1
Ind. Eng. Chem. Fundam. 1082, 21, 1-12
Studies on the Gasification of a Single Char Particle Kyrlacos Zygourakls LBpartment of C h e m b l Engineering, University of Minnesota, Minneapolls, Minnesota 55455
LUIS Arrl and Neal R. Amundson' Department of Chemlcal Engineering, Cuilen College of Engineering, University of Houston, Houston, Texas 77004
The internal structure of a devolatilized char particle can be characterized by two populations of pores, the large spherical vesicles and the cylindrical micropores. A probabilistic model is developed for the evolution of the intemal surface area and pore volume during the gasification of a single char particle. The model is applied to study the reaction of chars with CO, or 0,in the region of kinetically controlled rates. Model predictions were found to be in good agreement with published experimental data.
Introduction The char particles resulting from the devolatilization of parent coals have a very complex pore structure. Experimental investigations of this structure (Lightman and Street, 1968; Street et al., 1969; Field, 1969; Smith and Tyler, 1972) have shown that char particles from low rank coals, especially, have a cellular pore structure, with many large interconnected vesicles, which are open to the exterior. These pores have very large diameters of the order of several microns. When the chars are subsequently gasified, it has been generally observed that chemical reaction takea place throughout the intemal surface area, the degree of its utilization depending on the particular reaction and the conditions (Smith, 1971; Smith and Tyler, 1972,1974). Therefore, a model of the gasification of char particles should include this important effect of internal chemical reaction by accounting for the evolution with time of such physical properties of the particle as internal pore surface area, porosity, size distribution of pores, etc. Petersen (1957) proposed a model for the evolution of internal surface area with reaction. He has used the very strong assumptions of uniform cylindrical pores and that the pores do not coalesce as they enlarge. However, the pore size distribution of char particles is anything but unimodel and coalescence of the large vesicles and the micropores becomes the dominant factor after the initial stages of gasification. Since the enormous number of such entities makes a deterministic approach very difficult, a probabilistic model will be used. The general approach followed by Hulburt and Katz (1964) for the modelling of particle systems will be involved here. The problem of obtaining a closed set for the moments of the populations considered will be handled by the use of simplifying assumptions about the coalescence process. Such assumptions have been suggested by Hulburt and Katz (1964) and they introduce model parameters that must be estimated. Such an approach has been used by Hashimoto and Silveston (1973) to model the gasification of devolatilized anthracite by COP Their model involves ten adjustable parameters and the results are very sensitive to their values. More recently, Simons (1979) h& proposed a different random pore model that describes the combination of pores during gasification. Several empirical relations, however, are used to relate the important physical properties of chars to the fundamental pore statistics. A more refined random pore model has been formulated by
Gavalas (1980) that succeeds in predicting accurately a small set of the available experimental data. In the next section the general model is developed by applying Hulburt's approach to our particular system. In the third section the general model is applied to the gasification of a char particle, whose pore structure is comprised of the populations and which is gasified under chemical reaction control. Such modelling of the pore structure of chars, using a population of large spherical vesicles which are interconnected through a few large neck pores and another population of micropores, is suggested by experimental evidence (see Dutta et al., 1977a; Street et al., 1969; Smith and Tyler, 1972). In the final section, model predictions are compared to experimental data published in the literature and the effect of model parameters is investigated. The modelling of pore coalescence is presented in detail in Appendices A and C, while in Appendix B the assumption of chemical reaction control is investigated for the coal chars considered. Development of a Mathematical Model Consider a population of entities, where an entity can be either a spherical vesicle (bubble), a cylindrical neck pore (macropore), or a cylindrical micropore. At any time t the state of any entity may be completely characterized by (i) a characteristic length r, representing the entity dimension (internal coordinate) and (ii) the spatial coordinates x of the geometric center of the entity (external coordinates). Each entity in our processing system has ita geometric center fixed in space. However, the internal coordinate r will vary in time due to the chemical reaction taking place. Therefore the evolution of the coordinates of any entity in time may be described by differential equations of the form dxi/dt = 0; i = 1, 2, 3 (1) dr/dt = G(c(x,t),fl(x,t),r)
(2)
We assume now that the number of entities in our processing system is so large that their states at any time t represent a continuum fiiing the appropriate subspace of the phase space. Under this assumption we may define a number density function F(x,r,t) for the entities, where F(x,r,t)Idrlldxlis the number of entities in the system at time t that have phase coordinates in the range (x-dx/2, x dn/2), (r - dr/2, r + dr/2), where in the above definition ldrlldxl is a shorthand notation for the differential
+
0196-4313/82/1021-0001~01.25/0 0 1982 American Chemlcal Society
2
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
volume in the phase space. As the chemical reaction proceeds, the characteristic length of the entities will increase with time, until they are large enough to "collide" with neighboring entities. At the moment of collision we will assume that new entities are formed that have a new characteristic internal coordinate r. This coalescence results in generation of entities with coordinates in a new range of the phase space, as well as in destruction of entities with coordinates in another region. In order to model this process, we introduce a function H(x,r,t) where H(x,r,t)ldxlldrl is the net rate (at time t ) at which entities with phase coordinates in the range (x- d x / 2 , x d x / 2 ) , (r - dr/2, r + dr/2) are introduced into the system. The conservation equation for the entities in the phase space will lead to a differential equation in the number density function F(x,r,t) of the form
+
at
a + -(GF) = H(x,r,t) ar
(3)
The details for all the possible systems may be found in the before-mentioned work of Hulburt and Katz (1964). We only note here that eq 3 is the version of the classical Liouville equation pertinent to our reactive system. In order to find a working approximation to eq 3, we first assume that the growth function G may be split into two factors, one depending on the entity environment and the other on the particle dimensions. Hence we write G(c,e,r) = g(c,8)d49 (4) and eq 3 reduces to (5)
Formally eq 5 can be transformed into an equation for the moments pn of F where pn(Xtt) =
Lm
r"F(x,r,t) dr
(6)
Strictly speaking the integration must be over all possible values of r, which range from a positive lower limit to a finite upper limit. However, an appropriate extension of the density function F will permit us to take the limits as indicated in eq 6. Thus by multiplying both sides of eq 5 by r" and integrating, we obtain the following equation for the moments pn of F
