- Email: [email protected]
Microstructural evolution of char under oxidation induced by uneven heating
Microstructural Evolution of Char under Oxidation Induced by Uneven Heating ISAAC I. KANTOROVICH
and EZRA BAR-ZIV*
Department of Mechanical Engineering Ben-Gution University of the Negev, P.O.B. 653, Beer-Sheva (I.I.1y; E.B.-Z) Nuclear Research Center-Negev (E.B.-Z.), P.O.B. 9001, Beer-Sheua, Israel
Single spherical char particles were intentionally irradiated nonuniformly in an electrodynamic chamber, in the temperature range 600-1000 K. The char particles were irradiated from one side (bottom) and consequently heated unevenly. Nonuniform shrinkage of an initially spherical char particle has been observed during oxidation. The features of nonuniform shrinkage are rather peculiar: (1) Up to 40%-60% conversion, the particle shrank uniformly. Then spatial preferential consumption initiated, indicating the threshold nature of the phenomenon. (2) Preferential consumption, in most experiments, was observed to start from the top of the particle. (3) Above the threshold conversion, a disk structure became clear. (4) At very high conversion the particle became like a center-hole doughnut. A model was developed to explain these features by nonuniform transformation of the micropore structure due to oxidation of the unevenly heated particle. The threshold nature of the phenomenon was also explained in terms of the dynamic stability of a particle. The threshold of nonuniform shrinkage corresponds to the transition of the particle to a stable position. After initiation of nonuniform shrinkage, the macroporosity distribution inside the particle becomes nonuniform. Macroporosity was shown to increase near the particle’s center; this eventually creates a hole at high conversion. The good agreement of modeling results with experimental observation confirms the notion that nonuniform shrinkage in regime I (kinetically controlled) is an indication of the fundamental microstructural transformations in the course of oxidation.
NOMENCLATURE b d dll f”
G h
k,,
k,
kr
Lmic Lmic,
1
LO,ic,
1
m no
P
4
defined by Eq. 21, dimensionless. diameter of a particle, m. initial value of d,m. activation energy, J/mol. defined by Eq. 30, dimensionless. defined by Eq. 41, dimensionless. heat exchange coefficient, W/ (K m*>. defined by Eqs. 15 and 16, dimensionless. preexponential factor at the inverse time of half conversion [2], s-l. mean length of microcrystals, m. mean length of large microcrystals, m. initial mean length of large microcrystals, m. mass of a particle, kg. density of active sites [5], m-*. probability for the crystal edge to be revealed on the surface of a macropore, dimensionless. absorbed radiation flux density, W/(K m*>.
* Corresponding author. OOlO-2180/96/%15.00 SSDI OOlO-2180(95)00168-6
Q ii
Ki
4
Ro
RP
Rmic
RO,ic 4 R zo
R,, R-
coefficient related to absorbed radiation flux density, W/(K m4>. radial spherical coordinate, m. radial cylindrical coordinate, m. radius of the disk, m. gas constant, J/(mol K). radial cylindrical coordinate corresponding to an undistorted particle, m. radius of a particle, m. mean radius of microcrystals, m. initial mean radius of microcrystals, m. radius of a horizontal cross-section of a sphere at coordinate z, m. initial radius of a horizontal crosssection, m. radii of curvature in the top and the bottom of a particle, m. temperature, K. mean temperature of a particle, K. ambient temperature, K. thickness, m. cylindrical coordinate, m. position of the center of charge, m. position of the center of mass, m. cylindrical coordinate corresponding to an undistorted particle, m. COMBUSTIONAND FLAME 105: BO-91(1996) Copyright 0 1996 by The Combustion Institute Published by Elsevier Science Inc.
MICROSTRUCTURE X &I -%
OF CHAR
conversion, dimensionless. conversion before particle distortion, dimensionless. conversion in the center of a particle, dimensionless.
Greek Symbols ratio of the change of the diameter to length of a microcrystal due to its consumption, dimensionless. shrinkage factor, dimensionless. defined by Eq. 39, dimensionless. maximum difference of temperature within the particle, K. macroporosity, dimensionless. initial macroporosity, dimensionless. angle between the radius vector and the vertical axis, dimensionless. angle 8 for the coldest spot or rr - 8 for the hottest spot. effective thermal conductivity of a particle, W/(mK). thermal conductivity of gas, W/(mK). cos 0. apparent density, kg/m3. defined by Eqs. 15 and 16, dimensionless. defined by Eq. 42, dimensionless. INTRODUCTION The physicochemical properties of a highly porous char during oxidation in regime I (kinetically controlled) change in a manner strongly deviating from the traditional behavior. Shrinkage of a gasifying particle is an unexpected phenomenon in regime I [l]. Moreover, during oxidation highly porous chars exhibit unexpected development of (1) apparent density, (2) total internal surface area, and (3) rate of conversion [l, 21. Another example is the severe change of thermal conductivity during conversion [3]. Kantorovich and Bar-Ziv [4, 51 showed that these phenomena are determined by peculiar development of the micropore structure. They proposed a model for the pore structure of highly porous chars which enabled them to account for shrinkage and some other phenomena. Pore structure, being
81 an important factor affecting the reactivity of carbonaceous materials, is of crucial importance for highly porous coals and chars. The conventional way of studying the evolution of the pore structure by surface characterization methods is extremely difficult and not yet developed to enable the understanding and elucidation of these changes. At present, the perspective way that can shed light into the mutual influence of the pore structure and reactivity is to analyze the changes in physicochemical properties, caused by structural changes. As shown before [4, 51, the advantage of this indirect way is the possibility to point out tiny details of pore structure evolution, whose direct measurement is not feasible. The physical properties closely connected to the pore structure are: density, porosity, total surface area, external shape, thermal conductivity, and optical properties. The chemical properties of importance are: active surface area, active sites, and adsorption-desorption. Most of these can be measured during experiments conducted in an electrodynamic chamber (EDC) [61. The main advantages of the EDC are the ability to characterize the particle prior to reaction, to monitor the important quantities needed for understanding kinetics in real time and to study the oxidation of individual particles in well-controlled conditions. The porous nature of a particle under oxidation can reveal itself most noticeably in experiments conducted under nonequilibrium conditions, in the presence of strong temperature and concentration gradients, which can be intentionally imposed on a particle. The results can serve to achieve a more detailed characterization of the internal structure of carbonaceous materials. One example of such an experiment is the uneven heating of a particle, resulting in nonuniform shrinkage of the particle. Nonuniform shrinkage of unevenly heated synthetic char (Spherocarb) particles (diameter 150-200 pm) during oxidation in an electrodynamic chamber in the temperature range 600 to 1100 K has been observed by Weiss and Bar-Ziv [7]. A single char particle was levitated in the electrodynamic balance and heated from its bottom by a focused laser beam. As oxidation was shown to proceed all over the internal surface, the change in shape cannot be at-
82
I. I. KANTOROVICH
tributed to preferential consumption on the overheated external surface exposed to radiation. As shown below, nonuniform shrinkage is explained by nonuniform rearrangements of the microstructure. Figure 1 illustrates the evolution of the shape of an initially spherical particle during oxidation as observed by Weiss and Bar-Ziv [7]. Figures la-le relate to various conversions and Fig. Id, which appears twice, is the disk from perpendicular sides. Rather peculiar features of nonuniform consumption can be seen: (1) nonuniform consumption reveals a threshold nature, initiating at 40%-60% of conversion, (2) preferential consumption in most experiments starts from the particle’s top, despite heating from the bottom, (3) above threshold conversion a particle takes on a disk shape, and (4) at very high conversion a center-holed doughnut is formed. These features are explained below employing a model of the pore structure developed before [4, 51. THEORY Temperature Distribution In these experiments, a particle is heated from the bottom by a laser beam, the light flux
AND E. BAR-ZIV
density has a TEM& mode (doughnut shape) in the particle’s axis to ensure more uniform light absorption. A parabolic distribution of absorbed radiation over the lower hemisphere was assumed. The distribution of temperature T can be described by d2T
-;;;i-+;;+;;[(l-P9$]=o, (1)
d
with the boundary conditions: r = R,; A$
+ h(T - TJ
+ Qf(r, /.d = 0, (2)
r=O;
T
(3)
where
f(r,p) =
0; i r2j&
O
(4)
-lI/.LSO.
