Elutriation from fluidized beds

Powder

Technology.

38 (1984)

Elutriation

from

M. COLAKYAN

and 0. LEVENSPIEL

Chemical Engineering (Received

Fluidized

Department.

March 22,1983;

223

223 - 232

Beds

Oregon State University.

in revised form September

SUMMARY

Etutriation characteristics of widely different solids (density from 920 to 5900 kg/m3) were measured in fluidized beds (up to O-9 m in size) having high freeboard (7-5 m), using gas velocities up to 4 m/s. The experimental findings were compared with previously reported results and all the variables were well correlated with a simple empirical expression

Coz-~dlis.

OR 97331

(ti-A-4.)

7.1983)

sion based on the terminal velocity of the elutriating solids and superficial gas velocity_ Earlier work

Summaries of previous invest.igations can be found in Wen and Chen [ 11, Colakyan [ 2], Bachovchin [3], and Kunii and Levenspiel [4]; Appendix A lists some of the proposed literature correlations.

APPARATUS

AND

ESPERDIEKTAL

PROCEDURE

INTRODUCTION

Apparatus

In fluidized beds with wide distribution of particle sizes, the fluidizing gas velocity often exceeds the terminal velocity of the smaller particles_ This results in elutriation of these fines from the bed_ The loss of bed material is associated with a variety of problems, eg. pollution, reaction in cyclone and recycle lines, plugging of filters and changing fluidization characteristics of the bed_ Existing correlations on elutriation rates lead to widely different predictions_ Possible reasons for this are that some of these correlations were based on experiments in small diameter beds, while others only explored a narrow range of fluidizing velocities, particle sizes or densities. Also, in some cases, data were taken with very small freeboard, thus gi-ving an overestimate of the elutriation rate. In order to develop a correlation for elutriation of broad generality, we did experiments with solids of widely different densities and size under a wide range of gas velocities in both large- and small-sized beds having high freeboard_ The results were then compared with the literature predictions and finally were 211 compactly represented with a simple expres-

Three different fluidized beds were used in this study. The largest was a steel and plywood unit, square in cross-section, O-92 m on a side and 7.5 m high. The medium-sized unit was also square in cross-section, 0.3 m on a side, smoothly connected to a 0.34 m steel freeboard pipe 7.5 m high. The small bed consisted of a 0.1 m glass tube flanged to a 0.1 m X 7.5 m steel freeboard pipe_ Pairs of perforated steel plates each sandwiching a 100 mesh screen were used as distributor plates for all three beds. The plates for the 0.9 m bed mere punched with 6-6 mm holes on 19 mm centers in a square pitch, giving 6_7% open area. The O-3 m bed had 5.6 mm holes on 23 mm centers while the O-1 m bed had 2.8 mm holes on 11.5 mm centers_ The pressure drop across these distributors was sufficiently high to achieve equal flows of air through t.he openings_ Fluidizing air was supplied by a Roots positive displacement blower powered by a Caterpillar 325 HP diesel engine_ The exit gas stream from the O-9 m bed carrying elutriated particles was distributed into four cyclone separators operating in parallel, whereas the O-3 m bed used only one cyclone_ The particle-free gas leaving the @ Elsevier SequoialPrinted

in The Netherlands

224

cyclone passed through either a Venturi meter or an orifice meter. The 0.1 m bed used a smaller cyclone and its own orifice meter placed at its air inlet.

Experimental

RESULTS

Analysis

and was found to be -1 mm for polyethylene, -0.8 mm for silica, and -0.3 mm for zirconia. For the 0.9 m and 0.3 m beds, the added fines had a wide size distribution, the coarse and fine material were charged to the bed simultaneously, the whole bed was gently fluidized for a short time, a few seconds, after which the flow of air was rapidly increased to the desired value for the run. For the 0.1 m bed with its small volume, narrow cuts of fine solids were added, one cut per run, to the coarse material. The solids were mised by hand and then air was introduced at the desired rate_ Runs lasted about 1 h for the 0.9 m and the 0.3 m beds, but only 10 min in the 0.1 m bed. Elutriated solids were removed from the cyclone downcomer at various time intervals, from 10 s at the beginning of the run to 30 min at the end, by means of a butterfly valve. The collected samples from the 0.9 m and 0.3 m beds were then sieved for 15 min through No. 20,28,35,42,48,60,65,80, 100, 115, 150, 170, 200, 270, 325 and 400 mesh Tyler screens and weighed, while the samples from 0.1 m bed were weighed directly without sieving. The variables examined in these runs were the column diameter, the bed -weight, the particle density and the operating velocity. Appendix C lists all the combinations of these variables which were studied.

