Suppose an experimental determination of particle collection ef®ciency takes place during a de®nite time interval, t . Let us de®ne the effective inlet distribution,.
Aerosol Science and Technology 31:194] 197 s 1999. Q 1999 American Association for Aerosol Research Published by Taylor and Francis 0278-6826 r 99 r $12.00 q.00
Multivariate Fractional Ef® ciency J. S. Ontko FEDERAL ENERGY TECHNO LOGY CENTER, U. S. DEPARTMENT OF ENERGY, 3610 COLLINS FERRY ROAD, MO RGANTOWN, WV 26505, USA
ABSTRACT. The fractional ef® ciency of any particle separator operating on arbitrary particulate distributions is de® ned in this article. Some consequences of this de® nition for the interpretation of experimental data are examined using an example. These considerations show that the customary assumptions should not be made without critically scrutinizing data to which they are supposed to apply.
All particle separators, regardless of operating principle, share two related characteristics. First, they capture a fraction, h , of incoming particles and allow the remainder, n , to escape; h qn s 1. Second, they segregate the incoming particulate distribution into captured and escaped distributions. The fractional ef® ciency, h , describes the variation of collection ef® ciency with particle size and other prope rties. Frequently, unwarranted assumptions are made in experimental determinations of this quantity. For example, it is generally assumed that at low particulate loading h is independent of the effective cumulative inlet distribution, F , denote d brie¯ y by h / h s F .. A recent experimental investigation of cyclone separator performance s Ontko 1996. has shown that at moderate loading s ; 10 y2 kg particles r kg gas. h / h s F . is inconsiste nt with the data. Anothe r example is the arbitrary use of the various equivalent diameters to simplify data reduction. This is questionable for distributions of particles with dissimilar shapes. These assumptions, used
without experimental veri® cation, may lead to predictions which fail when applied to othe r data for which they are ostensibly valid s Abrahamson 1981.. A general de® nition of fractional ef® ciency is required to deal with these issues and is given in this article. It may be used with any number of variates, continuous or discrete, and allows the indepe ndence hypothe sis, also de® ned rigorously herein, to be tested. Its use is illustrated by an example and some rami® cations for the interpre tation of experimental data are then brie¯ y discusse d.
ANALYSIS Suppose an experimental determination of particle collection ef® ciency takes place during a de® nite time interval, t . Let us de® ne the effective inlet distribution, F s X1 , X 2 , . . . , X n ., of the random variables Xi , ai F X i F bi s i s 1, 2, . . . , n ., as the cumulative distribution function s CDF. of all particles which could be captured or escape during t . F may be constructe d in practice
Aerosol Science and Technology 31:2 ] 3 August] September 1999
Multivariate Fractional Ef® ciency
by mixing h parts of captured material with n parts of that which escapes. We require that h s X1 , X2 , . . . , Xn . be a continuous function, with range 0 F h F 1, or some subset of this interval. The total ef® ciency is the Riemann-Stie ltjes integral
h s
H H bn
an
bn y 1
H
???
an y 1
b1
a1
h s u1 , u 2 , . . . , u n .
= dF s u 1 , u 2 , . . . , u n .
AN EXAMPLE Suppose the particles to be removed consist of right circular cylinders of radius r and height h, all made of the same material. If these lengths are distributed continuously between r0 F X r F rm ax and h 0 F Xh F h m ax , the captured distribution is given by 1
G s r, h . s
r
h0
r0
HHh
h
s 1.
h
s u1 , u 2 .
= f s u 1 , u 2 . du1 du 2
s 4.
The captured distribution is given by
and the escaped distribution by
G s x1 , x2 , . . . , xn .
F s r, h . s
1
s
h
xn
xn y 1
H H H an
an y 1
x1
a1
s 2.
where the xi are ® xed values in the domain of the X i . De® ning the fractional penetration as n s 1 y h , the escaped distribution is given by F s x1 , x2 , . . . , xn . s
1
n
xn
H H an
xn y 2
an y 1
???
H
x1
a1
n s u1 , u 2 , . . . , u n .
= d F s u1 , u 2 , . . . , u n .
1
h
r
h0
r0
HHn
n
s u1 , u 2 .
= f s u 1 , u 2 . du1 du 2
h s u1 , u 2 , . . . , u n .
= d F s u1 , u 2 , . . . , u n .
195
s 3.
