Extractive Fermentations

Designand MathematicalDescription
of DifferentialGontactorsUsed in
ExtractiveFermentations
Stephen R. Roffler, Charles R. Wilke, and Harvey W. Blanch
Department of Chemical Engineering, University of California,
Berkeley, Californ ia 94720
Accepted for publication July 9, 1987
The per f or m anceo f d i ffe re n ti a cl o n ta c to rsfo r use i n extrac t iv e f er m enta ti o n i s c o m p l i c a te d b y th e effects of
pr oduc t f or m at i o n i n th e c o n ta c to r.Wh e n p roduct form a t i o n i s s i g n i f i c a n ta
, p p r o x i m a t ea n a l y t i c a sl o l u t i o n s
a r e pr es ent edf o r th e p e rfo rma n c eo f th e c o ntactorfor
two limitingcases:high and low substrateconcentrations.
W hen pr oduc t s a re fo rme d a t a c o n s ta n t ra te , there i s
a m i n i m u m r a f f i n a t es o l u t e c o n c e n t r a t i o nt h a t c a n b e
o bt ained,in c on tra s tto th e b e h a v i o ro f a c o l u mn i n the
a b s e n c eo f p r o d u c t f o r m a t i o n . G e n e r a le q u a t i o n sd e scr ibingt he beh a v i o ro f th e s y s te m fo r p ro d u ct formati on wit h bac k mi x i n gi n b o th p h a s e sa re p re s e nted.The
ca s eof a s t r ippin gfa c to rn o t e q u a lto u n i ty i s c onsi dered.
INTRODUCTION
Many important industrial fermentationsare adversely
affected by the accumulation of toxic metabolitesin the
broth during fermentation. Most strains of Saccharomyces
yeast, for example, are completely inhibited by ethanol
concentrationsabove I l0 glL.t In the acetone-butanol
fermentation,a butanolconcentrationof only l0-15 glL totally
inhibits product production and cell growth of Clostridium
acetoburylicum.2'3''
Product inhibition decreasesfermentor
productivity, limits final product concentrationsand requires the use of dilute substratesin the fermentor. Large
fermentors are thus required and product separationand
stillagehandling costs are high.
The effects of product inhibition can be reduced by removing toxic metabolitesfrom the broth during fermentation. One versatilemethod of in sira product removal is
extractive fermentation. Here, an organic solvent is contacted with the broth during fermentation;toxic products are
extractedinto the solventand product inhibition is reduced.
The production of acetoneand butanol by extractive fermentationhas recently been demonstratedin laboratory
scalefermentors.t-tFermentorproductivity was increased,
concentratedsubstratescould be used, and final butanol
concentrationin the extractionsolventwas higher than can
be obtained in regular batch fermentation.
Extractive fermentation can be carried out in two basically different ways: extractionsolventcan be addeddirectly
to the fermentor or whole broth can be cycled to an external
vesselwhere solvent is contactedwith the broth. In both
(1988)
B iot ec hnology
and Bi o e n g i n e e ri nVgo,l .3 2 ,Pp .1 9 2 -204
O 1988J ohnW ile y& S o n s ,In c .
methods of extractive fermentation, the design of the processis complicatedby the continual formation of products
during extraction.Fournier8and Kollerup and Daugulise
have presentedmodels describingextractive fermentation
processesin which the extractionsolvent is addeddirectly
to the fermentor. In production-scalefermentors, however,
high agitationis requiredto dispersethe extractionsolvent
in the broth. The high shearrate generatedat the impeller
can create stable emulsionsof solvent in the broth6leading to difficult separationof solvent and broth in later unit
operations.It may thus be preferableto contactthe solvent
and broth in an external extractionvessel. In many types
of differentialextractiondevices, such as reciprocating-plate
or pulsedcolumns, shearrate is relatively uniform over the
cross-sectional
areaof the column. Agitation in the column
can be adjustedto provide good mass-transferwhile preventing the formation of stableemulsions.
In this article, approximateanalytical solutionsdescribing performanceof differential contactorsin extractive fermentationare derived. The influenceof different operating
variableson extractorperformanceis discussedand a simple
method of determining the significance of product formation
by the cells in the extractoris described.
MODELDEVELOPMENT
Figure I showsa schematicdiagramof an extractivefermentationprocess.Whole broth, containing viable cells,
is cycled from the fermentor to a differential contactor in
which extraction solvent and broth are contactedin countercurrent flow. Inhibitory products in the broth are extracted
into the solvent so that product inhibition is reduced.Extractedbroth is recycledto the fermentorwhile the products
containedin the solvent are recoveredby suitable means
such as distillation or back-extraction.
Mass balancesover a differential sectionof a contactor
with two phasespassingcountercurrentlyin one-dimensional
steady state flow and product production taking place in
the continuousphaseyield the following equations:
- dtr,
t,i7-
,, dr,
U,
"-
,
t
k q , o ( . . . -c I ) * r r , u u : 0
(l)
ccc 0006-3592t88t0201
92- 13$04.00
FERMENTOR
BROTH
RECYCLE
REGENERATED
SOLVENT
Figure 1. Schematicof an extractivefermentationprocess.
