Indeed, it was shown (Bade, 1994; Schr6ter, 1992) that the chemical reaction carried out singularly without comminution does not reach a complete conversion of the solid reactant silicon in the lower temperature region up to 600 K. It is expected that the inhibiting effect of FeC12 is cancelled, when the process runs under the influence of mechanical energy in a vibration mill. Furthermore, the chemical reaction [eq. (25)] thermally runs only approximately above 573 K, while in the reaction mill it was shown that the
0.020 0.018
with comminution
without comminution
0.016 0.014 0.012 >~0.010 0.008 experiment
•
O.OOfi
•
model
0.004 0.002
0.000= 0
/ , ,e,, ~, . . . . . . . 500
il . . . . . . . . . 1000
I ......... 1500 t/s
i~,,L 2000
.....
I ......... 2500
I ......... 3000
3500
Fig. 10. Modelling of the solid conversion in the reaction mill at 543 K with the shrinking-core model [eq. (9)]; k, = 3.5 x 10-4 s-l; k~ = 3.0x 10-4 s-l; k = 3.8x 10 6ms-l; CHCl= 23 molm-3; Ps = 91,039 molm-3; ~1/vl = 1/3,
2724
S. Bade and U. Hoffmann
0.020 with commlnutlon
0,018
without comrninution
O.OlO
0.014 0,012
~
0.010
0,008 0.008
1
~
experiment
0.004 !
0.002 i
0.000,~
; ;
.........
500
!= .........
1000
, .........
1500
= .........
2000
, .........
2500
, .........
3000
i
3500
t/s Fig. l 1. Modelling of the solid conversion in the reaction mill at 553 K with the shrinking-core model [eq. (9)]; k,=3.5xl0-'*s-1; k =3.0xl0-'*s-1; k=5.4xl0-6ms-1; C.cl=22.6molm-3; Ps = 91,039 molm-3; COl/V1 = 1/3.
reaction can take place at temperatures down to 373 K (Bade, 1994). Results of investigations concerning the comminution process have shown that the brittle non-porous solid M G silicon (feed size: 800-1000/zm) can be efficiently reduced to a particle size of about 5/~m in the vibration mill and that the particle size distributions can be described quite well with the logarithmic normal distribution [eq. (6)]. So here the first model, described above, was used to model the simultaneous comminution and chemical reaction. In Figs 10 and 11 experimentally observed conversions of the solid in the reaction mill at 543 and at 553 K, respectively, are shown. The courses of conversions are divided into two parts: (1) in the first part the chemical reaction runs simultaneously with comminution, (2) in the second part only the chemical reaction proceeds. In the first part the modelling was carried out using eq. (9) combined with eqs (11) and (14), in the second part the calculation was carried out with the particle size distribution parameters at the time when the comminution process was interrupted. The parameters of k, and k, were previously determined. They are functions of the amplitude, the numbers of revolution, the acceleration and the fillings of grinding balls and charging material of the vibration mill. It is seen that the experimental conversions can be described very well with the model that considers the comminution process with two kinetic parameters ku and k,. In the temperature region 543-553 K the influence of temperature is relatively high, while the influence of mechanical energy is not so important, so
that the chemical reaction is mainly dominated by the surface enlargement during the comminution process. In this connection, the fresh surfaces produced are so active that the chemical reaction starts at temperatures below 573 K, while with pure thermal excitation it only starts above 573 K (Schr6ter, 1992; Bade, 1994). With decreasing reaction temperature, the model used here predicts increasing deviations from the experimental conversions, because the influence of mechanical energy in the reaction mill becomes more and more dominating.
