key: cord-0825904-o3fdqwad
authors: Rodrigues, Alírio E.
title: Residence Time Distribution (RTD) revisited
date: 2020-10-04
journal: Chem Eng Sci
DOI: 10.1016/j.ces.2020.116188
sha: d12dd1dbc6cf6520b939ffe3f77b78f71ff6034c
doc_id: 825904
cord_uid: o3fdqwad

Residence Time Distribution (RTD) theory is revisited and tracer technology discussed. The background of RTD following Danckwerts ideas is presented by introducing “distribution” functions for residence time, internal age and intensity function and how to experimentally obtain them with tracer techniques (curves C and F of Danckwerts). Compartment models to describe fluid flow in real reactors are reviewed and progressive modeling of chromatographic processes discussed in some detail. The shortcomings of Standard Dispersion Model (SDM) are addressed, the Taylor-Aris model discussed and the Wave Model of Westerterp’s group introduced. The contribution of Computational Fluid Dynamics (CFD) is highlighted to calculate RTD from momentum and mass transport equations and to access spatial age distribution and degree of mixing. Finally smart RTD and future challenges are discussed.

Engineering (CRE) course given by Jacques Villermaux 1 at ENSIC (Nancy, France) when doing my doctoral work. I appreciated his "system dynamics" approach in the modeling of real reactors.

From those days I remember a seminar from Levenspiel 2 given on September 17, 1973 (I still keep my notes…) at the "Centre de Cinétique Physique et Chimique", the CNRS laboratory directed by P. Le Goff where I was working. Levenspiel mixed English and French in his beautiful and pedagogical presentation. I tell my students that if they read and understand Levenspiel's book 3 it is more than enough; the original is always better than photocopies… He stressed that the task of an engineer is to transform raw materials into products, hopefully with increasing value. He pointed out that the output of a reactor depends on the input type, kinetics, mode of contact, etc. We can summarize as shown in Figure 1 : In the case of fluid flow he mentioned two approaches: the one by hydraulics (civil engineering) based on correlations (practical but not general) and fluid mechanics approach (involving mathematics and theorems…). How to conciliate these lines was the challenge taken by Prandtl 4 in 1904: instead of solving equations he simplified them and that is the beginning of the Modern Fluid Mechanics.

The RTD theory is linked to the name of Danckwerts and his seminal paper 5 in 1953. I was reading recently the book "Life on the edge" 6 about P. V. Danckwerts. His chemistry background from Oxford ("we had laboratories in the cellars and all the kinetic results were said to be catalyzed by cigarette smoke" 7 ), the attendance of MIT course to learn the tricks of Chemical Engineering (ChE) is very well documented there. The simplicity and clarity of presentation are due may be to his non-mathematical background (as Danckwerts said "At the age of 30 I had never used a sliderule or heard about differential equations" 7 ).

In 1983 I spent a sabbatical year at the Université de Technologie de Compiègne (UTC) and as a result of the interaction with E. Brunier, G. Antonini and A. Zoulalian, RTD was tackled by solving the complete set of equations for flow and mass transport of a passive tracer 8 

with the approach mentioned in the book by Nauman and Buffham 9 . This problem was later solved in 2004 with new CFD tools (Fluent) and published with education goal 10 .

In the middle of 90's I came across a series of papers by Westerterp's group [11] [12] [13] [14] [15] about what they called "wave model" where the limitations of the SDM (Standard Dispersion Model) were discussed. My interest on this topic was renewed during ISCRE 22 in Maastricht when Westerterp delivered a talk to the representatives of the Working Party of EFChE on CRE 16 ; unfortunately Westerterp passed away on August 24, 2013. In more recent years a series of papers 17-21 use CFD to assess internal age distributions and degree of mixing.

