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Residence time distributions in systems governed by the dispersion equation
OXW-2%W/8l/M0X7-l1l$Ul~ll PerganonPress Ltd.
RESIDENCE TIME DISTRIBUTIONS GOVERNED BY THE DISPERSION
IN SYSTEMS EQUATION
E. B. NAUMAN Rensselaer Polytechnic Institute, Troy, NY 12181, U.S.A.
(Received 13 March 1980: accepted 10 December 1980) Abstrr~et-The theories of discrete and continuous random walks have been applied to systems governed by the dispersion equation. It is shown that residence time distributions can he defined and determined at a particle level of scrutiny and that the resulting distributions are identical to those obtained using continuum methods. Now, however, the appropriate boundary and initial conditions are apparent; and the resulting solutionscan be given a clear physical interpretation. The long-sought residence time distribution for dispersion in an open system is shown to be the same as that in a closed system. This result is consistent with Danckwerts’ formulation for the yield of first order reactions but invalidates recent results which were based on a transfer function approach to residence time distributions. INTRODUCTION
A
CI( - m) = Co and that C,( +m) is bounded, the steady state version of eqn (1) can be solved piecewise for each of the three sections shown in Fig. 1, and this
that
key problem
in chemical reaction engineering is simultaneous convection, diffusion and reaction in tubular vessels. The one dimensional dispersion model for this process gives rise to a partial differential equation
composite analysis
solution provided
completes
the
steady
state reactor
D and U are known.
the parameters
when one or more of the D vanish. If of concentration is lost at the origin; and we obtain the famous Danckwerts[2] boundary conditions for a system closed at the inlet Special cases arise
D, = 0, continuity
(1) Solutions to this equation for a variety of initial and boundary conditions have occupied much of the literature for the past twenty-five years. Figure 1 illustrates some general features of the type of system which has been studied. Equation (1) corresponds to reaction in the unsteady state, and this situation has rarely been approached directly. Instead, interest has focused on solving the steady state equation with reaction or the time dependent equation without reaction. Much of the work on steady state reactions has been devoted to finding physically appropriate boundary conditions for the system illustrated in Fig. 1 where reactant enters at x = --o) but reacts only over the section 0 < x < L. It is now generally agreed that the correct boundary conditions are those that show the concentration, C, and the particle flux lp = UC-D&
U,Co = UC(O+)-
In most of what follows, by open boundary we will mean the situation where D, = D > 0 or D, = D > 0 although many of the concepts can also be applied to open boundaries with D# D, > 0 or Df D2>0. Also, we explicitly treat only the case U, = U = Uz. The idea of using tracer experiments to estimate U and D dates to Levenspiel and Smith[3]), and from their work springs much of the interest in solving eqn (1) in the unsteady state. The conceptual experiment is to inject a tracer at point x =O, to measure the concentration at point x = L, and from this to deduce U, D and the residence time distribution for the test section, O-C
CES Vol. 36.No.6-A
Outlet sectm
Test or
reaction sectlon
Fig.
(3)
dC 22 I r=O.
dC
Inlet section
Of
If Dx = 0, the system is closed at the outlet and the exit boundary condition becomes
to be continuous at x = 0 and x = L[l]. When these boundary conditions are augmented with the knowledge
-
Dg
Inlet
Outlet
rnjectlon
sample
probe
probe
1. Tubular reactor with provision for tracer experiments. 957
-
E. B. NAUMAN
958
x < L. This rather simply stated problem has proved to be quite difficult for the general case where D,, D, and 4 are all positive. Many papers have discussed this problem, the most recent being those by Gibilaro[4] and by Kreft and Zuber [5]. One purpose of the present paper is to again examine those solutions to the dispersion equation which are reputed to represent the residence time distribution in an open system. At first glance, the distribution of residence times for material leaving the reactor seems an intuitively obvious concept for discrete entities such as molecules or Brownian particles. However, eqn (1) is usually interpreted to govern only the net flow of material and not the dynamics of individual particles which, subject to dispersion, can cross and recross a boundary many times. This interpretation has led Buffham[6] and Gibilaro[4] to despair of ever finding the residence time distribution at a molecular level of scrutiny. Instead, Gibilaro has redefined the residence time distribution from a continuum viewpoint as a transfer function which relates the output flux of material to the input flux via convolution:
where f(r) is the differential or residence time frequency function with first moment:
That eqn (5) defines a unique function f(t) for all input disturbances $(t,O) follows from the linearity of Equation 1. The appropriateness of basing eqn (5) on the flux 4 rather than the concentration C has been discussed by Gibilaro and by Kreft and Zuber, and the validity of eqn (6) has also been shown by them. Equation (5) shows how a transfer function, alleged to be the residence time frequency function, can be determined from time dependent solutions to the dispersion equation without reaction. An alternative method for determining f(t) is to use steady state solutions to eqn (1) with reaction. Danckwerts[2] showed that the yield of a first order reaction is related to the residence time distribution via Laplace transformation. C(+ m) = C, 0me-“f(l) dt I
(7)
where s is the reaction rate constant. Equations (5) and (7) provide two different definitions of f(t). It was long assumed that these definitions were consistent and that both conform to the intuitive notion of time spent in the system. Gibilaro[4] however has recently shown that the dispersion model gives different results depending on which defining equation is used. The present paper treats this discrepancy and other inconsistencies in the current literature. The paper has the following specific goals: (1) To demonstrate that residence time distributions can be defined and determined at a particle level of
scrutiny, be these particles molecules, Brownian particles or fluid points in the sense of Danckwerts (7). (2) To show that the dispersion equation can be interpreted in terms of probability distributions governing the location of individual particles as well as the more conventional interpretation in terms of net fluxes. (3) To show what boundary conditions are suitable for deducing residence time distributions for systems governed by eqn (1). (4) To present new results for various types of systems governed by eqn (1) and to resolve the current literature dilemma regarding residence time distributions in systems with reaction. Some of these concepts are well established in the physics and mathematics literature. Others are new. The general approach will be to examine the motion of individual particles which are subject to dispersion. This examination will begin with the theory of discrete random walks; but, by passage to a limit of many small steps, we will show that eqn (1) governs the location of individual particles in a probability sense. The boundary conditions appropriate to the study of discrete random walks will then be translated to those appropriate from a continuum viewpoint. The final results will be a method by which true residence time distributions can be deduced from eqn (1) using continuum mathematics. DEFINITION OF RESIDENCE TIME
When applied to distinct entities which can be tracked over time and space, the idea of residence in a system seems intuitive and uncomplicated. However, care must be taken when there are several locations where particles can enter or leave the system and when an individual particle can make multiple entrances and exists. The complete definition of a “residence time” should include specification of the inlet, the outlet and of what actions stop and start the clock. The following definitions, although by no means inclusive, are useful for analyzing systems governed by eqn (1). For ease in visualizing these definitions, reference is made to Fig. 2 which shows representative particle trajectories in doubly open (D, >O, D3 > 0) and doubly closed (0, = D3 = 0) systems. The inlet-to-outlet sojourn time represents the simplest trajectory between the points x = 0 and x = L. Time starts when the particle leaves x =0 in the positive direction and ends when the particle first reaches x = L without having revisited x = 0. Inlet-to-outlet sojourn times are represented by trajectories M and 7-g in Fig. 2. Note that path 5-6, the reverse transition, is not included in the definition. A first passage time is represented by the trajectory 14 in Fig. 2. Timing starts when the particle first enters the system and continues until it first reaches x = L. Note that this definition may include some time spent outside the system with Y CO if the inlet boundary is open. A lust passage time is represented by the trajectory 7-10. The clock starts when the particle last leaves the point x = 0 and continues until the particle exits at x = L, never to return. This definition may include time spent
959
Residence time distributions in systems governed by the dispersion equation
particle which enters the system at .x = 0 will eventually exit at x = L never to return. Often, it also implies a finite duration for the discrete random walk which approximates the continuous process.
I
I
DISCRETE
(a
) Doubly
I
I
x=0
x=L
open
system
I
(b I
I
Doubly closedsystem
Fig. 2. Representative particle trajectories in dispersive systems.
outside the system with x > L if the outlet boundary is open. The life span of a particle is the elapsed time between when it first enters at x = 0 until it finally leaves at x = L, never to return. The corresponding path is the entire trajectory l-10, and will generally include some time spent outside the system if there is an open boundary. The residence time of a particle is the same as the life span except that time spent outside the system is excluded. Thus for the doubly open system, time spent on the trajectories 2-3, 4-5, 6-7, and 8-9 is all excluded. In what follows, the symbol t will be used to denote any of the above times; and a reference to the adjacent text will be necessary to distinguish between them. All of times have frequency functions, f(t), and cumulative distribution functions, F(r), which reflect random variations between particles following similar trajectories. Thus f(t) df = Fraction of particles (which follow a specified trajectory with specified timing rules) which experienced a time between t and f +dr. Each time will also have a mean and higher moments which are defined in the usual manner for continuous probability distributions. We consider only those situations where the mean, < is finite. This implies that a
lt.4NDOM
WALKS
A microscopic model for the residence time distributions in a dispersive system can be developed using the probability theories of Brownian motion and diffusion. Einstein [8] interpreted Brownian motion in terms of probability distributions, and Smoluchowski[9] suggested that these probability distributions could be approximated using a discrete random walk. This basic approach has been extensively treated in the physics and mathematical literature [IO-131 and is closely related to the well known problem of a gambler’s ruin[l4]. King[lS] suggested single particle simulations as a modeling technique for diffusion, and Aris and Amundson[M] suggested that random walk concepts could have been used to obtain some of the results they derive by classical methods. Shinnar and coworkers [17191 used random walks and probability theory to treat reactions and residence times, primarily is stirred tank reactors, while Brenner and Gajdos [20-223 used the single particle concept in their study of generalized Taylor dispersion phenomena. The present paper applies these ideas to what is now a classical problem in residence time theory. Inherent in the concept of residence times is a notion that the fluid contains individual elements which can be tracked and timed as they flow through the system. Considering these entities to be molecules is the logical choice and is indeed appropriate when D in eqn (1) is the molecular diffusivity. Brownian particles can also be treated given a different interpretation of D, and fluid points in the sense of Danckwerts[7] are appropriate for dispersion caused by turbulence. We shall use the term particle to denote any of the above entities and will approximate one dimensional diffusion, Brownian motion or dispersion by discrete random walks of such particles on the real line ( - m, m). A discrete random walk is a stepwise process; and in the model to be used here, each step moves the particle exactly one unit length, Ax. There is probability p that the particle will move Ax to the right and probability q = 1 -p that the particle will move Ax to the left. When p = q = f, the walk is symmetric and the corresponding physical process is pure diffusion in a stationary medium. Asymmetric walks with p > q will be used to model diffusion in a medium having convective velocity U, and it is possible to introduce a third probability r to reflect termination of the walk by reaction. This random walk model may not appear particularly realistic since real particles will obviously have a distribution of step lengths and of particle velocities. Chandrasekhar[ll] describes more complicated models which reflect this; but, in the limit of a large number of smail steps, the final results are insensitive to the exact form of the random walk. The random walk begins with the introduction of a single particle at some position jAX on the real line.
