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Variation of mineral matter distribution in indivudual pulverized coal particles: Application of the “URN” model
Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 1289-1296
VARIATION OF MINERAL MATTER DISTRIBUTION IN INDIVIDUAL PULVERIZED COAL PARTICLES: APPLICATION OF THE "URN" MODEL L. E. BARTA,* F. HORVATH,** J. M. BEI~R AnD A. F. SAROFIM Department of Chemical Engineering Massachusetts Institute of Technology Cambridge MA. 02139, USA
Introduc~on The mineral species concentration distribution in coal ash is of considerable interest to designers of pulverized coal combustion plant because of the relationship between the slagging-fouling characteristics of the flyash and the amount and type of the mineral matter in the coal. For this purpose it is usual to rely on data of standard (ASTM) bulk chemical analyses of the coal ash. The view is gaining acceptance, however, that the reason for the poor correlation between results of bulk chemical analyses of the coal mineral matter composition and its fouling propensity is due partly to the particle to particle variation of the mineral matter composition of the coal. It is considered that coal ash deposition on heat exchange surfaces in boilers is a strongly non linear function of the concentrations of the potentially damaging mineral species in the fly ash, which, in turn, points to the importance of particle to particle variation of these mineral concentrations in the coal. Mineral matter occurs in coals in three principal forms: a) incorporated in the coal matrix-having been part of the originally coalified organic matter, b) as mineral inclusions--having gotten into the coal by sedimentation during the coalification process, and c) as extraneous ash. This category includes mineral matter liberated from the coal by grinding. Although the organically bound minerals--mainly alkali metals-, and the extraneous ash also play important roles in the formation of fouling deposits, the present discussion focuses on the distribution of the included mineral matter (category b), which is usually the most abundant of the above mentioned three forms of coal minerals. The development of combined scanning electron *EGI Budapest Ostrom u. 25. Hungary. **VEIKI, Budapest Zrinyi u. 3. Hungary.
microscopy (SEM), automated image analysis (AIA) and energy dispersive X-ray spectroscopy (EDAX), pioneered by Huggins and Huffman8 for the analysis of mineral inclusions in coal, makes it possible to analyze a statistically significant number of mineral inclusions (several thousands) in a coal sample at an affordable expense in effort, for determining their distribution in major mineral groups (e.g. Quartz, Mixed Silicates, Pyrites, IUites, etc.) and in different size fractions (e.g. <2, 5 Ixm, 2, 5-5 I~m, 5, 0-10.0 Ixm, 10.0 0-2.0 ;xm, 20.0-40.0 I~m, >40.0 Ixm). However, the magnification applied (300-600 x) in previous studies was not high enough to determine the mineral matter distributions in individual coal particles, and the information so obtained was not sufficient for predicting the interactions of the mineral matter particles during the burnout of the coal. Using higher magnification (1000x), the data obtained from the microscopic examination of the sectioned areas of individual coal particles can be used as input to probability calculations to estimate the real variation of the mineral matter composition in individual coal particles. In the following, the "urn" model, a probabilistic method capable of yielding the random distribution (including the mean value and the variance) of mineral matter species in individual coal particles, is presented, and applied for an actual pulverized coal sample of a Texas lignite. Results of microscopic analyses (SEM-AIA-EDAX) serve as inputs to these calculations. A comparison is then shown of predicted and experimentally determined mineral matter concentration distributions for a narrow size fraction of the sample coal. As input to the calculations experimental data are needed (SEM-AIA-EDAX) of the distributions of mineral inclusion types and sizes in a pulverized coal sample of a narrow particle size range, together with those of the bulk analyses of the coal ash chemical composition. The "urn" model is used first for a narrow size range of coal particles (urns) and monosize mineral inclusions. These calculations lead to a Poisson dis-
1289
1290
MINERAL MA'IWER AND ASH
tribution for the number distribution of mineral particles in individual coal particles. Next, calculations are extended to the consideration of the sum of polysize mineral inclusions. To permit experimental verification model parameters were chosen for the calculations whose distribution can be tested experimentally. Such parameters are the volume fraction, surface fraction or density of the mineral inclusions. While the distributions of these convoluted random variables do not satisfy conditions directly for Poisson distributions, they can be obtained by their Fourier transforms. 1 Further, by expanding the Fourier transforms in Taylor series we can obtain more easily handled approximate expressions for the distributions of the convoluted random variables. These have normal distributions, yield mean values and variances which can then be directly compared with and tested by experimentally determined data. The Urn
Model and the Poisson Law
In the "urn" model it is assumed that balls of different sizes are randomly thrown into a number of urns of uniform size. In the application of the urn model to our problem the urns represent coal particles of a given size while the balls stand for the mineral inclusions of different types and sizes (e.g. quartz of 2.5 to 5.0 Ixm).
P (~qj(m, n) = ms , j = O, 1 . . . . . n) m! n!
-
(3)
n
m" [-[ (j!)~jmfl
j=o It can be shown ~ that the expected value of ~j(m, n) is
9n.__._~ 't ( 1 ) J ( 1 ) E(~ (m, n)) = m j!(n - j)!"
"-j
1-
(4)
which, in the limit can be given as J
lim E(~lS(m, n)) = m ' - - e -h,
.....
jr
(5)
if n and m tend to infinity in such a way that n/m = h remains finite. The second term in the product on the RHS of Eq. (5) is the expression of the Poisson distribution: Pj = hJ/j[ e -x with expected value of k. According to Kolchin et al. 6 the deviation of "qj(m, n) from its expected value: (m* Pj) for any given value of j and h in the limit of m, n ---> oo can be given as
lim p Cqs(m' n) - m Ps
~m-
) x
e-U~/~'~ du (6)
Distribution of Monosize Mineral Matter in a Narrow Size Range of Coal Particles: Let us consider a random distribution of "n" number of balls in "m" number of urns to yield the probability of 1/m for any one ball falling into any one urn. Let ~j(m, n) be the number of urns with
In the present application the value of the ratio:
total volume of a single type of included mineral with diameter "d"
n. da'tr
total volume of coal particles
m.6.V
'7" number of balls, where '7" can take any integer between o and n. The possible values of "qs(m, n) denoted by ms, are related to m and n by:
falls between zero and unity. It follows therefore that 0 < n / m < oo, which, in turn, satisfies the above mentioned condition.
