martin (2000) interpretation of resicence time distribution data.pdf

Chemical Engineering Science 55 (2000) 5907}5917

Interpretation of residence time distribution data

A. D. Martin*North West Water Ltd, Dawson House, Liverpool Road, Warrington WA5 3LW, UK

Received 26 November 1998; received in revised form 15 November 1999; accepted 27 March 2000

Abstract

`Tracera or Residence time distribution (RTD) studies are commonly exploited as a means of developing an understanding of the`mixinga status of vessels of various types. The e!ort involved in the setting up of such plant studies can be considerable and theexecution of the experiment its self is often a test of endurance. In the past full interpretation of the results has not been easy and asa consequence super"cial treatments have been employed. This paper presents an alternative method for interpreting RTD data,which is relatively easy to use and addresses some of the weaknesses of more conventional methods. An extention to the `tanks inseriesa concept is presented (ETIS) and united with the `reactor networka formulation. The suitability and appropriateness of themodel is discussed and compared with the `closeda dispersion model ( 2000 Published by Elsevier Science Ltd. All rights reserved.

Keywords: Residence time; Tanks in series; `Closeda system; Networks; Comparison; Case studies

1. Introduction

Diagnosis of the operational ills and the characterisa-tion of new equipment at both pilot and full scale arecommon activities in the process industries. An impor-tant component in the diagnosis or characterisation pro-cess is an understanding of the vessel hydrodynamics,which at a global or `black boxa level may be gainedfrom the interpretation of the vessel residence time distri-bution (RTD). One of the many drivers for the character-isation arises from the need to simulate process responsesto unusual operating conditions. Many reactor modelsencoded within dynamic simulation packages such asSTOATTM GPSXTM and SPEEDUPTM are capable ofusing information regarding the RTD of the vessel underconsideration to improve the "delity of the model. Typi-cally these models employ parameters such as the Pecletnumber (Pe), the dispersion number (N

D) or the number

of tanks in series (nT) in conjunction with the mean

residence time (¹R) to describe deviations from the ideal

continuous stirred tank reactor (CSTR) or the plug-#owreactor (PFR). The reactor models avoid the necessity ofemploying complex computational #uid dynamics byrestricting their consideration to a single characteristic

*Now at: Environmental Technology Centre, Department of Chem-ical Engineering, OMIST, Manchester M60 1QD, UK. Tel.: #44-161-200-4340.

E-mail address: [email protected] (A. D. Martin).

length. These models are consequently one dimensionalin form.

2. One-dimensional models

There have been many models of this type designed tointerpret deviations from the two ideal extremes. Theentire family of models may be said to describe dispersedplug-#ow reactors (DPFR). The magnitude of the disper-sion as quanti"ed by N

Dincreases from zero in the PFR

to in"nity in the CSTR. The `opena and `closeda disper-sion models fall into this family as does the tanks in series(TIS) model.

The **open++ dispersion model considers the axialmotion of a #uid element to be made up of two compo-nents:

1. The convective component arising from the bulkmotion of the #uid.

2. The di!usive component arising from the randommotion of the element in response to the decay ofturbulent eddies.

This concept is expressed mathematically by the con-ventional `di!usion with bulk #owa equation (Eq. (1)).

LC

Lh"G

D

u¸HL2CLz2

!

LC

Lz, (1)

0009-2509/00/$ - see front matter ( 2000 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 0 8 - 1

where the dimensionless group

D

u¸"N

D"

1

Pe(2)

and

h"t

¹R

"

QRt

<R

"

ut

¸

(3)

The òpena boundary conditions from which this modeldraws its name de"ne the #ow condition at the reactorinlet and outlet. The òpena condition is physicallyachieved when the #ow is undisturbed at the inlet and theoutlet. In 1957 Levenspiel and Smith published the ana-lytical solution to Eq. (1) for the òpena boundary condi-tions (Levenspiel & Smith, 1957). This solution is shownin Eq. (4).

E(h)"C

Cd

"

1

2SPe

pheG~Pe(1~h)2

4h H(4)

Where Cd

is the concentration that would have beenobtained had the `dyea been evenly dispersed through-out the vessel under study.

With mean and variance

hM0"1#

2

Pe, (5)

p20"

2

Pe#

8

Pe2, (6)

It can be seen that the mean of this distribution isa function of the dispersion and at high dispersion (lowPe) the mean is substantially '1. This result appears tobe inconsistent with the mass balance but may be ex-plained by the di!usion of `dyea upstream of the injec-tion point. Thus, the pulse injected at the inlet is not thesame as the pulse which would have been measured atthe inlet. Casual observation of the inlet to and exit froma high-dispersion reactor would also con#ict with theassumed òpena boundary conditions. The rigorousmathematical de"nition of the òpena boundary condi-tions makes this model more di$cult to use than thetanks in series formulation as the user is often left in somedoubt as to the degree to which the òpena condition isachieved in the system under investigation.

