Probabilistic design approach for filtration systems

Powder Technology, 67 (1991) 175-185

Probabilistic

175

design approach

for filtration systems*

Y. Zimmels Department of Civil Engineering, Technion NT, Haifa 32000 (Israel) (Received

August 3, 1990; in revised form January

15, 1991)

Abstract Probabilistic aspects of filtration systems that capture particles along a filtration path are considered. The use of the Geometric, Exponential and Poisson distribution functions in probabilistic representation of filters is reviewed and an extended, multiple capture, probabilistic scope of the filtration process

is introduced. The effect of random variables that characterize the filter as well as those pertaining to the particle population are considered. A design approach, in which filter performance at a prescribed level of confidence can be achieved, is described, followed by presentation of design equations for the filter length and for the filter parameter. Extended design equations for systems involving poissonic filtration processes is developed. Finally, probabilistic analysis of multiple filter units connected in parallel

is discussed,

followed

by analysis

of a specific

Introduction Filtration processes in which the filtered entities (e.g., particles) flow, along a filtration path, past collection sites where they are either captured or not, can be described by probabilistic models. A comparison of deep bed filtration models is provided by Tien [l], who reviewed the phenomenological equations and the current empirical, stochastic and rational models. In such systems, the filtration mechanism involves random variables that pertain to the filter on one hand and to the particle population on the other. Thus, the filter length required to effect capture of a single particle, the characteristic filter parameter, the number of particles entering the filter and their characteristic (e.g., size) distribution can be, and often actually are, random variables depending on the filtration process or else on the model used to describe it. A well-known measure of filter performance is its grade efficiency [2], or alternatively, the probability of capture of a particle of size x. Since this grade efficiency is expected to be a function of the characteristic filter as well as particle random variables, it is also a random variable. This leads to a design concept where the filter performance is evaluated, subject to constraints of prescribed level of confidence that at least this performance will be achieved. Thus, the design values *Presented at the 5th Nice, June 5-4, 1990.

World

Filtration

example.

of the filter parameters must be those which guarantee the required performance, or else efficiency, at an acceptable confidence level. The theory underlying this design approach is outlined for single- and multiple-unit filtration systems. This paper provides probabilistic models for the analysis and design of filtration systems in which particles are captured along a filtration path. These models are not concerned with build up of mechanically captured particles on filter media and the subsequent formation of filter cakes,

Theory Probabilistic representation of filter performance

We define a filter or a filtration system as one consisting of filtration elements which can be either real or conceptual. The entities which are filtered are collectively defined here as particles. Particles which flow across a filtration element can be either captured or not captured. We denote the event ‘the particle was captured by the ith filtration element’ by Ai and the complimentary event, i.e., ‘the particle was not captured by the ith filtration element’ by L&. If the probability of capture of one particle that flows through the filter is denoted by pl, then

Congress,

where the P in

0 1991 -

Elsevier Sequoia,

Lausanne

176

denotes probability and n denotes the number of filtration elements from which the filter is made or else through which the particle flows. The symbol 6 stands for a generalized

i-l

union operator,

F,(n) =F(N
z.e.,

ICIAi=Al~AZUA3...~A..

i-l

The theorem given by eqn. evaluation of pl:

(2) can be used

for

(2) where

;I

i-l

= 1 - [l -g(x)T

(7)

Equation (7) satisfies the cumulative geometric distribution where the action of each filtration element is defined as an experiment having two possible outcomes: capture or no capture of the particle. F,(n) gives the probability that N
denotes the generalized operator for the simultaneous occurrence of all n events, which in this case are Ai, i=l, 2, ..q, n. The probability that a particle will be captured by the ith filtration element is defined as the grade efficiency gi(x) of this element with respect to X, where X is the physical property that constitutes the basis of filtration. The randomvariable X is referred to, henceforth, as denoting particle size, but it may equally well denote other properties such as electric charge, electromagnetic susceptibility and combinations thereof. PM) =&)