n
E ms = m,
(I)
j=o n
j=o
jm s = n
(2)
According to Von Mises 1 if values of "mr' satisfy equations (1), (2), the joint probability of "qj(m, n), for given values o f j = 0, 1. . . . n can be given as
Distribution of Polysize Mineral Matter in a Narrow Size Range of Coal Particles: In order to extend the "urn" model to an additional size range of minerals, another random variable, v, is introduced which represents the number of balls in one urn. It's distribution function is binomial: 9
MINERAL MATTER DISTRIBUTION IN COAL PARTICLES P(v-j)=
[I].(1)J(I-1)':P
(7)
1291
Relationship Between SEM-AIA-EDAX Data and the Convoluted Random Variables Distribution: The convoluted random variables of
forj = 0, 1. . . . . n. Comparing Eqs. (4) and (7) we can write
1
m P(v = ,~ = E(~li(m, n)),
(8)
which represents the close connection between the two event-spaces ('qj and 1)). Corollarily,
x~ lim P(1) = j) = ~ e -x,
(9)
n, rn--->~ n/m---,X
which is the known Poisson approximation of binomial distribution. Further random variables: vb k = 1, 2 .... t representing other size ranges can be constructed, which stand for other sizes.
"VN,
1 )
- - " us, - - " 1)v calculated according to the forgoing VF VF discussion can be directly related to microscopic (SEM-AIA) measurement data obtained on sectioned coal particles, with corrections made for considerations of stereology.7 The distribution function of these random variables can be obtained by the application of Fourier transforms. It has been proven I that the Fourier transform (FT) of the sum of mutually independent random variables, can be calculated as a product of their FTs. Therefore, ff we determine the FT of each original random variable, the inverse FT of their product provides the distribution function for the sum of the original random variables. The respective FT formulas can be deduced as:
Determination of Convoluted Random Variables" Distributions:
- - VN (0) = exp VF
( fo
fN(D)
The convolution form of such a variable can be given as 1)N =
1)1(m, nx) + 1)2(m, nz) + ... + vt(m, nt)
(10)
where "t" is the number of size classes. The P(1)i(m, ni) = ji) can be .approximated by
(
Poisson distribution \ e -ndm
I:;)
VF
1)s = d2"lT 1)1 + d22"ff 1)2 + . . . + d t2"ff 1)t
(11)
or
d~=
fN(D)
(14)
/
function of Eq. 6 for the values of ji = 0, 1 . . . . . n~ and i = 1, 2 . . . . . t. The 1)N = 1)N (m, Y~=I ni) stands for the total number of mineral particles in a single coal particle, irrespective of their size. If the 1)i variables are mutually independent Poisson random variables, then their sum, 1)N, is also a Poisson random variable with the expected value of If,~=1 ni/m. It should be noted, however, that the sum of additive terms in the convoluted expressions of
d~=
-~
(i| D2qr ) 9e x p \ VF --1 dD
with the error
ji!
vo : - - ~ - 1 ) 1
- - us (O) = exp
dc~1)
+ - - ~ - 1 ) ~ + ... + - 6 1)t
(12)
which are products of lyi and a constant, does not satisfy the condition for a Poisson distribution,
VF Vv (O) = exp
fN(D) ( D37r 9exp iO 6VF
1) dD
(15)
where Nt is the total number of mineral particles and fN (D) is the number based size distribution of the mineral matter to be determined experimentally. In order to approximate the distribution functions of those random variables which have the above Fourier transforms we expand the FT transforms in Taylor series by using the following equality: z~ ~ e iz = 1 + iz - ~ +
e ~ z, real and lal -< 1
This procedure is applied, as an example, for the mineral volume ratio variable:
MINERAL MATTER AND ASH
1292 V--FVv(O) = exp iOM
* R, where
fN(D) Dan dD M = -Nt - s174 m 6VF tr2 R=exp(
Ntm
f~q(D)\ ~ F /
NtiO-a f ~
(16) (17)
dD and
(18)
(/txoDa~r~)
m 6 Jo fN/D/exp \
~-~F/ dD (19)
The first term in the product on the RHS of Eq. (16) is the FT of the Gaussian distribution with the mean value of M and variance of o~. The distribution function of the mineral volume fraction can be approximated therefore by a normal distribution in case the "R" function is small enough compared to the first term in Eq. (16). Similar results can be obtained also for the other two convoluted random variables. It should be noted that the approximation of the distribution of compound Poisson distribution can also be achieved by gamma distribution depending on the actual values of parameters (Nt, m, fN(D), VF) in Eq. (19). Experimental
tioned for the measurement. In the SEM measurements a backscattered electron detector was used to differentiate between mineral inclusions and the coal matrix as the electron beam scans the sample. The electron beam directed by the Particle Recognition Program 12 on the Tracor Northern computer scans a field of 78, 8 by 78, 8 p~m area at a magnification of 1000x until a mineral particle is found, i.e. the brightness of the backscattered radiation falls between the preset minimum and maximum limits. Once a mineral inclusion is found its sectioned area, perimeter, minimum and maximum dimensions are determined, and the shape factor, defined as (perimeter)~/4ar area, and average diameter are calculated. After these measurements an energy dispersive X-ray spectrum (EDAX) was obtained from the center of the particle during a period of 10 seconds and the raw data were evaluated by a computer cede.lZ This program converts the X-ray counts into mineral oxide representations and classifies the particles into 15 mineral classes using their chemical composition. The classification relies on published informationa and divides the mineral particles into classes within which the concentration of individual mineral compounds have small variations. In the following we shall refer to the experimental microscopic analysis and the data processing as "CCSEM" (COmputer Controlled Scanning Electron Microscopy).
