Indeterminacy in the estimation of flow rate and transport functions from tracer experiments in closed circulations

far. J. En~n~ sci..

1972, Vol. IO, pp. I 153- 1174.

Pergamon

Press.

Printed in Great ~dain

INDETERMINACY IN THE ESTIMATION OF FLOW RATE AND TRANSPORT FUNCTIONS FROM TRACER EXPERIMENTS IN CLOSED CIRCULATIONSt P. NAORS. R. SHINNAR and S. KATZ$ Department of Chemical Engineering, The City College, City University of New York, New York, N.Y. 1003 1 Abstract-The i~te~re~tion of tracer experiments in closed circulating systems is studied from the viewpoint of applied probability theory. It is shown that basic quantities and characteristics of the system, such as mean round time and its variability as well as the average volumetric flow rate cannot be extracted from the tracer response even if two concentration histories-one at the injection site and one at an arbitrary location on the same Sow path-are simult~eously recorded. The observed tracer responses are compatible with many feasible underlying round-time dis~butions (transport functions) and a many-one correspondence rather than a one-one correspondence between circulation time probability density and concentration history is shown to exist. This indete~ina~y is inherent and stems from the recirculation within the system. A value that can be determined is the maximum flow consistent with the measurements; any value below this maximum leads to feasible round-time distribution functions. The theory presented in this study leads to estimation procedures-pertaining to maximum feasible flow-which are completely free of prior distribution assumptions. I. INTRODUCTION

fields of applied science and enginee~ng tracer experiments have been used for a long time for the purpose of determining flow rates and associated physical quantities. The typical open-loop tracer experiment can be represented-in general terms by the following scheme: At some u~~rre~~ location the tracer material is injected into the moving fluid under study. The injection rate is assumed to be controllable and/or observable; in particular, an impulse injection-to be represented by a delta-function multiplied by a known constant, the quantity of tracer material-is considered to be realizable. The tracer material is assumed to get homogeneously admixed into its immediate environment, that is the (statistical) future of the tracer particles is indistinguishable from that of the encompassing ffuid particles. At some ~o~~srreu~ location that concentration history of the tracer in the surrounding fluid is observed and recorded. This observed function c(t) (0 s t < a;) may be given two complemen~ry interpretations: (1) The fiow system between the upstream injection point and the downstream observation point may be considered to possess the characteristics of a time-invariant, linear system within an arbitrary but steady fluid motion pattern. The concentration history c(6) is the ~e~g~~j~g~~~c?~~~ or impulse response function of the linear system under investigation. (2) The length of time, taken by tracer particles (as well as by the particles of the su~ounding fluid) to traverse the system between the upstream injection point and the downstream observation point is a random variable. The observed c(t) is proportional to the frequency of passages lasting t time units, hence it is also proportiona to the probability density of the above random variable. Let it be mentionIN MANY

Whis work was supported

by Grant HE 12495-01 from the National Institutes of Health, U.S. Public

Health Service. One of the authors (P.N.) was also supported by a research grant from the City University of New York and the Esso Education Foundation. SBoth Professor Naor and Professor Katz met an untimely death since the paper was submitted. The third author is grateful for having been able to enjoy their cooperation. II53

P. NAOR,

1154

R. SHINNAR

and S. KATZ

ed in passing, that the second viewpoint exhibited here takes for granted that no backdiffusion is experienced at the (downstream) site of observation and hence that the concepts of passage and first passage are identical. While we concern ourselves with ideal situations in the present study, the ‘no-back-diffusion’ assumption is also one that is frequently realized-at least to a very good approximation-in experimental situations. If the various assumptions enumerated above are indeed representative of reality, open-loop tracer experiments are greatly informative. Let m represent the quantity of tracer injected ~nstantaneousiy (t = 0) at the upstream site;f(t) is the probability density associated with the random variable ‘passage time between upstream and downstream sites’; w is the (average) volumetric flow rate. The experimentally observed function c(t) is measured in units of tracer quantity per unit volume of the encompassing fluid. The quantity of tracer material passing the downstream observation site during the narrow interval (t, t + At) may then be represented either as m$X(t)At,or as wc(t)At, that is WC(l) = mf(t) .

Since integration off(t) w from

(1)

(between 0 and m) yields unity we obtain the average flow rate m

w=

(2)

$3oc(t)dt 0

We note that the quantities on the right hand side of (2) are experimentally controllable and/or observable. Hence relation (2) is capable, in principle, of serving in the evaluation of the flow rate. Combination of (1) and (2) yields f(f) =

z c(t) = dr)

sp 0

(3)

c(t)dt

which contains (in principle, again) all the information regarding the random variable ‘passage time between injection and observation sites.’ In particular, the first moment p of the random variable, the expected passage time, is immediately given by

cc

P=

I

tc(r) dr =

I‘_wt c(f) dt

0 I 0

c(t)

.

