The dynamic response of a single catalyst pellet

Letters to the Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . The dynamic response of a single catalyst pellet (Recerved 26 February 1979) Dear Srs, In a recent artu9e[l] Burghardt and Smith have analyzed m detail the unsteady state performance of a Wcke Kallenbach type apparatus which can be used to measure effective dlffuslon coefficants The mathematical formulas developed m 011s piece of work nicely summarrze the moments of unpulse responses for many possible expenments that can be conducted with th1.s type of apparatus However there are several pomts which should be made m relation to this paper Two mam mcentives extst for adopting an unsteady state testmg technique. the first IS that a potentmlly major part of the catalytic actrvlty assocmted with a given catalyst pellet may reside m Its dead ended pores Thus dead ended network plays no part and hence IS not reflected m any steady state dlflusron expertment, m prmclple, therefore, dlffuslvlhes determined by transient means are to be preferred over theu steady state counterparts Secondly, a transient testmg technique frees the expenmenter from the confinmg geometry necessanly assoctated with a Wicke Kallenbach type cell, for single pellets thts freedom has seldom been utlhzed, noteable excepttons bemg an early expenmental paper by Gomng and DeRosset[2] or the general theoretlcal treatment of Greco et al [3,4] If neither of these mcentives for adopting an unsteady state dtiusion experiment are recognized and exploited. there IS little pomt m conducting the transient test, conventional steady state diffusion measurements are far easier to carry out In Ref (11 the authors have certamly not exploited the potentud geometnc freedom whrch 1s avadable, the catalyst pellet IS shll housed m a cell: whtch effectively seals the curved surface of the pellet Commercial pellets, particularly If they are small, cannot be tested m this fashion Though expressions for the first moments of the system impulse response are generated m[l] Burghardt and Smith observe that evaluatmg a dtiusron coefficient from a zeroth moment “could offer some advantages over usrng the normahzed first moment, smce zero moments are usually more accurately measured than first moments” This procedbre 1s more stralghtfonvard, of that there IS no doubt, but m domg tlus It must be reahzed that the authors have &scarded the one remammg advantage of a transient test, namely that it reflects the presence of a dead ended porous network That this IS so may not be tmmedrately evident, but It can be readily appreciated d eqns (2 3) and (2 5) of [l] are wntten for the steady state Adoptmg Burghardt’s and Smrth’s notation we have

F, C,, - F, CA, -I-AD (CAL- CA,) = L

-

&CA,

-

m

ccAL;cAm) =0

where KI = (AD)IF,L

cA,,(l+K3G~

(2)

where (CAL - C&L is the steady state concentration gradient Ehmmatmg either C,, or CAL allows the followmg equations to be wntten C,, IS an as yet unspecfied forcmg fun&ton

=E

K

l+K,+K2

’

F2L

When C, IS a umt step and IS therefore set equal to unrty m eqns (3) and (4) the expresslons for C,, and CA, wdl ave the steady state values of the unit step response functions from the upper and lower chambers of the dlfiuslon cell, these values are of course the values of the zeroth moment generated m [l] an observation confirmed by notmg that the first two entnes m Table 1 of Ill for zeroth moments are exactly the expresslons (3) and (4) mven above when the normaltzabon factor M 1s umty What the mathemaacs IS telling us 1s that the amount of tracer which actually dtiuses through the pellet (I e the zeroth moment of C,‘) 1s mdependent of whether dead ended pores are present Hrlthrn the pellet However the average time at wluch the tracer emerges having d&used through the pellet (1 e the mean of C,,) IS dependent on any dead endedness If ddfusion coefficients determmed independently from zeroth and first moments are m close agreement It IS a sure sign that the dead ended pores play no stgmlicant role m the transient dlffuslon test Conversely d values dtier appreciably the dead ended structure 1s havmg a slgnlficant effect on the system response, m the latter case one must question the vahdity of a mathematical model employmg only a smgle dlffuslon coefficient and possibly resort to a more complex model for instance of the form proposed by Gtbdaro et aI [51 Incidentally both the papers of Grbdaro et al (51 and Gorrmg and DeRosset[Z] predate Refs l-3 m (11 which Burghardt and Snuth suggest were the o-al development of the theory behind single pellet transient d&fusion testmg techmques, and papers published by Goodmght et a! [6,7l m 1960 and 1961 contam ideas closely allied to the above devetopment SIMON P WALDRAM MILORAD P DUDUKOVIC

Bpartment of Chemrcal Engmeenng Washutgton Unwerslty St Lows. MO 63130, US A NOTATJON pellet cross sectional area CA, mlet concentration of dlusmg species to upper chambers CAL outlet concentration of dlfFusmg species from lower chamber of dlifusmg species from upper CA. outlet concentration chamber D effecttve dtlfusion coefficient FI flow rate through upper chamber FZ flow rate through lower chamber A