E. B. NAUMAN
960
Define Pi(k. n) as the probability that this particle will occupy position kAX after exactly n steps. Then the difference equation Pj(k,n+l)=pPi(k-l,n)+qP,(k+l,n)
(8)
holds for all k in an unrestricted random walk. However, we are primarily interested in random walks of finite duration which occur over the interval (0.L). In the discrete model, let k = 0 correspond to the reactor inlet (x =0) and let k = z correspond to the outlet (x = L). Then distributions for the various residence times defined earlier can he found by solving eqn (8) subject to boundary conditions which specify what happens to the particle should it reach the points k = 0 or k = z. Barriers and boundary conditions Three basic types of boundary conditions have been studied in the physics and probability literature in connection with difference eqn (8). Corresponding to these boundary conditions are three types of hypothetical barriers which can be erected at various points along the real line (---M,m) and which can thus interfere with the random walk. With an absorbing barrier at point I, the particle is absorbed if it ever reaches Z; and the random walk is terminated. With a reflecting barrier at point z, a particle reaching point z at step n of the walk is returned to point z - 1 at step n + I. The transmitting barrier is a special case with neither reflection nor absorption so that the random walk is unrestricted. The values of p and q may change at such a barrier, however. Composite barriers which combine the properties of absorption, reflection and transmission can be constructed by assigning probabilities to these three outcomes as can barriers which have different properties when approached from one side than from the other. A formal treatment of dispersion in closed systems would require such composite barriers to model the process with a discrete random walk. However, most such mathematical complexities can largely be avoided by passage to the limit of continuous motion. The mathematical boundary condition which corresponds to an absorbing barrier at point z is Pj(Z*I? + I) =pPj(Z-
1, fl)
P,(z - 1, n + 1) = pP,(z -2, n)
(9)
while reflection at point z gives Pi(Z, n + 1) = pP,(z - 1, n) (10) Solution of difference eqn (8) also requires an initial condition. For the purposes of calculating residence time distributions, it is logical to start the particle at point k = 0 at step n = 0 (or at k = 1 and n = 1 if there is an absorbing barrier at the origin). However, the probability literature typically utilizes an arbitrary starting point, 0 I k 4 z, with the corresponding initial condition
Pj(k, no) = 1 if
j= k
Pj(k, no)=0
j# k.
if
(11)
With this initial condition and with boundary conditions selected for k = 0 and k = z, it is now possible to solve difference eqn (8) in a manner exactly analogous to the solution of partial differential eqn (1) when it is subject to appropriate initial and boundary conditions. Before doing this however, it is useful to consider how the results of the solution can be interpreted in terms of the various residence time distributions defined earlier. All random walks have a clear starting point due to initial condition 11. Absorbing barriers give them a clear stopping point which aids in the mathematical analysis. Consider a random walk which starts at k = 1 and which has absorbing barriers at k = 0 and k = I. Suppose P,(k, n) is the solution to difference eqn (8) subject to the absorbing barriers. Then P,(z,n) is the probability of absorption at z after exactly n steps. It is also seen to be the unnormalized distribution of inlet-to-outlet sojourns. This distribution is directly expressed in terms of an integral number of steps, n; but multiplication by the step duration r gives the distribution in terms of time
t = nr. Normalization of P,(k, n) is necessary since some of the particles will be absorbed at point k = 0 rather than k = z. The normalization factor is J?(z) = i: Pi(Z, n) n-1
(12)
which is the probability that a particle starting at k = j will eventually be absorbed at k = z. The normalized inlet-to-outlet sojourn distribution is thus P,(z, n)/P,(z). We note in passing that not all n are possible; P,(z, n) > 0 only if n 2 z and if n and c have the same parity (both odd or both even). The absorbing boundaries at 0 and z are merely mathematical conveniences for determining the intet-tooutlet sojourn time distribution. The distribution itself depends only on the dispersion mechanism within the region O< k < z; that is, on p and q and not on the boundary conditions at 0 and I. The same inlet-to-outlet distribution would be obtained with reflecting or transmitting barriers, but this distribution would no longer be identical to P, (k, n) obtained by solving eqn (8). Restated, this means that the inlet-to-outlet sojourn time distribution is invariant with respect to boundary conditions but that specific boundary conditions may be useful in mathematically determining this invariant distribution. The combination of a transmitting barrier at k = 0 and an absorbing barrier at k = Z allows determination of the first passage distribution for systems with an open boundary at the inlet. The particle is introduced at k =0 with no = 0, and the walk continues until the particle is eventually absorbed at k =I. The distribution of first passage times is seen to be independent of the boundary conditions at the reactor outlet but does depend on the inlet boundary condition. With a reflecting barrier at k = 0 and an absorbing barrier at k = z, the solution of
Residence time distributions in systems governed by the dispersion equation eqn (8) yields the distribution of first passage times in systems with a closed boundary at the inlet. The normalization probability of eqn (12) is unity in these cases since, with p k 4, all particles will eventually be absorbed at k = z. First passage times can also be determined for systems with an absorbing barrier at the inlet, but these are identical to the inlet-to-outlet sojourn times discussed above. At first glance, it appears that the distribution of last passage times in systems with an open exit boundary can be found by solving eqn (8) with absorption at k = 0 and transmission at k = z. However, particles introduced into such a system will either be absorbed at the origin and hence will not have a last passage or else will undergo a random walk of infinite duration. With p > 4, some particles will walk forever; and it is these particles that we would like to query regarding their last passage times. This is possible mathematically. Oneapproachis to assume a particle is at k = z and is leaving the system for the last time. We wish to learn how long it has been since the particle entered the system at k = 0. Consideration of Fig. 2 will show that this question is identical to asking information on first passage times in a system with an open inlet. Thus, the last passage distribution in systems with an open outlet boundary is equal to the first passage distributions in the same system with an open inlet boundary. Similarly, the distribution of last passage times in a system with a closed boundary at the outlet is identical to the distribution of first passage times in a system with a closed boundary at the inlet. Last passage time distributions are invariant with respect to outlet boundary conditions. We have so far avoided random walks of infinite duration but this will no longer be possible for the consideration of life span distributions. Suppose eqn (8) is solved for a walk beginning at k = 0 and no = 0 with transmitting barriers at 0 and Z. The domain of this walk is the entire real line; but, since p > q, we can expect the sum over all n of P,(z, n) to converge as the most probable location for the particle gradually shifts to the right. The ratio of P&, n) to this convergent sum will satisfy the necessary conditions to be a frequency function, and we thus speculate that this ratio is the differential probability distribution for life spans f(n)
=
_Pokn) _ Pok n) 2 P&n) n-1
PO(Z).
