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Integral transform technique in particle sizing
J. Aerosol Scl., Vol. 20, No. 8, pp. 1075-1077, 1989. Printed in Great Britain.
0021-8502/89 $3.00 + 0.00 Pergamon Press plc
INTEGRAL TRANSFORMTECHNIQUEIN PARTICLE 51ZING
Janusz Ycnczka Ph.D. I n s t i t u t e of Electrical Metrology Technical University of Wroclaw ul. bybrzete Wyapla~sklego 27 50-370 Wroclaw, Poland
I. INTRODUCTION Optical methods for particle size meeseuremente have significant advantages over convontlonal means based on sampling. They are non-tntrumlve and rapid. The instruments for ensad01e analysis of the particle size by means of the scattered l i g h t gonerelly use a solution of the Fredholm integral equation of the f i r s t kind. There exist two general methods of finding solution of this equation. The f i r s t method involves ¢apresentlng the kernel with an appcoxlmate analytical expression. In most cases Frauchofer d i f f r a c t i o n theory is used. Accocdtng to this theow the intensity of l i g h t scattered by a particle of a detenctnad radium con be found from the kL~j formula. In this case, the sclutton of an lntegcal equation does not depend on the pacttcle refraction indices and on the surrounding medium. The second method involving expressing the integcal kernel in the fore of numerical function knoan from 14te theow, requires the knowledge of real and imaginary parts of a refractive index. These valuea as y e l l as experimentally measured velume scattering function for the determined angles allow for the estimation of particle size distribution. 2. INVERSION OF FREDHOLMINTEGRALEQUATIONS The particle sizing problem requires that equation 40
i(e) : - ~I °
J312(xe)a2f(a)da. 0
(1)
be inverted, I.e. to provide information on the unknown p a r t i c l e size distribution, f(a) using measurements of scattered l i g h t which can be related to 1(8) Ill. No. the number distribution, f(a) Is not necessarily the function of interest, rather i t amy be some R]mont such as the area or volume distribution function that is desired. I f the area d i s t r i b u t i o n function was of interest, i t could obvtoumly be calculated after the fact fcom the solution, f ( a ) , obtained by the tnversionprncess, but i t is our objective here to consldsr the possib i l i t y that use of some moment of f(a) d i r e c t l y In Fredholm equations might tmrove the overa l l perfomence of the inversion 121. There are a nu~er of methods for inverting Integral equations in general, and Fredholm equation in particular. These can be divided into: reduction to a discrete linear system by numerical quadrature after T~amy 131; methods using functional analysls involving expansion of both the measured and urknoM~ functions In teems of etgenfunctton after 6ertero and Pike 141; singular functions after Berte¢o at. el. 151, or integral trensfocme i . e . analytical inversion of Fredholm equation. 3. INTEGRALTRANSFO~TECHNIQUE In the paper we consider the s o l u t i o n of the I n t e g r a l transform. Found from the equation (1) the function, eo
0 where: A(x8) = x831(xe)Yl(X8)
1075
(3)
J. MROCZKA
1076
<,, Yl(Xa) - 8essel f u n c t i o n of the second Kind makes possible the determination of moments As from the relationship As = / aSf(a)da 0
(5)
This proceckwe requires the integral transformation. I t can be avoided and one can determine the moments A on the basis of function f ( a ) . Doing integration of equation (2) multiplied e a r l i e r by a Sfactor as-x we obtain
,
,= c~2
0
<,>
0
As i t can be observed the l e f t side of the equation (6) i s the r e s u l t of NnlZin's tcanefocm8 t i o n of function I f ( a ) , whereas the r i g h t side i s the N e l l i n ' s t r a n s f o r ~ t i o n of the equation (2). For t h i s traneformation oO
g(s) = /f(x)
xS-ldx ,
(7)
0 at oo
£(x) = x~/~p fi(~) f2 (P) dp
(e)
0 the relatlonehip
gCs) = iil(S.~) g2(i-s-~-p )
(91
i s true. Hence, the e x p r e s s i o n d e s c r i b i n g As is GO
oO
f..-, f,, o<.> :, (~J . ,,(~),
,~,~ As = - C'-'~"
0
~.
<,o>
0
Assuming cC ,, O, p
1, f l
Y1
l~"~"~'J
e'~z(e)
=~
= O(e)
and oonsiderlng
f,(,,<)xS-l<,,< : rs fm<),is-ldx , 0
(n)
0
we detain
As = -
2~s_ 2 c77S_] R(s)
0o
fo(e) B1-s
dg ,
(12)
0 utmre oO
R(s) = f 3 1 ( x ) 0
Yl(x)xS-ldx •
(13)
Integral transform technique in particle sizing
1077
Using the Nellln's trunsformtlon for the derivative of the equatlon (10) we obtaln cO
As = - ~ 2~s-2
( l - s ) R(s)
lice)
83-sde .