(3)

P(ki) = 1 -gi(X)

(4)

The generalized reads [3]

certainty, i.e., P(&) = 1. Note that eqn. (1) gives the cumulative distribution function F,(n) that the number of filtration elements which is required for capturing one particle of size x is less than or equal to n. Thus, if the random variable which denotes the number of filtration elements is named N and if a specific value assigned to it is denoted by n, then combining eqns. (l), (2), (4), (5) and (6) gives

theorem of conditional

W=klr)

=L(k) k=l,

= [1 -&W%),

2, . . . . n

(8)

If for each experiment, i.e., for each filtration element, there exists a different grade efficiency gi(x), then an extended version of F,(n) applies:

-iooLl --&)I

(9)

E(n) = 1

It can readily be shown that the pdf corresponding to F,(n) is given by L(k) -fW

= klr)

k-l

probability

=j~[l-gi(x)]gk(x)

k=L

2,

so.,

n,&(x)=0

(10)

Assuming that g,(x) =g(x) = const., i = 1, 2, . . . , n and that,a particle can be captured only once gives P

k=l,

2, . . . . n

where P@,&) = P(A,) since the grade efficiencygo of the entrance section to the filter, where there are no filter elements, vanishes and hence A0 is a

Equations (9) and (10) can be defined as representing a generalized geometric distribution for which 1, as f,(k)>O, l
F&z, t) = 1

-,Doll -g&

r, 01

(11)

k-l f,(k,

t) = iFo [I

-&s

r, t)k(x?

r1 t>

(12)

177 For a particle population having a size distribution Q,(x),the cumulative grade efficiency, G(x) = G(X
J\rr[l-(l-

y)l]=l-em”)’

where F,(l) tion of the of a single The pdf of is obtained

s

IFl(x) dx=l

By letting x+ 00, eqn. (13) yields the expectation of F&r), i.e., G( 01) =E[F,(n)], where E denotes the expectation operator. The expectation E(N) and variance V(N) of N, i.e., of the number of filter elements, each having the sameg(x), that are required for capture of a particle is given by E(N)=

(15)

L g(x) E-

OGX<

(21)

CQ

(14)

0

v(N>

denotes the cumulative distribution funclength L, which is required for capture particle rated at a size x, i.e., F(L
L(l) =f(L = $9 = a(x) e - a(xy

OD

(20)

l-d4

(16)

Mx)S

The expectation and variance being E(L) = l/a(x) and V(L) = ll[a(x)]“, respectively. If I&(X)is the pdf of x, then, assuming that o(x) is a monotonous function of x, its pdf should satisfy

Furthermore, f,(l) and F,(l) are also distributed. Assuming that they also are monotonous functions of x, the corresponding probability density functions are readily obtained as Jl(x) err@)’

If, however, the grade efficiency g,(X) varies along the filtration path, i.e., from one element to another, then eqns. (17) and (18) apply.

E(N)=~~~n~~~[l-gi(*)lg”(*)

(17)

The expectation and variance of N can be used for evaluation of the filter performance as well as for probabilistic design purposes, where confidence levels of performance must be met. In the following uniform and continuous filters of finite length are considered. Uniform and continuous filters For filters that are continuous and uniform the grade efficiency of a filtration element having an infinitesimal filtration length dl can be defined as

(22)

fK(r>1 = (1 - cY(x)l)aa(x)/av

(23) The expectations of fx(l) and of F,(l) are given by eqns. (24) and (25), respectively.

