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Liquid-liquid mass transfer and its retardation by macromolecular adsorption
ooo4-2H)9/79/0501~1/strz
LIQUID-LIQUID MASS TRANSFER BY MACROMOLECULAR B J R SCH0LTENS.t Laboratory
S BRUIN*
w/o
AND ITS RETARDATION ADSORPTION and B H BIJSTERBOSCH
for Physical and Collard Chemistry, Agricultural The Netherlands
Umverslty,
De Dre.l]en 6, Wagenmgen,
(Recewed 13 June 1978, accepted 18 September 1978) Abstract-The transfer of KCI from water to I-butanol has been measured in a Lewis cell, under lammar flow con&Ions This system appears to extublt no slgmficant mterfaclal resistance The effect of spread macromolecules on the mass transfer 1s shown to be prlmardy of a hydrodynarmc ongm On basis of the boundary layer theory and the convective diffusion equation a simple quantltatlve estimate can be made of the mfluence of a (partial) rlgichfication of the tnterface on the transfer process INTRODUCTION
has been studled extensively durmg the last few decades, confllctmg views have been presented on the existence of an appreciable mterfacml resistance and on the mechamsm by which surfactants retard mass transfer Usually[l4] the resistance to transfer of a substance A from phase 1 to phase 2 IS described by a series of partial resistances (two film theory) Although
solvent
extractIon
(RzA) = (rIAm2,* + r_*m*,*
+ r+*)
(1)
where (IZzA)(s m-‘) 1s the average overall resistance with respect to phase 2, rIA, r2A and I-,^ are, respectively, the two hquld phase resistances and the mterfaclal the dlstnbutlon reastance, while mZIA represents coefficient of A between phases 2 and 1 Actually, the resistances equal the ratlo of the relevant concentration difference and the resultmg mass flux Whereas the dlffuslonal hquld phase resistances can be predicted for a number of systems with simple geometry and boundary layer flow, the value of r,^ IS unknown Q pnon and IS simply neglected m many cases [l-3] In pnnclple, Its value can be obtamed from comparison of with the calculated values of r,* and r2* In @2%x practice, however, this procedure 1s only successful when the expertmental accuracy IS extremely high (static systems) or when the boundary layer flow IS exactly known These prerequlsltes are almost never met l61 Yet, regularly studies appear m which. whether correctly or assumed to play a, someor not, r,^ IS determmed times even dommant, part (e g [7-201) Several different explanations have been proposed for the retardmg influence surfactants may have on the transfer process These are based on possible changes in either the hydrodynamics near the Interface, affecting
*Present address Central Laboratory DSM, Fundamental Polymer Research Department, P0 Box 18, Geleen, The Netherlands *Department of Food Process Engmeenng, Agncultural Unlverslty, De DrelJen 12, Wagenlngen, The Netherlands $In pnnclple, an excess InterfacIal shear v~scoslty may also alter the flow, but tins effect IS always of neghglble magmtude [2] 641
r,* and rzA (e g [l-3,6]), or m r,“, which may be caused by several effects (e g [ll. 16.21-25]), or m both The hydrodynamics near the Interface always play a very important role m the Interphase transfer process Therefore It IS worthwhde to mvestlgate tf and how these may be changed by adsorption The mere presence of a clean hqmd-hquld interface has a pronounced influence on the boundary condltlons of both hquld flows at the interface the tangential velocity components are equal m both phases, whtle the normal velocity components vanish, m ad&Ion, the momentum flux perpendicular to the interface IS contmuous Adsorption may influence the hydrodynamics by Its effect (pursued by a decrease m mterfaclal tenslon, y) on these boundary condltlons I The mterfacml tension can only influence the boundary layer flow when the interface IS curved (capillary force), (by an or when Its value vanes along the interface uneven dlstnbutlon of surfactant molecules) This uneven dlstnbutlon may be caused by a boundary layer flow and m this case the induced gradient m y will oppose Its