- Jo’);
As the Grashoff number is negligibly small for a particle of 200 pm diameter, heat exchange with the surrounding gas is governed by heat conduction; therefore for a spherical particle, h can be represented as h = h,/R,, where h, is the gas thermal conductivity. The solution of Eqs. l-4 can be expressed as a Legendre polynomial series:
I
/
, (a)
d/do=l,
X=0
(5)
(d) d/d,,=0.65, X=0.9
(b) d/d,=O& X=0.6
(d) w/d,,=O.33, X=0.9
(c) d/d,=0.8, X=0.7
(e) d/d,=O.54, X=0.95
Fig. 1. Redrawn shadowgraphs presenting the development of the preferential consumption of a particle. (a-e) relate to various conversion and Cd), which appears twice, is the disk from perpendicular sides.
The second term in the square brackets exceeds the third one by factor of three, the third one by factor of 7, the fourth one by factor of 20, etc. As Eq. 5 is an alternating series, one can limit it to the first three terms. lindrical coordinates z = pr and R = Usin 1 - p2, it is found that the temperature can r s”” be represented as T = T,, -t
(7; - T,,) 1 - 5(h
g
“::),
P
4z(2z2 - 3R2)h, +
5R;(3h + he)
I
’
(6)
MICROSTRUCTURE
OF CHAR
83 to which a cylindrical shape is attributed for simplicity, are considered as randomly distributed and interconnected. According to the “subskeleton” mechanism [4], shrinkage can be explained as the continuous process of break and restoration of the internal joints of the subskeleton containing the largest microcrystals. The relative change of volume (the shrinkage factor) y can be expressed by
(10) Fig. 2. Isotherms within a spherical unevenly heated particle. The temperature interval between consecutive isotherms is 5 K.
where (7) is the mean temperature of the particle. Figure 2 demonstrates isotherms that characterize the temperature distribution over a vertical cross section. The hottest and coldest spots are located on the particle’s surface symmetrically, and the angle for the spherical coordinate 0 for the coldest spot equals:
The theory of the evolution of the porous structure [8] enables these quantities to be represented as a function of conversion, X. The small microcrystals form the fine structure of the micromedium, and can coalesce to large ones, not changing the shape of the subskeleton. Coalescence of the small microcrystals is the reason for the change of the material’s reactivity during conversion. The evolution of the rate of conversion can be expressed 151by:
(l - X)(R,i, + (YL,ic)nak, dx
X exp( -E/R,T)
-= dt
Lank
(8) X
which is 40” for small A,/h. The maximum difference of temperature, AT, within the particle is AT=
(9)
For h,/A = 0.05, AT reaches 60 K for a typical particle temperature of 800 K. Pore Structural Transformation and Shrinkage To estimate the change in the dimensions and shape of a particle caused by such a temperature distribution, the theory of shrinkage and reactivity evolution developed before [4, 51 will be applied. In the theory a system of micropores and microcrystals is considered, which is referred to as a micromedium. Microcrystals,
0.5
/0
Rmic
LmicRmic fi (1 - X)(R,i, + aL,i,)n,(X)
’ (111
where IZ,, the density of active sites, evolves with conversion according to the law obtained in Ref. 5. As the temperature difference within the particle is small in comparison with the mean temperature, Eq. 11 yields the following change in conversion:
a!X=exp
E(T - T) R T2 dX,, g ( I
(12)
with the local temperature T, where the subscript c denotes a parameter at the center of a particle. Figure 3 shows the change in diameter of Spherocarb synthetic char particles for differ-
84
I. I. KANTOROVICH
180 7 16041 .s 140Jz 2 1200 loo5 ‘6 8060 I
8.
0
8.
1.
f,
f,
8.
0,
I
1000 2000 3000 4000 5000 6000 7000 time [s]
Fig. 3. Diameter of a reacting particle versus time.
ent temperatures, calculated by Eqs. 10 and 11. Temperature clearly affects quite strongly the reduction of diameter with time. A temperature variation of 60 K makes a significant difference in the shrinkage rate.
Force Balance
An irradiated particle in the electrodynamic chamber is balanced by a combined action of forces from different sources: gravity, electrodynamic, photophoretic and free convection. For the same light intensity the force distribution depends on the particle’s properties, but normally the gravitational and electrodynamic forces are the larger ones. The centers of mass and charge may not coincide with each other. The location of the center of each force depends mainly on the shape of the particle and uniformity of the porous structure. In fact, dimensions, shape and density distribution all change nonuniformly under uneven heating. The force imbalance may cause dynamic instabilities or rotation, even for a particle located at the null point of the EDC. The imbalance of centers of forces may exist initially, even before conversion started or it would form shortly after conversion started. Rotation of a particle is likely to occur at small separations of centers of forces and indeed was observed. In general, the conditions of instability are directly connected with peculiarities of uneven structural rearrangement inside the particle. If these conditions continue to exist after a revolution, the particle will continue to rotate around its horizontal axis. As a result, laser
AND E. BAR-ZIV
radiation will be scanning uniformly over the external surface, yielding uniform heating and shrinkage. Nonuniform shrinkage is apparent only above 40%-60% conversion, indicating the threshold nature of the phenomenon. Clearly, the threshold conversion is connected to the dynamic stability of the particle; i.e., the transition from continuous rotation of the particle to l nonrotation. In order to determine the threshold conversion corresponding to nonuniform shrinkage for an initially spherical char particle, consider the equilibrium of a particle under the action of the two main forces applied to the particle: gravity and electric. The weaker free convection and photophoretic forces are neglected here for simplicity. The centers of mass and charge for a homogeneously spherical particle with a uniform distribution of charge over the external surface are both at the center of the i sphere. After a slight nonuniform consumption, the centers of forces shift to different positions along the z axis. If the position of the center of mass z, is above the position of the center of charge z,, the net force will impose a nonzero moment that will cause rotation or flippage of the particle, as can be seen from Fig. 4. To analyze the stability condition zt? - z, > 0,
(13)
consider the shift of the centers of mass and charge after a small conversion dX and use as parameters the coordinates z0 and R,, corresponding to the undistorted particle. Using Eqs. 6 and 12, the change of dX with coordinate
Fig. 4. Rotation of the particle after a deviation of a small angle cp from the unstable equilibrium position; F, is the electrostatic force; Fg is the gravitation force.
MICROSTRUCTURE
OF CHAR
8.5
becomes
will now be divided into the upper and lower hemispheres.
dX(z,,R,)
I
= exp
x120
The Upper Hemisphere
7
P
(14) where
k,E(F- T,)h,
xl = R,F2(A + AJ
(15)
’
k,E(F - T,)A, x2 = R,T2(3A
(16)
+ A,) ’
Here, k, and k, are defined by the distribution of absorbed radiation over the lower hemisphere. For the parabolic distribution given in Eq. 4, k, = 1.6 and k, = 0.8.
dz(z,,
Center of Mass .The position obtained by R,
In accordance with the temperature distribution (see Fig. 2), in each layer of the upper hemisphere, shrinkage is a maximum at the center, but is minimal in the periphery. In order to keep the integrity of a layer, the relative change of its radius should be equal to the minimal shrinkage factor in the periphery, most probably retaining the plane shape of the layers. Local shrinkage of the micromedium decreases with distance from the center, being maximal at the center. Therefore, to maintain the shrinkage factor independent of coordinates within the layer, the decrease in micromedium volume near the center should be compensated by the increase in the macroporosity. The thickness of a layer and change of its radial coordinate R do not depend on the initial value R,, being functions of t , and initial radius of the layer R,, = P R, - z. , or
of the center
dm
/ -R, d.2
dz
(19)
(17)
R,,) = dR,(z,)[l
- bdX(z,,
RJI,
where b=
where zo, R,)R2(zo,
XdNz,, R,).
RJ,
(20)
’
dm -= dz
- bdX(z,,
dz,
0
=
= dR,(z,)[l
of mass can be dR(z,,
--z(zJZ
R,)
1 dlny __3 dX
(21)
x=x,’
X0 is the conversion before distortion of the particle. The apparent density distribution for the upper hemisphere is
R,)
(18)
Nonuniform contraction of different fractions of the microporous medium leads to a strain field inside the particle, which causes the growing of macropores and generation of new ones. This result is in contrast to the case of uniform shrinkage, where macroporosity is unchanged during conversion [5]. To clarify the character of the change in the macroporosity, separate the particle into thin horizontal layers of thickness, dz,. Discussion on contraction
p = p”[l - X(z,, &,)I ~(z,,
R,,)
’
(22)
The Lower Hemisphere Temperature is minimal in the center of a layer, the layer is constricting to its center and, to ensure integrity, macroporosity grows from center to periphery. As conditions in the periphery no more require the same rate of the constriction of each part of the layer in the radial direction, the shrinkage factor at each
86
I. I. KANTOROVICH AND E. BAR-ZIV
point depends on radial coordinate; therefore, the change in the radial coordinate dR can be represented as
the increase of the position of the center of mass.
dR(z,,
Center of Charge
R,)
= dR,Cq,M
- bdX(zO,
&)I. (23)
As the smallest shrinkage factor is achieved in the center, the peripheral part of the layer is now free to shift in a vertical direction. The character of this distortion depends on details of initial local distribution of macropores, which can change randomly within an ensemble of particles. Thus an initially planar layer can be deformed in the course of conversion. Most probably the curvature of the bottom increases after deformation. However, we further assume, for simplicity, that the plane shape is retained during conversion. Then the thickness of a layer dz is determined by the minimal shrinkage factor achieved in the center, or dz(z,,
R,)
= dR,(z,,)[l
- bdX(z,,O)l.