and Amundson (1966) have pointed out, the coalescence process is analogous in some respects to a second-order chemical reaction and the coalescence frequency for the discrete case may be obtained by alluding to this mechanism. Following their argument (see also Curl, 1963) we consider two distinct collections of entities Sl and S2,which have associated with them characteristic coalescence frequencies c1 and cF Entities from the sets S1and S2may coalesce with one another or an entity may coalesce with a member of its own set. Hence the rate of coalescence of one entity from SI with another one from S2is
which reduces to c1S1qS2. This is extended to the continuous case by letting S1 = F(x,rl,t) and S2= F(x,r2,t) in which case the coalescence frequency becomes C(x,rl,rz,t) = ~~(x,~l,t)~~(x,r~,t)F(x,rZ,t)F(x,rl,t) (9) It is assumed now that cl(x,rl,t) = P(x,t)h(rl) c2(x,r2,t) = P(x,t)h(r2) and then C(x,rl,r2,t) = P2(~,t)h(rl)h(r2)F(x,rl,t)F(x,r2,t) (10) The simplest collision model will assume that h(r) is independent of the size coordinate r. Going one step further, one may argue that since larger entities will sweep out a larger volume than their smaller counterparts, h(r) will be proportional to the size coordinate r. In order to obtain a closed set of moment equations it will be assumed that the product h(rl)h(r2) is proportional to the average value of the population internal coordinate P to a small integer power (1. for a cylinder, 2. for a sphere), as will be established in a later section. Thus we set a(x,t) = O2(x,t)h'(P) (11) Note also that according to eq 8 an entity of size r is formed by coalescence of two entities of sizes ( r / K )- p and p. However, entities of size r are removed from the system through coalescence with entities of any size. Then H(x,r,t) takes the form H(x,r,t) =
apn
- - ng (r"-Q(r)) =
Jm rnH(x,r,t) dr (7) at 0 where in shorthand notation pn = (r") = S t rnF dr and n = 0, 1, 2, ... . The leading moments, which in general are functions of time, and the external coordinates will provide us with an approximate description of the reactive system, since they are directly related to physical properties of the system, as we will show later. We now turn to the problem of defining an appropriate expression for the function H(x,r,t). We will assume that if p1 and p2 are the internal (size) coordinates of the two coalescing entities, then the size coordinate r of the resulting entity is given by r = K(pl + p2) (8) where K is a constant. The questions on the validity of this assumption for our system, the choice of K in each case,and the accuracy with which the evolving new entity can be characterized for a period of time after the coalescence are addressed in Appendix A. Coalescence of entities will be described in terms of a basic coalescence frequency C(x,rl,r2,t). It will be assumed that only two entities coalescence at a time. As Valentas
(12)
where the factor 'I2 is necessary in order not to count twice the coalescences of entities of sizes (r/K) - p and p. Using eq 12, eq 7 becomes
where the dependence of F(r) on x and t is implicitly assumed. Consider now each integral term of eq 13 separately, letting
a(x,t)roJmFF(r) dr = a ( x , t ) w g With the change of variables r' = ( r / K )- p and p' = p, the
Ind. Eng. Chem. Fundam.. Vol. 21, No. 1, 1982 3
second integral of the right-hand side of eq 13 becomes
Table I property
relation
total length
Micropores pPLp QfmP(r)dr = Q c l 0
total surface area
ppSw
total volume
p p V p nPlmraP(r)dr = n Q p 2
total number
Bubbles ppNb f m B ( R )d~ = mo
total length
p&b
2JmRB(R)dR = 2m,
total surface area
Ppsgb
4nJmR2B(R) d R = 4nm2
total volume
ppVb (4n/3)jmR3B(R)dR = (4n/3)m3
0
2na/"rP(r) dr = 2nQp1 0
0
= /2Kn a(x ,t)jmF(r'). 0
(n)r'i [ p 'n-iF(p I ) dp '
1=ot
dr'
0
0
0
Hence eq 10 yields
0
where @ 1 1. In the next section we will see that time can be replaced as the independent variable by the extent of reaction 5, so the t tf is equivalent to 5 1. Application to Gasification with Chemical Reaction Control The theoretical model developed in the previous section may now be applied to the gasification of coal chars under chemical reaction control. The pore structure of these chars can be described as follows: (i) The macropores consist of large spherical vesicles interconnected by narrower neck pores. These vesicles are generally several microns in diameter. The macropores may be thought of as the main transport arteries, making the internal surface area of char particles accessible to reacting gas molecules and they usually make up for most of the pore volume of the particles. (ii) The micropores, generally less than a few tens A in diameter, start from the walls of the macropores and extend between them. The micropores account for almost all of the measured internal surface area of the char particles. Because of the lack of published experimental data which would permit a complete characterization of the macropore system, the neck pore volume and surface area is lumped with that of the spherical vesicles or bubbles. Therefore, only two populations are considered (1)the spherical bubbles with a number density distribution B(R,t) and (2) the cylindrical micropores with a number density distribution P(r,t). In order to keep the model as simple as possible, it is assumed that each entity can be characterized by a single internal coordinate. It is assumed that the micropores have a uniform length E. This need not correspond directly to the actual pore length, but it is an average equivalent length defined by the pl moment. The moments of the density distribution functions for the two populations are directly related to the physical p r o p erties of the solid (which are measurable) according to the relations presented in Table I. An assumption of cylindrical shape for the micropores leads to the approximate expression
-
Q u a t i o n 14 is the general form of moment equations appropriate to the reactive systems under consideration. In the special case of chemical reaction control, where temperature and reactant concentration are uniform throughout the system, the momenta pn are functions of t alone. Then eq 14 becomes a system of ordinary differential equations in the p J t )
dCcn- ngJmP-i@(r)F(r)dr = dt
o
Ekplicit forms for a(t) for the system considered here will be obtained later. However, one can obtain a priori knowledge of the asymptotic behavior of a(t) by considering the equation for the leading moment p,,, which denotes the total number of entities per unit volume. This equation reads (16) Define now the reduced time
s = Jta(t? dt' and then eq 16 becomes
the solution of which is
With 2 - K > 0 (see Appendix A), we note that for long times the total number of entities moves inversely as the reduced time s. For the reactive system under consideration, the number of entities should go to zero as the reaction proceeds to completion. If the reaction is completed at some time t = tf and since &(tf) 0 we note that s ( t f ) = .fta(t )' dt'must go to i n f i n i t y in accordance with eq 15. Therefore the function a(t) must have a specific form, if the model is to predict the correct asymptotic behavior of the reactive system as t tfi The simplest form that a(t) may have in order to satisfy this condition is
-
-
-
which eliminates the generally not measured property L,. All physical properties in Table I are per unit volume of the char particle. Note that 4 i r gives ~ the approximate surface area of bubbles, since it does not account for the openings of bubble-pore intersections. The error will be negligible, however, since the contribution of bubbles to the total internal surface area is generally small. The reaction to be considered is the gasification of carbon with