E..4:2* u
",
-Yn * *
dz,
dz
''ux*'
k o - a ( c ,c-f ) : o
(z)
where E,E, are effective turbulent diffusion coefficients and
16o6is the rate of product formation in the continuous phase.
In order to solve eqs. (1) and (2) the kinetics of product
formation must be known. Assuming that Monod kinetics
describe cell growth and that sugar is the limiting substrate,
the specific growth rate p is given by
Itr^Ct
tL:vffi,
(3)
For constantyieldsof cell massandproduct,the volumetric
rate of productformationcan be represented
by
Ypt, ltrrncsc,n
7P''*:ffi
Yprsftr,ncm
When the concentrationof substrateis high (c, ) K,),
the rate of product formation in the extractioncolumn is
constantand is given by eq. (6). Assuminga linear equilibrium relationshiprepresented
by
cl : mc,
eqs.(1) and(2) can be expressed
in dimensionless
form.ro
d2X
E
^ ^dX
- ,,ti-
#,.
^ n/r7 - rz\ ,
NI,P,B(X
Y)+
urt
rr
8,1cffi:
O
r , t q r + F N o , P y B- ( Y
x ): o
(11)
wherethe dimensionless
concentrations
are given by
(5)
r - -c 1
t,- m
- ccr i- m
^u -- cm
yc',
(r2)
u_cr-cl
(13)
_m(cr-cj)
m-1:@
Eliminationof Y betweeneqs.(10)and(11)gives
d4x
d3x
^dzx
dx
E-a#-BE-yE-6:0
(6)
u,
(9)
(10)
Assuming that the concentration of cell mass is constant
throughout the extraction column, the rate of product formation can be expressed as a zero-order reaction
TPrd:
Case l: High substrate concentrations
e)
Two usefulsolutionsto eqs.(1) and(2) canbe obtainedby
consideringlimiting casesof this kineticexpression.
When
the substrateis in excess(c, ) K,), the rate of productformationis given by
'*d:T
LIMITING CASES
(14)
where
where
,,:
Ypt,ltr^C^
3;:
const.
(7)
The secondlimiting caseis for low substrate
concentrations
(c, ( K,) whereproductproduction,againassuminga constantcell massin the extractor,is proportionalto substrate
concentration
ux
r n o o: 4 c ,
(8)
Equations(1) and (2) canbe solvedfor the two limiting
casesrepresented
by eqs.(6) and(8). The caseofhigh substrateconcentrationwill be consideredfirst.
a,:
B(P* - Pr)
(1s)
F:N*B(P,+FPy)+P,PyBZ
(16)
l:No,P,P,B2(I-F')
(17)
^
o :
v,LZFP.BNo,
EJco,
mct)
(18)
In eq. (18), 6 representsa dimensionless
rate of product
formation.The relative importanceof productformation
andmasstransferin the contactoris indicatedby the magnitudeof e, - 6ly.At largevalueof e,, concentration
changesin the contactorare due primarily to productfor-
ROFFLER,
WILKE,AND BLANCH:DIFFERENTIAL
CONTACTORS
193
mation while at small values of e,, concentration changes
are due to mass transfer. The solutions to eq. (10), (11),
and (14) are given by
x : A, * Are\zz * Are\tz * Aoex+z- a,z
wherek : 0, 120, and240ofor trr, trr, andtrorespectively
and
("\'
Y : A, + Arare\2z + Araret'tz * Aoaoexoz
/
1 \
- ".\, *
o : \ t *, TB
(28)
qB
n : /o\'
G u+rT) + t
(2e)
(19)
Q0)
W)
,,:
(30)
cos-,[",)
where the tr, are the roots of the characteristic equation
L3-41,2-BI-7:0
(21)
and the ai are given by
tr,
Ii
Qr: a
r *,
"o @
(22a)
or,o
dd i :
_F(l
-
I\I/P,B)
L + I\t/PyB)
(22b)
The coefficientsA, areobtainedfrom the boundaryconditionsgivenby MiyauchiandVermeulen.ttTheseboundary
conditionsassumestep changesin the concentrations
of
the continuousand dispersedphasefeedsas they enterthe
columndue to backmixing,while the phasesleavingthe
extractorexhibit no concentration'Jumps". The coefficientsA, aregiven by
i
) I",a,A, : e,
.'-
(23)
This solutionis valid if q' - p'< 0 which is normally
satisfiedin this application.The full solutionto eqs. (19)
and(20) canbe simplifiedby taking advantageof the propertiesof the roots tr,. tr, is large and positive, 1., is large
and negativeand .tr.is small and can be positive or negative.r0In orderto obtainsimplifiedsolutionsfor Xr andI0,
eqs. (23) to (26) are solvedfor the A, andsubstitutedinto
eqs. (19) and (20). On dividing numeratorand denominator by exp(),r),all terms containingeither exp(-l,r) or
exp(),r)are neglected.After simplifying the equations,one
obtainssimple approximationsfor the concentrationsof
soluteexistingthe extractor.
