Residence Time Theory - Industrial & Engineering ... - ACS Publications

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Residence Time Theory E. Bruce Nauman The Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180

The intellectual roots of residence time theory date back to 1908 and, thus, span the 100-year history of Industrial & Engineering Chemistry. The theory was created, developed, and extended by chemical engineers. It permeates chemical engineering in general and chemical reaction engineering in particular. It also has found widespread utility in the geosciences, environmental engineering, medicine, and biology. This paper provides an overview of the theory and gives some new results here and there. Introduction

Qinain ) Qina(0+) - DAc

Residence time theory and associated measurements are standard tools for understanding and analyzing flow systems. These concepts permeate chemical engineering and a host of other disciplines. The idea of residence time distribution (RTD) can be found in 6000 papers, and “residence time” by itself occurs in 40 000 papers. I am fortunate to have followed the development of residence time theory throughout much of its post-Danckwerts history. Even so, it would be foolhardy to attempt a review without carefully restricting the scope. This review covers the fundamentals and points to applications but does not attempt any meaningful survey of the vast experimental literature beyond showing that the concepts of residence time theory have been applied throughout the physical sciences. Avoided are numerous mathematical embellishmentssand sometimes still-ongoing argumentssthat may add rigor but make only minor contributions of a practical nature. Papers that ignore antecedents also are avoided, unless the new paper makes a significant additional contribution. With these restrictions, the review is of manageable length, although, undoubtedly, there are significant omissions. I apologize for them but note that the guidelines that I applied have excluded many of my own papers. The Beginnings Residence time theory began 100 years ago with the first nontrivial model of a chemical reactor. The axial dispersion model is normally attributed to Danckwerts1 and his pioneering paper that introduced residence time theory, but the model is actually due to Irving Langmuir.2 With minor changes in notation, Langmuir’s model is described as

uj

d2a da ) D 2 + RA dz dz

(1)

subject to the boundary conditions that * Tel.: 518-276-6726. Fax: [email protected].

518-382-2912. E-mail address:

[dadz]

0+

(2)

and

[dadz] ) 0 L

(3)

The reader will recognize that the boundary conditions apply to a closed system and are usually attributed to Danckwerts.1 Langmuir envisioned the reactor to have porous plugs at the inlet and outlet, which was a clever way to at least partially justify the boundary conditions. Langmuir’s model languished for 45 years. In the interim, reactor models generally assumed piston flow in a tubular reactor (a PFR), and perfect mixing in a continuous stirred tank reactor (CSTR) or batch reactor. Purging calculations for one or more stirred tanks in series were published by Ham and Coe,3 McMullin and Weber,4,5 and Kandiner.6 These papers calculated, for stirred tanks in series, what is now called the residence time washout function, which is the response of the system to a turn-off step change of a tracer. The early articles did not generalize the concept of RTD to nonideal reactors. The modern concept of residence time theory sprang newly born from the same paper where Danckwerts1 reintroduced the axial dispersion model. We briefly summarize the basic concepts of residence time theory without undue rigor or complications. Many simplifying assumptions are useful, although most of them will be relaxed at some point in this review. The starting assumptions are: (1) The flow system is at steady state. (2) There is a single inlet and a single outlet. (3) The inlet and outlet are closed, so that flow across the system boundaries is unidirectional. (4) The system is homogeneous (i.e., single-phase). (5) Inert tracer experiments can be performed on the system without disturbing the flow. (6) When the system is a reactor, it is isothermal. The concept of residence time applies to any conserved entity that enters a flow system. We suppose that the entity will

10.1021/ie071635a CCC: $40.75 © 2008 American Chemical Society Published on Web 04/23/2008

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eventually leave the system, never to return. The time that it spends inside the system boundaries then is considered to be its residence time (t). Typical conserved entities are molecules, portions of molecules, colloids, Brownian particles, or anything that maintains a recognizable identity as it flows through the system. We will call them particles and, occasionally, molecules. Except for the special case of piston flow, velocity profiles and molecular diffusion will cause a distribution of residence times. The distribution can be characterized in several ways, but the conceptually simplest manner is the washout function:

W(t) ) fraction of molecules leaving the system that remained in the system for a time greater than t (4) Consider a vessel where saltwater has been flowing for such a long time that the inlet and outlet concentrations have become equal. Normalize the salt concentration, so that cout(0) ) 1. At t ) 0, switch to freshwater and monitor cout(t). The result of this turn-off step change is the washout function: W(t) ) cout(t). Turn-off step changes of a tracer are relative easy to achieve, and the long-term response is known: cout(∞) ) 0. Thus, W(0) ) 1 and W(∞) ) 0. The cumulatiVe distribution function, which is defined as F(t) ) 1 - W(t), can be measured by a turn-on step change.

F(t) ) fraction of molecules leaving the system that remained in the system for a time less than t

(5)

The experimental disadvantage of a turn-on step change is that the long-term response may not be known in advance of the experiment. The differential distribution function is defined as

f(t) dt ) fraction of molecules leaving the system that remained in the system for a time between t and t + dt (6) This is sometimes called the residence time density function or the residence time frequency function, and because of Levenspiel’s popular book,7 it is sometime denoted as E(t). Regardless of name or notation, f(t) is the response of a closed-flow system to a Dirac delta function, δ(t), that is applied to the reactor inlet at time t ) 0. The delta function is also known as a unit impulse. The various distribution functions are related by

f(t) )

-dW dF ) dt dt

(7)

Central moments of the distribution are given by

µn )

∫0∞ tnf(t) dt ) n ∫0∞ t n-1W(t) dt

(8)

The zeroth moment is 1. This, and non-negativity, are the only restrictions on f(t). The restrictions on W(t) are that W(0) ) 1, W(∞) ) 1, and that W(t) is non-increasing. The first two moments of the RTD can usually be determined with some accuracy and are generally useful. The first moment is the mean residence time,

ht )

∫0∞ tf(t) dt ) ∫0∞ W(t) dt

(9)

Thus, ht can be found by integrating under an experimental washout function. It can also be found from measurements of the system inventory (Fˆ V) and steady-state throughput (FoutQout ) FinQin), because

Fˆ V ht ) F Q out out

(10)

which becomes just ht ) V/Q when the density is constant. Agreement of the ht values calculated by these two methods provides a good check on the experimental accuracy. Occasionally, an experimental washout function is used to determine an unknown volume or an unknown density. For a piston flow reactor (PFR),

W(t) ) 1

(for t < ht )

W(t) ) 0

(for t > ht )

and

f(t) ) δ(t - ht ) For a CSTR,

t W(t) ) exp - t h

( )

and

1 t f(t) ) t exp t h h

() ()

The space time is a semi-archaic measure of residence time. It is defined as

space time )

V Qin

and is equal to ht when the density is constant. The second moment of the RTD is usually replaced by the Variance, which is the second moment about the mean:

σt2 )

∫0∞ (t - ht )2f(t) dt ) -th2 + 2 ∫0∞ tW(t) dt

(11)

where the subscript t denotes a variance with units of (time)2. The variance is usually made dimensionless:

σ ) 2

σt2 2 ht

(12)

This gives σ2 ) 0 for a PFR and σ2 ) 1 for a CSTR. Welldesigned reactors in turbulent flow will have 0 < σ2 < 1, but poorly designed reactors and laminar flow reactors can have σ2 > 1. The theory, to this point, is that given in the classic paper by Danckwerts,1 and he went slightly further with his discussion. However, the next intellectual breakthrough was due to a subsequent paper by Danckwerts8 and a beautiful follow-up by Zwietering.9 Micromixing Consider a homogeneous, first-order reaction that is occurring in an isothermal reactor. The probability that a molecule will react is dependent only on the time that the molecule spends in the system: PR ) exp(-kt). Then, for a distribution of molecules,

P hR )

aout ) ain

∫0∞ exp(-kt)f(t) dt

(13)

where a denotes the concentration of the reactive component. The yield of a homogeneous, first-order reaction in an isothermal reactor is uniquely determined by the RTD. This result can be

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Figure 2. Zwietering’s examples of complete segregation and maximum mixedness.