Thus h and n partition F into G and F; the limitations on h guarantee that G and F are CDFs. Consider a particle separator operating at ® xed conditions. Let F s X1 , X2 , . . . , Xn . and F *s X1 , X2 , . . . , X n . denote arbitrary distributions de® ned on domains D and D*, h the fractional ef® ciency when operating on F and h * the fractional ef® ciency when operating on F *. The fractional ef® ciency is said to be independent of effective inlet distribution, h / h s F ., if and only if, for all points s x1 , x2 , . . . , xn . g s D j D*., h s x1 , x2 , . . . , xn . s h *s x1 , x2 , . . . , xn . for any F and F *. Othe rwise, h s h s F ..
s 5.
where f denotes the probability density of F . To illustrate the application of Equations s 1. ] s 3. with a mixed distribution, suppose the cylinders may have either of two distinct densities, r 1 - r 2 , any one of three dielectric constants, « 1 - « 2 - « 3 , and these prope rties are relevant to particle capture. Choose any r 0 - r 1 , r 3 ) r 2 , « 0 - « 1 , and « 4 ) « 3 . Then r 0 - Xr - r 3 and « 0 - X« « 4 . De® ning the unit step function
w
0, Is u. s 1,
u-0 uG 0
we have, if r
0F
s 6.
r Fr
3
and «
ij
s u1 , u 2 .
0F
« F « 4,
F s r , « , r, h . s
h
HH h0
r
3
p
p
2
f
r0 js 1 is 1
= I s r y r i . I s « y « j . du1 du 2
s 7.
The captured distribution is given by G s r , « , r, h . s
1
h
HH
= f
h
h0
ij
r
p
3
p
2
r0 js 1 is 1
h s r i , « j , u1 , u 2 .
s u1 , u 2 . I s r y r i .
= I s « y « j . du 1 du 2
s 8.
196
Aerosol Science and Technology 31:2 ] 3 August ] September 1999
J. S. Ontko
and the escaped distribution by F s r , « , r, h . s
1
n
HH
= F
h
h0
ij
r
p
3
p
2
r0 j s 1 is 1
n s r i , « j , u1 u 2 .
s u1 , u 2 . Is r y r i .
= I s « y « j . du1 du 2
FIGURE 1. Level contours C( z ) for Zs C
s 9.
.
DISCUSSION It is common practice to use particle sizers which reduce a distribution of nonsphe rical particles to a single variate, Z. For example, Z might denote the diameter, D, of a sphere having the same volume as a randomly chosen particle. A more re® ned approach uses the sphericity c , the ratio of the surface area of a sphere to that of a
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Multivariate Fractional Ef® ciency
particle having the same volume, and de® nes Z s C s c D. Such equivalent diameters are frequently combine d with othe r particulate prope rties in proposing scaling laws. But for a reduction to be scalable, all particles having the same ® xed value of Z must have the same h . This may be demonstrated using the continuous case of the previous example; here D s s 6 r 2 h .1 r 3 and C s 3rh r s r qh .. Consider a reduction Z s f s r, h ., de® ned and continuous for all r0 F rF rm ax and h 0 F h F h max such that Z takes on every value between a and b , a - b . Let z denote a ® xed value of Z and C s z . a locus. For a speci® c example, see Figure 1. Let As z. denote the region ZF z and de® ne
HH 1 Q s z . s HH h Ps z. s
As z .
f s u 1 , u 2 . du 1 du 2
As z .
s 10.
h s u 1 , u 2 . f s u 1 , u 2 . du1 du 2 s 11.
h às z. s h
dQ r dz dP r dz
sh
dQ dP
s 12.
Note that h à , not h , is the quantity determined experimentally from reduced data. But h à s h if and only if, for all z, a - z - b , dQ r dP is de® ned and h s r, h . is constant for all points s r, h . g C s z ., by Equation s 12.. The relationship between h à and h depends on the Z chosen and cannot be unique ly determined from reduced data.
197
To illustrate what is required to show h / h s F ., let us extend our example by allowing Xr and X« to vary continuously on r 0 F Xr F r 3 and « 0 F X« F « 4 . By de® nition, h / h s F . only if, for ® xed operating conditions, h s r , « , r, h . is the same for any F at any given point s r , « , r, h . in this augmented domain s h may of course vary from point to point.. In particular, this condition must be satis® ed in both previous examples, assuming implicit values of r and « for the continuous case. It has frequently been found in ste adystate experiments s Strauss 1966. that h is a monotonically incre asing function with h s a1 , a 2 , . . . , an . s 0 and h s b 1 , b 2 , . . . , bn . s 1. In this case h s x1 , x2 , . . . , xn . may be interprete d as the cumulative probability of capture at s x1 , x2 , . . . , xn .. In general h s h s F . under these circumstance s, since the locations of the extreme values of h depend on the bounds of the domain, which in turn depend on F . References
Abrahamson, J. s 1981.. Mechanisms of Dust Collection in Cyclones. In Progress in Filtration and Separation 2, R.J. Wakeman, ed. Elsevier Scienti® c, Amsterdam, The Netherlands, pp. 6 ] 11. Ontko, J. S. s 1996.. Cyclone Separator Scaling Revisited, Powder Technol. 87:93] 104. Strauss, W. s 1966.. Industrial G as Cleaning, Pergamon Press. Oxford, U.K., Ch. 5 ] 10. Received April 2, 1998; accepted April 2, 1999.