own orlpn Only for the most mterestmg case from a technological pomt of view, VIZ turbulent flow, a quantitative theory has been proposed[3,26] to explam the retarding effect of monolayers Although the complex hydrodynamics were slmphfied considerably m thts at least quahtafive predlctlons have been analysls, confirmed by expenments On the other hand it has been suggested that the rnterfacral resistance may be increased by adsorption by (partly) blockmg the mterface (sieve effect) or by a chemical mteraction of A with the surfactant or the Interface Itself (free energy barrier) [ 11, 16,2 l-251 The precise mechamsm of retardation determmes which of the three terms at the right hand side of (1) changes most as compared to the surfactant-free case This Implies that any observed retardation can only be explamed exclusively by, e g a change m r,^, provided the second term of (1) 1s at least comparable in magnitude to the two other terms (bottleneck principle) The controversial views on the importance of r,^ and on the mechamsm of retardation by adsorptlon can be studled under deliberately chosen expenmental conditions such that the hquld phase resistances are relatlvely small and still amenable to calculations To meet
662
B J R
SCHOLTENS et
these demands we developed a transport vessel wrth a flat hquld-hqmd Interface and centro-symmetic lammar tangential flow Smce model measurements of this type may be severely dlsturbed by spontaneous (Marangom) mstablhtles [U-29], theu absence must be checked thoroughly dunng the transfer process In a recent commumcatlon [30] It was tentatively suggested that vmation of the hydrodynamics may enable the overall resistance to be separated mto the sum (T,%~,~ + rzA) quantitative and r,^rn2, * In this paper we present arguments m favour of this approach, whzch provides indeed a key to a better understanding of the mterfaclal phenomena
The overall resistance was measured for the KC1 transfer (starting at a concentration of 0 1 mol 1-I) from with I-butanol) to I-butanol water (always saturated (saturated with water) The transport vessel (Fig l), rather similar to that of Nltsch et 01 [31,32], contamed large propellers to obtwn relatively high overall flow rates with as little turbulence as possible These were counterrotattBecause KCI was the only electrolyte under conslderatlon, we refram from using the mdex KCI when mentlonmg (I&), cb or
al
mg, pumping the llquld m inward, mainly radial. duection along the interface which was flat and without slgmficant waves Draft tubes with (SIX) baffles guaranteed a systemattc fiow near the interface and optimum partlclpatlon of all hquld m the transfer process In order to achieve comparable hydrodynamics at both sides of the interface, we always chose equal values of Re for both phases (&/AL = ub/vy = 2 78) Reproducible results were obtained by always venfymg (with a cathetometer) the uutlal posrtlons of all parts that could Influence the hqtnd flow and adlusting d necessary For that same reason RO wire netting was used, m contrast to[31,32] In order to suppress any mterfenng vlbratlons, the supporting construction, carrying two identical vessels, was mounted on a heavy concrete slab that rested on rubber plugs The glass jackets were thermostatted at 25 OO+ 0 01°C (&)t was determined from the time-dependent KC1 concentration m the butanol phase, which was measured conductometncally In order to convert the conductance m-l) to the KC1 concentration cb, the expenKb (i-i-’ mental relation cb = 1 201(#&)’ 1304mol I-’ was used, which held true for the concentration range 0 I16 mmol 1-l (mb, = 172 X lo-*) The value of (&) was obtained usmg eqn (A6) (see the Appendix) By applying moderate sturmg speeds (40 < & C 90 mm-‘) a lammar
SupportIng construction
I