Figure 1. Segregated and perfectly mixed continuous stirred tank reactors (CSTRs).

quite easily extended to reactions of order other than first order,

aout )

∫0∞ abatch(t)f(t) dt

(14)

but the result is no longer unique. For kinetics that are more complicated than first-order, the reaction probabilities are dependent not just on the time spent in the reactor but also on the interactions with other molecules. The concentration environment of a molecule that has been in a flow reactor for a time t may be different than what it would experience in a batch reactor at time t. Danckwerts8 defined a “point” as a collection of molecules that is small, compared to the system as a whole, but large enough to eliminate statistical variations in concentration when the molecules undergo reaction. We suppose that a point somehow has a traceable identity. The age of a point R is the time that has elapsed since the point entered the system, and this age is possibly different from the ages of the molecules inside the point at any instant. Danckwerts asked the following question: do molecules that enter together in a point stay together, or do they mix with molecules than entered at different times? One possibility is that there is no mixing, and the points are the entities to which the measured function f(t) applies. Such a system is said to be completely segregated. Equation 14 gives the flow-average exit concentration from a completely segregated reactor. Complete segregation is a theoretical possibility for any RTD, even the exponential distribution of a CSTR. This is illustrated in Figure 1a, where the circulating points have an exponential distribution of residence times and where there is no mixing between points. The other possibility is that some mixing can occur between points. If the mixing between points is quick and complete, the molecules have an exponential distribution of residence times, and the vessel is perfectly mixed, regardless of the RTD of the points. The question posed and answered by Zwietering9 was this: how much mixing can occur between points without altering the residence time distribution? A point in the system at some instant will have an age of R, a residual life of λ, and will eventually emerge with a residence time of t:

R+λ)t

(15)

Mixing between points that have different values of R is called micromixing. The extent of this mixing is limited for a fixed

value of f(t). The exponential RTD gives the broadest limits. It allows both zero micromixing and complete micromixing, the latter possibility corresponding to the usual idea of a perfectly mixed CSTR, as illustrated in Figure 1b. In principle, a CSTR can operate anywhere between the limits of zero mixing between points, provided that the points have an exponential distribution of residence times, and complete mixing between the points. The limiting case of complete mixing between points is possible only for a CSTR. A real reactor will have something less than complete mixing but something more than the limiting case of complete segregation. Zwietering9 defined the upper limit on molecular-level mixing that is possible with a given RTD. He called this limit maximum mixedness, and the yield associated with this limit is found by solving Zwietering’s differential equation:

f(λ) da [a - a(λ)] + RA ) 0 + dλ W(λ) in

(16)

The solution does not use the inlet concentration of reactant, ain, as a boundary condition. Instead, the usual boundary condition associated with eq 16 is

lim

λf∞

da )0 dλ

(17)

In this formulation, λ is the residual life of a molecule, although it can be treated as a dummy variable of integration. Solutions to eq 16 admit a number of special cases. The solutions can be rather subtle and even contrary to physical intuition. Consider, for example, two equal volume CSTRs in series. The greatest amount of mixing that is possible for this physical arrangement occurs when both CSTRs are perfectly mixed. Yet, the solution of eq 16 gives a different yieldslower for a second-order reactionsthan that for two perfectly mixed CSTRs in series. It is theoretically possible to build a reactor with the RTD corresponding to two CSTRs in series that has more micromixing than two perfectly mixed CSTRs in series. The reaction yields predicted using eqs 14 and 16 provide bounds on the yield possible with a given RTD for a broad class of reactions. Further discussion is given in the later section on the bounding theorem. Zwietering9 derived these equations from simple models of systems that could replicate an arbitrary RTD. The model in Figure 2a has side exits distributed along the length of a piston flow reactor (PFR). All the feed enters at the reactor inlet with age R ) 0 and leaves the reactor at various ages as needed to duplicate the RTD of the reactor being modeled. There is no mixing between points with different values of R, so the system is completely segregated. A

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Figure 4. (a) Axial dispersion models for (a) a closed system and (b) an open system.

Figure 3. Conversion limits for a second-order reaction with rate constant kthain.

component balance leads to eq 14. If nothing is known about a system other than its RTD, complete segregation is always possible and usually provides one limit on reactor performance. Zwietering’s model for maximum mixedness is shown in Figure 2b. The side exits have been replaced by side entrances. An arbitrary RTD can still be replicated by arranging the entrances, but the internal mixing state is now quite different from that in Figure 2a. Zwietering showed that the model in Figure 2b gave the maximum amount of micromixing possible with a given RTD. A component balance on the maximum mixedness reactor gives eq 16. If nothing is known about a system other than its RTD, maximum mixedness is always possible and usually provides one limit on reactor performance. More-detailed models of flow systems can have inherent levels of micromixing. Rippin10 showed that a PFR in a recycle loop is a maximum mixedness reactor, provided the recycle stream mixes completely with the incoming feed. A system of PFRs in parallel can model any RTD but is completely segregated. Convective diffusion models, such as the axial dispersion model, show behavior intermediate between complete segregation and maximum mixedness.11 Micromixing theory extends the number of ideal flow reactors to three: a piston flow reactor, a perfectly mixed CSTR, and a completely segregated CSTR. Figure 3 illustrates the yield behavior for these ideal reactors for a second-order reaction with RA ) -ka2. The difference between the perfectly mixed and completely segregated CSTRs is fairly small and has largely eluded efforts to measure it using premixed reactants. The perfectly mixed case is usually a good approximation, except for fast reactions with unmixed components. Bounding Theorems, Weak and Strong Calculations show that the yields predicted by eq 14 with f(t) ) (1/th) exp(t/th), e.g., for a second-order reaction, are different than what is calculated for a perfectly mixed CSTR. Thus, knowledge of the RTD does not allow prediction of reaction yields except for first-order reactions or the special case of a PFR. Zwietering9 showed that complete segregation and maximum mixedness were limits on micromixing, as he defined those terms. He suggested, but did not prove, that completely segregated and maximally mixed reactors provide conversion limits for single reactions. This assumption turns out to be untrue if RA(a) is a sufficiently complex function of the single

concentration a. Instead, restrictions on RA must be imposed. Chauhan, Bell, and Adler12 considered reaction rates that are either concave up (d2RA/da2 > 0) or concave down (d2RA/da2 < 0). They then attempted to prove that the states of complete segregation and maximum mixedness provide conversion limits. This conjecture has been called the strong bounding theorem. They were able to prove it, but only when mixing itself was restricted in nature. Specifically, they assumed that molecules with the same residual life λ cannot unmix. Some have argued that this restriction is imposed by thermodynamics, but it has been proven untrue when diffusion is important.13 Nauman and Buffham14 removed the assumption of no unmixing, but at the expense of imposing a more severe restriction, namely, that RA must be a monotonic function of a. Their theory is known as the weak bounding theorem, because it is more restrictive, with respect to the allowable RA. Their proof is relative simple and the restrictions on RA are satisfied by most single reactions. A recent but more complicated proof15 shows the strong boundary to be correct. Thus, the states of complete segregation and maximum mixedness do impose absolute conversion limits on the yield of a homogeneous reactions in isothermal reactors, provided that RA is a function of a single concentration or stoichiometrically linked concentrations and that RA is either concave upward or concave downward. Open and Closed Systems Figure 4 compares open and closed forms of the axial dispersion model. The closed form illustrated in Figure 4a is governed by the Danckwerts boundary conditions and has been widely used to model processing equipment. The RTD can be measured by transient response experiments; and, for a closed systems, these measurements can be used to determine the mean residence time, ht ) L/uj, and the axial dispersion coefficient, D. The governing relationships between tracer experiments and model parameters were found by solving an unsteady-state version of the axial dispersion model. Brenner16 applied a turnoff step change to the model and obtained a series solution for W(t) that can be fit to experimental data to obtain ht and the axial Peclet number, Pe ) ujL/D. Levenspiel and Smith17 used Laplace transform methods to relate the dimensionless variance of an impulse experiment to Pe:

σ2 )

2 2 [1 - exp(-Pe)] Pe Pe2

(18)

The use of tracer impulse experiments was popularized in Levenspiel’s introductory book.7 Bischoff18 showed how to