4
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
-
carbon dioxide which may be represented as A(gas) + B(so1id) 2C(gas) where B stands for the reactive solid. If fi is the rate of reaction per unit carbon area, then the differential equation describing the evolution in time of the internal coordinate r of a bubble or a micropore becomes
According to the nomenclature introduced in the previous section
It is now assumed that pore-pore coalescence takes place when a pore of radius p1 touches a pore of radius pz, regardless o€the orientation o€the cylinder, since, no matter what the orientation is, coalescence results in loss of internal pore surface area. The equivalent radius of the new cylinder is given by r = lh + r ~ ) (244 where l is of course a lumped constant and, as it is shown in Appendix A, may vary between 0.5 and 1.0. Using the same argument as in the previous kind of Coalescence, it is deduced that 1 u , ( [ ( t ) ) = 167rtEg-
*(f)
+(r) = 1 and g(c,O) is the same for both bubbles and micropores. In general R remains constant throughout the reaction (see Wicke, 1955) so that g is n o t a function of time. It is also shown in Appendix B that R is uniform throughout the char particle, since the gasification takes place in the chemical reaction control regime. The extent of reaction is now defined as CB
i=1--
CB"
Then
or
and by use of eq 19 we may eliminate t from the moment equations and use E as the independent variable. Due to the fact that the system contains two distinct sets of entities (bubbles and micropores) there are three possible outcomes of the coalescence process. An entity may coalesce with a member of its own set or a bubble may coalesce with a micropore. This last form of coalescence will couple the moment equations for the two populations. Since, however, the radius of a bubble is much larger than that of a pore, it is reasonable to assume that a bubble-pore coalescence results only in the destruction of the pore, since the bubble will not enlarge appreciably. In order to cope with the problem of the orientation of the pore with respect to the bubble, it is assumed that a bubble-pore coalescence occurs when the geometric center of the pore moves inside the expanding bubble. In this way we hope to average the continuous loss of pore surface area due to expanding neighboring bubbles. However, the contribution of this form of coalescence to the destruction of pores is not large, since it is proportional to the ratio of the bubble over the pore surface area. By simple geometric arguments, we deduce that ,
where t and E are the average pore radius and length at time t. Bubble-bubble coalescence presents no problem of orientation and a simple geometric argument leads to ab(t(t)) = 32rR'g-
*(E)
(26)
where is the average bubble radius. The equivalent radius of the bubble formed by coalescence of two others is again given by R K(R1+ Rz) (24~ with K a lumped constant parameter. Further details are given in Appendix C. According to our assumptions, the differential equations for the momenta of the bubble population are given directly by eq 15. The moment equation for the pores will be coupled to the first moment of the bubble population, since in that case
where abp = - = 47rg-
1
R2 *(E) Now using the fact that 4 ( r ) = 1 in this case
(28)
Using eq 22, we obtain the set of moment equations with
E as the independent variable. These equations
(29-35)
are given in Table 11. We introduce now dimensionless moments mi ri = mio Pi
Yi
Then where subscript bp refers to bubblepore coalescence and R is the radius of the bubble with which the pore is coalescing. Equation 23 results from considering the probability that a pore will have its center in the volume "added" to the bubble between t and t + d t and hence collide with it in this time interval. The function $(E) will be a strong function of [ and its presence is necessitated by eq 11 and 18.
1
=Pio
i = 0, 1, 2, 3
(36)
i = 0, 1, 2
(37)
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982 5
Table 11. Moment Equations
dm f ( m z , ~ l=cgm, + ab(KZ- l)mom, dm f(m2,pl)---?= 2gm, t ab(K3 - l)m,m, + d€ abK3mlz dm f ( m 2 . p 1 e 3 =3gm2 + a b ( ~ 4 -l)m,m, t dC 3abK4mlmz where f(mz,rl)=
PB
g(4nm2 t 2nQr1)
PP
Table 111. Dimensionless Moment Equations
with the extent of gasification. Our main objective will be to predict the evolution of the internal surface area and consequently the net rate of gasification, for which experimental results are available. Equations 41 and 45 can also be used to approximate the evolution of micropore and macropore volume, respectively. It should be pointed out, however, that strong assumptions about the coalescence of pores were introduced in order to close the set of moment equations. These assumptions seem to restrict the validity of eq 41 and 45. Coalescence of entities belonging to the same set (cylinders or spheres) will not change the corresponding void volume. Similarly a collision between a spherical bubble and a cylindrical micropore will not produce a change in the total void volume (micropore plus macropore porosity). The physical constraint that the volume must be conserved in the coalescence process has been applied by Simons (1979) in his model derivation and is amenable to the mathematical description of our model. Let HpT(x,r,t),the net rate of micropore generation, be split into the sum of Hp(x,r,t), the rate of micropore generation by cylinder-cylinder coalescences, plus Hpb(x,r,R,t), the rate of micropore "generation", (actually destruction) by cylinder-sphere coalescences. Similarly, let for the bubble population HbT(X8,t) Hb(x&T) + Hbp(X&r,t) Therefore the conservation of volume implies that r2Xmr2HP(x,r,t) dr = 0
and
The last expression after substituting into it the previous two, gives 4rsmRSHbp(x,R,r,t) 3 0 dR + raSmr2Hpb(x,R,r,t) 0 dr = 0
By dividing eq 29-35 by g and letting up* = a /g, ab* = ab/g, ab,* = abp/g, we obtain the equations for the dimensionless moments (36-45), which are given in Table 111. The nonlinear system of differential equations (39)-(45) may now be solved numerically. The coefficients of the equations can be computed from initial values of the physical properties of the char particles, once the relation between average radius and momenta are substituted into up*, ubp* and ab*. Note that if it is assumed that the average length C of the pores remains constant throughout the gasification, then this parameter does not appear in the coefficients of equations. This assumption is made since the length of the pores is not usually measured directly. The moment equations (39)-(45) will be used to predict the evolution of the physical properties of the char particles
The above relations indicate that the sum of the second and third terms of the right-hand side of eq 41 should be zero. The constraint is only approximately satisfied by our model. The numerical computations have shown that the relative error is small for small values of 5, but it increases to about 20% for values of 5 larger than 0.8. The same is also true for the corresponding terms of eq 45. Also, a collision between a bubble and a micropore was described as resulting in the disappearance of the micropore without an increase of the radius and consequently the volume of the bubble. The coefficient of the term describing this collision process was given as