The Mineral Matter in the Experimental Coal:
Microscopic Examinations: The mineral inclusions in pulverized coal particles were examined by a Scanning Electron Microscope (SEM) and the data analyzed by a Joel 733 Super Probe that was backed by a Tracor Northern TN 5500 minicomputer as a controller. A sample of pulverized coal particles (0.9 g weight and 63-75 micron size) embedded in epoxy resin was see-
The experimental coal was a Texas lignite of 6375 I~m particle size. The measured mean mineralogical composition of the coal inclusions is given in the Tables I. The mineral matter consists mainly of quartz and illite clay resulting in 76.66% SiO2 and 15.2% A1203 contents. The calculated standard deviations for each oxide show large particle to particle variations. From Table I, it can be seen that
TABLE I Average mineral composition. Mineral types Mixed silicate Quartz Rutile Illite Pyrite/Macrasite Halite Apatite/Evensite Gypsum Baryte Kaolinite Jarosite
Number %
Surface %
Volume %
Mass %
30.33 41.04 0.09 25.65 0.05 0.16 0.06 0.53 0.03 1.28 0.79
17.79 57.01 0.05 22.66 0.05 0.23 0.01 0.31 0.00 1.51 0.38
8.54 73.18 0.01 16.45 0.02 0.17 0.00 0.09 0.00 1.40 0.13
8.05 73.10 0.02 17.05 0.03 0.14 0.00 0.08 0.00 1.40 0.13
MINERAL MATrER DISTRIBUTION IN COAL PARTICLES the mineral class of mixed silicate, due to its smaller size, plays more important role in the number and surface distributions than it does in the cases of the distributions of particle volume and mass.
I.O
_m ~ 0.6
Measured mineral inclusions size distributions form inputs for the calculations of the convoluted random variables' distributions. These latter functions can then be tested by comparison with further processed experimental data. The sequence described below is followed for the measurements and treatment of the CCSEM data.
-d
classes. 3. Obtain a functional relationship for the volume based size distribution by a lognormal distribution function, n
Convoluted Random Variables" Distributions. Comparison with Calculations: The distribution of the volume based total number of mineral particles, (vN/VF) can be calculated by using CCSEM data for each individual coal particle and application of inverse Abelian transformation, s According to our previous discussion (Section 3) this distribution function can be approximated also by a compound Poisson distribution with vN/ VF as a random variable. The best fit of such a distribution function calculated for a 3478 Ixms frame volume using Eq. (13) and the X2 test, I~ is shown in Fig. 1. The fit is accepted at 5% significance level. By taking the mean value of the latter random variable, it is possible to calculate the mean value and variance of the total number of minerals in the coal particle of size of 69 micrometer. The result for both the mean value of the volume based total number of mineral inclusions and its variance is 761. To determine whether the Gaussian approximation with the lognormal number size distribution can be used to approximate the distributions of other convoluted random variables such as the mineral volume fraction (Eq. 16), the relative magnitude of the R(O) function has to be checked. The evaluation of the real and the imaginary parts of the R(@)
I
I
I
I
I
0.8
Measurement of Mineral inclusions Size Distributions:
1. Determine size distribution of mineral particles in a sectioned plane of a coal particle. It is assumed that the mineral particles are spherical. 2. Use inverse Abelian transformation to obtain the volume based particle size-, surface- and volume distributions from the measured area based distributions.T M The particle mass distribution can then be calculated from the sizevolume distribution using values of densities corresponding to the chosen mineral matter
I
1293
~ 0.2
O~
O. .0
I
I
I0.0
20.0
I
30.0
40.0
TOTAL NUMBER OF PARTICLES Flc. 1. Total number of mineral particles in the
3478 I~ma frame volume. Measured: solid line; calculated (X2 fitted) by using Poissou distribution: stepped line. Poisson parameter - 15. The fit it is accepted by • test at 5% significance level. function for this particular coal shows that the value of R is small compared to the first term in the Eq. (16) and its effect on the lrl" of the mineral volume ratio random variable can therefore be neglected. The same applies also for the mineral surface and mass/volume ratio random variables. In order to determine whether the CCSEM data can be deduced from normal distirbutions, • tests were applied. The calculated normal and measured distribution functions for the mineral volume as a fraction of the coal volume can be seen in Fig. 2. The fit can be accepted at 5% significance level. The mean value of this random variable is 23.2% while its standard deviation is 7.9%. The relatively high mean value corresponds to the high ash content of the coal and the standard deviation shows the ash content variation from particle to particle. This distribution function can be used to determine the coarse limit size distribution function of the fly ash assuming no fragmentation for the coal (one fly ash particle from one coal particle) and may also serve as an input for the calculations at a given coal fragmentation rate. According to the theory of stereology7 the total mineral surface area in a given coal particle can be calculated by measuring the perimeter of the sectioned mineral particles in the observed area. Using the urn model, normal distribution is predicted for this random variable, also. The comparison of calculated (x2 fitted) and measured distributions of the coal volume based mineral surface area (Fig. 3.) show good agreement. The total mineral surface area in a coal particle is indicative of the probability of
MINERAL MATTER AND ASH
1294
calculate the density distribution of the coal. By taking different mineral classes that have the same specific weight and calculating the weight of the mineral particles in unit coal volume (density) we can deduce their distribution function assuming 1.2 g/cm 3 density for the pure coal matrix. The result of the normal distribution approximation is given in Fig. 4. With reference to expressions 17,zs we can give relationships between the mean values and the variances of the different random variables. By taking lognormal approximation for the number based size distribution of mineral inclusions, Eq. 20 and 21, respectively, are such examples for the case of the mineral volume ratio:
#
Z
o o.e
O0
#
' J ' ~
'
'
,
,
L
M. . . . . exp m 6VF
9 0.0 20.0 40.0 60.0 MINERAL VOLUME F R A C T I O N •
FIC. 2. Mineral volume fraction distribution in 69 I~m TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean-23.2%, standard deviation - 7.9%. The fit it is accepted by X~ test at 5% significance level. interaction (sintering and coalescence) between the included and the organically bound mineral matter in the coal during combustion. The volume fraction distribution of the mineral particles can be used to 1.0
o'2 .
.
l
l
i
i
i
i
i,-'9"-11
~
F
-
i
.
(20)
exp (18 o'~log.+ 6 Mlo~)
(21)
m
Equation (20) shows that having larger number of minerals in the coal particle and coarser included mineral matter size, the mineral volume fraction also tends to be larger. This trend is shown also for the variance, ~z, in Eq. 21. Similar result can be obtained for every other convoluted random variable, including the mineral surface ratio and the coal
LO i
.
Cr]o~+ 3 Mlo~
I
---~---:l
:-:
i
....
i /
~ 0.8
g
0.8
o.6
~
0.4
/
.