Fig 1 Vertical cross section of one transport vessel and the supportmg construction, 1, conductance cell, 2, upper stu-rmg shaft (#J = 0 60 cm) with a three-blade. nght-hand propeller ($ = 5 1 cm, 2 7 cm above the interface). both stainless steel, 3, stamless steel shaft connected to a baffle of the draft tube, 5, thermostat& double-walled Pyrex glass vessel (mslde 9 = 8 08 c 0 02 5 90 cm, outside 4 = 6 36 cm, he@ = 2 50 cm) with SIX bafRes (2 60 x 0 75 mterface, 7, ldentlcal lower draft tube. reshng with three baffles m holes m stainless steel plug (4 = 5 8 and 1 5 cm, height = 0 6 and 3 4 cm) with a hutyl
4, PVC cover with butyl rubber O-nng, cm), 6, upper draft tube (mslde $ = cm’), the lower edge 1 2 cm above the the plug 8, butyl rubber gasket ring, 9, rubber 0-ring, fixed by a PVC nut (20),
10, lower stvrmg shaft, ldentlcai to 2, 11, toothed belt (Synchroflex), tooth = 0 5 cm, 12, toothed pulley (& = 3 cm), 13, Teflon shde rmg, 14, brass shde bearmg, fixed on I5 with msulatmg material. IS, upper ;tlummmm plate
(thickness 0 6cm), 16, lockmg ring, 17, brass column (9 = 1 Ocm, he& = 20cm). 18, shaft with nut to fix the cover (length = IS cm). 19, lower alummmm plate (33 x 22 x 0 3 cm’) with two insulating rubber mats. 20, PVC nut, 21, msulatmg PVC columns, 22, alumuuum support (33 x 20 x 0 3 cm’) Both propellers and draft tubes are smutted symmetncally with regard to the Interface, whde VW = 225 and V, = 210 cm3
Llqtud-bquld
mass transfer and Its retardation
velocity profile along the interface was guaranteed with 500 > Re, = Reb c 1200 Although theoretically Marangem mstabrhtres are possible for the present transfer dIrectIon, we checked very thoroughly that they were absent at the chosen concentration dfierenceE61 In addition we studied the effect of three, only water soluble, polymenc substances on the KCI transfer PVAH, B2 and B4 The first 1s a completely hydrolysed polyvmyl alcohol (vrscoslmetnc mean molar mass = 118 kg mol-‘), while the other two are p-ally reacetylated copolymers denved from PVA-H, contammg 6 8 and 21 3 mole% vinyl acetate respectively These substances were spread at the interface from a concentrated solution, usmg a micrometer syringe Further detmls of the expenmental procedures, the completely automated determination of (&) and all materials are given extensively eIsewhere [6]
by macromolecular
v, = -a/r
N, = constant
I
‘1
v,=-f3z/r
W w il
r
0
@
(2)
Since u, pomts to r= 0 m the experimental set-up, we only consider negative constants, which may stil depend on z The followmg two cases ~111 be examined (see Fig 2) v,=
-fflr
(3)
and (4)
v, = -@IT
where a(m’ s-‘) and @(m s-l) are stants For the diffusion of A out of the lammar boundary layer, the equation (e g [33], p 559) can be approximated
ah ar
tl,-=D,4-7.
thus
posltlve
con-
interface into the of continuity of A by
a2cA az
provided a pseudo-steady state approxlmatlon can be made (&,/at = 0) and the velocity profile IS not affected by the dlffuslon process The boundary conditions for the solution of (5) ~th (3) or (4) read
I,
@
,2--Bd-A r0
663
In the followmg we shall solve a slmphfied form of the contuuuty equation for the transfemng component which IS applicable for the two ultunate cases (I e the completely moblle resp ngdified interface) From these solutions the concentration profiles can be evaluated and from these, m turn, the hqmd phase resistances In addition, for all intermediate cases, VIZ a partly covered Interface, the solution will be shown to be a combmatlon of the two ultimate situations Consider an axially symmetic, flat liquid-hqutd Interface at z = 0, extending between r. and rl (rO > r,), wth a lammar boundary layer for z Z=0 If we assume that the flow 1s stationary (au/at = 0), that the density remams constant and that only a radial velocity component (u,) 1s present along the interface, the continuity equation for the flowing hquld (e g [33]. p 83) can be readily mtegrated to
ANALYSIS