EK(l)l =

jiM4CIX

(24)

0

Similarly,

the expectation

of E(L)

is given by (26)

g(x) = a(x) dl

=- +)l n where (Y(X) is a function uniform dl could be defined gives dl+0 and g(x) +O, eqns. (7) and (19) at the lim F,(n) = F.(l) n-r-

(19 of x. Since the filter is as dl =lln, so that n + m as required. Combining limit when n -+ 01 gives

Unifonn jilters connected in series Consider n filters connected in series. The length and parameter of the ith filter i= 1, 2, . . . . n, are li and w(x), respectively. The probability that a particle flowing through the ith filter will not be captured is e-ui(x)e. The probability that the total filtration length 1=Z$_,,& will not be sufficient to capture the particle is then

(32) where &= C?(X)and @denotes the standard cumulative normal distribution. If, for design purposes, it is required that the probability of particle capture will exceed a target value of 1 - p, 0
”

(Y(X)= xCYi(X)li/l, or if

a(X)

is continuous

i-l

F[~(x), Z]=l-e-*X)‘>l-p

I e(x)

=

f

s

a(x)

(33)

which gives

dl.

0

(r(x)> +ln If the filters are of equal i=l, 2, ..,, n, then C(X)

=

t

length,

i.e., li=Zo for

P[ti(x) > as] = 1 -

,$(Yi(X)

is the mean. For a large n, t?(x) is normally distributed having expectation E[&(x)] =E[a(x)] and Variance q&(x)] = l/n~u(x)], the corresponding standard deviation being u&= CT,/&. Note that the above expectation and variance relate to the population of a(x) which, for a given x is a random variable that corresponds to the set [ai( Q(X), . . . . an(x)]. Furthermore, C?(X)would still be normally distributed even if li = lo, i = 1, 2, . . . , n, would not be satisfied. In this case, with li being a random variable, (Y(x) would be the sum of random variables ai(x)Zi/l which, by the central limit theorem, would confer on it the properties of a normally distributed variable. The counterpart of eqns. (20) and (21) for a distributed C+(X)is given by eqns. (28) and (29), respectively. F(&, I)= l-e-“)

(28)

fx( Cu,I) = &(x)e- ‘W’

(29)

Since for a given x, C(x) is a random variable, so too are its monotonous functions F,(&, 1) and f,(ai, I). Thus, invoking the same theorem used to derive eqns. (22) and (23) for this case gives

=E[cr(x)].

The cumulative distributions of C?(X) and of fX(&,I) and F,(&, I) have the same value once e(x) is set. Thus, FL&(&

ffp-P @C ffCi 1

01 =Wx(& 41

(35)

hence,

1

@(y)=Y

or Z,=

(36) -ffa-Y a&

i Substitution of CY~from eqn. (34) in eqn. (36) and solving for I gives the following design equation: 1+ --!Z+e+P

In f 0

0o

(37)

Equation (37) shows that the design criterion for the filter length depends on the expectation and standard deviation of C?(X),on the accepted error level y and on the probability of no capture j3, For example, if it is required that the probability of capture be greater than 99% at a confidence level of 99%, then the error levels are /3= 0.01 and y=O.Ol. From standard tables of the normal distribution Z O.ol= - 2.33. Hence, @=O.Ol,

p=E[&(x)]

(34)

=1-y

I-

where

1 =crp 0P

y=O.Ol) =

In 100 (/~-2~33u~)

(38)

provided that p- 2.33u,> 0. Since for acceptable error levels Z, 0 is larger by a factor of l/(1 +&.a,/~) as compared with the deterministic case where f&=0. Equation (37) also provides the minimum error level y and its attendant maximum confidence level 1 - y, which are set by the condition I> 0 that is satisfied if ZuuJp> - 1. Thus, for uJp= 1, Z,> - 1 and hence y> 0.158 7 must be satisfied. This implies a maximum obtainable confidence level which is less than 84.13%.