(13)
To see that this is so, consider a particle which is at the point z. There is probability p that the particle will move to z t 1 on the next step; and once there, the probability is Q that the particle will never revisti z. An explicit formula for Q can be found by considering a random walk which starts at point k + 1 and which has an absorbing boundary at z; but for now we need only note that pQPe(z, n) is the probability that a particle at point z will exit the system never to return. The life span frequency function is thus (14)
961
where N is a normalization factor such that f(n) sums to unity. Performing this summation and noting that p. Q and N are all independent of n gives N = PQ C PO@,n)
(I9
from which eqn (13) follows directly. Equation (13) was derived with transmitting barriers at 0 and t and thus gives the life span distribution in a doubly open system. The case with a closed inlet but open outlet can be treated similarly but with Po(z, n) found from eqn (8) subject to a reflecting barrier at k = 0 and transmission at k = z. The life span distribution in systems which are closed at the outlet requires special consideration. With pure reflection at k = z, the random walk would not only be unending but the sum of the Pj(z, n) would diverge. The probabilities P;(z, n) will then approach an equilibrium distribution which, with p > q. is skewed toward the reflective barrier and which is independent of both n and i. This form of random walk has been used to model sedimentation of Brownian particles against the bottom of a vessel[l3], but is not applicable to a dispersive system where there is some probability of leaving the vessel via convection. Instead, we will use a composite barrier which has probability r for absorption and probability 1 - E for reflection where E = p - (l/2) = (l/2) - q represents the net convective flow of material. We shall later see that CJAX
l=2o
(16)
is a good choice for l . With this type of barrier, all walks will eventually terminate via absorption at k = z. Thus Pc,(z, n) directly gives the life span distribution for systems closed at the outlet. Table 1 summarizes the various cases considered so far and also indicates whether literature solutions exist for the discrete and continuous versions of the random walk. Before discussing these solutions, however; we will first treat the relationship between life spans and residence times in dispersive systems. Consider a discrete random walk with a reflecting barrier at k = 0 and a composite barrier of the type described above at k = z. A particle is introduced with i = 0 and no = 0, and the walk continues until the particle is eventually absorbed at I. It may of course be reflected at z many times before absorption. Statistics on the duration of this random walk will approximate the residence time distribution in a doubly closed system. These same statistics will also give the residence time distribution in an open dispersive system since time spent outside (0, z) does not contribute to the residence time. To see this, note that a particle leaving the system at k =0 will eventually return. If time spent in the regions k CO is not included in n, then exactly when the particle returns is unimportant, and it may as well return immediately via refkction. At the exit end of the system, a particle may be leaving for good or it may return. In
962
E. B. NAUMA~ Table 1. Summary of discrete random walks
BARRIER
AT 0
BARRIER
AT Z
TYPE OF RESIDENCE
ABSORBING
ABSORBING
INLET-TO-OUTLET
TRANSMITTING
ABSORBING
FIRST
ABSORBING
TRANSMITTING
LAST PASSAGES
SYSTEM BOUNDARIES
TIME
SOJOURNS
PASSAGES
LITERATURE DISCRETE
SOLUTION ? CONTINUOUS
ANY/ANY
YES
YES
OPEN/ANY
YES
YES
YES'
YES*
NO
NO
ANY/OPEN CLOSED/ANY
REFLECTING
ABSORBING
FIRST
ABSORBING
REFLECTING**
LAST PASSAGES
TRANSMITTING
TRANSMITTING
LIFE
SPANS
REFLECTING
TRANSMITTING
LIFE SPANS
TRANSMITTING
REFLECTING**
LIFE SPANS
OPEN/CLOSED
REFLECTING
REFLECTING**
LIFE SPANS
CLOSED/CLOSED
REFLECTING
REFLECTING
RESIDENCE
l
THIS SOLUTION
*+ TSESE BARRIERS VIA CONVECTION *** LaPLACE
IS READILY
OBTAINED
ARE PRIMARILY
TRANSFORM
PASSAGES
to discrete
NO*
OPEN/OPEN
YES
YES
CLOSED/OPEN
YES
NO
EITHER CLOSED
TIMES
FROM THE ONE IMMEDIATELY
REFLECTING
BUT MUST
INCLUDE
OPEN
OR
YES*
NO*
NO
YES***
NO*
YES***
PROCEEDING
A PROBABILITY
B OF ABSORPTlON
ONLY
either case, the clock stops when the. particle crosses k = t; and if the particle returns, it may as well do so immediately via reflection. We have thus determined that the residence time distribution in a dispersive system is independent of whether the system boundaries are open or closed. This conclusion, which seems obvious in retrospect, was masked for many years by the fact that open and closed systems behave quite differently when subjected to tracer experiments. When the residence time distribution is desired, the proper mathematical manipulation is that corresponding to closed boundaries even though the system being modelled may be open. Solutions
NO*
ANY/CLOSED
random
walks
In this section some solutions to difference eqn (8) are presented for boundary conditions of the type shown in Table 1. Several of these solutions can be derived from one basic solution, due to Feller [14] among others, for a random walk between two absorbing barriers. The barriers are located at points 0 and z, and the particle is introduced at point j with initial step n,. Then the probability of absorption at the origin after exactly n steps is
(17) where m = II -no. The probability of eventual absorption at the origin is
The probability of adsorption at .z after exactly n steps may be found from eqn (17) by setting j = z - 1 and interchanging p and q. If we also normalize, the result is
fitrn)
=
pi(Zt ml _ II- (4/P)‘l(4Pq)“/2(q/p)“~*2’
pi(z) z-1
2 (- I)‘+’ cos
m_-l
xi
f
rri
. lrij
sm 1 stn 7
(19)
With j = 1 and no = 1, eqn (19) gives the inlet-to-outlet sojourn time distribution for a discrete dispersive system with any of the boundary conditions: open, closed or absorbing. The discrete distribution of first passage times in a system with an open inlet can also be found from eqn (17) without need to resolve eqn (8) subject to a transmitting barrier at k = 0. The approach is to take the limit of eqn (17) as z becomes large so that the random walk covers the infinite interval (0, m) with absorption at the origin. We then interchange p and q and suppose that the particles enter at point j = z with n, = 0 where z now has its previous, finite value. The result is equivalent to a random walk which starts at the origin with a transmitting barrier at the origin and with an absorbing barrier at point z. One form (14) for the resulting distribution is p*(z,
I
n)
=
yp(n-z1/2q~n+lm
1
0
while the probability of eventual absorption at z is P,(Z)= t - P,(O).
Al -(qILJy’l
COY’ mu sin PU sin ?rza de.
This is equivalent to the computationally
(20)
simpler form
Residence time PO(z,n) =
distributions in systems governed by the dispersion equation
zrl!
963
L=zAx
.(~)!(~)!P(“+=~*q(“-=)‘*(21)
t = nr where we now explicitly see that n P z and that z and n must be either both odd or both even for the particle to exit after exactly n steps. Otherwise, P&, n) = 0. With p > 4, all particles will eventually be’absorbed at z so that eqns (20) and (21) are already normalized. Thus these eqpations give the discrete distribution of first passage times in systems which are open at the inlet. By the arguments presented in the previous section, they also provide the distribution of last passages in systems which are open at the outlet. Few literature solutions have been found for any of the discrete cases with reflection. A series solution for reflection at the origin with absorption at z can be expressed in terms of the P1(O, n) and &(z, n) obtained for the doubly absorbing case, but this solution has not been summed in a convenient form. Kac[l3] has treated the case with reflection and transmission. His solution is applicable to life span distributions in systems with one open and one closed boundary, but the form of his discrete solution is complex and will not be repeated here. Feller[l4] has solved difference eqn (8) for two purely reflecting barriers; but the sum in eqn (12) obviously diverges with pure reflection and the system assumes an equilibrium spatial distribution. The final discrete random walk to be considered is that corresponding to open boundaries at both ends of the reactor. This gives an unending walk over the entire real line ( -a, a), and the solution to eqn (8) is a form of the binomial distribution
where we have set no = 0. In this expression not all n are possible, and Pi(z, n) = 0 unless n and z have the same parity and unless R 3 z-j. Normalization of this result wiil be considered after passage to the limit. CONTlNUQUSRANDOMWALKS
Passage to the limit This section considers the limiting behaviour of discrete random walks as z becomes large but Ax and the duration of a step, 7, become small. The probabilities p and q will approach (l/2) in this limit but in such a way that allows for a net convective velocity, U, to be imposed on the system. Specifically, let
to = nor
(23)
and suppose that both Ax and 7 approach zero in such a manner that the ratio Ax’ -_=D 27
(24)
remains constant. Then, the discrete transition probabilities Pi(k, n) and difference eqn (8) can be written as C,(x, t + T) = pC,(x -AX, t) t qC,(x t AI, t)
(25)
where C(x, t) is the probability that a particle will be at position x at time t. Expanding this equation in a Taylor series using a first order approximation for the time dependence and a second order approximation for the spatial dependence gives
substituting for q, p and AX% gives
+Jg+D~
(27)
which is identical to eqn (1) with r(c) = 0. Indeed, had a reaction probability been added to difference eqn (8), an r(c) term would have appeared in eqn (27). Thus, in the particular limit indicated by eqns (23) and (24). the one dimensional random walk becomes a one dimensional dispersion process. A key requirement that this limit be reasonable from a physical viewpoint is that the instantaneous particle velocity, AX/T, must be large compared to the convective velocity, CJ.In the mathematical limit, AXITbecomes infinite while U and Ax’lr are held constant. The various solutions to difference eqn (8) can also be subjected to the limiting process. This, of course, is an alternative to solving partial differential eqn (27). The appendix gives asymptotic approximations which are useful in making this passage to the limit. Applying these to eqn (19) gives sinh uL 4 e-uqt-to)‘4D $, (- l)i+‘i
j,(t) = (y)
sin ~
,-~*i2D(~-r~lLz
(8)
L
Setting y = to = 0 gives the inlet-to-outlet sojourn time distribution for a system governed by the dispersion equation: x=kAx y=jAx
4T2D2 UL f(t) = m sinh 20 e-
UW4D
,$,
(_
l)‘+‘i*
e--r2i2DliL2,
(29)
964
E. B. NMJMAN
To obtain the distribution of first passages, eqns (20) or (21) could be taken to the limit directly. However, a physically more transparent approach is to apply the limiting process to eqn (17) giving