DISCUSSION

of results

The elutriation of particles of size i from a fluidized bed can be represented by the rate espression

procedure

Batch runs using ambient air were made with beds of coarse non-elutriable solids of size distribution shown in Appendis B, plus a small amount, never more than lo%, of fines which were to be elutriated. The surface average diameter of the coarse charging misture was calculated according to the expression

AND

-- =i dt

A-;AXi = -

c&X,

(1)

w

where Kf and Xi are two forms of the elutriation rate constant, and Xi is the mass fraction of solids of size i in the bed of weight U7 and cross-sectional area A. In a batch experiment, if the total weight of solids in the bed is taken to be approximately constant over the period of the esperiment and if attrition and formation of fines is negligible, then integration of eqn. (1) gives, in terms of the weight of particles of size i which have elutriated from the bed Wie, Wi,

= W,,(l

-

ewKi’)

(2)

where IZ’i, is the initial weight size i in the bed. For friable solids one has generation of fines of size i in this situation, the differential representing batch operations -

dWi -= +KilYi dt

of particles of a continuous the bed, Fi- In equation (1) is modified to

Fi

which on integration leads to W,, = Wi,(l

-

e-“i’)

+ Fit -

The first term on the right-hand side represents the removal of solids of size i which are initially present in the bed. The last two terms represent the removal from the bed of particles of size i which are generated in the bed by attrition since time t = 0. Figure 1 compares the prediction of eqns. (2) and (4). Note that a non-zero limiting slope on the elutriation curve shows that attrition is taking place in the bed and allows determination of the rate of generation of fines of size i in the bed. The elutriation and attrition rate constants Xi and Fi were calculated by forming an objective function S:

g

represented the variation of elutriation with particle size and gas velocity:

with attrilion F #O. equation 4 /

t

rate

Iimiling slope Fi gives generation rate of fines We

time. t

Fig. 1. Removal operations_

of fines from a bed. during batch

14, 12.t

10 1

t 5s

t

___*-----______--z-d” ____Q-----

slope

_._y;b;G;---‘---------yti,*

z*

an atiril~ sorid P z

.-

li 08, P 7

r

1

_.

~t..,_.~~~~_~~~~~~~~_~~~___~_ ___; _._.____~_____i 40 lime

adopt

and

estend

this

equation

form

in

the present study.

without altrition Fi = 0. equation 2

60

Effect of bed diameter The earlier experiments were repeated here with the same materials in all three sizes of bed_ The results for all sizes of elutriated silica sand particles were then correlated in the form of eqn_ (5) and are displayed in Fig. 3. The slopes of the lines of best fit are 32.2 for the 0.9 m bed, 31.6 for the 0.3 m bed and 28.9 for the 0.1 m bed. Statistically this variation in slope was found to be not significant at the 5% level. Hence, although we may suspect that elutriation is depressed slightly in small diameter beds, and that it may be worthwhile looking further into this matter, still, until further esperimental evidence is obtained we will conclude that the elutriation in beds from 0.1 m up in size is independent of bed size.

[min]

i;ig.

2. Difference in the amount of fines collected from attriting and non-attriting solids.

S = C [ IVi,_,,(esperiment) n=i -

Wi,_ .I(equation)]2

where Wj,(equation) was chosen to be either eqn. (2) or eqn. (4) and by minimizing S by use of a Gauss-Newton technique with respect to the parameters of interest, Ki and Fi. Figure 2 represents two sets of esperimental data taken in the 0.9 m bed, one for sand, the other for polyethylene. The dashed lines represent the fitted model. Clearly, sand is an attritable solid while polyethylene is not. The values of Ki were transformed into Kf = K,CV/A, a quantity which should be independent of bed geometry. This elutriation constant was then correlated with the variables of the system. Effect of gas velocity In an earlier study made with silica sand in the 0.9 m bed alone, Colakyan et al. [5] found that the following correlation well

Fig. 3. Variation

and gas velocity non-dependence

of elutriation

rate with particle size for silica sand particles. showing of elutriation rate on bed diameter.