179

Characteriration of the captured and ove$ow fractions If I@) is the pdf of X, then the probability that for particles entering the filter X will fall between x and x+ dx is Jl(x)dx. For particles not captured (i.e., those reporting to the overflow), this probability is e-q”)‘Jl(x)ch, while [l-e-*xy]Jl(x) ch applies for those captured. The probability density functions corresponding to the above probabilities can be obtained by applying appropriate normalization factors. fe(x) = K.e-q*o’llr(x) K,=

m s

(39)

1 e - +)‘+(x) ch

0

Distributed number of particles entering the filter at fied levels of x and of c(x) In this section, the number of particles Y entering the filter is a random variable. The filtration process of each particle consists of a Bernoulli trial which results in either the event ‘capture’ or the complimentary event ‘no capture’. It is assumed that the probability of capture p1 of one particle is constant and satisfies eqn. (28), and hence P1=l_e-““”

(44)

The probability that y particles will be captured in such Y Bernoulli trials satisfies the binomial distribution, having YpI and Ypr(l - pr) as its expectation and variance, respectively. Denoting E(Y) = p and recalling thatpi is constant givesE(Yp,) = p1 P. Hence, E(YpI) = (1 -e-*+)I*

(45)

E[Y( 1 -pl)] = e-*+lr.

(46)

UY(I -PI)1 = (1 -P1)‘WI

K,=UK--1)

= (e - Way)2 (40) where subscripts e and c denote the overflow and captured fractions, respectively. The cumulative distribution functions are readily obtained as

s-

where ay is the standard deviation of Y. It follows that the standard deviation of the number of particles captured by the filter aycl _plj = {V[Y(l -pI)]}‘” is given by (48)

.I

F,(x)=K

e +)‘flx)

dx

(41)

0

(42) where Fr(x) = ro$(x) dx is the cumulative distribution of x which enters the filter. If either C?(X)or $(x) are distributed, for example due to variation of $(x) and the type of feed to the filter with time, then, f=(x), fc(x), F,(x) and F,(x) are also distributed. Hence, evaluations involving any of these functions should include the required level of confidence. For example, a filter functions to remove particles of size larger than x and it is required that in the overflow F,(x) > 1 -p, 0 x)

< j3] =P[l -F,(x) =1-y

(47)


Substitution of F,(x) from eqn. (41) in eqn. (43) gives a design equation from which 1 can be evaluated as a function of p and y.

If the function of the filter is to limit the number of particles reporting to the overflow below a prescribed level yo, at a confidence level of 1- y, then the filter length required to this end can be evaluated as follows:

Solving eqn. (49) first for y, and then for 1 gives yo=(l+Z1-,~)e-*“~~

(SO)

I= --& In[ IL(/zk7+1)]

(51)

If ay=O, then c=O, and eqn. (51) reduces to yol CL=e-q@‘. Recognizing that in this case yolk is (for a large CL)also the probability of no capture P(L > I>, eqn. (27) is obtained as expected. Note that in using eqn. (49) it is implied that the central limit theorem can be invoked or else it is assumed that Y is normally distributed. The design eqns. (SO) and (51) show that if l>O, then higher values (as compared with the case c=O) of y. (given I) and of I (given yo) are obtained, depending on the required confidence level. Assuming that,yolp < 1, as is the case in most practical cases, 1 can be expressed as

180

r=I(l=0)+1(po, =[Z+q]@=0) q=@-0,

=

P(L > llr) = 1 - P(L
Z1_,)

r-l

=iz e-‘(AlV)‘/i!

(52)

zl-,)/l(r=o)

W&-,+

1)

(53)

In e 0Yo q stands for the dimensionless contribution to 1 due to the constraint imposed by the confidence level. q increases with f and with Z1 _ y and asyo approaches CL.The significance of q is expected to be enhanced in cases involving materials which are difficult to filter and difficult filtration conditions, where p1 is likely to be relatively low. A significant surge in the number of particles Y entering the filter, which (given pi) results in higher levels of y,, also increases the significance of 7. For example, if yoIp=O.l, 5= 1 and 1 - y= 0.975 (Zo,,,5 = 1.96), then VJ= 0.47. Thus, a 47% increase in Z([= 0) is required if a 97.5% confidence level is to be maintained. Note that q is still significant even when highly efficient filters are considered. For example, if yoIp is decreased by two orders of magnitude to 0.001, then q decreases to X.7%, which is by no means insignificant. Poissonic filtration processes

