Homogeneous sedimentation in the centrifugal field

Pam&r

TeehnaZagy-l3sevia Sequoia

Homogeneous LADISLAV

JITKA

Sedimentation

Laxanne-

in

Rimed

in the Centriftxgal

in Pow&r Technology. Uniremr~

of

Field

Brndjord

Bradford. Yorkx {GI. Br~xainj

FIUEDOVA

hfanagemem Cenrre, Uniwrsit~ of Bradford, Bradyord, Yorks. (GL (Receiwd

the Nc’3xrlands

SVAROVSKY School of Stiies

POS:grahaXe

SA_

Septcmbcr

2.1371)

Summary EFaluation

of

sedimentation

Brrram)

reszdts

of

homogeneous

with variable

centrifgaal

time operation

may

be

carried out either by us@ some approximation for soIving the basic drferentiai equation or by assuming the shape of the distribution curue. A method ir described here in which the unknown distribution curz is assumed to follow a three-parameter equation and the parameters are fourul by means of a direct search cunz fitting technique applied to the measured concentrations_ The result@ cumuIatir;e size distribution in terms of a function is very convenient for further use and aaluation. A measure offt is used to test the validity of the arsumption. The we of the method extends to the case of both modes of operation, variable time and zzzriable height, combined if a scanning system is used. The radial scannbzg speedr up tko analysiri and the suggested method is at present the only known way of evaluation in such case2s_

at a fued Ievel and difierent times in a centrifuge to their particIe size distribution was derived by Kamack3 and has no exact general solution of practical utility. It can only be solved either by approximation or by assuming the shape of the unknown distribution curve. Of the available approximation methods the one due to Kamackks is the most widely used as, for instance, with the Simc& centrifuge_ The analysis must, however, go down to only a few per cent of the initial concentration since a straight fine interpolation from zero is assumed for the first point of the lowest concentration, and the results would otherwise be subject to a higher error. If a multiple-parameter function is expected to tit the distribution curve, the basic equation may be solved and the resulting function fitted to the measured concentrations so that the unknown parameters are found_ A method of doing this is described in the fo!lowing_

2 THEORY

1. INTRODUCTION

Information on the particle size distribution of submicron powders may be obtained using sedimentation in a centrifuge_ The method based on the change in powder concentration at a given level, called the incremental variable time method is experimentally very convenient and, therefore, often used_ Ifthe change in mass concentration is detected, as for in?mce in the X-ray’ or the Simca3 centrifuges, the homogeneous technique has to be used where the powder is initially dispersed into a suspension and the sedimentation starts from all points within the centrifuge. The equation relating the concentration of solids Powda TechoL.

5 (1971/X!)

Many routine analyses of different powders with the Simcar and the new X-ray centrifuge (based on the principle described earlier*)evaluated by the Kamacks approximation resulted in size distri-

butions that follow the three-parameter suggested by Harris’ I y=f(x)=l

-

IL’ -

equations

_-. air (Z) 1

where y = f(x) is the cumulative size distribution, xisparticleGze,kisthe

maximum

particle size and

m,rareparameters This equation contains in fact three parameters m, r and ir, and the Schuhman equation6 (for r= 1)

274

L. SVAFCOVSKY, J. FRIEDOVA

or the Gaudin-Meloy equation’ (for m= 1) are its special cases. An attempt was made to assume the shape of the distribution curve in this form and use it for solving the basic equation, which may be written in the form’

The integral in eqn. (11) can be solved resulting in :

co

j 0

5 (--2nf)p

x

D

C -=

p=o -- P!

(2)

f’(x)d(x)dx

where A(x)=exp(-2ax’t)

(3) D is the size of the largest particle in the reference zone, f’(x)=dy/dx is the unknown distribution function, c is powder concentration, co is the initial concentration and CYis a constant for a particular solid, liquid and centrifuge. Both functions y (eqn. 1) and A(x) (eqn. 3) that appear in eqn. (2) can be rewritten in the terms of the Taylor series or, in this case, in the terms of a more special type, the MacLaurin series, since

(4) (5)

D(‘~=+*P) 4m+-Jp

(12)

This solution may be rearranged into a more convenient form for the two following cases: Z-2 Variable time mode of operation For the variable time mode of operation, where the radius of the measurement zone R is a constant, the following relationship may be derived from Stokes’ Law :

(3 (9’

(13)

where t is the time of settling of a particle having size D and tk is the time of settling of a particle having size k_ By means of eqn. (13) the solution in eqn. (12) may be rewritten for two ranges of the ratio r,frk: For t/tk< 1, Le. Ji= D

where x1 = -22axzt and x2

=

Xrn

-

0 resulting in f(x)= and

(6)

k

’

n=r

(7)

For t/tk > 1, Le. k > D

-1m-r 2 4’9 g=r (i)(-lP

2 =

G)*

p (-2LYx%)P c _, (9) p-m p=o Y: it can be shown tha: both series quickly converge for any values of x &-=I0, k and the limit signs may, therefore, be dropped ‘in the following. Differentiation of eqn. (8) leads to : A(x)=lim

(10) Both eqns. (9) and (10) may be now substituted into eqn. (2) : C

-=

CF

D

Q

oqgl w+‘(;),x(x

Ponder

p 1 qm xc-P=. p! qmf2p

Technd,

5 (1971/72)

2 (-2ax2rLc p=o P!