Effect of particlc density ar;d the final correlation Figure 4 shows the elutriation data obtained for the three different solids in the three different sized beds. The equations representing the best lines through these data are as follows:

226

0 (l-

J

J

02

---&w-L--

10

Or3 .

(,_!3i]2

$1’

uo

Fig. 4. Correlation for three different solids in three different

-_-

bed sizes.

2

for zirconia

(6)

for silica sand

(7)

for polyethylene

(8)

Fig. 5. General expression for the elutriation constant for all densities of solids, ali particles sizes. and all gas velocities. ---.

‘or----------

2

2

The constants in these equations are close to proportional to the density of the solids. Accounting for this factor, the final correlation for the elutriation constant accounting for all sizes and densities of particles is

Kf=O

Fig. 6. Comparison of the correlation of Yagi and Aochi with the present data.

when Llti2 ~0 (9)

Figure 5 displays all the data taken (29 runs, 118 values of Kf, for three types of particle in three sizes of bed) and the final recommended correlation, eqn. (9). About 85% of the data fall within 35% of the recommended equation. Larger deviations are observed at lower values of the velocity group (1 - Uti/Uo), hence for the largest sizes of particles being elutriated. Appendix C tabulates all the elutriation rate constants obtained and other pertinent variables. Appendix D indicates how the terminal velocity L(tiwas evaluated. Fitting the data to the iiterature correlations Figures 6 to 10 show how all the data of this study fitted the correlations of Yagi and

( $i;&)o-725

~,i”3”[&-+q’~’

Fig_ 7_ Comparison of the correlation of Wen and Hashinger, eqn. (A2). with the present data.

Aochi [6], Wen and Hash’mger [ 71, Tanaka et al. [ 83, Merrick and Highley [9 J and the modified Geldart et aZ_ [lo], which are tabulated in Appendix A.

?

Fig. 8. Comparison of the correlation et al_. eqn. (A3). with the present data.

equ~icm A4. from kkrr;d( ard -7

02

’

Fig. 10. Comparison of the modified correlation oi Geldart et al.. wherein pE is replaced by pa (see eqn. (AS)), with the present data.

of Tanaka

Highfey

--be=

oa ’

04

IO

Fig. 9. Comparison of the correlation of Merrick and Highley, eqn. (A4), with the present data.

Examination of these graphs shows that the earliest of these universal correlations, that of Yagi and Aochi 16 3, fits the data reasonably well (see Fig. 6), with no systematic variation with particle size, density or bed size. The recommended correlation of Geldart et al. TABLE

[lo] was not tested because of difficulties in estimating or calculating pe_ However, a modification of this correlation in which pr was replaced by pp was tested (see Fig. lo), gave a reasonably good fit, but showed a systematic variation with particle density. To evaluate more precisely the goodness of fit of all these recommended correlations and of eqn. (S), their correlation coefficients were evaluated and are displayed in Table 1_ This shows that eqn. (9) fits the data best, the equations of Yagi and Aochi [S] and the modified equation of Geldart et al. [lo] give a reasonable fit, certainly better than the remaining espressions.

CONCLUSIOxS

The present work offers a simple correlation which can be used with beds of very high freeboard and above 0.1 m in diameter,

1

Goodness

of fit of the various correlations

to the data in this study

Investigators

Correlation coefficient R

R2 a measure of the variation removed by the correlation

Yagi and Aochi (1955) Wen and Hashinger (1960) Tanaka et al. (1972) Merrick and Highley (1974) Geldart et al. (1979)* Present work

O-75 0.36 O-24 -0-49 -0_79 0.95

0.56 0.1-l 0.05 O-24 0.62 0.91

*Modified

from the original by the use of pp in place of pe_

228

LIST OF SYMBOLS

K:=O.ollps(l-~~[f]

A cd

when uti < u0

dpi

when uti > u. As shown in Table 1, this correlation fits the data of this study better than the recommended correlations in the literature. Let us examine the form and extremes of this expression. First of all, when the particle size is small enough such that Uti 4 uo, then the volumetric elutriation constant Ki*