4
~ 0.2 I
O0 .--~/"I
0
,
,
t
i
i
I
t
I
I
0.0 0.2 0.4 0.6 MINERAL SURFACE/COAL VOLUME (I//~m) FiG. 3. Mineral surface/coal volume distribution
in 69 Ixm TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean - 0.211 (1/ ixm), standard deviation - 0.077 (1/Ixm). The fit it is accepted by X2 test at 5% significance level.
0.0
---'--L
I.I
~,~
i
i
i
I
I
I
I
I
1.3 1.5 1.7 1.9 2.t COAL PARTICLE DENSITY ( g / c m 3 )
FIG. 4. Coal particle density distribution in 69 Ixm TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean - -1.537 (g/l~m3), standard deviation - 0.11 (g/Ixm3). The fit it is accepted by • test at 5% significance level.
MINERAL MAq'TER DISTRIBUTION IN COAL PARTICLES
1295
TABLE II Comparison of parameters of predicted and measured distribution functions
Number of minerals in mean the frame volume stand, dev.
Confidence interval at 95% level Student and X2 test for
Measured values
mean
variance
Calculated values by using predicted distribution functions
15.38 3.93
(14.58-16.18) --
-(3.4-4.5)
15" X/15*
Mineral volume fraction
mean stand, dev.
0.23 0.079
(0.215-0.247) --
-(0.069-0.092)
0.22 0.16
Mineral surface per coal volume ratio (l/~,m)
mean stand, dev.
0.211 0.077
(0.201-0236) --
-(0.0767-0.102)
0.219 0.088
Coal density
mean
1.537
(1.515-1.559)
--
1.548
stand, dev.
0.11
(g/cm a)
--
(0.096-0.128)
0.121
*Selected values.
particle density. The calculated confidence intervals for the means and variances of these four random variables are shown in Table II. The calculated data show good agreement with results of measurements. The variance for the volume fraction distribution is, however, overestimated. This may be due to the bias in the stereological calculation caused by the relatively small number of coal particles analyzed. It should also be noted that any uncertainty in the determination of the mineral size distribution is amplified in the calculations of mean values and variances due to the exponential terms in the expressions of (20) and (21).
scanning electron microscopic measurements yielded data on a Texas Lignite for comparison with the above mentioned calculated distributions. These comparisons gave good agreement for the distribution functions of mineral number, volume and mass, including the statistical parameters of these distributions: the mean values and variances.
Nomenclature m~ m n
Conclusions Analytic solution can be obtained to the problem of the variations of mineral inclusion size and composition distributions from coal particle to coal particle by the application of probability theory. The number distribution of polysize minerals in a narrow size range of coal particles can be described by a Poisson distribution which extends to a compound Poisson distribution for the case of convoluted random variables (e.g. mineral volume fraction in a single coal particle), In order to approximate parameters which are experimentally measurable such as the volume or surface fraction of mineral matter per coal particle, Fourier transforms of compound Poisson random variables have been obtained. Statistical laws of particle stereology applied to computer controlled
~(m, n) h
number of urns containing ' f ' balls number of urns number of balls random variable; n u m b e r of urns containing ")~" balls. mean value of number of balls in one Urn.
Pj trj
di V v
vi(m, VN
vs
hi)
value of poisson distribution standard deviation mineral particle diameter of ith size class (middle value) volume of one coal particle random variable; number of balls in one urn random variable; number of balls of 'T'th size per urn random variable; number of balls in one urn regardless of their sizes (convolution of vi's) random variable; total surface area of balls in one urn.
1296 vv VF AF 1 F - - " VN(| VF
MINERAL MATTER AND ASH random variable; total volume of balls in one urn. frame volume measured frame area Fourier transform of random vari1 able V--FVN
N, Nt
total number of mineral particles number based size distribution of mineral particles (Mlog~, ~rlog~) lognormal distribution's parameters M mean value of mineral volume ratio random variables o~ variance of mineral volume ratio random variable
Acknowledgments The research was supported by the United States DepartmentrOf Energy (PETC) under a Grant RA 22-86 PC90751 (subcontract from Physical Science, Inc.), and a Contract D E / A C 22-89 PC88654 (subcontract from Combustion Engineering, Inc.). Two of the authors (Laszlo E. Barta and Ferenc Horvath) thank their respective institutes in Hungary (EGI and VEIKI) and the U.S. Fulbright Foundation for financial support. The authors thank Shin Gyoo Kang and Steve Recca for their assistance in the combustion experiments and the microscopic analyses respectively, and Peter Hajnal for his advice in the theoretical part of our study. Bonnie Caputo's valuable assistance with the preparation of this manuscript is gratefully acknowledged.
REFERENCES 1. VON MISES, R.: Mathematical Theory of Probability and Statistics, New York, Academic Press, 1964. 2. WICKSELL, S. D.: The Corpuscle Problem, Biometrika, 17, p. 84, 1925. 3. MORAN, P. A. P.: The Probabilistic Basis of Stereology. Suppl. Adv. Appl. Prob. 69-92, 1972. 4. SNEDDON, I. N.: Mixed Boundary Value Problems in Potential Theory, Amsterdam, North Holland, Pub. Co., 1966. 5. GREEN, C. D.: Integral Equation Methods, New York, Barnes & Noble, 1969. 6. KOLCHIN, V. F., SEVASTYANOV,n. A., AND CHISTYAKOV, V. P.: Random allocations, Washington, V. H. Winston, 1978. 7. UNDERWOOD, E. E.: Quantitative Stereology, Addison-Wesley Publishing Company, 1970. 8. HUGGINS, F. E., KOSMACK,D. A. HUFFMAN, G. P., AND LEE, E. R. L.: Coal Mineralogies by SEM Automated Image Analysis. Scanning Electron Microscopy, Vol. I. p. 531, 1980. 9. JOHNSON, N. L., AND KOTZ, S.: Urn Models and their Applications. New York, Wiley, 1977. 10. HAIGHT, F. A.: Handbook of the Poisson Distribution. New York Wiley, 1967. 11. KOLMOGOROV,A. N.: l]ber das logritmisch normale verfeilungs-gesetz der dimensionen der teilchen bei zerstfickelung. Compts Rendus (Doklady) de' Academic des Sciences de I'URSS, 1941, Vel. xxxi. No. 2. 12. LOEHDEN, D. O. : The formation of fouling and slaging Deposits in Pulverized Coal Combustion. theses. MIT Chem. Eng. Dep. 1987.