where f,(r) dt is the probability that a particle introduced at point y will be absorbed at the origin before reaching point z and that this absorption will occur within the time interval t to t +dr. To find first passages from this, we set U = - U, take the limit as L -+ m and then set y equal to the original, finite L. Feller [14] gives this result as (31) where f(t) d T is now interpreted as the probability that a particle introduced at the origin will first reach point x = L during the time interval between t and r + df. Note that the old origin of eqn (30) has become an absorbing barrier at point L while the other absorbing barrier has been shifted to --co. None of the particles will reach this barrier so that eqn (31) is automatically normalized. It is clear from the original derivation of eqns (20) and (21) or from the above development of eqn (31) that a particle introduced at x =0 is free to move to locations with x
(32)
where C(x, r) dx is the probability that the particle will be located between x and x tdx at time r. The units on C(x, t) can be interpreted as particles per unit length and are thus seen to be appropriate concentration units for a one dimensional process. To obtain the life span distribution, eqn (32) is normalized according to the continuous analog of eqn (13) f(t)
=
_
c(x7 I) C(x, 1) d f
This result is the same as that obtained by solving partial differential eqn (27) subject to an impulse function at the inlet[3]. Thus the life span distribution in a doubly open system is identical to the impulse response function. Direct solution to the continuous case Conventional solutions to partial differential eqn (27) give results identical to those obtained at a particle level of scrutiny provided that appropriate boundary and initial conditions are used. The continuum counterpart of an absorbing barrier at L is the absorbing boundary conditions, C(L, r) =O. The reflecting barrier at L is equivalent to the closed boundary, C(L, I) = 0 while a transmitting barrier corresponds to the open boundary. With these correspondences established, all the forms of residence time distribution listed in Table 1 can be found using continuum mathematics although literature solutions have not been located for each case. The open and closed boundaries are familiar to the chemical engineer; but the use of absorbing boundaries is rare, at least in the context of residence time distributions. Their use will be illustrated by a direct solution to the inlet-to-outlet sojourn time distribution for a continuous system governed by eqn (27). The discrete solution to this problem used absorbing barriers at points 0 and z, and individual particles were introduced at intermediate point i. The corresponding boundary conditions’ for the continuous case are C(0, t) = C(L, r) = 0 while particles are introduced using the initial condition C(x, 0) =6(x-y). A solution to eqn (27) that satisfies these conditions is
(34) The sojourn time distribution is found by normalizing the flux at the outlet, the procedure recommended by Petho[23] being correct in this case. The normalization factor is
(35) The series in eqn (35) can be summed using tabulated results [24], and the normalized flux becomes
-DE
fm = -_ N I
r_2rDsinhg
_
L -ae
UZ1/4D
20
which is identical to eqn (28) with to = 0. Taking the limit as y +O gives eqn (29) as the frequency function for inlet-to-outlet sojourns. As mentioned above, not all the cases listed in Table I have reported solutions. What must be considered the most important case without an explicit solution is the
Residence time distributions in systems governed by the dispersion equation
distribution of life spans in a doubly closed system since this is also the long sought residence time distribution in a doubly open system. This case has, however, been analyzed using the Laplace transform technique. The transform of the residence time frequency function can be obtained from van der Laan’s[25] general result by setting D, = D, = 0 to give qJ
ewL12DXI-m
(37) f(S)=(1+8)‘_(1-B)‘e’-UUD’B where fi = v/(1 +4sa UL) and where s is the Laplace transform parameter. Equation (37) can also be derived from the steady state version of eqn (1) for a first order reaction with r(C)= SC using the Danckwerts’ boundary conditions; eqns (3) and (4) or by using Bischoff’s (1) boundary conditions for an open system. AU three derivations give the same result, eqn (37), and the dilemma raised by Gibilaro has vanished. Eqn (7) can indeed be used to obtain f(t) from reaction data; and eqn (5). while it gives an interesting function with mean r= Ll U, does not give the residence time distribution. Equation (37) is the Laplace transform of the residence time frequency function for a doubly open system, a doubly closed system or for either combination of half open and half closed. It does not happen to be the residence time distribution for a system with absorbing boundaries. Such systems can show unusual behavior with regard to means residence times, giving an apparent contradiction to Buffham’s 1261 generalization that the mean residence time in both flow and non-flow systems is equal to the steady state holdup divided by the transmission rate through the system. This topic has been discussed separately [27].
965
future results since the mathematical techniques are simpler and more widely known. However, all the discrete solutions shown as missing in Table 1 are also missing in the continuous case. Hopefully, some of these will appear in the near future, particularly an explicit form for the doubly closed case. The historic difficulty in deriving residence time distributions from the axial dispersion model was due to the backwards particle flow which is an inherent part of that model. If particles did not flow upstream against the convective velocity, all differences between open and closed systems would vanish. As recently summarized by Sundaresan, Amundson and Aris [28], backwards flow of particles can be expected in systems where mixing is driven by concentration gradienEs with a diffusion-like mechanism. Molecular, Brownian, and turbulent diffusion all fall into this category as does Taylor dispersion. If these phenomena are the major cause of mixing, dispersion models can be expected to fit the real system even with respect to fine structure. Most real systems would require a two or three dimensional model for precise results, but the basic dzerences between open and closed systems would still prevail. If, however, reverse flow does not occur, then dispersion models cannot be expected to fit the fine structure of real systems; and the residence time distributions predicted in this paper could not be expected to hold in detail. Indeed, Sundaresan et a1.[28] have shown that no existing model accurately predicts all significant phenomena in a packed bed at high Reynolds number. Even for packed beds, however, the axial dispersion model is adequate for many purposes; and if a prediction of the residence time distribution is needed, then the one corresponding to closed boundary conditions is clearly the best to choose. These boundary conditions at least prohibit reverse flow at the inlet and outlet.
CONCLUSIONS NOTATION
It is
apparent from the foregoing results that dispersive systems can be scrutinized on the scale of individual particles without intrinsic difficulties. Sojourn time and residence time distributions are equally applicable to single particles as they are to large numbers of particles. Although the dispersion equation is conventionally interpreted in terms of net fluxes, it can also be interpreted in terms of probability distributions for individual particles. Indeed, the individual particle viewpoint is necessary to obtain clear and unambiguous definitions of residence time. It is now apparent that both open and closed systems have the same residence time distribution when time spent outside the system boundaries is excluded from the total. This residence time distribution agrees with the one deduced from reaction yields. It disagrees with the result of Gibilaro[4], and his result is now seen to be the distribution of first passages. Although the major results in this paper were obtained with discrete random waIks and with passage to the limit of continuous motion, the conventional approach using the dispersion equation is equally valid when the appropriate boundary and initial conditions are used. This continuum approach will probably be most useful for
u
C Co D f F
i j k L m n n0 N p P,(k, n) PI(k)
q Q r f(c) s
dummy variable of integration concentration of particles inlet concentration dispersion coefficient residence time frequency function residence time distribution function index of summation discrete initial position discrete spatial variable length of reactor discrete step count, n-no number of steps in random walk initial step number normalization factor probability of moving in positive direction discrete transition probability discrete normalization factor probability of moving in negative direction probability of not revisiting point z reaction probability reaction rate expression reaction rate constant and Laplace transform parameter
E. B. NAUMAN
966
t continuous time variable
to initial value of t u drift velocity of particles x I
continuous spatial variable discrete reactor length
Greek symbols 0 d/(1 + SLYLx) y continuous initial position
1261 But&m B. A., Nature1978 274 879. (271 Nauman E. B., Chem. Engng Commrcn. 1981 8 53. 1281 Sundaresan S., Amundson N. R. and Aris R., A.1.Ch.E. J. 1980 26 529. APPENDIX Limiting forms In the following results, the indicated limit refers to that process governed by eqns (23) and (24).
E incremental probability 0 flux of particles 7 duration of a step REFERENCES I
[I]
Bischoff K. B., Chem. Engng Sci. 1960 12 69. [2] Danckwerts, P. V., Chem. Engng Sci 1953 2 1. [31 Levenspiel 0. and Smith W. K., Chem Engng 5%. 1957 6 221. [4] Gibilaro L. G., Chem. Engng Sci. 1978 33 487. [5] Kreft A. and Zuber A., Chem. Engng Sci. 1978 33 1471. [61 Buffham B. A., Chem. Engng Sci. 1969 24 1386. [7] Danckwerts P. V., Chem. Engng Sci. 1958 8 93. [8] Einstein A., Inuestigotions on the Theory of fhe Brownian Mouemenf, (Edited by Fuerth R.). Dover, New York 1956. [9] Smoluchowski, M., Phys. Zeif. 1916 17 557. IlO] Uhlenbeck G. E., and Omstein, L. S., Phys. Rev. 1930 36 823. 1111 Chandrasekar S., Rev. Mod. Phys. 1943 15 1. I121 Wang M. C., and Uhlenbeck G. E., Rev. Mod. Phys. 1945 17 323. 1131 Kac, M., Am. Math. Monthly 1947 54 369. [141 Feller W., An Introduction to Probability Theory and its Applications, (3rd Edn), Vol. I. Wiley, New York 1968. [IS] King G. W., Ind. Engng Chem. 1951 43 2475. 116) Aris R., and Amundson N. R., A.LCh.E. 1. 1957 3 280. [17] Shinnar R., and Naor D., Chem. Engng Sci. 1967 22 1369. [18] Shinnar R., Naor P. and Katz S., Chem. Engng Sci 1972 27 1627. [19] Shinnar R., Glasser D., and Katz S., Chem. Engng Sci. 1973 28 617. [20] Brenner H. and Gajdos L. J., /. Colloid Interface Sci. 1977 58 312. 1211 Gajdos L. J. and Brenner H., Sep. Sci. Technol. 1978 13 215. [22] Brenner H., Int. J. Phys. Chem. Hydra. 1980 191. [23] Petho A., Chem. Engng Sci. I%8 23 807. 1241 Jolley L. B. W., Summation of Series, p. 104. Dover, New York 1%1. [25] van der Laan E. T., Chem. Engng Sci 1957 7 187.
lim
sjn
Y = sin !T, L
2
For walks of finite duration ending via absorption at I, lim y
= lim e
= f(t). I
For unending random walks, the particle concentration is found by a limiting process on the spatial dependence of Pj(, n) ,im
[email protected]) = C(*, I)
2Pi(z)AX
where the factor of two arises from the fact that Pi(z, n) = 0 for every other value of .z The asymptotic approximations used in the limiting process do not show these zero values and thus must be divided by two to obtain the correct spatial average for C(x, t).