Xt -
Y':
(24)
r - +P,B
I
t l
.
++-l
,^,t,:
",lr
*,r)
No,F
P,B )
3",(t
a
194
cos(ul3+ k)
e^oc,* c,
(32)
Cr : Ftrrar(),, - tro)
(33)
T^
- trr)
7a3aa(lto
(34)
r / l
*F\";*
(26)
(27)
(31)
-']
c,:('-rft("+
il
es)
whereet : 1 and trr : 0. The A, canbe obtainedfor particular problemsby solving eq. 23-26 numerically.The
lengthof extractorrequiredfor a givenseparation,however,
must be found by iteration sincelength is containedin all
tr, terms. Approximateanalyticalsolutionsfor the outlet
concentrations
Xr and Y0, however,can be obtainedby using a methoddescribedby Hanland and Mecklenburgh.r2
The perfonnanceof extractioncolumnsin extractivefermentationcan then be estimatedwithout the needfor numerical computation.The length of contractorcan also
be calculated,althoughstill by iteration, in a straightforward manner.
The roots of the characteristicequation),r, ),r, and Lo
are given byro
t r , : T * 2!p
e^ocr(e,C,* 1) * e, CrCu + FCz
Cz:
i=2
*('-*")o,:
e,C2Ca
wherethe constantsC, to Cuarc given by
a
) L,e^e,- e,
o ^ o C t l e , C r1* - 1 / F l l
Co= (l
/
Cr:{1
\
Cu:(1
l
1 \
ry* r*,)
-q,)(1.#) .#*
(3s)
(36)
*)(^:il tr (37,
-r{r-",G+.i)}
- t\*/ 1 -
I
r \
*
,,u ,*r)
(38)
Equations(31) and (32) give the outlet concentrations
of
productin the continuousand dispersedphasesfrom the
extractor.Theseequationsare thus useful for simulating
the performanceof extraction columns in extractive fermentation.Theseequationscan also be usedfor extractor designby making contactorlength explicit. Length is
madeexplicitby substituting),,I : L, andB - Lf Ho,into
V OL.
B I O T EC H N O L OGY
A N D B IO E N GIN EE R IN
G, 32,JU LY1988
eqs.(31)-(38)and eqs. (15)-(18).The ),i are foundby
solvingan equationanalogous
to eq. (21).
Whenproductproductionis negligible,eq. (31) reduces
to the solutionsgiven by Pratt.lo
,:irL
|rraoQt'a- Ij)Xt
"'lrtlj(L; tr;
r ; ) ( x ' - 1+
Equation(44) describesthe concentrationprofile of substrateover the lengthof the contractorwhen substrateconcentration
is low. Combination
of eqs.(44), (1), ( 2), and
(8) andconversionto dimensionless
form leadsto the following equations
(3e)
| /F)
dzx
il
E-r,Ur-No,P,B(x-Y)+
The contactor length can thus be estimated without iteration when the rate of product formation in the column is
small relative to the rate of product mass transfer.
dY2
*+
Case ll: Low Substrate Goncentration
t(dre^tt * dre*zz):g
(49)
F N o P y B ( x -r ) : g
(50)
dY
rrt;r+
where
When the concentrationof growth limiting substrateis
Yr,rkrc?
low (c, < K,), the rate of productformation,represented
(51)
t,
(
r
l - ^r')
by eq. (8), dependslinearly on the concentration
of substratein the continuousphase.It is thus necessary
to esti- The solutions to eqs. (49) and (50) are as follows
mate the substrateconcentrationprofile throughoutthe
x : A, * Are\zz* Are\zz* Aoe\tz* Bre^rz * Bre^zz
length of the contactor.Assumingthat the substrateis not
(s2)
extractedinto the dispersedphaseand cell yield and cell
massare constantover the contactorlength, a massbalY : A, + Ararexzz+ Ararexzz* Aoaoexoz
anceover a differentialsectionof column vields the fol* Brbre^tz + Brbre^r'
(53)
lowing equation
where the a, and 1,,are given by eqs. (22) and (27), a, B
are given by eqs. (15) to (17) and the B, and b, are
given by
+-+c*:o
8 .dz'
1 - u"' . dz
KrYr,r"'
( 4 0 ) and y
"'
form as
Equation40 can be expressedin dimensionless
d.s
d2.s
b i : r +No"
+ - ANorPrB
, r - € 4BiNo,PrB
(41)
E-',t;,-k,s:o
@? - qm? - F*? - y*,)
8; : {d,lPrB(FNo,- m,) - *?l
where
s :c+;
(42)
U,L,
*, ' :
(43)
KE,Y^
The solutionto eq. (a1) is givenby
(4s)
m'-PrB*-kr:o
andd, andd, are obtainedby substitutingeq. (44) into the
boundaryconditionsgiven by Miyauchi and Vermeulen.rr
e^ocr(l- r/F)
e^oc,* c,
* crrt) + en4cra,l
+' +
Lu,yr''1e^ocre,
* cr?r'
e^oc,
(s7)
vo-e^ocr+FC,
e^oct+ cz
*'
PrBe^2
.,
l
2
^ \ n , l C r 1 e ' , O*; o , ) * e ^ o C , , r l t , l
e^oC, T wz i=l
- mr)
(s8)
(46)
d2
where C, and C, are defined in eqs. (33) and (34) and
PrBe^r
e^t(P,B - mr) - (*r/*r)e^2(P,B - mt)
(47)
where
o;:1-
,,: *,{!(o,fttr,t7t2:
*r,t
t
+l@,8)'
(56)
(44)
wherem, andm2 areroots of the characteristicequation
e^2(P,B - mr) - (*rl*r)t*t(P,B
(5s)
Approximatesolutionsto eqs. (52) and (53) can be obtainedin the sameway that eqs. (19) and(2) weresimplified. Upon simplification,one obtains
Xt-
S : dp^rz * dze^22
dr:
?9t
D Pi
(s4)
+
ktfi/z
(48)
L^z
- Fb \, ("*7, P-B
mi
1l
(se)
D - # r * , - u+ )( r\- b )
DIFFERENTIAL
CONTACTORS
ROFFLER,
WILKE,AND BLANCH:
(60)
195
,,:(,
;)we#.#-,]
,u,,
-F)
-b,Ch.