VARIATION OF MINERAL MATTER DISTRIBUTION IN INDIVIDUAL PULVERIZED COAL PARTICLES: APPLICATION OF THE "URN" MODEL L. E. BARTA,* F. HORVATH,** J. M. BEI~R AnD A. F. SAROFIM Department of Chemical Engineering Massachusetts Institute of Technology Cambridge MA. 02139, USA
Introduc~on The mineral species concentration distribution in coal ash is of considerable interest to designers of pulverized coal combustion plant because of the relationship between the slagging-fouling characteristics of the flyash and the amount and type of the mineral matter in the coal. For this purpose it is usual to rely on data of standard (ASTM) bulk chemical analyses of the coal ash. The view is gaining acceptance, however, that the reason for the poor correlation between results of bulk chemical analyses of the coal mineral matter composition and its fouling propensity is due partly to the particle to particle variation of the mineral matter composition of the coal. It is considered that coal ash deposition on heat exchange surfaces in boilers is a strongly non linear function of the concentrations of the potentially damaging mineral species in the fly ash, which, in turn, points to the importance of particle to particle variation of these mineral concentrations in the coal. Mineral matter occurs in coals in three principal forms: a) incorporated in the coal matrix-having been part of the originally coalified organic matter, b) as mineral inclusions--having gotten into the coal by sedimentation during the coalification process, and c) as extraneous ash. This category includes mineral matter liberated from the coal by grinding. Although the organically bound minerals--mainly alkali metals-, and the extraneous ash also play important roles in the formation of fouling deposits, the present discussion focuses on the distribution of the included mineral matter (category b), which is usually the most abundant of the above mentioned three forms of coal minerals. The development of combined scanning electron *EGI Budapest Ostrom u. 25. Hungary. **VEIKI, Budapest Zrinyi u. 3. Hungary.
microscopy (SEM), automated image analysis (AIA) and energy dispersive X-ray spectroscopy (EDAX), pioneered by Huggins and Huffman8 for the analysis of mineral inclusions in coal, makes it possible to analyze a statistically significant number of mineral inclusions (several thousands) in a coal sample at an affordable expense in effort, for determining their distribution in major mineral groups (e.g. Quartz, Mixed Silicates, Pyrites, IUites, etc.) and in different size fractions (e.g. <2, 5 Ixm, 2, 5-5 I~m, 5, 0-10.0 Ixm, 10.0 0-2.0 ;xm, 20.0-40.0 I~m, >40.0 Ixm). However, the magnification applied (300-600 x) in previous studies was not high enough to determine the mineral matter distributions in individual coal particles, and the information so obtained was not sufficient for predicting the interactions of the mineral matter particles during the burnout of the coal. Using higher magnification (1000x), the data obtained from the microscopic examination of the sectioned areas of individual coal particles can be used as input to probability calculations to estimate the real variation of the mineral matter composition in individual coal particles. In the following, the "urn" model, a probabilistic method capable of yielding the random distribution (including the mean value and the variance) of mineral matter species in individual coal particles, is presented, and applied for an actual pulverized coal sample of a Texas lignite. Results of microscopic analyses (SEM-AIA-EDAX) serve as inputs to these calculations. A comparison is then shown of predicted and experimentally determined mineral matter concentration distributions for a narrow size fraction of the sample coal. As input to the calculations experimental data are needed (SEM-AIA-EDAX) of the distributions of mineral inclusion types and sizes in a pulverized coal sample of a narrow particle size range, together with those of the bulk analyses of the coal ash chemical composition. The "urn" model is used first for a narrow size range of coal particles (urns) and monosize mineral inclusions. These calculations lead to a Poisson dis-
1289
1290
MINERAL MA'IWER AND ASH
tribution for the number distribution of mineral particles in individual coal particles. Next, calculations are extended to the consideration of the sum of polysize mineral inclusions. To permit experimental verification model parameters were chosen for the calculations whose distribution can be tested experimentally. Such parameters are the volume fraction, surface fraction or density of the mineral inclusions. While the distributions of these convoluted random variables do not satisfy conditions directly for Poisson distributions, they can be obtained by their Fourier transforms. 1 Further, by expanding the Fourier transforms in Taylor series we can obtain more easily handled approximate expressions for the distributions of the convoluted random variables. These have normal distributions, yield mean values and variances which can then be directly compared with and tested by experimentally determined data. The Urn
Model and the Poisson Law
In the "urn" model it is assumed that balls of different sizes are randomly thrown into a number of urns of uniform size. In the application of the urn model to our problem the urns represent coal particles of a given size while the balls stand for the mineral inclusions of different types and sizes (e.g. quartz of 2.5 to 5.0 Ixm).
P (~qj(m, n) = ms , j = O, 1 . . . . . n) m! n!
-
(3)
n
m" [-[ (j!)~jmfl
j=o It can be shown ~ that the expected value of ~j(m, n) is
9n.__._~ 't ( 1 ) J ( 1 ) E(~ (m, n)) = m j!(n - j)!"
"-j
1-
(4)
which, in the limit can be given as J
lim E(~lS(m, n)) = m ' - - e -h,
.....
jr
(5)
if n and m tend to infinity in such a way that n/m = h remains finite. The second term in the product on the RHS of Eq. (5) is the expression of the Poisson distribution: Pj = hJ/j[ e -x with expected value of k. According to Kolchin et al. 6 the deviation of "qj(m, n) from its expected value: (m* Pj) for any given value of j and h in the limit of m, n ---> oo can be given as
lim p Cqs(m' n) - m Ps
~m-
) x
e-U~/~'~ du (6)
Distribution of Monosize Mineral Matter in a Narrow Size Range of Coal Particles: Let us consider a random distribution of "n" number of balls in "m" number of urns to yield the probability of 1/m for any one ball falling into any one urn. Let ~j(m, n) be the number of urns with
In the present application the value of the ratio:
total volume of a single type of included mineral with diameter "d"
n. da'tr
total volume of coal particles
m.6.V
'7" number of balls, where '7" can take any integer between o and n. The possible values of "qs(m, n) denoted by ms, are related to m and n by:
falls between zero and unity. It follows therefore that 0 < n / m < oo, which, in turn, satisfies the above mentioned condition.