If a filter works so that along the filtration path the filtered entity, such as particles, aerosols, colloids, microorganisms and molecules, can jump from one site to another, then the filtration operation involves poissonic processes. The transfer mechanism from one site to another involves repeated capture and release of the filtered entities. Such a mechanism may, for example, be due to replacement processes, where the landing of one entity (e.g., particle) on an occupied site releases a captured entity back into the filtered stream. Poissonic filtration mechanisms can be also part of processing the particles, as is the case in adsorption-desorption processes that differentially retard the motion of solutes moving through a chromatographic column. If the specific number of captures (associated with the jumps from site to site) occurring per unit filtration length is denoted by A, then the probability of j captures in a filtration length 1 is eWM(Aly’/j!. In the following, it is assumed that A is constant and 1 is a random variable. Fixed A and variable 1

The length of a filter which is required for r captures to occur has a Gamma distribution. Thus, the probability that a filter of length L > 1 will be required for r captures to occur is given by

(54)

For r= 1, and upon replacing A by h(x), eqn. (27) is obtained. Equation (54) expresses the probability of an event that is equivalent to no capture. This applies, since the condition of occurrence of r capture and release cycles, within I, is not satisfied as the particle escapes the filter once r- 1 such cycles, or less, are completed. Invoking the same reasoning that led to eqns. (50) and (51) and replacingpi from eqn. (44) by P(L > Zlr) gives the following extended design equation: r-l yo=(l

+Z1_-T[)~x i-0

e-“[(My/i!

(55)

A numerical solution of eqn. (55) would provide 1 at the required confidence level 1 -‘y. If r is larger, then the sum of the right-hand side of eqn. (54) can be evaluated using the normal approximation procedure as follows:

(56)

where r/A and rfA2 are the expectation and variance of the Gamma distribution (with parameters A and r) respectively. Using eqn. (56) instead of eqn. (54) for the derivation of the design equation gives Yo=u+zl-,i)P[

l-@(Y)]

.I= GZl-a+r A

/?= (1 +zY,o_,S)r

(58)

Note that here 1 -p denotes the probability of occurrence of r captures and 1 - y is the confidence level with which l-/3, or else J3, is obtained. The case of fixed Y and 1 and variable A is considered next. Fired Y and 1 and variable A

If due is fixed, entering parameter and the particles

to geometrical constraints the filter length then for a fixed number of particles Y the filter, the design variable is the filter A which depends on the filtration method filter materials used. Assuming that the are uniform in size,

W>r)=l-i~

r eeN(N)' 0

7

(5%

where R denotes the number of captures in 1. If r is large, then the Poisson distribution can be ap-

181

proximated

by the normal

wr)=l-i~ El-@

distribution

as follows:

r eeN(NV)

-jj---

0

=EEFi(x)l

r-,N

( 1 -

hi

If it is required that P(R >r) > 1 - p, where /3 is set b the designer, then q(rAl)l&] Q p, (r-N)/ $ N < 2, and A can be determined from the following quadric equation:

A=

the minimum

required

A’- z, + &pq2 4

Since & are random variables, the operation of the filter assembly can be described by a joint distribution function J(&, t2, . . . . &,) which, assuming that the filtration units are independent, takes the following form:

u=J;i

uz+Zgu-r=O

Hence,

(62)

1

level of h is

(61)

Equation (61) shows that A increases with a decrease in p (which results in an increase in -2,) and with an increase in r. For example, fixing p at 0.13% and r at a level of 20 per cm gives 2, = - 3.0 and A = 38.65 captures/cm. Increasing p to 15.87% sets Z,= - 1.0 and A=25 captures/cm. Equation (61) shows that the significance of the required level of p is enhanced in filters that operate at lower levels of r. Note that if the number of particles entering the filter is distributed, then eqn. (59) can be used as a design equation for A at the required levels of p and y. The theory presented hitherto is concerned with the probabilistic design approach for single filters. The extension of the probabilistic approach to specific filtration systems that operate, simultaneously, more than one filtration unit is considered in the following section.

