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Stochastic modeling of transient residence-time distributions during start-up
Pergamon
Chemical Enoineerin0 Science, Vol. 50, No. 2, pp. 211 221, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
001D-2509(94)00224-X
STOCHASTIC MODELING OF TRANSIENT RESIDENCE-TIME DISTRIBUTIONS DURING START-UP L. T. FAN, t B. C. SHEN* and S. T. C H O U Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, KS 66506, U.S.A. (Received 20 January 1994; accepted in revised form 6 July 1994)
Abstract--The theory of residence-time distribution, RTD theory in short, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. The creation of the modern, systematic RTD theory has been attributed to Danckwerts. As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residence-time distribution. While Danckwerts steered the development of the RTD theory essentially along the path of deterministic physics, obviously, the description of RTD is better couched in the statistical or stochastic parlance. Stochastic modeling visualizes the fluid in a flow system as being composed of discrete entities. This visualization reveals a greater insight into the underlying mechanism than deterministic modeling, thereby facilitating our understanding of the flow and mixing characteristic of the system. In the present work, an attempt has been made to derive a unified mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Specifically, the expressions for RTDs of molecules, fluid particles or any flowing entities passing through continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions.
I. INTRODUCTION The theory of residence-time distribution (RTD theory), one of the theories unique to chemical engineering, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. Moreover, the utility of different aspects of the RTD theory has long been recognized in fields as diverse as physiology, pharmacology, chemistry, and civil engineering. Nevertheless, the creation of the modern, systematic RTD theory has been attributed, probably rightly so, to Danckwerts (1953). As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residencetime distribution. Danckwerts, however, steered the development of the RTD theory essentially along the path of deterministic physics. Very likely, he was sensitive to the fact that the mathematics of stochastic processes had not been included in the repertoire of the chemical engineering profession, and, thus, the acceptance of the RTD theory by the chemical engineering community would be inhibited if the theory were couched purely in the parlance of stochastic processes. It appears that this has hindered the establishment of a firm or self-consistent foundation for the RTD theory based on the first principles.
tAuthor to whom correspondence should be addressed. *Present address: Research Department, Miles Inc., New Martinsville, WV26155, U.S.A.
Efforts to remedy this situation by resorting to stochastic processes have made substantial headway in recent years. Some earlier efforts are those by Shinnar and his coworkers. Naor and Shinnar (1963) have introduced the concept of intensity function. By viewing the turbulent chemical reactors as a network of stirred tanks and taking the interstage flows to be stationary Markov processes, Krambeck et al. (1967) have developed the stochastic models of RTDs of turbulent flow and mixing systems based on the random walk assumption. Adoption of a broader class of Markov processes, birth~leath processes, has enabled Shinnar and Naor (1967) to propose a more general method of calculating the residence-time distributions for systems with internal reflux. By introducing the concept of joint probability distribution for the number of recycles and the residence time, the total regional residence-time distribution for a continuous recycle system involving either a single unit or a cascade has been analyzed and characterized (Mann et al., 1979; Mann and Rubinovitch, 1981). Rubinovitch and Mann (1983a, b, 1985) have presented a Markov chain model for analyzing particulate processes. Nauman (1981) has applied the theories of discrete and continuous random walks to the open systems governed by the dispersion equation. Fan and his colleagues have treated the arbitrary complex networks of stirred tanks by a general continuous-time compartmental model (Fan et al., 1982, 1985) and by the master equation approach (Fox and Fan, 1984). What differentiates these works from the conventional deterministic approach is the definition of ran211
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dom variables characterizing the random or fluctuating nature of the flow and mixing of molecules or fluid particles, and the subsequent applications of the theories of probability and stochastic processes. Research on the RTD theory has mainly focused on steady-state systems, even though some attempts have been made to develop a theory of the unsteady-state RTD. Cha and Fan (1963) have considered the transient age distributions of a completely mixed tank or tanks-in-series with a pulsating feed having an arbitrary entrance-age distribution. In his work on the RTD of a system with unsteady hydrodynamics, Nauman (1969) has systematically defined the system's inlet life-expectation and outlet residence-time distribution functions. He has further pointed out that these functions are generally not identical under unsteady-state flow conditions. Based on these conceptual developments, Nauman (1969) has developed the RTD theory for an unsteady-state stirred-tank reactor. This theory has been generalized by Chen (1971) to the unsteady-state flow reactors of arbitrary geometric shape through the population balance approach. In an attempt to achieve the optimal constant RTD control, Nauman (1970) has shown that a system of stirred-tank reactors can be subjected to changes in working volume and throughput while maintaining a stationary residence-time distribution. Nevertheless, these approaches are essentially deterministic. Fan et al. (1979) have developed a stochastic model of the unsteady-state RTD; its applications can be found in Fahidy (1987). The objective of the present work is to derive a mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Parallel to Nauman's (1969, 1981) conceptual development cast in deterministic parlance, the unsteady-state age distributions of fluid particles are introduced in the present work as the probabilistic functions. Moreover, the expressions for RTDs of molecules, fluid particles or any flowing entities in continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions; some of these expressions have been unavailable hitherto. 2. PRELIMINARIES
A flow system with one entrance and one exit is considered to contain an incompressible fluid with a finite volume of V. The fluid in the system is visualized in the present work as being composed of N discrete fluid particles that are conserved. These particles can be atoms, molecules, Brownian particles or any other fluid elements [see e.g. Nauman and Buffham (1983)]; nevertheless, by analogy, any conserved flowing objects or entities such as solid particles, liquid droplets, and gas bubbles passing through a flow system can be regarded as fluid particles in analyzing their RTDs. The inflow fluid enters the system through the entrance and ultimately departs from the
exit. In other words, none of the fluid particles will remain permanently in the system. The volumetric flow rate of the inflow fluid, v, corresponds to a flow of n discrete fluid particles per unit time. Moreover, the inflow fluid particles are assumed to be fresh at the moment when they enter the system, i.e. their ages are all zero initially and increase at a rate equal to the time spent in the system. The N fluid particles originally residing in the system are the inventory fluid particles. After the onset of the process, these inventory fluid particles are continually replaced by the inflow fluid particles entering the system at a constant rate of n fluid particles per unit time. Hence, the compositions of both the inventory and inflow fluid particles in the system vary as functions of time even though the overall mean flow rate of the outgoing fluid particles, consisting of both inflow and inventory fluid particles, remains constant at n. Obviously, the number of inventory fluid particles will continually decrease, and that of inflow fluid particles will continually increase inside the system. Eventually, the inventory fluid particles will be totally replaced by the inflow fluid particles. At this moment, the age distribution in the flow system, comprising N inflow fluid particles, attains a steady state. Note that the initial age distribution of the inventory fluid particles in the system affects its unsteady-state age distribution. For simplicity, however, only the situation where the initial ages of the inventory fluid particles are uniformly zero is considered at the outSet.
Let t denote the time elapsed from the onset of the process, i.e. the time of the process, in brief; and ,7, the time when a fluid particle enters the system. The inventory particles can be considered as having entered the system instantaneously at time v/of 0 [Fig. l(a)], while the inflow particles as having entered the system during 0 < ,7 ~< t [Fig. 1(b)]. Normally, the life expectation of an inventory or inflow fluid particle entering the flow system at time ~/, i.e. the probable duration before the particle's exit from the system, cannot be exactly predicted or determined due to the
I
t
I
o ,1=0
t
t4-At
a. Inventory particles entering the system at q=O.
,I
t-n
-I
-I
I
I
t
0
q
t
b. Inflow particles entering the system during O
Fig. 1. Time scales of the process.
Stochastic modeling of transient RTD
213
complexity of the flow pattern and behavior in the system. Thus, the possible value of the particle's life expectation entering the system at time r/ is better represented by a continuous random variable, denoted by O(r/). The definition of the random variable, ®(t/), is twofold [see e.g. van Kampen (1981)]. The first is the set of possible values or realizations that ®(t/) can assume, i.e. the sample space, which spans [0, oo ). The second is the probability distribution over this sample space represented by the following cumulative distribution function:
ticles may exit from the system at any subsequent moment. The random variable representing the life expectation of an inflow particle entering the system at time )/is ®(0 < q ~< t), denoted hereafter by ®2(r/) for convenience. Again from eq. (la), its cumulative distribution function is
Fo~(O) - Pr [t0(r/) ~< 0]
Fo2~.~(O) is also customarily called the inlet life-expec-
= Pr [a fluid particle entering the system at time r/has a life expectation of 0 or less, i.e. it will exit from the system prior to or at time (q + 0)] (la) or
1 - Fo~(O) =- Pr [to(r/) > 0] = Pr [a fluid particle entering the system at time ~/will remain there at time (r/+ 0), i.e. it will exit from the system after time (r/+ 0)]. (lb) The cumulative distribution functions of inventory and inflow fluid particles will be treated separately in what follows. For convenience, fluid particles will henceforth be termed particles. In other words, inflow fluid particles will be termed inflow particles; inventory fluid particles, inventory particles; and outflow fluid particles, outflow particles. 2.1. Inventory particles The inventory particles are the ones residing in the system at the onset of the process. They may be considered as entering the system instantaneously at r / = 0, as mentioned earlier. Hence, the random variable representing the life expectation of an inventory particle can be written as O ( r / = 0). For convenience, O ( r / = 0) is henceforth written as ®1. From eq. (la), the cumulative distribution function of O1 is
Fo,(O) = Pr(®l ~< 0) = Pr (an inventory particle has a life expectation of 0 or less).