(11)

&

(-lp”

(i)

x

(+)W’2

p1 c-

x P

p=. p! qm+2p

(-2W

(15)

if$=at, - kz = In R/R, = wnst, (16) where R is the radius of the measurement zone and R, is the radius of the free surface. The relationship in cqn. (16) is the Stokes’ Law applied to centrifugal sedimentation’.

22 Scanning made of operation Since an analysis of a fine powder using the variable time mode of operation may take several hours before sufficient data are obtained, this can be shortened by adopting a scanning technique, where both time and height of fall are changing. Scanning is usually started after some time t, of stationary measurement (variable time) in order to utilise the greatest height of fall for an accurate measurement

HOMOGENEOUS

SEDIMENTATION

IN THE

CEKI-RIFUGAL

FIELD

of settling of the coarsest particles in the system_ An assumption is made here that the time t, when the scanning is started is longer than the time fi of settling of the top size particle IL This means that eqn. (15) may be nsed for :/f&c l_ During scanning eqn. (13) cannot bc applied since the radius R is no longer constant_ A more general application of Stokes- Law has to be used here:

TABLE

275 1

A tbwr&caI distribution hariq r,=O582 min for 8=0-415 coqxrcd the approximation

ptamcfuz nith rszdu

m=r=Z and obraincd from

(17) where /3 = In -

R

&

and

is variable with time

(18)

Rl j3, = In is constant. RO

(19)

since R, is the initial radius of the measurement zone (for time 0 < t < t 1)_ The solution in eqn. (12) may be now rewritten using eqn. (17) for the scanning mode of operation : C c,

=

5

(-1)4”

(i)(

$grz

x

q=1

p 1 xc-q’u p=l-Jp! qmi2p

(-2fl)P

(20)

Equation (15) is a special case of eqn. (20) for 8=/I,. Parameter f3 is now a function of ?ime and depends on what kind of scanning is adopted. If a constant speed radial scanning is used, then: R = R, -~(t--fl)

except for the last point where a straight line interpolation from zero was assumed by Kamack. This invariably gives too high a value tc the pcrccntage ?ines-T i.e. undersize the smallest s’%e at which the concentration is determined experimentally_ A direct use of eqns. (14). (15) and (20) for evaluation of the measured data can be achieved b> adopting some of the direct search curve fitting techniques. The problem is a multivariate one and the method due to Hooke and Jee\esR was used for this purpose. It comprises a combination of exploratory and pattern moves starting from a base point The method requires an objective function to be f

ST.CRT MASTER SE!Z.!!NT I PEAD

ml-*

\

/ I

(21)

where t is the speed of scanning and this may be substituted into eqn. (18) for evaluation of fi_

fls%rr

GENERAiION

OF TIME I

I ! E”ALLmnOLI

3F

SETA

1

I I

3TESTS

CALL

MINHJ

txq

Xl I

I

In order to test the solutions (14) (15) a theoretical distribution with parameters m = T= 2 and tk = 0.382 min was used and six values of y and c/c0 were evahrated from eqns (1) and (15) for constant /I=O_415, taking the first four members of the Ma&am-in series and negIecting the rest The calculated values of c/c0 were then evaluated by Kamacks approximation. A good agreement between the two sets of values of y can be seen from the results in Table 1. The results obtained using Kamack’s approximation agree closely with the theoretical results, Ponder

TechwL,

5 (1971/72)

I 3sALuAnow

OF DIAMETERS

I

I

I EVALLlAllON

OF WWLILATlVE

-ON

I w2rrEFDLLaJTFuT I

\ I

Fig l_ Flow &al-t diagram for the compulcr are the inits valua of the tbra palam&~): values of the same.

program used x4 arc the computed

I_ SVAROVSKY.

276 TABLE

J. FRIEDOVA

2

An analJrsis of ‘rare earth oxide” in the X-ray centrifuge without scanning (~9=0 5709) evaluated the proposed method (m= 131, r=Z35. r&=036 min) Diameter

lime (mW

c/co measured

y jKamL7cli)

C/CDfated

(%I

(%I

(%I

by Kamack%approximation and

by

J fproposed merkod] (%I

1 2 4 8 16 32

LO15 1425 1.008 O-713 0.504 0356

TABLE

57-74 4185 2850 19-00 12.58 7.82

8001 6070 4180 27.44 18.39 16 17

57.74 41.80 2860 18.99 lz39 8.00

8155 60.34 41.92 28.10 18.45 1196

3

An ztr~al>sis of ?-are earth oxide” in the X-ray centrifuge with radial constant speed scanning switched on after 15 miq evatuated by the proposed method(m=1.17, r=L07, r,=O_34)