The objective of the filtration is to perform (at a prescribed level of confidence) in the range defined by t> &, where 0 Q & < 1 is the overall target grade efficiency set by the designer or else by the operator. For reasons of convenience a new variable is defined by tO=m&, thus transforming the above objective into Xzr &> to or rnt> to. The problem is defined here as one of multidimensional probabilistic integration of

\i-1

/

(i.e., of the kth moment of 6 with respect to J) in the space of 4 and subject to the condition m&> to. As an example, consider the two-dimensional case m = 2, t= (& + t2)/2, J= & s2. For this case, ‘letting k= 1, the design objective reads

W25>&)>1-P O-=P=Gl

(64) I

n(&%

50) =

f!

!(I;

+

S2)J

d&

d52

(65)

A

Multiple unit filtration systems

In this section, we limit the discussion to a system of filters which are connected in parallel. In this system, the filters can function as a separate independent unit or else be a part of a bundle or assembly that provides the required filtration capacity. Consider m filtration units that are connected in parallel. The grade efficiency Fi(X) of the ith unit is a random variable that depends on the quality of the filter and on the effect of time. Assuming that the inlet stream, which is characterized by a size distribution $(x), is equally divided between the m filtration units, the total captured fraction 4 in the filter assembly is the sum depending on individual contributions 6 from each unit, i.e., t= i i$l[i where 5i is given by

where A is the domain of integration 25‘~5, and I is given by

= ~(c3

as defined by

v35)

Consider the case in which & as well as & are uniformly distributed in the (a, 1) range, Oda~ 1. Hence, they have the same pdf which is given by l/(1 -a), and invoking their statistical independence gives J(&, e2) = l/(1 -a)‘. The Figure shows the geometric details of the problem. The domain of integration, which is defined by a Q Z;,g 1 for i= 1, 2, is divided into two subdomains that’are denoted

182 domain of integration a15,11,

n(2t’b)=

a15,Sl

I

2a 15 5 l+a,Mdenotes

II

l+a
,

g,+

5,~

5,

denotes 5, + 5,> 5,

(l+a);l-n)2

__$to2a+

503

3

-&g2+

(1+a)(1-a)2

W2&

50) = $1’ (I =P

‘I”(& (I

1

I, then applying

+ &) &

d52 d&

(503/3 - &12u+ 4a3/3)

I(1 -a)”

-&2+

$

(71)

!g

(l+a)(l-a)2

by I and II. If &a is in subdomain eqn. (65) gives

3

Thus, if a 1 -p confidence level is required, then the effective design value of &, can be evaluated by eqns. (72) and (73) in subdomains I and II, respectively.

$

Fig. 1. Details of a two-dimensional design problem of two filtration units connected in parallel.

(”

=p

2u<&)<1+a

(72)

l+a<&<2

(73)

$ =l-@

Alternatively, if to satisfies either eqn. (72) or eqn. (73), then there is at least a 1-p probability that fi will satisfy the condition 25>to. For example, suppose that each of the two filter units operates in the 90% to 100% grade efficiency range, i.e., a = 0.9. If /3 is set at a level of l%, then, using eqn. (72), .$,,= 1.814 5 (l= 0.907 2) is obtained, while for increases to to = 1.845 6 p= 10% the result (&=0.922 8). Thus, at a 99% confidence level, the effective design value of the grade efficiency 90.72% is close to the lower bound of the range of variation of 5;:, i.e., to a, whereas at a 90% confidence level it increases to 92.28%. The use of &= 1.9 (r= 0.95) provides a 49.12% confidence level, which usually is considered too low. Analysis of eqns. (72) and (73) shows that, at a given confidence level, 50deviates less from a as the latter increases, i.e., the system becomes less flexible designwise as a approaches its upper limit of 1.