(2)
The probability density function of O1, fo,(O), is defined as d Fo, (0) fo,(O) = dO (3) Naturally, fo,(0)A0 represents the probability that an inventory particle has a life expectation in interval (0, 0 + A0), or, in terms of deterministic parlance, the fraction of the inventory particles exiting from the system during time interval (0, 0 + A0). 2.2. Inflow particles After the onset of the process, the inflow particles enter the system continuously and any of these par-
Fo~.)(O) = Pr [02(r/) ~< 0] = Pr (an inflow particle entering the system at time r/has a life expectation of 0 or less). (4) tation distribution. The probability density function of O2(r/), fo2(.)(0), is defined as
f°2~")(O) --
dFo2tnl(0) dO
(5)
Clearly, fo2t,~(O)AO represents the probability that an inflow particle entering the system at time r/has a life expectation in interval (0, 0 + A0) or, in terms of deterministic parlance, the fraction of the inflow particles entering the system at time r/has life expectations in interval (0, 0 + A0). In general, Fo:t~)(O) is a function of 0 as well as r/prior to the establishment of a steady state in the system. Nevertheless, more often than not, a hydrodynamic or flow-pattern steady state is essentially established instantaneously regardless of when the inflow particles enter the system, i.e. r/; this is especially the case if the fluid particles are incompressible and the physical and transport properties, such as the sizes, densities and viscosities, of the inflow and inventory particles are nearly identical. Under this situation, the life-expectation distribution of the inflow particles will remain invariant until the corresponding compositional steady state is established; therefore,
Fo2~n~(O) = Fo2(O)
(6)
fo~t~)(O) = fo~(0).
(7)
As will be delineated in the succeeding section, Fo2(O) is equal to the cumulative residence-time distribution function at the exit of the system under steady-state conditions, F(0); andfo2(0), the residence-time distribution function at the exit of the system under steady-state conditions, E(0), i.e.
Fo2(O) = F(O)
(8)
f02(O) = E(O).
(9)
It is worth mentioning that F(O) and E(O), and consequently Fo2(O) and fo2(O), can be determined experimentally by a tracer technique [see e.g. Danckwerts (1953) and Wen and Fan (1975)]. 3. STOCHASTICPOPULATION BALANCE Under hydrodynamic steady-state conditions described in the preceding paragraph, the total number of particles in the system and the rate of inflow or outflow in terms of the number of particles will be invariant at N and n, respectively, throughout the
L. T. FAN et al.
214
process. The conservation of these numbers gives rise to the stochastic population balances governing the compositional variations of the inventory and inflow particles.
Dividing both sides of this expression by nat, taking the limit as At---}0, and substituting eqs (3), (5), (7), and (13) into the resultant expression yield
The total number of particles in the system at time t, N, comprises the contributions from the following. (i) Inventory particles remaining in the system at time t. The expected number of such particles is NIl
- Fa,(t)].
(I0)
(ii) Inflow particles entering the system in time interval (0, t) and remaining there at time t. The expected number of such particles is n
]" o
[1 - Fo2(t - r/)] dr/.
(11)
Since the total number of particles is conserved as stated,
f,
N=N[1--fe,(t)]+n
[1-Fo2(t-r/)]dr/.
(12)
do
The mean residence time, z, is N ~ =
-.
(13)
n
Hence, eq. (12) becomes Fol(t) = ~lfo~ [1 - Fo~(t
-
r/)]
dr/.
(14)
This is equivalent to the population balance equation of the particles in the system. 3.2. Population balance around the exit of the system Since the number flow rate, n, remains invariant throughout the process, the total number of particles exiting from the system in time interval (t, t + At) is nat; it consists of the contributions from the following. (i) Inventory particles exiting from the system in time interval (t, t + At). The expected number of such particles is N[Fo~(t + At) - Fo,(t)].
(15)
(ii) Inflow particles entering the system in time interval (0, t + At) and exiting from the system in time interval (t, t + At). The expected number of such particles is n
U
Fo2(t + At - rl) dr~ - n
0
f
2(t - r/) dr/= 1.
zfo,(t) +
3.1. Population balance over the system
Fo2(t - r/) dr/.
(18)
This expression is equivalent to the population balance equation of particles around the exit of the system. 3.3. Internal age distribution Let l(OI t)AO be the probability of a particle, either inventory or inflow, inside the system to have an internal age, i.e. the time spent by the particle inside the system since its entrance, between 0 and (0 + A0) at time t. In other words, l(Olt) is the probability density function of the internal age. In conventional or deterministic parlance, I(Olt)AO is the number fraction of particles inside the system, each having an age between 0 and (0 + A0), at time t; thus, I(OI t) is often termed the internal age distribution function of the particles based on the number or, simply, the internal age distribution. If the sizes and densities of all particles can be regarded as identical, I(Olt) based on the number will be the same as that based on the mass. At any time t, the number of particles in the system having ages between 0 and (0 + A0), Nl(OIt)AO, is composed of the following two contributions. (i) Inventory particles in the system having ages between 0 and (0 + A0). The expected number of such particles is NIl - Foj(t)] {U(t - O) - U [ t - (0 + A0)]} (19) where U is the unit step function defined by 0,
U(t-O)=
1,
t< 0
t>~O.
(20)
(ii) Inflow particles in the system having ages between 0 and (0 + A0). The expected number of such particles is f0n[1 -- Fo2(t _ r/)] × {U[(t
- r/) - O] -
U[(t
-
CO + AO)]}
r/) -
d~.
(21) The expected number of particles in the system with ages between 0 and (0 + A0) at time t is the sum of the expected numbers of the inventory and inflow particles with such ages; therefore,
dO
(16)
NI(OIt)AO
= N I l - Fol(t)] {U(t - 0 ) - U[t - (0 + A0)]}
Thus, nat = N [ F o , ( t + At) - Fo,(t)]
]
+
~ t +A!
+ n
f'
n [1 -- Fe2(t -- r/)]
0
Fo~(t + At - r/)dr/
x {u[(t
O
- - n f o t Fo~(t - -
q)dr/.
(17)
-
r/) -
o]
- (0 + A0)] } dq.
-
u[(t
-
,I)
(22)
Dividing both sides of equation by NAO, taking the
Stochastic modeling of transient RTD continuous limit, and noting that u ( t - O) - u [ t - (0 + A0)]
3(t - 0) = lim ~o~o
A0
(23)
where Ne is the expected number of fresh particles entering the system in a small time interval of (r/, r/+ At/) and exiting from the system in time interval (t, t + At), q ~< t, i.e.
where 3 is the unit impulse function defined as 6(t - 0) = ~0, ( oo,
t :# 0
Ne = n A q [ F o 2 ( t + A t - q) - Fo~(t - r/)].
(24a)
t=O
and o~b(t
0) dt = 1
215
(24b)
(28b)
The expected number of particles exiting from the system with ages between 0 and (0 + A0) in time interval (t, t + At) is the sum of the expected numbers of inventory and inflow particles having ages between 0 and (0 + A0) given in (i) and (ii); hence, E[Z I(t, At}]
yield the general internal age distribution of particles in the system at time t:
1
~
{U(Ot~ - O)
i=t
-
HOlt) = [1 - Fo~(t)]f(t - O)
+ - [1 -- Fo2(0)] U(t - O)
nAtE(OIt)AO =
u[01~
-
(0 +
A0)]} +
U(02,
-
O)
i=
(25)
U[02,-
T
where z = N / n and U is the unit step function indicating that the maximum possible internal age is t. The first and second terms on the right-hand side of this equation represent the contributions from the inventory and inflow particles, respectively.
(0 + A0)] }).
Dividing both sides of this equation by At and A0, taking the continuous limits as both At --. 0, At/~ 0, and A0 ~ 0 lead to the expression for the general exit-age distribution function, E(Olt) = z fol(t) 6(t - O) + fo2(O)U(t - 0).
3.4. Exit-age or residence-time distribution Let E(OIt)AO be the probability of a particle, either inventory or inflow, at the exit of the system to have an age between 0 and (0 + A0) at time t, i.e. E(Olt) is the probability density function of the exit age of this particle. In conventional or deterministic parlance, E(OIt)AO is the number fraction of particles at the exit, each having an age between 0 and (0 + A0) at time t; thus, E(Olt) is usually termed the exit-age or residence-time distribution function of the particles. The number of particles exiting from the system in time interval (t, t + At) having exit ages between 0 and (0 + A0), n(At)E(Olt)AO, is composed of the following two contributions. (i) Inventory particles exiting from the system and having ages between 0 and (0 + A0). From eq. (15), the expected number of inventory particles exiting from the system in time interval (t, t + At), E [Zl(t, At)], is E [ Z a ( t , At)] = S [ F o ~ ( t + A t ) - Fo~(t)]
(26)
of which E [ Z i(t, At}]
{U(O~,-O)-
U [ O ~ , - ( O + AO)]}
(27)
i-I
have ages between 0 and (0 + A0). In the above expression, 0~i is the age of inventory particle i, i = 1, 2..... E [Z t (t, At)]. Since all inventory particles exiting from the system in time interval (t, t + At) will have ages between t and (t + At), we have t <~ OH < t + At, i = 1,2 ..... E [ Z 1 ( t , At)]. (ii) Inflow particles exiting from the system and having ages between 0 and (0 + A0). The expected number of such particles is U(02i - O) - U[02~ - (0 + A0)] i=
(28a)
(29)
(30)
Substituting eq. (19) into this expression gives rise to E(Olt) = [1 - Fo~(t)]3(t - O) + fo2(O)U(t - 0).
(31) 3.5. Steady-state aoe distributions When time t approaches infinity, 6(t - O) ~ 0 and U(t - O) -~ 1; hence, eqs (25) and (31) reduce, respectively, to l ( O l t - - , oo) = I(0) = _1[1 - Fe~(0)]
(32)
E(Olt ~ ~ ) = E(O) =fo2(0).
(33)
and
The former is the expression for the steady-state internal-age distribution function in terms of Fe2(0); ~ind the latter, the steady-state residence-time distribution function in terms of fo2(O). Either is solely in a function of Fo2(O) or fo~(0) since only the inflow particles remain inside the system at a steady state. Obviously from eq. (33), the steady-state, residence-time distribution, E(O), is identical to the probability density function for the life expectation of an inflow particle, fo2(O). Consequently, the corresponding cumulative residence-time distribution function at the steady state, F(O), is F(O) = Fo2(O).
(34)
This expression verifies the validity of eq. (9). Substituting eq. (34) back into eq. (32) yields I(0) = _1[1 -- F(0)].
(35)
T
This is the celebrated result given by Danckwerts (1953).