(4)

dt

The outcome on the extreme right hand side of (4) is exhibited in the given fashion in order to manifest that this result (as well as the formulas pertaining to the other moments) does not even depend on knowledge of the correct propo~ionality factor connecting f(t) and c(t); it is sufficient that proportionality exists-the factor cancels out eventually when the moments are evaluated. The volume V associated with the system between the two sites is immediately gained as

Flow rate and transportfunctions in closed circulations

1155

The rather lengthy and detailed discussion relating to open-loop experiments has been presented here since, at first sight, the subject matter of this communicationclosed-loop tracer experiments within closed circulations-appears to be a minor modification of open-loop experiments performed on source-to-sink flows. Intuitively one would assume that here again both the flow rate and the density~(~) can be uniquely determined. This is incorrect. It is the leitmotif of the present study that in spite of many analogies between open-loop and closed-loop experiments there are far-reaching differences of substance between these two modes of experimentation. In p~icular we shall show that under the most ideal conditions of expe~mentation the probability density f(t) is essentially indeterminate and can be ascertained only if the flow rate is known independently. The same statement holds, of course, with respect to the moments associated with the densityf(t). If the flow rate is not known a priori, only a maximum flow rate consistent with the observed tracer response can be extracted. The single most important area, where closed-loop tracer experiments are of paramount importance is the system of blood circulation (see, for instance, the historical survey and critical theoretical evaluations presented by Zierler in[ 1,211. However such expe~ments are in no way limited to physiological systems and analogous problems appear in a variety of fields in enginee~ng as for example in the study of recirculation patterns in fluidized bed crackers. It is definitely not the inren~ of this communication-by bringing out the basic indeterminacy involved in a closed loop tracer experiment-to attempt the inv~idation of conclusion drawn from such experiments. Rather it is our purpose to stress the necessity of clearly articulating prior knowledge and assumptions regarding the blood circulation system as well as of designing experimentation in such a manner that the appropriate combination of observations drawn from separate sources will yield maximum information. We must bear in mind, that if we view the system of blood circulation in very sharp detail (i.e. during extremely short intervals) it cannot be considered time-invariant. One effect of the heart pumping is the introduction of a periodic oscillation in the system characteristics; in addition there are random changes in the nature of the heart-beat and this again affects the system. Likewise on viewing the system over very long periods (slow) drifts in the character of the system may be discerned. Yet for many purposes-pa~icularly in the analysis of phenomena whose evolution stretches neither over a very short time nor over a very long time - the assumption of time-invariance is a valid approximation. It is within the general frame of time-invariance that we demonstrate the existence of a basic indeterminacy. If the assumption of time-invariance is not realized additional indeterminacies of different character are introduced and must be taken into account. 2. THE BASIC EQUATIONS

We have noted in the preceding section that open-loop tracer experiments may be interpreted in two distinct complementary fashions. The same holds in the present case of closed-loop experiments. Indeed it is possible to develop the theory of closed-loop tracer experiments on one track only without having to resort to the use of probability

1156

P. NAOR,

R. SHiNNAR

and S. KATZ

concepts. However we find it both informative and advantageous for presentation to introduce probability ideas in the very early phases of the argument. Consider the typical tracer experiment in a system of closed circulation. An injection site A and an observation site B are chosen: the quantity in a tracer material is injected at A(t = 0) and the concentration history c(t) at B unfolds itself and is recorded. We immediately note one difference with respect to the experiment previously described. The notions ‘upstream’ and ‘downstream’ used before are no longer welldefined in the present context. The two well-separated points A and B have no clear upstream-downstream relationship due to the very fact that they are placed within a closed circulation system. Fortunately this notion which was usefully employed before is not needed in the present del~berations.~ Another difference of import between open-loop and closed-loop experiments is the ~ymptoti~ development of the concentration curve c(t). In the open-loop tracer experiment we were concerned with one ptlssage of a particle (tracer or encompassing fluid); hence the function c(t) turned out to be exactly proportional to the probability density of the passage time. After passing through the observation point tracer particles would never return and in time the concentration would necessarily decrease and asymptotically approach the value c(m) = 0. In the closed-loop experiment tracer particles are continually recirculated and the concentration is smoothed in time until it reaches the asymptotic value

where V is the total volume of the circulating fluid. Since m is known and c(m) is experimentally observabfe (in principle) we have almost immediately gained knowledge of the volume. We are made to realize at the same time that the concentration function at some finite time t is partly due to first, second,. . . . . k-th, passages of tracer particles in some apparently complex time-dependent fashion. In the open-loop experiment f(r) was proportional to c(t). What probability function bears the analogous relationship to the closed-loop concentration function c(t)? Indeed such a function exists-the renewal density. This function was originally developed in a context completely unconnected with the present subject matter. We shall not discuss the original subject area in which the ideas of renewal theory originated-the interested reader may wish to consult the monograph by D. R. Cox[3]. Rather we shall proceed in this discussion in terms of tracer experiments. Consider site B at some narrow time interval (t, t + At) after injection (into A) of the tracer. Let us fix attention on some given particle and pose the following problem: Derive the mean number of passages of this particle which may be expected in the tWe should add two reservations to this statement: (a) We have assumed in this discussion that the points however this is not necessarily the case and we may encounter situations where these points are infinitesimally close to each other. Under such circumstances the upstream-downstream relationship is indeed well-defined. This fact will be utilized later in this section. (b) In many systems-among them the system of blood circulation-there is a distinguished site, the pump (the heart, in our case). which causes the fluid motion to occur. This site (i.e. a heart chamber) which may or may not coincide with one of the two sites can be selected as a reference point for the whole system and by virtue of such a definition the upstre~-downstream relationship may be established. This might be useful for some purposes: thus. for instance, this reference point would define the point of inflow into an organ as being upstream with respect to the point of outflow.

A and f? are well-separated;

Flow rate and transport functions in closed circulations

1157

narrow time interval. The interval being narrow, this mean number is just the probability that the given particle passes under the observation point during (t, t + At). Were we to concern ourselves with $rst passages only, the answer would be~~~(~~A~,where f’&) is the open-loop probability density associated with the passage time from A to B. If we interest ourselves in second passages only, the proper answer is the convolution (f,&fRB) (r)At wheref,, is the open-loop probability density associated with the first passage time from B back to itself; this is sometimes referred to as the round-time density with respect to B. And if we deal with k-th passages exclusively the probability is the convolution (f&f&-“)*) (r)At, where the exponent of fBH signifies the (k- 1)th convolution of this function with itself. But the original problem did not impose any conditions as to the number of passages of the particle. Since k-th and k’-th passages of the particle are disjoint events if k # k’, the (unconditional) probability, ~~~(~)A~,that the particle comes under observation in the narrow interval (8, t + Ar f is given by

= [f~~(~$.fA~+f~~+...f’~-~‘*+f~*+...)l(t)A~=

(f:R$fk&)(f)A~.