’ = ~l’~(z,, 0)y2’3(zo, R,) ’
1
z, =
Rp R(zo)z(zO)-$dz, 0
-%J
/
RpR(z,); -%
d-K*
For Spherocarb char, b is positive. Therefore, the nonuniform heating by irradiation leads to
(27)
dz, 0
The change in the element of the arc length s of the outline of the axial cross-section of the particle can be expressed by
Substitution of Eqs. 19-21, 23, 24, and 28 into Eq. 27 yields 53x2 280
-
-
i
21 x1 x2 80
-
Now that z, and z, have been deduced, the condition of stability can be determined. Stability Criterion Employing Eqs. 22, 26, and 29 we can represent the condition for stability of a particle Eq. 13 as
(25)
In order to simplify the expressions for the centers of forces we expand the exponent in Eq. 14, using terms up to second order, in respect to X, and up to first order in respect to X2. After substitution of Eqs. 18-25 into Eq. 17 one can obtain with an accuracy of the first order of &CCthe position of the center of mass:
X1 + $1 - X)
/
(24)
The shrinkage factor of different fragments of a layer is determined by (1) contraction in the z direction, which is equal in all parts of the layer, and (2) two-dimensional contraction along the plane, which depends on the radial coordinate R,. Therefore, the apparent density distribution in lower hemisphere can be written as pO(l -X>
The center of charge can be determined by
x1 --
d In P
5
dx
I
x=x”
+f>
0,
(30)
where f represents the terms, first order in X2 and second order in X,: f=
+x2(% - +).
(31)
The second term in Eq. 30 is an order of magnitude less than the first one for X, = 1, = 0.3, which corresponds to h/h, = 0.05 x2 and the temperature range of regime I. With this accuracy we can write the condition of stability of the position of the particle as d In p
-
a!x
> 0.
(32)
’
MICROSTRUCTURE
o e
87
OF CHAR
0.8 Q 0.6
0.2
0.4
0.6
0.8
1.0
Conversion X Fig. 5. Apparent density p and regions of stability. Circles -experimental data of Dudek [l].
RESULTS AND DISCUSSION Stability of Particle Position The inequality 32 is the condition of the transition from uniform to non-uniform shrinkage, which defines the threshold conversion X, as the minimum of the function p(X). This condition does not include k, or k,, i.e., does not depend on the distribution of heat sources over the external surface, provided radiation is absorbed only on the lower hemisphere. The stability condition is affected neither by the mechanism of chemical reactions nor by thermal conductivities of char and its surroundings. It is exclusively the result of structural transformations. The function p(X) obtained for the “subskeleton” mechanism of shrinkage [4] is shown in Fig. 5. Note that weaker forces were neglected. In this case, the behavior of a particle under oxidation would be as follows: Before conversion reaches its threshold value, the particle is unstable and rotates continuously. Due to this continuous rotation, it is being heated uniformly and thus exhibits a uniform symmetric shrinkage. Above the threshold conversion, corresponding to the minimum of p(X), the particle ceases to rotate and nonuniform shrinkage starts. The calculated threshold conversion is 0.69, which is higher than observed values. To explain the discrepancy, note that the rate of transition from unstable to stable equilibrium is proportional to the derivative of the density with respect to conversion. The region where the weaker forces (photophoretic and free convection) are influencing the most is at the transition. The drag force from free
convection favors the increase of the stability for the distorted particle. This force ensures a stable equilibrium before the difference between the centers of main forces reaches a certain value, then the particle flips. It can be expected that instability of the particle is resolved in a succession of flips, rather than in a continuous rotation. The frequency of the flips is large at small conversions, where the derivative of density with respect to conversion is large. When the change in density is low, as it is near the threshold, the frequency of flips is small and the particle has enough time to be consumed preferentially between two successive flips. As the minimum of p(X) is not sharp, preferential consumption actually starts before the threshold conversion. Flips, indeed, were observed in our experiments [3]. Shrinkage Mechanism Nonuniform shrinkage can serve as a means to distinguish between the various mechanisms for the evolution of the porous structure. Kantorovich and Bar-Ziv [4] suggested three possible pathways for shrinkage: The “no-coalescence” mechanism (model 1 in Ref. 41, in which changes occur deterministically, i.e., the microcrystals do not change in shape, number, or position in respect to the structure, just in dimensions. The “stack” mechanism (model 2 in Ref. 4) involves the solid microcrystals of the porous medium undergoing random displacement and coalescence with conversion. The “subskeleton” mechanism (model 3 in Ref. 4) represents a combination of the deterministic pathway of the large microcrystals and the random process for the small ones. Clearly, the threshold of nonuniform shrinkage depends essentially on the mechanism of shrinkage, which determines the behavior of p(X). For the “no-coalescence” mechanism, nonuniform shrinkage is not expected because according to this mechanism the apparent density decreases monotonically with conversion. On the other hand, for the “stack” mechanism, nonuniform shrinkage should begin from zero conversion. The reason is that the apparent
88
I. I. KANTOROVICH
density, in the “stack” mechanism, practically does not change with conversion, except at very high conversions. What is clear is that the temperature differential within the particle is not sufficient to account for nonuniform shrinkage; it requires also a proper behavior of the change of density with conversion. The “subskeleton” mechanism is the only mechanism that can produce conditions to satisfy the stability criterion, and generate nonuniform shrinkage. “Preferential Consumption from the Top” Using Eqs. 19, 20, 23, and 24 it can be shown that after small consumption, the radii of curvature, R, and R_, in the top and bottom halves of the particle can be expressed after the threshold conversion as R,=
R,[l
- beT(X1-X2)dX,].
(33)
Equation 33 shows that the radius of curvature at the top of the particle is larger than that at the bottom. Such behavior produces a visual effect of the initiation of the preferential consumption from the top of the particle, though actually the rate of consumption is greater at the lower hemisphere. This effect can be easily understood if he distribution of temperature in Fig. 2 is employed. One can see that the maximum and the minimum rates of consumption occur not in the bottom and top, but in the slopes of external surface where the maximum and minimum temperatures are. As a result, the slopes of the particle are consumed more intensely than the bottom in the lower hemisphere and less intensely than the top in the upper hemisphere. This result leads to the visual effect of further flattening of the top and sharpening of the bottom as is illustrated in Fig. 6, curve b. It follows from Eq. 33 that the difference in the curvatures of the top and bottom is affected by the distribution of the radiation absorbed in the bottom of a particle and also by the effective thermal conductivity. Curve a in Fig. 6 shows the spherical particle at the threshold conversion. From curve c in Fig. 6 it can be seen that the particle gradually takes on the shape close to a disk. At this later stage the shape is still asymmetric
AND E. BAR-ZIV
Fig. 6. Evolution of the particle shape: a-the spherical shape at threshold conversion; b-the particle flattened in the top and sharpened in the bottom; c-the particle takes on the shape close to disk.