216
L. T. FAN et al.
3.6. Expressions of age distributions in terms of steady-state counterparts The expressions of l(OIt) and E(Olt) can be written in terms of their steady-state counterparts, I(O) and E(0), respectively. Substituting eqs (32) and (14) into eq. (25) gives rise to the general internal age distribution in terms of I(O) as shown below:
I(Olt) =
(f7)
I(z) dz 6(t - O) + I(0) U(t - 0). (36)
Combining eqs (33) and (34) with eq. (31) leads to the following exit-age or residence-time distribution in terms of E(0): e(olt) =
q7 ) E(~) d ,
a(t-O)+E(O)U(t-O). (37)
These two equations are identical to the expressions derived on the basis of the physical insight [see e.g. Cha and Fan (1963) and Wen and Fan (1975)]. The significance of eq. (36) and that of eq. (37) are illustrated in Figs 2 and 3, respectively. The second term on the right-hand side of eq. (36) or (37) represents, respectively, the steady-state internal or exit-age
l(o~)
impulse [area =
I;l(o)a(o)
Jtl(O)dO ]
7 " 2- 7- .._.
0 o
0
Fig. 2. Illustration of the general internal age distribution at time t.
E(alt)
~timpubse[area = J t E ( O ) d O
]
E(O)dO
IK// 7"/. t
o
Fig. 3. Illustration of the general residence-time distribution at time t.
distribution function generated by the inflow particles in the system having ages less than t. The first term on the right-hand side signifies the contribution from the inventory particles residing in the system at the outset of the process. Note that since all inventory particles have the age of t, their fractional amount can be represented by the fractional impulse function with a magnitude of St~ I(0) dO or S[ E(O) dO that is equal to the cross-hatched area in Figs 2 or 3, respectively. This area is actually the fraction of fresh particles that would have internal or exit ages greater than t if the distribution were of the steady state. When t approaches infinity, all the inventory particles inside the system will eventually be replaced by the inflow particles. As a result, the impulse term vanishes, and only the second term remains. In other words, l(Olt) or E(Olt) reduces to the steady-state, internal-age distribution function, I(0), or the steady-state, exit-age distribution function, E(O), as t ~ oo.
4. M A R K O V
PROCESSES
l(Olt) and E(Olt) are expressed in terms of the steady-state counterparts, I(0) in eq. (36) and E(O) in eq. (37), respectively. I(0) and E(O) can be either experimentally determined by tagging the particles by a tracer or stochastically modeled by visualizing passage of a single particle through the system. For simplicity, we assume that the stochastic process of a particle through the system is Markovian, i.e. only the last state occupied by the particle is relevant in determining its future behavior [see e.g. Chiang (1980)]. In other words, the probability of making a transition to each state of the process depends only on the state presently occupied by the particle. Various Markov processes, e.g. pure-death and birth~leath processes, are available, which may be suitable for different flow systems or models. For example, the particle passage through a single vessel or a tanks-in-series system may be modeled by the pure-death processes, as will be delineated in the succeeding two subsections, respectively. 4.1. Pure-death processes in a single vessel The flow of particles through a single vessel with arbitrary geometry involves continual entrance of the inflow particles into the vessel and their subsequent continual exit from it. In the parlance of stochastic processes, this process can be regarded as a birth~leath process; the birth is signified by the entrance of a particle, and the death, by the exit of a particle [see e.g. Seinfield and Lapidus (1974)]. Nevertheles, the former can be treated as a nonrandom or deterministic event since the rate of entrance of the inflow particles is usually predetermined, specified or well-controlled. Only the latter remains as a random or stochastic event, thereby giving rise to a pure-death process [see e.g. Fan et al. (1979)]. 4.1.1. Steady-state age distributions Now, we are to derive I(0) and E(O) of the steady-state flow
Stochastic modeling of transient RTD through a single vessel by identifying it as a puredeath process. Unlike the transient state considered in some of the previous sections, in which both the inventory and inflow particles are involved and consequently the population balance over the entire system has to be considered in deriving the general transient age distributions, only the inflow particles are involved in the process. The probability of a randomly behaving inflow particle to have a life expectation longer than 0, i.e. to remain in the system at age 0, is [1 - Fo2(0)]. The probability that it will exit from the system, in the succeeding age interval, (0, 0 + A0), is denoted as [#2(0)A0 + o(A0)], where/~2(0) is the exit intensity of this inflow particle [see e.g., Naor and Shinnar (1963)-I, signifying the propensity of exit during a unit time interval. Thus, the probability for this particle to exit, Fo2(O), is governed by the following equation: dFo2(O) dO
[1 --
Fo2(0)]112(0).
217
4.1.3. Two ideal cases. Depending on the hydrodynamic characteristics or flow conditions of the system, various representations or models can be considered for #2(0). Visualized below are two idealized systems. Plugflow system. For this system, the particles exiting from the system during the start-up are all inventory particles initially residing in the system, i.e. residing in the system at t = 0. At t = z, all the inventory particles will exit from the system; eventually, a steady state is attained. At this moment, all the inflow particles at the exit of the system will have the identical age or residence time of z. Hence, the exit or replacement intensity of an inflow particle,/t2(0), is ~2(0) =
{O'ooO<~z , 0>'~.
(45)
Substituting eq. (45) into eqs (43) and (44) results, respectively, in
(38)
Solution of this equation subject to the initial condition of Fo2(0) = 0 is
Fo2(O)= l - e x p [ -
ffu2(q)d~l].
t>T
(39)
(40)
(46)
)'6(t - 0),
t~
E(Olt) = (6(z - O) C(t - 0),
t>Z.
(47)
Substituting this expression, in turn, into eqs (32) and (33) leads, respectively, to
As t--, oo, eq. (46) reduces naturally to the wellknown uniform internal age distribution function at the steady state [see e.g. Danckwerts (1953)],
l(O)= ~ e x p I - f]m(q)dq]
(41)
I(0) = - U(z - 0).
E(O)= l~2(O)exp[ - f~#2(q)d~l].
(42)
1
These expressions are identical to the published results (Naor and Shinnar, 1963). 4.1.2. Transient age distributions. Substituting eqs (41) and (42) into eqs (36) and (37) yields, respectively, the unsteady-state or transient internal-age distribution, l(OIt), and the transient exit-age distribution, E(O[t), of particles flowing through a single vessel; they are given below:
Completely stirred tank system. For this system, the chance of an inflow particle inside the system exiting from it is independent of its history [see e.g. Danckwerts (1953)], i.e. #2(0) is independent of age 0 and equal to n/N; hence, we can write 1
#2(0) =/~2 = ;t = - . T
;- ] ~2(~)dt; U(t - O)
(49)
Substituting this expression into eqs (43) and (44), respectively, yields
HOlt) = exp( - t/z)6(t - O) + -1exp ( - 0/~) U(t - O)
,(Olt)=(!ff{exp[-fol~2(q)dq]})6(t-O) + -' e x p[
(48)
T
(50) 1
(43)
E(OIt) = exp( - t/z)fi(t - O) + - e x p ( - O/z) U(t - 0). T
(51) When a steady state is reached, the above expressions, in turn, yield +
o,.
(44) Note that eqs (43) and (44) are in accord with the available results (Fan et al., 1979).
E(0) = !exp ( - !).
(53)
L.T. FAN et al.
218
These are the celebrated completely random internal and exit-age distributions of Danckwerts (1953). 4.2. Pure-death processes in a tanks-in-series system A flow system composed of m well-mixed tanks-inseries has frequently served as a model for flow accompanied by axial dispersion of the particles. In this system, an inflow particle always enters the first tank, followed by its passage through the succeeding ( m - 1) tanks and eventually by its exit from the system; moreover, a particle in any tank will never return to the preceding tank. This is analogous to considering that no circulation of the fluid takes place between any pair of adjacent tanks in the conventional treatment of the subject.
1
E(0)
-
-
(m
-
(mO~'-'
1
= - e x p ( -- mO/z) ~
1)!
4.2.2. Transient age distributions. Substituting eqs (58) and (59) into eqs (36) and (37), respectively, leads to
I(Olt) =
exp( -- mt/'r)i:l ~ ~=ofl.= • +[~exp(-mO/r)~
¢5(t--0)
l
(mO~i-1 j
× U(t - O) 4.2.1. Steady-state age distributions. Under a steady-state condition, only the inflow particles are involved in the process that is no longer a function of t. Let #: be the transition intensity of a single inflow particle under such a condition; then,
E(Olt) = / e x p ( - rot~z)
(60) 1
/'rot\i-I]
o0.,] x U(t - 0).
]A2A0 + o(A0)
= Pr [an inflow particle in tank Si, i = 1, 2..... m, prior to age 0 will be in tank Si+, prior to age (0 + A0)]. (54) Note that #2 of the inflow particle is independent of its age, 0, since each tank is well mixed as in the completely stirred tank system. On the average, each tank contains N/m particles and the number flow rate through it is n particles per unit time; therefore, the transition intensity, #2, can be identified to be n
#2 = N/m
nm
= --.
(55)
N
Let P~(O) be the probability that an inflow particle initially entering tank 1 will be in tank i, i = 1, 2. . . . . m, at age 0. Then, ( # 2 0 ) ~- 1
Pl.i(O)=--exp(-#20), (i-
1)r
i = 1 , 2 . . . . . m. (56)
Hence, the probability that the inflow particle will exit from the system after residing in it for a duration of 0 is
Fo2(O) = 1 - ~ i=l
Pl,i(O) (#20)'-'
i=, ~ !
e x p ( - - #20).
(57)
By substituting this expression and eq. (55) into eqs. (32) and (33) and noting that z = N/n, we obtain the following well-known, steady-state age distribution functions for the tanks-in-series system [see e.g. Wen and Fan (1975) and Nauman and Buffham (1983)]:
I(0)=! Z Pl,i(O) ~i=l
(58)
(61)
When m ~ ~ , eqs (60) and (61) become eqs (46) and (47), respectively, i.e. the distributions for the plug flow. When m = 1, eqs (60) and (61) reduce to eqs (50) and (51), respectively, i.e. the distributions for the completely stirred tank. 5. DISCUSSION
In the previous work, only a single subclass of discrete-state Markov processes, the pure-death process, has been exploited in mathematically deriving the transient RTDs of relatively simple flow systems. There is no reason, however, that other named Markov processes are incapable of analytically or semianalytically yielding the expressions for the transient RTDs of more complex flow systems. Such Markov processes include the renewal processes, Poisson processes, pure-birth processes, birth-death processes, and queuing processes in the category of the discretestate processes and Liouville processes, Wiener processes, and Ornstein-Uhlenbeck processes in the category of the continuous-state processes. In fact, the steady-state RTD of the tanks-in-series system with recirculation or backmixing between any two adjacent tanks has been derived by regarding the particle behavior in the system as a birth-death process [see e.g. Fan et al. (1982)]. 5.1. Moments and fluctuations The mean residence time or first moment of the RTD of particles at any time t, zm(t), is
rm(t) = ~oOE(OI t) dO.