The objective of this paper is to revisit RTD theory and tracer experiments to understand the fluid flow through the reactor. The paper will be organized as follows:

First the background of RTD theory will be presented and tracer experiments will be discussed;

ii) Models to describe fluid flow in reactors will be reviewed and applied in real reactors (e.g.,NetMix) and chromatographic processes;

iii) Shortcoming of Standard Dispersion Model (SDM) as presented by Danckwerts will be discussed, the Taylor-Aris model revisited and the Wave Model of Westerterp's group introduced;

iv) RTD calculations from momentum and mass transport equations and CFD modeling will be addressed; v) Finally we will discuss how to access age distribution and degree of mixing, smart RTD and future challenges.

Danckwerts 5 (1953) approached the study of fluid flow in reactors in a brilliant and simple way: "introduce a pulse of tracer into the fluid entering the reactor and see when it leaves". The normalized outlet tracer concentration versus time, E(t) is the Residence Time Distribution (RTD).

Here the normalized outlet concentration is simply the tracer outlet concentration, c out normalized by the area under the curve c out (t) versus time , i.e.,

The study of RTD of flowing fluids and its consequences can be put under the umbrella of tracer technology. This is important for chemical engineers 22, 23 , researchers in the medical field 24 , environment 25 , etc to diagnose the reactor ill-functioning, drug distribution in the body, dispersion of pollutants in rivers, etc. When I taught this subject at the University of Virginia in 1988 students "saw" the application of RTD in a Department Seminar where Michael J. Angelo from Merck

Chemical Manufacturing Division talked about pharmacokinetic models 26 ! Danckwerts built a theory based on the characterization of fluid elements of the population inside the reactor using two characters: age or internal age, , and life expectation, , and of the population leaving the reactor: residence time, . Obviously . Then he introduced the = + "distribution" relative to each character; the residence time distribution E(t) defined as ( ) being the fraction of fluid elements leaving the reactor with residence time between and . + Similarly the internal age distribution , a decreasing monotonic function, was defined as ( ) ( ) being the fraction of fluid elements inside the reactor with age between and . + When I taught RTD in undergraduate CRE course I made an analogy with the population in a country 27 ; assuming that the life of n 0 babies born today will follow the same pattern of those babies born before… the number of elements will decrease with time and after 110 years or ( ) so all "left" the system. If decreases suddenly near time zero it means high birth mortality rate ( ) in a given population or a by-pass (short-circuit) in reactors.

The intensity function was defined as being the fraction of fluid elements Λ( ) Λ( ) inside the reactor that, in each class of internal ages, has zero life expectancy (it means that those fluid elements will leave the reactor immediately after). These theoretical "distribution" functions are related by

where is the space time of the fluid flowing through the reactor volume with a volumetric = 0 flowrate . This is a result from the fact that in a time interval dt, the amount of fluid leaving the 0 reactor with residence time between and , is the fraction of the amount of + [ 0 ( ) ] fluid inside the reactor with age between and , which has zero life expectancy, 

The next question is how to experimentally have access to . This brings the tracer ( ) technology to the center of the arena. The response of the reactor to an impulse of tracer, c out normalized by a reference concentration, , is the experimental C curve of Danckwerts, 0 = ( )

. It should be noted that n is the amount of tracer injected which should appear at the outlet and therefore . The experimental C curve is directly related with the RTD, by: is the unit Heaviside step function, the normalized reactor response is the F curve of ( )

. Furthermore, as a consequence of system linearity, the response to an ( ) = 0 impulse is the derivative of the response to a step input since the Dirac delta function is the derivative of the step function, i.e, and therefore the RTD, can be obtained from ( ) = The mass conservation of the tracer in a tracer experiment where the inlet concentration is a step of magnitude c 0 is:

which leads to a relation between the experimental F curve of Danckwerts and the internal age distribution I(t):

where is the purge function, response of the system to a negative step going from 1 -( ) = ( ) an initial tracer concentration c 0 to zero.