RESIDENCE TIME DISTRIBUTIONS GOVERNED BY THE DISPERSION
IN SYSTEMS EQUATION
E. B. NAUMAN Rensselaer Polytechnic Institute, Troy, NY 12181, U.S.A.
(Received 13 March 1980: accepted 10 December 1980) Abstrr~et-The theories of discrete and continuous random walks have been applied to systems governed by the dispersion equation. It is shown that residence time distributions can he defined and determined at a particle level of scrutiny and that the resulting distributions are identical to those obtained using continuum methods. Now, however, the appropriate boundary and initial conditions are apparent; and the resulting solutionscan be given a clear physical interpretation. The long-sought residence time distribution for dispersion in an open system is shown to be the same as that in a closed system. This result is consistent with Danckwerts’ formulation for the yield of first order reactions but invalidates recent results which were based on a transfer function approach to residence time distributions. INTRODUCTION
A
CI( - m) = Co and that C,( +m) is bounded, the steady state version of eqn (1) can be solved piecewise for each of the three sections shown in Fig. 1, and this
that
key problem
in chemical reaction engineering is simultaneous convection, diffusion and reaction in tubular vessels. The one dimensional dispersion model for this process gives rise to a partial differential equation
composite analysis
solution provided
completes
the
steady
state reactor
D and U are known.
the parameters
when one or more of the D vanish. If of concentration is lost at the origin; and we obtain the famous Danckwerts[2] boundary conditions for a system closed at the inlet Special cases arise
D, = 0, continuity
(1) Solutions to this equation for a variety of initial and boundary conditions have occupied much of the literature for the past twenty-five years. Figure 1 illustrates some general features of the type of system which has been studied. Equation (1) corresponds to reaction in the unsteady state, and this situation has rarely been approached directly. Instead, interest has focused on solving the steady state equation with reaction or the time dependent equation without reaction. Much of the work on steady state reactions has been devoted to finding physically appropriate boundary conditions for the system illustrated in Fig. 1 where reactant enters at x = --o) but reacts only over the section 0 < x < L. It is now generally agreed that the correct boundary conditions are those that show the concentration, C, and the particle flux lp = UC-D&
U,Co = UC(O+)-
In most of what follows, by open boundary we will mean the situation where D, = D > 0 or D, = D > 0 although many of the concepts can also be applied to open boundaries with D# D, > 0 or Df D2>0. Also, we explicitly treat only the case U, = U = Uz. The idea of using tracer experiments to estimate U and D dates to Levenspiel and Smith[3]), and from their work springs much of the interest in solving eqn (1) in the unsteady state. The conceptual experiment is to inject a tracer at point x =O, to measure the concentration at point x = L, and from this to deduce U, D and the residence time distribution for the test section, O-C
CES Vol. 36.No.6-A
Outlet sectm
Test or
reaction sectlon
Fig.
(3)
dC 22 I r=O.
dC
Inlet section
Of
If Dx = 0, the system is closed at the outlet and the exit boundary condition becomes
to be continuous at x = 0 and x = L[l]. When these boundary conditions are augmented with the knowledge
-
Dg
Inlet
Outlet
rnjectlon
sample
probe
probe
1. Tubular reactor with provision for tracer experiments. 957
-
E. B. NAUMAN
958
x < L. This rather simply stated problem has proved to be quite difficult for the general case where D,, D, and 4 are all positive. Many papers have discussed this problem, the most recent being those by Gibilaro[4] and by Kreft and Zuber [5]. One purpose of the present paper is to again examine those solutions to the dispersion equation which are reputed to represent the residence time distribution in an open system. At first glance, the distribution of residence times for material leaving the reactor seems an intuitively obvious concept for discrete entities such as molecules or Brownian particles. However, eqn (1) is usually interpreted to govern only the net flow of material and not the dynamics of individual particles which, subject to dispersion, can cross and recross a boundary many times. This interpretation has led Buffham[6] and Gibilaro[4] to despair of ever finding the residence time distribution at a molecular level of scrutiny. Instead, Gibilaro has redefined the residence time distribution from a continuum viewpoint as a transfer function which relates the output flux of material to the input flux via convolution:
where f(r) is the differential or residence time frequency function with first moment:
That eqn (5) defines a unique function f(t) for all input disturbances $(t,O) follows from the linearity of Equation 1. The appropriateness of basing eqn (5) on the flux 4 rather than the concentration C has been discussed by Gibilaro and by Kreft and Zuber, and the validity of eqn (6) has also been shown by them. Equation (5) shows how a transfer function, alleged to be the residence time frequency function, can be determined from time dependent solutions to the dispersion equation without reaction. An alternative method for determining f(t) is to use steady state solutions to eqn (1) with reaction. Danckwerts[2] showed that the yield of a first order reaction is related to the residence time distribution via Laplace transformation. C(+ m) = C, 0me-“f(l) dt I
(7)
where s is the reaction rate constant. Equations (5) and (7) provide two different definitions of f(t). It was long assumed that these definitions were consistent and that both conform to the intuitive notion of time spent in the system. Gibilaro[4] however has recently shown that the dispersion model gives different results depending on which defining equation is used. The present paper treats this discrepancy and other inconsistencies in the current literature. The paper has the following specific goals: (1) To demonstrate that residence time distributions can be defined and determined at a particle level of
scrutiny, be these particles molecules, Brownian particles or fluid points in the sense of Danckwerts (7). (2) To show that the dispersion equation can be interpreted in terms of probability distributions governing the location of individual particles as well as the more conventional interpretation in terms of net fluxes. (3) To show what boundary conditions are suitable for deducing residence time distributions for systems governed by eqn (1). (4) To present new results for various types of systems governed by eqn (1) and to resolve the current literature dilemma regarding residence time distributions in systems with reaction. Some of these concepts are well established in the physics and mathematics literature. Others are new. The general approach will be to examine the motion of individual particles which are subject to dispersion. This examination will begin with the theory of discrete random walks; but, by passage to a limit of many small steps, we will show that eqn (1) governs the location of individual particles in a probability sense. The boundary conditions appropriate to the study of discrete random walks will then be translated to those appropriate from a continuum viewpoint. The final results will be a method by which true residence time distributions can be deduced from eqn (1) using continuum mathematics. DEFINITION OF RESIDENCE TIME
When applied to distinct entities which can be tracked over time and space, the idea of residence in a system seems intuitive and uncomplicated. However, care must be taken when there are several locations where particles can enter or leave the system and when an individual particle can make multiple entrances and exists. The complete definition of a “residence time” should include specification of the inlet, the outlet and of what actions stop and start the clock. The following definitions, although by no means inclusive, are useful for analyzing systems governed by eqn (1). For ease in visualizing these definitions, reference is made to Fig. 2 which shows representative particle trajectories in doubly open (D, >O, D3 > 0) and doubly closed (0, = D3 = 0) systems. The inlet-to-outlet sojourn time represents the simplest trajectory between the points x = 0 and x = L. Time starts when the particle leaves x =0 in the positive direction and ends when the particle first reaches x = L without having revisited x = 0. Inlet-to-outlet sojourn times are represented by trajectories M and 7-g in Fig. 2. Note that path 5-6, the reverse transition, is not included in the definition. A first passage time is represented by the trajectory 14 in Fig. 2. Timing starts when the particle first enters the system and continues until it first reaches x = L. Note that this definition may include some time spent outside the system with Y CO if the inlet boundary is open. A lust passage time is represented by the trajectory 7-10. The clock starts when the particle last leaves the point x = 0 and continues until the particle exits at x = L, never to return. This definition may include time spent
959
Residence time distributions in systems governed by the dispersion equation
particle which enters the system at .x = 0 will eventually exit at x = L never to return. Often, it also implies a finite duration for the discrete random walk which approximates the continuous process.
I
I
DISCRETE
(a
) Doubly
I
I
x=0
x=L
open
system
I
(b I
I
Doubly closedsystem
Fig. 2. Representative particle trajectories in dispersive systems.
outside the system with x > L if the outlet boundary is open. The life span of a particle is the elapsed time between when it first enters at x = 0 until it finally leaves at x = L, never to return. The corresponding path is the entire trajectory l-10, and will generally include some time spent outside the system if there is an open boundary. The residence time of a particle is the same as the life span except that time spent outside the system is excluded. Thus for the doubly open system, time spent on the trajectories 2-3, 4-5, 6-7, and 8-9 is all excluded. In what follows, the symbol t will be used to denote any of the above times; and a reference to the adjacent text will be necessary to distinguish between them. All of times have frequency functions, f(t), and cumulative distribution functions, F(r), which reflect random variations between particles following similar trajectories. Thus f(t) df = Fraction of particles (which follow a specified trajectory with specified timing rules) which experienced a time between t and f +dr. Each time will also have a mean and higher moments which are defined in the usual manner for continuous probability distributions. We consider only those situations where the mean, < is finite. This implies that a
lt.4NDOM
WALKS
A microscopic model for the residence time distributions in a dispersive system can be developed using the probability theories of Brownian motion and diffusion. Einstein [8] interpreted Brownian motion in terms of probability distributions, and Smoluchowski[9] suggested that these probability distributions could be approximated using a discrete random walk. This basic approach has been extensively treated in the physics and mathematical literature [IO-131 and is closely related to the well known problem of a gambler’s ruin[l4]. King[lS] suggested single particle simulations as a modeling technique for diffusion, and Aris and Amundson[M] suggested that random walk concepts could have been used to obtain some of the results they derive by classical methods. Shinnar and coworkers [17191 used random walks and probability theory to treat reactions and residence times, primarily is stirred tank reactors, while Brenner and Gajdos [20-223 used the single particle concept in their study of generalized Taylor dispersion phenomena. The present paper applies these ideas to what is now a classical problem in residence time theory. Inherent in the concept of residence times is a notion that the fluid contains individual elements which can be tracked and timed as they flow through the system. Considering these entities to be molecules is the logical choice and is indeed appropriate when D in eqn (1) is the molecular diffusivity. Brownian particles can also be treated given a different interpretation of D, and fluid points in the sense of Danckwerts[7] are appropriate for dispersion caused by turbulence. We shall use the term particle to denote any of the above entities and will approximate one dimensional diffusion, Brownian motion or dispersion by discrete random walks of such particles on the real line ( - m, m). A discrete random walk is a stepwise process; and in the model to be used here, each step moves the particle exactly one unit length, Ax. There is probability p that the particle will move Ax to the right and probability q = 1 -p that the particle will move Ax to the left. When p = q = f, the walk is symmetric and the corresponding physical process is pure diffusion in a stationary medium. Asymmetric walks with p > q will be used to model diffusion in a medium having convective velocity U, and it is possible to introduce a third probability r to reflect termination of the walk by reaction. This random walk model may not appear particularly realistic since real particles will obviously have a distribution of step lengths and of particle velocities. Chandrasekhar[ll] describes more complicated models which reflect this; but, in the limit of a large number of smail steps, the final results are insensitive to the exact form of the random walk. The random walk begins with the introduction of a single particle at some position jAX on the real line.