(62)
Substituting eq. (37) into the above expression gives rise to Zm(t) =
OE(O)dO + t
E(O)dO.
(63)
Stochastic modeling of transient RTD
for realizing the transient' internal-age and residencetime distributions of a flow system, I(01 t) and E(OI t); it is as follows:
A s t --* o o , z,~® = z =
l i m ~m(t) t~3
= foOE(O) dO.
(64)
Comparison of eqs (63) and (64) indicates zm(t) ~< z.
(65)
This result implies that the mean residence time during the start-up does not exceed that at the steady state. In general, the kth central moment of the residence-time distribution, Mk(t), is defined as
Mk(t) = fo[O - #(t)]kE(OIt)dO.
(66)
By substituting eq. (37) into this expression, we have
Mk(t) =
219
S
[0 -- ~(t)]kE(O)dO + [t - ~(t)] k
E(O)dO. (67)
The second-order central moment is the variance, denoted by tr2; the third-order central moment is the skewness, denoted by ),a; and so on. For a continuous fluid-flow system with overall steady-state flow pattern, the magnitude of RTD's fluctuations due to the random movement of the fluid particles inside the system or exiting from it at any moment is often negligible since it is of order N~, 1/2, where Nm is the number of fluid particles or molecules inside the system or that exiting from the system at time moment; the magnitude of Nm is of the order of the Avogadro number under relatively low temperatures and normal or high pressures [see e.g. van Kampen (1981)-]. In other words, the shapes of the RTDs measured in various experimental runs under the same macroscopic flow condition should be almost identical. Nevertheless, for various flow systems, the sizes of entities passing through them, such as relatively large solid particles, liquid droplets or gas bubbles, are rather large. Thus, the number of the entities in any of such systems is far less than N,, and the instantaneous flow rate of the flowing entities or particles always fluctuates around its mean value even though the mean value may be a constant with respect to process time. Consequently, the shapes of the RTDs obtained in various experimental runs under the same macroscopic flow condition tend to be appreciably different. In other words, fluctuations of any RTD due to the inherent random nature of the flow system, specifically, the random variations of small populations of the finite-size particles inside and at the exit of the system, can be substantial. It is imperative, therefore, that such fluctuations be stochastically analyzed or modeled. 5.2. Procedure for evaluation and verification It should be of practical interest and of convenience to describe in a stepwise manner a possible procedure
(1) Ascertain if the flow system satisfies the conditions set forth in Section 2; specifically, ascertain if the transient stage of the process essentially involves only the compositional variations of the inventory and inflow particles with an overall steady-state flow pattern maintained. (2) Adopt the general expressions for l(OIt) and E(OIt) in terms of I(0) and E(O), respectively, derived in Section 3, specifically, eq. (36) for l(OIt) and eq. (37) for E(OIt). (3) Determine the steady-state age distributions, I(0) and E(O), through stochastic modeling as a Markov process. The procedure is given in Section 4. (4) Measure I(0) and E(O) experimentally by a tracer technique. The procedure is straightforward. The flow system under investigation may first be kept running until the compositional steady state is reached. At this stage, 1(0) and E(O) are measured by tagging the flowing entities or particles with a tracer [see e.g. Nauman and Buff'ham (1983)]. (5) Compare the mathematically derived and experimentally determined I(0) and E(O) to verify the stochastic models and to recover the parameters of the models. To evaluate these parameters, various moments or central moments of I(0) and E(O) derived from the modeling, e.g. mean residence time, variance, and skewness, may be compared with those obtained experimentally. (6) Substitute I(0) and E(O) obtained in the previous steps into eqs (36) and (37), respectively, thereby giving rise to the expressions for the transient internal-age distribution at time t, I(OIt), and the transient residence-time distribution, E(Olt). For a relatively simple flow system whose flow pattern is readily predictable, e.g. a flow vessel with a simple shape and intensive agitation, the stochastic modeling can be accomplished based only on the available empirical evidence for a given system configuration and under specific operation conditions. For a system in which the flow pattern is highly complicated so that it cannot be modeled analytically through the stochastic approach, I(Olt) and E(OIt) can be numerically determined by substituting the experimentally determined I(0) and E(O) directly into eqs (36) and (37), respectively, as mentioned earlier.
5.3. Chemical reactions By definition, the RTD theory and various expressions of the residence-time or age distributions derived from the theory of stochastic processes are applicable to systems, such as communities of living
220
L. T. FAN et al.
organisms, human societies, and continuous-flow process vessels, involving flow of discrete entities that age progressively. Nevertheless, the most prominent utility of the RTD theory and residence-time distributions is in the characterization of flow chemical reactors. As in the case of the steady-state RTD, the unsteady-state RTD can describe the characteristics of the macromixing in a flow system, i.e. the duration of a fluid particle's stay inside the system. The unsteadystate RTD, or even the steady-state RTD, however, cannot predict the exact pathway of a particle as it moves through the system. For a first-order reaction under isothermal conditions, the knowledge of the unsteady-state RTD is sufficient to determine the extent of the reaction at the exit of the system since the extent of the isothermal first-order reaction depends only on the time the particles spend in the vessel under an isothermal condition. For a nonlinear reaction, i.e. a non-first-order reaction or a nonisothermal firstorder reaction, however, knowledge of the unsteadystate RTD does not suffice because the extent of reaction depends not only on the particles' residence times, but also on their pathways through the system. In this case, supplementary information on the behavior of the particles in a local environment in the system, i.e micromixing, is indispensable [see e.g. Nauman (1969), Chen (1971) and Wen and Fan (1975)]. 6. C O N C L U D I N G REMARKS
The present work shows that the stochastic approach is capable of systematically and rigorously deriving various age or residence-time distributions of a flow system during unsteady-state operations and that the resultant expressions reduce to those of the well-known steady-state distributions. Moreover, the efficacy of the approach has been amply demonstrated by several examples. The theory of RTD and the explicit expressions of RTDs for some flow systems based on it have been derived in the present work for the transient stage of a process undergoing only compositional variations with the overall steady-state flow pattern. Nevertheless, the theory can be readily extended, by analogy, to the processes with variable flow rates.
E(O)
E(010 Fo, (0)
fo,(O) Fo~(~)(O)
NOTATION residence-time distribution function of the particles at the exit of the system under steady-state condition residence-time distribution of the particles at the exit of the system at time t probability that an inventory particle has a life expectation of 0 or less probability density function that an inventory particle has a life expectation of 0 or less probability that an inflow particle entering the system at time r/has a life expectation of 0 or less
fo2t.)(0)
Fo(~)(O)
F(O)
I(0)
l(OIt) m Mk(t ) n N Ne
t U V Z l(t, At)
probability density function that an inflow particle entering the system at time q has a life expectation of 0 or less probability that a fluid particle entering the system at time r/has a life expectation of 0 or less, i.e. it will exit from the system prior to or at time (q + 0) cumulative residence-time distribution function at the exit of the system under steady-state condition internal age distribution of the particles in the system under steady-state condition internal age distribution of the particles in the system at time t number of well-mixed tanks kth central moment of the residence-time distribution rate of inflow particles entering the system total number of particles in the system number of inflow particles entering the system in a small time interval of (q, q + A~/)and exiting from the system in (t, t + At) time elapsed from the onset of the process until step function volume of a flow system number of inventory particles exiting from the system in time interval (t, t + At)
Greek letters fi unit impulse function r/ time when a particle enters the system 0 realization of ®(q), ®l or ~)2(q) O(r/) random variable representing the life expectation of a particle entering the system at time r/ Ox random variable representing the life expectation of an inventory particle ®2(q) random variable representing the life expectation of an inflow particle entering the system at time r/ /~2(0) exit intensity of an inflow particle r mean residence time ( = N/n) ~..(t) mean residence time of particles at time t
REFERENCES
Casella, G. and Berger, R. L., 1990, Statistical Inference, p. 55. Wadsworth & Brooks/Cole, Pacific Grove, CA. Cha, L. C. and Fan, L. T., 1963, Age distributions for flow systems. Can J. chem. Engn# 41, 62-66. Chen, M. S. K., 1971, The theory of micromixing for unsteady state flow reactors. Chem. Engng Sci. 26, 17-28. Chiang, C. L., 1980, Introduction to Stochastic Processes and Their Applications, pp. 271-299. Robert E. Krieger, New York. Danckwerts, P. V., 1953, Continuous flow systems: distribution of residence times. Chem. Engng Sci. 2, 1-11. Fahidy, T. Z., 1987, An application of the Fan-Nassar model
Stochastic modeling of transient RTD to the unsteady-state age distribution of a class of nonideal flow. Chem. En#n# ScL 42, 1513-1514. Fan, L. T., Fan, L. S. and Nassar, R. F., 1979, A stochastic model of the unsteady state age distribution in a flow system. Chem. Engng Sci. 34, 1172-1174. Fan, L. T., Too, J. R. and Nassar, R., 1982, Stochastic flow reactor modeling: a general continuous time compartmental model with first order reactions, in Residence Time Distribution Theory in Chemical Engineerin# (Edited by A. Peth6 and R. D. Noble), pp. 75-102. Verlag Chemie, Weinheim, West Germany. Fan, L. T., Too, J. R. and Nassar, R., 1985, Stochastic simulation of residence time distribution curves. Chem. Engng Sci. 40, 1743-1749. Fox, R. O. and Fan, L. T., 1984, A master equation formulation for stochastic modelling of mixing and chemical reactions in inter-connected continuous stirred tank reactors, in Proceedings of lSCRE 8, The 8th International Symposium on Chemical Reaction Engineering, pp. 561-568. Pergamon Press, Oxford. Krambeck, F. J., Shinnar, R. and Katz, S., 1967, Stochastic mixing models for chemical reactors. Ind. Engng Chem. Fundam. 6, 276-288. Mann, U. and Rubinovitch, M., 1981, Characterization and analysis of continuous recycle systems: II. Cascade. A.I.Ch.E.J. 27, 829-836. Mann, U., Rubinovitch, M. and Crosby, E. J., 1979, Characterization and analysis of continuous recycle systems: I. Single unit. A.I.Ch.E.J. 25, 873-882. Naor, P. and Shinnar, R., 1963, Representation and evaluation of residence time distributions. Ind. Engng Chem. Fundam. 2, 278-286.
