Verification of a two-phase holdup relation

403 symmetnc and the maxlmum of dC*/dr, as shown m Fig 2, would not be at the same r for vartous values of Pe, The efiluent time, 7 YS Pe, or 7 vs H,* at C* = 0 S would not be a straight honzontal line for erther case A curved hne instead of a straight hne would be expected In Fm 3 and 4, particularly for C* = 0 5 A no&near adsorptlsn isotherm wouid probably have the same effect on the concentration profile and the effluent time as dls cussed above for these coefficients

IN-JAE CHUNG HSIEN-WEN HSUS

Department

of Chemwal and Metallurg~cat Engmeenng The Unruersrty of Tennessee Knoxodle, TN 37916 USA

Pf Pei

constant concentration of solute m the solution of mobile phase, moles//cm3-solution concentration of solute In the particle pores, moles/cm3 solution mitral concentration of solute, moles/cm3-solution dispersion coefficient, cm*/sec dlffuslon coefficient, cm*/sec theoretlcai plate height, cm mass transfer coefficient, 1 /set mth term of Hermlte polynomial adsorptlon rate constant, 1 /set desorptlon rate constant, I /set phase equlhbrlum constant adsorptlon equlbbrtum constant as defined m eqn (29) column length, cm concentration of solute adsorbed, moles/cm’-solution Peclet number of column as defined m eqn (14) Peclet number of adsorbent particles as defined m eqn (14) distance from the center of the partrcle m the radial dIrectIon, cm variable of Laplace transformatton of I time, set mterstlhal velocity of solution, cmlsec axial distance from the entrance of column, cm

*Author

to whom

quantity

Subscript

a

C, D D, H H, H,,, (x1 k, ko K K, L

Q void volume fraction of the column c standard devlatron 4 overall porosity III column as defined m eqn (29) q Gausslan probabihty density fun&Ion as defined m eqn (15) @ Poisson probabdlty density function as defined m eqn (20) 7 dImensIonless time as defined m eqn (14) Superscrpt * dlmenslonless

NOTATION

c,,,

Greek symbols ls porosity of packed sphere

correspondence

should be addressed

D m i R

delay time mobile phase stationary phase the moment of order k retention time REFERENCES

111Buffham B A, Glbllaro L G and Rathor M N , A ICh E J 1970 16(2) 218

PI Glddmgs J C , Dynamrcs

of Chromatography, Part I Marcel Dekker, New York 1%5 [31 Gluekauf E , Trans Faraday Sot 1955 51 34 [4l Ham&on P B , Bogue D C and Anderson R A, Analy Che I%0 32 1782 [5] Khnkenberg A and S]enltzer F , Chem Engng Scr 1956 5

[6] kzera E , J Chromatogr 1965 19 237 [7] Laptdus L and Amundson N R , J Phys Chem 1952 56 984 [S] Snyder L R , J Chromat 1968 34 455 [9] Snyder L R , J Chromat So 1%9 7 352 [IO] Van Deemer J J , Zulderwerg F J and Khnkenberg A ,

Chem Engng Scr 1956 5 271 [II] Hsu H W and Chung I J , J Chromatogr 1977 138 267 [12] Gelringer H , Muthematrcal Theory of Probabrhty and Statrstrcs Academic Press, New York (1964) [13] Chung I J , Ph D Dlssertatlon, Umverslty of Tennessee, Knoxville (1976) [14] Grushka E , / Phys Chem 1972 76 2586 [15] Glddmgs J C , J Phys Chem 1955 59 416

Verification of a two-phase holdup relation (Received 11 August 1977, accepted 18 August 1977)

Recently a general phase flow [ 11

holdup

where the dlstrlbutlon

equation

coefficient 1 I

has been derived

for two

However the derlvatlon wtmh was presented tn the development of relation (1) was not ngorous m the sense that it appeared that Ro,, vr3 and A, were arbltartly introduced into the working wlthout adequate Justficatlon The result was that It appeared that the final relation 11) was forced Into the form

C,, was defined as

(pL - &G)rG ti dA

C,= &_-&Ti

j

rG dA]

[k

A3 and the mltlal function

B was defined as

IcdA]

‘2)