E. B. NAUMAN
960
Define Pi(k. n) as the probability that this particle will occupy position kAX after exactly n steps. Then the difference equation Pj(k,n+l)=pPi(k-l,n)+qP,(k+l,n)
(8)
holds for all k in an unrestricted random walk. However, we are primarily interested in random walks of finite duration which occur over the interval (0.L). In the discrete model, let k = 0 correspond to the reactor inlet (x =0) and let k = z correspond to the outlet (x = L). Then distributions for the various residence times defined earlier can he found by solving eqn (8) subject to boundary conditions which specify what happens to the particle should it reach the points k = 0 or k = z. Barriers and boundary conditions Three basic types of boundary conditions have been studied in the physics and probability literature in connection with difference eqn (8). Corresponding to these boundary conditions are three types of hypothetical barriers which can be erected at various points along the real line (---M,m) and which can thus interfere with the random walk. With an absorbing barrier at point I, the particle is absorbed if it ever reaches Z; and the random walk is terminated. With a reflecting barrier at point z, a particle reaching point z at step n of the walk is returned to point z - 1 at step n + I. The transmitting barrier is a special case with neither reflection nor absorption so that the random walk is unrestricted. The values of p and q may change at such a barrier, however. Composite barriers which combine the properties of absorption, reflection and transmission can be constructed by assigning probabilities to these three outcomes as can barriers which have different properties when approached from one side than from the other. A formal treatment of dispersion in closed systems would require such composite barriers to model the process with a discrete random walk. However, most such mathematical complexities can largely be avoided by passage to the limit of continuous motion. The mathematical boundary condition which corresponds to an absorbing barrier at point z is Pj(Z*I? + I) =pPj(Z-
1, fl)
P,(z - 1, n + 1) = pP,(z -2, n)
(9)
while reflection at point z gives Pi(Z, n + 1) = pP,(z - 1, n) (10) Solution of difference eqn (8) also requires an initial condition. For the purposes of calculating residence time distributions, it is logical to start the particle at point k = 0 at step n = 0 (or at k = 1 and n = 1 if there is an absorbing barrier at the origin). However, the probability literature typically utilizes an arbitrary starting point, 0 I k 4 z, with the corresponding initial condition
Pj(k, no) = 1 if
j= k
Pj(k, no)=0
j# k.
if
(11)
With this initial condition and with boundary conditions selected for k = 0 and k = z, it is now possible to solve difference eqn (8) in a manner exactly analogous to the solution of partial differential eqn (1) when it is subject to appropriate initial and boundary conditions. Before doing this however, it is useful to consider how the results of the solution can be interpreted in terms of the various residence time distributions defined earlier. All random walks have a clear starting point due to initial condition 11. Absorbing barriers give them a clear stopping point which aids in the mathematical analysis. Consider a random walk which starts at k = 1 and which has absorbing barriers at k = 0 and k = I. Suppose P,(k, n) is the solution to difference eqn (8) subject to the absorbing barriers. Then P,(z,n) is the probability of absorption at z after exactly n steps. It is also seen to be the unnormalized distribution of inlet-to-outlet sojourns. This distribution is directly expressed in terms of an integral number of steps, n; but multiplication by the step duration r gives the distribution in terms of time
t = nr. Normalization of P,(k, n) is necessary since some of the particles will be absorbed at point k = 0 rather than k = z. The normalization factor is J?(z) = i: Pi(Z, n) n-1
(12)
which is the probability that a particle starting at k = j will eventually be absorbed at k = z. The normalized inlet-to-outlet sojourn distribution is thus P,(z, n)/P,(z). We note in passing that not all n are possible; P,(z, n) > 0 only if n 2 z and if n and c have the same parity (both odd or both even). The absorbing boundaries at 0 and z are merely mathematical conveniences for determining the intet-tooutlet sojourn time distribution. The distribution itself depends only on the dispersion mechanism within the region O< k < z; that is, on p and q and not on the boundary conditions at 0 and I. The same inlet-to-outlet distribution would be obtained with reflecting or transmitting barriers, but this distribution would no longer be identical to P, (k, n) obtained by solving eqn (8). Restated, this means that the inlet-to-outlet sojourn time distribution is invariant with respect to boundary conditions but that specific boundary conditions may be useful in mathematically determining this invariant distribution. The combination of a transmitting barrier at k = 0 and an absorbing barrier at k = Z allows determination of the first passage distribution for systems with an open boundary at the inlet. The particle is introduced at k =0 with no = 0, and the walk continues until the particle is eventually absorbed at k =I. The distribution of first passage times is seen to be independent of the boundary conditions at the reactor outlet but does depend on the inlet boundary condition. With a reflecting barrier at k = 0 and an absorbing barrier at k = z, the solution of
Residence time distributions in systems governed by the dispersion equation eqn (8) yields the distribution of first passage times in systems with a closed boundary at the inlet. The normalization probability of eqn (12) is unity in these cases since, with p k 4, all particles will eventually be absorbed at k = z. First passage times can also be determined for systems with an absorbing barrier at the inlet, but these are identical to the inlet-to-outlet sojourn times discussed above. At first glance, it appears that the distribution of last passage times in systems with an open exit boundary can be found by solving eqn (8) with absorption at k = 0 and transmission at k = z. However, particles introduced into such a system will either be absorbed at the origin and hence will not have a last passage or else will undergo a random walk of infinite duration. With p > 4, some particles will walk forever; and it is these particles that we would like to query regarding their last passage times. This is possible mathematically. Oneapproachis to assume a particle is at k = z and is leaving the system for the last time. We wish to learn how long it has been since the particle entered the system at k = 0. Consideration of Fig. 2 will show that this question is identical to asking information on first passage times in a system with an open inlet. Thus, the last passage distribution in systems with an open outlet boundary is equal to the first passage distributions in the same system with an open inlet boundary. Similarly, the distribution of last passage times in a system with a closed boundary at the outlet is identical to the distribution of first passage times in a system with a closed boundary at the inlet. Last passage time distributions are invariant with respect to outlet boundary conditions. We have so far avoided random walks of infinite duration but this will no longer be possible for the consideration of life span distributions. Suppose eqn (8) is solved for a walk beginning at k = 0 and no = 0 with transmitting barriers at 0 and Z. The domain of this walk is the entire real line; but, since p > q, we can expect the sum over all n of P,(z, n) to converge as the most probable location for the particle gradually shifts to the right. The ratio of P&, n) to this convergent sum will satisfy the necessary conditions to be a frequency function, and we thus speculate that this ratio is the differential probability distribution for life spans f(n)
=
_Pokn) _ Pok n) 2 P&n) n-1
PO(Z).