(67) In region

II, see cross-hatched

area in Fig. 1, Conclusions

_+($to2+3 Z(1 -a)

l+a6&<2

(68)

The result of adding eqns. (67) and (68), for 50= 1 +a, is unity and solving for I gives z=1+a Hence, W&50)=

(69) in subdomain 1 (l+a)(l_a)2

and in subdomain

II,

I, b3 3 -502a+

4a3 3

(70)

The probabilistic representation of filters facilitates the use of distribution functions, rather than of deterministic models, for analysis and design of filters so as to comply with prescribed confidence levels of performance. An extended scope of the filtration process is defined as one which involves a desirable number of particle captures, jumps, or else capture and release cycles, along the filtration path. The commonplace single-capture filtration process is a special case of the above extended definition. In this work, design equations are provided for filtration systems in which particles flow along a filtration path. Different design equations are provided for random variables that characterize the

183

design problem. The following categories of design equations were developed for single-path filters: (1) Design variable - filter length (i) Distributed filter parameter for single (eqn. (37)) and multiple, poissonic, capture processes (eqn. (58)). (ii) Distributed number of particles entering the filter at fixed particle size and fixed filter parameter (eqn. (51)). (2) Design variable - filter parameter Multiple capture poissonic processes at fixed length of filter and fixed number of particles entering the filter (eqn. (61)). (3) Design objective - level of efficiency Multiple unit filtration systems that are connected in parallel (eqn. (64)). These categories give a spectrum of design options at prescribed levels of confidence for filter performance. The general single-capture filtration process which takes place along a filtration path can be described by the geometric distribution. For continuous filters, the exponential distribution applies as the limit of the geometric distribution. These well-known distributions provide the probability that a certain number of filtration elements, or else filtration length, will be necessary to effect the first success of the filtration process, i.e., the occurrence of a particle capture. This probability is often defined as the grade efficiency of the filter, which is a function of X, i.e., of the variable that constitutes the basis for the filtration process. Thus, if x is distributed, then the geometric or else the exponential distribution functions are also distributed as a function of x. If x is tied and the parameter of the exponential distribution is a random variable which, being the sum of random variables, satisfies the normal distribution, then the cumulative distribution of the grade efficiency can also be evaluated by the normal distribution. The design approach in this case is to require that the probability of particle capture exceeds a target level at a prescribed confidence level. The result is a design equation relating the required filter length 1 to the target level of probability of capture (or alternatively, the probability of no capture) and to the prescribed level of confidence, that this target value will be achieved. The effect of increasing the required level of probability of capture or the related confidence level is to increase the design value of the filter length. Furthermore, the design equation provides values of 1 which can be significantly larger than those provided by deterministic models, i.e., those characterized by a vanishing standard deviation of C?(X).In some cases, the maximum obtainable level of confidence is also limited and cannot be exceeded, however large the value of 1 is set. These constitute

cases in which the filter cannot operate to the required levels of efficiency and must therefore be redesigned, discarded or replaced. If X, e.g. the particle size, as well as C(X) are fixed whereas the total number Y of particles entering the filter is a random variable, then a different design equation applies. In this case, 1 is a function of l= cry/p, &+-, and of the confidence level with which the number of particles not captured,y,, is estimated. The relative increase in the design value of 1, with respect to the case where uy=O, is enhanced for larger values: of 5, of the confidence level, and as p/y0 approaches unity. A similar design equation is presented for poissonic filtration processes where the filter length is a function of: the prescribed number of the particle jumps or else captures r, the specific probability of capture A, yO, 5, p and of level of confidence that the required r will be achieved. In cases where 1 is fixed, a design equation for A is also provided. Evaluation of h facilitates the selection of the type of filter that can comply with the required efficiency as well as with the imposed geometrical constraints of a fied 1. The extension of the theory to a filtration system consisting of several filtration units connected in parallel involves multidimensional integration of the sum of the individual grade efficiencies in the sample space of these variables. This integration, which provides the normalized expectation of the overall grade efficiency of the system, often requires the partition of the sample space into subdomains. Analysis of a two-dimensional problem of two filtration units, each, having a uniform pdf over a range of grade efficiencies and which are connected in parallel, provides an insight into the theory. The results of this analysis show that if high confidence levels of performance are to be maintained, the effective design value of the overall grade efficiency should be close to the lower limits of the sample space of the individual grade efficiencies. This constraint becomes more stringent for smaller sample spaces, of the individual grade efficiencies, that have unity as their upper bound.