4. DISCUSSION The described models for calculating the conversions of a solid reactant during the influence of mechanical energy have shown that they can represent the experimentally observed conversions, in particular in the higher temperature region, where the dispersity of the solid educt plays a dominating role. To validate the models, higher conversions of the solid have to be determined experimentally. The discussion of the limiting cases (very slow chemical reaction; very slow comminution) leads to the conclusion that a reaction mill is only interesting in those cases where the chemical reaction and the comminution have comparable rates. Otherwise the two processes can be separately carried out without getting into any disadvantage. The two models for the comminution process show some differences: the first model is a more empirical model and describes the macroscopic effect of comminution, while the second model is a more theoretical one and is based on the physical properties of the
2725
Simultaneous comminution and chemical reaction solid in dependence on the particle size. So the first model only needs two kinetic parameters ku and k,, while the second model needs (n - 1) rates of breakage and (n(n - 1)/2) distribution parameters. In the described example above the relationship between the kinetic parameters for comminution and chemical reaction in the first model is 2 : 1, in the second model it is 46 : 1. So it is necessary to consider the comminution problem: if it is easy to handle (use of conventional particle size distributions) the first model is a good and simple approach, if it is unusual (e.g. bimodal or other distribution that cannot be described with conventional functions) one has to determine as many rates of breakage and distribution parameters as possible to use in the second model. This determination requires a very high experimental effort because very narrow particle size fractions have to be comminuted for very short times and the analysis of the particle size distributions has to be very exact. In general, the first comminution model is preferable because of its ease of application and the minor effort needed to determine the two desired kinetic parameters. The limit of both comminution models is that the influence of the surrounding gaseous medium cannot be considered; but the gaseous medium influences the comminution characteristics as well as the temperature. So in combining the two kinetics of comminution and reaction the influence of the surrounding medium on particle breakage is assumed to be negligible. In the above represented modelling of the shrinking-core model with particle size distributions, only
the technically most important case of reaction control was considered (Assumption 4). Let us now discuss the other two cases of ash diffusion and film diffusion control. The expression of the conversion for film diffusion control can be written as (Szekely et al. 1976): Xi
3koCAt vl psR
(2) 1
(26)
The conversion expression of ash diffusion control is represented by (Szekely et al. 1976) 1-3(1-Xi)
2/3+2(1-Xi)=
t
-
(ol 6DeCAt Vl psR 2 (27)
which can be rearranged as X'=l-[sin(~sin-l(
1
~°l2DeCa~t\'~sR2 vl ) ) +~]3 (28)
These conversion expressions can be used to calculate the overall conversion with eq. (7), if a particle size distribution of the solid educt exists. Again the restrictions that were made for evaluating the integral of eq. (7) must be paid attention. In Fig. 12 the conversions for the three limiting cases and for the same time of complete conversion of the mean particle size are represented. If eqs (26) and (28) are used with a simultaneously running comminution process, some differences appear compared to the case of reaction control. In the case of film diffusion control, it is possible that
O.g O.B 0.7 0.6
ash dif
X 0.5 0.4 0.3 0.2
film diffusion
0.1 0
"t"~llllll[ll,l,llll]llltllll,]tl~l,t,,lltllllllllllll,
100
200
H t,lltllll=lll]ll,,l=ll
300
400
500
600
I
700
800
t/s Fig. 12. Comparison of the isothermal conversions calculated with the shrinking-core model incl. a LND for chemical reaction, ash and film diffusion control [eq. (7)]; # = 500#m; ~r = 1; CA = 200molm 3; o;l/v~ = 1/3; k = 5.58 x 10 -4 m s- 1; D~ = 4.65 x 10-8 m 2 s- 1; kg = 1.86 × 10 -4 m s- 1; z(~) = 1224 s.
2726
S. Bade and U. Hoffmann
reduction of the particle size by the eomminution process results in a change in the controlling mechanism: The larger particles can react according to film diffusion control, but in the reaction of the finer particles the resistance of the external gas film diminishes so that the chemical reaction controls the conversion. Here it is possible that one has to fix a temperaturedependent limiting particle size diameter and this separates the film diffusion and reaction control. So one has to operate with two equations for integration to evaluate the total conversion. In the case of ash diffusion control Assumption 2 is no longer valid. Assumption 2 has now to be formulated so that the particles react with constant radius because an ashlayer that consists of solid product $2 is formed. Reduction of particle size is only caused by comminution, so that eq. (7) has to be combined with eqs (10) and (11) to model the simultaneous comminution and chemical reaction. If the ash of product $2 has the same comminution characteristics as the solid educt S~, the kinetic parameters ku and k, do not have to be determined under reaction conditions; but if there is a difference in the behavior, which is more probable, the kinetic parameters of comminution have to be determined for the case of simultaneous chemical reaction. The second model of comminution can also be discussed with the two other limiting cases. The time constants zi for film diffusion control can be derived from eq. (26) with Xi = t/% (Szekely et al. 1976). It can be written analogous to eq. (20): Zi
vx Ps (Ri -- Ri+ 1). cox 3k~Ca
(29)
So the combination of eq. (29) with eq. (22) yields the system of differential equations that can be used to calculate the courses of the mass proportions and the conversion of the solid. If there exists a change in the controlling mechanism with decreasing particle size (from film diffusion to chemical reaction control) a combined calculation with eqs (29) and (20) is necessary. In the case of ash diffusion control, time constants cannot be defined because the particles do not shrink with time according to the proceeding chemical reaction. So the material balance (Fig. 7) is simplified because the size reduction occurs only by comminution. But now an accumulation takes place in the fraction with i = n and conversion of unity is given when all particles in this fraction are converted to the solid product $2. Here eq. (24) for determining the solid conversion is no longer valid. Use of eq. (7) is universal. It can be applied to all particle size distributions, e.g. logarithmic normal distribution, normal distribution, Gates-Gaudin-Schuhmann distribution, RRSB (Rosin, Rammler, Sperling and Bennet) distribution and all other particle size distributions, that can be used in special cases. Furthermore, other conversion expressions Xi can be used in eq. (7), which result from other models, e.g.