CES 50:2-E
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Nauman, E. B., 1969, Residence time distribution theory for unsteady stirred tank reactors. Chem. Enong Sci. 24, 1461-1470. Nauman, E. B., 1970, Maintenance of product quality during transients in stirred tank reactors. Chem. Engng Sci. 25, 1595-1603. Nauman, E. B., 1981, Residence time distributions in systems governed by the dispersion equation. Chem. Engn# Sci. 36, 957-966. Nauman, E. B. and Buflham, B. A., 1983, Mixing in Continuous Flow Systems, pp. 3-52. Wiley, New York. Rubbinovitch, M. and Mann, U., 1983a, Single-particle approach for analyzing flow systems: I. Visits to flow regions. A.I.Ch.E.J. 29, 658-662. Rubinovitch, M. and Mann, U., 1983b, Single-particle approach for analyzing flow systems: II. Regional residence times and local flow rates. A.I.Ch.E.J. 29, 663-668. Rubinovitch, M. and Mann, U., 1985, A single-particle approach for analyzing flow systems: IlL Multiple fluids. A.I.Ch.E.d. 31, 615-620. Seinfeld, J. H. and Lapidus, L., 1974, Mathematical Methods in Chemical Engineering, Vol. 3, Process Modelino, Estimation, and Identification, pp. 197-288. Prentice-Hall, Englewood Cliffs, NJ. Shinnar, R. and Naor, P., 1967, Residence time distributions in systems with internal reflux. Chem. Enong Sci. 22, 1369-1381. van Kampen, N. G., 1981, Stochastic Processes in Physics and Chemistry, pp. 76-283 North-Holland, Amsterdam. Wen, C. Y. and Fan, L. T., 1975, Models for Flow Systems and Chemical Reactors, pp. 9-50. Marcel Dekker, New York.
Chemical Enoineerin0 Science, Vol. 50, No. 2, pp. 211 221, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
001D-2509(94)00224-X
STOCHASTIC MODELING OF TRANSIENT RESIDENCE-TIME DISTRIBUTIONS DURING START-UP L. T. FAN, t B. C. SHEN* and S. T. C H O U Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, KS 66506, U.S.A. (Received 20 January 1994; accepted in revised form 6 July 1994)
Abstract--The theory of residence-time distribution, RTD theory in short, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. The creation of the modern, systematic RTD theory has been attributed to Danckwerts. As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residence-time distribution. While Danckwerts steered the development of the RTD theory essentially along the path of deterministic physics, obviously, the description of RTD is better couched in the statistical or stochastic parlance. Stochastic modeling visualizes the fluid in a flow system as being composed of discrete entities. This visualization reveals a greater insight into the underlying mechanism than deterministic modeling, thereby facilitating our understanding of the flow and mixing characteristic of the system. In the present work, an attempt has been made to derive a unified mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Specifically, the expressions for RTDs of molecules, fluid particles or any flowing entities passing through continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions.
I. INTRODUCTION The theory of residence-time distribution (RTD theory), one of the theories unique to chemical engineering, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. Moreover, the utility of different aspects of the RTD theory has long been recognized in fields as diverse as physiology, pharmacology, chemistry, and civil engineering. Nevertheless, the creation of the modern, systematic RTD theory has been attributed, probably rightly so, to Danckwerts (1953). As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residencetime distribution. Danckwerts, however, steered the development of the RTD theory essentially along the path of deterministic physics. Very likely, he was sensitive to the fact that the mathematics of stochastic processes had not been included in the repertoire of the chemical engineering profession, and, thus, the acceptance of the RTD theory by the chemical engineering community would be inhibited if the theory were couched purely in the parlance of stochastic processes. It appears that this has hindered the establishment of a firm or self-consistent foundation for the RTD theory based on the first principles.
tAuthor to whom correspondence should be addressed. *Present address: Research Department, Miles Inc., New Martinsville, WV26155, U.S.A.
Efforts to remedy this situation by resorting to stochastic processes have made substantial headway in recent years. Some earlier efforts are those by Shinnar and his coworkers. Naor and Shinnar (1963) have introduced the concept of intensity function. By viewing the turbulent chemical reactors as a network of stirred tanks and taking the interstage flows to be stationary Markov processes, Krambeck et al. (1967) have developed the stochastic models of RTDs of turbulent flow and mixing systems based on the random walk assumption. Adoption of a broader class of Markov processes, birth~leath processes, has enabled Shinnar and Naor (1967) to propose a more general method of calculating the residence-time distributions for systems with internal reflux. By introducing the concept of joint probability distribution for the number of recycles and the residence time, the total regional residence-time distribution for a continuous recycle system involving either a single unit or a cascade has been analyzed and characterized (Mann et al., 1979; Mann and Rubinovitch, 1981). Rubinovitch and Mann (1983a, b, 1985) have presented a Markov chain model for analyzing particulate processes. Nauman (1981) has applied the theories of discrete and continuous random walks to the open systems governed by the dispersion equation. Fan and his colleagues have treated the arbitrary complex networks of stirred tanks by a general continuous-time compartmental model (Fan et al., 1982, 1985) and by the master equation approach (Fox and Fan, 1984). What differentiates these works from the conventional deterministic approach is the definition of ran211
212
L. T. FAN et al.
dom variables characterizing the random or fluctuating nature of the flow and mixing of molecules or fluid particles, and the subsequent applications of the theories of probability and stochastic processes. Research on the RTD theory has mainly focused on steady-state systems, even though some attempts have been made to develop a theory of the unsteady-state RTD. Cha and Fan (1963) have considered the transient age distributions of a completely mixed tank or tanks-in-series with a pulsating feed having an arbitrary entrance-age distribution. In his work on the RTD of a system with unsteady hydrodynamics, Nauman (1969) has systematically defined the system's inlet life-expectation and outlet residence-time distribution functions. He has further pointed out that these functions are generally not identical under unsteady-state flow conditions. Based on these conceptual developments, Nauman (1969) has developed the RTD theory for an unsteady-state stirred-tank reactor. This theory has been generalized by Chen (1971) to the unsteady-state flow reactors of arbitrary geometric shape through the population balance approach. In an attempt to achieve the optimal constant RTD control, Nauman (1970) has shown that a system of stirred-tank reactors can be subjected to changes in working volume and throughput while maintaining a stationary residence-time distribution. Nevertheless, these approaches are essentially deterministic. Fan et al. (1979) have developed a stochastic model of the unsteady-state RTD; its applications can be found in Fahidy (1987). The objective of the present work is to derive a mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Parallel to Nauman's (1969, 1981) conceptual development cast in deterministic parlance, the unsteady-state age distributions of fluid particles are introduced in the present work as the probabilistic functions. Moreover, the expressions for RTDs of molecules, fluid particles or any flowing entities in continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions; some of these expressions have been unavailable hitherto. 2. PRELIMINARIES
A flow system with one entrance and one exit is considered to contain an incompressible fluid with a finite volume of V. The fluid in the system is visualized in the present work as being composed of N discrete fluid particles that are conserved. These particles can be atoms, molecules, Brownian particles or any other fluid elements [see e.g. Nauman and Buffham (1983)]; nevertheless, by analogy, any conserved flowing objects or entities such as solid particles, liquid droplets, and gas bubbles passing through a flow system can be regarded as fluid particles in analyzing their RTDs. The inflow fluid enters the system through the entrance and ultimately departs from the
exit. In other words, none of the fluid particles will remain permanently in the system. The volumetric flow rate of the inflow fluid, v, corresponds to a flow of n discrete fluid particles per unit time. Moreover, the inflow fluid particles are assumed to be fresh at the moment when they enter the system, i.e. their ages are all zero initially and increase at a rate equal to the time spent in the system. The N fluid particles originally residing in the system are the inventory fluid particles. After the onset of the process, these inventory fluid particles are continually replaced by the inflow fluid particles entering the system at a constant rate of n fluid particles per unit time. Hence, the compositions of both the inventory and inflow fluid particles in the system vary as functions of time even though the overall mean flow rate of the outgoing fluid particles, consisting of both inflow and inventory fluid particles, remains constant at n. Obviously, the number of inventory fluid particles will continually decrease, and that of inflow fluid particles will continually increase inside the system. Eventually, the inventory fluid particles will be totally replaced by the inflow fluid particles. At this moment, the age distribution in the flow system, comprising N inflow fluid particles, attains a steady state. Note that the initial age distribution of the inventory fluid particles in the system affects its unsteady-state age distribution. For simplicity, however, only the situation where the initial ages of the inventory fluid particles are uniformly zero is considered at the outSet.
Let t denote the time elapsed from the onset of the process, i.e. the time of the process, in brief; and ,7, the time when a fluid particle enters the system. The inventory particles can be considered as having entered the system instantaneously at time v/of 0 [Fig. l(a)], while the inflow particles as having entered the system during 0 < ,7 ~< t [Fig. 1(b)]. Normally, the life expectation of an inventory or inflow fluid particle entering the flow system at time ~/, i.e. the probable duration before the particle's exit from the system, cannot be exactly predicted or determined due to the
I
t
I
o ,1=0
t
t4-At
a. Inventory particles entering the system at q=O.
,I
t-n
-I
-I
I
I
t
0
q
t
b. Inflow particles entering the system during O
Fig. 1. Time scales of the process.