One fundamental requirement to analyse the mass transfer process and to understand the effect of adsorption quantitatively, 1s the theoretical estimate of both hqurd resistances m the absence and presence of adsorbed molecules Thus, the innermost part of the boundary layer flow should be known precisely for these situations Flow visuahzation showed a rather systemauc, mamly radial flow pattern m both phases near the mterface[6] For the clean hquld-hquld Interface we shall assume a radial plug flow (Ag 2a), since It cannot resist any tangential shear stress Later this approximation ~111 be shown to mtroduce no serious error m the estimate of both liquid phase resistances In the presence of su&zctunt the radial flows will drive the adsorbed molecules to the centre of the interface, until the induced mterfacld tension gradient 1s able to oppose the two shear stresses Since adsorptlon of the macromolecular substances used was shown to be u-reverslble[6], they could not desorb from the centre into the aqueous solution Ultimately, the interface may be completely covered by an adsorptlon layer, the surface excess mcreasmg towards the centre, so that the interfacial velocity is zero everywhere, Just as for a hquld-sohd interface For the mnermost part of the boundary layer in this situation. a radially symmetnc velocity 1s assumed, mcreasmg linearly with the distance from the interface (Fig 2b)
I
adsorptIon
CA = CA,
at
z>O
and
r=ro
c.4 = CA,
at
z=O
and
rlsrsro
CA = CA,
at
z=m
and
r,srcr,
(6) 1
In order to solve (5) with (3) and (6) we introduce dlmenslonless vanabIe
the
z I
4
r
I_ 0
Frg 2 Schemattc representatton of the two hydrodynannc boundary layers along the Interface whrch extends between rO and r,
Inserting
into (5) leads to d*c, CdcA _ dk+2dC-O
(8)
664
B J R SCHOLTENS
while the boundary
conditions
can now be formulated
as
eta1
for r, s r s r2 and mobile for r2 < r c r. (Fig 3a) In the mobile, region the flow may be approximated by (3) and in the inner region by (4) For the outer region the calculations are identical to those for the clean interface, but now with ri = r2 However, for the inner region the solution of the completely rigid interface does not apply since the boundary conditions at r = r2 are now given by (10) and not by the appropriate form of (9a) However, we can still use solution (16) by applying a mathematical trick consider another, hypothetical interface completely immobilized for r, s r c r3 (with r+ < r3 < r& as drawn in Pig 3b We choose r3 m such a way that at r = r2 the concentration penetration depth for this case IS equal to that calculated with (10) for the outer region Obviously, (rA) for the partly covered interface is then given by outer,
CA = CA,
at
(9a)
CA = CA0
at
i=O
(9b)
The solution
of (8) satisfying
these boundary
conditions
IS
cAo-cA = CA.7- CA.=
I
c
0
exp (- 6’14) dr = PW2,
I)
(10)
_ exp (- t’l4) dS I II
where P(1/2,5) is an incomplete I-function (e g [34], p 260) From this concentration profile, the local molar flux 42, and the average mass transfer resistance (rA) can be evaluated with standard methods
(18) and
h) =
(CA- - c.&&r(ro2 - r,a) = I(1/2)(rclZ - r**)“* 2(2aD,)‘” (ro’ - rtZ)“2 = 1 5%(aD,)“’
(12)
Since the determination of r3 as a function of r,,, rz, (I, p and DA IS tedious-the concentration profiles both contain incomplete P-functions-the approximate method of Von Karman and Pohlhausen (e g [U]. p 187) was used As a trial solution for the concentration profile we used (for z s S (r)) CA
Analogously, the solution of (5) with (4) and (6) can be obtained by introduction of
’=‘(o,($Inserting
r2))“’
(13)
(C&T-- CA&i - z/s(r))’ +
=
CA,
(19)
where the concentration boundary layer 6(r) is only a function of r With this relation eqn (5) was solved again with (3) or (4) and (6) This resulted in S(r) = (6DA(ro2 - r’)/a)“’
into (5) gives
(20)
and 42, = (&$+)j’m while the same boundary conditions (9) now hold when 5 IS substituted for 5 The solution of (14) then reads CA0 -
CA
CA, - CA,
= P(1/3,5)
for the completely
mobile
(c*, interface,
- c,,)
(21)
and in
(15)
so that
(16) and (rA)
=
2U1/3)(ro+- r,*)“’ = (ro’- r,Y 3(6/3D,,‘)“’
Now we consider only partly covered
1 017(BDA2)‘”
(17)
the same interface, but this time by surfactant molecules ngidified
Fig 3 SchematIc representanon of the hydrodynamrc dlffus#on (8) boundary layer for a hqmd-hquld mterface r, and rlr llgrdlfied for r, G r < r2
(A) and between
Llquld-llquld mass transfer and Its retardanon by macromolecular adsorptIon S(r) = 18D*(r**