(14)
0 Hence, a f t e r further transformations we obtain for s=l, A1 = ~
~2 Ca
and
Zl.
[ e3z(e) ] ,
(i5)
U ~
oo
0
A3
/" - - - - ~ j Z ( 8 ) aO.
= 3.l
(1~)
0 In fact, the function I(8) Is deter~innd for angle Brain, @max' so from the metrolnglcal polnt of vleu the evaluation of the Influence of integration limits change in equations (15) and (16), (17) Is lmortent. The alternetton of l m 11sit of integration is relevont here. To sinlmlze this error s linear extrq~olstton of the function I(8) can be done st the onole range f m 9,4 . to O. An average particle size as well as function f(s) distribution ere lmport~--t h e m - - '
REFERENCES 1. Nroczko 3., Particle size deten[Lnatlon thmuOh mmll m i l e l i g h t e~rtl~rtnO u a s u r n m t . PToccodLngs of the l l t h Triennial World Congress of the International ~ a s u r m m t s t t o n Confederation, INEKO, pp. ~ - 4 2 , 16-21 Octob~ 1986, H~ston~ Texas, USA. 2. Nroczka 3 . , The method of , m a n t a In p a r t i c l e sizing. ASTN 2nd Symoslua on Liquid Perttcls Stzo Memremnt Technlqum, 9-11 Nmmm~ 1988, Atlwd~, Georgia, USA. 3. TNmmy S., On the nmmrtcel aslutlon of Frndholm integral equations of the f i r s t kind W the lnveroion of the l l n N r systm produced by quadrature. 3. Assoc. Cuput. Nech. 10 (196)) 97-101. #. Borte~ M., Ptke E.R., Panicle size d~st~Ibutlons f m Freum~er diffractions an ~yttc sigmfunctton approach. Opt. Acts )0 (198~) 1043-1049. $. h r t o r e M., st. o1., PortlrJm size (Listrlbutiom f r m Fraunhofer dtffrectton~ the singulsr-vslue spectrun. Inverse Peobleas 1 (1985) 111-126.
0021-8502/89 $3.00 + 0.00 Pergamon Press plc
INTEGRAL TRANSFORMTECHNIQUEIN PARTICLE 51ZING
Janusz Ycnczka Ph.D. I n s t i t u t e of Electrical Metrology Technical University of Wroclaw ul. bybrzete Wyapla~sklego 27 50-370 Wroclaw, Poland
I. INTRODUCTION Optical methods for particle size meeseuremente have significant advantages over convontlonal means based on sampling. They are non-tntrumlve and rapid. The instruments for ensad01e analysis of the particle size by means of the scattered l i g h t gonerelly use a solution of the Fredholm integral equation of the f i r s t kind. There exist two general methods of finding solution of this equation. The f i r s t method involves ¢apresentlng the kernel with an appcoxlmate analytical expression. In most cases Frauchofer d i f f r a c t i o n theory is used. Accocdtng to this theow the intensity of l i g h t scattered by a particle of a detenctnad radium con be found from the kL~j formula. In this case, the sclutton of an lntegcal equation does not depend on the pacttcle refraction indices and on the surrounding medium. The second method involving expressing the integcal kernel in the fore of numerical function knoan from 14te theow, requires the knowledge of real and imaginary parts of a refractive index. These valuea as y e l l as experimentally measured velume scattering function for the determined angles allow for the estimation of particle size distribution. 2. INVERSION OF FREDHOLMINTEGRALEQUATIONS The particle sizing problem requires that equation 40
i(e) : - ~I °
J312(xe)a2f(a)da. 0
(1)
be inverted, I.e. to provide information on the unknown p a r t i c l e size distribution, f(a) using measurements of scattered l i g h t which can be related to 1(8) Ill. No. the number distribution, f(a) Is not necessarily the function of interest, rather i t amy be some R]mont such as the area or volume distribution function that is desired. I f the area d i s t r i b u t i o n function was of interest, i t could obvtoumly be calculated after the fact fcom the solution, f ( a ) , obtained by the tnversionprncess, but i t is our objective here to consldsr the possib i l i t