Stochastic modeling of transient RTD
213
complexity of the flow pattern and behavior in the system. Thus, the possible value of the particle's life expectation entering the system at time r/ is better represented by a continuous random variable, denoted by O(r/). The definition of the random variable, ®(t/), is twofold [see e.g. van Kampen (1981)]. The first is the set of possible values or realizations that ®(t/) can assume, i.e. the sample space, which spans [0, oo ). The second is the probability distribution over this sample space represented by the following cumulative distribution function:
ticles may exit from the system at any subsequent moment. The random variable representing the life expectation of an inflow particle entering the system at time )/is ®(0 < q ~< t), denoted hereafter by ®2(r/) for convenience. Again from eq. (la), its cumulative distribution function is
Fo~(O) - Pr [t0(r/) ~< 0]
Fo2~.~(O) is also customarily called the inlet life-expec-
= Pr [a fluid particle entering the system at time r/has a life expectation of 0 or less, i.e. it will exit from the system prior to or at time (q + 0)] (la) or
1 - Fo~(O) =- Pr [to(r/) > 0] = Pr [a fluid particle entering the system at time ~/will remain there at time (r/+ 0), i.e. it will exit from the system after time (r/+ 0)]. (lb) The cumulative distribution functions of inventory and inflow fluid particles will be treated separately in what follows. For convenience, fluid particles will henceforth be termed particles. In other words, inflow fluid particles will be termed inflow particles; inventory fluid particles, inventory particles; and outflow fluid particles, outflow particles. 2.1. Inventory particles The inventory particles are the ones residing in the system at the onset of the process. They may be considered as entering the system instantaneously at r / = 0, as mentioned earlier. Hence, the random variable representing the life expectation of an inventory particle can be written as O ( r / = 0). For convenience, O ( r / = 0) is henceforth written as ®1. From eq. (la), the cumulative distribution function of O1 is
Fo,(O) = Pr(®l ~< 0) = Pr (an inventory particle has a life expectation of 0 or less).
(2)
The probability density function of O1, fo,(O), is defined as d Fo, (0) fo,(O) = dO (3) Naturally, fo,(0)A0 represents the probability that an inventory particle has a life expectation in interval (0, 0 + A0), or, in terms of deterministic parlance, the fraction of the inventory particles exiting from the system during time interval (0, 0 + A0). 2.2. Inflow particles After the onset of the process, the inflow particles enter the system continuously and any of these par-
Fo~.)(O) = Pr [02(r/) ~< 0] = Pr (an inflow particle entering the system at time r/has a life expectation of 0 or less). (4) tation distribution. The probability density function of O2(r/), fo2(.)(0), is defined as
f°2~")(O) --
dFo2tnl(0) dO
(5)
Clearly, fo2t,~(O)AO represents the probability that an inflow particle entering the system at time r/has a life expectation in interval (0, 0 + A0) or, in terms of deterministic parlance, the fraction of the inflow particles entering the system at time r/has life expectations in interval (0, 0 + A0). In general, Fo:t~)(O) is a function of 0 as well as r/prior to the establishment of a steady state in the system. Nevertheless, more often than not, a hydrodynamic or flow-pattern steady state is essentially established instantaneously regardless of when the inflow particles enter the system, i.e. r/; this is especially the case if the fluid particles are incompressible and the physical and transport properties, such as the sizes, densities and viscosities, of the inflow and inventory particles are nearly identical. Under this situation, the life-expectation distribution of the inflow particles will remain invariant until the corresponding compositional steady state is established; therefore,
Fo2~n~(O) = Fo2(O)
(6)
fo~t~)(O) = fo~(0).
(7)
As will be delineated in the succeeding section, Fo2(O) is equal to the cumulative residence-time distribution function at the exit of the system under steady-state conditions, F(0); andfo2(0), the residence-time distribution function at the exit of the system under steady-state conditions, E(0), i.e.
Fo2(O) = F(O)
(8)
f02(O) = E(O).
(9)
It is worth mentioning that F(O) and E(O), and consequently Fo2(O) and fo2(O), can be determined experimentally by a tracer technique [see e.g. Danckwerts (1953) and Wen and Fan (1975)]. 3. STOCHASTICPOPULATION BALANCE Under hydrodynamic steady-state conditions described in the preceding paragraph, the total number of particles in the system and the rate of inflow or outflow in terms of the number of particles will be invariant at N and n, respectively, throughout the
L. T. FAN et al.
214
process. The conservation of these numbers gives rise to the stochastic population balances governing the compositional variations of the inventory and inflow particles.
Dividing both sides of this expression by nat, taking the limit as At---}0, and substituting eqs (3), (5), (7), and (13) into the resultant expression yield
The total number of particles in the system at time t, N, comprises the contributions from the following. (i) Inventory particles remaining in the system at time t. The expected number of such particles is NIl
- Fa,(t)].
(I0)
(ii) Inflow particles entering the system in time interval (0, t) and remaining there at time t. The expected number of such particles is n
]" o
[1 - Fo2(t - r/)] dr/.
(11)
Since the total number of particles is conserved as stated,
f,
N=N[1--fe,(t)]+n
[1-Fo2(t-r/)]dr/.
(12)
do
The mean residence time, z, is N ~ =
-.
(13)
n
Hence, eq. (12) becomes Fol(t) = ~lfo~ [1 - Fo~(t
-
r/)]
dr/.
(14)
This is equivalent to the population balance equation of the particles in the system. 3.2. Population balance around the exit of the system Since the number flow rate, n, remains invariant throughout the process, the total number of particles exiting from the system in time interval (t, t + At) is nat; it consists of the contributions from the following. (i) Inventory particles exiting from the system in time interval (t, t + At). The expected number of such particles is N[Fo~(t + At) - Fo,(t)].
(15)
(ii) Inflow particles entering the system in time interval (0, t + At) and exiting from the system in time interval (t, t + At). The expected number of such particles is n
U
Fo2(t + At - rl) dr~ - n
0
f
2(t - r/) dr/= 1.
zfo,(t) +
3.1. Population balance over the system
Fo2(t - r/) dr/.
(18)
This expression is equivalent to the population balance equation of particles around the exit of the system. 3.3. Internal age distribution Let l(OI t)AO be the probability of a particle, either inventory or inflow, inside the system to have an internal age, i.e. the time spent by the particle inside the system since its entrance, between 0 and (0 + A0) at time t. In other words, l(Olt) is the probability density function of the internal age. In conventional or deterministic parlance, I(Olt)AO is the number fraction of particles inside the system, each having an age between 0 and (0 + A0), at time t; thus, I(OI t) is often termed the internal age distribution function of the particles based on the number or, simply, the internal age distribution. If the sizes and densities of all particles can be regarded as identical, I(Olt) based on the number will be the same as that based on the mass. At any time t, the number of particles in the system having ages between 0 and (0 + A0), Nl(OIt)AO, is composed of the following two contributions. (i) Inventory particles in the system having ages between 0 and (0 + A0). The expected number of such particles is NIl - Foj(t)] {U(t - O) - U [ t - (0 + A0)]} (19) where U is the unit step function defined by 0,
U(t-O)=
1,
t< 0
t>~O.
(20)
(ii) Inflow particles in the system having ages between 0 and (0 + A0). The expected number of such particles is f0n[1 -- Fo2(t _ r/)] × {U[(t
- r/) - O] -
U[(t
-
CO + AO)]}
r/) -
d~.
(21) The expected number of particles in the system with ages between 0 and (0 + A0) at time t is the sum of the expected numbers of the inventory and inflow particles with such ages; therefore,
dO
(16)
NI(OIt)AO
= N I l - Fol(t)] {U(t - 0 ) - U[t - (0 + A0)]}
Thus, nat = N [ F o , ( t + At) - Fo,(t)]
]
+
~ t +A!
+ n
f'
n [1 -- Fe2(t -- r/)]
0
Fo~(t + At - r/)dr/
x {u[(t
O
- - n f o t Fo~(t - -
q)dr/.
(17)
-
r/) -
o]
- (0 + A0)] } dq.
-
u[(t
-
,I)
(22)
Dividing both sides of equation by NAO, taking the
Stochastic modeling of transient RTD continuous limit, and noting that u ( t - O) - u [ t - (0 + A0)]
3(t - 0) = lim ~o~o
A0
(23)
where Ne is the expected number of fresh particles entering the system in a small time interval of (r/, r/+ At/) and exiting from the system in time interval (t, t + At), q ~< t, i.e.
where 3 is the unit impulse function defined as 6(t - 0) = ~0, ( oo,
t :# 0
Ne = n A q [ F o 2 ( t + A t - q) - Fo~(t - r/)].
(24a)
t=O
and o~b(t
0) dt = 1
215
(24b)
(28b)
The expected number of particles exiting from the system with ages between 0 and (0 + A0) in time interval (t, t + At) is the sum of the expected numbers of inventory and inflow particles having ages between 0 and (0 + A0) given in (i) and (ii); hence, E[Z I(t, At}]
yield the general internal age distribution of particles in the system at time t:
1
~
{U(Ot~ - O)
i=t
-
HOlt) = [1 - Fo~(t)]f(t - O)
+ - [1 -- Fo2(0)] U(t - O)
nAtE(OIt)AO =
u[01~
-
(0 +
A0)]} +
U(02,
-
O)
i=
(25)
U[02,-
T
where z = N / n and U is the unit step function indicating that the maximum possible internal age is t. The first and second terms on the right-hand side of this equation represent the contributions from the inventory and inflow particles, respectively.
(0 + A0)] }).
Dividing both sides of this equation by At and A0, taking the continuous limits as both At --. 0, At/~ 0, and A0 ~ 0 lead to the expression for the general exit-age distribution function, E(Olt) = z fol(t) 6(t - O) + fo2(O)U(t - 0).
3.4. Exit-age or residence-time distribution Let E(OIt)AO be the probability of a particle, either inventory or inflow, at the exit of the system to have an age between 0 and (0 + A0) at time t, i.e. E(Olt) is the probability density function of the exit age of this particle. In conventional or deterministic parlance, E(OIt)AO is the number fraction of particles at the exit, each having an age between 0 and (0 + A0) at time t; thus, E(Olt) is usually termed the exit-age or residence-time distribution function of the particles. The number of particles exiting from the system in time interval (t, t + At) having exit ages between 0 and (0 + A0), n(At)E(Olt)AO, is composed of the following two contributions. (i) Inventory particles exiting from the system and having ages between 0 and (0 + A0). From eq. (15), the expected number of inventory particles exiting from the system in time interval (t, t + At), E [Zl(t, At)], is E [ Z a ( t , At)] = S [ F o ~ ( t + A t ) - Fo~(t)]
(26)
of which E [ Z i(t, At}]
{U(O~,-O)-
U [ O ~ , - ( O + AO)]}
(27)
i-I
have ages between 0 and (0 + A0). In the above expression, 0~i is the age of inventory particle i, i = 1, 2..... E [Z t (t, At)]. Since all inventory particles exiting from the system in time interval (t, t + At) will have ages between t and (t + At), we have t <~ OH < t + At, i = 1,2 ..... E [ Z 1 ( t , At)]. (ii) Inflow particles exiting from the system and having ages between 0 and (0 + A0). The expected number of such particles is U(02i - O) - U[02~ - (0 + A0)] i=
(28a)
(29)
(30)
Substituting eq. (19) into this expression gives rise to E(Olt) = [1 - Fo~(t)]3(t - O) + fo2(O)U(t - 0).