- rZ)I/3p3
were found 9~ 10m3 m Smce no slgmficant differences within this volume, these measurements provided us with the average radial bulk velocities at r = 2 5 x lop2 and /zJ = 6 x 10e3 m A curve fitting analysis ylelded
(22)
and
dir =
(fj$ppJ”(c,,
-
CA_)
(23)
for the completely covered Interface Notice the close agreement with the exact solutions From (20) and (22) r3 can now easdy be derived as an lmphclt function of the other variables 2(D/+(ro2 - r*+)/a)’ = 3(D,4(r,2 - r2+)/j3)2
(24)
Making use of experimental values (see results), it can be shown that insertion of r2 instead of the evaluated r3 in (18) for this system results m a value for (rA) that IS maximally 1% too low-as exemplified rn Table 1 for an mtermedlate Nb Therefore It 1s Justified to replace r3 by r2 in (18), resulting in
which reduces to (12) and (17) for the two ultimate cases, Equation (25) VlZ r2 = r, and r2 = ro. respectively enables us, with a simple Iterative method, to determine rz and so the fractional coverage of the Interface, provided all other expenmental values are known Therefore it may provide us with a method to study the dynamic behavlour of adsorbed macromolecular layers in a vanable shear field varlatlon of hydrodynamics (a and /3) will induce a different (rA) from which the new r2 can be evaluated RESULTS In both phases hquld velocity measurements were performed by photographing suspended polystyrene partlcles[6] At four different stlmng speeds their average velocity was determined m the volume confined 2x10-*
665
vbr
=
-63x
lo-‘(N,
mm)14m
VW = - 2 3 x lo-’ (iVb mm)’ 4 m
K?r,t
(m)
35 0 300 2s 0 20 0 15 0 100 50
35 1 30 2 25 4 20 7 16 1 11 8 82
10%
+ MM)*
/3 = 5 7 x 10e3 (vrp”/(6
(27)
x 10-4~)“z m
s-’
(29)
where the numerical constant In the numerator has the dlmensron m1’2 In Table 2 we have collected all relevant hydrodynamic parameters for the experimental conditions the hydroSince mbw IS very low (1 72 X lo-‘), dynamics in both phases are comparable (Re, = Reb) and Db < D, (q, = 2 87X 10p3, vW = 1 2 x lop3 Pa s), It wrll be clear that for the present system the sum of the hquld phase resistances 1s completely controlled by rb The values of r. and rl are given by the geometry of the set-up (they amount to 4 04 x lo-’ and 0 m, respectively), so that the only unknown quantity whtch thwarts the
108L(d&
l@&(M
(m’ s-l)
(m’ s-‘)
(m’ s-‘)
34 45 53 59 63 66 67
I 12 0 89 0 67 048 0 30 0 15 004
I 14 093 0 73 0 54 0 37 021 008
A%” 06 09 11 11 11 09 06
Wetermmed from (24) with rO= 0 0404(m), Q, = 3 7x 10m4(m2 s-l), 0 34 (m s-l), DbKc’= 3 x IO-” (m* s-‘) SThe sum of the integrals m the denommator of (18). with r, = 0 3The second rntegral evaluated with the relevant rS value lIThe second mtegral evaluated with the relevant r2 value *he relative increase In the sum of the two mtegrals by usmg r2 Instead of rS CEsVd34NoS-E
s-’
For the completely ngd Interface p,, and & were estimated from the analogy of the flows with that m the vlcuuty of a stagnatlon pomt (e g [353, p 88), smce at the penphery of the mterface the two flows approach lt at nght angles The actual flow IS radtally away from the “stagnation cucle” in inward direction. so that the solution for plane flow seems reasonably appropriate From the solution of this veloczty profile we obtamed, using (4) at the afore mentioned coordmates
addition the Integrals of (18) are evaluated, the effect of using rz Instead of ra can be Inferred from the last four columns
(m)
(26)
The radial mterfaclal velocity, z),,, of a completely mobde Interface was estimated with relations derived by Lock[36] Assuming the water and butanol streams to be parallel and lammar, thrs resulted in v,, = 0 77 x v,, = -4 8 x lop5 (Nb mln)’ a and so, using (3) at r = 2 5 X lo-’ m
Table 1 The value of r, as a function of r2 at N* = 60 nun-‘, as calculated from (24) In
lO%,
s-’
&, =
B I R SCHOLTENS et al
666 Table 2 Approximate Nil (mm-‘) 40 50 60 70 80 90
values for tbe hydrodynamic
10&t cm s-‘)
- 1oLt (m SK’)
1I
04
15 19 24 29 34
05 07 09 11 13
Nb (mm-‘) 40 50 60 70 80 90
(rt.)