y that use of some moment of f(a) d i r e c t l y In Fredholm equations might tmrove the overa l l perfomence of the inversion 121. There are a nu~er of methods for inverting Integral equations in general, and Fredholm equation in particular. These can be divided into: reduction to a discrete linear system by numerical quadrature after T~amy 131; methods using functional analysls involving expansion of both the measured and urknoM~ functions In teems of etgenfunctton after 6ertero and Pike 141; singular functions after Berte¢o at. el. 151, or integral trensfocme i . e . analytical inversion of Fredholm equation. 3. INTEGRALTRANSFO~TECHNIQUE In the paper we consider the s o l u t i o n of the I n t e g r a l transform. Found from the equation (1) the function, eo
0 where: A(x8) = x831(xe)Yl(X8)
1075
(3)
J. MROCZKA
1076
<,, Yl(Xa) - 8essel f u n c t i o n of the second Kind makes possible the determination of moments As from the relationship As = / aSf(a)da 0
(5)
This proceckwe requires the integral transformation. I t can be avoided and one can determine the moments A on the basis of function f ( a ) . Doing integration of equation (2) multiplied e a r l i e r by a Sfactor as-x we obtain
,
,= c~2
0
<,>
0
As i t can be observed the l e f t side of the equation (6) i s the r e s u l t of NnlZin's tcanefocm8 t i o n of function I f ( a ) , whereas the r i g h t side i s the N e l l i n ' s t r a n s f o r ~ t i o n of the equation (2). For t h i s traneformation oO
g(s) = /f(x)
xS-ldx ,
(7)
0 at oo
£(x) = x~/~p fi(~) f2 (P) dp
(e)
0 the relatlonehip
gCs) = iil(S.~) g2(i-s-~-p )
(91
i s true. Hence, the e x p r e s s i o n d e s c r i b i n g As is GO
oO
f..-, f,, o<.> :, (~J . ,,(~),
,~,~ As = - C'-'~"
0
~.
<,o>
0
Assuming cC ,, O, p
1, f l
Y1
l~"~"~'J
e'~z(e)
=~
= O(e)
and oonsiderlng
f,(,,<)xS-l<,,< : rs fm<),is-ldx , 0
(n)
0
we detain
As = -
2~s_ 2 c77S_] R(s)
0o
fo(e) B1-s
dg ,
(12)
0 utmre oO
R(s) = f 3 1 ( x ) 0
Yl(x)xS-ldx •
(13)
Integral transform technique in particle sizing
1077
Using the Nellln's trunsformtlon for the derivative of the equatlon (10) we obtaln cO
As = - ~ 2~s-2
( l - s ) R(s)
lice)
83-sde .
(14)
0 Hence, a f t e r further transformations we obtain for s=l, A1 = ~
~2 Ca
and
Zl.
[ e3z(e) ] ,
(i5)
U ~
oo
0
A3
/" - - - - ~ j Z ( 8 ) aO.
= 3.l
(1~)
0 In fact, the function I(8) Is deter~innd for angle Brain, @max' so from the metrolnglcal polnt of vleu the evaluation of the Influence of integration limits change in equations (15) and (16), (17) Is lmortent. The alternetton of l m 11sit of integration is relevont here. To sinlmlze this error s linear extrq~olstton of the function I(8) can be done st the onole range f m 9,4 . to O. An average particle size as well as function f(s) distribution ere lmport~--t h e m - - '
REFERENCES 1. Nroczko 3., Particle size deten[Lnatlon thmuOh mmll m i l e l i g h t e~rtl~rtnO u a s u r n m t . PToccodLngs of the l l t h Triennial World Congress of the International ~ a s u r m m t s t t o n Confederation, INEKO, pp. ~ - 4 2 , 16-21 Octob~ 1986, H~ston~ Texas, USA. 2. Nroczka 3 . , The method of , m a n t a In p a r t i c l e sizing. ASTN 2nd Symoslua on Liquid Perttcls Stzo Memremnt Technlqum, 9-11 Nmmm~ 1988, Atlwd~, Georgia, USA. 3. TNmmy S., On the nmmrtcel aslutlon of Frndholm integral equations of the f i r s t kind W the lnveroion of the l l n N r systm produced by quadrature. 3. Assoc. Cuput. Nech. 10 (196)) 97-101. #. Borte~ M., Ptke E.R., Panicle size d~st~Ibutlons f m Freum~er diffractions an ~yttc sigmfunctton approach. Opt. Acts )0 (198~) 1043-1049. $. h r t o r e M., st. o1., PortlrJm size (Listrlbutiom f r m Fraunhofer dtffrectton~ the singulsr-vslue spectrun. Inverse Peobleas 1 (1985) 111-126.