(31) 3.5. Steady-state aoe distributions When time t approaches infinity, 6(t - O) ~ 0 and U(t - O) -~ 1; hence, eqs (25) and (31) reduce, respectively, to l ( O l t - - , oo) = I(0) = _1[1 - Fe~(0)]
(32)
E(Olt ~ ~ ) = E(O) =fo2(0).
(33)
and
The former is the expression for the steady-state internal-age distribution function in terms of Fe2(0); ~ind the latter, the steady-state residence-time distribution function in terms of fo2(O). Either is solely in a function of Fo2(O) or fo~(0) since only the inflow particles remain inside the system at a steady state. Obviously from eq. (33), the steady-state, residence-time distribution, E(O), is identical to the probability density function for the life expectation of an inflow particle, fo2(O). Consequently, the corresponding cumulative residence-time distribution function at the steady state, F(O), is F(O) = Fo2(O).
(34)
This expression verifies the validity of eq. (9). Substituting eq. (34) back into eq. (32) yields I(0) = _1[1 -- F(0)].
(35)
T
This is the celebrated result given by Danckwerts (1953).
216
L. T. FAN et al.
3.6. Expressions of age distributions in terms of steady-state counterparts The expressions of l(OIt) and E(Olt) can be written in terms of their steady-state counterparts, I(O) and E(0), respectively. Substituting eqs (32) and (14) into eq. (25) gives rise to the general internal age distribution in terms of I(O) as shown below:
I(Olt) =
(f7)
I(z) dz 6(t - O) + I(0) U(t - 0). (36)
Combining eqs (33) and (34) with eq. (31) leads to the following exit-age or residence-time distribution in terms of E(0): e(olt) =
q7 ) E(~) d ,
a(t-O)+E(O)U(t-O). (37)
These two equations are identical to the expressions derived on the basis of the physical insight [see e.g. Cha and Fan (1963) and Wen and Fan (1975)]. The significance of eq. (36) and that of eq. (37) are illustrated in Figs 2 and 3, respectively. The second term on the right-hand side of eq. (36) or (37) represents, respectively, the steady-state internal or exit-age
l(o~)
impulse [area =
I;l(o)a(o)
Jtl(O)dO ]
7 " 2- 7- .._.
0 o
0
Fig. 2. Illustration of the general internal age distribution at time t.
E(alt)
~timpubse[area = J t E ( O ) d O
]
E(O)dO
IK// 7"/. t
o
Fig. 3. Illustration of the general residence-time distribution at time t.
distribution function generated by the inflow particles in the system having ages less than t. The first term on the right-hand side signifies the contribution from the inventory particles residing in the system at the outset of the process. Note that since all inventory particles have the age of t, their fractional amount can be represented by the fractional impulse function with a magnitude of St~ I(0) dO or S[ E(O) dO that is equal to the cross-hatched area in Figs 2 or 3, respectively. This area is actually the fraction of fresh particles that would have internal or exit ages greater than t if the distribution were of the steady state. When t approaches infinity, all the inventory particles inside the system will eventually be replaced by the inflow particles. As a result, the impulse term vanishes, and only the second term remains. In other words, l(Olt) or E(Olt) reduces to the steady-state, internal-age distribution function, I(0), or the steady-state, exit-age distribution function, E(O), as t ~ oo.
4. M A R K O V
PROCESSES
l(Olt) and E(Olt) are expressed in terms of the steady-state counterparts, I(0) in eq. (36) and E(O) in eq. (37), respectively. I(0) and E(O) can be either experimentally determined by tagging the particles by a tracer or stochastically modeled by visualizing passage of a single particle through the system. For simplicity, we assume that the stochastic process of a particle through the system is Markovian, i.e. only the last state occupied by the particle is relevant in determining its future behavior [see e.g. Chiang (1980)]. In other words, the probability of making a transition to each state of the process depends only on the state presently occupied by the particle. Various Markov processes, e.g. pure-death and birth~leath processes, are available, which may be suitable for different flow systems or models. For example, the particle passage through a single vessel or a tanks-in-series system may be modeled by the pure-death processes, as will be delineated in the succeeding two subsections, respectively. 4.1. Pure-death processes in a single vessel The flow of particles through a single vessel with arbitrary geometry involves continual entrance of the inflow particles into the vessel and their subsequent continual exit from it. In the parlance of stochastic processes, this process can be regarded as a birth~leath process; the birth is signified by the entrance of a particle, and the death, by the exit of a particle [see e.g. Seinfield and Lapidus (1974)]. Nevertheles, the former can be treated as a nonrandom or deterministic event since the rate of entrance of the inflow particles is usually predetermined, specified or well-controlled. Only the latter remains as a random or stochastic event, thereby giving rise to a pure-death process [see e.g. Fan et al. (1979)]. 4.1.1. Steady-state age distributions Now, we are to derive I(0) and E(O) of the steady-state flow
Stochastic modeling of transient RTD through a single vessel by identifying it as a puredeath process. Unlike the transient state considered in some of the previous sections, in which both the inventory and inflow particles are involved and consequently the population balance over the entire system has to be considered in deriving the general transient age distributions, only the inflow particles are involved in the process. The probability of a randomly behaving inflow particle to have a life expectation longer than 0, i.e. to remain in the system at age 0, is [1 - Fo2(0)]. The probability that it will exit from the system, in the succeeding age interval, (0, 0 + A0), is denoted as [#2(0)A0 + o(A0)], where/~2(0) is the exit intensity of this inflow particle [see e.g., Naor and Shinnar (1963)-I, signifying the propensity of exit during a unit time interval. Thus, the probability for this particle to exit, Fo2(O), is governed by the following equation: dFo2(O) dO
[1 --
Fo2(0)]112(0).
217
4.1.3. Two ideal cases. Depending on the hydrodynamic characteristics or flow conditions of the system, various representations or models can be considered for #2(0). Visualized below are two idealized systems. Plugflow system. For this system, the particles exiting from the system during the start-up are all inventory particles initially residing in the system, i.e. residing in the system at t = 0. At t = z, all the inventory particles will exit from the system; eventually, a steady state is attained. At this moment, all the inflow particles at the exit of the system will have the identical age or residence time of z. Hence, the exit or replacement intensity of an inflow particle,/t2(0), is ~2(0) =
{O'ooO<~z , 0>'~.
(45)
Substituting eq. (45) into eqs (43) and (44) results, respectively, in
(38)
Solution of this equation subject to the initial condition of Fo2(0) = 0 is
Fo2(O)= l - e x p [ -
ffu2(q)d~l].
t>T
(39)
(40)
(46)
)'6(t - 0),
t~
E(Olt) = (6(z - O) C(t - 0),
t>Z.
(47)
Substituting this expression, in turn, into eqs (32) and (33) leads, respectively, to
As t--, oo, eq. (46) reduces naturally to the wellknown uniform internal age distribution function at the steady state [see e.g. Danckwerts (1953)],
l(O)= ~ e x p I - f]m(q)dq]
(41)
I(0) = - U(z - 0).
E(O)= l~2(O)exp[ - f~#2(q)d~l].
(42)
1
These expressions are identical to the published results (Naor and Shinnar, 1963). 4.1.2. Transient age distributions. Substituting eqs (41) and (42) into eqs (36) and (37) yields, respectively, the unsteady-state or transient internal-age distribution, l(OIt), and the transient exit-age distribution, E(O[t), of particles flowing through a single vessel; they are given below:
Completely stirred tank system. For this system, the chance of an inflow particle inside the system exiting from it is independent of its history [see e.g. Danckwerts (1953)], i.e. #2(0) is independent of age 0 and equal to n/N; hence, we can write 1
#2(0) =/~2 = ;t = - . T
;- ] ~2(~)dt; U(t - O)
(49)
Substituting this expression into eqs (43) and (44), respectively, yields
HOlt) = exp( - t/z)6(t - O) + -1exp ( - 0/~) U(t - O)
,(Olt)=(!ff{exp[-fol~2(q)dq]})6(t-O) + -' e x p[
(48)
T
(50) 1
(43)
E(OIt) = exp( - t/z)fi(t - O) + - e x p ( - O/z) U(t - 0). T
(51) When a steady state is reached, the above expressions, in turn, yield +
o,.
(44) Note that eqs (43) and (44) are in accord with the available results (Fan et al., 1979).
E(0) = !exp ( - !).
(53)
L.T. FAN et al.
218
These are the celebrated completely random internal and exit-age distributions of Danckwerts (1953). 4.2. Pure-death processes in a tanks-in-series system A flow system composed of m well-mixed tanks-inseries has frequently served as a model for flow accompanied by axial dispersion of the particles. In this system, an inflow particle always enters the first tank, followed by its passage through the succeeding ( m - 1) tanks and eventually by its exit from the system; moreover, a particle in any tank will never return to the preceding tank. This is analogous to considering that no circulation of the fluid takes place between any pair of adjacent tanks in the conventional treatment of the subject.
1
E(0)
-
-
(m
-
(mO~'-'
1
= - e x p ( -- mO/z) ~
1)!
4.2.2. Transient age distributions. Substituting eqs (58) and (59) into eqs (36) and (37), respectively, leads to
I(Olt) =
exp( -- mt/'r)i:l ~ ~=ofl.= • +[~exp(-mO/r)~
¢5(t--0)
l
(mO~i-1 j
× U(t - O) 4.2.1. Steady-state age distributions. Under a steady-state condition, only the inflow particles are involved in the process that is no longer a function of t. Let #: be the transition intensity of a single inflow particle under such a condition; then,
E(Olt) = / e x p ( - rot~z)
(60) 1
/'rot\i-I]
o0.,] x U(t - 0).