1 17 1 12 109 108 1 05 104
100 8 55 7 58 6 76 6 17 5 68
116920 14 9552009 8 242007 732?005 646kOO4 5 88kOO3
-r 1
(m2 s-‘)
08 12 15 19 22 26
(&Lx/(&l
lo-‘(G)th (s m-l)
to-‘UW,, (s m-‘)
s-‘)
10
m the vlcmlty
101lx,
- lo%,, (m
theoretical estimation of the butanol resistance with (12) or (17) IS 0, Its value was denved from the hmltmg molar conductance of KC1 m the butanol phase, using the Nernst equation for completely dlssoclated electrolytes (which was shown to be apphcable[6]) Making allowance for the low actlvlty coefficient m the butanol phase[6] and assummg equal amontc and catiomc calculated equivalent conductances, we La= 3 x lo-” m2 s-l for the relevant concentration range Experimental values for (I&.) and theoretical values for (rb) (calculated with 12 and 28), both In the absence of any polymenc mater&, are collected in Table 3 as a function of Nb ( = 2 78 NW) The effect of the apphed (co)polymers was diverse, depending on the total amount spread, S (which vmed between 0 2 and 2 x ld mg me2), the acetate content and Nb Even at the highest Se”*’ no slgnficant mfluence was observed at any h$, On the other hand, the two copolymers caused a defimte retardation of the Table 3 Companson of expenmental (RI,) and theoretlcal values as a function of the sturmg speed
parameters
21 29 37 47 56 66
(m
of the Interface
Bb s-‘)
0 15 0 24 0 34 047 063 0 80
(m
BW s-‘)
006 009 0 13 0 17 0 23 0 29
Fig 4 The expenmental retardation &a* for different Nb as a functton of the amount of B2 spread at the mterface, 0, Nb = 40, & Nb = 50, Cl, Nb = 60, and 7, Nb = 70,80 and 90 mu-’
transport rate, that can be expressed as the quotient of the overall resistances wtth and wlthout macromolecular matenal present and IS mdrcated by QN = Experunental (Ra) ,,.,~co~~Ymer/(R~) at N,, = N mm-’ values of QN vs S can be found m Fig 4 for B2 and m Fig 5 for B4 These values are the average of at least four different experunents (whde the (Rb) values are based on approximately 45 runs) For both co-polymers the retardation at constant S Increases with decreasmg stimng speed Whrle for B2 the maximum retardation seems not yet to be reached, at the highest SH4 It IS attamed for most to 6 1 *O 2) Since several QN values Nb (amountmg could be determmed dunng one experlmental run, we also studled the effect of the hlstory of the spread layers on QN a senes of measurements with Na successively 40, 90, 40, 90 and 4Onun-’ showed QW and QW to be essentially constant m all cases
102
lo3 rng m-2
Fig 5 The expenmental retardation QN” for dtierent Nb as a function of the amount of B4 spread at the interface, 0, Nb = 40, & Nb = 50, [7. Nb = 60. a, Nb = 70, A. Nb = 80. and n, Nb = 90 nun-‘, V, represents a comcldmg point for at least two of the four lowest rotation speeds and v for at least two of the four highest rotation speeds
Llqutd-hquld
mass transfer and Its retardation
by macromolecular
adsorptron
667
DISCUSSION
the expertments m the absence of any macromolecufar material, a comparison of expenmental overall with theoretical liquid phase resistances 1s mven in column 4 of Table 3 Although on first sight these results might indicate a small r, a closer mspection of eqns (I), (12) and (28) reveals that the relative conmbutlon of r, must grow with Nb, while column 4 shows the opposite trend Insertion of (28) m (12) shows that (rblfh should be proportIona to Nb-“’ If an mterfaclal resatance were operative, (Rb)ex would be approximately proportional to N to a less negative power However, linear regression analysis of (Rb)cx and log Nb yielded -0 84 as the power, mdlcatmg the absence of an mterfacial resistance We think that the observed devtatlons from theoretical maccuracy m predictions are not due to any (systematic) CY_or & [6] Rather