The **closed++ dispersion model treats the system inexactly the same fashion as the òpena dispersion model.Eq. (1) is solved with the `closeda boundary conditions.The `closeda boundary conditions relate to the physicalsituation in which the #ow approaches the inlet to thereactor in idealised plug #ow (Pe"R), transforms todispersed #ow within the reactor and returns to idealisedplug #ow at the exit. This situation is very closely ap-proximated in many real reactors even those which arethemselves low dispersion devices. Thomas and McKee(1944) published the analytical solution with `closeda

boundary conditions in 1944. Their solution, publishedin a dimensional form, is reproduced here in non-dimen-sional form (Eq. (7)) to maintain consistency with Leven-spiel and Smith's òpena solution (Eq. (4)). Yagi andMiyauchi reproduced the Thomas and McKee solutionin 1953 with an alternative condensation of the terms(Yagi & Miyauchi, 1953).

E(h)"C

Cd

"2n/=+n/1

PeanePe@2M1~h(a2n`1)@2N

Pe(a2n#1)#4 Gan cosA

Pe

2anB

#sinAPe

2anBH, (7)

where an

is given by the positive roots of Eq. (8).

tanAPe

2anB"

2an

(a2n!1)

. (8)

Levenspeil (1972) published expressions for the mean andvariance of the `closeda system RTD though did notreport the analytical solution for the RTD itself. Leven-spiel's mean and variance can be shown to be equal tothose of Eq. (7) thus con"rming their association. Theseresults are shown below Eq. (9) for the mean,

hMC"4

+n/=n/1

Kn

Pe2(an#1)2

+n/=n/1

Kn

Pe(an#1)

"1 (9)

and Eq. (10) for the variance.

p2C"32

+n/=n/1

Kn

Pe3(an#1)3

+n/4n/1

Kn

Pe(an#1)

!1

"

2

PeG1!(1!e~Pe)

Pe H, (10)

where Kn

is given by Eq. (11)

Kn"

2Pean

Pe(a2n#1)#4

. (11)

Satisfactory enumeration of Eq. (7) becomes increasinglydi$cult at higher values of Pe. This di$culty derivesfrom the relative magnitudes of the early terms in theseries with respect to the later terms and the "nal sum.When h"0 the series itself is non-convergent with suc-cessive terms oscillating as the following.

limn?4

(Sn)"2(!1)(n~1)ePe@2 . (12)

At a practical upper limit of Pe"33 a conventionaldouble precision summation will yield a residual of lessthan 0.00005 at h"0.001 but will require some 300 termsto achieve this.

5908 A. D. Martin / Chemical Engineering Science 55 (2000) 5907}5917

Fig. 1. Exit age distributions of the tanks in series model for 1 and2 tanks in series.

Fig. 2. Comparison of the tanks in series model with the closed disper-sion model with a common variance of 0.5.

The tanks in series model seeks to describe the #ow ina reactor system by considering it to be discretised into ofa strand of equal-sized hypothetical CSTRs. Each hypo-thetical CSTR is independent of those preceding or fol-lowing it. The number of tanks in series n

Tdescribes the

dispersion with nT"1 representing in"nite dispersion

and being equivalent to Pe"0. Integration of a simpledynamic mass balance around the strand of reactorsreadily yields the system RTD (Eq. (13)). Eq. (13) is alsothe de"nition of the Erlang distribution.

E(h)"C

Cd

"

nnTT

(nT!1)!

h(nT~1)e~nTh. (13)

With mean and variance

hMT"1, (14)

p2T"

1

nT

. (15)

Conceptually, the development of this model is easy tofollow. It also has the advantage that the precise de"ni-tion of the inlet and exit boundary conditions is notrequired. Similarly concerns regarding the method of`dyea injection and measurement do not arise. Phys-ically, this model is at its best when the number of tanksin series is low and concerns over the appropriateness ofeither `opena or `closeda boundary conditions are attheir height. This model however has a signi"cant draw-back when n

Tis small due to the integer constraint.

Fig. 1 shows that the E curve for nT"1 di!ers very

signi"cantly from that resulting from nT"2. Many real

CSTRs exhibit RTDs which lie in this range and aretherefore only characterised very approximately by thetanks in series model. Frequently in the interpretation ofRTD data n

Thas been related to Pe or N

Dvia the

variances of the RTDs (Eqs. (10) and (15)). This is tanta-mount to assuming that the tanks in series model and the`closeda dispersion model are equivalent. Elgeti (1996)develops an alternative relationship between the twoRTD forms by following the progress of an arbitrary

reaction and expanding the dispersion equation inTaylor series. This analysis leads to the following equiva-lence relationship for Pe:

Pe"2(nT!1). (16)

Kramers and Alberda (1953) also proposed this equiva-lence. Fig. 2 shows quite clearly that for the varianceequivalence, n

T"2 and Pe"2.557, respectively,

the RTDs are quite di!erent. A plot generated for the`Kramers, Alberda and Elgetia equivalence shows verysimilar behaviour though the co-location of the peaks isslightly poorer than illustrated in Fig. 2. The `closedadispersion model exhibits a considerably higher peakvalue than the tanks in series model. This di!erencereaches a maximum at n

T"2, Pe"2.557. The `closeda

dispersion model can also be seen to exhibit a region witha positive second derivative at h(1 which is absent fromthe tanks in series model. At values of n

T'15,

Pe'28.97 the di!erence between the two maxima is lessthan 5% and is likely to be di$cult to resolve experi-mentally. Thus for practical purposes the two models aresu$ciently similar to be regarded as the same. The tanksin series and `closeda dispersion models however di!ersigni"cantly from the `opena dispersion model under thesame conditions.