where X = s g b o / s and this ratio has very small values for all chars consicfLred. Consequently, the predicted effect of bubble-micropore coalescence is very small except for the highest values of 5. Obviously, the error introduced by the above approximation will be negligible for small initial microporosities of the particles. In the derivation of our model, the collisions between entities of the same set were characterized by the generation of a new member of the set, whose size was averaged for all collisions by the two parameters t and K (for the cylinders and spheres, respectively). These parameters
6
Ind. Eng. Chem. Fundam., Vd. 21, No. 1, 1982
I 1.2
HYDRANE CHAR
1
150
I w 2.4
B
v \
IGT CHAR # IS5
I - -
C.075
8:IS
,934.C
a\ I 0
I 02
0 4
06
0.8
10
E - CONVERSION
Figure 1. Gaeification rate VB.conversion curves for Hydraue char No. 150 in COz (B = 1.5, f = 0.7): 0,experimental data; -, present model predictions.
O4I
i 0
0.2
0.4
06
E
0.8
3
Figure 2. Gasification rate w. conversion curves for IGT char No. HT155 in COz (B = 1.5, f = 0.75): 0 ,experimentaldata; -, present model predictions.
should be determined experimentally and their effect on the model predictions is presented in the next section. The previously discussed constraints for the conservation of pore volume during the coalescence process may be used to modify eq 41 and 45. The succea9 of any model, however, should be judged by comparing the model predictions to experimental data giving the total void volume as a function of the extent of gasification by a relation of the form
v, + v,
= (V,"
+ V,O) + [V,O
where VBois the initial volume occupied by the reacting solid.
Results and Discussion In this section the gasification rates predicted by the present model are compared to the experimental data published by Dutta et aL (1977a) for various chars obtained from different devolatilization processes. The influence of the model parameters on the predicted values is also investigated. In the above-mentioned experiments the various chars and their parent coals were gasified by C02 at temperatures of 840-1100 "C. The gasification process was divided into two distinct stages. The first one is a pyrolysis stage, during which the chars rapidly release the remaining volatile matter. The second stage is a slow reaction between the reactive solid and COP The first stage was completed after about 1.5 min, while the reaction stage was much longer. Typical computed times for 0.5 conversion (tOb)for that stage varied between 50 and 80 min at temperatures about 900 "C for the different chars. The physical properties of the chars after the completion of the pyrolysis stage were measured and presented in the same study (Dutta et al., 1977a) along with the micropore size distribution. These data were used in the present study to estimate the initial values of the moments, and the predictions of the model were compared to the experimental data for the second stage of gasification. The neceesary micropore specific volumes were estimated from the pore size distributions, assuming that the micropores have a cylindrical shape. The numerical resulta for the gasification rate are compared with the experimental data in Figures 1to 4 for three different chars at various temperatures. The three chars have entirely different pore structures. The IGT char No. 155 and the Hydrane char No. 49 come from Illinois No.
l E - CONVERSION
Figure 3. Gasification rate vs. conversion curves-higher temperatures: -, 0 , predicted and experimental values for IGT char No. HT155; - - -,A, predicted and experimental values for Hydrane char No. 150.
C -CONVERSION
Figure 4. Gasification rate vs. conversion curves for Hydrane char No. 49 in COP(B = 1.2, f = 0.5): 0,experimental data; -, present model Predictions.
6 coaL They both have very large values of internal surface area (measured by BET nitrogen adsorption) and, as ex-
Ind. Eng. Chem. Fundam., Vol. 21,
pected, most of it is contributed by pores smaller than 15 A in radius. On the other hand, the Hydrane char No. 150 comes from Pittsburgh HVab coal and has a more open pore structure. It gives a small value of BET surface area and has no pores with radius smaller than 15 A. The general behavior of the computed rate of gasification agrees with physical reasoning and other data from the literature (see Kawahta and Walker (1962) for a study on the gasification of anthracite). Initially the internal surface area increases, as the pores enlarge due to the chemical reaction, and the gasification rate increases also. As the reaction proceeds, however, more and more pores will coalesce with neighboring ones. Soon the coalescence of pores becomes the dominant factor, driving down the total internal surface area and consequently decreasing the rate of gasification. Therefore the evolution of internal surface area (and hence of the gasification rate) may be described in terms of two competing processes: that of pore growth and that of pore coalescence. Since most of the surface area for all chars lies in pores less than 50 A in diameter, the evolution of the micropore surface area will be the dominant factor here. Consider now eq 40 for the second moment of micropore population, which reads
Table IV. Surface Areas and Micropore Volumes of Chars BETsurface Vkl, V P ~ ~VPT, , devolatilized samples area, m2/g cm3/g cm /g cm3/g IGT No. HT 155 Synthane No. 122 Hydrane No. 49 Hydrane No. 150 Pittsburgh HVab Coal
0
The first term of the right-hand side above describes the growth process, while the other two terms model the coalescence process. The coefficient of the growth term is inversely proportional to the product of the true density of the solid and the micropore volume. Therefore it is expected that Hydrane char 150, which has a lower density and lower micropore volume than the IGT char, will exhibit a more rapid growth of micropore area in the initial stages of gasification than the IGT char. This is indeed the trend exhibited by the experimental data, as we will see now. In Figure 1 the gasification rates for the Hydrane char No. 150 are plotted as the conversion for the two lowest temperatures. The experimental data show a maximum rate a t about 5 = 0.4. The computed curves approximate the experimental data very closely. The parameter values were the same for both temperatures. The IGT char exhibits a different behavior. The maxima are less pronounced and occur earlier during the reaction. Again there is good agreement between computed and experimental data for the same value of parameters at both temperatures (see Figure 2). In Figure 3 results for the two chars are shown for higher temperatures. The agreement between computed and experimental data is not so good as in the previous cases, indicating that intraparticle diffusional resistance may have become important. In Figure 4 the corresponding results are plotted for the Hydrane char No. 49. This char exhibits similar behavior to that of the IGT No. 155. Numerical simulations were not performed for the Pittsburg HVab devolatilized coal, since mercury porosimetry data were not available. This char exhibits the more pronounced maxima at lower reaction temperatures. This fact is in agreement with the previous observation that the chars with the smaller micropore volume are more likely to exhibit pronounced maxima in their gasification
No. 1, 1982 7
424 28 1 172 18 16
02
0.098 0.052 0.032 0 0
04 06 08 E CONVERSION
0.075 0.094 0.067 0.032 0.009
0.173 0.146 0.100 0.032 0.009
IO
Figure 5. Comparison of present model predictions to Dutta et al. (1977a) empirical fit. IGT No. HT155: 0,experimental data; -, present model prediction; - - -,empirical fit. Hydrane No. 150 A, experimental data; -, present model predictions; - - -,empirical fit.