n
E ms = m,
(I)
j=o n
j=o
jm s = n
(2)
According to Von Mises 1 if values of "mr' satisfy equations (1), (2), the joint probability of "qj(m, n), for given values o f j = 0, 1. . . . n can be given as
Distribution of Polysize Mineral Matter in a Narrow Size Range of Coal Particles: In order to extend the "urn" model to an additional size range of minerals, another random variable, v, is introduced which represents the number of balls in one urn. It's distribution function is binomial: 9
MINERAL MATTER DISTRIBUTION IN COAL PARTICLES P(v-j)=
[I].(1)J(I-1)':P
(7)
1291
Relationship Between SEM-AIA-EDAX Data and the Convoluted Random Variables Distribution: The convoluted random variables of
forj = 0, 1. . . . . n. Comparing Eqs. (4) and (7) we can write
1
m P(v = ,~ = E(~li(m, n)),
(8)
which represents the close connection between the two event-spaces ('qj and 1)). Corollarily,
x~ lim P(1) = j) = ~ e -x,
(9)
n, rn--->~ n/m---,X
which is the known Poisson approximation of binomial distribution. Further random variables: vb k = 1, 2 .... t representing other size ranges can be constructed, which stand for other sizes.
"VN,
1 )
- - " us, - - " 1)v calculated according to the forgoing VF VF discussion can be directly related to microscopic (SEM-AIA) measurement data obtained on sectioned coal particles, with corrections made for considerations of stereology.7 The distribution function of these random variables can be obtained by the application of Fourier transforms. It has been proven I that the Fourier transform (FT) of the sum of mutually independent random variables, can be calculated as a product of their FTs. Therefore, ff we determine the FT of each original random variable, the inverse FT of their product provides the distribution function for the sum of the original random variables. The respective FT formulas can be deduced as:
Determination of Convoluted Random Variables" Distributions:
- - VN (0) = exp VF
( fo
fN(D)
The convolution form of such a variable can be given as 1)N =
1)1(m, nx) + 1)2(m, nz) + ... + vt(m, nt)
(10)
where "t" is the number of size classes. The P(1)i(m, ni) = ji) can be .approximated by
(
Poisson distribution \ e -ndm
I:;)
VF
1)s = d2"lT 1)1 + d22"ff 1)2 + . . . + d t2"ff 1)t
(11)
or
d~=
fN(D)
(14)
/
function of Eq. 6 for the values of ji = 0, 1 . . . . . n~ and i = 1, 2 . . . . . t. The 1)N = 1)N (m, Y~=I ni) stands for the total number of mineral particles in a single coal particle, irrespective of their size. If the 1)i variables are mutually independent Poisson random variables, then their sum, 1)N, is also a Poisson random variable with the expected value of If,~=1 ni/m. It should be noted, however, that the sum of additive terms in the convoluted expressions of
d~=
-~
(i| D2qr ) 9e x p \ VF --1 dD
with the error
ji!
vo : - - ~ - 1 ) 1
- - us (O) = exp
dc~1)
+ - - ~ - 1 ) ~ + ... + - 6 1)t
(12)
which are products of lyi and a constant, does not satisfy the condition for a Poisson distribution,
VF Vv (O) = exp
fN(D) ( D37r 9exp iO 6VF
1) dD
(15)
where Nt is the total number of mineral particles and fN (D) is the number based size distribution of the mineral matter to be determined experimentally. In order to approximate the distribution functions of those random variables which have the above Fourier transforms we expand the FT transforms in Taylor series by using the following equality: z~ ~ e iz = 1 + iz - ~ +
e ~ z, real and lal -< 1
This procedure is applied, as an example, for the mineral volume ratio variable:
MINERAL MATTER AND ASH
1292 V--FVv(O) = exp iOM
* R, where
fN(D) Dan dD M = -Nt - s174 m 6VF tr2 R=exp(
Ntm
f~q(D)\ ~ F /
NtiO-a f ~
(16) (17)
dD and
(18)
(/txoDa~r~)
m 6 Jo fN/D/exp \
~-~F/ dD (19)
The first term in the product on the RHS of Eq. (16) is the FT of the Gaussian distribution with the mean value of M and variance of o~. The distribution function of the mineral volume fraction can be approximated therefore by a normal distribution in case the "R" function is small enough compared to the first term in Eq. (16). Similar results can be obtained also for the other two convoluted random variables. It should be noted that the approximation of the distribution of compound Poisson distribution can also be achieved by gamma distribution depending on the actual values of parameters (Nt, m, fN(D), VF) in Eq. (19). Experimental
tioned for the measurement. In the SEM measurements a backscattered electron detector was used to differentiate between mineral inclusions and the coal matrix as the electron beam scans the sample. The electron beam directed by the Particle Recognition Program 12 on the Tracor Northern computer scans a field of 78, 8 by 78, 8 p~m area at a magnification of 1000x until a mineral particle is found, i.e. the brightness of the backscattered radiation falls between the preset minimum and maximum limits. Once a mineral inclusion is found its sectioned area, perimeter, minimum and maximum dimensions are determined, and the shape factor, defined as (perimeter)~/4ar area, and average diameter are calculated. After these measurements an energy dispersive X-ray spectrum (EDAX) was obtained from the center of the particle during a period of 10 seconds and the raw data were evaluated by a computer cede.lZ This program converts the X-ray counts into mineral oxide representations and classifies the particles into 15 mineral classes using their chemical composition. The classification relies on published informationa and divides the mineral particles into classes within which the concentration of individual mineral compounds have small variations. In the following we shall refer to the experimental microscopic analysis and the data processing as "CCSEM" (COmputer Controlled Scanning Electron Microscopy).