P,

---

with units

m3 of solids elutriated m2 of bed-s

g Ki

K; N

I

levels off to a limiting value of O-011, meaning that the flux of fines from the bed is independent of particle size and only depends on the fraction of these solids in the bed. This limit also shows that the flux of these very fine solids is independent of the gas velocity uo_ This result suggests that the limiting phenomenon in the elutriation of these very fine solids is related somehow to the upwelling of fresh solids to the bed surface followed by the release or tearing away of fines from the surface layer of solids. For the particular particle size where uti = uo, eqn. (9) predicts that the elutriation constant should be zero, meaning that there is a limiting particle size which can elutiate and that this limiting size is directly dependent on the gas velocity. For even larger particles (where Uti > uo), eqn. (9) predicts that no elutriation will occur. This is the reason for adding the inequality condition to eqn (9). This finding of zero elutriation for these large particles differs from the findings of Geldart et al. [lo] _ One possible reason for this difference is that our experiments used low concentrations of elutriable solids ((10%) while they used very high concentrations. Because of this, the gas-solid ‘fluid’ leaving their bed had a very high solid loading and thus behaved as a fluid of much higher mean density than in our experiments_ ACKNOWLEDGEMENT

research was carried out in the context of research financed by a grant from the National Science Foundation (USA_) No_ CPE 8026799. This

Fi

t %nf UO uti

W Wi wie

wio xi

cross-sectional area of the bed, m drag coefficient, diameter of particles of size i, the mean of two adjacent screen openings, m rate of generation by attrition of particles of size i, kg/s gravitational acceleration constant, m/s* elutriation rate constant for particles of size i, s-’ elutriation rate constant for particles of size i, kg/(m* s) number of samples of elutriated solids collected in a run time, s minimum fluidization velocity, m/s superficial gas velocity, m/s terminal velocity of particles of size i, m/s

total bed weight, kg mass of the particles of size i in the bed at any time, kg total mass of particles of size i elutriated until time t, kg initial mass of the particles of size i in the bed, kg mass fraction of particles of size i in the bed gas viscosity, kg/(m -s) gas density, kg/m3 solids density, kg/m3 sphericity

REFERENCES C. Y_ Wen and L_ H_ Chen. AZChE J.. 28 (1982)

117_

M-SC. Thesis, Oregon State University, 1980, p_ 3_ D. M. Bachovchin,M.Sc. Thesis, Massachusetts Instituteof Technology. 1978, p_ 31. D. Kunii and 0. LevenspieI, FZuidizalion Engineering. Krieger Publishing Co_. 1977. p_ 304_ M_ Colakyan, N_ Catipovic, G_ Jovanovic and T. J. Fitzgerald, AZChE Symp_ Ser., 77, No_ 205 (1981) 66. S. Yagi and T_ Aochi, cited in Kunii, D. and 0. Levenspiel, FZuidization Engineering. Krieger Publishing Co.. 1977. p_ 315. C. Y_ Wen and R. F. Hashinger. AZChE J_, 6 (1960) 220. I. Tanaka, H. Shinohara, H. Hirosue and Y_ Tanaka, J. Chem. Eng. Japan, 5 (1972) 51_ D. Merrick and J. Highley, AIChE Symp. Ser., 70, No. 137 (1974) 366_ D. Geldart, J. Cullinan. S. Georghiades, D. Gilvray and D. J. Pope. Trans. In&n. Chem. Engrs., 57

M_ Colakyan,

6

7 8 9 10

(1979)

269.

229 11 E. S. Pettyjohn and E. B. Christiansen, Chem. Eng. s+0gr_. 44 (1948) 157_ 12 C. Y- Wen and Y. H. Yu, AIChE J.. I2 (1966) 610.

=

(A31

APPENDIX

A

Literature

correlations

Yagi and Aochi

for estimating

K;

Merrick and Highley

[S]

Ki* ~ P&o

Kfdpi’

191

=B+130ex

(A41 where B = 0.0001 Wen and Hashinger

173

Kif

i

(P,

%i)

Pe(UO= 1.52

23.7

0_0015_

2)

exp(-5_4

(AS)

‘CPibO

where pi is the solids loading the exit gas_

(“.i~~~72s[(uO~tf)2~-I

x 10-s

-

Geldart et al. IlO] Ki’

of the ith size

in

This study P,

x

t

-Pp PP

l-l5

G-1

)

Tanaka et al. [S]

when Uti < u.