(7)

The functionfO,$(r) = 8(t) which makes its appearance in (7) is (Dirac’s) delta-function. As noted earlier, h,,(t)Allt simultaneously represents the probability that the particle comes under observation during (t, t + At) and the mean number of times it does come under observation during this interval. The function ~~~(~) is accordingly the mean rate at which a given particle appears at the observation point. The function h,,(r) in the terminology of Cox’ mono~aph - the iodized renewal denser. The definining equation (7) may be rewritten as

This is referred to as the ~~~eg~u~eq~~f~o~ of renewal theory. Next we relate the renewal density to the concentration history. We proceed analogously to the argument which led to equation (1). If w8 represents the average volumetric flow rate at site B the following equation holds

WBCAB(~) = mh,,(t). The left-hand side and the right-hand side of (9) are two different ways of looking at one physical process. Both represent (after multiplication by A?) the quantity of tracer material which passes site B during the narrow interval (t, tt At). We observe that the step which led from (1) to (2)-and hence to the determination of the flow rate-cannot

1158

P. NAOR,

R. SHINNAR

and S. KATZ

be repeated in a closed circulation system since the integral J,“ph,,(r) dl (unlike the integral Jtfft) dt) diverges. We proceed to make use of the opportunitywhich certainly exists in principle and frequently in practice as well-of observing the concentration history at a point infinitesimally close to, and slightly upstream of, site A. The equations (7) and (8) are then transformed into h,,(t) = C f% (t) = j=l

fa.&)+~;X&) h/&f-u)

du.

(10)

The (thought) experiment was performed at a site slightly upstream of A (and not precisely at A) in order not to introduce a delta function into (10). The function hAA presents -again in the terminology of Cox’ monograph- the ordinary renewal density. The analogue of equation (9) is

While the real experiment AA may be quite intricate the notions introduced in the thought experiment AA are simpler than in the corresponding experiment AB. It is easy to verify (and we shall do so in the next section) that a one-to-one correspondence exists between~~~(~) and h,,(t); these two functions imply each other. This is not the case in an AB experiment: any given function h&f) is compatible with a great many pairs of feasible fAB(f) and f&f). Hence the analysis of the AA experiment must precede the investigation of the AB experiment. Furthermore in the present study we shall not deal with the most general AB experiment. Rather we shall engage in the study of a specialized AB experiment, to wit, the experiment where the injection point A and the observation point B are assumed to be in the Sclme pOw bard. A and B are defined to be in the same how path if any fluid particle leaving A (or B) cannot return to A (or B) without passing the other point. This is signified by

.fMft) =“fBaw= tf~BBfBA)tf) = f(f)

(12)

and wA= wB =

w.

(13)

Let equation (10) be multiplied by m/w and be combined with (1 l), (12), and (13). This results in

CA/l(f)= A corresponding

development

$f(r)+ fC&ff

it).

(14)

leads to

(15) Integral equations (14) and (15) were derived here from probability considerations. It is entirely feasible to obtain them from considerations of material balance. We may

Flow rate and transport functions in ctosed circulations

Il.59

consider c,&)(= c&f)) to be weighting functions of the linear (closed-loop) system under investigation and present the argument without any probability interpretations. It was stated before in the programme of this paper that we prefer to make progress through employing both types of ~~rnent with emphasis on the probability (in this context: renewal) interpretation. 3. INDETERMINACY

OF THE

FLOW

RATE

We begin this section by introducing the Laplace transform of functions studied in the paper. If we have any function ~(8) (0 c t < 03) its Laplace t~nsfo~ p(s) provided it exists -is evaluated as

I

$0) = ; e-” p(t) dt.

(16)

Integral equation ( 10) is reduced to

which immediately proves a statement made before: A probability density and its associated ordinary renewal density imply each other. Now what we observe in an (ideal) tracer experiment is not h,,(t) but rather the function cAA(l) which is proportional to it (by equation (11)). To proceed we define the concept of the reduced concentration

CAA(=)

m

CAA(f)

=

h,,(t) -hAA

=

~Aah.&) = phz,&)

(18)

which is (a) observable (unlike the renewal density), and (b) standardized in the sense that it tends asymptotically to the value 1. The quantity p (or /_LAA or, indeed, pBB) is the expected value of the random variable associated with the probability densityf(t) (=&A(t)) or, alternatively and equivalently, with the renewal density h,,(t). In terms of RAA(I) the integral equation (IO) and the associated Laplace transform (equation ( 17)) read RAA(f)

=

elf

+

(RA*Af)(t)

(19)

and

ii,,(s) = ^f&d

1--f(f)

(20)

Suppose now that a solution&(t) with expectation p. = J,” tfo(t) dt has been found which obeys (19) (and, of course (20)). We can immediately construct another valid solution {ff(t), pcLI)with an expectation arbitrarily larger than p0 I*1

=;

and whose probability density is given by

(0 < p1 < 1)

(21)

1160

P. NAOR, .m

=P*.Mt)+p*(l

R. SHINNAR

-Pl)f;*(t)+*

and S. KATZ

* *-tp,(l

-p,)‘-Y;*(t)+.

..