o;:(1 V,""G-*")
r]
(62)
o,:u,(r.H-'('
H
solutionsfor low substrateconcentrations
exhibitedsimilar
behavior.The curvesin Figure 2 weregeneratedusing the
parameters
listed in Table I. In general,the approximate
solutionsdeviatedless than SVofuomthe exact solutions
whencontactorlengthwas greaterthan4 feet and No,) Z.
In practice,the contactorlength is usually much greater
than4 feet andthe differencebetweenthe approximateand
exactsolutionsis negligible.The approximate
solutionsare
thus generallyaccuratefor practicalcasesin which Z >
4 feet andNo" ) 2.
(63)
-(,-b,)
r,:*,1#('*('
;)]
H
(64)
Equations(57) and (58) give the concentrations
of productsin the continuousanddispersedphasesleavingthe contactorfor the caseof low substrateconcentration(c, < K").
For contactordesigncalculations,the contactorlength requiredto removea given amountof productfrom the continuousphasecanbe found by iterationafter \iL - ),, and
B : Lf Ho. are substitutedinto eqs. (57) to (64), eqs. (15)
to (18)andeq. (21).
DrscusstoN
The equationsderivedin the previoussectionare for the
generalcasein which backmixingoccursin both phases
and the strippingfactor F doesnot equal 1. If backmixing
is absentfrom either phaseor F : I then the solutions
presentedarenot valid. In practice,thereis normally some
degreeof backmixing in both continuousand dispersed
phases.Furthermore,the optimum strippingfactor occurs
at ca. 0.7 or 1.4 dependingon the directionof masstransferr3so that the solutionspresented
are usuallyapplicable.
For casesin which either P, ot P, - m or F - 1, appropriate solutionscan be obtainedby using the limits given
by Pratt.ro
Accuracy of Approximate Solutions
Approximateanalyticalsolutionsgiving the dimensionlessconcentrations
leavingthe contactor,Xr and Io, were
derivedfor conditionsof low and high substrateconcentration. The accuracyof the approximatesolutionswas tested
by comparingpredictionsof Xr and I0 using the approximatesolutionsto valuesgeneratedby the exact solutions.
The deviationof the approximatesolutionsfrom the exact
solutions.definedas
(6s)
(66)
wascalculatedfor a varietyof operatingconditions.Figure2
showsthat the approximatesolutionsfor high substateconcentrations[eq. (31) and(32)l approachthe exactsolutions
as the length of the contactorincreases.The approximate
Performance and Design of Differential
Contactors in Extractive Fermentation
The significanceof productformationon the designand
performanceof differentialcontactorsdependson the magnitude of severaloperatingand physical parameters.In
general,the effect of product formation on contactorperformanceincreasesas rr : E/7, which is a measureof the
relativeimportanceof productformationand masstransfer
in the contactor,increases.e, is proportionalto the rate of
productproductio,rrs
t)xt andthe continuousphaseresidence
time in the contactor.At long residencetimes (small (J, or
largeL), the contactoracts as anotherfermentorand significant productproductioncan take placein the extractor.
The significanceof productformationon the performance
of a differentialcontactorcan be gaugedby comparingthe
contactorperformancewith and without productformation
takingplaceinsidethe contactor.The deviationbetweenthe
estimatedconcentrationof soluteleaving a contactorwith
and without internalproductformationwas calculatedby
AXr:t*(%9
LYo:tn(#)
(67)
(68)
whereXf andI$ arethe outletconcentrations
whenproduct
formation is neglectedand Xj and Yooarethe outlet concentrationswhenproductformationis accountedfor. In all
cases,comparisonswere madeusingthe solutionsfor high
substrateconcentrationsand the model parametersshown
in TableII.