]A2A0 + o(A0)
= Pr [an inflow particle in tank Si, i = 1, 2..... m, prior to age 0 will be in tank Si+, prior to age (0 + A0)]. (54) Note that #2 of the inflow particle is independent of its age, 0, since each tank is well mixed as in the completely stirred tank system. On the average, each tank contains N/m particles and the number flow rate through it is n particles per unit time; therefore, the transition intensity, #2, can be identified to be n
#2 = N/m
nm
= --.
(55)
N
Let P~(O) be the probability that an inflow particle initially entering tank 1 will be in tank i, i = 1, 2. . . . . m, at age 0. Then, ( # 2 0 ) ~- 1
Pl.i(O)=--exp(-#20), (i-
1)r
i = 1 , 2 . . . . . m. (56)
Hence, the probability that the inflow particle will exit from the system after residing in it for a duration of 0 is
Fo2(O) = 1 - ~ i=l
Pl,i(O) (#20)'-'
i=, ~ !
e x p ( - - #20).
(57)
By substituting this expression and eq. (55) into eqs. (32) and (33) and noting that z = N/n, we obtain the following well-known, steady-state age distribution functions for the tanks-in-series system [see e.g. Wen and Fan (1975) and Nauman and Buffham (1983)]:
I(0)=! Z Pl,i(O) ~i=l
(58)
(61)
When m ~ ~ , eqs (60) and (61) become eqs (46) and (47), respectively, i.e. the distributions for the plug flow. When m = 1, eqs (60) and (61) reduce to eqs (50) and (51), respectively, i.e. the distributions for the completely stirred tank. 5. DISCUSSION
In the previous work, only a single subclass of discrete-state Markov processes, the pure-death process, has been exploited in mathematically deriving the transient RTDs of relatively simple flow systems. There is no reason, however, that other named Markov processes are incapable of analytically or semianalytically yielding the expressions for the transient RTDs of more complex flow systems. Such Markov processes include the renewal processes, Poisson processes, pure-birth processes, birth-death processes, and queuing processes in the category of the discretestate processes and Liouville processes, Wiener processes, and Ornstein-Uhlenbeck processes in the category of the continuous-state processes. In fact, the steady-state RTD of the tanks-in-series system with recirculation or backmixing between any two adjacent tanks has been derived by regarding the particle behavior in the system as a birth-death process [see e.g. Fan et al. (1982)]. 5.1. Moments and fluctuations The mean residence time or first moment of the RTD of particles at any time t, zm(t), is
rm(t) = ~oOE(OI t) dO.
(62)
Substituting eq. (37) into the above expression gives rise to Zm(t) =
OE(O)dO + t
E(O)dO.
(63)
Stochastic modeling of transient RTD
for realizing the transient' internal-age and residencetime distributions of a flow system, I(01 t) and E(OI t); it is as follows:
A s t --* o o , z,~® = z =
l i m ~m(t) t~3
= foOE(O) dO.
(64)
Comparison of eqs (63) and (64) indicates zm(t) ~< z.
(65)
This result implies that the mean residence time during the start-up does not exceed that at the steady state. In general, the kth central moment of the residence-time distribution, Mk(t), is defined as
Mk(t) = fo[O - #(t)]kE(OIt)dO.
(66)
By substituting eq. (37) into this expression, we have
Mk(t) =
219
S
[0 -- ~(t)]kE(O)dO + [t - ~(t)] k
E(O)dO. (67)
The second-order central moment is the variance, denoted by tr2; the third-order central moment is the skewness, denoted by ),a; and so on. For a continuous fluid-flow system with overall steady-state flow pattern, the magnitude of RTD's fluctuations due to the random movement of the fluid particles inside the system or exiting from it at any moment is often negligible since it is of order N~, 1/2, where Nm is the number of fluid particles or molecules inside the system or that exiting from the system at time moment; the magnitude of Nm is of the order of the Avogadro number under relatively low temperatures and normal or high pressures [see e.g. van Kampen (1981)-]. In other words, the shapes of the RTDs measured in various experimental runs under the same macroscopic flow condition should be almost identical. Nevertheless, for various flow systems, the sizes of entities passing through them, such as relatively large solid particles, liquid droplets or gas bubbles, are rather large. Thus, the number of the entities in any of such systems is far less than N,, and the instantaneous flow rate of the flowing entities or particles always fluctuates around its mean value even though the mean value may be a constant with respect to process time. Consequently, the shapes of the RTDs obtained in various experimental runs under the same macroscopic flow condition tend to be appreciably different. In other words, fluctuations of any RTD due to the inherent random nature of the flow system, specifically, the random variations of small populations of the finite-size particles inside and at the exit of the system, can be substantial. It is imperative, therefore, that such fluctuations be stochastically analyzed or modeled. 5.2. Procedure for evaluation and verification It should be of practical interest and of convenience to describe in a stepwise manner a possible procedure
(1) Ascertain if the flow system satisfies the conditions set forth in Section 2; specifically, ascertain if the transient stage of the process essentially involves only the compositional variations of the inventory and inflow particles with an overall steady-state flow pattern maintained. (2) Adopt the general expressions for l(OIt) and E(OIt) in terms of I(0) and E(O), respectively, derived in Section 3, specifically, eq. (36) for l(OIt) and eq. (37) for E(OIt). (3) Determine the steady-state age distributions, I(0) and E(O), through stochastic modeling as a Markov process. The procedure is given in Section 4. (4) Measure I(0) and E(O) experimentally by a tracer technique. The procedure is straightforward. The flow system under investigation may first be kept running until the compositional steady state is reached. At this stage, 1(0) and E(O) are measured by tagging the flowing entities or particles with a tracer [see e.g. Nauman and Buff'ham (1983)]. (5) Compare the mathematically derived and experimentally determined I(0) and E(O) to verify the stochastic models and to recover the parameters of the models. To evaluate these parameters, various moments or central moments of I(0) and E(O) derived from the modeling, e.g. mean residence time, variance, and skewness, may be compared with those obtained experimentally. (6) Substitute I(0) and E(O) obtained in the previous steps into eqs (36) and (37), respectively, thereby giving rise to the expressions for the transient internal-age distribution at time t, I(OIt), and the transient residence-time distribution, E(Olt). For a relatively simple flow system whose flow pattern is readily predictable, e.g. a flow vessel with a simple shape and intensive agitation, the stochastic modeling can be accomplished based only on the available empirical evidence for a given system configuration and under specific operation conditions. For a system in which the flow pattern is highly complicated so that it cannot be modeled analytically through the stochastic approach, I(Olt) and E(OIt) can be numerically determined by substituting the experimentally determined I(0) and E(O) directly into eqs (36) and (37), respectively, as mentioned earlier.
5.3. Chemical reactions By definition, the RTD theory and various expressions of the residence-time or age distributions derived from the theory of stochastic processes are applicable to systems, such as communities of living
220
L. T. FAN et al.
organisms, human societies, and continuous-flow process vessels, involving flow of discrete entities that age progressively. Nevertheless, the most prominent utility of the RTD theory and residence-time distributions is in the characterization of flow chemical reactors. As in the case of the steady-state RTD, the unsteady-state RTD can describe the characteristics of the macromixing in a flow system, i.e. the duration of a fluid particle's stay inside the system. The unsteadystate RTD, or even the steady-state RTD, however, cannot predict the exact pathway of a particle as it moves through the system. For a first-order reaction under isothermal conditions, the knowledge of the unsteady-state RTD is sufficient to determine the extent of the reaction at the exit of the system since the extent of the isothermal first-order reaction depends only on the time the particles spend in the vessel under an isothermal condition. For a nonlinear reaction, i.e. a non-first-order reaction or a nonisothermal firstorder reaction, however, knowledge of the unsteadystate RTD does not suffice because the extent of reaction depends not only on the particles' residence times, but also on their pathways through the system. In this case, supplementary information on the behavior of the particles in a local environment in the system, i.e micromixing, is indispensable [see e.g. Nauman (1969), Chen (1971) and Wen and Fan (1975)]. 6. C O N C L U D I N G REMARKS
The present work shows that the stochastic approach is capable of systematically and rigorously deriving various age or residence-time distributions of a flow system during unsteady-state operations and that the resultant expressions reduce to those of the well-known steady-state distributions. Moreover, the efficacy of the approach has been amply demonstrated by several examples. The theory of RTD and the explicit expressions of RTDs for some flow systems based on it have been derived in the present work for the transient stage of a process undergoing only compositional variations with the overall steady-state flow pattern. Nevertheless, the theory can be readily extended, by analogy, to the processes with variable flow rates.
E(O)
E(010 Fo, (0)
fo,(O) Fo~(~)(O)
NOTATION residence-time distribution function of the particles at the exit of the system under steady-state condition residence-time distribution of the particles at the exit of the system at time t probability that an inventory particle has a life expectation of 0 or less probability density function that an inventory particle has a life expectation of 0 or less probability that an inflow particle entering the system at time r/has a life expectation of 0 or less
fo2t.)(0)
Fo(~)(O)
F(O)
I(0)
l(OIt) m Mk(t ) n N Ne
t U V Z l(t, At)
probability density function that an inflow particle entering the system at time q has a life expectation of 0 or less probability that a fluid particle entering the system at time r/has a life expectation of 0 or less, i.e. it will exit from the system prior to or at time (q + 0) cumulative residence-time distribution function at the exit of the system under steady-state condition internal age distribution of the particles in the system under steady-state condition internal age distribution of the particles in the system at time t number of well-mixed tanks kth central moment of the residence-time distribution rate of inflow particles entering the system total number of particles in the system number of inflow particles entering the system in a small time interval of (q, q + A~/)and exiting from the system in (t, t + At) time elapsed from the onset of the process until step function volume of a flow system number of inventory particles exiting from the system in time interval (t, t + At)
Greek letters fi unit impulse function r/ time when a particle enters the system 0 realization of ®(q), ®l or ~)2(q) O(r/) random variable representing the life expectation of a particle entering the system at time r/ Ox random variable representing the life expectation of an inventory particle ®2(q) random variable representing the life expectation of an inflow particle entering the system at time r/ /~2(0) exit intensity of an inflow particle r mean residence time ( = N/n) ~..(t) mean residence time of particles at time t
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