(22

To prove that {fi(t), p,} is indeed a solution we first observe that the functionf,(t preserves the required property of non-negativity. Secondly integration off,(t) yields I ,:fi(t)dr=p, whereas, thirdly, the expectation

ii: (l-Q= j=n

integral

@*i(t)dr = p1 Po ij(l

CL1=

(23)

1

-p#-1

(24)

= F

.i=l

shows consistency

with (2 1). Finally we observe that the Laplace transform -2

f*(s) = Plf,,W+pl(l

-p,)f()(s)

+ . . . p,( 1 -p,)J_’

-. f:,(s)

+ . .

P&JW =

(25)

1 -(l-p,)]&)

on insertion into the right-hand side of (20) generates P1.h)

fi

CLJIW = l-f,(d

Pl

1 -_(l-PI).?&)

1_

Pl”f&) 1 -(I

=p k&l(~) 1 - jm

(26)

-P,if&)

This completes the proof that {fi(t), p,}is a solution if {fo(t), pO} is a solution. If any value of p is consistent with the integral equation ( 19) so is any larger value of I*. We have presented our argument in a rather formal fashion. However, we may directly develop a probabilistic argument in terms of the renewal process (i.e. the process of consecutive passages) described earlier. Suppose we desire to dilute p,, in the ratio l/p, where 0 < p1 < 1. Then we need only fail to recognize, with probability (1 -pl), the appearance of a tracer particle at A. The renewal density hAA will be lowered in the ratio pl, and the reduced function RAA(t) will not be affected at all. We therefore see that the probability density generated by this imaginary process would be indistinguishable in the final concentration curve from the original probability density. To recognize the appearance of a tracer particle only with probability p1 is precisely the transformation exhibited in (22). We have shown that the two first-passage time densities&(t) andf,(t) lead to identical linear responses. The conclusion is accordingly that if, for a known function R,,(t), a value of p leads to a valid (i.e. non-negative) probability density./(t), then any larger value of p leads also to a valid probability density on utilizing transformation (22). We note that while the transformation formally works (in the sense that it leaves Raa(t 1 unchanged) for a reduction in p, that is, for p, > 1 it does not necessarily lead to valid probability densities in this case, since the terms on the right hand side of (22) alternate

I161

Flow rate and transport functions in closed circulations

in sign. Further, for an ordinary well-behaved function R.&f) it seems clear that sufficiently large values of y will surely work, since equation ( 19) for such values is reduced dominated by the leading termf(r) = ~~~(~)/~ and so leads to a positivef(the concentration R being positive). Finally, since if a value of p works, so does every larger value it follows that if a value of p fails, so does every smaller value. We are thus led to the conclusion that the set of valid p’s (for which equation (19) can be properly solved) is bounded from below. The lower bound is designated p,, as of now: the associated density isf,(t). So far we have tried to extract maximum information from an AA type experiment (carried out under ideal conditions). We have seen that such experiments are indeterminate, in principle, in the sense discussed above. The most such a tracer experiment can attain is bounding the expectation Al,from below and generation of the associated bounding density. Since the volume of the system can be obtained from the tracer experiment, setting the lower bound of p is equivalent to establishing the upper bound of the volumetric flow rate, w, with which the data of the tracer experiment are consistent. The problem arises how supplementary information can be utilized. If independent precise information regarding the expectation p-or alternatively about the flow rate w-is available, the tracer experiment in the closed-loop system yields exactly the same results that can be attained from a tracer experiment in an openloop system. indeed the closed-loop experiment represents a test-under these circumstanceswhether or not the independent CL(or w) is consistent with the data. Even if the supplementary information is not defined in precise terms of equalities, independent bounds on k and w may be sharpened and the associated probability densities as well as bounds for the higher moments may be established as a result of anAA type tracer experiment. If the additional information is simply an additional concentration history recorded simultaneously at a point B (on the same IIow path as A) &Sebasic indeterminacy in PAA (and in /AAB)is not removed. This may be demonstrated as follows: We return to (8) and rewrite it as RABff)

=

(27)

&w(f)+ t%WM)

where R&t) is defined - analogously to (18) -as RAdf) =

s =p

CAB(t) = *

AB

= AB

The application of the Laplace transformation

i?,,(s) =

on (27) yields

$g.

We shah consider (27) (and equivalently (29)) as supplementing from (19) and (20). Let (20) be divided by (29); this renders ii,,(s) AAh)

_

(28)

phAB(t)*

DC1

fb) jkB@)’

(29) information

avaiIable

(30)

1162

P. NAOR,

But it is a simple consequence

R. SHINNAR

and S. KATZ

of ( 12) that this ratio equals (31)

If we return to the time domain (take the inverse transform) this reads as

R4‘4w= tRA*B&4) 0).

(32)

The interpretation to be attached to (32) (or (3 I)) is rather interesting: A tracer is injected in A and the concentration histories at A and at B are simultaneously observed. Application of (32) enables us, in principle, to gain complete information - probab~iity density, moments and, in particular, the expectation y BA-regarding the partial system BA. This is a slightly strange result at first sight. Whether or not ii is practicable to to actually experiment in such a manner must remain the subject of a separate study. Unfortunately this is the only unique result that can be derived from the two simultaneous recordings. Since by (20)f(s) may be expressed as (33) equation (30) may be reshaped into (34) Suppose we have the (minimal) solution {f&),&.&s)] (already Laplace-transformed) with {y,, poAB}; that is these functions are simultaneously consistent with both concentration histories C.&I) and cAB,(t). It is easy to verify that the pair {f;(s), j;,,(s)} with {pcLI, ~~~~ is a solution wheref,(s) and p1 are given by (25) and (21), and analogously fi.&) and PM are represented by

L3w =

P1foaeb)

(35)

l- (1-P1LhAS) l--P,

I%43 = CLOAB + I_Lo-.