List of symbols a Ai, ki

&Cl)],

lower bound of variation of grade efficiency events denoting capture and no capture of a filtered entity expectation operator E[F,OI, E[Fi(x)], E(L), E(N), E[&)],

E[441, W3 expectations ables

of the respective

vari-

184 f,(k), F,(n)

f,(k,

t), F,(n,

flx6 r)l, flF,t6 01 FL&t4 01. WA4 01 f&)3 f&>

Fe(x), F&l

F@)

go(x) G(x) Z i J

10

&Y= 01, 45> 0, a-7)

m

n

probability density and cumulative distribution functions of the number k and n of filter elements required to effect capture at a given level of r)Time-dependent L(k) and FJn), respectively probability density and cumulative distribution functions that a length L =I and L ~1 will be required to effect capture at a given level of x C(x)-dependent fx(l) and F,(l), respectively probability density functions of f,(&, I) and Fx(6, 1) respectively cumulative distribution functions of f,(G, I) and F’(&, 1), respectively probability density functions of x in overflow and captured fractions, respectively cumulative distribution functions of x in overflow and captured fractions, respectively grade efficiency of ith filtration unit grade efficiency of a filtration element, of ith, kth and nth elements, respectively size-, time- and position-dependent grade efficiencies of ith and kth filtration elements, respectively grade efficiency of entrance section to filter cumulative grade efficiency, i.e., for xO at confidence level 1 - y number of filters connected in parallel integer, index, subscript and also used to denote number of filter elements

Pl

P

t

WI, Y4x)l, v[m1

x, x Yo

probability of capture of a single particle a symbol denoting probability number of captures, jumps or capture and release cycles time variance of N, of a(x), and of C(x), respectively random variable that constitutes basis for filtration (e.g., particle size) and its specific value number of particles reporting to overflow, i.e., not captured tabulated independent variable of standard normal distribution corresponding to cumulative probabilities /3, 1 -p, y, and 1- -y, respectively

Greek symbols a(x)*

ai

parameters of exponential distribution of a filter and of ith filter, respectively expectation of e(x) lower limit of E(x) that depends on 1 and p probability of error (error level) and also used as a subscript probability of error (error level), i.e., 1 -confidence level ratio of standard deviation of number of particles entering filter to their expectation relative increase in design value of 1 as compared with I(l=O) parameter (captures per unit length) of Poisson distribution and of a filter characterized by poissonic processes expectation of G(x), and also used for expectation of number of particles entering filter overall grade efficiency of a filtration system consisting of m units connected in parallel expectation of grade efficiency (with respect to x) of ith filter unit minimum required level of grade efficiency =m&, integration domain boundary standard cumulative normal distribution probability density function ofx, e.g., size distribution first moment probabilisticintegration function

185

References 1 C. Tien, Proc. Nth World Filtration Congr., 22-25 April 1986, Ostend, Belgium, Technologisch Instituut-K. VIV Mechanica Separation and Particle Technology, Antwerp, Belgium, Vol. 8, Part II, pp. 13.1-13.9.

2 L. Svarovsky, in L. Svarovsky (ed.), Solid Liquid Separation, Chemical Engineering Series, Butterworths, London, 1977, pp. 31-57. 3 P. L. Meyer, Introductory Probability and Staristical Applications, Addison-Wesley, New York, 1970.