It is also interesting to note that the RTD is the inverse Laplace transform of the transfer function G(s), i.e., The variance of the RTD, E(t), indicates how broad is the RTD and is calculated by

, or centered second moment of E(t). 2 1 The experimental first moment or mean residence time can be used for reactor diagnosis.

it indicates the presence of short-circuit and the fraction of tracer by-passed is ; on > 1the other hand if the system has dead zones and the fraction of the reactor volume occupied < by dead zones is . These tools have been applied by the author to an industrial polymerization 1reactor where a short-circuit around 6% of the feed flowrate was observed 27 .

For step input of tracer c in =c 0 H(t) the data treatment is also simple; first the outlet signal is normalized by c 0 leading to the Danckwerts F(t) curve and the RTD is just the time derivative of

When the inlet tracer concentration c in (t) is not a perfect Dirac impulse, both inlet and outlet signals c in (t) and out c out (t) are recorded (or two signals at different points along the reactor) and these two signals are related with the RTD, E(t) through the convolution integral or [13] 

The RTD, E(t) is obtained by deconvolution or as the inverse of the transfer function ,i.e., the ratio of Laplace transforms of the outlet and inlet signals, as indicated above in Eq (8) .

The interpretation of RTD data has been discussed in many publications; for example Leclerc et al 33 analyzed data from measurements in industrial processes and the potential of the method of moments in chromatography has been discussed in detail by Qamar and Seidel-Morgenstern 34 .

A class of models (compartment models) can be built by assembling nuclei elements in order to represent tracer experiments in real reactors. Those nuclei elements are: i) perfectly mixed reactor;

ii) plug flow reactor; iii) laminar flow reactor, iv) recycle; v) by-pass or short-circuit; vi) dead zones and vii) stagnant zones. Nuclei-elements are shown in Figure 5 .

Typical models of this class such as Cholette and Cloutier 35 , Adler and Hovorka 36 , plug flow with recycle 37 , tank in series with recycle 38 and unified time delay model 39 are shown in Figure 6 . The transfer function is:

and the RTD is a series of impulses with decreasing strength:

In the above equations R is the recycle ratio: ratio between the recycle flowrate (RQ 0 ) and feed flowrate (Q 0 ).

(d) The model of tanks in series with volumes and with recycle is shown in (1 -) 0 Figure 6d . The system is a second-order system with dumping factor and a time =

where the space time of the reactor is . The RTD is now: 

where is the lateral flowrate in each discrete cell) for a delay time distribution in lateral zones.

The transfer function is:

where and is the transfer function of the lateral zone with an exponential time delay = 0 ( ) distribution .

A recent review on compartmental modeling can be find elsewhere 40 . 

The NETmix ® reactor is shown in Figure 7 . It is a network of chambers connected by channels 41, 42 . It is a technology for static mixing which has been patented 43 and successfully used industrially for the production of hydroxyapatite nanoparticles 44, 45 . The model assumes that the cambers are perfectly mixed reactors and the channels are plug flow systems. The RTD can be derived easily for such model

In Equation (21) It is interesting to note the patented HEART design of the mixers elements. 

The more used models to describe fluid flow in packed bed columns are the staged model and the standard dispersion model (SDM) or plug dispersion flow model.

One 

is the feed flowrate, is the fluid solute concentration, is the adsorbed phase 0 * concentration at the particle/fluid interface in equilibrium with , is the average particle 〈 〉 concentration , is the particle specific area and is the mass transfer coefficient between bulk fluid and particle.

By putting the transfer function is: 

For a packed bed with porous particles the mass balance for inert, passive tracer in a bed volume element is:

and at the particle level the mass balance writes as:

Obviously in a tracer experiment in a bed with porous particles what we are measuring is an "apparent axial dispersion"

where the interstitial velocity is , the ratio of fluid in in the particle pores and outside particles is and the intraparticle diffusion time constant is . Alternatively one can write where the parameter is the ratio of the time constants for diffusion and convection. Only = when intraparticle diffusion is very fast and goes to zero we are measuring axial dispersion free of intrusion of pore diffusion. This is illustrated in Figure 10 showing the apparent Peclet number, versus . 