(36)

171

We have already shown-through the use of (26)-that R,,(t) is invariant under the transformation (25). Next consider (29); on insertion of (35) we obtain ,. PlfOA&)

PlflABtS) = ? 1- t 1- Pl)_fO3d = fh$ku!@ 1-_f-at4 ’ ??a) PJXS) I_

1--(I -Pl)jhw That is function RA&), too, is inv~ant

under the transfo~ation

(35).

(37)

1163

Flow rate and transport functions in closed circulations

In effect it is impossible to determine eitherf,,(t) orf(r) even if two. simultaneous records c&t) and c,&) are available. The basic indeterminacy is not removed. This does not mean that there is no advantage in having the additional observation cAB(f). First through employing c,&) we may be able to obtain a modified (and hence better) lower bound for CL.Secondly a lower bound for pAB may be stated which, of course, was not possible before. Thirdly the value of pBAis attained-in principle. We may make use of the information that is contained in cABin a somewhat indirect fashion. Suppose that some prior knowledge is available regarding the volume distribution within the system. As an example this may be stated as an inequality U
possessing

(38) some physical interpretation.

Relation (38)

E&t 2 1-K CL

(39)

which, in turn, renders an upper bound for ~1 (40) The quantities on the right hand side of (40) are fixed and either a priori known (1 -K) or determinable by two simultaneous tracer experiments (pBA). If the left inequality in (38) is really an equality (so that we know the volume between A and B as well as the total volume V), then (40) becomes an equality also, and we have in effect determined the flow rate. Without such prior knowledge the indeterminacy that exists with respect to two points on the same flow-path, may be done away with only if three simultaneous tracer experiments are carried out and their concentration histories cAB, cBA and cAA(=cBB) are available. If A and B are not on the same flow path the triple experiment is not sufficient. 4. DILATATION

AND

FEASIBLE

CONTRACTION DENSITY

OF THE

ROUND-TIME

It has been established in the preceding section that a particular solutionf(t) with expectation p generates additional solutions possessing larger expectations. We recollect that -for the AA case-a solution is an identification of a round-time probability density f(t) which is compatible with the observed reduced concentration RAA(r). In the AB case two probability densitiesf&) andf(t) represent a solution- that is they are compatible with the two observed functions R.&t) and RAA(t)- and again on using this solution as a starting point it is possible to obtain solutions possessing larger expectations. The transformation generating the new solutions from the old is essentially that given in equation (22) in the pure AA case. In the AB case it is again transformation (22) together with

(41)

1164

P. NAOR, R. SHINNAR

and S. KATZ

which is the inverse Laplace transform of (35). We refer to these two transformations as dilatations of the probability density and it has been proven that dilatations of a solution are always possible. This is an extension of known properties of renewal functions (Daley[4]). Furthermore since prescription of the expectation generates the renewal density (whereas before we dealt with the reduced concentration only) there exists not more than one solution with a designated mean CL.The following problem arises then in a natural way: Given a solution f(t ); it is desired to decide whether it is of the f,(t) variety - i.e. dilated from a solution with a smaller expectation-or whether it is&t). Fu~hermore if it is of typefrft) how far backwards can this solution be contracted in order to generatef,(t)? Now formally such a contraction takes place if a value exceeding 1 is chosen for p1 in transformation (22). Iiowever, non-negativity of the density thus generated is not assured, as was explained above. This problem of the feasible contraction to minimum round time-and hence the determination of the feasible maximum flow rate possesses several facets not all of which will be treated here. We shall illustrate the theory given above with some mathematical examples, that is with models where explicit distribution assumptions of the round-time are introduced. In order to stress the independence of the theory presented here on specific distribution assumptions we shall also outline an approach using a purely numerical algorithm. The treatment of data from real physiological experiments-where additionally available information should be utilized-is deferred to a later report. As our first example consider a single (ideally) mixed stage. The round-time density-observable if an open-loop type experiment were carried out-is negative exponential with mean p

The associated reduced concentration constant value 1.

is easily evaluated as the function possessing the

Indeed this is what may be observed in a closed-loop experiment. But this (observable) function is compatible with any negative exponential round-time density whatever its expectation. This is indeed corroborated on applying transformation (22). fl(t~=pl~e-t~~+p,(l-pp,)_tie-t/~+. Ir,

f . . .= ;

e-f/h

mif1 -pdtl~

z

j=.

(j- l)!pJ

. * +PIU

= ae_ p

tj-I

--PIP

tw e”-P”“t””

l),Crje-t"

cj_

=

pr

e-Pit/P

El,

*

(44)

The dilatation transfo~ation (44) applied on density (42) has generated a new exponential density which possesses an expectation (&,) exceeding p since p1 was taken from the (open) interval (0, 1). But it is immediately obvious that contractions, too, are feasible. If p1 is made to exceed the value 1, non-negativity of the transformed function fi(t) is assured; indeed it is again an exponential density with mean (&p,)(< p). The

116.5

Flow rate and transport functions in closed circulations

transformation (44) represents a feasible contraction in this case. This collates with the statement made before that the reduced concentration (43) is compatible with any exponential density. A further consequence is that the lower bound ,uOof the feasible expectations possesses the value 0. In simpler physical language: the flow rates compatible with this experimental evidence have no upper bound: all positive flow rates give consistent results with the experimentally observed function (43) -the form of the round-time density remains negative exponential. Next we consider systems made up of several-n, say-ideally mixed stages of equal size which are arranged in series. Let each stage possess an expected residence time q-‘. It is possible to furnish a mathematical proof (which will be presented elsewhere) that the residence time density of such a system- (~~~~-l~~~- l)!)e-SL-cannot be contracted. In other words: Suppose we observe the reduced concentration history of such a system in a closed-loop type experiment or, ~ternatively, we are presented with the appropriate formula; if we search now- numerically or analytically-for the residence time density with minimum expectation which is consistent with the data we recover the function f(t) = (r)F’l(n - I))!e- rlt. Any attempt to contract beyond this function will yield a density possessing negative values in some intervals, i.e. a non-feasible solution. Within this class the case n = 2 is chosen as the first example. Since the expectation pcl equals (2/q) we can write = $-te+t

j&t)

Application of the tr~sformation

=

-$ e-2tllm.