with respect to the horizontal plane: the particle has- a flattened top and- a cone-shaped bottom. This shape has not been observed in experiments. The reason is that the particle occasionally flips and flattens the sharp edges and eventually gets into a disk configuration. Pore Structure CollapseCenter-Holed “Doughnut” The oxidation of carbonaceous materials at very high conversions attracts great interest due to its importance in combustor operation and pollutant emission. In conditions of regime I, most carbons with low and intermediate porosity revealed percolation breakdown. The links between solid fractions break due to excessive consumption of the pore walls. However, percolation breakdown of highly porous carbon, exhibiting shrinkage, has never been observed. The model of shrinkage proposed in Ref. 4 provides an explanation for this behavior. One can roughly view a porous particle as a two phase “micromedium-macropore” system. The micromedium is represented by microcrystals and micropores, which are playing a principal role in the process of oxidation. The micromedium can be viewed as “solid” walls of the macropores. The consumption of the walls can be the reason for the breakdown at high conversions. It was shown in Ref. 4 that shrinkage occurs inside the micromedium. Therefore, the change in the volume of the macropores just follows the behavior of the micromedium. Thus, for an evenly heated particle the macroporosity remains unchanged, ensuring the particle’s integrity up to full burnout. For particles heated unevenly, as was discussed above,
MICROSTRUCWRE
OF CHAR
89
macroporosity depends on time and on the spatial coordinate, changing in different manners for the upper and lower hemispheres. At very high conversions, when the particle’s shape is close to a disk, most of the particle has the macroporosity distribution characteristic of the upper hemisphere; i.e., macroporosity increasing to the center. In order to consider the evolution of the macroporosity for a disk-shaped particle, it is necessary estimate first the temperature distribution. To simplify the problem, assume that the thickness w of the disk is much smaller than its radius R,. Note that at high conversion, when the particle takes on a disk shape, frequent vibrations in horizontal direction were observed. In this case, as the laser beam moves randomly over the bottom of the particle, the distribution of absorbed radiation can be taken as uniform, though the intensity of the laser beam is not distributed uniformly. Then the heat transfer problem can be written as
racy. Then, the relative volume of the micromedium fraction, or (1 - E,,J, is also proportional to (1 - X). Note that the macroporosity E,,~ in the periphery of the disk can be taken as the initial macroporosity E&,, which is unchanged during conversion, because shrinkage in the periphery determines shrinkage of the disk as a whole and thus does not require the change in the macroporosity. Therefore the macroporosity at distance R from the center can be represented in terms of conversion at distance R and in the periphery as
q&R)
- T,,) = 0,
dt
d
go,
R=R,;
AZ
(35) = - 2’7
- To).
(36)
The solution of Eqs. 34-36 to first-order accuracy of the small ratio (h,R,)/(hw) can be expressed as
T(R)
=
To + [T(R,)
- To]&
E:,,).
(38)
a
(1
_
x)~~-WRJ(JV,
where 6 = 0.757. Taking into account that the temperature difference within the particle is small in comparison to the temperature in the periphery T(R,) and employing Eqs. 37-39 one can obtain an expression for the distribution of macroporosity over the particle:
(34) R = 0;
ll-;;d;(1 -
To simplify the dependence of conversion on distance use the following correlation between rate of conversion, given by Eq. 11, and conversion, which is valid for conversion higher than 0.5: dx
&(T
= 1-
q,,,,(R) = 1 - (1 - E:,,) 1 _
G~~“[‘-(R/R,,)*]
.
where (371 For h,/h = 0.05 and a mean temperature of 800 K the maximal difference of temperature within the particle is 12 K. As can be seen from Fig. 2 such a small difference of temperature can cause appreciable change in the volume of the micromedium and, thereby, in the macroporosity. At conversions above 0.5 the shrinkage factor, represented by Eq. 10, can be considered to be proportional to (1 - X1, with good accu-
1 - E&, l-
1-S
I
i I - &%C
G=
ew -
1 - EZ,, I-&’
(41)
( I - &C 1 W=
E[T(R,) - T,,l A, R,[T(Rd)122* ’
.$,,, is the macroporosity particle.
(42)
in the center of the
I. I. KANTOROVICH AND E. BAR-ZIV
90
0)
(4 o$o
012
014
018
0:s
110
Fig. 8. Schematic view of the macropore structure before (a) and after (b) the percolation breakdown in the center.
rncI
Fig. 7. The particle.
distribution
of macroporosity
within
the
The conversion X averaged over the particle’s volume can be obtained using Eqs. 38 and 39 and has the form:
x=
1 -
(1
-x,)/ol(l - Ge”(‘-~‘)l’(l-S)&, (43)
where X,, is the initial conversion for Eq. 39, i.e., the value of conversion at which the disk shape is regarded as formed. Equations 40 and 43 can be considered as the dependence of macroporosity on conversion expressed in the parametrical form with G as a parameter. Figure 7 shows the distribution of the macroporosity over the particle for h,/h = 0.05, T(R,) = 800 K and X,, = 0.5. The initial macroporosity of Spherocarb is 0.44 [8]. The macroporosity cannot be larger than the threshold value that corresponds to the percolation breakdown, which can be taken as 0.8 according to [9]. From Fig. 7 it is seen that the maximum macroporosity is achieved in the center. Thus the formation of the hole in the center is the indication of a very peculiar phenomenon, which has not been reported before-local percolation breakdown taking place only in the center. Figure 8 shows schematically how the nonuniform macroporosity distribution leads to collapse in the center. The conversion at which the hole is formed is found to be 0.95 for the above-mentioned temperature, initial conversion and thermal conductivity. This result is in agreement with the observed values. Note that the correlation between macroporosity, conversion and relative radial coordi-
nate does not depend on any dimensions of the particle. The determining parameters are particle temperature, initial conversion X0 and effective thermal conductivity of the particle. Fortunately, the form of the correlation is not sensitive to T and X0, the latter being not clearly defined in Eq. 43, in a reasonable ranges of values. The parameter with the greatest influence on macroporosity is the effective thermal conductivity. Figure 9 demonstrates the change of the macroporosity in the center of a particle with conversion for different values of the effective thermal conductivity. The strong influence of the thermal conductivity is evident. Thermal conductivity is a powerful tool that provides detailed information on the microstructure, particularly on intercrystal joints and other details of the internal topology. In Ref. 10 we have developed a model of the evolution of thermal conductivity with conversion. In an earlier publication [3] a new independent experimental method for the measurement of thermal conductivity of hundred-micron sized
0.8 0.7 i ow 0.6 0.5 0.4"'
0.5
a I " 0.6 0.7
n . " 0.8 0.9
I ' 1.0
Conversion X Fig. 9. Plot of macroporosity in the center of the particle versus conversion.
MICROSTRUCTURE
OF CHAR
particles was developed. A further development of the thermal conductivity analysis, combined with the study of nonuniform shrinkage and local percolation breakdown can contribute to the elucidation of the nature of microstructural transformation in highly porous materials. SUMMARY AND CONCLUSION It has been shown that all specific features of nonuniform shrinkage due to uneven heating (the threshold character; observation of “preferential consumption from the top”; formation of disk, formation of a hole in the center at high conversions) are the result of severe pore structural transformation inside the particle. These phenomena are consistent with the “subskeleton” mechanism of the pore structure evolution. Highly porous particles can exhibit percolation fragmentation, but in a very peculiar form: local fragmentation, occurring only in the particle center. As for the delay of preferential consumption, since real particles are not ideally spherical and initially homogeneous, the value of the threshold conversion may differ from the discussed ideal case. Because at low conversions the center of mass goes up faster than the center of charge, at some threshold conversion the centers may coincide and the particle would lose stability. Although to this moment the particle may take on a nonspherical shape, up to the next threshold (corresponding to stability) it would be heated uniformly, due to
91 rotation or frequent flips, and it would shrink uniformly. In contrast, at a conversion corresponding to a stable state, shrinkage is anisotropic and the particle shape changes continuously. This work was partially supported by the Belfer Foundation and the Israel Ministry of Science and Technology.
REFERENCES 1. Dudek, D. R., Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (1988). 2. D’Amore, M. D., Tognotti, L., and Sarofim, A. F., Proceedings of the Symposium on Gasification Mechanisms, American Chemical Society, Div. Fuel Chem.,
36(3):934 (1991). 3. Weiss, Y., Ben-Ari, Y., Kantorovich, I. I., Bar-Ziv, E., Krammer, G., Modestino, A., and Sarofim, A. F., Twenty-Fifih Symposium
(International) on Combustion
The Combustion Institute, Pittsburgh, 1994, pp. 5199525. 4. Kantorovich, I. I., and Bar-Ziv, E., Combust. Flame 97:61 (1994).
5.
Kantorovich, I. I., and Bar-Ziv, E., Combust.
Flame
97:79 (1994).
6.
Bar-Ziv, E., and Sarofim, A. F., Prog. Ener. Combust. Sci. 17:l (1991).
7. Weiss, Y., and Bar-Ziv, E., Combust.
Flame 95:362
(1993).
8.