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correct for imperfect injection of the tracer pulse. These methods work when the system is closed and in turbulent flow, although the physical basis of the axial dispersion model becomes questionable for small Pe where the model predicts high levels of mixing, approaching that of a perfectly mixed CSTR. A special difficulty encountered with laminar flow systems is discussed in a subsequent section. Figure 4b illustrates an open system. Tracer experiments can be performed using the indicated injection and detection points, but they no longer measure the RTD.19,20 This fact, perhaps obvious in retrospect, confused the chemical engineering literature for more than 20 years. A molecule ceases to accumulate residence time if it temporarily leaves the reaction zone, 0 < z < L, but continues to accumulate time between injection and detection, and in fact may be detected many times before finally leaving the system, never to return. Temporary excursions outside the system boundaries are possible in open systems (for example, by dispersion upstream of the injection point) but are impossible in closed systems. The first moment of an impulse response experiment made on an open system includes time spent outside the system boundaries and is larger that the mean residence time in the reaction zone: µ1 > ht ) L/uj. The first moment was determined by van der Laan:21

µ1 )

[

L Din + Dout 1 1 ) ht 1 + + + 2 uj (Pe) (Pe) uj in out

]

(19)

Note that the Peclet numbers used in this formulation are based on the length of the test sections. The inlet section upstream of the injection point and the outlet section downstream of the detection point are theoretically infinite in length, so a Peclet number based on the length of these sections has no meaning. The transient response function that corresponds to impulse injection into a uniformly open system with Din ) D ) Dout was derived by Levenspiel and Smith:17

g(t) )

[

]

-Pe(t - ht )2 Peth exp 4πt 4t

x

(20)

where ht ) L/uj is the actual value of the mean residence time in the reactor section. The first moment of g(t) is µ1 ) ht [1 + (2/Pe)], which agrees with eq 19 when Din ) D ) Dout. The dimensionless variance of g(t) is given by

σ2 )

8 + 2Pe 4 + 4Pe + Pe2

(21)

where σ2 has been normalized by µ12, which is the apparent mean determined by impulse injection into a uniformly open system. Equations 20 and 21 can be used to estimate parameters for the axial dispersion model when the model applies to the entire system, including connections. This approach is suitable for estimating velocities and Peclet numbers in turbulent pipeline flows. Use of eq 18 is incorrect for open systems, but the error drops to 5% for Pe > 16. It remains to calculate the yield for a reaction in an open system. This can be done by solving the axial dispersion model for the three regions: -∞ < z < 0, 0 < z < L, and L < z < ∞, with reaction only in the central region. The concentration far upstream is ain, and the concentration far downstream is bounded: da/dz ) 0. Concentrations and fluxes are matched at the boundaries (z ) 0 and z ) L) of the reaction zone. An analytical solution was found by Wehner and Wilhelm22 for a first-order reaction, and the result is identical to that of Langmuir2 and Danckwerts1 for a closed system. Equation 13

shows that the fraction unreacted for a first-order reaction with rate constant k is given by the Laplace transform of f(t) with transform parameter k. The unique relationship between a function and its Laplace transform shows that the RTD in the open system is identical to that in the closed system. The RTD within the reaction section is independent of temporary excursions outside the section. Comments on the Boundary Conditions The behavior of an open system governed by the axial dispersion model causes few conceptual difficulties. The boundary conditions for an open systems are those used by Wehner and Wilhelm.22 They allow both concentration and flux to be continuous at the boundaries of the reaction zone. The reactant diffuses forward in the inlet section, because of the downstream reaction. The reactant concentration gradually declines from ain at z ) -∞ to some lower value at z ) 0. Upon entering the reactor, the concentration continues to decline until a minimum is reached at the reactor outlet. Flux is continuous at the outlet, as well as at the inlet. The diffusive component of flux in the outlet zone is zero because the reaction has stopped and there is no concentration gradient. The concentration is also continuous at the outlet because there is no further reaction. This forces the solution in the reactor zone to have zero slope as well. The Langmuir-Danckwerts boundary conditions for a closed system are somewhat counter-intuitive, but I here quote Nauman and Buffham:14 “Partially because Langmuir and Danckwerts were right and partially because many other people were wrong, these boundary conditions have now achieved the exalted status of being obvious.” The concentration profile of a reactant will have a discontinuity at the reactor inlet and zero slope at the outlet. This can be readily understood for the limiting case Pe f 0, because the reactor then behaves as a CSTR, and CSTRs have an inlet discontinuity where the concentration plunges from ain to aout. CSTRs also have zero slope within the reactor, because the concentration within the reactor is everywhere equal to aout. At high Pe, the axial dispersion model approaches piston flow, and the inlet discontinuity vanishes. A first-order reaction in a PFR gives a(z) ) ain exp(-kz/uj), and this concentration profile does not have zero slope at the outlet. The axial dispersion model at high but finite Pe closely approximates the PFR curve and only shows zero slope immediately before the end of the reaction zone. The size of the inlet discontinuity and the length of the region near the outlet where the slope is small and vary inversely with Pe. At intermediate values of Pe, there is an inlet discontinuity and zero slope at the outlet, and this represents intermediate behavior between that of a CSTR and a PFR. The concentration at z ) 0 is the same for the closed model as for the open model: the effect of forward diffusion toward the reaction zone of an open system exactly matches the inlet discontinuity of a closed system. Within the reaction zone itself, the concentration profiles are identical and, thus, conversion is identical. Over the years, there have been many attempts to prove the Danckwerts boundary conditions with a more rigorous argument than given in the previous paragraph. The most convincing demonstrations are due to Wehner and Wilhelm,22 Pearson,23 and Choi and Perlmutter,24 who justified the Danckwert’s boundary conditions by comparing open and closed systems. Proofs based on thermodynamics25 and random walk theory19 have also been presented. The literature contains contrary

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assertions as well, but most of these have been refuted. Occasionally, alternative boundary conditions have been proposed on the grounds of numerical convenience. Whenever RTD theory is applied to a new field, e.g., geoscience, the practitioners of the new field find it necessary to question the LangmuirDanckwerts boundary conditions but eventually conclude these conditions are correct.26 There are only two known exceptions to the standard boundary conditions. One exception is a compressible fluid in a system with a significant pressure drop. Choong, Paterson, and Scott27 used boundary conditions based on mole fractions, rather than concentrations, to account for this situation. Another exception is a nominally open system, where the inlet and outlet regions are short.28 In the limit of shortness, the system is closed.

yield, but improvement ceases when there is too much diffusion. For large values of Dth/R2, radial concentration gradients have already been removed and further increases in Dth/R2 cause axial mixing to become important. This case has typically been ignored, because the situation does not occur in conventionally sized equipment.33 Axial diffusion in micrometer-scale reactors causes yields to become worse than for laminar flow without diffusion and approach those of a CSTR.34 The axial dispersion model is a crude approximation of a laminar flow reactor that is applicable only in a limited range. More-realistic models allow a velocity profile in the crosschannel direction. Examples are laminar flow between flat plates or in a tube:

( )

(

)

∂a 3uj y2 ∂a ∂2a ∂2a 1- 2 + ) D 2 + 2 + RA ∂t 2 Y ∂z ∂y ∂z

Laminar Flow In the absence of molecular diffusion, the washout function corresponding to a parabolic velocity profile in a tube is

W(t) ) 1 2

t W(t) ) h 2 4t

(for t > 2ht )

( )

(22)

(23)

These criteria set both upper and lower bounds on the diffusion group:

Dt L . 2h . 0.25 Rx48 R

(

)

∂a r2 ∂a 1 ∂a ∂2a ∂2a + + RA (26) + 2uj 1 - 2 )D + ∂t r ∂r ∂r2 ∂z2 R ∂z

This distribution has a slowly converging tail that varies algebraically as t-2. The integral for the first moment, ht , converges as t-1, and the integral for the second moment diverges, giving an infinite variance. A divergent variance is a general property of diffusion-free laminar flows.29 This asymptotic behavior results from the velocity being a linear function of distance near a zero-slip boundary. In a real system, molecular diffusion will become important at long times so that the washout function will have an exponential tail and finite moments. This asymptotic behavior is a general property of all real flow systems.29 However, depending on the diffusivity, σ2 > 1 is possible in laminar flow, so that a variance measured experimentally can exceed the upper limit of the axial dispersion model. Taylor30 and Aris31 concluded that the axial diffusion model can be used to model laminar flow if the reactor is sufficiently long so that radial diffusion effectively removes radial concentration gradients. The criteria for a circular tube have been given by Taylor:32

L R2 L . . x3 R 4R Dth

or

( )

(for t < 2ht )

(25)

(24)