(45)

(22) renders

fiV)=pI.Mt)+p,~l-pI)f~2(t)+~~

.+p,(l-p,)(j-llf,*(j-l)+.

q2jt2*-’ -r)t

q4t3 _~tf~~-p,(l-p,)j-l-e

=plq2te-vt+pI(1-p1)

ye

=$$y+( m-z+

bM=z)“+.

]!

--rltsinh (qt-)

(2j-I)!

. . . ht~d2’-’

3! = ,$&

..

+ *. .

(Zj- I)! e-2t~Pnsinh (2~~).

(46)

It is not difficult to verify that (46) is meaningful-as a probability density-if and only if p1 does not exceed 1, i.e. the transformation can operate as a dilatation only. If the transformation is tried out as a contraction (pr > 1) the function generated is fi(f)

=

2p1 e-Zt/ikl sin (“7).

FOG3

(46’)

This function is negative in some intervals oft and hence not admissible as probability density. This state of affairs is exhibited in Fig. la. The density (45) is presented together with two dilated densities (pi equals 0.5 and O-7, respectively) and one non-

IJES: Vol. IO, No. 12-L

P. NAOR,

1166

R. SHINNAR

and S. KATZ

Fig. I .(a) Dilatation and contraction of the first passage time density ff relating to two stirred tanks of equal size connected in series (Values of p,: 0.5.0.7, 1.1) (b) Reduced concentration in a dosed system for the first passage time densities given in Fig. l(a).

feasible contracted density (pi equals 1.2). The reduced concentration associated with all these densities is exhibited in Fig. lb. The function R,,(t) is derived (e.g. from~~(~)).

RA4V)=

Mt) = h (t) = f E&(t) +fp(t)+ rl 0

-j&.&)

2

+2fe-““+~e-“‘+. = 2e-nt qt +9l.$+ I

2+2j-1

* -“/y(t) + * *]

. 2L_._e-~t+.

*

(Zj-l)!

. . . __!$Z]

= 2e-W eV’~eWn’=; 1_ e-!BJt= 1_ e-4tILLo.(47)

Flow rate and transport functions in closed circulations

1167

The very same reduced concentration history may be obtained from the density presented in (46) where 0 < p1 < 1 and even from the density (46’) with which, of course, no physical meaning can be associated. The system tt = 3 is treated analogously and exhibited in Fig. 2a and Fig. 2b . The appropriate analytical expressions are somewhat more complex than those of the preceding system. However the qualitative features - e.g. the development of the concentration history-of these two cases are rather similar. One additional system belonging to this class-to wit, n = lo- was evaluated in numerical detail and the results are exhibited in Fig. 3a and Fig. 3b. The reduced concentration associated with this sytem surges upwards and oscillates before approaching the value 1. This is indeed what

Fig. Z.(a) Dilatation and contraction of the first passage time densityf, relating to three stirred tanks of equal size connected in series (Values ofp,: 0*5,0*7,1-l). Reduced concent~t~on in a closed system for the first passage time densities given in Fig. Z(a).

IX68

P. NAQR,

R. SHINNAR

and S. KATZ

Fig. 3.(a) Dilatation and contraction of the first passage time densityf, relating to ten stirred tanks of equal size connected in series (Values of pi : 0.5,0-T, 1.1). (b) Reduced concentration in a closed system far the first passage time densities given in Figure 3a.

should be expected from a residence time density whose coefficient of variation is relatively small-of the order 30 per cent- One is tempted to associate the first peak in the reduced concentration history (Fig* 3b) with the first round-time, the second peak with the second round-time, etc, While this may be the case it is just as correct - prima facie, in the absence of additional evidence-and hence possibte, to generate a feasible round-

Flow rate and transport functions in closed circulations

1169

time density with both mean and coefficient of variation greatly exceeding the values furnished by intuition. This may be observed in Fig. 3a. The case II + m may be ascribed physical meaning if at the same time r) is assumed to tend to infinity whereas the ratio n/q = CL,,approaches a finite, non-zero constant. A system possessing such characteristics is said to have plug-flow and the appropriate probability “density” is given by

w-

_/a) =

(48)

PO).

This is essentially a deterministic situation and hence the reduced concentration is not smoothed asymptotically towards the value one. Yet even under such exceptional circumstances the application of the dilatation transformation (22) to fo(t) produces additional densities

flu) = Z/1(1

all of which are compatible with the reduced concentration

R&l(f) =

(49)

-pJk-‘60-&J

PO

Ii 80 -k/.4.