Hurt, R. H., Ph.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1987. 9. Dutta, S., Wen, C. Y., and Belt, R. J., Ind. Eng. Chem. Process Des. Deu. 16:20 (1977). 10. Kantorovich, I. I., and Bar-Ziv, E., submitted.. Received 23 August 1994; revised 7 July 1995
and EZRA BAR-ZIV*
Department of Mechanical Engineering Ben-Gution University of the Negev, P.O.B. 653, Beer-Sheva (I.I.1y; E.B.-Z) Nuclear Research Center-Negev (E.B.-Z.), P.O.B. 9001, Beer-Sheua, Israel
Single spherical char particles were intentionally irradiated nonuniformly in an electrodynamic chamber, in the temperature range 600-1000 K. The char particles were irradiated from one side (bottom) and consequently heated unevenly. Nonuniform shrinkage of an initially spherical char particle has been observed during oxidation. The features of nonuniform shrinkage are rather peculiar: (1) Up to 40%-60% conversion, the particle shrank uniformly. Then spatial preferential consumption initiated, indicating the threshold nature of the phenomenon. (2) Preferential consumption, in most experiments, was observed to start from the top of the particle. (3) Above the threshold conversion, a disk structure became clear. (4) At very high conversion the particle became like a center-hole doughnut. A model was developed to explain these features by nonuniform transformation of the micropore structure due to oxidation of the unevenly heated particle. The threshold nature of the phenomenon was also explained in terms of the dynamic stability of a particle. The threshold of nonuniform shrinkage corresponds to the transition of the particle to a stable position. After initiation of nonuniform shrinkage, the macroporosity distribution inside the particle becomes nonuniform. Macroporosity was shown to increase near the particle’s center; this eventually creates a hole at high conversion. The good agreement of modeling results with experimental observation confirms the notion that nonuniform shrinkage in regime I (kinetically controlled) is an indication of the fundamental microstructural transformations in the course of oxidation.
NOMENCLATURE b d dll f”
G h
k,,
k,
kr
Lmic Lmic,
1
LO,ic,
1
m no
P
4
defined by Eq. 21, dimensionless. diameter of a particle, m. initial value of d,m. activation energy, J/mol. defined by Eq. 30, dimensionless. defined by Eq. 41, dimensionless. heat exchange coefficient, W/ (K m*>. defined by Eqs. 15 and 16, dimensionless. preexponential factor at the inverse time of half conversion [2], s-l. mean length of microcrystals, m. mean length of large microcrystals, m. initial mean length of large microcrystals, m. mass of a particle, kg. density of active sites [5], m-*. probability for the crystal edge to be revealed on the surface of a macropore, dimensionless. absorbed radiation flux density, W/(K m*>.
* Corresponding author. OOlO-2180/96/%15.00 SSDI OOlO-2180(95)00168-6
Q ii
Ki
4
Ro
RP
Rmic
RO,ic 4 R zo
R,, R-
coefficient related to absorbed radiation flux density, W/(K m4>. radial spherical coordinate, m. radial cylindrical coordinate, m. radius of the disk, m. gas constant, J/(mol K). radial cylindrical coordinate corresponding to an undistorted particle, m. radius of a particle, m. mean radius of microcrystals, m. initial mean radius of microcrystals, m. radius of a horizontal cross-section of a sphere at coordinate z, m. initial radius of a horizontal crosssection, m. radii of curvature in the top and the bottom of a particle, m. temperature, K. mean temperature of a particle, K. ambient temperature, K. thickness, m. cylindrical coordinate, m. position of the center of charge, m. position of the center of mass, m. cylindrical coordinate corresponding to an undistorted particle, m. COMBUSTIONAND FLAME 105: BO-91(1996) Copyright 0 1996 by The Combustion Institute Published by Elsevier Science Inc.
MICROSTRUCTURE X &I -%
OF CHAR
conversion, dimensionless. conversion before particle distortion, dimensionless. conversion in the center of a particle, dimensionless.
Greek Symbols ratio of the change of the diameter to length of a microcrystal due to its consumption, dimensionless. shrinkage factor, dimensionless. defined by Eq. 39, dimensionless. maximum difference of temperature within the particle, K. macroporosity, dimensionless. initial macroporosity, dimensionless. angle between the radius vector and the vertical axis, dimensionless. angle 8 for the coldest spot or rr - 8 for the hottest spot. effective thermal conductivity of a particle, W/(mK). thermal conductivity of gas, W/(mK). cos 0. apparent density, kg/m3. defined by Eqs. 15 and 16, dimensionless. defined by Eq. 42, dimensionless. INTRODUCTION The physicochemical properties of a highly porous char during oxidation in regime I (kinetically controlled) change in a manner strongly deviating from the traditional behavior. Shrinkage of a gasifying particle is an unexpected phenomenon in regime I [l]. Moreover, during oxidation highly porous chars exhibit unexpected development of (1) apparent density, (2) total internal surface area, and (3) rate of conversion [l, 21. Another example is the severe change of thermal conductivity during conversion [3]. Kantorovich and Bar-Ziv [4, 51 showed that these phenomena are determined by peculiar development of the micropore structure. They proposed a model for the pore structure of highly porous chars which enabled them to account for shrinkage and some other phenomena. Pore structure, being
81 an important factor affecting the reactivity of carbonaceous materials, is of crucial importance for highly porous coals and chars. The conventional way of studying the evolution of the pore structure by surface characterization methods is extremely difficult and not yet developed to enable the understanding and elucidation of these changes. At present, the perspective way that can shed light into the mutual influence of the pore structure and reactivity is to analyze the changes in physicochemical properties, caused by structural changes. As shown before [4, 51, the advantage of this indirect way is the possibility to point out tiny details of pore structure evolution, whose direct measurement is not feasible. The physical properties closely connected to the pore structure are: density, porosity, total surface area, external shape, thermal conductivity, and optical properties. The chemical properties of importance are: active surface area, active sites, and adsorption-desorption. Most of these can be measured during experiments conducted in an electrodynamic chamber (EDC) [61. The main advantages of the EDC are the ability to characterize the particle prior to reaction, to monitor the important quantities needed for understanding kinetics in real time and to study the oxidation of individual particles in well-controlled conditions. The porous nature of a particle under oxidation can reveal itself most noticeably in experiments conducted under nonequilibrium conditions, in the presence of strong temperature and concentration gradients, which can be intentionally imposed on a particle. The results can serve to achieve a more detailed characterization of the internal structure of carbonaceous materials. One example of such an experiment is the uneven heating of a particle, resulting in nonuniform shrinkage of the particle. Nonuniform shrinkage of unevenly heated synthetic char (Spherocarb) particles (diameter 150-200 pm) during oxidation in an electrodynamic chamber in the temperature range 600 to 1100 K has been observed by Weiss and Bar-Ziv [7]. A single char particle was levitated in the electrodynamic balance and heated from its bottom by a focused laser beam. As oxidation was shown to proceed all over the internal surface, the change in shape cannot be at-
82
I. I. KANTOROVICH
tributed to preferential consumption on the overheated external surface exposed to radiation. As shown below, nonuniform shrinkage is explained by nonuniform rearrangements of the microstructure. Figure 1 illustrates the evolution of the shape of an initially spherical particle during oxidation as observed by Weiss and Bar-Ziv [7]. Figures la-le relate to various conversions and Fig. Id, which appears twice, is the disk from perpendicular sides. Rather peculiar features of nonuniform consumption can be seen: (1) nonuniform consumption reveals a threshold nature, initiating at 40%-60% of conversion, (2) preferential consumption in most experiments starts from the particle’s top, despite heating from the bottom, (3) above threshold conversion a particle takes on a disk shape, and (4) at very high conversion a center-holed doughnut is formed. These features are explained below employing a model of the pore structure developed before [4, 51. THEORY Temperature Distribution In these experiments, a particle is heated from the bottom by a laser beam, the light flux
AND E. BAR-ZIV
density has a TEM& mode (doughnut shape) in the particle’s axis to ensure more uniform light absorption. A parabolic distribution of absorbed radiation over the lower hemisphere was assumed. The distribution of temperature T can be described by d2T
-;;;i-+;;+;;[(l-P9$]=o, (1)
d
with the boundary conditions: r = R,; A$
+ h(T - TJ
+ Qf(r, /.d = 0, (2)
r=O;
T
(3)
where
f(r,p) =
0; i r2j&
O
(4)
-lI/.LSO.