These criteria were used by Taylor to determine when the spread in concentration of an injected compound can be used to measure the diffusion coefficient of that compound. Too little diffusion means that the spread is dominated by hydrodynamics (i.e., the parabolic velocity profile). Too much diffusion and the width of the pulse becomes significant, compared to the length of the apparatus. Within these limits, the tracer concentration is governed by the unsteady-state axial dispersion model. The usual case is too-little diffusion, so that steady-state reaction yields approach those for laminar flow without diffusion. Increasing diffusion within the range of eq 24 improves the

If the axial diffusion term is retained, these models do not readily accommodate closed boundary conditions. The existence of the velocity profile means that both radial and axial diffusion will occur in the inlet and outlet of an open system, as well as in the reaction zone. Thus, the concentration at the inlet plane to the reactor, as well as the velocity, will be a function of spatial position on the plane. Danckwerts-type boundary conditions become problematic both mathematically and physically. For example, the porous plugs used conceptually by Langmuir2 would force a hydrodynamic entrance region inside the reactor. Actual reactor designs that approximate a closed system can be analyzed using computational fluid dynamics (CFD). The problems that are associated with determining a RTD from CFD data are discussed in a subsequent section of this paper. The current section deletes the axial diffusion terms in eqs 25 and 26 but later returns them when the system is open. If the reactor is at steady state, if the reaction is first order, and if the axial diffusion term is neglected, eqs 25 and 26 give Graetz-type problems, for which an analytical solution should be possible. Then, according to eq 13, the differential RTD, f(t), can be found by inverse Laplace transformation. Prior to the current publication, only moments of the RTD appear to have been calculated because the necessary series is difficult to sum.35 In contrast, numerical solutions of steady-state reaction problems are relatively easy even when the axial diffusion term is retained. These solutions for the first-order case can be then be used to obtain f(t) by numerical inversion of eq 13. An alternative approach in the absence of axial diffusion is a numerical washout experiment. Equations 25 and 26 become wave equations when D ) 0. Care must be taken to avoid numerical diffusion, so the wavefront that is associated with the step change in concentration is sharp, and that an accurate result for W(t) is obtained in the limit of small D. One method to achieve this is the random walk simulation discussed in a subsequent section. Another method involves direct solution of the governing partial differential equation by a technique that maintains a sharp wavefront when there is no diffusion. Figure 5 shows washout functions that correspond to eq 25, but without the axial diffusion term. Open boundary conditions with axial diffusion are physically possible. The reaction zone could be a catalytic bed or the

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Figure 5. Wash-out functions for the convective diffusion model.

reaction could be induced by actinic radiation that is limited to the reaction zone. Numerical solutions for the steady-state case remain feasible, and f(t) can be obtained by inverse Laplace transformation. Results for micrometer-scale reactors that operate with waterlike fluids show significant effects of axial diffusion, unless the reactors are quite long.34 There is an awkward size, with a characteristic channel width of ∼100 µm, that can be fabricated by soft lithography. These devices are in deep laminar flow but are too large for diffusion to quickly remove cross-channel concentration gradients. Special types of static mixers have been designed to enhance cross-channel mixing in these devices.36 Tracer Injection and Detection The problem caused by open boundaries is that an injected particle may leave the system temporarily before leaving finally and that the time spent during temporary exits affects the impulse response function but not the RTD. This effect has a tendency to be small, and the theoretical openness of a system often can be ignored. Also, the inlet and outlet piping of processing equipment typical has small diameters and operates in turbulent flow so that Din and Dout have a tendency to be small. Assuming the velocity profile to be flat, the spread of an injected impulse can be estimated. Suppose that a perfect Dirac injection is made at location -Z in the system. Then define Pein ) ujZ/Din and calculate σ2 using eq 18. An estimate of Din can be obtained from the literature.7 The duration of the pulse, when it occurs, will be ∼2σZ/uj. Add an estimate of the time required for the injection. The sum is the approximate spread of the pulse when it enters the system. Typically, it will be small, compared to ht for the system. If it is not small, the possibility of using the imperfect pulse method of Bischoff18 remains. The openness of a laminar flow system is caused by axial diffusion and can be neglected in most cases. Other problems occur when there is a velocity profile in the inlet piping. Suppose that the inlet to a system is a circular tube with a parabolic velocity profile and that diffusion is minimal. There are two difficulties in injecting a pulse upstream of the inlet: achieving a uniform distribution of tracer particles across the injection plane and maintaining a reasonably sharp impulse at the downstream entrance to the system, z ) 0. The first is probably the more difficult, but it can be resolved in theory by an instantaneous reaction (for example, by radiation) that converts molecules in a small region ∆z to tracer particles. If the injection is done from an external source, it must be done uniformly across the tube at rates proportional to the local velocity. We

suppose that this is possible. (See Turner37 and Buffham38 for details.) The second problem is of axial dispersion of the pulse before it reaches the inlet. Such dispersion will be caused by the velocity profile but can be minimized by locating the injection plane close to the system inlet. Detection of the tracer signal as it leaves the system (at z ) L) poses another problem that exists for laminar reactors, even at steady state. Material near the wall will have long residence times and will react more than material near the centerline. The resulting composition distribution can be sampled in two ways. One way is the mixing cup average, which is indeed the appropriate measure of conversion. The mixing cup average is what the product manager has to sell. It is also the appropriate measure of concentration for tracer experiments, because RTDs are concerned with the flow of particles, and mixing cup averages are weighted by volumetric flow. A flow-weighted average can be obtained by sampling the entire outlet stream. This is common practice in measuring RTDs in plastic extruders. Returning to the steady-state operation of a laminar reactor, some analytical techniques measure the spatial average composition, for example, using ultraviolet (UV) absorption. The spatial average composition is also called the resident composition or the through-the-wall composition. It corresponds to suddenly freezing the outlet stream and sampling a cross-section. This sample will give a different result than obtained by sampling the flowing output from the system, and the difference can be large.39 This is the same sampling issue that occurs during tracer measurements.36,37 Tracer Particles, Random Walks, and Computational Fluid Dynamics (CFD) Tracer experiments are the preferred experimental method for determining RTDs. They are directly useful in closed systems, and most engineering systems are at least approximately closed. Furthermore, it is generally possible to find an inert tracer that is innocuous and capable of being detected in low concentration, so that its injection does not significantly disturb the flow field. Molecules enriched in 2H (deuterium), 13C, and 16O can be detected with good accuracy and are environmentally innocuous at the low concentrations that are typical of tracers.40 In open systems, transient response experiments such as impulse injection can be used to fit model parameters that, in turn, can be used to deduce the RTD. We turn now to the problems of determining RTDs from hydrodynamic models using the concept of tracer particles. The basic approach dates to Wein and Ulbrecht.41 The inlet plane to a flow system is divided into several small area elements ∆Ai. A single point on the area element is used to define the start of a streamline. The velocity at the point of entry multiplied by the area of the element is approximately the volumetric flow rate that is associated with the element (∆Qi). Follow the streamline through the system and record the elapsed time (ti), when it exits the system. Enter paired values of ∆Qi and ti into a table and then sort to achieve ti+1 e ti. The partial sum,

W(tn) )

1

n

∑ ∆Qi

Q i)1

(27)

then approximates the washout function. An often-repeated blunder is to weight the parameter ti by the area and not by volumetric flow rate. The consequence of this, for a laminar flow system, is that the calculated first moment of the distribution (by area) diverges. (See Nauman42 for details and examples of the blunders.)