Ki*

when Uti 2 u.

P&o-Q) APPENDIX

(9) B

Size distribution Tyler mesh

of bed materials Zirconia

Polyethylene Coarse (W)

Fines (90)

Coarse (S)

Silica sand Fines (%)

Coarse (5)

6/1O 10114 14116 16/20

3-75 27.05 2573 22-64

20128 28132 32135

16.11 3.83 o-79

1-O 17.0 16.0

43-42

0.19

35j42 42148

O-06 0.04

0.38 l-71

23.6

OS3

11.1 l-1.0 a.0 6.3

0-o-I 0.02 O-01 0.01

48160 60165 65 180 aojloo 100 /I15 115/150 150 j170 170 j200 2ooj250 250/270 2701325 3251400 c400

Fines (Z)

0.16 6.62 39-33

14.16 21-63 23.19 x7-4 11.12 6.32 2.66 O-43 0.81

9.26

0.69 2.45 14.32 19.9s 13-45 s3s 4.77 5.60 7.74 l-84 20.78

1-5s 9.90 28-99 17.45 19.2s 8.15 5.97 3-46 2.00 3-22

230 APPENDMC Experimentaiconditionsandfindings (a)The0.9 m bed (A =0.846 m2) d,;ofelutriated particles (W) 2-x 227 192 161 134 384 323 271 227 192 384 323 2'71 227 192 261 134 192 161 134 11-i 96 81 67 58 48 40 19 271 227 192

126 96 81 192 161 126 96

81 271 227 192 161 126 96 81 271 227 192 161 126 96

PS (kg/m3)

920 920 920 920 920

920

920 920 920 920 920

920 920

920 920 920 920 5900 5900 5900 5900 5900 5900 5900 5900 5900 5900 5900 920 920 920

2750 2750 2750 2750 2'750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750

Bed weight, 1%' (kg) 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 350 350 350 350 350 350 350 350 350 350 350 50 50 50 620 620 620 620 620 620 620 620 620 620 620 620 620 620 620 670 670 670 670 670 670

utiofelutriated particles (m/s) 0.91 0.91 0.91 0.91 0.91 1.52 1.52 1.52 1.52 1.52 l-52 1.52 1.52 1.52 1.52 1.52 1.52 2.13 2-13 2.13 2.13 2-13 2.13 2.13 2.13 2-13 2.13 2.13 1.52 1.52 1.52 9-91 0.91 0.91 1.52 1-52 1.52 1.52 1.52 2.13 2.13 2.13 2.13 2.13 2-13 2.13 2.74 2-74 2.74 2-74 2.74 2.74

0.71 0.59 0.48 0.40 0.32 1.06 0.87 O-71 0.59

O-48 1.06 O-87 0.n 0.59 O-48 0.40 0.32 2.06 1.68 1.37 1.13 O-93 0.77 0.62 0.52 0.43 0.35 0.15 0.71 0.59 O-48 0.73 0.50 O-38 l-24 l-02 O-73 0.50 0.38 l-81 l-50 1.24 1.02 0.73 0.50 0.38 1.81 1.50 1.24 1.02 0.73 0.50

O-48 0.92 1.80 2.70 3.28 O-45 l-17 2.21 2.86 3.82 0.46 1.22 2.18 3-21 3.85 4.69

4.60 1.45 3.38 5-29 10-75 15.64 20.88 24.74 32.32 38.11 44.32 45.62 3.86 3.33 3-45 3.05 7.15 9-64 2-48 5-04 8.85 12.21 15.72 2.82 4-82 7-83 8.55 13.80 19.53 21.17 4-31 7-05 9.95 12-78 17-55 23.35 (continued)

"31 (a)The

0.9 mbed

(continued)

dpi ofelutriated

PS

Bed weight, IV

UO

particles (Pm)

(kglm3)

(kg)

(m/s)

Uti ofelutriated particles (m/s)