- Jo’);
As the Grashoff number is negligibly small for a particle of 200 pm diameter, heat exchange with the surrounding gas is governed by heat conduction; therefore for a spherical particle, h can be represented as h = h,/R,, where h, is the gas thermal conductivity. The solution of Eqs. l-4 can be expressed as a Legendre polynomial series:
I
/
, (a)
d/do=l,
X=0
(5)
(d) d/d,,=0.65, X=0.9
(b) d/d,=O& X=0.6
(d) w/d,,=O.33, X=0.9
(c) d/d,=0.8, X=0.7
(e) d/d,=O.54, X=0.95
Fig. 1. Redrawn shadowgraphs presenting the development of the preferential consumption of a particle. (a-e) relate to various conversion and Cd), which appears twice, is the disk from perpendicular sides.
The second term in the square brackets exceeds the third one by factor of three, the third one by factor of 7, the fourth one by factor of 20, etc. As Eq. 5 is an alternating series, one can limit it to the first three terms. lindrical coordinates z = pr and R = Usin 1 - p2, it is found that the temperature can r s”” be represented as T = T,, -t
(7; - T,,) 1 - 5(h
g
“::),
P
4z(2z2 - 3R2)h, +
5R;(3h + he)
I
’
(6)
MICROSTRUCTURE
OF CHAR
83 to which a cylindrical shape is attributed for simplicity, are considered as randomly distributed and interconnected. According to the “subskeleton” mechanism [4], shrinkage can be explained as the continuous process of break and restoration of the internal joints of the subskeleton containing the largest microcrystals. The relative change of volume (the shrinkage factor) y can be expressed by
(10) Fig. 2. Isotherms within a spherical unevenly heated particle. The temperature interval between consecutive isotherms is 5 K.
where (7) is the mean temperature of the particle. Figure 2 demonstrates isotherms that characterize the temperature distribution over a vertical cross section. The hottest and coldest spots are located on the particle’s surface symmetrically, and the angle for the spherical coordinate 0 for the coldest spot equals:
The theory of the evolution of the porous structure [8] enables these quantities to be represented as a function of conversion, X. The small microcrystals form the fine structure of the micromedium, and can coalesce to large ones, not changing the shape of the subskeleton. Coalescence of the small microcrystals is the reason for the change of the material’s reactivity during conversion. The evolution of the rate of conversion can be expressed 151by:
(l - X)(R,i, + (YL,ic)nak, dx
X exp( -E/R,T)
-= dt
Lank
(8) X
which is 40” for small A,/h. The maximum difference of temperature, AT, within the particle is AT=
(9)
For h,/A = 0.05, AT reaches 60 K for a typical particle temperature of 800 K. Pore Structural Transformation and Shrinkage To estimate the change in the dimensions and shape of a particle caused by such a temperature distribution, the theory of shrinkage and reactivity evolution developed before [4, 51 will be applied. In the theory a system of micropores and microcrystals is considered, which is referred to as a micromedium. Microcrystals,
0.5
/0
Rmic
LmicRmic fi (1 - X)(R,i, + aL,i,)n,(X)
’ (111
where IZ,, the density of active sites, evolves with conversion according to the law obtained in Ref. 5. As the temperature difference within the particle is small in comparison with the mean temperature, Eq. 11 yields the following change in conversion:
a!X=exp
E(T - T) R T2 dX,, g ( I
(12)
with the local temperature T, where the subscript c denotes a parameter at the center of a particle. Figure 3 shows the change in diameter of Spherocarb synthetic char particles for differ-
84
I. I. KANTOROVICH
180 7 16041 .s 140Jz 2 1200 loo5 ‘6 8060 I
8.
0
8.
1.
f,
f,
8.
0,
I
1000 2000 3000 4000 5000 6000 7000 time [s]
Fig. 3. Diameter of a reacting particle versus time.
ent temperatures, calculated by Eqs. 10 and 11. Temperature clearly affects quite strongly the reduction of diameter with time. A temperature variation of 60 K makes a significant difference in the shrinkage rate.
Force Balance
An irradiated particle in the electrodynamic chamber is balanced by a combined action of forces from different sources: gravity, electrodynamic, photophoretic and free convection. For the same light intensity the force distribution depends on the particle’s properties, but normally the gravitational and electrodynamic forces are the larger ones. The centers of mass and charge may not coincide with each other. The location of the center of each force depends mainly on the shape of the particle and uniformity of the porous structure. In fact, dimensions, shape and density distribution all change nonuniformly under uneven heating. The force imbalance may cause dynamic instabilities or rotation, even for a particle located at the null point of the EDC. The imbalance of centers of forces may exist initially, even before conversion started or it would form shortly after conversion started. Rotation of a particle is likely to occur at small separations of centers of forces and indeed was observed. In general, the conditions of instability are directly connected with peculiarities of uneven structural rearrangement inside the particle. If these conditions continue to exist after a revolution, the particle will continue to rotate around its horizontal axis. As a result, laser
AND E. BAR-ZIV
radiation will be scanning uniformly over the external surface, yielding uniform heating and shrinkage. Nonuniform shrinkage is apparent only above 40%-60% conversion, indicating the threshold nature of the phenomenon. Clearly, the threshold conversion is connected to the dynamic stability of the particle; i.e., the transition from continuous rotation of the particle to l nonrotation. In order to determine the threshold conversion corresponding to nonuniform shrinkage for an initially spherical char particle, consider the equilibrium of a particle under the action of the two main forces applied to the particle: gravity and electric. The weaker free convection and photophoretic forces are neglected here for simplicity. The centers of mass and charge for a homogeneously spherical particle with a uniform distribution of charge over the external surface are both at the center of the i sphere. After a slight nonuniform consumption, the centers of forces shift to different positions along the z axis. If the position of the center of mass z, is above the position of the center of charge z,, the net force will impose a nonzero moment that will cause rotation or flippage of the particle, as can be seen from Fig. 4. To analyze the stability condition zt? - z, > 0,
(13)
consider the shift of the centers of mass and charge after a small conversion dX and use as parameters the coordinates z0 and R,, corresponding to the undistorted particle. Using Eqs. 6 and 12, the change of dX with coordinate
Fig. 4. Rotation of the particle after a deviation of a small angle cp from the unstable equilibrium position; F, is the electrostatic force; Fg is the gravitation force.
MICROSTRUCTURE
OF CHAR
8.5
becomes
will now be divided into the upper and lower hemispheres.
dX(z,,R,)
I
= exp
x120
The Upper Hemisphere
7
P
(14) where
k,E(F- T,)h,
xl = R,F2(A + AJ
(15)
’
k,E(F - T,)A, x2 = R,T2(3A
(16)
+ A,) ’
Here, k, and k, are defined by the distribution of absorbed radiation over the lower hemisphere. For the parabolic distribution given in Eq. 4, k, = 1.6 and k, = 0.8.
dz(z,,
Center of Mass .The position obtained by R,
In accordance with the temperature distribution (see Fig. 2), in each layer of the upper hemisphere, shrinkage is a maximum at the center, but is minimal in the periphery. In order to keep the integrity of a layer, the relative change of its radius should be equal to the minimal shrinkage factor in the periphery, most probably retaining the plane shape of the layers. Local shrinkage of the micromedium decreases with distance from the center, being maximal at the center. Therefore, to maintain the shrinkage factor independent of coordinates within the layer, the decrease in micromedium volume near the center should be compensated by the increase in the macroporosity. The thickness of a layer and change of its radial coordinate R do not depend on the initial value R,, being functions of t , and initial radius of the layer R,, = P R, - z. , or
of the center
dm
/ -R, d.2
dz
(19)
(17)
R,,) = dR,(z,)[l
- bdX(z,,
RJI,
where b=
where zo, R,)R2(zo,
XdNz,, R,).
RJ,
(20)
’
dm -= dz
- bdX(z,,
dz,
0
=
= dR,(z,)[l
of mass can be dR(z,,
--z(zJZ
R,)
1 dlny __3 dX
(21)
x=x,’
X0 is the conversion before distortion of the particle. The apparent density distribution for the upper hemisphere is
R,)
(18)
Nonuniform contraction of different fractions of the microporous medium leads to a strain field inside the particle, which causes the growing of macropores and generation of new ones. This result is in contrast to the case of uniform shrinkage, where macroporosity is unchanged during conversion [5]. To clarify the character of the change in the macroporosity, separate the particle into thin horizontal layers of thickness, dz,. Discussion on contraction
p = p”[l - X(z,, &,)I ~(z,,
R,,)
’
(22)
The Lower Hemisphere Temperature is minimal in the center of a layer, the layer is constricting to its center and, to ensure integrity, macroporosity grows from center to periphery. As conditions in the periphery no more require the same rate of the constriction of each part of the layer in the radial direction, the shrinkage factor at each
86
I. I. KANTOROVICH AND E. BAR-ZIV
point depends on radial coordinate; therefore, the change in the radial coordinate dR can be represented as
the increase of the position of the center of mass.
dR(z,,
Center of Charge
R,)
= dR,Cq,M
- bdX(zO,
&)I. (23)
As the smallest shrinkage factor is achieved in the center, the peripheral part of the layer is now free to shift in a vertical direction. The character of this distortion depends on details of initial local distribution of macropores, which can change randomly within an ensemble of particles. Thus an initially planar layer can be deformed in the course of conversion. Most probably the curvature of the bottom increases after deformation. However, we further assume, for simplicity, that the plane shape is retained during conversion. Then the thickness of a layer dz is determined by the minimal shrinkage factor achieved in the center, or dz(z,,
R,)
= dR,(z,,)[l
- bdX(z,,O)l.