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As a practical matter, good convergence to ht ) V/Q has proven difficult for typical CFD codes. It seems that the finiteelement scheme used in most CFD codes does not lend itself to the necessary accuracy near zero-slip boundaries. Also, the spatially uniform distribution of tracer particles that is typically used in CFD studies is not ideal for determining ht . A report about FLUENT43 shows reasonably good agreement with the exponential distribution of a single stirred tank, but it remains unclear whether the code weights the residence times by area or by volumetric flow rate. (The latter is required.) The simulation of diffusion poses a special problem for CFD codes. Numerical diffusion confounds real diffusion. One approach is to carefully map the time-average velocity profiles in the system and then to add random displacements to the particle position. The particle thus samples a variety of streamlines during its sojourn through the system. The random displacements can be fixed in size as in a classic random walk or they may be sized according to a probability distribution (a Gaussian distribution with zero mean and variance proportional to D1/2 is commonly used). (See, for example, the work by Mackley, Saraiva, and Neves,44 who used this approach to study mixing in a complex flow field but did not determine a RTD. The random walk technique can be illustrated by determining the RTD for flow between flat plates. Using dimensionless variables, the model is

( )

∂c Dt ∂2c ∂c 3 + (1 - y2) ) h2 2 ∂τ 2 ∂z H ∂y

(28)

Rather than using a predetermined set of starting points, particles are introduced at random locations that have been chosen to be uniformly distributed, with respect to volumetric flow rate. This means that they are uniformly distributed with respect to the washout function, so that simulation of a large number of particles directly simulates the washout function. A value for W is chosen from a uniform distribution with a range of 0-1. The starting point, p, for a particle then is found from

W)

∫0p 23(1 - y2) dy

(29)

The simulated particle moves along a line of constant y for a time increment of ∆τ. It is then displaced sideways by an increment ∆y. Classic random walk theory uses a constant magnitude for ∆y, where only the sign of the displacement is chosen randomly, and this approach is adequate for illustrative purposes. We nominally relate ∆y to the diffusion coefficient by

∆y )

x( ) 2

Dth ∆τ H2

(30)

Reflective boundary conditions are used at solid boundaries and axes of symmetry. Results are similar to those in Figure 5. Application of the random walk technique to geometries other than flat plates can be complicated. For laminar flow in a circular tube, the counterpart of eq 29 is simple:

r ) x1 - x1 - W R

(31)

Use of a constant displacement ∆r is still possible, but the probability of positive and negative displacements must be adjusted to ensure constant density. For ∆r < r < R - ∆r, the probability of a step in the negative r direction must be decreased, because there is less available area at small values of r:

Pneg )

2r - ∆r 4r

(32)

More-complicated adjustments are needed near the wall and centerline, but the importance of the adjustments vanishes as ∆r f 0. Stagnancy, Bypassing, and More Distribution Functions Industrial residence time measurements, which are largely unpublished, have been used to examine reaction and separation systems that behave differently than designed. Pathological flow phenomena, including stagnancy and bypassing, can be detected by such experiments. Stagnancy and bypassing are judged relative to the mean residence time in the system and also relative to the nature of the system. A qualitative definition of stagnancy is that some portion of the fluid has a much longer residence time than would be expected, given the overall nature of the flow. A stirred tank reactor exhibits stagnancy when the tail of the washout function declines more slowly than exp(-t/ ht ). Tubular reactors in turbulent flow show stagnancy when any portion of the flow has a residence time more than a few times larger than ht . Bypassing is the compliment to stagnancy. Bypassing in a stirred tank means that the washout function initially declines more rapidly than exp(-t/th). Because the integral under the washout function must be ht , bypassing at small t necessarily means that some portions of the washout function must lie above exp(-t/th) at longer times. Bypassing in a tubular reactor can occur when the viscosity near the wall is substantially higher than at the centerline, giving an elongated velocity profile. The above definitions are qualitative. In an attempt to more precisely define stagnancy, Noar and Shinnar45 defined the residence time intensity function as

Λ(t) )

f(t) W(t)

(33)

The intensity function measures the probability that a particle will escape from the system. When Λ(t) is a decreasing function of t, the escape probability is decreasing. Noar and Shinnar took this as a definition of stagnancy. For a CSTR, Λ(t) ) 1/th. The escape probability is constant, and there is no stagnancy. A laminar flow reactor without diffusion has Λ(t) ) 2/t and shows stagnancy for all t > ht /2. A real laminar flow reactor will have an exponential tail and that portion of the fluid will not exhibit stagnancy according to the Noar and Shinnar definition, although that fluid is the last to leave the system. A widely used measure of dispersion is the dimensionless variance of the RTD, σ2. Its natural limits are 0 to 1 for welldesigned systems, but σ2 > 1 is possible. A CSTR with stagnancy or bypassing will have σ2 > 1 and there is no theoretical upper limit. Buffham and Mason46 have advocated the use of the internal age holdup function:

1 H(R) ) t h

∫R∞ W(t) dt

(34)

The holdup function is determined by tracking the amount of original material that still remains in the system during a washout experiment. Buffham and Mason proposed several measurement methods for the measurement of H(R), including tomography. Such measurements should be especially sensitive to stagnancy, because the inventory in the vessel is measured directly. When measured directly, the holdup function provides a more accurate determination of higher moments because the

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Figure 6. Reactor in a loop with instantaneous recycle.

power on the weighting factor t in the integral for the moments is lower:

µn ) n(n - 1)th

∫0∞ tn-2H(t) dt

(for n g 2)

(35)

Compare this to eq 8. An extreme form of stagnancy is lost Volume. Integration under the washout function estimates ht . (See eqs 9 and 10.) Comparing the measured value of ht to the expected value provides a good check on the accuracy of a residence time experiment. If the difference is larger than can be ascribed to experimental error, lost volume is possible. As a real example, a reactor in a polymer plant showed diminished performance. A washout experiment showed the reactor to have lost almost half its volume. This result justified an unscheduled maintenance shutdown that found the reactor to be badly plugged with crosslinked polymer, as the result of an earlier but unreported upset. An interesting side excursion of residence time theory is the treatment of cycle time distributions47 in circulating systems, of which physiological systems are the most interesting. A tracer injection into a recycle system that has zero net throughput can be used to determine system parameters. One injection point but multiple detection points, all inside the system, are preferred. Such measurements should provide insight for agitator design and could be used to validate CFD codes, but these ideas have not been realized very much in practice. The concept of cycle times has been extended to stirred tanks with some net throughput as well as internal circulation.48 First Appearance Times, Recycle Systems, and Surge Dampening

Consider the system with instantaneous recycle shown in Figure 6, and note that the recycle does not change ht . Note that the recycle flow does not change ht but will decrease the first appearance time normalized by ht . As the recycle flow increases, the overall system will approach an exponential RTD, regardless of the nature of the flow element. This statement is subject only to the provision that there is no portion of the system through which flow is bounded when the recycle rate is increased at constant net throughput.52 The exponential distribution is also approached when the net throughput is decreased at constant recycle flow. As a practical matter, q/Q ) 100 represents a very high recycle rate that is rarely achieved in an industrial design. This recycle rate would give τfirst ≈ 0.01. Recycle systems provide a good way to dampen surges in concentration. The outlet response to a time-varying input of a nonreactive component is

Cout(t) )

∫-∞t Cin(θ)f(t - θ) dθ ) ∫0∞ Cin(t - θ)f(θ) dθ

Suppose that the input concentration is disturbed by a sinusoidal wave of amplitude ∆C and frequency ω and that f(t) ) (1/th) exp(-t/th), which corresponds to a CSTR or a recycle system with a high recycle rate. Equation 36 then gives

[C - C h ]max )

tfirst ht

It ranges from τfirst ) 0 for a CSTR to τfirst ) 1 for a PFR, but no real system reaches either limit. First appearance times in diffusion-free laminar flow are known for a variety of geometries. They can range from ∼0.1 in a static mixer designed to approximate an exponential distribution of residence time49 to >0.85 for laminar flow in a helically coiled tube with periodic changes in the direction of the axis of the helix.50,51 The indicated geometry is four coils of the helix followed by a 90° change in the direction of the axis, another four coils and so on. Short, fat reactors can give τfirst < 0.5, because of entrance effects. Turbulent flow reactors give τfirst ≈ 0.8 according to the 1/7 power law. First appearance times are pertinent in sterilization and disinfection processes, where it is necessary to treat all portions of the fluid for some minimum time and where treatment of some portions of the fluid has no bad consequences. Current CFD codes are fairly accurate for predicting first appearance times.