&

A3 which

Zubcr

and

Fmdlay[Z]

= f( G3) and

others[3]

(4) had

found

to fit

Shorter Commumcatlons

404

expernnental data This IS not the case as shown by a more detailed working of the derivation By equatmg the mass velocity flux through an area S m the mixture region, Nguyen and Speddmg(l] showed that

(pr.-&)ro6dS-

(PL-~GO)QGS=

Is

s In the first term on moderate pressures that fi dS = fin dS cross-sectlon of the becomes

I s Integration

p’U’dS

(5)

the right hand srde pr. - & = a constant at and can be taken out of the Integral Note where n IS the unit vector normal to the flow condurt S, and 0 n = VT Thus this term

(PL--PGC)~GU~S=P~-&

I

rs vTTsdS

(6)

s

by parts leads to

VTs, Eos and As into eqn (1) IS amply justified The distnbunon coefficient G of eqn (11) ~111 shll possess the same value as the orlgmal C, of eqn (2) but IS cast IIIa new form The dlstrtbutton defined origmally as the ratio becoefficient C, was tween the mean value of the product of density difference, gas phase structural parameter and velocity and the product of their mean values The distribution parameter 4 on the other hand 1s tn place density dltrerence the ratio of the mean to that of the mean delivered density difference plus a correction for the holdup variation with flow Thus m the hmlt C$ wtll be 1 0 and as the integral term can be expected to be negative C’$ wdl m general be larger than 1 0 Acknowfedgemenl-One D S I R of New course of this work

Deportment of Cbemlcal and Materials .??ngJneenng Unlversrty of Auckland New Zealand

(7)

notmg that &

= (l/A,)

I ro dA, substltutlon A,

gives

(8)

Thus eqn (5) becomes

@L - 6~)

QGs

=

(pr - 600)

-31

AsRosdVTs

-

E

s

-

&x As &s

-r--i p U dS

(9)

.s

dlvldlng by (PL - boo), R, s and As gives (10) where Cjj IS the dlstrlbutlon

coeflictent

q=PL-PO

k-b0

(11)

of us (J J J C ) would hke to thank the Zealand for financial support durmg the

P

L

SPEDDING J J J CHEN

NOTATEON

conduit cross-sectlon area, Lz mttlal function defined eqn (3). L/T dlstrlbutlon parameter defined eqn (2) dlstrlbution parameter defined eqn (11) umt vector volumetric flow rate, L3/T local holdup or structural parameter time average holdup surface m nuxture region, L* velocity vector at a point, L/T fluctuation vector of velocity at a point, L/T time-average, area-average velocity, L/T fraction density, M/L’ time average denstty, M/L‘) fluctuation component of density hi/L’ gas density as mass rate/time average volume

rate, M/L3

Subscripts 1 all gas region 2 all hquld region 3 mixture region volume z gas L hquld area ii surface m mixture region REFERENCES

s

III Nguyen V T and Speddmg

and the uutlal fun&Ion B IS unchanged and 1s given by eqn (3) The rest of the orlgmal derlvatlon LS the same and leads to a form slmdar to eqn (1) m which Cz replaces C, In the above derlvatlon the seemmgly arbitary mcluslon of

P L, Chem Engng Scr 1977 32 1003 [2] Zuber N and Fmdlay J A , Trans A S M E 1%5 87C 453 [3] Walhs G B , One DJmensronai Two-Phase Row McGrawHII], New York I%8

ChemrcnlEnsrneenn~ Scrrnce 1978 Vol 33 pp 404-407 Pergamon Press Prmted I” Great Bntam

The design of experiments for the determination of kinetic parameters depending on the type of laboratory reactors used (Recerved

30 June 1977, accepfed

For the determmatlon of kinetic parameters (e g actlvatlon energy E and preexponentlal factor A) many authors have used stattstlcally desIgned experlments[l-81 In these studies, a kmetlc model was taken prlmarlly Into conslderatlon Attention was not

20 July 1977)

focused on the d&rent types of reactors m which the kmetlc studies were performed In addttlon to plug-flow tubular reactors (PFTR) or stIrred tank batch reactors contmuous stirred tank reactors (CSTR) and recycle reactors are being mcreasmgly used