rate vs. conversion curves. In Table IV the micropore volumes of all chars are given, as estimated from their cumulative surface area as pore radius curves, under the assumption that the pores are cylindrical in shape. Here V p , is the volume associated with micropores of radius less than about 15 A, V p is the volume associated with pores of radius in the 15 to 50 A range, and V p T the total microscope volume. A different assumption for the micropore shape will introduce another geometric factor in the computation of volumes, but the relative order of the micropore volume values for the various chars will remain the same. The model predictions based on the micropore volume values given in Table 1V approximate well the experimentally observed rates, indicating that the micropore volume is a significant factor in determining the shape of the gasification rate vs. conversion curve for any char considered. In Figure 5 the present model is compared to the empirical equation proposed by Dutta and his co-workers for the evolution of the relative available specific area per unit weight of solid. Both models approximate well the experimental data for conversions larger than 0.2. The present model predicts the existence of a maximum rate at an intermediate value of the conversion, while the empirical fit predicts a continuous decrease of the rate of gasification. The empirical fit model fails to predict the drastic decrease of surface area, as the reaction proceeds to near completion. The evolution of internal surface area predicted by the present model is presented in Figure 6. The two chars considered are the IGT and the Hydrane char No. 150, for which almost identical values of the adjustable parameters { and /3 gave a good fit to the experimental data. Hence the results plotted show only the effects of the measured physical properties of the two chars. The relative specific
1
8
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
7.0
8
1
p
me
31
X
I
w
.IO
l
'\ \
'I
\
1
It I
O
\ \ \
L 01
' 02
1
'
04
03
'
1
06
05
07
08
E -CONVERSION
Figure 8. Effect of values of parameter 0 on gasification rates of Hydrane char No. 150.
I
0
02
04
E
06 08 CONVERSION
10
Figure 6. Evolution of A relative specific surface area per unit weight of solid available to reaction vs. conversion and B: total relative surface area available for reaction: -, Hydrane char No. 150; - - -,IGT char No. HT155.
/I
lot K.085
r
i.08
1
;.a9
K:080
\
I'
I
RELATIVE HICROPORE
2; I
i lr O'
dl
02
d3
0'4
05
06
E -CONVERSION
d7
08
'
Figure 7. Effect of values of parameter j- on gasification rates of Hydrane char No. 150.
area per unit weight of solid for the Hydrane char passes through a maximum a t 5 = 0.8. The IGT char exhibits a much lower maximum a t 5 = 0.7. The total internal surface area available for reaction passes through a maximum at a much lower conversion and decreases fast afterwards, as the solid is consumed by the chemical reaction. In Figures 7 and 8 the influence of the adjustable parameters l and @ on the predicted gasification rates for Hydrane char No. 150 is illustrated. From the model equations it is evident that large values of Z (always less than 1)will tend to diminish the significance of the coalescence process. Then the growth process dominates for a longer period of time, with the result that the internal surface area reaches a higher maximum value than in the case of small {values. This maximum also is attached at a later time in the gasification process. However, the differences between gasification rates for different f are generally small for small conversions (less than 15% up to .$ = 0.4 for the case of Figure 7). The introduction of the function Jl(.$) may be thought of as an approximate quantification of the fact that the probability that two entities will coalesce increases rapidly as the gasification proceeds. Toward the completion of the reaction the rate at which remaining entities coalesce
I
+
0
I
1
I
01
02
03
04
E
05
06
07
08
CONVERSION
Figure 9. Predicted effect of parameter drane char No. 150.
I(
on the porosity of Hy-
becomes very large, ultimately driving their number toward zero. The justification for the introduction of $(E) and of its form was presented in eq 18 and 19. Figure 8 shows that the error in the gasification rate for @41,2]is between 3 5 % for 5 C 0.4, but then it increases significantly. In short, one may say that { is a major factor in determining the internal surface area at low conversions. However, @ becomes the dominant factor at high conversions. The influence of the bubble combination constant K on the gasification rate is negligible for all chars and under all conditions considered. However, K is the dominant factor in determining the bubble volume predicted and therefore the porosity. Unfortunately, no data for the evolution of porosity (differentiating between micro- and macroporosity) are given in Dutta's study. The effect of K on the predicted porosity is illustrated in Figure 9. Low values of K underestimate severely the radius of bubbles formed by coalescence. Hence, the porosity will pass through a maximum and then decrease, following closely the corresponding curve for the micropore relative volume, also given in Figure 9. On the other hand, high values of K will overestimate bubble pore volume, resulting in
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
0
0
02
I
. HYORANE CHAR#150
04 06 08 CONVERSION
9
IGT C H A R X I 5 5
IO
E
Figure 10. Gasification rate vs. conversion curves for Hydrane No. 150 char in O2 = 0.7, @ = 1.5): 0,experimental data; -, present model predictions.