grainy-pellet model or crackling-core model. But when using these expressions with a simultaneous comminution process attention must be paid to the fact that the assumed structure that develops in the solid particles during chemical reaction, strongly is destroyed by particle breakage. So it seems that these complicated gas-solid models are not so suitable to calculate conversions of the solid, because the effect of destroying structures in the solid dominates over the effect of developing structures. Naturally, eq. (7) can be used with other particle geometries, e.g. cylinder or plate, but here the geometries that result from chemical reaction and comminution, have to match.
5. SCOPE The problem discussed here to model the simultaneous running processes of comminution and chemical reaction in a reaction mill is a new field of modelling heterogeneous systems especially for gas-solid reactions. But other heterogeneous noncatalytic systems are conceivable, too: liquid-solid, solid-solid and gas-liquid-solid systems are the potential for reaction mills. Here the use of other models for the chemical reaction is necessary. Particularly the reaction between two solids requires a more complicated model for the chemical reaction, because the reaction is difficult to investigate without the influence of comminution and yet a model does not exist that describes the reaction between two solids. The case of gas-solid reactions in a reaction mill is of technical interest for reaction control only. Film diffusion control is not desirable and ash diffusion can be excluded because the ash layer is removed by the influence of mechanical energy. But the input of mechanical energy influences not only the dispersity of the solid but also the reactivity of the solid: it is increased because of a mechanical activation. So the chemical reaction proceeds at lower temperatures with increased reaction rates. In this connection it is necessary to modify the chemical kinetics: formation, relaxation and chemical reaction of active centres that are generated in the solid reactant have to be considered. The reaction rate is now proportional to the number of active centres (Heinicke, 1984; Bade, 1994). This approach is particularly useful in the lower temperature region, while in the higher temperature region the approach with the surface-related rate can be used. The selection of a system of substances for use in a reaction mill demands a brittle solid, which is even more brittle at elevated reaction temperatures and in a mill for fine grinding. This is because a mechanical activation of a solid can only be observed if it is comminuted to a particle size of several pm (Heegn, 1989). Vibration mills have a very high effectiveness with regard to formation of fine particles. But other mills, e.g., ball mill and impact mill are also suitable as reaction mills.
Simultaneous comminution and chemical reaction Vibration mills have the advantage to have a very good transport of heat because of the multiple contacts between particles and reactor wall, grinding balls as well as other particles. So exothermic reactions can be carried out almost isothermally. The behavior of the solid particles in the reaction mill can be described similar to an aerosol state in a fluidized bed reactor (Bade, 1994; Bock, 1994) and it is thinkable that the modelling of the simultaneous comminution and chemical reaction can also be carried out using fluidized bed models (Bade, 1994). A further development is the construction of continuous operating reaction mills. The advantage for modelling of the two processes is that in the steady state the particle size distribution is time independent but dependent on the reactor length. So the simple eq. (7) can be used and the determination of the kinetic parameters of the comminution process is not necessary. Finally, it can be mentioned that the understanding of a comminution process and, in particular, its effects on a proceeding chemical reaction is just in its beginning and the modelling of a simultaneous chemical reaction and comminution mainly suffers from a lack of comprehension of the processes that are caused by the comminution in the microstructure of the solid.