3. Extended tanks in series

If the tanks in series model is merely regarded asa residence time distribution function, whose form de-pends solely on the value of the parameter n

T, and is

freed from the arti"cial need to have a physical manifes-tation, it becomes possible to address the quantisationissue arising in the analysis of high-dispersion systems.Simply introducing the concept of a non-integer numberof hypothetical tanks in series will achieve the desiredresult. The exit age distribution or E curve of the

A. D. Martin / Chemical Engineering Science 55 (2000) 5907}5917 5909

Fig. 3. Thread and knot concept for the tractor network model.

extended tanks in series (ETIS) model is given by a subsetof the gamma distribution family (Eq. (17)).

E(h)"nnTT

!(nT)hnT~1e~nTh. (17)

When nT

is integer it is identically equal to the Erlangdistribution (Eq. (11)). As with the Erlang distribution themean and variance are given by Eqs. (14) and (15). TheETIS model removes the problem of quantisation whichoccurs as n

Ttends to 1 in the tanks in series model.

4. Reactor network structure

The models described in the previous sections are allone dimensional in their nature and are consequentlyunable to describe gross structure in the #ow withina reactor. To describe such large scale structures in the#ow pattern it is necessary to introduce appropriatelarge-scale structure to the model. Many model struc-tures have been proposed to describe a range of physical#ow con"gurations (Levenspeil, 1972; Monteith& Stephenson, 1981; Smith, Elliot & James, 1993). Todate the fundamental building blocks or `threadsa ofthese reactor networks have been limited to either idealCSTRs or ideal PFRs. Algebraic expressions for theC and E curves of some of the simple con"gurations havebeen derived (Levenspeil, 1972; Monteith & Stephenson,1981). The technique for generating the model RTD forthe majority and the more complex networks involvesthe numerical integration of the dynamic CSTR massbalance Eq. (18) (Smith et al., 1993).

dC

dt"

(Ci!C)

¹

(18)

Clearly to describe real DPFRs using simple CSTR orPFR building blocks requires large numbers of the fun-damental unit and the numerical integration of reactornetwork models becomes inordinately cumbersome asthe extent of the network grows. The use of the CSTR asthe basic building block also constrains the networkdescription to integer values of n

T. Employing the ETIS

model to characterise the basic building block signi"-cantly reduces the complexity of network required whilstsimultaneously relaxing the integer n

Tconstraint. To-

gether these advantages dramatically improve the tracta-bility of the data analysis problem. The reactor networkstructure is developed further in the following sections tofacilitate the description of `reala vessels exhibiting com-plex composite dispersion behaviour with bypassing andstagnant zones.

The network dexnition: The reactor network structure isconstructed from two component types.

1. Threads,2. Knots.

These components serve to de"ne the volume elementsof the reactor and their connectivity, respectively. The`threadsa receive #ow from an up stream or source`knota and discharge to a down stream or sink `knota.The characteristics of the `threadsa are de"ned by the#ow through the thread its hypothetical volume and exitage distribution (Q, <, E). The `knotsa receive #ows fromsource `threadsa and distribute the summed #ow to thesink `threadsa and are de"ned as zero-volume blendersplitters. The `knotsa are characterised by a single-#owsplit fraction parameter ( f

s). A number of assumptions

are implicit in this basic de"nition of the network.

1. Individual `threadsa are assumed to be fully seg-regated from each other.

2. The exit age distribution of an individual `threada isgiven by the convolution of the composite E curvepassed by the source `knota.

3. The composite E curve passed at any `knota is givenby the linear #ow weighted sum of the E curves of thecontributing `threadsa.

4. There is no dispersion through `knotsa.

Assumptions 1 and 4 are likely to be the most conten-tious, particularly when considering reaction kinetics inconjunction with the RTD. They are however critical tothe formulation of a manageable problem. Clearly withinthe context of assumption 4 the choice of model for thecharacterisation of dispersion is restricted to the `closedadispersion model, the TIS model and the ETIS model.Despite the reservations expressed above the ETIS modelis used to illustrate the concept of the network.Fig. 3 shows a network of three `threadsa connectedtogether in to two #ow paths or `strandsa by three`knotsa. This type of network is a common result ofinterpreting the RTD of a large shallow packed bedreactor. This type of network di!ers from those employedby previous authors (Levenspeil, 1972; Monteith& Stephenson, 1981; Smith et al., 1993) in that all`threadsa accommodate dispersed #ow.