(r
predicted porosities larger than 1. The model predicts, also,that most of the bubble coalescences occur during the late stages of the reaction, and, as a consequence of that, the porosity increases sharply toward the end. It is important to notice that this incorrect behavior (porosity greater than one, existence of maximum, etc.) is predicted for conversion values larger than 0.7 where the validity of the model is doubtful (Gavalas, 1980) as the particle starts to break into fragments (Dutta et al., 1977a). Therefore the fact that our model does not strictly satisfy the principle of conservation of volume is of no apparent consequence and the proposed correction is unwarranted. Dutta and Wen (1977b) have also measured the rates of gasification of the same chars in an oxygen-nitrogen atmosphere. The chars were previously devolatilized in a nitrogen atmosphere. For the C02gasification experimenta the chars had been devolatilized under the same conditions but in a C02 atmosphere, to determine their pore characteristics and densities. Dutta et al. state that the solids undergoing reaction with O2 are likely to have almost identical physical characteristics with the COz gasified chars. However, Street et al. (1969) in their microscopic study of heat treated coal particles both in air and nitrogen have noticed significant differences of the particle pore structure in the two cases. The particles treated in a nitrogen atmosphere had a more open internal structure with larger pores than the air-treated ones. This raises the possibility that the devolatilization atmosphere (reactive or inert) may indeed affect the internal structure and consequently the pore characteristics of the chars. The experimental data for Hydrane char 150 are compared with model predictions in Figure 10. The same initial values of the physical properties and the same values of { and 0 were used for these simulations. The agreement between predicted and experimental values is very good, indicating that the internal area evolves in a similar way as in the case of C02 gasification. However, the same values of physical properties and parameters gave poor agreement for the other two chars (IGT No. 155 and Hydrane No. 49). These chars were found to have very large internal surface areas when analyzed after devolatilization in a C02 atmosphere. Also these chars come from a different parent coal (Illinois No. 6 as opposed to Pittsburgh HVab for Hydrane char No. 150). The rate vs. conversion curves for oxidation of these two chars exhibit a different behavior than the corresponding curves for gasification with C02. They pass through sig-
E CONVERSION
Figure 11. Gasification rate vs. conversion curves for IGT char No. HT155 in O2 = 0.75,fl = 1.5): 0, experimental data; -, model predictions for lower values of surface area; - - -,model prediction for same values of surface area and micropore volume as in the C02case.
(r
nificantly higher maxima and the maxima are also shifted toward higher conversions. Since the oxidation reaction is highly exothermic we investigated the possibility that the increase of rate may be due to the fact that the temperature of the particles rises above the ambient one; this difference increases as the reaction rate (and correspondingly the heat generation) increases. Our numerical simulations have shown that the maximum temperature difference between gas and reacting solid is generally less than 3 K. This small difference cannot explain the large increase of the gasification rates observed and this is in agreement with the estimates presented by Dutta and Wen (197713). If one assumes that the different environments during the devolatilization process prior to reaction significantly affects the pore characteristics, then it would be expected that the chars devolatilized in N2 would have a more open internal structure. This is in accordance with the qualitative observations of Street et al. (1969). Then the internal surface area and consequently the micropore volume of the chars would be smaller than in the C02case. Gasification rates predicted by the model are in good agreement with experimental values, if lower values for pore surface and micropore volume are used, with the rest of the parameters having the same values as in the C02 case. Such results are shown in Figure 11for the IGT char No. 155 for two different temperatures. The value of the pore area used was 80 m2/g and that of micropore volume 0.05 cm3/g (instead of the 424 m2/g and 0.17 cm3/g, respectively, measured after devolatilization in a C02 atmosphere). Similar results have been obtained for the Hydrane No. 49 char. The possible catalytic effects of inorganic impurities on the gasification reactions of chars have also received considerable attention (see Walker et al., 1968; Hippo and Walker, 1975; Tomita et al., 1977; Mahajan and Walker, 1979). It is believed that Na, Ca, and Fe may catalyze the reac$ion of carbon with C 0 2 and O2under certain conditions. Of these, Ca a d as catalyst for both reactions. Also, Fe is a good catalyst for carbon gasification but its oxides are not. It is expected, however, that Fe would be oxidized to Fe304in the presence of COPin the temperature range 880-1OOo "C considered here and in the presence of oxygen (Walker et al., 1968). One way that the catalytic activity of these impurities may vary with carbon burn-off is if the dispersion of the
10
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
catalysts changes as the reaction proceeds. Then the fact that gasification by C02 is carried out at temperatures much higher than those for the O2 reaction can be significant, since higher temperatures facilitate sintering of the catalysts and a corresponding decrease of char reactivity. Most of the inorganic impurities present as discrete minerals, however, have a low initial dispersion and the degree of dispersion will not change significantly during the gasification process. If the inorganic impurities are responsible for the differences observed under the two reactive atmospheres, they must be highly dispersed initially within the carbon matrix (see Tomita et al., 1977), but this fact has not been established experimentally. It must be noted also that Tomita et al. (1977) have observed significant effects of demineralization treatment on the reactivity of some chars. The reaction rate vs. conversion curves for the original and the demineralized chars exhibit different characteristics. There is some similarity between this observation and the differences in reactivity of the IGT and Hydrane No. 49, when they are gasified by C02 or OF However, the demineralization process profoundly affects the N2 internal surface area of the chars, so that the demineralized char has only about l / l o o the surface area of the original char. Our model can predict the correct change in the evolution of reactivity, when such a drastic change of the value of this internal surface area takes place. We have seen that in the case of the IGT char (Figure 11). We are currently investigating different probabilistic models that will be able to describe more accurately the coalescence of pores. Preliminary results indicate that the values of micropore surface area and micropore volume are very important parameters in describing the evolution of reaction rates. The trends exhibited are the same as those predicted by the present model. SmaJl values of micropore volume result in gasification rate vs. conversion curves with pronounced relative maxima occurring at high conversions, while large values of surface area and micropore volume have the opposite effect. In conclusion, the model succeeds with a minimum number of parameters to approximate well the evolution of physical properties of chars with quite different pore structures, when the initial values of those properties are known. One set of parameter values for each char gave satisfactory results for the whole range of temperatures considered. Considering the fact that the description of the coalescence process was rather simplistic, the performance of the model has been encouraging. The fact that the reactivity of some chars evolves in a different way in the two reacting environments may be explained by a significant change in the internal structures of chars, caused by the different devolatilization atmosphere in the two cases. These discrepancies may also be due to the catalytic effects of inorganic impurities, if their dispersion changes dramatically during the reaction or they are deactivated for some chemical reason. The answer to this question has to come from experiments. Appendix A In the model developed above we have assumed that the entity resulting from the coalescence of two others may be approximated as an entity of the same geometric shape (sphere or cylinder, respectively). Of course the actual situation is much more complicated since this idealized picture is certainly not valid. In this section we will consider the simple problem of the coalescence of two entities and address the problem of how accurately the assumption of an equivalent size coordinate (as defined in eq 24a and
Table V. Combination Constants K and 5 P K (spheres) 5 (cylinders) 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0
0.630 0.693 0.759 0.804 0.836 0.858 0.889 0.909
___
0.707 0.745 0.791 0.825 0.850 0.869 0.896 0.914
TRUE VALUES
APPROXIMATION
K.O 75 ICORRESPONDING TO p
i
31 /
TIME (ARBITRARY UNITS)
0
Figure 12. True and equivalent-sphere model volumes for twosphere vesicles: -, true values; - - -,approximation.