Effect of Product Formation Rate on Contactor
Performance
The rate that productsare formed during fermentation
varieswidely dependingon the organismand processconfigurationused.In batchculture,Clostridiumacetobuelicum
producesbutanolat a volumetricproductivityof ca. 0.5 g/
L h.5 Butanolproductivitycan reach3 g/f h when this
samestrainis grown in fed-batchextractivefermentation.6
In batchculturesof yeastgrowingon molasses,
ethanolproductivityis normally 4-6 g/L ht while ethanolproductivities of over 8 glLh havebeenreportedfor the continuous
cultureof yeaston molasses.to
Much higherproductivities
havebeenobtainedby concentratingcells in the fermentor
B I O T E C H N OL OGY
A N D B]OE N GIN EE R IN
V OL.
G, 32,JU LY1988
No
o\
:
\,li
- b u
u
x
- v
li
.Fl
+-)
-1
l.
li
I
[:r..
a
p-*,40
.C
ux : 1 0
t:l:
ti,
C-)
6
X
r:
[: nvo
Tr
li
an
t\
-)
|i
f
!
t\,
ctsr
tr20
-tJ
-
\
U)
o 1 0
\
,
-
.
li
\.r.++qo*.rtrrrsr
X
S
\..'\...
i
n
*
-O
+
(tsr
'6"4t4:'<'^''-""""
-f\/
/,'
4','
tr
.
+)
F<
't -20
AX'
-30
Contactor Length, ft
Figure 2. Comparisonof approximateand exact solutionsfor the caseof high substrateconcentrations [eqs. (31) and (33)]. Parametervaluesare given in TableI.
with ultrafiltration membranes.For example,ethanolproductivitycanreacha0 g/L h in fermentorsthatrecycleyeastr
and ethanolproductivitiesof 67 glLh havebeenobtained
in a membranereactorusing cells of Khryueromyces
fragilis
grown on whey.tsSimilar productivitieswere reportedfor
Table I.
lactic acidproductionfrom Lactobacillusdelbreuckiigrown
in a fermentorusingultafilration to maintain high cell denTable II.
Parameters
co,
Parameters used in model comparisons.
Parameter
Value
Units
cl
l0
0
I
0 .7
40
20
20
100
3
0 .5
s/L
cl
F
m
ct"
m
F
Ho,
L
E*
Ey
vr
U,
uy
Urtn
F
cm
ft
cmzf s
cmzf s
slL h
cm/s
cm/s
Parameters
usedto generatecurvesshownin Figures2 and3.
U,
uy
Ho,
F,
Ey
ux
&
c!
c2
lL
Yrr,
Ypr,
Value
10
0
0.7
1.0
0.5
0.714
80
20
100
a
t0
3
10
1.0
0.1
0.5
ROFFLER,
WILKE,AND BLANCH:DIFFERENTIAL
CONTACTORS
Units
e/L
slL
cm/s
cm/s
cm
cmzfs
cm2fs
e/Lh
s/L
s/L
s/L
h-1
s/s
o t o
o/o
DID
197
sities.t6The rateof productfermentation,u,, canthusrange
from lessthan I g/t h in batchcultureto 50-60 g/L h in
cell recyclesystems.
Figure 3 showsthat for evenlow productivityfermentations, productformationcan significantlyaffectthe performanceof a differentialcontactor.For example,by neglecting
productformationwhen u, is 5 g/Lh, the estimatedoutlet
concenffations
of solutefrom the contactor,Y0 andXl, are
low by 10 and 37Vo,respectively.The continuousphaseis
affectedby product formation to a greaterextent than is
the dispersedphasebecauseproductiontakesplace in the
continuousphase.
The designof differentialcontactorsfor usein extractive
fermentationis also complicatedby product formation in
the column. Normally, for a linear equilibriumrelationship
anda strippingfactorlessthan 1, the concentrationof solute
in the continuousphaseleaving the contactor(raffinate)
canbe reducedto any arbitraryconcentrationby increasing
the contactorlength. Figure4 showsthat this is not ffue in
extractivefermentation.At a given volumetricproductivity
and set of operatingconditions,thereis a minimum solute
concentrationwhich canbe obtainedevenwith an infinitely
long contactor.For example,at the operatingconditions
shownin TableII anda volumefficproductivityof l0 g/th,
the concentrationof solute in the raffinate cannotbe reducedbeyondXr : 0.255. Furthermore,the minimum
concentrationof solutein the raffinatedoesnot occurat infinite contactorlength;the raffinatefrom an infinitely long
bi
(-1
P
rtl
il -10
lri
4J
--,
c
l-r
0-', -20
b,0
-
{J
a)
a!
Z -3 0
X
a)
d
O
c2
T)
o -40
t-{
-r-?
tr
)-i
>r
Volumetric Production Rate, g/Il-lnr
Figure 3.
198
Effect of product formation during extractive fermentation.
BIOTECHNOLOGY
AND BIOENGINEERING,
VOL. 32, JULY 1988
€
cFr
j
rL)
rU\J
o!