. The variance is obtained by replacing by . The response to a step ( ) = 1 + 1/ 2/ input of tracer is 5

Danckwerts used this result to experimentally show for the first time, according to Amundson 52 , that was close to 2.

The proper boundary conditions for the SDM have been discussed for a long time 53 The mass balance for the solute in a volume element of the tube is: 

When axial diffusion is neglected compared to axial convection, the asymptotic behavior is reached for long-time distribution of the solute ,i.e., or allowing the quasi-≫ area-averaged solute concentration. The concentration profile is:

The area-averaged convective flux relative to the moving observer is:

and using eqs (39) and (40) we get:

The 1D macrotransport equation in terms of the fixed reference frame z governing the areaaveraged solute concentration is ( , )

where u*=u and the Taylor-Aris dispersion is the result obtained by Aris 59 .

The validity of Taylor dispersion imposes or . In the region and ≪ The area-averaged concentration profile is:

The limiting case of pure convection holds if the time constants for radial diffusion, and where is the adsorption equilibrium parameter for linear isotherms.

The For a passive tracer the "lumped" diffusion/convection model for the particle with slab geometry is: 

A practical application of this concept is on the chromatographic separation of proteins. The pore velocity can be estimated from the equality between relative pressure drop at bed and particle scales assuming that Darcy's law is valid; the result is where is the ratio of particle and bed 0 = permeabilities. The Van Deemter equation for conventional packings was modified by Rodrigues 71, 72 where the C term shows the effect of intraparticle convection. 

The SDM described by Eq (30) and Danckwerts boundary conditions, eqs (30a) and (30b) has raised doubts concerning several points, which may be considered as drawbacks of the model.

Westerterp 16 listed some of those concerns, namely:

i) The basic idea of SDM is that axial dispersion is superimposed to plug flow; however plug flow means that radial dispersion is infinitely fast but in packed beds the Bodenstein number for radial dispersion is not zero;

ii)

The physical significance of boundary conditions has been questioned along the years 73 .

The inlet boundary condition means a sudden fall more pronounced for small Peclet numbers although this fall certainly happens in a perfectly mixed reactor by definition;

iii) In packed beds backmixing can occur as shown by Hiby 74 Assuming no chemical reaction and equal cross section areas occupied by both streams mass balances for each stream are:

(52a)

where k is the exchange coefficient between both streams (relaxation time constant) expressed in sec -1 .

The average concentration c and the dispersion flux j d are given by c=(c 1 +c 2 )/2 and j d =v(c 1 -c 2 )/2, respectively; the conservation equation for mass becomes: discrepancies are more significant for fast processes with steep temperature and concentration profiles along the reactor. 

Equations 57-60 were solved for and together with appropriate boundary 0 < < 0 < < conditions. At the inlet a parabolic profile was imposed and the flowrate through the = ( ) reservoir is and the mean residence time is = ∫ 0 ( -) = 3 

where is a flowrate parameter and e is the depth of the reservoir. This problem was solved later with modern tools (Fluent) 10 . It is much better to visualize the movement of the tracer inside the reactor 90 although the information contained in or internal age distribution is the same as in ( ) the "old" time. Figure 

The exit concentration of a reactor can in general be obtained by solving the equations for flow and tracer mass balance; for incompressible flows they are:

From equation (61) we can get the spatial distribution of the mean age (averaged over time locally)

a(x) under steady flow 91, 92 . More interesting is to obtain the spatial distribution of the instantaneous fluid particle age a(x,t) as introduced by Sandberg 93 . The balance of the fluid age is then

This was called by Simcik et al 94 the Smart RTD (SRTD). While RTD only gives the distribution of the internal age for steady flow, the SRTD gives the space-time distribution of the ( ) instantaneous fluid age a(x,t) for unsteady state flow in general. This approach was also discussed by Liu and Tilton 20 , Baléo and Le Cloirec 95 and Liu 19 who performed 2D simulations in a single phase reactor.