’ = ~l’~(z,, 0)y2’3(zo, R,) ’
1
z, =
Rp R(zo)z(zO)-$dz, 0
-%J
/
RpR(z,); -%
d-K*
For Spherocarb char, b is positive. Therefore, the nonuniform heating by irradiation leads to
(27)
dz, 0
The change in the element of the arc length s of the outline of the axial cross-section of the particle can be expressed by
Substitution of Eqs. 19-21, 23, 24, and 28 into Eq. 27 yields 53x2 280
-
-
i
21 x1 x2 80
-
Now that z, and z, have been deduced, the condition of stability can be determined. Stability Criterion Employing Eqs. 22, 26, and 29 we can represent the condition for stability of a particle Eq. 13 as
(25)
In order to simplify the expressions for the centers of forces we expand the exponent in Eq. 14, using terms up to second order, in respect to X, and up to first order in respect to X2. After substitution of Eqs. 18-25 into Eq. 17 one can obtain with an accuracy of the first order of &CCthe position of the center of mass:
X1 + $1 - X)
/
(24)
The shrinkage factor of different fragments of a layer is determined by (1) contraction in the z direction, which is equal in all parts of the layer, and (2) two-dimensional contraction along the plane, which depends on the radial coordinate R,. Therefore, the apparent density distribution in lower hemisphere can be written as pO(l -X>
The center of charge can be determined by
x1 --
d In P
5
dx
I
x=x”
+f>
0,
(30)
where f represents the terms, first order in X2 and second order in X,: f=
+x2(% - +).
(31)
The second term in Eq. 30 is an order of magnitude less than the first one for X, = 1, = 0.3, which corresponds to h/h, = 0.05 x2 and the temperature range of regime I. With this accuracy we can write the condition of stability of the position of the particle as d In p
-
a!x
> 0.
(32)
’
MICROSTRUCTURE
o e
87
OF CHAR
0.8 Q 0.6
0.2
0.4
0.6
0.8
1.0
Conversion X Fig. 5. Apparent density p and regions of stability. Circles -experimental data of Dudek [l].
RESULTS AND DISCUSSION Stability of Particle Position The inequality 32 is the condition of the transition from uniform to non-uniform shrinkage, which defines the threshold conversion X, as the minimum of the function p(X). This condition does not include k, or k,, i.e., does not depend on the distribution of heat sources over the external surface, provided radiation is absorbed only on the lower hemisphere. The stability condition is affected neither by the mechanism of chemical reactions nor by thermal conductivities of char and its surroundings. It is exclusively the result of structural transformations. The function p(X) obtained for the “subskeleton” mechanism of shrinkage [4] is shown in Fig. 5. Note that weaker forces were neglected. In this case, the behavior of a particle under oxidation would be as follows: Before conversion reaches its threshold value, the particle is unstable and rotates continuously. Due to this continuous rotation, it is being heated uniformly and thus exhibits a uniform symmetric shrinkage. Above the threshold conversion, corresponding to the minimum of p(X), the particle ceases to rotate and nonuniform shrinkage starts. The calculated threshold conversion is 0.69, which is higher than observed values. To explain the discrepancy, note that the rate of transition from unstable to stable equilibrium is proportional to the derivative of the density with respect to conversion. The region where the weaker forces (photophoretic and free convection) are influencing the most is at the transition. The drag force from free
convection favors the increase of the stability for the distorted particle. This force ensures a stable equilibrium before the difference between the centers of main forces reaches a certain value, then the particle flips. It can be expected that instability of the particle is resolved in a succession of flips, rather than in a continuous rotation. The frequency of the flips is large at small conversions, where the derivative of density with respect to conversion is large. When the change in density is low, as it is near the threshold, the frequency of flips is small and the particle has enough time to be consumed preferentially between two successive flips. As the minimum of p(X) is not sharp, preferential consumption actually starts before the threshold conversion. Flips, indeed, were observed in our experiments [3]. Shrinkage Mechanism Nonuniform shrinkage can serve as a means to distinguish between the various mechanisms for the evolution of the porous structure. Kantorovich and Bar-Ziv [4] suggested three possible pathways for shrinkage: The “no-coalescence” mechanism (model 1 in Ref. 41, in which changes occur deterministically, i.e., the microcrystals do not change in shape, number, or position in respect to the structure, just in dimensions. The “stack” mechanism (model 2 in Ref. 4) involves the solid microcrystals of the porous medium undergoing random displacement and coalescence with conversion. The “subskeleton” mechanism (model 3 in Ref. 4) represents a combination of the deterministic pathway of the large microcrystals and the random process for the small ones. Clearly, the threshold of nonuniform shrinkage depends essentially on the mechanism of shrinkage, which determines the behavior of p(X). For the “no-coalescence” mechanism, nonuniform shrinkage is not expected because according to this mechanism the apparent density decreases monotonically with conversion. On the other hand, for the “stack” mechanism, nonuniform shrinkage should begin from zero conversion. The reason is that the apparent
88
I. I. KANTOROVICH
density, in the “stack” mechanism, practically does not change with conversion, except at very high conversions. What is clear is that the temperature differential within the particle is not sufficient to account for nonuniform shrinkage; it requires also a proper behavior of the change of density with conversion. The “subskeleton” mechanism is the only mechanism that can produce conditions to satisfy the stability criterion, and generate nonuniform shrinkage. “Preferential Consumption from the Top” Using Eqs. 19, 20, 23, and 24 it can be shown that after small consumption, the radii of curvature, R, and R_, in the top and bottom halves of the particle can be expressed after the threshold conversion as R,=
R,[l
- beT(X1-X2)dX,].
(33)
Equation 33 shows that the radius of curvature at the top of the particle is larger than that at the bottom. Such behavior produces a visual effect of the initiation of the preferential consumption from the top of the particle, though actually the rate of consumption is greater at the lower hemisphere. This effect can be easily understood if he distribution of temperature in Fig. 2 is employed. One can see that the maximum and the minimum rates of consumption occur not in the bottom and top, but in the slopes of external surface where the maximum and minimum temperatures are. As a result, the slopes of the particle are consumed more intensely than the bottom in the lower hemisphere and less intensely than the top in the upper hemisphere. This result leads to the visual effect of further flattening of the top and sharpening of the bottom as is illustrated in Fig. 6, curve b. It follows from Eq. 33 that the difference in the curvatures of the top and bottom is affected by the distribution of the radiation absorbed in the bottom of a particle and also by the effective thermal conductivity. Curve a in Fig. 6 shows the spherical particle at the threshold conversion. From curve c in Fig. 6 it can be seen that the particle gradually takes on the shape close to a disk. At this later stage the shape is still asymmetric
AND E. BAR-ZIV
Fig. 6. Evolution of the particle shape: a-the spherical shape at threshold conversion; b-the particle flattened in the top and sharpened in the bottom; c-the particle takes on the shape close to disk.