The Mineral Matter in the Experimental Coal:
Microscopic Examinations: The mineral inclusions in pulverized coal particles were examined by a Scanning Electron Microscope (SEM) and the data analyzed by a Joel 733 Super Probe that was backed by a Tracor Northern TN 5500 minicomputer as a controller. A sample of pulverized coal particles (0.9 g weight and 63-75 micron size) embedded in epoxy resin was see-
The experimental coal was a Texas lignite of 6375 I~m particle size. The measured mean mineralogical composition of the coal inclusions is given in the Tables I. The mineral matter consists mainly of quartz and illite clay resulting in 76.66% SiO2 and 15.2% A1203 contents. The calculated standard deviations for each oxide show large particle to particle variations. From Table I, it can be seen that
TABLE I Average mineral composition. Mineral types Mixed silicate Quartz Rutile Illite Pyrite/Macrasite Halite Apatite/Evensite Gypsum Baryte Kaolinite Jarosite
Number %
Surface %
Volume %
Mass %
30.33 41.04 0.09 25.65 0.05 0.16 0.06 0.53 0.03 1.28 0.79
17.79 57.01 0.05 22.66 0.05 0.23 0.01 0.31 0.00 1.51 0.38
8.54 73.18 0.01 16.45 0.02 0.17 0.00 0.09 0.00 1.40 0.13
8.05 73.10 0.02 17.05 0.03 0.14 0.00 0.08 0.00 1.40 0.13
MINERAL MATrER DISTRIBUTION IN COAL PARTICLES the mineral class of mixed silicate, due to its smaller size, plays more important role in the number and surface distributions than it does in the cases of the distributions of particle volume and mass.
I.O
_m ~ 0.6
Measured mineral inclusions size distributions form inputs for the calculations of the convoluted random variables' distributions. These latter functions can then be tested by comparison with further processed experimental data. The sequence described below is followed for the measurements and treatment of the CCSEM data.
-d
classes. 3. Obtain a functional relationship for the volume based size distribution by a lognormal distribution function, n
Convoluted Random Variables" Distributions. Comparison with Calculations: The distribution of the volume based total number of mineral particles, (vN/VF) can be calculated by using CCSEM data for each individual coal particle and application of inverse Abelian transformation, s According to our previous discussion (Section 3) this distribution function can be approximated also by a compound Poisson distribution with vN/ VF as a random variable. The best fit of such a distribution function calculated for a 3478 Ixms frame volume using Eq. (13) and the X2 test, I~ is shown in Fig. 1. The fit is accepted at 5% significance level. By taking the mean value of the latter random variable, it is possible to calculate the mean value and variance of the total number of minerals in the coal particle of size of 69 micrometer. The result for both the mean value of the volume based total number of mineral inclusions and its variance is 761. To determine whether the Gaussian approximation with the lognormal number size distribution can be used to approximate the distributions of other convoluted random variables such as the mineral volume fraction (Eq. 16), the relative magnitude of the R(O) function has to be checked. The evaluation of the real and the imaginary parts of the R(@)
I
I
I
I
I
0.8
Measurement of Mineral inclusions Size Distributions:
1. Determine size distribution of mineral particles in a sectioned plane of a coal particle. It is assumed that the mineral particles are spherical. 2. Use inverse Abelian transformation to obtain the volume based particle size-, surface- and volume distributions from the measured area based distributions.T M The particle mass distribution can then be calculated from the sizevolume distribution using values of densities corresponding to the chosen mineral matter
I
1293
~ 0.2
O~
O. .0
I
I
I0.0
20.0
I
30.0
40.0
TOTAL NUMBER OF PARTICLES Flc. 1. Total number of mineral particles in the
3478 I~ma frame volume. Measured: solid line; calculated (X2 fitted) by using Poissou distribution: stepped line. Poisson parameter - 15. The fit it is accepted by • test at 5% significance level. function for this particular coal shows that the value of R is small compared to the first term in the Eq. (16) and its effect on the lrl" of the mineral volume ratio random variable can therefore be neglected. The same applies also for the mineral surface and mass/volume ratio random variables. In order to determine whether the CCSEM data can be deduced from normal distirbutions, • tests were applied. The calculated normal and measured distribution functions for the mineral volume as a fraction of the coal volume can be seen in Fig. 2. The fit can be accepted at 5% significance level. The mean value of this random variable is 23.2% while its standard deviation is 7.9%. The relatively high mean value corresponds to the high ash content of the coal and the standard deviation shows the ash content variation from particle to particle. This distribution function can be used to determine the coarse limit size distribution function of the fly ash assuming no fragmentation for the coal (one fly ash particle from one coal particle) and may also serve as an input for the calculations at a given coal fragmentation rate. According to the theory of stereology7 the total mineral surface area in a given coal particle can be calculated by measuring the perimeter of the sectioned mineral particles in the observed area. Using the urn model, normal distribution is predicted for this random variable, also. The comparison of calculated (x2 fitted) and measured distributions of the coal volume based mineral surface area (Fig. 3.) show good agreement. The total mineral surface area in a coal particle is indicative of the probability of
MINERAL MATTER AND ASH
1294
calculate the density distribution of the coal. By taking different mineral classes that have the same specific weight and calculating the weight of the mineral particles in unit coal volume (density) we can deduce their distribution function assuming 1.2 g/cm 3 density for the pure coal matrix. The result of the normal distribution approximation is given in Fig. 4. With reference to expressions 17,zs we can give relationships between the mean values and the variances of the different random variables. By taking lognormal approximation for the number based size distribution of mineral inclusions, Eq. 20 and 21, respectively, are such examples for the case of the mineral volume ratio:
#
Z
o o.e
O0
#
' J ' ~
'
'
,
,
L
M. . . . . exp m 6VF
9 0.0 20.0 40.0 60.0 MINERAL VOLUME F R A C T I O N •
FIC. 2. Mineral volume fraction distribution in 69 I~m TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean-23.2%, standard deviation - 7.9%. The fit it is accepted by X~ test at 5% significance level. interaction (sintering and coalescence) between the included and the organically bound mineral matter in the coal during combustion. The volume fraction distribution of the mineral particles can be used to 1.0
o'2 .
.
l
l
i
i
i
i
i,-'9"-11
~
F
-
i
.
(20)
exp (18 o'~log.+ 6 Mlo~)
(21)
m
Equation (20) shows that having larger number of minerals in the coal particle and coarser included mineral matter size, the mineral volume fraction also tends to be larger. This trend is shown also for the variance, ~z, in Eq. 21. Similar result can be obtained for every other convoluted random variable, including the mineral surface ratio and the coal
LO i
.
Cr]o~+ 3 Mlo~
I
---~---:l
:-:
i
....
i /
~ 0.8
g
0.8
o.6
~
0.4
/
.
4
~ 0.2 I
O0 .--~/"I
0
,
,
t
i
i
I
t
I
I
0.0 0.2 0.4 0.6 MINERAL SURFACE/COAL VOLUME (I//~m) FiG. 3. Mineral surface/coal volume distribution
in 69 Ixm TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean - 0.211 (1/ ixm), standard deviation - 0.077 (1/Ixm). The fit it is accepted by X2 test at 5% significance level.