The mean age is the first moment of age calculated as:

The denominator is invariant and equal to the ratio of the total amount of tracer in the pulse to the volumetric flowrate. Liu presented a method to predict the tracer concentration and mixing in CFSTRs with mean age distribution 18 . The mean age theory was extended to multiphase systems by Russ and Berson 96 . CFD solutions of mean age allow spatial and time resolution within the flow field.

The study of mixing processes needs somehow a parameter to quantify the "goodness of mixing" based on experimental measurements 29, 97, 98 . Liu 19 presented a method for the calculation of the degree of mixing in steady continuous flow systems. The age  of a molecule is the time that has elapsed since the molecule entered the reactor; the average age of all molecules in the flow is .

The variance of the ages of all molecules will be: (65) var = ( -) 2 and the variance of mean age all over the whole reactor volume is

The degree of mixing of Zwietering 29 was defined as

Therefore from RTD one calculate the distribution of internal ages, and obtain the variance ( ) of the ages of all molecules var . However, to get one still needs the spatial distribution of mean age to calculate the variance of mean age var a. In a CSTR the variance of mean age is zero and ; for completely segregated system . The calculation of the degree of mixing was = 0 = 1 possible with the work of Liu and Tilton 20 who derived the governing equations for all moments of age from the time dependent concentration equation and equation for the mean age

where is the n th moment of age calculated from the tracer concentration at every spatial position x:

The Zwietering degree of mixing can now be written as

CFD is a powerful tool to help process development in chemical engineering and examples in stirred tanks and static mixers have been discussed by LaRoche 99 , Liu 17 , Khapre et al. 100 . One example at hand is the study of mixing and its mechanisms addressed in detail in the book by Eng. Sci., 192, 199-210 (2018) with permission from Elsevier)

The concept of RTD is spread through many areas from solid processing (continuous blenders, extruders, rotary drums, fluidized beds) used in continuous manufacturing of chemicals, plastics, polymers, food, catalysts and pharmaceutical products 104 , to hydrodynamic modeling of real systems (liquid-liquid systems 105 , simulated moving bed 106, 107 , cascades of stirred tanks 108 , continuous flow polymerization 109 ).

Tracer technology 38, 60, 110 has many applications outside the traditional chemical engineering 111 and extensions of the traditional RTD have appeared, 112, 113, 114 and still experimental improvements are reported 30 .

The recent push of continuous flow chemistry in pharmaceutical industry and the development of milli-and microreactors renewed the interest on RTD for the characterization of the flow behavior in such devices 115, 116 .

I am particularly interested in less conventional applications of RTD theory. I will mention two topics. One of those areas is Perfume Engineering 117 with the aim of predicting the trail of perfumes or sillage 118, 119 . Briefly a perfume is commonly viewed as a mixture of top notes, middle notes and base notes in a solvent. Some perfumers will say there are no middle notes and what matters is the diffusivity of top and base notes containing fast and slow diffusing molecules 119 . Therefore measurements of diffusivities of perfumery raw materials (PRM) in air are required to address the effect of molecule 120 and skin adsorption.

A second area is related to archeology concerning the application of diffusion-adsorption-decay models for dating bones 121 . But in our everyday life the measurement of RTD is routinely made in our main research activity on Cyclic separation/reaction processes.

This work was financially supported by: Base Funding -UIDB/50020/2020 of the Associate Laboratory LSRE-LCM -funded by national funds through FCT/MCTES (PIDDAC).

Part of the manuscript was written at LPA of UFC (Brazil) as a Visiting Researcher of the program "Ciência sem Fronteiras".

The help of Dr Elson Gomes and Dr Rui Faria with the preparation of figures is gratefully acknowledged.

-specific area of the particle, m -1 a(x,t)-age of a fluid particle at position x and time t 

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