defined as

(50)

k=l

Again the only interpretation that can be given to (49) is that of a dilatation; a contraction (pl > 1) yields non-feasible densities. The above signifies that a simple (and seemingly deterministic) function such as that given in (50) is still associated with indeterminacy and, in principle, the underlying open-loop residence time distribution cannot be determined in a unique fashion unless additional information is available. As our final synthetically constructed example we view a cascade made up of six mixed stages (not all equal) arranged in series. Point A where the injection is carried out precedes four small mixed stages each possessing the intensity parameter 47. Point B where concentration history AB is being recorded is located at this juncture and it is followed by two large mixed stages each possessing the intensity parameter 7). The second large stage is linked up with the first small stage at point,4 and this closes the loop. A second concentration history -AA -is being recorded at this location slightly upstream of injection point A. No analytical expressions will be offered here -graphical representation is preferred. Figure 4a represents the two (open-loop) residence densities fOAA(t) and fOAB(f) associated with the flow mechanism postulated above. Figure 4b depicts the observable reduced concentration histories of this model. Figure 4c and Fig. 4d represent feasible dilations offOAA(t) andf,,,(t) @I equals O-5 and 0.7, respectively) brought about by the two transformations (22) and (41). These pairs of functionsf,,.&) andA,, are as compatible with the observable reduced concentration as the originally postulated foita(t) and foAB(f). Hence they represent valid feasible solutions. In Fig. 4e we observe a contraction (pi equals 1.1) of the original densities; the new densities thus generated are non-feasible. In each of the above examples we embarked on our route by postulating some simple flow mechanisms (e.g. mixed stages in series). The second phase was the derivation - analytical or numerical -of the associated reduced concentration history. Finally we

1170

P. NAOR, R. SHINNAR

and S. KATZ

demonstrated in each example that there existed a ~u~y-o~e correspondence between underlying open-loop probability density functions on the one hand and observable closed-loop reduced concentration histories on the other. One rather outstanding feature of the famiiity of eligible probability densities is the minimum feasible expected passage time or equivalently, the maximum feasible volumetric flow rate. In a genuine experimental situation we start out with the observed reduced concentration history and typically we have no compelling theoretical reason to assume the existence of compartments, their sizes and configurations; again very little is a priori known about the flow and mixing mechanism within and between these compartments. The theory

t/&a

Fig. 4(a).

t//J%

Fig. 4(b). Fig. 4(a) First passage time densities in a system comprised of four small stirred tanks of equal size (intensity parameter 47) connected in series with two large stirred tanks of equal size (intensity parameter r)). (A scheme of the structure of the system will be introduced in Fig. 4(a)). (b) Reduced ~oncen~ations in a closed system for the first passage time densities given in Fig. 4(a).

Flow rate and transport functions in closed circulations

1171

presented in this paper enables one to pursue an analysis which is completely free of prior distribution assumptions. The only tool necessary (and sufficient) for such an analysis is knowledge of the value of the reduced concentration to a very high degree of precision for any desired time r. The attainment of such numerical precision is, of course, no mean task in the typical experimental situation. Assuming that such precision is available one proceeds by successively solving a discrete version of equation (19). The time axis is divided into small intervals of equal A, denoting the average values of R across the intervals as a,, u2,. . . ., the correspondIng integrals offas 7r1,p2 . . . and setting down the discrete renewal equation ai =

E

7Tf +

f-l

x

UjTi-j

i= 1,2,..

.

(51)

j=l

I 0

I

0.5

,

1.0

1.5

1.0

1.5

VP. Fig. 4(c).

I!!bPEH t/A+. Fig. 4(d).

1172

P. NAOR,

, 0

R. SWINNAR

and S. KATZ

1 c&s

I 1.Q

%.5

t/*0

Fig. 4(e). Fig. 4(c) Dilatation of the first passage time densities given in Pig. 4(a} (p, = O-5).(d) Dilatation of the first passage time densities given in Fig. 4(a) fp, = O-7). fe) Cont~ctio~ of the first passage time densities given in Fig. 4(a) (p, = l-1).

with an assumed p. Since the ai are known, numerical solutions for the ni can readily be carried out with appropriate. trial values of p/A. Indeed, rTTican be written down explicitly as a polynomial in a,, az, . . . ai the coef5cients being given by number-theoretic considerations invotving the partition of i, although it may in practice be more convenient simply to pick off the values nl, 7r2, . . . etc., recursively from (5 1). In any cases a check on the numerical procedures is given by the fact that any solution of (5 I ) satisfies the conditions xCi=l

(52)

CiVi=,’ i

t53)

for any choice of e. Now that a procedure for calculating the ?T~is in our possession, we next recognize that assumed values of p leading to all probabilities mi 2 0 are at or above the minimum and hence are feasible while values of p leading to some ri < 0 are below that minimum and hence are non-feasible. Suitable search schemes can then readily be developed to estimate the minimum CL.

In a simplified form the results of this study can be summarized as follows: A single tracer experiment in the circulation does not lead to a unique estimate of flow rate. While in an open system an accurate tracer experiment would lead to an accurate measurement of both flow and volume, tracer experiments in closed-loop systems have an inherent indeterminacy due to the recirculation. This indeterminacy is not removed by simultaneously measuring the tracer response at the point of injeetion. What can be determined from a tracer response is the maximum flow rate which is

Flow rate and transport functions in closed circulations

1173

compatible with the observed cha~cte~stics of the system under investigation. This maximum flow rate can be uniquely determined provided precise values of the concentration history- the tracer response-are available for any desired time f after injection. There is no a priori cogent argument leading to the identification of the maximum feasible flow rate with the actual flow rate. Hence the pertinent purposes of closedloop tracer experiments must be: (a) the bounding of feasible values of physical properties possessed by the system such as the moments of the round-time distribution, the flow rate etc.; (b) the blending of prior knowledgenot necessarily well-articulated -with tracer experiment data in order to attain deeper understanding of the system properties. Whether or not these objectives are easily attainable depends on the concrete circumstances of experimentation. Thus, for instance, it may be possibIe to design the experiments such that utilization of inequality (40) leads to a relatively narrow interval within which p is to be found. Indeed, with knowledge of the geometry of the system or with independent measurement of the flow rate the tracer experiments lead to (almost) unique first-passage time probability densities (or, equivalently, to almost unique evaluation of the system transfer function). It is also interesting to observe that overspecification of the system may lead to erroneous conclusions. Thus, for example, gamma densities which are commonly used by the engineer for plant modelling will lead to unique unmerited estimates of w. In several further communications we shall discuss the interpretations of data from actual physiologic~ tracer experiments, approximation methods and some further theoretical ramifications. Ack~owle~g~~nts-The authors had a number of fruitful discussions with Professor Walter L. Smith of the University of North Carolina at Chapel Hill and with Professor James B. Bassingthwaighte of the Mayo Foundation, Rochester, Minnesota. The authors’ appreciation is hereby expressed. The authors wish to thank Mr. Thomas Ackerman for performing numerical computations and for preparing the graphical representations as exhibited in the Figures of this study.

