The residence time distribution for laminar flow in a helical tube

628

Letters to the Editors

Deparrment

of Chemrcal Engmeenng Un&srty ofRochester Rochester, NY 14627, US A

TERRENCE

L DONALDSON

[3] Donald son T USA

L

and Qumn 1

1974 714995

A.

[41 Donaldson T L , Ph D Ihssertauon.

RRPgRENCEs 111 Tsao G T , Chem Engng Set 1972 27 1593 [21 Kernohan J C , Blochlm Bophys Acta I%5 % 304

vania t974 [Sl Danckwerts P V , Gas-Lqwd New York I970 WI Lee Y Y and Tsao 0 T , C&m

Proc

No:

Umversrty

Reactrons

Acad

Scr

of Pennsyl-

McGraw-Hdl,

Engng SCI 1972 27 1601

Carbon dioxide absorptfon in the presence of carbonic anhydrase (Recerved 29 July 1977) Dear

sirs,

Tsao, in Ls orrgmal artmle[l], speculated on two posslbdthes a “layer model” and a k, value of about one reciprocal second, tn explanung the discrepancy between the lack of enhancement m the lammar let expenments and the appreciable enhancement observed m the stared cell Donaldson[Z] IS apparently m favor of the second explanation One perhaps should repeat the experunents with enzyme preparations of known punty and kmetlcs constants Since the ks value In Ref [2] 1s only an order-of-magnitude esttmate. It cannot be considered a reliable basts for dlscnmmatmg the “layer model” from that of Danckwerts The range of uncertamty of the k2 value IS much larger than the range of the expernnental enhancement factors (varmd from 1 to 2 1) For instance. d the ks value were 0 22 x IO-’ Instead of 1 I x IO-’M-’ set-’ as Donaldson estuhated, the “predicted” enhancement factors at 67 rpm would become I 17 and 138 instead of I 68 and 2 35, respectively (see Table 1 of Ref [2]) In that case,

we would mdeed need sometlung hke the “layer model” to explam the experunental enhancement factors There has been no reported kmetics study on the carbonic anbydrase Iunetrcs In phosphate buffers Tsao’s data[l] were intended only for comparison among themselves It IS a dangerous business m elaboratmg on data from another laboratory without a full knowledge of the source and purity of the enzyme preparatron mvolved

School of Chemrcal Engmeenng Purdue Universrtv West L.ajayette. iN 47907 USA

GEORGE

T

TSAO

-cEs

T , Chem Btgng Scr 1972 27 1593 [21 Donaldson T L , Chem Engng SCI 1977 33 627

[I] Tsao G

The residence time dfstribution for hminar flow in a helical tube (Recetved 28 Aped 1977, accepted 26 September 1977) Dear sm. In a recent paper Nauman[l] has cnuclzed an earlier calculation by the present author[2]. of the residence time dtsmbuhon for Ideal lamutar flow ma helix The results of these calculations are compared tn Table 1, which also includes the results of an even earlier Independent calculation of the E(B) funchon by GalIoway[3] The agreement between the three calculatrons may been seen to be remarkably close, part~cularty m view of the fact that m Nauman’s analysu the F function IS calculated first and the E function IS then obtamed by dtlferentlatlon of a fitted fuactlon whereas, m the calculauons of Ruthven and Galloway, the E fun&on was obtamed due&y from the numerical scheme Whether or not the dtierence represents, a “sIzeable error”, as clalmed by Nauman, depends on one’s perspective The range of the earher calculation of the present author (0 < 0 c 2 2) covers about 96% of the F curve and within this range the approximate expressions (19< 0,613 E(8) = F(8) = 0, F(g) = 1 - 0 25ge2 “‘) provide a fan @>0613 E(e)=o7058-3”’ representation of the computed curves Indeed the error in these expressions does not significantly exceed the margm of uncertamty mhcrent III the numerical analysis That deviations become srgmflcant if these cxpresslons are extrapolated beyond the range of the calculahons from which they are derived IS not surprising Such an extrapolation IS lmpbcd m the calculation of the first and second moments The norrnahzation criterion

suggested by Nauman (l; gE(g) de = 1) IS therefore clearly mapproprmte unless the E(g) curve IS avadable over the entrre range of B (Perhaps the lmutat~on on the range of B should have been stated more exphculy m the ongmal paper although It IS quite obvious from Fig 2) For many practmal purposes tbe tad of the RTD curve for 8 > 2 IS of minor importance Thus, for example, the dimensionless holdmg times required for 50% and 9046 conversion m a first order system, calculated from the approximate E(B) function given by the present author and the (presumably) more accurate expression Oven by Nauman, are essentmlly the same It IS dlfiicult, f not unposslble, to make rehable tracer measurements of the till of the RTD curve smce, In Uus region, the effects of molecular ddIuslon become Increasingly important in any real system Over the experimentally accesstble region the results of Trrvedl and Vasudeva[4] show good agreement with the sunphtied theory In summary, over the range 0 613 < 0 < 2 the recent calculation of Nauman gwes results wluch are in substantml agreement with earher calculauons (which were hmrted to this range) The tmproved numtrtcal scheme adopted by Nauman made it possible to extend the calculauon to somewhat h&e.r values of 0 (F(8) > 0 %) and, in this reson, the results show, as IS to be expected, some devmhon from a snnple extrapolation based on Ruthven’s approximate expression However, in view of the inherent hmi-