5. Experimental data interpretation

Data Gathering. There are four general types of experi-mental protocol for the conduct of `dye tracera


experiments. The classical experimental technique in-volves the injection of a Dirac d function pulse of `dyeafollowed by the measurement of its concentration as itemerges in the outlet stream over an extended period oftime. This method produces a readily interpretable resultand maximises the potential for resolving the detailedstructure of the experimental exit age distribution.A common alternative technique employs a `stepa input.This technique generates the reactor F curve, which maybe di!erentiated to yield the E curve. Direct di!erenti-ation of experimental data results in the ampli"cation ofnoise arising from sampling and analytical methods.Curve "tting to the integrated form or F curve may alsolead to poorer estimates of the parameters. This arisesbecause detailed features are less distinctive on theF curve than on the E curve. A third experimental proto-col involves the injection of a `blocka input. This methodis considerably inferior to the two previous methods. Theexperimental exit age distribution obtained from thistechnique is a hybrid lying between the classical E andF curves. The duration of the `blocka tends to producea broad peak and damp out the detailed structure, whichwould have been revealed by a d function pulse input.A "nal technique involves the analysis of a natural ran-dom signal in the input stream. This technique has theadvantage of not altering the process stream but su!ersfrom the same disadvantages as the `blocka input tech-nique. The AWWA guide (Teefy, 1996) o!ers advice onthe conduct of `dye tracera tests in water and waste-water treatment plants particularly with respect to theselection of suitable `dyesa.

Data processing. Historically the interpretation ofexperimental RTD data has been rather super"cial(Tomlinson & Chambers, 1979). Often this interpretationhas been limited to extraction of the mean and varianceof the data set (Eqs. (19) and (20)).

¹e+tM

e"

+Cntn*t

n+C

n*t

n

, (19)

p2e+

+Cnt2n*t

n+C

n*t

n

!tM 2e. (20)

The sequential nature of the data gathering guaranteesthe collection of biased data sets. So whilst for randomlygathered data fairly modest sample sizes are su$cient toobtain good estimates of the population mean and vari-ance this is not the case for RTD data. It can be seen thatEq. (19) is acutely `tail sensitivea, thus early truncation ofthe data set leads to serious under estimation of thepopulation mean or achieved mean residence time. Toovercome this problem protracted measurement periodsare usually prescribed. Typical measurement periods arechosen to be of the order of 3}5 times the mean residencetime with the need for longer periods coinciding withhighly disperse systems. Furthermore tM h is an estimate ofthe mean of the RTD. It is therefore only an estimate of

the mean residence time when the `closeda dispersion,TIS or ETIS models are appropriate.

The estimates obtained from Eqs. (19) and (20) haveoften been related to the dispersion number (N

D) or the

number of tanks in series (nT) via Eqs. (10) and (15) with

little regard to the overall shape of the RTD. A number ofadditional empirical relationships have also been de"nedto characterise the dispersion in reactors:

gH"

t10tM

, (21)

IM"

t90t10

, (22)

I4#"1!

tmtM

. (23)

Smith et al. (1993) examined these indices and foundthem to be mutually inconsistent when used to describetheir data. These indices may be consistently related tothe variance of the RTD only in circumstances when thedistribution is symmetrical, i.e. very low dispersion orplug #ow. This situation rarely coincides with the occa-sions when the departure from plug #ow is important.

6. Selected case studies

The ETIS model has been used in the following casestudies to provide the quanti"cation of the dispersionwithin an individual `threada. This has been done be-cause of the ease of application and despite concernsregarding strict validity at intermediate values of Pe. Thecurve-"tting process has been conducted on MicrosoftEXCEL 97t using the built in SOLVER and mathemat-ical functions. The built in functions were also supple-mented with visual basic for applications (VBA) codedeveloped by the author.

Simulated very large CSTR. The reactor under studyconsisted of a nominal 1500 m3 cuboid vessel witha multi-ported inlet manifold located at the bottom of thevessel and an outlet weir located on the opposite wall.Fig. 4 shows a schematic elevation of the vessel whichwas nominally 19.5 m in the axial direction by 19.5 mwide by 4 m deep. The inlet manifold directed the feed#ow across the #oor of the reactor underneath a grid ofaeration equipment. The process feed rate was0.0523 m3/s. The reactor was simulated using a commer-cial CFD package and an `experimentala RTDgenerated using a particle tracing technique. Several ap-proximations were made in the construction of the CFDmodel of this reactor:

1. The vessel was assumed to be semi-in"nite (normal tothe direction of #ow).

2. The feed manifold was modelled by a continuousaxially pointing slot at the foot of the feed wall.


Fig. 4. Schematic vertical section through the simulated very largeCSTR.

Fig. 5. Early period exit age distribution for the simulated very largeCSTR showing contributions from the component threads.

Fig. 6. Complete exit age distribution for the simulated very largeCSTR.