-
O
-1
bBo
p
O
rr(
ri
(,3 60
0
0.0
0.2
0.4
0.6
0.8
X1, Continuous Phase Oullet Concentration,
Figure 4.
o ^ ' o r+"* )r t' +
r .L
. -uI:\ fi G
f ii
Ci-
1.2
Effect of contactor length on exit concentration of product (solute). No minimum exit solute concentration exists.
contactor will contain more solute than will most shorter
columns. This behavior is due to the counteracting effects
of mass transfer and product formation in the contactor.
While the rate of product formation is independent of contactor length at high substrate concentrations (c, ) K,),
the rate of mass transfer decreasesas contactor length increasestransfer decreasesas contactor length increases. As
contactor length increases from L : 0, the solute concentration in the raffinate at first decreasesin the normal fashion as more transfer units are added. As length increases
further, however, the rate of mass ffansfer decreasesuntil
the average rate of mass transfer in the contactor equals the
average rate of product formation. This is the length of
contactor giving the minimum raffinate solute concentration, X). Any increase in contactor length beyond this
point results in a higher raffinate concentration as the average mass-transfer driving force further decreases. As the
contactor length goes to infinity, the raffinate concentration approachesa constant limiting value.
The contactor length L* giving the minimum raffinate
concentration at high substrateconcentrations can be calculated iteratively from
Ci
1.0
dimensionless
- C)
* ;I-i - rt
whereCiaregivenbysubstitutingli
L:
in eqs. (33) to (36) and ej is given by
e', :
(70)
E,P,(l - f) (cl - mcl)
The minimumraffinateconcentration,
Xtr, canbeestimated
by substitutingL, into eq. (31). The concentration
of solute in the raffinatefrom an infinitely long column,Xl, is
givenby
Xl :
X' - e',Ci.
lgg
(7r)
Figure5 showshow Xl, Xt* andL, dependon the rate of
product formation in the contactor.When u, : 0, an infinitely long contactorgivesthe minimumraffinateconcentration. At finite valuesof u,, however,the concentration
of solutein the raffinateis lower in columnsof finite length.
For example,whenv,: 30 g/L h, Xt- :0.9 whileXt
from a 9 foot column is only 0.5. At very high ratesof
productformation such as thoseobtainedin cell recycle
systems,the concentrationof solutecan increasethroughout the length of the contactor.Solute concentrationwill
increasein the contactor,regardlessof contactorlength,
wheneverthe volumetricproducqionrate,v,, is greaterthan
a critical value given by
-8,P,(c!
(6e)
Ir,andB: L/Ho,
vrFHs,
Ho,F
F ) \ ' 4 G | - C t ) + ( F ' - l )|
+q
C:
(72)
ROFFLER,
WILKE,AND BLANCH:
DIFFERENTIAL
CONTACTORS
199
X-
t A
1.*
+J
o
1.2 !
c
q)
P
q-i
O
(-)
i B o
-{J
1
b0
+_:
O
-
-
iJ
a)
J
t
v)
U)
0.8 q
al
+J
o 6 0
-
6
+i
tr
c-)
v)
-
X1
=
---
J
"-/'
40
q)
U.6
fi
r-
--?
><
-.'t/a---=---r>
0.4
7
0
1
0
2
0
3
ux
0
-
4
0
5
0
6
0
e\,-nr
'
Figure 5. Effect of the rate of product formation on minimum solute concentrations and on minimum
tor length.
Operation in the critical range can be avoided by increasing the ratio of extraction solvent to feed in the contactor
(decreasingF). Thus, when highly productive fermentations are coupled with extraction, the optimal stripping factor in the extraction column will shift to values lower than
the standard stripping factor of 0.7.
It is apparent from Figures 4 and 5 that Xr can be greater
than 1 when v, ) 0; more solute can leave the contactor in
the raffinate than entered in the feed. This behavior has
been observed during the extractive fermentation of acetone and butanol produced by Clostridium acetobutylicum.l
The concentration of acetone in the broth leaving the differential extraction column was often greater than its concentration entering the column. Figure 6 shows that the
concentration of solute in the continuous phase can also
reach a maximum inside the contactor. At large values of
u,, the rate of product formation can exceed the rate of
mass transfer over a section of the contactor resulting in a
concentration maximum inside the column. The concentra-
200
tion jump at the continuous phase feed inlet can also be
positivewhen u" is large. For example,with u,: 30 g/Lh,
the continuous phase solute concentration jumps from Xo :
1.0 in the feed to X0 : 1.06 inside the column.
Product formation inside a differential contactor must be
accountedfor when u, is large. Product formation will also
be important when the continuous phase superficial velocity in the column is small
Effectof ContinuousPhaseVelocityon Contactor
Performance
The superficial velocity of the continuous phase, (J*, affects contactor performance in two ways. First, the continuous phase Peclet number is proportional to U, so that the
effects of backmixing increase as the velocity of the continuous phasedecreases.Second,the continuousphaseresidence time in the contactor is inversely proportional to U,;
more product is produced per pass through the column at
B I O T E C H N OL OGY
A N D BIOE N GIN EE R IN
G, 32,JU LY1988
V OL.