0.0
---'--L
I.I
~,~
i
i
i
I
I
I
I
I
1.3 1.5 1.7 1.9 2.t COAL PARTICLE DENSITY ( g / c m 3 )
FIG. 4. Coal particle density distribution in 69 Ixm TEXAS lignite. Measured: dashed line; calculated (X2 fitted) by using normal distribution: solid line. Normal parameters: mean - -1.537 (g/l~m3), standard deviation - 0.11 (g/Ixm3). The fit it is accepted by • test at 5% significance level.
MINERAL MAq'TER DISTRIBUTION IN COAL PARTICLES
1295
TABLE II Comparison of parameters of predicted and measured distribution functions
Number of minerals in mean the frame volume stand, dev.
Confidence interval at 95% level Student and X2 test for
Measured values
mean
variance
Calculated values by using predicted distribution functions
15.38 3.93
(14.58-16.18) --
-(3.4-4.5)
15" X/15*
Mineral volume fraction
mean stand, dev.
0.23 0.079
(0.215-0.247) --
-(0.069-0.092)
0.22 0.16
Mineral surface per coal volume ratio (l/~,m)
mean stand, dev.
0.211 0.077
(0.201-0236) --
-(0.0767-0.102)
0.219 0.088
Coal density
mean
1.537
(1.515-1.559)
--
1.548
stand, dev.
0.11
(g/cm a)
--
(0.096-0.128)
0.121
*Selected values.
particle density. The calculated confidence intervals for the means and variances of these four random variables are shown in Table II. The calculated data show good agreement with results of measurements. The variance for the volume fraction distribution is, however, overestimated. This may be due to the bias in the stereological calculation caused by the relatively small number of coal particles analyzed. It should also be noted that any uncertainty in the determination of the mineral size distribution is amplified in the calculations of mean values and variances due to the exponential terms in the expressions of (20) and (21).
scanning electron microscopic measurements yielded data on a Texas Lignite for comparison with the above mentioned calculated distributions. These comparisons gave good agreement for the distribution functions of mineral number, volume and mass, including the statistical parameters of these distributions: the mean values and variances.
Nomenclature m~ m n
Conclusions Analytic solution can be obtained to the problem of the variations of mineral inclusion size and composition distributions from coal particle to coal particle by the application of probability theory. The number distribution of polysize minerals in a narrow size range of coal particles can be described by a Poisson distribution which extends to a compound Poisson distribution for the case of convoluted random variables (e.g. mineral volume fraction in a single coal particle), In order to approximate parameters which are experimentally measurable such as the volume or surface fraction of mineral matter per coal particle, Fourier transforms of compound Poisson random variables have been obtained. Statistical laws of particle stereology applied to computer controlled
~(m, n) h
number of urns containing ' f ' balls number of urns number of balls random variable; n u m b e r of urns containing ")~" balls. mean value of number of balls in one Urn.
Pj trj
di V v
vi(m, VN
vs
hi)
value of poisson distribution standard deviation mineral particle diameter of ith size class (middle value) volume of one coal particle random variable; number of balls in one urn random variable; number of balls of 'T'th size per urn random variable; number of balls in one urn regardless of their sizes (convolution of vi's) random variable; total surface area of balls in one urn.
1296 vv VF AF 1 F - - " VN(| VF
MINERAL MATTER AND ASH random variable; total volume of balls in one urn. frame volume measured frame area Fourier transform of random vari1 able V--FVN
N, Nt
total number of mineral particles number based size distribution of mineral particles (Mlog~, ~rlog~) lognormal distribution's parameters M mean value of mineral volume ratio random variables o~ variance of mineral volume ratio random variable
Acknowledgments The research was supported by the United States DepartmentrOf Energy (PETC) under a Grant RA 22-86 PC90751 (subcontract from Physical Science, Inc.), and a Contract D E / A C 22-89 PC88654 (subcontract from Combustion Engineering, Inc.). Two of the authors (Laszlo E. Barta and Ferenc Horvath) thank their respective institutes in Hungary (EGI and VEIKI) and the U.S. Fulbright Foundation for financial support. The authors thank Shin Gyoo Kang and Steve Recca for their assistance in the combustion experiments and the microscopic analyses respectively, and Peter Hajnal for his advice in the theoretical part of our study. Bonnie Caputo's valuable assistance with the preparation of this manuscript is gratefully acknowledged.
REFERENCES 1. VON MISES, R.: Mathematical Theory of Probability and Statistics, New York, Academic Press, 1964. 2. WICKSELL, S. D.: The Corpuscle Problem, Biometrika, 17, p. 84, 1925. 3. MORAN, P. A. P.: The Probabilistic Basis of Stereology. Suppl. Adv. Appl. Prob. 69-92, 1972. 4. SNEDDON, I. N.: Mixed Boundary Value Problems in Potential Theory, Amsterdam, North Holland, Pub. Co., 1966. 5. GREEN, C. D.: Integral Equation Methods, New York, Barnes & Noble, 1969. 6. KOLCHIN, V. F., SEVASTYANOV,n. A., AND CHISTYAKOV, V. P.: Random allocations, Washington, V. H. Winston, 1978. 7. UNDERWOOD, E. E.: Quantitative Stereology, Addison-Wesley Publishing Company, 1970. 8. HUGGINS, F. E., KOSMACK,D. A. HUFFMAN, G. P., AND LEE, E. R. L.: Coal Mineralogies by SEM Automated Image Analysis. Scanning Electron Microscopy, Vol. I. p. 531, 1980. 9. JOHNSON, N. L., AND KOTZ, S.: Urn Models and their Applications. New York, Wiley, 1977. 10. HAIGHT, F. A.: Handbook of the Poisson Distribution. New York Wiley, 1967. 11. KOLMOGOROV,A. N.: l]ber das logritmisch normale verfeilungs-gesetz der dimensionen der teilchen bei zerstfickelung. Compts Rendus (Doklady) de' Academic des Sciences de I'URSS, 1941, Vel. xxxi. No. 2. 12. LOEHDEN, D. O. : The formation of fouling and slaging Deposits in Pulverized Coal Combustion. theses. MIT Chem. Eng. Dep. 1987.