Fig. 7. Reactor network to model the exit age distribution of thesimulated very large CSTR.

3. A 300 mm wide #ow section with `periodica boundaryconditions was assumed to be su$ciently large tomodel the expected #ow structure.

Preliminary inspection of the early period results(Fig. 5) shows clearly a delay of approximately 20 minbefore a signi"cant concentration of `dyea is found in theexit. The delay is followed by a steady increase to a max-imum at approximately 100 min elapsed time. These gen-eral characteristics suggest a candidate frameworkcapable of describing the exit age distribution andE curve. Additional threads are added to the network byanalysis of the residual after attempting to "t the modelto the data. The analysis of the residual must be under-taken with some care as the degrees of freedom aresuccessively reduced with the addition of each newthread. Equally, the inherent noise in the data arisingfrom measurement errors must be considered. The solidline in Figs. 5 and 6 presents the E curve generated asa result of this process. The broken lines show the #owweighted E curves of the individual threads. Fig. 7 illus-trates the "nal network structure synthesised to describethe data.

The theoretical hydraulic residence time of this vesselTR

is 484 min and the mean of the "tted residence timedistribution ¹

Ris found to be 509 min. The estimate of

¹R

obtained from the èxperimentala data set tM is foundto be 476 min. The data set from which this estimate ismade extends to 1620 min or 3.4 residence times. The "ve`threadamodel proposed in Fig. 7 satisfactorily describesthe èxperimentala data. The dominant #ow strand

consists of a plug-#ow `threada and a large CSTR`threada. Together these account for the vast majority ofthe vessel volume. Two small additional `threadsa arerequired to describe the small by-pass #ow around thelarge CSTR volume. The "fth `threada describes the`deada volume. This technique is unable to order con-secutive `threadsa. However, with knowledge of the reac-tor geometry a considered assignment may be made. Inthis instance the plug #ow section is placed "rst and itscharacteristics are attributed to the region de"ned by thedecaying feed `jeta underneath the aeration grid. FromFig. 7 it can be seen that `threadsa 1 and 4 are verysimilar in character. This suggests that thread 4 is a por-tion of the feed `jeta which by-passes the well-mixedregion. This may be considered to be a portion of the#uid that travels the full length of the reactor underneaththe aeration grid and then ascends the end wall to the exitweir. Such a #ow strand would have failed to interactwith the air bubbles in the reactor and would thereforeremain un-converted. A similar argument may be ad-vanced to describe `threada 3. Together `threadsa 3 and5 represent 5% of the #ow through the vessel. Thus,based on the above argument this reactor would beunlikely to achieve greater than 95% conversion.

The mean of the èxperimentala data tM is remarkablyclose to the value of T

R. This situation is di$cult to


Fig. 8. Schematic vertical section through the packed bed reactor.

Fig. 9. Exit age distribution for the packed bed reactor.

Fig. 10. Reactor network to model the exit age distribution of thepacked bed reactor.

support given the truncation of the data set at 3.4 resi-dence times. Integration of the complex model E curveover 1620 min (3.4 residence times) reveals an expected98% `dyea recovery. It also provides an estimate of themean based on a truncated portion of the E curve. Thisestimate is found to be 473 min which is in close agree-ment with tM . On the basis of these consistency checks509 min estimated from the consideration of the whole ofthe "tted distribution is likely to be reliable. Practically,neither ¹

R'T

Rnor negative dead volume fraction

(`threada 5) can be supported and may be indicative of anerror in either the #ow or volume speci"cation of themodel region. Such errors may arise from the numericalhandling of symmetrical or periodic boundary condi-tions. The CFD model also predicts a signi"cant fractionof the vessel to exhibit an occluded volume exceeding5%. This is attributed to the presence of bubbles. Theexpected e!ect is to reduce the value of ¹

Rrelative to

TR

since the reactor network interpretation of the bubblecloud is as dead volume. Hence the expected dead vol-ume fraction is of the order of 5% rather than the !5%estimated by the network model analysis.

Packed bed reactor. Fig. 8 shows a schematic verticalsection through this reactor, which consists of a 5 m by5 m by 4.150 m deep cuboid vessel with 2.2 m depth ofbouyant granular packing operated in up #ow. The feedenters the lower distribution chamber, then passes upthrough the packed bed and hold down plate into anupper collection chamber before discharging from thereactor via wall mounted weirs. A second reactant isintroduced into the base of the packed bed and #ows inco-current mode through the packing. The dumpedpacking has a void volume fraction of approximately55%.