CI
0
6
c4
1n
a
-J
p
(-,
- Y
{ I H
C)
-
p
'-l
tL
06
4J
O
c-)
^
O
a
A
O
n
V .AT
a
a)
fr
a
0,2
><
0.0
0
o.2
0.4
Z
0.6
0.8
1.0
FLactional Lengfh
Figure 6. Concentrationof solutein the continuousphaseas a functionof volumetricproductformationrate (v,).
small phase velocities. Both effects act to increase the importance of product formation at low U,. Figure 7 shows
the effect of the continuous phase velocity on the concentration of solute in the raffinate from a column operating at
the conditions shown in Table II. Although product formation significantly alters the concentration of solute in the
raffinate at all superficial phase velocities, product formation is especially significant at low U,. For example, when
U, : 0.1 cm/s, X' without product productionis ca. 0.29
while Xr increasesto 0.64 when u, -- 3,0 g/L h in the
contactor.
Figure 8 shows the strong dependenceof the continuous
phase concentration profile on U,. When the continuous
phase passes rapidly through the contactor, the effects of
backmixing and product formation are minimized, e" is
small, and the concentration profile is steep. As U, de-
creases, however, backmixing increases and the concentration profile flatens out. At very low continuous phase
superficial velocities, product formation becomes very significant (e, ) 1). The concentration at the feed inlet jumps
upward (Xo > 1), the concentration profile reachesa maximum inside the contactor, and the outlet concentration is
higher than the feed concentration(X' > Xo: 1). Thus,
even relatively low rates of product formation can affect
contactor performance when the continuous phase velocity
in the column is low.
coNGLUstoNs
The perfofinance and design of differential contactor for
use in extractive fermentation is complicated by the effects
of product formation inside the contactor. In general, when
ROFFLER,
WILKE,AND BLANCH:DIFFERENTIAL
CONTACTORS
201
P
0.0
Continuous Phase Peclet Number
0.8
1.2
1.6
?.0
-10
m
n
g r.z
-
A
._{
-+J
m
VX1
t1
C)
-/tI
\-/
. r-.i
''?-l
\J
t
n
1
-30
-rJ
c)
l-'
rz-i
v
A
v
+?
Fr
n
rtl
{'ir
tr
O
_40
a0
g
O.B
-p
a)
q)
d
O
-cu
-NJ
"t-?
hn
w
(J
X
-o
E 0 6
-60
m
8
a \
v
-
/t{
,..q
rv
A
v
a
-{
- (Bnv
x o.4
-{
L
-
'i-i
L
L
tr
v
X1 with product production
-{-i
g
-80
Q
l.
L
kl
- ^
X
o,2
T<
-90
X1withourorro".;;tt
-100
0.2
U*
Figure 7.
o.4
0.6
r.2
r.4
Continuous Phase Superficial Velocity, crn/s
The effect of continuousphasevelocity on the concentrationof solute in the raffinate.
substrateconcentrationis high (c, > K,), product formation is significantwheneverer : u,FLf (J,(1 - F')(rl *cfi is of order unity or higher. Parametere, is increased
by high ratesof productformation, long continuousphase
residencetimes, low productconcentrations,and poor exfractionconditions(strippingfactor -+ 1).
When productformation is significant,the performance
of differential contactorscan be predictedfrom eqs. (31)
and (32) at high substrateconcentrations
or from eqs. (57)
and (58) at low substrateconcentrations.
Although these
equationsare approximations,they give resultswithin 5Zo
of the exactsolutionswhen contactorlengthis greaterthan
4 f andff0, > 2. For practicaloperatingand designconditions, the differencebetweenexact and approximatesolu-
202
0.8
tions is negligible.when designinga differentialcontactor
in which productsare formed at a constantrate inside the
contactor,it is importantto note that there is a minimum
raffinatesoluteconcentrationthat can be obtained.In contrast to the behaviorof a column without productproduction, the minimum concentrationof solute in the raffinate
from a column with internalproductformationoccursat a
finite contactorlength. The contactorlengthgiving the minimum concentrationof solute in the raffinate decreasesas
e, increasesand for highly productivefermentations,the
concentrationof solutecan increasethroughoutthe contactor. For a given set of operatingconditions,the length of
contactorgiving the lowestpossiblesoluteconcentrationin
the raffinatecan be estimatedfrom eq. (69).
BIOTECHNOLOGY
AND BIOENGINEERING,
VOL.32,JULY1988
d
':-1 t.2
-.
/fi
l.{
+?
tr
C)
H
.4
1
u* : 0.075
e* : 1.58
p
()
P-J
O
a
O.B
/r{
d
n
-
I
06
d
. r-l
{--?
(-
O
U": 0'30
€* -: 0.395
0.4
T<
U * : 1' 0
e* : 0.118
0
0.0
0.1
o.2
o.4
0 .3
Z
0.5
0.6
o.7
0.8
0.9
1.0
Ffactional LengLh
profile.