24b) approximates the evolving true new entity. Consider first two spheres. At the moment they touch each other their combined surface area and volume are given by 4a V, = -(R13 3
+ R23)
Let R2 = pR1, and define the radius of the equivalent new sphere as
Re, = K(R1 + R2) Let us now match the volumes of the equivalent and actual spheres at the moment of coalescence 4a 4?r -Req3 = -(R13 3
3
+ R23)
or
K=
(1
+ p3)1/3
l + P
(A-1)
We chose to match the volumes because this gives the best approximation of both surface area and volume of the true two-sphere vesicle for a period of time after the collision. The value of the combination constant K is a function of the ratio of radii, p . Values of K for various p are given in Table V. Since K varies only between 0.63 and 0.90, one may argue that an average value of K will give reasonably good approximations of the evolving surface area and volume at the two-sphere vesicle. Some results are presented in Figures 12 and 13, where the equivalent sphere surface area and volume are plotted and compared to true values. The agreement is reasonably good and generally much better for the surface area.
Ind. Eng. chem.Fundam., Vol. 21, No. 1, 1982 11 -
2.0
Table VI.
- TRUEVbUIES __-- APPROXIMATKN
-
R,' rm
T,"C
@
100 200 500 100 200 500 100 200 500
900 900 900 1000 1000 1000 1100 1100 1100
1.09 x 2.16 X 5.41 X 4.17 X 8.34 x 0.208 0.132 0.263 0.658
lo-' lo-' lo-' 10-a 10-
w 1 ( = 0 7 5 ( p = 31 MbXIMUYERRORSt I % .-14%
I
I
0
TIME ( ARSITMRY UNITS)
Figtam 13. "me and equivalent-sphere model surface areas for two-sphere vesicle: -, true values; - - -,approximatiod.
The problem of pore-pore coalescence is much more complicated because of the question of the orientation of the axes of the two pores. If it is assumed that the axes are parallel, then a similar analmay be performed and the combination constant 1d e f i e d as
Values of 1are given in Table V for various values of
p.
Again 1liea in a rather narrow region between 0.7 and 0.91 for pe[l.O, 10.01. An average value of {approximates the
evolving surface area of the two-cylinder pore with about the same accuracy as the average value of K did in the previous case. The value of this simple analysis is that it pointa out the physically feasible range in which these two parameters may lie. This becomes particularly useful in predicting values of K and 1, when one considers that porosimetry data suggeat that most of the micropores and bubbles have radii in small ranges, usually within a factor 2 or 3 of their average values. Appendix B In this section the assumption that the gasification of char particles with C02proceeds under chemical reaction control is investigated. Wicke (1955) and Rossberg and Wicke (1956) have investigated the reaction of carbon with C02for various typea of carbon of widely different porosity. Assuming a first-order reaction, they give an activation energy value of 84 f 3 kcal/g-mol. The frequency factor depends on the type of carbon gasified. Since our goal here is to estimate the reaction regime in the worst case we will consider the higher of those values so that
k
= 2.6
X
lo7 exp(-84000/RT)
m/s
(B-1)
For a firsborder reaction, the rate of gasification is given by i = PPJcg [b-mol/(ms)(s)~ 03-2) Then the Thiele modulus for a spherical particle is defined as
0
Poeitiom of spheres (xl,Rl)and (x&J which coalesce in (t, t + at).
Figure 14.
The char particlea used in Dutta's study are between 250 and 500 pm in diameter. The reaction temperatures we will consider are in the range 900 to lo00 "C. Rough estimates of the effective diffusion coefficient may be obtained from the random pore theory. Under these assumptions values of 4 for the IGT char, which has the b i g h t internal surface area of all char considered, are given in Table VI. We conclude that the effectiveness factor for the small particles considered here is very close to 1 for the temperature range of interest.