Visual inspection of the experimental data (Fig. 9)shows a delay in the detection of the `dyea in the outletstream and a substantial tail on the RTD. This suggeststhat the reactor requires a minimum three `threadamodel for characterisation. This model is shown inFig. 10. The reactor con"guration however suggests thata four `threadamodel may be more suitable from a phys-ical standpoint. Both models were constructed to ascer-tain the improvement in "delity between a two `threadamodel and a three `threada model. The overall improve-ment was found to be insigni"cant with respect to the "tto the experimental data. However, comparison of thecharacteristics of the individual threads o!ered furtherinsights into the possible structure of the operating bed.The low-dispersion `threada,1 in each model, was un-changed between the 3 and 4 `threada models but thehigh dispersion `threada of the 3 `threada model splitinto 2 CSTR `threadsa in the 4 `threadamodel. It may beinferred that the low dispersion `threada 1 models theregion in the reactor that is occupied by the packed bed.However, the void volume of the packed depth is consis-tently underestimated by approximately 20%. The

volume estimates of the high-dispersion regions areequivalently high. The description of the high-dispersionregion by the 4 `threada model is particularly interestingsince the over estimate of the volume is con"ned toa single `threada. The placement of the `knota between`threadsa 2 and 3 coincides with the operating volume ofthe upper collection chamber. By elimination the overestimated `threada is assigned to the lower distributionvolume. The relative under and over estimates of thevolumes in this part of the reactor may be attributed to2 causes:

1. Failure of the model to adequately describe the condi-tions at the interface between the distribution volumeand the packed region.

2. Interaction between the lower portion of the packedbed and the distribution volume such that a propor-tion of the bed is disturbed.

The reactor network model assumes that there is nodispersion through `knotsa which is likely to be valid


Fig. 11. Schematic plan of the internally recirculating CSTR.

Fig. 12. Exit age distribution for the internally recirculating CSTR.

Fig. 13. Reactor network to model the exit age distribution of theinternally recirculating CSTR.

within the reactor if either the feed or discharge `threadsaexhibit low dispersion. Whilst this assumption is imper-fect at Pe"17 the error introduced is insu$cient todescribe the phenomenon observed in the analysis. Thecorollary of this is that the placement of the model`knota does not coincide with the expected physicalboundary between the packed bed and the distributionvolume. This was con"rmed by the observation of in-stabilities at the interface between the packed bed and thelower distribution volume. Particles occupying the lower300 mm of the packing were observed shearing o! thebed and temporally dispersing in the distribution vol-ume. Thus both the 3 and 4 thread models have identi"edthe stable packed bed only as a low-dispersion region.Since the objective of this study was to characterise theactive portion of the reactor, the 3 `threada model isadequate.

Internally recirculating CSTR. This reactor consistsof a 12 m wide by 3.1 m deep oval channel with acentre line length of 236 m and an operating volume ofapproximately 8200 m3. Fig. 11 shows a plan view sche-matic of the vessel. The feed is introduced through theouter perimeter wall and the discharge exits via theinterior perimeter wall approximately 1

3of a lap down

stream of the feed position. Clockwise internal circula-tion is mechanically induced by a system of surfacepaddles.

The experimental procedure involved the injection ofa delta function pulse into the feed followed by a long tail.This is indicated in Fig. 12 by the closed circles. The longtail arises from the entrapment of `dyea in an externalrecycle. Entrapped `dyea passes through a number ofexternal unit operations before returning to the feed. Theoriginal delta function input is very heavily damped andthe intermittent peaks observed in this trace cannot beattributed to this source. It is more probable that thesepeaks arise as a result of random #ushing of residual`dyea from the reactor inlet chamber where the originalinjection was made.

The experimental output trace, open circles in Fig. 12,shows two distinct patterns of behaviour. The early peri-od behaviour extending from h"0.0}0.6 shows clearperiodicity with decaying amplitude. The late periodbehaviour extending from h"0.6 onward shows approx-

imate exponential decay. Based on the form of the inputtrace and the observed response the experimental datawas modelled in two parts.

1. The response to the delta function input was modelledin detail by the network shown in Fig. 13.

2. The response to the long tail was modelled solely onthe aggregated system behaviour, i.e. an exponentiallydecaying RTD.

A calculation method for this type of reactor networkmodel is described by Battaglia et al. with reference toland "ll leachate (Battaglia, Fox and Pohland, 1993). Thehigh dispersion of the Battaglia et al. system requiredfewer than 10 terms of their series solution to achievea satisfactory result. In this work the development fol-lowed a route equivalent to Battaglia et al. with the seriessummation being implemented in (VBA). The recycle wasopened out and the circulating #ow considered to bemade up of a number of parallel `threadsa. Each `threadawas considered to represent the fraction of #uid enteringthe reactor and completing a constant fraction of a lapplus an integer number (s) of additional laps before exit-ing. The general concept is illustrated in Fig. 14 togetherwith the recurrence relationships for the `threada para-meter values. The numerical subscripts in Fig. 14 relateto the parameters in Fig. 13. The total number of parallel


Fig. 14. Numerical equivalent network for the evaluation of the inter-nally recirculating CSTR model.

threads considered (u) was determined by a precisioncriterion given by Eq. (24).

u"1#IntGlog

10(p)

log10A1!

¹2#¹

3¹

RBH, (24)

where the numerical subscripts refer to the `threadanumbers in Fig. 13.