Figure 8. Effect of the superficialcontinuousphasevelocityon thecontinuousphaseproductconcentration
The equationspresentedin this paperare for the general
casein which backmixingoccursin both phasesand the
strippingfactornot equalto 1. Althoughthe equationswere
cast such that product formation occursin the continuous
phase,the equationsare equallyvalid whenproductformation takesplace in the dispersedphase.In addition, the
equationsare not limited to extractionbut can be usedto
simulateor design any type of differential contactorin
which the kinetics of product formation are zero or first
order in substrateconcentration.
NOMENCLATURE
Ai
a
d;
coefficientin eqs.(19), (20), (52), and(53)(dimensionless)
inbJertacial
areaper unit contactorvolume, 7z17t - 7-r
definedby eqs. (22a) and(22b) (dimensionless)
B
Bi
bi
cm
cs
cx
cy
Ci
di
Ej
L/Ho, (dimensionless)
coefficientsin eqs. (52) and (53), dimensionless
definedby eq. (54), dimensionless
concenftationof cells in X phase(M/L3)
concenffationof substratein X phase(M/L3)
concentrationof solutein X phase(M/Lt)
concenffationof solutein I phase(MlLt)
definedby eqs.(33) to (38)
definedby eqs. (46) and (47) (dimensionless)
effectivelongitudinaldiffusioncoefficientin thejth phase(L2/T)
F
stripping(extraction)factor,#
uy @i^"nsionless)
("true")
heightof an overall
transferunit basedon X phase(Z)
constantdefinedby eq. (43) (dimensionless)
overall masstransfercoefficientbasedon X phase(L/T)
sugarsaturationconstant(MlLt)
total length of contactor(l)
slopeof equilibriumline, dc,f dc, (dimensionless)
roots of characteristiceq. (45)
Ho,
h
ko,
K,
L
m
nti
DIFFERENTIAL
CONTACTORS
ROFFLER,
WILKE,AND BLANCH:
203
No"
Pj
S
Uj
X
Y
Yp6
Yr,
Z
z
number of "true" overall transfer units based on X phase
(dimensionless)
Peclet number of the jth phase, UjHo,f Ej (dimensionless)
generalized substrate concentration in the X (feed) phase
(dimensionless)
superficial velocity of jth phase (t/I)
generalized solute concentation in X (feed) phase (dimensionless)
generalized solute concentration in Y (extractant) phase
(dimensionless)
yield of product (solute) from substrate (M /M , dimensionless)
yield of cells from subsffate, M/M (dimensionless)
zf L, fractional length within column (dimensionless)
length within column measured from X phase inlet (Z)
Greek letters
d
B
y
AXr
Afo
defined by eq. (15)
defined by eq. (16)
defined by eq. (17)
deviation between approximate and exact solutions for X phase
outlet concenfration, dimensionless
deviation between approximate and exact solutions for I phase
outlet concentration (dimensionless)
definedby eq. (18)
6/7 (dimensionless)
definedby eq. (60)
4i
di
definedby eq. (56)
l,j
roots of characteristiceq. (21)
specificgrowth rate of cells (T-I)
lL
maximumspecificgrowth rate (?-t)
Pvx
rate of productformation,definedby eq. (7) (M/L3/f)
definedby eq. (51)
€
cfi
definedby eq. (63)
o, definedby eq. (59)
*, definedby eq. (64)
o, definedby eq. (62)
Q)i
definedby eq. (61)
VX' definedby eq. (67)
vy0 definedby eq. (68)
6
Ex
Superscripts
0
I
*
feed inlet end, outsidecolumn (Z : 0)
exffactantinlet end, outsidecolumn (Z = 1.0)
value at equilibriumwith other phase
Subscripts
A
204
E
i
j
M
m
/V
s
x
y
exactsolutionforX or Y
numberof root of characteristicequationand of coefficientsin
solutionfor X or Y
XorYphase
value at minimum soluteconcenffationin raffinate
cells
solutionto X or I neglectingproductformation
subsftate
X phase(feed)
Y phase(extractant)
References
1. B.L. Maiorella, Ph.D. thesis,Universityof California,Berkeley,
cA, 1993.
2. J. J. H. Hastings, in EconomicMicrobiology (Academic,New york,
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I (1987).
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11. T. Miyauchiand T. Vermeulen,Ind. Eng. Chem.processDes. Develop.,14,74 (1975).
12. S. HartlandandJ. C. Mecklenburgh,The Theoryof bacbnixing;The
designof ContinuousFlow ChemicalPlnnt with Backmixing(Wiley,
New York, 1975).
13. C. J. King, Separation
(McGraw-Hill, New york, 2ndEd.,
Processes
1980).
14. H. R. Bilford, R. E. Scalf,W. H. Stark,andp. J. Kolachov,lnd.Eng.
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Prod. J. (February9, 1985).
16. E. Ohleyer,C. R. Wilke, andH. W. Blanch,Appl. Biochem.Biotechnol., 11,457(1985).
approximatesolution for X or Y
BIOTECHNOLOGY
AND BIOENGINEERING,
VOL. 32, JULY 1988