Acknowledgements
The authors very much appreciate the support of this project within the scope of the Sonderforschungsbereich 180 by the Deutsche Forschungsgemeinschaft.
n q3 rc t w x xi
2727
number of particle size fractions, dimensionless particle size distribution (related to mass), m - 1 radius of unreacted core, m time, s rate of breakage, s- 1 diameter of particle, m limiting diameter of class i, m mean diameter of the particle size distribution, m
Greek letters median of a particle size distribution, m vi stoichiometric coefficients of gaseous reactants, dimensionless Ps molar density of solid, mol m - 3 a standard deviation of a particle size distribution, dimensionless z time constant in the shrinking-core model for total conversion, s z~ time constant in the shrinking-core model to reach the next fraction, s @ stoichiometric coefficients of solid reactants, dimensionless Subscripts 1, 2 0 oc i, j
refer to components refers to t = 0 refers to t = mark the number of particle size fraction REFERENCES
A,S
CA De
AH o
Ns Q3 R
Ri Xi X bij d g h k ko
k. k~ mi
NOTATION gaseous and solid reactants, respectively concentration of gaseous component A, molto -3 effective diffusion coefficient, m 2 s standard enthalpy of reaction, J m o l - 1 moles of solid component S, mol cumulative particle size distribution (related to mass), dimensionless radius of particle, m mean radius of particle in class i, m conversion of solid particle with radius Ri, dimensionless over all conversion of solid, dimensionless distribution parameter, dimensionless diameter of reaction chamber, m acceleration due to gravity = 9.81 m s -2 height of reaction chamber, m rate constant of the chemical reaction, ms-I mass transfer coefficient, m s rate constant related to the median #, s rate constant related to the standard deviation a, s-1 mass proportion of particle size fraction i, dimensionless
Austin, L. G. and Bhatia, V. K. (1971/1972) Experimental methods of grinding studies in laboratory mills. Powder Technol. 5, 261-266. Austin, L. G. and Luckie, P. T. (1971/1972) The estimation of non-normalized breakage distribution parameters from batch grinding tests. Powder Technol. 5, 267-271. Bade, S. (1994) Einsatz einer Reaktionsschwingmiihle zur simultanen Zerkleinerung und chemischen Reaktion yon Ferrosilicium mit Chlorwasserstoff. Ph.D. thesis, Technical University Clausthal. Bock, U. (1994) Leistungsaufnahme und Kinetik der Fiillung in einer Schwingmiihle bei grofAen Beschleunigungen und Amplituden. P h . D . thesis, Technical University Clausthal. Gardner, R. P. and Austin, L. G. (1962) A chemical engineering treatment of batch grinding. Symp. Zerkleinern, pp. 217-248. VDI, Diisseldorf. Heegn, H. (1989) Uber den Zusammenhang von Feinstzerkleinerung und mechanischer Aktivierung. Aujbereitungstechnik 30, 635-642. Heinicke, G. (1984) Tribochemistry. Akademie, Berlin. Klimpel, R. R. and Austin, L. G. (1970) Determination of selection-for-breakage functions in the batch grinding equation by nonlinear optimization. Ind. Engng Chem. Fundam. 9, 230-237. Mular, A. L. (1970) The determination of selection elements and lumped parameters for grinding mills. Bull. Canad. Mining & Metall. 821 826.
2728
S. Bade and U. Hoffmann
Reid, K. J. (1965) A solution to the batch grinding equation. Chem. Engng Sci. 20, 953-963. Sch6nert, K. (1971) Mathematische Simulation von Zerkleinerungsprozessen. Teil 1: Das allgemeine Modell und der station/ire Sonderfall. Chem. Ing. Technik. 43, 361-367. Schr6ter, M. (1992) Reaktions- und zerkleinerungstechnische Untersuchungen zur Trichlorsilansyn-
these fiir die Entwicklung einer Reaktionsmfihle. Ph.D. thesis, Technical University Clausthal. Senna, M. and Okamoto, K. (1989) Rapid synthesis of Ti- and Zr-Nitrides under tribo-chemical condition. Solid State lonics 32/33, 453-460. Szekely, J., Evans, J. W. and Sohn, H. Y. (1976) Gas-Solid Reactions. Academic Press, New York.