these should be found m differences between expenmental and theoretlcal flow profiles, such as end effects Indeed, flow vlsuahzatlon studles[6] showed the radlaf flow along the mterface to be less effective for ra 3 5 x lo-* m and particularly for r =S 2 X lo-’ m than for the regton in between, especially at low Nb The effects became less distinct with mcreasmg Ns, consistent with the findings in Table 3 and also with results of transfer expenments with a very thm plate at the centre or rmg at the penphery of the interface161 In the calculation of (rb)* plug flow (Fig 2a) was assumed whale m fact a small velocity gradient au,/a_z exists at z = 0 (as depicted at the left of Fig 6) Using results of Beek and Bakker[37& It can easdy be shown, however, that this approxlmatlon does not mtroduce errors larger than 2% Having considered all aspects thoroughly, we conclude that the mterfacml resistance for the present system IS very small d at all present, so that the butanol phase resistance LS the only rate determining step m the transfer process Because of the neghpble tnterfactal resistance, the eflect of the spread copolymers on the transfer process must obviously be an increase m the butanol phase resistance Elsewhere[6] It was shown that there IS a one-to-one correspondence between adsorption (mterfacial excess) and retardation for this system, that Db IS unaffected by macromolecular layers at the interface and that no excess interfacial shear vlscosrty can be attnbuted to these layers Therefore, any increase m (Rb) can only be caused by dynamic utteractions between the Bow along the Interface and the compressed macromolecular layer, resultmg m (partly) nsddicatlon Actually, the flow along the ngd interface is not of the structure indicated m Fig 2, but rather hke that at the nght of Fig 6 However, since Scb = 11 x lo’, only the very innermost part of the hydrodynanuc boundary layer plays a part m the mass transfer process, for whrch the former profile IS a good approxlmatron The effect that complete immobdlzatton of the interface can have on the butanol phase resistance was calculated from (12) and (17) Insertron of (28) and (29) yldds that QNnv” should be independent of Nbr amountmg to 4 9, which IS close to our expenmental results for For
Fig
6
Boundary layer flow along an Interface that extends between r, and r, and IS covered for r, s r s r,
very high Se4 The quantttative difference m Q”” IS attnbuted to the Imperfectness of the actuaI flow, smce only part of the interface IS flown along effectively due to curvature of the streamlines In the absence of surfactant, this mterfacml flow wdl induce (less effecovely) hqtud movements along the parts that are not reached directly However, for a rtgtd interface this indirect flow mechanism wdl be stall further dimuushed by fmctlonal forces, resulting m a hrgher expertmental Q”” From the expenmental results we may mfer that PVAH desorbs completely after spreadmg, while Bz 1s only weakly surface active usual values lur the surtace excess of macromolecules amount to 0 5-2 mg mm2 Only at high Se4 a plateau value for QN1s found for most Na. but even for this sample apparently a great deal desorbs after spreadmg These findings on the surface activity are in qualitative agreement with interfacial tenslon measurements, as shown m Fig 7 Obviously, the activity increases with acetate content suggesting that the acetate groups are anchor segments at the interface The effects of acetate content and its mtra-molecular dlstrtbutlon have been the subJect of a systematic study on the surface activity of polyvinyl alcohol-acetate copolymers [6,381 While the reduction of v, by an interfacial tension gradient could be vlsuallzed[6] there exists yet another yw,