0

(3)

1361

KI K2 L iif

2::: length’of Pellet normahzation factor

1362

Letters to the Editors REFERENCES

[l] Burghardt A and Smrth J M , Chem Engng Scf 1979 34 267 [2] Gorrmg R L and DeRosset A J , J Catuf 1964 3 341 [3] Greco G Jr, Iono G , Tola G and Waldram S P , Tram Instn Chem Engr 1975 53 55 [4] Iorlo G , Greco G Jr and Waldram S P , Trans bstn Chem

[5] Glbdaro L G , Gloia F and Greco G Jr , Chem Engng J 1970 185 [6] Goodmght R C , Khkoff W A Jr and Fatt I, J Phys Chem 191% 64 1162 [‘I] Goodntght R C and Fatt I, J Phys Chem l%l 65 1709

Engrs 1976 54 199

Reply to Waldram and Dudukovw (Recerved 4 June 1979) Dear Sus, Waldram and DudukoviC m their letter have added helpful mformatlon concerning dynamic studies of Muston m single catalyst pellets Whde we agree that a potential advantage of dynarmc measurements IS the evaluation of dead-end pores in catalyst pellets, this advantage has not, to our knowledge, been effectively exploited Perhaps this IS because dead-end pores constitute but a very mmor part of the pore volume A more likely explanatron may be that it IS d&cult to make expenmental studies accurate enough to dlstmguish between ddTuslvltles obtamed m steady-state and transient expenments Therefore, up to the present this mcenhve for dynamrc measurements appears to be of limited value Our experience with dlffuslon measurements have indicated that the advantages to using a dynamic techmque are of a

ddferent sort Dynamic experiments can be camed out very rapidly so that much more mformatron IS obtamed m the same amount of expenmental tune than with steady-state studies For example, both eqmhbnum and rate parameters can be obtamed from the same experimental mformahon Further, dynamic experiments require much less material than steady-state experunents These advantages were not clearly stated m our paper We regret that the pertment papers mentioned m the letter of Waldram and DudukovlE were not noted m our paper

Department of Chemrcal Engmeenng Vnwersrty of Cahfomia Davrs, CA 95616, V S A

Droplet diameters in agitated liquid-liquid

J M SMITH

systems

(Received for pubhcataon 25 June 1979) Dear sirs. The recent note by McManamey[l] has summarlsed published data on droplet sizes in stvred tanks by a Kolomogoroff relahonshrp of the form &=K

go6

go4p02

The constant K was shown[l] to be approximately 0 22 when the specific power dissrpation q was based on the Impeller’s swept volume, m which all the drop break-up was assumed to occur This relatlonshrp was vahd for low hold-ups at which coalescence effects were neglmble The above value of the constant K IS of slmdar magnitude to the value of 0 36 obtamed by Baud and Lane[Z] for hqmd droplets formed In a reclprocatmg plate extractron column Break-up was assumed to occur m the entire column volume m this case[2] Recent measurements by Khemangkom et af [3] of the droplet sizes m a pulsed perforated plate extraction column are consistent with K = 0 18 Kuble and Gardner141 found that the maximum droplet sues m turbulent hquld-hquld pipe flow were correlated by a Kolmogoroff-type equation with a dzmensionless constant of 0 725, srmdar to the value observed by Clay[S] for hquld-liquid dlsperslon m a rotatmg cyhnder apparatus If It 1s assumed that d,, - 0 5d,.,, these results[4, S] would mdrcate that K-036

It therefore appears that the parameter K In eqn (1) has values m the range of 0 18-O 36, I e the same order of magmtude regardless of the means of turbulent amtatlon

Chemrcal Engmeenng Department MeMaster Vnrversrty Hamilton, Ontuno Canada

M H I BAIRD

NOTATION

d d msi K * (r P

Sauter mean droplet diameter maximum droplet drameter dImensionless constant power chssrpatlon per unit volume interfacial tension density of conhnuous phase

REFERENCES [l] McManamey W J , Chem Engng Scr 1979 34 432 [2] Bsurd M H I and Lane S J , Chem Engng Sx 1973 28 947 [3] Khemangkom V , Muratet G and Angehno I-I , Pmc Zntl Solvent fitn Conf, Toronto, 1977, paper 2% (Canad Inst Mm and Met, to be published 1979) [4] Kutue J and Gardner G C , Chem Engng Scr 1977 32 185 [S] Clay, P H , Pmc Sect Scr K ned A&ad Wet 1940 43 852