(13)
To see that this is so, consider a particle which is at the point z. There is probability p that the particle will move to z t 1 on the next step; and once there, the probability is Q that the particle will never revisti z. An explicit formula for Q can be found by considering a random walk which starts at point k + 1 and which has an absorbing boundary at z; but for now we need only note that pQPe(z, n) is the probability that a particle at point z will exit the system never to return. The life span frequency function is thus (14)
961
where N is a normalization factor such that f(n) sums to unity. Performing this summation and noting that p. Q and N are all independent of n gives N = PQ C PO@,n)
(I9
from which eqn (13) follows directly. Equation (13) was derived with transmitting barriers at 0 and t and thus gives the life span distribution in a doubly open system. The case with a closed inlet but open outlet can be treated similarly but with Po(z, n) found from eqn (8) subject to a reflecting barrier at k = 0 and transmission at k = z. The life span distribution in systems which are closed at the outlet requires special consideration. With pure reflection at k = z, the random walk would not only be unending but the sum of the Pj(z, n) would diverge. The probabilities P;(z, n) will then approach an equilibrium distribution which, with p > q. is skewed toward the reflective barrier and which is independent of both n and i. This form of random walk has been used to model sedimentation of Brownian particles against the bottom of a vessel[l3], but is not applicable to a dispersive system where there is some probability of leaving the vessel via convection. Instead, we will use a composite barrier which has probability r for absorption and probability 1 - E for reflection where E = p - (l/2) = (l/2) - q represents the net convective flow of material. We shall later see that CJAX
l=2o
(16)
is a good choice for l . With this type of barrier, all walks will eventually terminate via absorption at k = z. Thus Pc,(z, n) directly gives the life span distribution for systems closed at the outlet. Table 1 summarizes the various cases considered so far and also indicates whether literature solutions exist for the discrete and continuous versions of the random walk. Before discussing these solutions, however; we will first treat the relationship between life spans and residence times in dispersive systems. Consider a discrete random walk with a reflecting barrier at k = 0 and a composite barrier of the type described above at k = z. A particle is introduced with i = 0 and no = 0, and the walk continues until the particle is eventually absorbed at I. It may of course be reflected at z many times before absorption. Statistics on the duration of this random walk will approximate the residence time distribution in a doubly closed system. These same statistics will also give the residence time distribution in an open dispersive system since time spent outside (0, z) does not contribute to the residence time. To see this, note that a particle leaving the system at k =0 will eventually return. If time spent in the regions k CO is not included in n, then exactly when the particle returns is unimportant, and it may as well return immediately via refkction. At the exit end of the system, a particle may be leaving for good or it may return. In
962
E. B. NAUMA~ Table 1. Summary of discrete random walks
BARRIER
AT 0
BARRIER
AT Z
TYPE OF RESIDENCE
ABSORBING
ABSORBING
INLET-TO-OUTLET
TRANSMITTING
ABSORBING
FIRST
ABSORBING
TRANSMITTING
LAST PASSAGES
SYSTEM BOUNDARIES
TIME
SOJOURNS
PASSAGES
LITERATURE DISCRETE
SOLUTION ? CONTINUOUS
ANY/ANY
YES
YES
OPEN/ANY
YES
YES
YES'
YES*
NO
NO
ANY/OPEN CLOSED/ANY
REFLECTING
ABSORBING
FIRST
ABSORBING
REFLECTING**
LAST PASSAGES
TRANSMITTING
TRANSMITTING
LIFE
SPANS
REFLECTING
TRANSMITTING
LIFE SPANS
TRANSMITTING
REFLECTING**
LIFE SPANS
OPEN/CLOSED
REFLECTING
REFLECTING**
LIFE SPANS
CLOSED/CLOSED
REFLECTING
REFLECTING
RESIDENCE
l
THIS SOLUTION
*+ TSESE BARRIERS VIA CONVECTION *** LaPLACE
IS READILY
OBTAINED
ARE PRIMARILY
TRANSFORM
PASSAGES
to discrete
NO*
OPEN/OPEN
YES
YES
CLOSED/OPEN
YES
NO
EITHER CLOSED
TIMES
FROM THE ONE IMMEDIATELY
REFLECTING
BUT MUST
INCLUDE
OPEN
OR
YES*
NO*
NO
YES***
NO*
YES***
PROCEEDING
A PROBABILITY
B OF ABSORPTlON
ONLY
either case, the clock stops when the. particle crosses k = t; and if the particle returns, it may as well do so immediately via reflection. We have thus determined that the residence time distribution in a dispersive system is independent of whether the system boundaries are open or closed. This conclusion, which seems obvious in retrospect, was masked for many years by the fact that open and closed systems behave quite differently when subjected to tracer experiments. When the residence time distribution is desired, the proper mathematical manipulation is that corresponding to closed boundaries even though the system being modelled may be open. Solutions
NO*
ANY/CLOSED
random
walks
In this section some solutions to difference eqn (8) are presented for boundary conditions of the type shown in Table 1. Several of these solutions can be derived from one basic solution, due to Feller [14] among others, for a random walk between two absorbing barriers. The barriers are located at points 0 and z, and the particle is introduced at point j with initial step n,. Then the probability of absorption at the origin after exactly n steps is
(17) where m = II -no. The probability of eventual absorption at the origin is
The probability of adsorption at .z after exactly n steps may be found from eqn (17) by setting j = z - 1 and interchanging p and q. If we also normalize, the result is
fitrn)
=
pi(Zt ml _ II- (4/P)‘l(4Pq)“/2(q/p)“~*2’
pi(z) z-1
2 (- I)‘+’ cos
m_-l
xi
f
rri
. lrij
sm 1 stn 7
(19)
With j = 1 and no = 1, eqn (19) gives the inlet-to-outlet sojourn time distribution for a discrete dispersive system with any of the boundary conditions: open, closed or absorbing. The discrete distribution of first passage times in a system with an open inlet can also be found from eqn (17) without need to resolve eqn (8) subject to a transmitting barrier at k = 0. The approach is to take the limit of eqn (17) as z becomes large so that the random walk covers the infinite interval (0, m) with absorption at the origin. We then interchange p and q and suppose that the particles enter at point j = z with n, = 0 where z now has its previous, finite value. The result is equivalent to a random walk which starts at the origin with a transmitting barrier at the origin and with an absorbing barrier at point z. One form (14) for the resulting distribution is p*(z,
I
n)
=
yp(n-z1/2q~n+lm
1
0
while the probability of eventual absorption at z is P,(Z)= t - P,(O).
Al -(qILJy’l
COY’ mu sin PU sin ?rza de.
This is equivalent to the computationally
(20)
simpler form
Residence time PO(z,n) =
distributions in systems governed by the dispersion equation
zrl!
963
L=zAx
.(~)!(~)!P(“+=~*q(“-=)‘*(21)
t = nr where we now explicitly see that n P z and that z and n must be either both odd or both even for the particle to exit after exactly n steps. Otherwise, P&, n) = 0. With p > 4, all particles will eventually be’absorbed at z so that eqns (20) and (21) are already normalized. Thus these eqpations give the discrete distribution of first passage times in systems which are open at the inlet. By the arguments presented in the previous section, they also provide the distribution of last passages in systems which are open at the outlet. Few literature solutions have been found for any of the discrete cases with reflection. A series solution for reflection at the origin with absorption at z can be expressed in terms of the P1(O, n) and &(z, n) obtained for the doubly absorbing case, but this solution has not been summed in a convenient form. Kac[l3] has treated the case with reflection and transmission. His solution is applicable to life span distributions in systems with one open and one closed boundary, but the form of his discrete solution is complex and will not be repeated here. Feller[l4] has solved difference eqn (8) for two purely reflecting barriers; but the sum in eqn (12) obviously diverges with pure reflection and the system assumes an equilibrium spatial distribution. The final discrete random walk to be considered is that corresponding to open boundaries at both ends of the reactor. This gives an unending walk over the entire real line ( -a, a), and the solution to eqn (8) is a form of the binomial distribution
where we have set no = 0. In this expression not all n are possible, and Pi(z, n) = 0 unless n and z have the same parity and unless R 3 z-j. Normalization of this result wiil be considered after passage to the limit. CONTlNUQUSRANDOMWALKS
Passage to the limit This section considers the limiting behaviour of discrete random walks as z becomes large but Ax and the duration of a step, 7, become small. The probabilities p and q will approach (l/2) in this limit but in such a way that allows for a net convective velocity, U, to be imposed on the system. Specifically, let
to = nor
(23)
and suppose that both Ax and 7 approach zero in such a manner that the ratio Ax’ -_=D 27
(24)
remains constant. Then, the discrete transition probabilities Pi(k, n) and difference eqn (8) can be written as C,(x, t + T) = pC,(x -AX, t) t qC,(x t AI, t)
(25)
where C(x, t) is the probability that a particle will be at position x at time t. Expanding this equation in a Taylor series using a first order approximation for the time dependence and a second order approximation for the spatial dependence gives
substituting for q, p and AX% gives
+Jg+D~
(27)
which is identical to eqn (1) with r(c) = 0. Indeed, had a reaction probability been added to difference eqn (8), an r(c) term would have appeared in eqn (27). Thus, in the particular limit indicated by eqns (23) and (24). the one dimensional random walk becomes a one dimensional dispersion process. A key requirement that this limit be reasonable from a physical viewpoint is that the instantaneous particle velocity, AX/T, must be large compared to the convective velocity, CJ.In the mathematical limit, AXITbecomes infinite while U and Ax’lr are held constant. The various solutions to difference eqn (8) can also be subjected to the limiting process. This, of course, is an alternative to solving partial differential eqn (27). The appendix gives asymptotic approximations which are useful in making this passage to the limit. Applying these to eqn (19) gives sinh uL 4 e-uqt-to)‘4D $, (- l)i+‘i
j,(t) = (y)
sin ~
,-~*i2D(~-r~lLz
(8)
L
Setting y = to = 0 gives the inlet-to-outlet sojourn time distribution for a system governed by the dispersion equation: x=kAx y=jAx
4T2D2 UL f(t) = m sinh 20 e-
UW4D
,$,
(_
l)‘+‘i*
e--r2i2DliL2,
(29)
964
E. B. NMJMAN
To obtain the distribution of first passages, eqns (20) or (21) could be taken to the limit directly. However, a physically more transparent approach is to apply the limiting process to eqn (17) giving