∆C

x1 + ω2ht 2

(37)

so the amplitude of the disturbance is attenuated. A (large) CSTR uses conventional equipment and provides good surge dampening across the range of frequencies. There is an alternative that is almost always better. Two piston flow elements in parallel can attenuate the disturbance completely. This fact is no doubt obvious to electrical engineers if the “piston flow element” is replaced by “zero-order lag,” but it seems to have been recognized only recently within the chemical engineering community.49 Even more, complete attenuation of any strictly periodic disturbance is possible. This innovative idea has been implemented industrially. It uses what Smith, Graham, and Palamara53 have called a “flat” RTD:

f(t) )

The first appearance time is the time required for an impulse injection at the inlet to appear at the outlet. We are usually concerned with the dimensionless first appearance time, τfirst:

τfirst )

(36)

1 R

f(t) ) 0

(for 0 < t < R)

(38a)

(for t > R)

(38b)

Their reasoning was based on frequency response concepts, but the effectiveness of the flat RTD is more easily demonstrated using a convolution integral. The period-average input (and output) is C h and, and Cin(t) is a periodic disturbance imposed on C h . We choose the right-hand side of eq 36 to give

Cout(t) )

(R1) ∫

R

0

Cin(t - θ) dθ )

(R1) ∫

t

t-R

Cin(θ′) dθ′ (39)

h . The where Cout(t) is the outlet disturbance imposed on C rightmost integral in eq 39 can be set to zero by choosing R ) tP, where tP is the period of the disturbance. This is complete attenuation. Smith, Graham, and Palamara53 applied their ideas to pressure swing adsorption. Their process has a complex concentration waveform, but the period of the wave is known and constant. Applying eq 9 to the flat RTD shows that ht ) R/2 ) tP/2. The washout function corresponding to eqs 38 is a straight line from W(0) ) 1 to W(2th) ) 0. The design problem is to construct a physical device that gives a uniform (flat) distribution of residence times over the range of 0-2th. Smith, Graham, and Palamara chose a design reminiscent of Zwietering’s maximum

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combinations. The first and still most important use of residence time measurements is to explore flow patterns within processing equipment. Simple models for approximating RTDs within the normal range of 0 < σ2 < 1 include the following: (1) The fractional tubularity model consists of a CSTR in series with a PFR. As perhaps the first exercise in fitting experimental tracer experiments, Gilliand and Mason71 applied the fractional tubularity model to bubbling fluidized beds. (2) The tanks-in-series model consists of CSTRs in series. (3) Recycle models feed a portion of the effluent back to the inlet, without separation and typically instantaneously. (4) Side capacity models have a CSTR that is fed from and empties into a CSTR. (5) Parallel models consist of two or more systems with different ht in parallel. There may or may not be transport between the streams. The last two models are capable of modeling stagnancy and bypassing, σ2 > 1. Figure 7. Comparison of noise dampening by a flat residence time distribution (RTD) and an exponential RTD.

mixedness reactor, in which an outlet pipe is fed at numerous points. A collection of tubes of varying diameters or varying lengths in parallel is an option. The flat RTD is tuned to a particular wavelength and will give complete attenuation for this wavelength and its harmonics. This is illustrated in Figure 7, along with a comparison with the dampening ability of the exponential RTD. The smooth curve in Figure 7 is a plot of eq 37. Almost everywhere, the flat RTD filter exceeds the surge dampening performance of a stirred tank. However, a variable volume stirred tank can at least partially accommodate surges in the flow rate, as well as surges in concentration.

Multiple Inlets and Outlets Residence time studies have applications in fields such as geoscience, environmental engineering, and medicine, where systems with multiple inlets and outlets are important. Studies of estuaries and aquifers are examples. Washout experiments on such systems are generally impractical. Instead, impulse injection is usually used.11 The information that is usually desired is a conditional density function, fij(t), that applies to particles that are entering the system at the inlet i and leaving at outlet j:

fij(t) dt ) fraction of i-to-j flow that has a residence time between t and t + dt

This function is determined by injecting an impulse at inlet i and measuring the response at outlet j:

Models of Residence Time Distributions The literature contains many models that have attempted to describe RTDs in a manner that is useful for the correlation of data, for design, or for the conceptual understanding of flow phenomena. Wen and Fan54 and Nauman and Buffham11 have given summaries, but new models and analyses of existing models continue to arise. The models can be grouped into two large groups: theoretical models and empirical models. The theoretical models are generally of the convectivediffusion type, such as those described in eqs 1, 25, and 26. In principle, the theoretical models contain no adjustable parameters, but compromises are sometimes needed, particularly in turbulent flow. Solutions to convective diffusion equations coupled to the equations of motion are obtained using CFD, but there is a remaining need to verify and validate the codes. In this context, Verification refers to proof that the code is actually solving the mathematical equations that constitute the model. Have the computations converged, with respect to solution technique and grid size? An annoying tendency among CFD practitioners is to use the finest grid that is possible, in terms of computer memory and time without showing that a somewhat courser grid produces the same results: One suspects that it does not. Validation asks whether the properly solved equations reflect physical reality. It fair to say that CFD has not yet emerged as a reliable means for a priori design. Empirical models have adjustable parameters used to fit experimental data. They typically consist of ideal reactor models (e.g., CSTRs and PFRs) connected in series and parallel

(40)

fij(t) )

cj(t)

∫0

∞

(41)

cj(t) dt

A complete analysis of a multiport system requires (nonsimultaneous) injection at each of the inlets and detection at each of the outlets. If this can be done for each outlet, the integrals in eq 41 provide a measure of the relative flow from inlet i to each of the outlets. Multiphase Systems The restriction to systems with a single fluid phase can sometimes be relaxed. An important practical example is the use of washout experiments to determine an unknown holdup in a multiphase system. Choose a tracer that is confined to a single phase and perform a washout experiment. Equations 9 and 10 then are used to determine the volume (more precisely, the inventory) of the phase. Generally, multiphase systems can be treated as multiport systems, although there can be practical complications regarding sampling and the choice of tracer. If a component of interest exists only in one phase, then its RTD can be determined using a tracer that is also confined to the single phase. If the component can transfer between phases, the tracer must behave similarly. Picking it might be difficult, although isotopically tagged molecules are a possibility. A review of two-phase flows in bubble columns confirms the trend to publishing data based on CDF studies.55 A study of capillary and monolith reactors

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discusses RTDs in these devices.56 Gas-solid systems have been studied extensively and are discussed in the applications section of this review. Unsteady Systems Thus far, we have assumed that the system is at steady state, with respect to overall flow, and that any tracer experiments did not affect that flow. However, a collection of particles leaving a systems at time θ will have a distribution of residence times, even if the flow systems are not steady state. The washout function for an unsteady system is defined as57

W(t|θ) ) fraction of particles leaving the system at time θ that remained in the system for a duration greater than t (i.e. that entered before time θ - t) (42) The limits of reaction yield that correspond to complete segregation and maximum mixedness have been applied to the unsteady distribution functions.57 For the case of a single CSTR operating with variable flow rates and volume, the unsteady washout function is

[

W(t|θ) ) exp -

Qin dθ′ V

θ ∫θ-t

]

(43)

The corresponding density function is

f(t|θ) )

Qin(θ - t) V(θ - t)

[

exp -

∫θθ-t

Qin dθ′ V

]

(44)

The theory was used to devise a control strategy for variablevolume CSTRs in series whereby product transitions could be made with an arbitrarily small amount of off-specification material.58 A study by Chen59 treated micromixing more thoroughly. Krambeck, Katz, and Shinnar60 considered the complications that can arise in nominally steady-state systems when there are fluctuating throughputs. A paper by Schork and co-workers61 has addressed unsteady residence time theory from a pedagogical viewpoint. Recent papers (e.g., that by Randhir and Starzak62) have emphasized stirred tanks in series, although this case was explicitly addressed in the original publication.57 To date, the theory has been confined mainly to stirred tanks and the almost trivial case of an unsteady PFR. For stirred tanks, the simpler approach is to assume perfect mixing and to solve the dynamic material balances for component concentrations and the system volume. Thus, it is unclear whether the extension of residence time theory to unsteady systems has yet to make a significant contribution to chemical engineering science. I say this as the originator of the concept, in the interest of provoking stimulating rebuttal. I also note that papers that prominently feature the words “stochastic, Markov, and chaotic” have made marginal contributions of a practical nature. Generalized Reaction Times The probability that a molecule reacts in an isothermal, homogeneous reactor is dependent on the time spent in the reactor and possibly on interactions with other molecules that are also in the reactor. If the reaction is first-order, the reaction probability is dependent solely on time; and as shown by eq 13, the conversion of a first-order reaction is uniquely determined by the RTD. Consider now the situation where the reactor is still homogeneous but not isothermal. For a single reaction with a single activation energy, define the thermal time as

tT )