REFERENCES, [l] K. L. ZIERLER, Circulation Res. 10,393 (1962). [2] K. L. ZIERLER, Handbook ofPhysiology (Edited by W. F. HAMILTON tion 2. American Physiological Society (1962). [3] D. R. COX, Renewal Theory. Methuen ( 1962). [4] D. J. DALEY, Proc. Camb. Phil. Sot. 61,5 19 ( 1965).

and P. DOW) Vol. 1, Sec-

R&um&L’interpr&ation de captages expt%imentaux dans des systbmes &circulation fermCe est dtudiie du poit de vue de la thtorie de probabilitk appliqube. 11est montrt5 clue les quantiths et caracteristiques fondamentales du systlime, tel que le temps moyen cyclique et sa variabiIitC, aussi bien que le dCbit voIum6trique moyen, ne peuvent pas &tre extraites du capteur de rksponses, m&me si deux 6voIutions de concentration,-l’une au site d’injection et I’autre & un emplacement arbitraire sur le m&me parcours de flux-, sont enregistrbes simultan6ment. Les reponses du capteur observees sont compatibles avec de nombreuses distributions rt5alisables de temps cyclique sous-jacent (fonctions de transport) et I’on dtmontre 1’6xistence d’une correspondance multiple-unique plutet qu’une correspondance unique-unique entre la densit de probabilid de temps de circulation et I%volution de cbncentration. Cette indetermination est inhCrente et provient de la recirculation dans le cadre du systtme. Une valeur qui peut itre d&enni&e est Ie d&bit maximal correspondant aux mesures; toute valeur endessous de ce maximum conduit &des fonctions de dist~bution r&&sables de temps cyclique. La theorie pr& sent&e dans cette Ctude conduit B des procedures d’appr~ciation-relines avec un flux maximal realisabte qui sont libres de toute pr~somption de distribution prbalable.

1174

P. NAOR, R. SHINNAR

and S.

KATZ

Z~~~~Die Deutung von Indikatorexpe~menten in geschlossenen Kreislaufsystemen wird unter dem Gesichtspunkt angewandter Wahrscheinlichkeitstheo~e untersucht. Es wird gezeigt, dass grundlegende Quantitiiten und Kennzeichen des Systems. wie mittlere Umlaufzeit und ihre Verlinderlichkeit, sowie die durchschnittliche Raumfliessgeschwindigkeit vom lndikatoransprechen nicht erhalten werden kijnnen, sogar wenn zwei Konzentrationgeschichteneine an der Einfilhrtmgszone und die andere an einem willkiirlichen Ott am selben Flussweg- gleichzeitig aufgezeichnet werden. Die beobachteten Indikatoransprechdaten sind mit vielen zugrundeliegenden Umlaufszeitverteilungen (Transportfunktionen) vereinbar und eine vieleeins Zuordnung, eher als eine eines-eins Zuordnung zwischen der Dichte der Umlaufszeitwahrscheinlichkeit und der Konzentrationsgeschichte wird als bestehend gezeigt. Diese Unbestimmtheit ist innewohnend und stammt vom Wiederumlauf innerhalb des Systems. Ein bestimmbarer Wert ist die mit den Messungen verein-

bare Maximalstromung; jeder Wet? unter diesem Maximum ftthrt zu annehmbaren Umlaufszeitverteilungsfunktionen. Die in dieser Arbeit vorgelegte Theorie fiihrt zu Schatzmethodenin Bezug auf die maximale annehmbare Stromung- die vollstPndig frei von vorhergehenden Verteihmgsannahmen sind. L’inte~re~zione di esperimenti con traccianti nei sistemi a circolazione chiusa viene studiato da1 punto di vista della teoria di pro~bilit~ applicata. Si dimostra the le caratteristiche e le quantita fondamentali de1 sistema, quali il tempo di circolazione medio e la sua variabilita, come pure il flusso volumetrico medio, non possono essere estratti dalla risposta de1 tracciante anche se si registrano simultaneamente due concentrazioni differenti, una nel punto d’iniezione e l’altra in un punto arbitrario sullo stesso percorso. Le risposte del tracciante osservate sono compatibili con molte distribuzioni di circolazione sottintese e fattibili (funzione di trasporto) e si dimostra l’esistenza di una corrispondenza multi-uno anziche una di uno-uno fra la densita probabile de1 tempo di circolazione e il tipo di concentrazione. Tale indeterminatezza e inerente ed e data dalla ricircolazione all’interno de1 sistema. Un valore determinabile e il flusso massimo consistente con le misurazioni; qualsiasi valore al di sotto di quest0 massimo porta a funzioni di distribuzione di circolazione fattibili. La teoria presentata in quest0 studio porta a procedimenti di valutazione-nei riguardi del flusso fattible massimo-the sono completamente liberi da assunzioni di distribuzione precedenti.

Sonmario-

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