Time

DICZl7lt?tZT

c/co measured

c/c, fitted

WI)

(PI

(%I

(%I

10 1.5 LO 25 3.0 35 4-o 15

2015 1646 1357 1.146 0 976 0 829 0 693 0558

57.74 48.58 41.00 36 10 32.50 27-87 24 10 19.53

57.74 4838 4152 36.25 31.84 27.87 2399 19 85

ttsed, which is in this case the criterion for the best lit given by an inte_gI-alof the squared error: T

F=

r -0

e2 (t)dt

!22)

Y (%I

B

78 86 67.19 5662 48.19 41.07 34.71 28.72

05709 0.5709 05175 04611 04013 03377 0.2698

2272

0 1970

F (eqn. 22), which is a measure of the lit Table 2 presents an example where a ?-are earth oxide” powder was analysed in the X-ray centrifuge with fl= 0.5709 at six times in a two to one progres-

wheree(r)=c,-c,,c,isthemeasuredconcentration and c, is the concentration evaluated by eqn. (14) or (1% A Fortran program was designed following the llow chart diagram in Fig. 1. It first generates times at which measurements were taken and evaluates jl from eqns. (18) and (19) The program then finds the three parameters m, r and tr: by fitting the best line through the measured concentrations c/co by means of the subroutine MINI-U. This is a standard subroutine for the minimalisation procedure due to Hooke and Jeeves8. It calculates the length of each step while the direction of it and the evaluation of the minimalised function (eqn. 22) are calculated by another two subroutines_ The master segment then computes values of y, D and prints out all calculated data including the fitted values of c/co. the optimum parameters,m,r,t,andthevalueofminimumsquares Ponder

Tecka&

5 (1971/72)

W

*mmz

W

stdres ‘pm, Fig 2 Comparison of results of two analof the same material scanning.TbCMLiICpI.CSUlprrscnrs analyserlwithandwithoutradiat the resulting cumulative size di5triiution without scanning while the dottedlineis the distribution obtaiaedwithscanning-

iiOMOGENEOUS

SEDIMENTATION

IN THE

CENTRIFUGAL

sionin time.TheanalysiswasevaluatedbyKamack’s approximation and by the proposed method for comparison_ The value of minimum squares F was O_oooO1,which indicated a very good fit between the measured and fitted concentrations. The resulting cumulative size distributions also agree very ciosely, except for the last point because the last measured concentration is still too high for Kamacks approximation_ The resulting parameters may be used in eqn. (1) for expressing the distribution in terms of a function. The top size k was evaluated from the parameter r, and Stokes’ Law (eqn. 13), and isequal to 3.36 /nn. Table 3 shows an analysis of the same sample as in the previous example, but this time the scanning mode of operation was used. Constant speed scanning was switched on after 1.5 min of variable time operation_ The analysis was again evaluated by means of the program described above; the value of F was OJlOOO9. The two analyses from Tables 2 and 3 may be compared in Fig. 2. They give nearly the same median size but the slope of the two curves is slightly different-This is due to the limited response time of the instrument, which causes delay in the output, and the effect is far more significant in the case of the fast scanning, where it slightly flattens the distribution curve. The top size was found equal :o 3.46 m in this case_ CONCLUSIOd The proposed method proves suitable in cases where the measured sizedistribution can be replaced

FIELD

277

by the function suggested by Harris Since this function contains three parameters, a satisfactory substitution is likely in most cases The suitability of the method is tested by the value of F. which gives a measure of the fit. Unlike the case of Kamacks approximation, the analysis need not be carried out down to a very low percentage with a stationary measuring point (which is often very timeconsuming). but radial scanning can be used in order to speed up the analysis of very fine powders The con\ enient form of results in a function allows an easy further mathematical treatment of results as. for instance, evaluation of different mean diameters and the top size or transfer from the mass distribution to the surface or number distributions

REFEREXCES Martin J. H. Broan P_ K de Brqn Submiaon sizing bJ X-n> talmiquc l.Xrr~$&e Purdckx \\5cy_ Kira York 1963_ 2 C Slata and I_ Cohen, A ccntrifqal pa&& six anzi>rrr. 3. Sri Instr_39 (1962) 614. 3 H. J. Kamack Par&It sin dc:crminztion -w antrih_al pipu sedlmemation. AML Chem. 23 (1951) 6

I J. I

3 T. Allm and L_ Smromk>. A new X-ral scdimcntomcrcr. J_ Ph?_r E Sci Imr_ 3 (1970) 158. 5 C C Harris. AIME Tram_ ?+I (1969) lS7_ 6 R Mmhman. AIM= Tech Pubi 1189 (19-W). 7 .k >I_ Gandin and T_ I’_ Melo>. AI_XiE Tr223 (I%?) 3% S R Hookc and T. -4 Jeacs -Direc! search- solution of numerical and statinical problems Assot= Compuring Mu&,~_a. 8 (1961) 21’