(

∫0t exp R-∆E T(R) g

)

dR

(45)

where R is the age of the molecule as it moves through the reactor. Nauman63 showed that the distribution of thermal times, fT(tT), can be used to predict the conversion of a first-order reaction and that limits on the conversion, according to the limits of maximum mixedness and complete segregation, also apply to homogeneous but nonisothermal reactions. Schemes to measure the distribution of thermal times have been proposed but never implemented. Theoretical calculations of thermal times have been used to explain the relatively good performance of plastic extruders as reactors.63 Fluid elements with low residence times have a tendency to have high temperatures, giving a result that is surprising uniform, i.e., that is similar to piston flow, with respect to thermal times, even though the RTD is typical of laminar flow. The general concept of temperature-history compensation, wherein short residence times are matched with high temperatures and conversely provides a conceptual framework for reactor optimization. The idea of temperature history distributions has also been utilized in food processing.64,65 Heterogeneous but isothermal reactors have also been modeled using the concepts of a generalized reaction time. (See Tayakout-Fayolle, Othman and Jallut66 for a discussion of approaches, primarily in the context of contact times.) The underlying idea of contact time is that the extent of reaction is dependent on the time that the molecules spend on a catalytic surface. Because surface concentrations are difficult or impossible to measure in industrial reactors, the almost universal approach is to use pseudo-homogeneous kinetics to model the reaction. The contact time is defined in this spirit:

tc )

j Fj

∫0t F dR

(46)

The factor of j/Fj provides the convenient property that ht c ) ht F, where ht F ) jV/Q is the time that molecules would spend in the fluid phase if there were no absorption. The conversion of a first-order, heterogeneously catalyzed reaction is

aout ) ain

∫0t exp(-ktc)fc(tc) dtc

(47)

where fc(tc) is the differential distribution of contact times. The first measurement of fc(tc) was done by Orcutt, Davidson, and Pigford,67 who realized that eq 47 is the Laplace transform of fc(tc) with transform parameter k. They varied k in a fluidizedbed reactor by varying temperature, assuming that this did not affect fc(tc), and obtained fc(tc) by inverse Laplace transformation. Nauman and Collinge68,69 devised a technique for measuring both fc(tc) and fF(tF), using paired tracer experiments, where the tracers were adsorbed on the catalyst surface to different extents. Here, fF(tF) is the density function for the fluid phase, ignoring any time spent in the adsorbed state. It can be measured independently, using a nonadsorbable tracer such as helium. This approach can be faulted on the grounds that the motion of the fluidized particles would affect the contact time. Although recognized by Nauman and Collinge in their original papers and undoubtedly true, the effect seems to be second-order and the practical differences seem obscure.63 The Nauman and Collenge68,69 methodology correctly predicted fF(tF), as measured independently using a nonadsorbed tracer. Although dual tracer experiments have been used industrially, few results have been published. The contact time distributions reported for an

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industrial-scale reactorsa large cold flow model of a gas fluidized bedsused the dual tracer methodology to estimate the contact time distribution; the results were plausible.70 Applications Most of the 6000 literature references to RTDs describe applications to real physical systems, either experimentally via tracer studies or increasingly by CFD studies. The results can be used in many ways. The obvious use is to obtain an overall understanding of flow patterns in the system. The washout function can be interpreted as a de facto velocity profile for the overall system. It can measure the holdup in a vessel that is not hydraulically full, and it can be used to fit a model of the system. The following paragraphs are far from comprehensive but illustrate the breadth of applications to which residence time measurements have been applied. I attempt to provide a recent citation for various application areas to indicate the current state of the art and again apologize for omitting many excellent papers. The first applications of residence time theory were to fluidized-bed reactors.71,72 Gilliland and Mason71 observed that the washout function had a sharp first appearance time, followed by an exponential tail. They accordingly modeled the system as a PFR in series with a CSTR. Models have grown far more sophisticated and include detailed treatments of such things as the RTD for solid particles in various portions of a circulating fluidized bed. Davidson and co-workers73 have used an in situ method for converting ordinary particles to tracer particles. Residence time measurements are useful for fluid-particle systems in general. Examples include spouted beds74,75 and a variety of polymerization reactors that involve solid particles of the polymer.76-78 The application of RTD theory to fluidsolid catalytic systems has been proven useful in trickle beds.79 There have been many industrial studies of RTDs in plastics extruders and similar devices (such as twin-screw extruders, Buss Koneaders, and Farrel continuous mixers). There is even a patent80 that claims a method for measuring RTDs in polymer extruders, and some publications discuss methodology.81,82 Relatively few of the experimental studies have appeared in the referred literature.83 Extruders rarely run hydraulically full, so washout experiments are one means of measuring in-process inventory. The other common means is to suddenly stop the machine and to “pull the screw,” but this is particularly difficult for twin-screw devices. Multiple products are typically run on the same processing line, so that purging experiments are clearly important. Extruders are used for manufacturing food products as well as plastic products, and, again, the RTD becomes very important.84 Another food processing application is continuous aseptic processing.85 There are many more applications for RTD theory in the process industries. Villermaux and co-workers86 studied an sulfur dioxide oxidation reactor, a superphosphate granulator, a chromium ore rod mill, a shaft furnace, a copper melting process, and a fluidized-bed calciner. Other applications include the detection of fouling in reverse osmosis modules87 and the RTD of methane in diamond epitaxy.88 Perhaps the largest volume industrial process to which CFD has been applied is a >500 000 m3 mechanically aerated lagoon used for wastewater treatment in the pulp and paper industry.89 CFD was suggested as a substitute for conventional tracer experiments to measure the RTD in this system. The ongoing urge to replace actual tracer experiments with CFD calculations is found in the production of potable water,90 but many systems remain so complex that experiments cannot be replaced.91 Earthworm

treatment had a substantial effect on the axial dispersion coefficient in soil filters.92 It has been recognized for many years4 that the “streamlining” of chemical processes requires good mixing in the direction perpendicular to flow. The goal of most static mixers is to promote radial mixing to reduce axial dispersion and more closely approach piston flow. The twisted pipes introduced by Saxena and Nigam50 are an example of elegant conceptual design.51,93 More common are proprietary devices that are inserted into tubes that have been widely studied, with respect to RTD.94-96 Biochemical processes use the same intellectual base of RTD theory as petrochemical applications, but there is a greater emphasis on separations rather than reactions.97,98 Chromatography is dependent on differences in mean residence time.99 Size exclusion separations exploit differences in the volume that is available to molecules of different sizes,100 giving mean residence times that are, more or less, proportional to molecular weight. Continuous fermentations are increasingly popular, and the airlift fermentor has been modeled with respect to RTD101 and axial dispersion.102 Within medicine, pharmocokinetics has embraced residence time theory and, indeed, the field can be defined as the study of input/output relationships in a living organism based on residence time theory.103,104 Some studies can be quite specific with a subject matter that is perhaps unexpected by chemical engineers.105 Yet, the authors of papers that are very much in the mainstream medical literature are sometimes affiliated with chemical engineering departments.106 The use of residence time theory in the geosciences is perhaps the furthest afield from classical chemical engineering. Naturally occurring passive tracers can sometimes be used, and the time scale is on the order of billions of years.107 Oceanographic studies involve shorter time scales but they are still thousands of years.108 The tools of residence time theory have also been applied to estuaries109 where the timescales are mere days and months. Tracer experiments are widely used to study residence times in river basins.110 Some analyses are quite sophisticated and involve unusual terminology.111 A second-order stream is a stream fed by two first-order streams, with these being streams that lack tributaries. The hyporheic zone is the region beneath a stream where groundwater and stream flow interact. This term is rare enough that even the cited reference by Haggerty et al.111 found it necessary to give a definition. The measured response to the pulse injection of a tracer had a tail that declined as t-1.28. The authors attribute this to the slow interchange between the stream and the hyporheic zone. They cite no papers in the chemical engineering literature, and it is unclear whether the response curve, although called the residence time distribution and denoted as f(t), satisfies the requirement that

∫0∞ f(t) dt ) 1

(48)

A chemical engineer might attempt to model the system as a PRF with limited mass transfer to a parallel PFR, but explaining the algebraic tail is an interesting problem. What Remains To Be Done Some years ago, a pioneer of the transition from unit operations to chemical engineering scienceswhose name most readers would instantly recognizessubmitted a proposal to the National Science Foundation to explore the remaining issues in residence time theory. Regrettably, the proposal was not