with respect to the horizontal plane: the particle has- a flattened top and- a cone-shaped bottom. This shape has not been observed in experiments. The reason is that the particle occasionally flips and flattens the sharp edges and eventually gets into a disk configuration. Pore Structure CollapseCenter-Holed “Doughnut” The oxidation of carbonaceous materials at very high conversions attracts great interest due to its importance in combustor operation and pollutant emission. In conditions of regime I, most carbons with low and intermediate porosity revealed percolation breakdown. The links between solid fractions break due to excessive consumption of the pore walls. However, percolation breakdown of highly porous carbon, exhibiting shrinkage, has never been observed. The model of shrinkage proposed in Ref. 4 provides an explanation for this behavior. One can roughly view a porous particle as a two phase “micromedium-macropore” system. The micromedium is represented by microcrystals and micropores, which are playing a principal role in the process of oxidation. The micromedium can be viewed as “solid” walls of the macropores. The consumption of the walls can be the reason for the breakdown at high conversions. It was shown in Ref. 4 that shrinkage occurs inside the micromedium. Therefore, the change in the volume of the macropores just follows the behavior of the micromedium. Thus, for an evenly heated particle the macroporosity remains unchanged, ensuring the particle’s integrity up to full burnout. For particles heated unevenly, as was discussed above,
MICROSTRUCWRE
OF CHAR
89
macroporosity depends on time and on the spatial coordinate, changing in different manners for the upper and lower hemispheres. At very high conversions, when the particle’s shape is close to a disk, most of the particle has the macroporosity distribution characteristic of the upper hemisphere; i.e., macroporosity increasing to the center. In order to consider the evolution of the macroporosity for a disk-shaped particle, it is necessary estimate first the temperature distribution. To simplify the problem, assume that the thickness w of the disk is much smaller than its radius R,. Note that at high conversion, when the particle takes on a disk shape, frequent vibrations in horizontal direction were observed. In this case, as the laser beam moves randomly over the bottom of the particle, the distribution of absorbed radiation can be taken as uniform, though the intensity of the laser beam is not distributed uniformly. Then the heat transfer problem can be written as
racy. Then, the relative volume of the micromedium fraction, or (1 - E,,J, is also proportional to (1 - X). Note that the macroporosity E,,~ in the periphery of the disk can be taken as the initial macroporosity E&,, which is unchanged during conversion, because shrinkage in the periphery determines shrinkage of the disk as a whole and thus does not require the change in the macroporosity. Therefore the macroporosity at distance R from the center can be represented in terms of conversion at distance R and in the periphery as
q&R)
- T,,) = 0,
dt
d
go,
R=R,;
AZ
(35) = - 2’7
- To).
(36)
The solution of Eqs. 34-36 to first-order accuracy of the small ratio (h,R,)/(hw) can be expressed as
T(R)
=
To + [T(R,)
- To]&
E:,,).
(38)
a
(1
_
x)~~-WRJ(JV,
where 6 = 0.757. Taking into account that the temperature difference within the particle is small in comparison to the temperature in the periphery T(R,) and employing Eqs. 37-39 one can obtain an expression for the distribution of macroporosity over the particle:
(34) R = 0;
ll-;;d;(1 -
To simplify the dependence of conversion on distance use the following correlation between rate of conversion, given by Eq. 11, and conversion, which is valid for conversion higher than 0.5: dx
&(T
= 1-
q,,,,(R) = 1 - (1 - E:,,) 1 _
G~~“[‘-(R/R,,)*]
.
where (371 For h,/h = 0.05 and a mean temperature of 800 K the maximal difference of temperature within the particle is 12 K. As can be seen from Fig. 2 such a small difference of temperature can cause appreciable change in the volume of the micromedium and, thereby, in the macroporosity. At conversions above 0.5 the shrinkage factor, represented by Eq. 10, can be considered to be proportional to (1 - X1, with good accu-
1 - E&, l-
1-S
I
i I - &%C
G=
ew -
1 - EZ,, I-&’
(41)
( I - &C 1 W=
E[T(R,) - T,,l A, R,[T(Rd)122* ’
.$,,, is the macroporosity particle.
(42)
in the center of the
I. I. KANTOROVICH AND E. BAR-ZIV
90
0)
(4 o$o
012
014
018
0:s
110
Fig. 8. Schematic view of the macropore structure before (a) and after (b) the percolation breakdown in the center.
rncI
Fig. 7. The particle.
distribution
of macroporosity
within
the
The conversion X averaged over the particle’s volume can be obtained using Eqs. 38 and 39 and has the form:
x=
1 -
(1
-x,)/ol(l - Ge”(‘-~‘)l’(l-S)&, (43)
where X,, is the initial conversion for Eq. 39, i.e., the value of conversion at which the disk shape is regarded as formed. Equations 40 and 43 can be considered as the dependence of macroporosity on conversion expressed in the parametrical form with G as a parameter. Figure 7 shows the distribution of the macroporosity over the particle for h,/h = 0.05, T(R,) = 800 K and X,, = 0.5. The initial macroporosity of Spherocarb is 0.44 [8]. The macroporosity cannot be larger than the threshold value that corresponds to the percolation breakdown, which can be taken as 0.8 according to [9]. From Fig. 7 it is seen that the maximum macroporosity is achieved in the center. Thus the formation of the hole in the center is the indication of a very peculiar phenomenon, which has not been reported before-local percolation breakdown taking place only in the center. Figure 8 shows schematically how the nonuniform macroporosity distribution leads to collapse in the center. The conversion at which the hole is formed is found to be 0.95 for the above-mentioned temperature, initial conversion and thermal conductivity. This result is in agreement with the observed values. Note that the correlation between macroporosity, conversion and relative radial coordi-
nate does not depend on any dimensions of the particle. The determining parameters are particle temperature, initial conversion X0 and effective thermal conductivity of the particle. Fortunately, the form of the correlation is not sensitive to T and X0, the latter being not clearly defined in Eq. 43, in a reasonable ranges of values. The parameter with the greatest influence on macroporosity is the effective thermal conductivity. Figure 9 demonstrates the change of the macroporosity in the center of a particle with conversion for different values of the effective thermal conductivity. The strong influence of the thermal conductivity is evident. Thermal conductivity is a powerful tool that provides detailed information on the microstructure, particularly on intercrystal joints and other details of the internal topology. In Ref. 10 we have developed a model of the evolution of thermal conductivity with conversion. In an earlier publication [3] a new independent experimental method for the measurement of thermal conductivity of hundred-micron sized
0.8 0.7 i ow 0.6 0.5 0.4"'
0.5
a I " 0.6 0.7
n . " 0.8 0.9
I ' 1.0
Conversion X Fig. 9. Plot of macroporosity in the center of the particle versus conversion.
MICROSTRUCTURE
OF CHAR
particles was developed. A further development of the thermal conductivity analysis, combined with the study of nonuniform shrinkage and local percolation breakdown can contribute to the elucidation of the nature of microstructural transformation in highly porous materials. SUMMARY AND CONCLUSION It has been shown that all specific features of nonuniform shrinkage due to uneven heating (the threshold character; observation of “preferential consumption from the top”; formation of disk, formation of a hole in the center at high conversions) are the result of severe pore structural transformation inside the particle. These phenomena are consistent with the “subskeleton” mechanism of the pore structure evolution. Highly porous particles can exhibit percolation fragmentation, but in a very peculiar form: local fragmentation, occurring only in the particle center. As for the delay of preferential consumption, since real particles are not ideally spherical and initially homogeneous, the value of the threshold conversion may differ from the discussed ideal case. Because at low conversions the center of mass goes up faster than the center of charge, at some threshold conversion the centers may coincide and the particle would lose stability. Although to this moment the particle may take on a nonspherical shape, up to the next threshold (corresponding to stability) it would be heated uniformly, due to
91 rotation or frequent flips, and it would shrink uniformly. In contrast, at a conversion corresponding to a stable state, shrinkage is anisotropic and the particle shape changes continuously. This work was partially supported by the Belfer Foundation and the Israel Ministry of Science and Technology.
REFERENCES 1. Dudek, D. R., Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (1988). 2. D’Amore, M. D., Tognotti, L., and Sarofim, A. F., Proceedings of the Symposium on Gasification Mechanisms, American Chemical Society, Div. Fuel Chem.,
36(3):934 (1991). 3. Weiss, Y., Ben-Ari, Y., Kantorovich, I. I., Bar-Ziv, E., Krammer, G., Modestino, A., and Sarofim, A. F., Twenty-Fifih Symposium
(International) on Combustion
The Combustion Institute, Pittsburgh, 1994, pp. 5199525. 4. Kantorovich, I. I., and Bar-Ziv, E., Combust. Flame 97:61 (1994).
5.
Kantorovich, I. I., and Bar-Ziv, E., Combust.
Flame
97:79 (1994).
6.
Bar-Ziv, E., and Sarofim, A. F., Prog. Ener. Combust. Sci. 17:l (1991).
7. Weiss, Y., and Bar-Ziv, E., Combust.
Flame 95:362
(1993).
8.
Hurt, R. H., Ph.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1987. 9. Dutta, S., Wen, C. Y., and Belt, R. J., Ind. Eng. Chem. Process Des. Deu. 16:20 (1977). 10. Kantorovich, I. I., and Bar-Ziv, E., submitted.. Received 23 August 1994; revised 7 July 1995