81 271 227 192 161 126 96

2750 2750 2750 2750 2750 2750 2750

670 670 670 670 670 670 670

2-74 3.66 3.66 3.66 3-66 3.66 3.66

O-38 1.87 1.56 l-24 1.02 0.73 0.50

20.85 699 11.21 11.90 16.93 1821 24.67

uti ofdutriated particles (m/s)

K,

l-81 1.50 l-24 l-02 0.77 O-63 O-50 2-56 2.21 l-81 1.50 1.24 l-02 O-77 O-63 l-50 1.24 0.93 0.77 0.63 0.38 2.50 2.06 1.68 l-37 l-13 0.93 o-77 O-62 O-52 0.43 0.35 0.15 l-57 l-29 l-06 087 o-71 O-59 0.48 0.40 O-20 o-07

l-34 2.94 4.79 6.88 10.25 12.01 1889 l-40 3.96 'i-31 10.34 11.08 16.91 21-57 26.47 2-68 -1.26 7.92 7.41 10.55 15.50 2.47 Q-94 9-57 15.66 21.68 26-38 33.17 40-04 46.90 52.46 54.00 56-62 0.36 O-75 l-50 2-46 3-89 5.59 7-12 8.35 9-65 10.91

(b)The0_3mbed

Kt (kg/(m*s))

(A=0.0929m2)

d,i ofelutriated

PS (kglm3)

Bed weight, W (kg)

UO

particles (w-n) 271 227 192 161 134 114 96 384 323 271 227 192 161 134 114 227 192 161 134 114 81 227 192 161 134 114 96 81 67 58 48 40 19 542 455 384 323 271 227 192 161 89 36

2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 2750 5900 5900 5900 5900 5900 5900 5900 5900 5900 5900 5900 5900 920 920 920 920 920 920 920 920 920 920

65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 24 24 24 24 24 24 43 43 43 43 43 43 43 43 43 43 43 43 19 19 12 19 19 19 19 19 19 19

1.83 1.83 1.83 1.83 1.83 l-83 l-83 3.35 3-35 3.35 3-35 3.35 3-35 3-35 3.35 1.83 1.83 l-83 1.83 l-83 l-83

(m/s)

2.71 2.71 2-74 2-i-Z 2.74 2.74 2-74 2.74 2.74 2.7-I 2.74 2-74 l-83 l-83 1.83 1.83 1.83 l-83 1.83 l-83 l-83 l-83

&g/(m=s))

232 (A

(c)TheO_lmbed dpi of elutriated

particles (pm) 114 96 81 161 134 192 134 134 134 114 114 96 96 134 114 96

= 0.00811m2)

PS (kg/m?

Bed weight, If &g)

2750 2750 2750 2750 2750 2750 2550 2750 2750 2750 2750 2750 2750 2750 2750 2750

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

uti

(mls)

_ ofelutrlated particles (m/s)

I;i* (kg/(m*s))

1.95 1.95 1.95 1.92 1.92 l-92 1.16 1.89 2-47 1.92 2-47 l-89 2.47 O-94 0.94 0.94

0.63 O-50 0.38 l-02 0.77 l-24 O-77 0.77 0.77 0.63 0.63 0.50 OS0 O-77 0.63 O-50

9.86 13.60 16-93 6.95 lo-56 4-85 l-64 12.00 18.08 13.85 20.35 11.71 20.96 0.90 l-64 2.88

UO

APPENDMD

Calcularion

Of

Uti

The termianl veiocity Uti of elutriated particles of size i was calculated and corrected for sphericity Gs according to Wen and Hashinger [ 73 : 0.153g0-71d,i1-14(~~ (Uti)spherical

=

and according [ll]: (%iX,

onspherical

-p,)“-71

&“-2gpo-43 to Pettyjohn

=

0.843

(Cl) and Christiansen

log tc21

The validity of the numerical results obtained from eqns. (Cl) and (C2) was verified by CakLdating (Uti)nonspheed from the charts of C, us. Ret of Pettyjohn and Christiansen [ll]_ They were found to be in good agreement with each other in the range of Reynolds numbers of interest. The sphericities of the particles were estimated using the correlation of Wen and Yu [12]. They were found to be 9, = 0.8 for sand, & = 0.6 for polyethylene particles, and & = 0.8 for zirconia particles, and were assumed to be constant for all sizes_