Appendix C 1. Bubble Coalescence. Consider at time t a sphere ( x 2 ,R2). The radius of the sphere grows due to the chemical reaction at a rate a
2
= g(x,t) dt This sphere will collide with another sphere (xl,R,) at time t + dt (or earlier) if < R1+ R2 + 2g dt R 1 + R2 < 11x2 The solution is 'depicted in Figure 14. The probability that a bubble of radius R1 will have ita center in the spherical shell is
Br(R1 + R2)2g d t B(RJ dR1 The total number of collisions between spheres of radii R1 and R2 will be
Br(R1 + Rd2.g dt B(RJ dR1 B(R2) d R 2 Thus the required collision frequency will be
Cb(RitR2) = 8 d R i +
B2(t)gB(Ri)B(RJ
and the function B2(t) must be introduced to predict the correct asymptotic behavior as discussed previously. Now setting R1 R2 = 2l?, where R is the average radius at time t
+
where R, is the radius of the char particles.
ab(t)
32rR2 B2(t)g
12
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
2. Cylindrical Pores. The collisions between cylindrical pores is a more complicated question since now the directions of the axes of the cylinders come into question. It will be considered that a collision takes place when a cylinder of radius p1 “touches” a cylinder of radius p 2 , regardless of the orientation of the cylinders, since, no matter what the orientation may be, a collision results in “loss” of internal surface area and pore volume. The analysis for the bubbles carries through, if we change the expression for the volume of the now cylindrical shell of thickness 2g dt and length 2C. Hence the required collision frequency is CP(P1,P2) = 4 7 h l
+ p2)C(2g)P2(t)P(p1)P(p2)
If as before p1 + p 2 = 2p, where p is again the average radius at time t
Greek Letters
a p ( t )= 16rpP2(t)g
P = coalescence constant defined by $([) = (1- [)@
3. Bubble-Pore Coalescence. We will consider that a bubble collides with a pore when the geometrical center of the pore moves inside the expanding bubble. Since the difference in radius between a bubble and a pore is very large, it will be assumed that at the moment of collision only the pore disappears, while the radius of the bubble remains practically unchanged. Therefore collisions of this kind will contribute only to the destruction function for the cylindrical pores (macro- or micropores). The coalescence frequency will be found using these assumptions. Consider at time t a sphere of radius R. The number of pores of radius between r + dr in the spherical shell 4?rR2g dt will be 47rR2g dt P(r) dr
The total number of collisions between spheres of radius R and pores of radius r is 4rR2 g dt P(r)dr B(R)
h(r) = function defined in eq 10 C = average length of micropores, m L = total length of pore system, m m .= jth moment of density distribution B dW, = molecular weight of reactive solid N = total numer of entities P(r,t) = number density distribution functions for micropores r = characteristic length of entity, micropore radius, m 5 = radius of spherical bubble, m R = rate of chemical reaction per unit carbon area, kgmol/ (8)(m2) S, = total surface area of pore population per unit weight of solid, m2/kg t = time, s V = total volume of pore population per unit weight of solid, m3/kg x = vector of spatial coordinates
dR
Hence Cbp(R,r)= 47rR2P2(t)gB(R)P(r) and thus ab,(t) = 4.rE72@2(t)g Nomenclature a(x,t) = collision frequency function defined by eq 11 a*(x,t) = function defined as a* = a/g B(R,t) = number density distribmion function for spherical bubbles c(x ,t) = concentration ci(x ,r,t) = Characteristic coalescenee frequency function C(x,r1,r2,t)drl dr2 = rate at which entities with characteristic length sizes in (rl,rl + dr,) and (r2,r2+ dr2) coalesce f(r2,y1)= function defined as f(r2,yl)= gpB(s,bor2 + S@Oyl) P(r2,p1) = function defined as f* = f/g F(x,r,t) = numer density function of entities g(c,B) = function defined in eq 4 G(c,O,r) = local rate of growth of characteristic entity length, g(c,O)+(r) H(x,r,t)ldxldr = rate at which entities with phase coordinates in (x- d x / 2 , x + dx/2, (r - dr/2, r + dr/2) are introduced into the system
B(x,t) = function defined in eq 10 Y,, = dimensionless moment, ~ l ~ / p , , O rn= dimensionless moment, m,,/nno e = porosity [ = combination constant for micropores, defined in eq 24a 0 = temperature, K p,, = nth moment of density distribution F(x,r,t),nth moment of micropore number density distribution P(r,t) = extent of reaction p~ = true density of reacting solid, kg/m3 pp = bulk density of reacting particle, kg/m3 +(r) = function defined in eq 4 $([) = coalescence function defined as $([I = (1- [)B
Sub-
scripts and Superscripts b = spherical bubble population B = reacting solid 0 = initial value p = cylindrical pore p = micropore T = total Literature Cited
Curl, R. L. AICM J . 1963, 9 , 175-181. Dutta. S.; Wen, C. Y.; Beit, I?.J. Ind. Eng. Chem. Process Des. Dev. 1977a, 16, 20-30. Dutta, S.; Wen, C. Y. Ind. Eng. Chem. Process Des. D e v . 1977b, 16, 31-37. Field, M. A. Combust. Flame 1969, 13. 237-252. Gavalas, 0. R. AIChE J . 1980, 26, 577-584. Hashlmoto, K.; Sihreston, P. L. AICM J . 1973, 19, 259-268. Hippo, E.; Walker, P. L., Jr. Fuel 1975, 54m, 245-248. Hulburt, H. M.; Katz, S. Chem. Eng. Sci. 1964. 19, 555-574. Kawahata. N.; Walker, P. L., Jr. “Proceedings. 5th Carbon Conference”; Pergamon Press: London, 1962. Lightman, P.; Street, P. J. Fuel 1968, 47. 7-28. Mahajan, 0. P.; Walker, P. L.. Jr. Fuel 1979, 58, 333-337. Petersen, E. E. A I C M J . 1957, 3 , 443-448. Rossberg, N.; Wlcke. E. Chem. Ing. Tedm. 1956, 28, 181-189. Simons, G. A. Combust. Sei. Ted?m/.1979, 19. 227-235. Smith, I. W. Combust. Fleme 1971. 17, 303-314. Smith, I. W.; Tyler, R. J. Fuel 1972, 51, 312-321. Smith, 1. W.; Tyler, I?.J. Combust. Sci. Technol. 1974, 9 , 87-94. Street, P. J.; Weight, R. P.; Lightman, P. Fuel 1969, 48.343-365. Tomlta. A.; Mahajan, 0. P.; Walker, P. L., Jr. Fuel 1977, 56, 137-144. Valentas. K. J.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1988, 5 , 533-542. Walker, P. L., Jr.; Shelef, N.; Anderson, R. A. “Chemistry and physics of Carbon”; Vol. 4; Marcel Dekker: New York. 1968. Wicke, E. "Proceedings, 5th Symposlum on Combustion”; Reinhold: New York, 1955.
Received for review August 19, 1980 Accepted June 29,1981