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funded. There may still be discoveries, basic and fundamental, lurking within the theory as outlined in this review. This possibility is reinforced by the recent discovery of new surge dampening techniques and particularly the concept of a flat residence time distribution (RTD). It is also interesting that the concept originated with industrialists, not academics. Both have their chance in future innovations. The field of residence time theory is mature but not moribund. As a practical matter, modern CFD codes are actually quite sloppy, with respect to calculating RTDs. In particular, they have difficulty satisfying eq 9 even when calculated residence times for tracer particles are properly weighted by volumetric flow rather than area. The probable cause is that finite-element codes seem to have difficulties with the low velocities near solid boundaries. I suppose the software vendors should be primarily accountable for any deficiencies. Still, users and particularly users who seek to publish their CFD studies have an obligation to both validate and verify their results. Large programs that no single person understands are becoming a problem in the scientific community. We must learn how to deal with them. There remain many interesting aspects of residence time theory that can be explored, but I caution against publications that are merely mathematical in nature and that do not provide genuinely new insights. My perception is that many attempted and actual publications fall into this category, and that some of the publications represent a failure of the reviewing process. Similarly, some submitted and even accepted publications fail to cite anticipating literature. With modern means for literatures searches, omission of prior publications should no longer be acceptable, at least when the relevant publications are in similar fields. Real challenges remain. Here are a few: (1) Suppose deep laminar flow, a fixed ht , and a constraint on the allowable pressure drop. Devise a static mixer that maximizes τfirst. Specifically, what internal configuration of a tube or duct optimizes this aspect of performance at small multiples of the pressure drop for laminar flow in a tube? (2) Modify CFD codes to allow easy, accurate, and semiautomatic determination of RTDs in laminar flow. As a next step, treat molecular diffusion in laminar flows and the combination of eddy diffusion and molecular diffusion in turbulent flow. (3) The heat-transfer literature is replete with designs for internals that increase heat transfer in tubes. Suppose instead that the desire is to achieve a uniform distribution of thermal times. How can this best be done? (4) There are literature studies that give empirical fits to turbulent velocity profiles in a tube, but eddy and molecular diffusion are also important. Combine these effects to predict the RTD as a function of the Reynolds number and the Schmidt number. Use the results to best-fit the axial dispersion model and compare the results to literature correlations for the dispersion coefficient. Literature Cited (1) Danckwerts, P. V. Continuous-flow systems. Distribution of residence times. Chem. Eng. Sci. 1953, 2, 1. (2) Langmuir, Irving The velocity of reactions in gases moving through heated vessels and the effect of convection and diffusion. J. Am. Chem. Soc. 1908, 30, 1742. (3) Ham, A.; Coe, H. S. Calculation of extraction in continuous agitation. Chem. Metall. Eng. 1918, 19, 663. (4) MacMullin, R. B.; Weber, M., Jr. The theory of short-circuiting in continuous-flow mixing vessels in series and the kinetics of chemical reactions in such systems. Trans. Am. Inst. Chem. Eng. 1935, 409. (5) MacMullin, R. B.; Weber, M., Jr. Determining the efficiency of continuous mixer and reactors. Chem. Metall. Eng. 1935, 42, 254.

(6) Kandiner, H. J. Sampling lag and purging time in mixing vessels in series. Chem. Eng. Progress 1948, 44, 383. (7) Levenspiel, O. Chemical Reaction Engineering; Wiley: New York, 1962. (8) Danckwerts, P. V. The effect of incomplete mixing on homogeneous reactions. Chem. Eng. Sci. 1958, 8, 93. (9) Zwietering, Th. N. Degree of mixing in continuous flow systems. Chem. Eng. Sci. 1959, 11, 1. (10) Rippin, D. W. T. The recycle reactor as a model of incomplete mixing. Ind. Eng. Chem. Fundam. 1967, 6 (4), 488. (11) Vatistas, N. Danckwerts’ degree of segregation for the axial dispersion model. Chem. Eng. Sci. 1991, 46 (1), 307. (12) Chauhan, S. P.; Bell, J. P.; Adler, R. J. Optimum mixing in continuous homogeneous reactors. Chem. Eng. Sci. 1972, 27 (3), 585. (13) Nauman, E. B. On the bounding theorem of micromixing theory. Chem. Eng. Sci. 1984, 39, 174. (14) Nauman, E. B.; Buffham, B. A. Mixing in Continuous Flow Systems; Wiley: New York, 1983. (15) Buffham, B. A.; Nauman, E. B. Extremes of conversion in continuous-flow reactors. Chem. Eng. Sci. 2004, 59, 2831. (16) Brenner, H. Diffusion Model of Longitudinal Mixing in Beds of Finite LengthsNumerical Values. Chem. Eng. Sci. 1962, 17, 229. (17) Levenspiel, O.; Smith, W. K. Notes on the Diffusion-Type Model for the Longitudinal Mixing of Fluids in Flow. Chem. Eng. Sci. 1957, 6, 227. (18) Bischoff, K. B. The general use of imperfect pulse inputs to find characteristics of flow systems. Can. J. Chem. Eng. 1963, 41, 129. (19) Nauman, E. B. Residence Time Distributions in Systems Governed by the Dispersion Equation. Chem. Eng. Sci. 1981, 36, 957. (20) Nauman, E. B. Transient Response Functions and Residence Time Distributions in Open Systems. AIChE Symp. Ser. 1981, 202, 77-87. (21) van der Laan, E. T. Notes on the Diffusion-Type Model for the Longitudinal Mixing of Fluids in Flow. Chem. Eng. Sci. 1958, 7, 187. (22) Wehner, J. F.; Wilhelm, R. H. Boundary conditions of low reactor. Chem. Eng. Sci. 1956, 6, 89. (23) Pearson, J. R. A. A note on the Danckwerts boundary conditions for continuous flow reactors. Chem. Eng. Sci. 1959, 10, 281. (24) Choi, C. Y.; Perlmutter, D. D. A unified treatement of the inlet boundary condition for dispersive flow models. Chem. Eng. Sci. 1976, 31, 250-252. (Also see Chem. Eng. Sci. 1978, 33, 1567.) (25) Standart, G. The thermodynamic significance of the Danckwerts’ boundary conditions. Chem. Eng. Sci. 1968, 23, 645. (26) Peters, G. P.; Smith, D. W. Numerical study of boundary conditions for solute transport through a porous medium. Int. J. Numer. Anal. Methods Geomech. 2001, 25 (7), 629. (27) Choong, T. S. Y.; Paterson, W. R.; Scott, D. M. Axial dispersion in rich, binary gas mixtures : model form and boundary conditions. Chem. Eng. Sci. 1998, 53 (24), 4147. (28) Novy, R. A.; Davis, H. T.; Scriven, L. E. Upstream and downstream boundary conditions for continuous-flow systems. Chem. Eng. Sci. 1990, 45 (6), 1515. (29) Nauman, E. B. The Residence Time Distribution for Laminar Flow in Helically Coiled Tubes. Chem. Eng. Sci. 1977, 32, 287. (30) Taylor, G. I. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. (London), A 1953, A219, 186-203. (31) Aris, R. On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. (London), A 1956, A235, 67-77. (32) Taylor, G. I. Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Soc. (London), A 1954, A225, 473-477. (33) Merrill, L. S., Jr.; Hamrin, C. E., Jr. Conversion and temperature profiles for complex reactions in laminar and plug flow. AIChE J. 1970, 16 (2), 194. (34) Nauman, E. B.; Nigam, A. On Axial Diffusion in Laminar Flow Reactors. Ind. Eng. Chem. Res. 2005, 44, 5031. (35) Farrell, M. A.; Leonard, E. F. Dispersion in laminar flow by simultaneous convection and diffusion. AIChE J. 1963, 9 (2), 190. (36) Stroock, A. D.; McGraw, G. J. Investigation of the staggered herringbone mixer with a simple analytical model. Philos. Trans. R. Soc. London, A 2004, 362, 971-986. (37) Turner, J. C. R. The interpretation of residence-time measurements. I. systems with and without mixing. Chem. Eng. Sci. 1971, 26, 549. (38) Buffham, B. A. Residence-time distribution for a system with velocity profiles in its connections with the environment. Chem. Eng. Sci. 1972, 27 (5), 987. (39) Kreft, A.; Zuber, A. On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Eng. Sci. 1978, 33 (11), 1471.

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ReceiVed for reView November 30, 2007 ReVised manuscript receiVed February 12, 2008 Accepted February 13, 2008 IE071635A