where f,(r) dt is the probability that a particle introduced at point y will be absorbed at the origin before reaching point z and that this absorption will occur within the time interval t to t +dr. To find first passages from this, we set U = - U, take the limit as L -+ m and then set y equal to the original, finite L. Feller [14] gives this result as (31) where f(t) d T is now interpreted as the probability that a particle introduced at the origin will first reach point x = L during the time interval between t and r + df. Note that the old origin of eqn (30) has become an absorbing barrier at point L while the other absorbing barrier has been shifted to --co. None of the particles will reach this barrier so that eqn (31) is automatically normalized. It is clear from the original derivation of eqns (20) and (21) or from the above development of eqn (31) that a particle introduced at x =0 is free to move to locations with x
(32)
where C(x, r) dx is the probability that the particle will be located between x and x tdx at time r. The units on C(x, t) can be interpreted as particles per unit length and are thus seen to be appropriate concentration units for a one dimensional process. To obtain the life span distribution, eqn (32) is normalized according to the continuous analog of eqn (13) f(t)
=
_
c(x7 I) C(x, 1) d f
This result is the same as that obtained by solving partial differential eqn (27) subject to an impulse function at the inlet[3]. Thus the life span distribution in a doubly open system is identical to the impulse response function. Direct solution to the continuous case Conventional solutions to partial differential eqn (27) give results identical to those obtained at a particle level of scrutiny provided that appropriate boundary and initial conditions are used. The continuum counterpart of an absorbing barrier at L is the absorbing boundary conditions, C(L, r) =O. The reflecting barrier at L is equivalent to the closed boundary, C(L, I) = 0 while a transmitting barrier corresponds to the open boundary. With these correspondences established, all the forms of residence time distribution listed in Table 1 can be found using continuum mathematics although literature solutions have not been located for each case. The open and closed boundaries are familiar to the chemical engineer; but the use of absorbing boundaries is rare, at least in the context of residence time distributions. Their use will be illustrated by a direct solution to the inlet-to-outlet sojourn time distribution for a continuous system governed by eqn (27). The discrete solution to this problem used absorbing barriers at points 0 and z, and individual particles were introduced at intermediate point i. The corresponding boundary conditions’ for the continuous case are C(0, t) = C(L, r) = 0 while particles are introduced using the initial condition C(x, 0) =6(x-y). A solution to eqn (27) that satisfies these conditions is
(34) The sojourn time distribution is found by normalizing the flux at the outlet, the procedure recommended by Petho[23] being correct in this case. The normalization factor is
(35) The series in eqn (35) can be summed using tabulated results [24], and the normalized flux becomes
-DE
fm = -_ N I
r_2rDsinhg
_
L -ae
UZ1/4D
20
which is identical to eqn (28) with to = 0. Taking the limit as y +O gives eqn (29) as the frequency function for inlet-to-outlet sojourns. As mentioned above, not all the cases listed in Table I have reported solutions. What must be considered the most important case without an explicit solution is the
Residence time distributions in systems governed by the dispersion equation
distribution of life spans in a doubly closed system since this is also the long sought residence time distribution in a doubly open system. This case has, however, been analyzed using the Laplace transform technique. The transform of the residence time frequency function can be obtained from van der Laan’s[25] general result by setting D, = D, = 0 to give qJ
ewL12DXI-m
(37) f(S)=(1+8)‘_(1-B)‘e’-UUD’B where fi = v/(1 +4sa UL) and where s is the Laplace transform parameter. Equation (37) can also be derived from the steady state version of eqn (1) for a first order reaction with r(C)= SC using the Danckwerts’ boundary conditions; eqns (3) and (4) or by using Bischoff’s (1) boundary conditions for an open system. AU three derivations give the same result, eqn (37), and the dilemma raised by Gibilaro has vanished. Eqn (7) can indeed be used to obtain f(t) from reaction data; and eqn (5). while it gives an interesting function with mean r= Ll U, does not give the residence time distribution. Equation (37) is the Laplace transform of the residence time frequency function for a doubly open system, a doubly closed system or for either combination of half open and half closed. It does not happen to be the residence time distribution for a system with absorbing boundaries. Such systems can show unusual behavior with regard to means residence times, giving an apparent contradiction to Buffham’s 1261 generalization that the mean residence time in both flow and non-flow systems is equal to the steady state holdup divided by the transmission rate through the system. This topic has been discussed separately [27].
965
future results since the mathematical techniques are simpler and more widely known. However, all the discrete solutions shown as missing in Table 1 are also missing in the continuous case. Hopefully, some of these will appear in the near future, particularly an explicit form for the doubly closed case. The historic difficulty in deriving residence time distributions from the axial dispersion model was due to the backwards particle flow which is an inherent part of that model. If particles did not flow upstream against the convective velocity, all differences between open and closed systems would vanish. As recently summarized by Sundaresan, Amundson and Aris [28], backwards flow of particles can be expected in systems where mixing is driven by concentration gradienEs with a diffusion-like mechanism. Molecular, Brownian, and turbulent diffusion all fall into this category as does Taylor dispersion. If these phenomena are the major cause of mixing, dispersion models can be expected to fit the real system even with respect to fine structure. Most real systems would require a two or three dimensional model for precise results, but the basic dzerences between open and closed systems would still prevail. If, however, reverse flow does not occur, then dispersion models cannot be expected to fit the fine structure of real systems; and the residence time distributions predicted in this paper could not be expected to hold in detail. Indeed, Sundaresan et a1.[28] have shown that no existing model accurately predicts all significant phenomena in a packed bed at high Reynolds number. Even for packed beds, however, the axial dispersion model is adequate for many purposes; and if a prediction of the residence time distribution is needed, then the one corresponding to closed boundary conditions is clearly the best to choose. These boundary conditions at least prohibit reverse flow at the inlet and outlet.
CONCLUSIONS NOTATION
It is
apparent from the foregoing results that dispersive systems can be scrutinized on the scale of individual particles without intrinsic difficulties. Sojourn time and residence time distributions are equally applicable to single particles as they are to large numbers of particles. Although the dispersion equation is conventionally interpreted in terms of net fluxes, it can also be interpreted in terms of probability distributions for individual particles. Indeed, the individual particle viewpoint is necessary to obtain clear and unambiguous definitions of residence time. It is now apparent that both open and closed systems have the same residence time distribution when time spent outside the system boundaries is excluded from the total. This residence time distribution agrees with the one deduced from reaction yields. It disagrees with the result of Gibilaro[4], and his result is now seen to be the distribution of first passages. Although the major results in this paper were obtained with discrete random waIks and with passage to the limit of continuous motion, the conventional approach using the dispersion equation is equally valid when the appropriate boundary and initial conditions are used. This continuum approach will probably be most useful for
u
C Co D f F
i j k L m n n0 N p P,(k, n) PI(k)
q Q r f(c) s
dummy variable of integration concentration of particles inlet concentration dispersion coefficient residence time frequency function residence time distribution function index of summation discrete initial position discrete spatial variable length of reactor discrete step count, n-no number of steps in random walk initial step number normalization factor probability of moving in positive direction discrete transition probability discrete normalization factor probability of moving in negative direction probability of not revisiting point z reaction probability reaction rate expression reaction rate constant and Laplace transform parameter
E. B. NAUMAN
966
t continuous time variable
to initial value of t u drift velocity of particles x I
continuous spatial variable discrete reactor length
Greek symbols 0 d/(1 + SLYLx) y continuous initial position
1261 But&m B. A., Nature1978 274 879. (271 Nauman E. B., Chem. Engng Commrcn. 1981 8 53. 1281 Sundaresan S., Amundson N. R. and Aris R., A.1.Ch.E. J. 1980 26 529. APPENDIX Limiting forms In the following results, the indicated limit refers to that process governed by eqns (23) and (24).
E incremental probability 0 flux of particles 7 duration of a step REFERENCES I
[I]
Bischoff K. B., Chem. Engng Sci. 1960 12 69. [2] Danckwerts, P. V., Chem. Engng Sci 1953 2 1. [31 Levenspiel 0. and Smith W. K., Chem Engng 5%. 1957 6 221. [4] Gibilaro L. G., Chem. Engng Sci. 1978 33 487. [5] Kreft A. and Zuber A., Chem. Engng Sci. 1978 33 1471. [61 Buffham B. A., Chem. Engng Sci. 1969 24 1386. [7] Danckwerts P. V., Chem. Engng Sci. 1958 8 93. [8] Einstein A., Inuestigotions on the Theory of fhe Brownian Mouemenf, (Edited by Fuerth R.). Dover, New York 1956. [9] Smoluchowski, M., Phys. Zeif. 1916 17 557. IlO] Uhlenbeck G. E., and Omstein, L. S., Phys. Rev. 1930 36 823. 1111 Chandrasekar S., Rev. Mod. Phys. 1943 15 1. I121 Wang M. C., and Uhlenbeck G. E., Rev. Mod. Phys. 1945 17 323. 1131 Kac, M., Am. Math. Monthly 1947 54 369. [141 Feller W., An Introduction to Probability Theory and its Applications, (3rd Edn), Vol. I. Wiley, New York 1968. [IS] King G. W., Ind. Engng Chem. 1951 43 2475. 116) Aris R., and Amundson N. R., A.LCh.E. 1. 1957 3 280. [17] Shinnar R., and Naor D., Chem. Engng Sci. 1967 22 1369. [18] Shinnar R., Naor P. and Katz S., Chem. Engng Sci 1972 27 1627. [19] Shinnar R., Glasser D., and Katz S., Chem. Engng Sci. 1973 28 617. [20] Brenner H. and Gajdos L. J., /. Colloid Interface Sci. 1977 58 312. 1211 Gajdos L. J. and Brenner H., Sep. Sci. Technol. 1978 13 215. [22] Brenner H., Int. J. Phys. Chem. Hydra. 1980 191. [23] Petho A., Chem. Engng Sci. I%8 23 807. 1241 Jolley L. B. W., Summation of Series, p. 104. Dover, New York 1%1. [25] van der Laan E. T., Chem. Engng Sci 1957 7 187.
lim
sjn
Y = sin !T, L
2
For walks of finite duration ending via absorption at I, lim y
= lim e
= f(t). I
For unending random walks, the particle concentration is found by a limiting process on the spatial dependence of Pj(, n) ,im
[email protected]) = C(*, I)
2Pi(z)AX
where the factor of two arises from the fact that Pi(z, n) = 0 for every other value of .z The asymptotic approximations used in the limiting process do not show these zero values and thus must be divided by two to obtain the correct spatial average for C(x, t).