Analysis of the experimental data resolved the peri-odicity in the early period to be 11.55 min (0.061¹

R).

However the expected "rst peak at 4.75 min (h"0.025) isalmost completely absent from the experimental data.The "rst strong peak is detected at 16.3 min (h"0.086).This result has two possible explanations.

1. The sampling frequency at the outlet was insu$cientlyhigh to resolve the peak.

2. The dispersion normal to the direction of #ow wasincomplete.

A simple test to check the possible sample frequencyexplanation indicates that at least two samples wouldhave been gathered which contained signi"cant concen-trations of `dyea. This test indicates that the samplescollected at 4 and 6 min (h"0.0211 and 0.0316) are thosea!ected and the expected concentration of `dyea is likelyto be double that observed in any of the samples prior tothe "rst signi"cant peak. Whilst this test is not conclusivethe result suggests that incomplete normal dispersion isthe more likely explanation. Based upon these deduc-tions the derived reactor network structure and para-meter values are given in Fig. 13. The aggregated or lateperiod behaviour of this model is as expected equivalentto a CSTR (n

T"1) with a mean residence time of

¹R"190 minutes. The convolution of the long tail of the

input trace with the CSTR characteristics describes thelate period behaviour of the experimental RTD verysatisfactorily.

7. Discussion

Suitability of ETIS model. From a theoretical stand-point the ETIS model may be criticised because of itslack of rigour with respect to the inlet and exit boundaryconditions. It is however this lack of rigour which makesthe model easy to use particularly in circumstances ofhigh dispersion. The extension of the classical tanks inseries model into the continuous (ETIS) form also con-tributes to the ease of use since it becomes possible to usesimple optimisation algorithms for the interpretation ofexperimental data. The ETIS model also eliminates thequantisation e!ects observed in the tanks in series modelat low n

T. The elimination of these e!ects is a signi"cant

advantage since it is at low nTor Pe that the tanks in

series analogy is at its most useful. To achieve an equiva-lent removal of quantisation using the tanks in seriesmodel requires the addition of further `threadsa to thenetwork with independent residence times.

Hitherto, the absence of a solution to the `closedadispersion model has been widely accepted. This solutionwas however published by Thomas and McKee in 1944.The rigorous handling of the inlet and outlet boundaryconditions is a signi"cant bene"t relative to the ETISmodel and unlike the `opena dispersion model does notadd to the di$culty of use. The series solution presentedin their paper is however numerically intensive in use andbecomes practically non-convergent at low values ofh and high values of Pe. Frequently in the past the tanksin series model has been assumed equivalent to the`closeda dispersion model. The equivalence has beenestablished by equating the respective variances (Eqs. (10)and (15)). This assumption has been shown to be falseexcept under the condition n

T"1 and Pe"0. Practic-

ally however the di!erences become small enough for thedistributions to be regarded as the same when n

T'15

and Pe'28.97. The di!erences between the two distri-butions are such that, for a chosen variance or equivalentpair of n

Tand Pe values, the reactor design based upon

the ETIS model yields a slightly larger volume than thatbased on the `closeda dispersion model. This arises fromthe more `peakya nature of the `closeda dispersionmodel RTD. The `closeda dispersion model predicts thatsigni"cantly more material is discharged from the reactorwith an exit age between 0.25 ¹

Rand ¹

R.

The ETIS model as applied in the three case studieshas provided very satisfactory "ts to the experimentaldata. There are few areas in which the application of themore numerically intensive `closeda dispersion modelwould practically bene"t the interpretation of the data.Thread 3 of the simulated very large CSTR may describethe data better if the `closeda dispersion model wereused. This `threada, though, accounts for less than 3% ofthe reactor throughput and the value of n

T(22) is toward

the upper end of the range in which the ETISand `closeda dispersion models di!er signi"cantly.


Application of the `closeda dispersion model to `threada1 of the packed bed reactor network may improve the "tto the experimental data in the region 0.2(h(0.3 andh"0.75. However, as can be seen from Fig. 9 the scopefor improvement is small.

Dispersion at knots. A basic assumption of the reactornetwork structure is that there is no dispersion throughthe `knotsa. This assumption is completely consistentwith the application of the `closeda dispersion model tocharacterise the interconnecting `threadsa. Such a highlevel of consistency cannot be claimed for the use of theETIS due to the lack of rigour in the de"nition of the inletand outlet boundary conditions. The implied assump-tions for the tanks in series model relate to the inlet andoutlet of a CSTR and would indeed be consistent withthe no dispersion through the `knotsa assumption. Thesame implicit assumption is made in the ETIS model butis conceptually di$cult when n

Tis not an integer.

The extent of dispersion through the `knotsa is onlylikely to be signi"cant where both source and sink`threadsa are themselves characterised by high disper-sion. The qualitative e!ect of removing the central `knotajoining 2 CSTR `threadsa is shown in Fig. 2. By analogywith the previous discussion regarding the suitability ofthe ETIS model for the characterisation of individual`threadsa it may be deduced that the no dispersionthrough `knotsa assumption holds within experimentalaccuracy when n

T'15 and Pe'28.97 in the more plug

#ow `threada. Quantitative assessment of the breakdownof the no dispersion through the `knotsa assumption isnot reported here but might be approached via consid-eration of the Kurtosis of the RTDs.