Table 1 Compartson

e

Nauman[l]

0 613 0 672 0826 1022 1 374 1 74 2 22

0 0 214 0548 0744 0884 0 93s 0%5

II

Letters to the Editors

629

of the rental RTD curves for ideal lammar flow m a hehcal tube F(O) Ruthven[2]

Nauman[l]

E(8) Ruthven [2]

4 37 3 II I 59 066 022 0096 004

46 32 146 0 65 021 0086 0031

0 0235 057 0764 0897 0947 0973

The definmons used here are those used by Levensptel[S] uses f(e) for E(B) and 1 -F for F(B) B IS the dtmenslonless tattons Imposed by the use of Dean’s velocrty profile and the mcreastng Importance of molecular dtffuston at large ttmes, the value of such calculatrons may be constdered somewhat academtc D M RUTHVEN

of Chemrcal Engmeenng Unrversrty of New Brunsmck Fredencton N B Canada Department

Galloway [3] 41 (e=o62) 3 12 1 45 064 022 -

F(B) = Jt E(B) de Nauman residence time REFERENCES

[I I Nauman E B . Chem Engng Scd 1977 32 287 [2] Ruthven D M , Chem Engng SCI 1971 26 1113 131 Galloway F M , M SC Thests, Case Insbtute of Technology, Cleveland, Ohto (t%2) [41 Tnvedt R N and Vasudeva K , Chem Engng Scr 1974 29 2291 [S] Levensptel 0 , Chemrcal Reactron Engmeenng, 2nd Rdn, p 257 Wdey, New York 1972

On the use of S&fan-Maxwell’s diffusion equations for expressing non-isobaric porous media (Recerved

6 Julv 1977 recetved forpublrcotron

Dear Sns In a recent art(cle[l] Hesse reports a solution for multt component gas dtffuston and chemtcal reactton with volume change m porous catalysts Two of the assumpttons whrch are consldered there are (I) the system IS (sobanc (2) total fluxes are gtven by tsobarlc Stefan-Maxwell diffusion equations Let us consider the first assumptton Bemg a catalyst pellet a system wrth constant volume control. any change m the net number of moles wtll naturally produce a pressure gradtent wrth an associated VISCOUSflux The effect of several parameters on the pressure gradrents established m the porous pellet as we8 as the non-tsobatc effecttveness factor have been analyzed for binary systems[2 31 Thus the expression “wrth volume change which IS quoted frequently tn related literature IS to be considered as a mere settled practice for takmg into account the pressure gradient which IS developed m the system Let us now constder the second assumption The mathematical frame for analyzing dtffusron transport m porous media seems now to be quite well estabhshed, both from theoretical and expertmental points of view [4 5-71 Restrtctmg ourselves to tsothermal condtttons and absence of surface flux the followmg set of equations IS a consistent model Laajleagst one of them) for analyzing mass transport in porous Total molar flux of a given speats diffusive and VISCOUStransport

results from the addition of

N, = N,O + N,”

(I)

Both partial fluxes are m turn gtven by the followmg constttunve equations

- x,N,~ -x,N,~ zz

,=I

ND

(2)

14 November

total fluxes in

1977)

NY,_Epx,dC w

dz

where Q, dnd D,,K are respectively the molecular and Knudsen effective coeffictents (n the so called “second approxtmatton ‘, B IS the permeabthty and p c and x, are respectrvely, pressure, molar concentratton, and molar fractton On the other hand an equation relating total fluxes may usually be wrnten for a gtven system For a catalyst pellet thts equation mvolves the reactton stotchlometnc coeffictents (see, however [S] I For the particular case of molecular dtffuston regtme (whrch IS the one cons&red tn Hesse’s analysts[l]), by mtroducmg eqn (3) mto eqn ( 1) and the result mto eqn (2). It arlses

wh(ch IS the generic Stefan-Maxwell dlffuston equation written wtth total fluxes So tt 1s a mere fortunate comctdence that Stefan-Maxwell equattons whrch are ualrd for systems devord of walls could be applied here For porous systems, these equattons are not of universal valdtty if tsobartc condrtrons are assumed This tmplies (a) 9, = constant, (b) dc,/dz = c dJdz Fortunately agam, unless the permeabthty be extremely low, the pressure vartatron developed m molecular dtffuston regtme IS not very rmportant, m spite of VISCOUSflux beotg an wnportant fractron of total flux [2] Consequently, m spite of lsobanc Stefan-Maxwell ddfoston equations being not m general valid for porous medta they should be applied tn those cases where molecular dlffuston regtme prevads and permeabtltty of porous soltd IS not very low