Ordering. Using the simple conservative `dyea -tracingmethods reported here it is not possible to order the`threadsa in a reactor network without additional struc-tural information. To make progress with respect toordering of model elements in the absence of structuralinformation it is essential to employ a non-conservative`dyea -tracer method with a second-order decay process.The application of such techniques will not be discussedhere.

8. Conclusions

The ETIS model in conjunction with the reactor net-work structure has been shown to be a versatile methodof describing the characteristics of a small but diversegroup of reactors. The ETIS model has been comparedwith the conventional tanks in series approach and hasbeen found to be superior due to the elimination of thequantisation which is inherent in the latter approach.The ETIS model has also been compared with theThomas and McKee `closeda solution to the dispersionmodel. The `closeda dispersion model has been shown todi!er signi"cantly from the ETIS model in the range0(Pe(28.97 and 1(n

T(15. The `closeda disper-

sion model is also found to have the advantage ofrigorously de"ned inlet and exit boundary conditions.Combination of the ETIS model with the reactor net-work structure permits a considerable increase in theversatility of networks with out the concomitant increasein numerical intensity which characterises networks ofCSTRs.

Notation

C dimensionless concentration DimensionlessD eddy di!usion coe$cient L2 t~1

E(h) exit age distribution function Dimensionlesse base of natural logarithms

(2.718...)Dimensionless

f #ow fraction DimensionlessI `mixinga index Dimensionless¸ characteristic length LN

Ddispersion number Dimensionless

nT

number of tanks in series Dimensionlessp precision DimensionlessPe peclet number DimensionlessQ liquid feed rate L3 t~1

S a term in the series solutionof the closed dispersionmodel

Dimensionless

¹ mean residence time tT Hydraulic mean residence

timet

t elapsed time ttM mean of the residence time

distributiont

u velocity L t~1

< volume L3

v volume fraction Dimensionlessz axial displacement L

Greek letters

! gamma functiong E$ciencyp 3.141592# strand dimensionless

residence timeh dimensionless time DimensionlesshM mean of the dimensionless

RTDDimensionless

p2 variance of the dimensionlessRTD

Dimensionless

Subscripts

10 pertaining to the recovery of10% of the injected `dyea


90 pertaining to the recovery of90% of the injected `dyea

c `closeda dispersion modele experimentalH hydraulici inletM merrilm modalo `opena dispersion modelR whole reactors thread indexsc short circuiting¹ pertaining to the TIS and ETIS

models

Acknowledgements

The author would like to acknowledge the assistanceof the following colleagues who contributed their experi-mental data for the case studies. Ms. Laura Burrows,Experimental data from the internally recirculatingCSTR. Mrs. Barbara Gray, Experimental data from thepacked bed reactor. Mr. Tony Robinson, Calculation ofthe simulated RTD for the Large CSTR.

References

Battaglia, A., Fox, P., & Pohland, F. G. (1993). Calculation of residencetime distribution from tracer recycle experiments. Water Research,27(2), 337}341.

Elgeti, K. (1996). A new equation for correlating a pipe #ow reactorwith a cascade of mixed reactors. Chemical Engineering Science,51(23), 5077}5080.

Kramers, H., & Alberda, G. (1953). Frequency response analysis ofcontinuous #ow systems. Chemical Engineering Science, 2, 173}181.

Levenspeil O. (1972). Chemical reaction engineering, (2nd ed), ISBN0-471-5306-6 New York: Wiley.

Levenspeil, O., & Smith, W. K. (1957). Notes on the di!usion-typemodel for the longitudinal mixing of #uids in #ow. Chemical Engin-eering Science, 6, 227}233.

Monteith H. D., & Stephenson J. P. (1981). Mixing e$ciencies infull-scale anaerobic digesters by tracer methods. Journal WPCF53(1).

Smith, L. C., Elliot, D. J., & James, A. (1993). Characterisation of mixingpatterns in an anaerobic digester by means of tracer curve analysis.Eccological Modelling, 69, 267}285.

Teefy S. (1996). Tracer studies in water treatment facilities: A protocol andcase studies. ISBN 0-89867-857-9. AWWA Research Foundationand American Water Works Association.

Thomas, H. A., & McKee, J. E. (1944). Longitudinal mixing in aerationtanks. Sewage Works Journal, 16(1), 42}55.

Tomlinson E. J., & Chambers B. (1979). The e!ect of longitudinalmixing on the settleability of activated sludge. TR122 WRc July1979.

Yagi, S., & Miyauchi, T. (1953). On the residence time curves of thecontinuous reactors. Chemical Engineering (Tokyo), 17(10), 382}386.


martin (2000) interpretation of resicence time distribution data.pdf

Documents