II
20
mN
1 m’
/ 75
1
If
I-
R
x-
82
o-
Cl
5
lb
Fe
i5
In,”
7 Interfaclal tension between water and I-butanol (= O), and between (co)polymer solutions m water (5g l-‘1 and lbutanoi as a funcuon of tune at 25 O’C Details can be found m Ref [6]. Chap
335
B J R
668
SCHOLTENS
quantitative verlficatlon of this mechamsm To that end we formulate the force balance for an mfimteslmal mterfacial area for which u,~ 1s zero
Insertion
of 7.w
=
-
and eqn (4) and mtegratlon 2t??bPb
+
%@w)/(rz
(31)
r)(avJar) between +
rd
=
rz and r, yields Ayh
-
rd
(32)
In Table 4 the required by values, as deternuned for the area probably flown along directly, are collected as a function of Nb Comparison with Fg 7 shows that the maxunum possible Ay, 1 7 mN m-l, IS of the same order as Ay required for a complete Immobdlzation for all Nb It also explams that PVA-H does not retard the transfer slgmficantly, that B2 cannot reach QMmax and that B4 causes maximum retardation for Nb up to 80 mln-’ For the explanation of the course of QN as a function of Nb and S we can confine ourselves to Rg 5, smce Fig 4 has the same shape as the very left of 5 From (32) Table 4 Estimated values for rnterfacnxl the tenston ddference reqmred to oppose the shear stresses of both flowmg phases
Nb
(mm-‘)
tCakulated
35x10-*and
SCalculated
4 x IO-’ and
Fig 8 The overall
r, = for r,=2xlO-*m r t =fg
mass
2*~ lO-‘
transfer
resistance
etai
we infer that at low Nb (and so at low fib and &.) ay/ar required to oppose both shear stresses wdl also be low Smce the mterfaclal excess was found to be attached n-reversibly at the mterface, It remains constant on varlatlon of Nb Therefore, a given amount of macromolecules 1s able to cover a larger area at lower Nb (rz Increases and consequently aylar decreases) This accounts quahtatlvely for the findmg QN 3 QN+c10at all S With Increase m S the total surface excess grows so that QN Increases, until the Interface IS completely covered for the lowest Na Any further mcrease m S at that Nb will dlmuush the absolute value of y(r) but not of aylar which is the relevant quantity For the higher Nb, however, this increase m S will still have effect until QN maxhas also been attamed for this sltuatlon From eqns (12), (17), (28) and (29) one can Infer that the dependence of (&) on Nb should be the same for a completely mobile and a completely ngd Interface This 1s venfied m Fig 8, where both ultimates show a lmear dependence AU mtermedlates are curved, caused by the change of r2 (and so the area covered) with Nb The deviations from the theoretical flow prevent apphcatlon of (25) to the present results However, modlficatlons of the experlmental set-up are under study m order to arrive at more perfect flow profiles, so that (25) may possibly be apphed to forthcommg results In hterature, Nltsch et al and Davies et al reported on quite slmdar mvestigatlons, so that a mutual comparison of results ESrequued Though Nltsch et al [31,32,39-42] apphed conslderably higher rotation speeds, they also assumed lammar flow and found tr - N” They used the analogy between mass and heat transfer to demonstrate the absence of an mterfacml resistance (for quite another system) Unfortunately, the effect of surfactants on these processes was of different magmtude the retardation of the heat transfer amounted to about a factor 2, the mass transfer was retarded 4-S times The reason of thus discrepancy IS not clear In addltlon they found (R) to be proportlonal to N-’ m most cases, which would imply that the coverage of the Interface is Independent of N, m contrast to eqn (32) and our findmgs Finally, they often used low molecular we&t surfactants, causing a short-crrcultmg
as a funcuon of Nb at Sa4 = O(A), 0 5(O), I( + ), 2(V) and 2 x ld(Cl) mg me2
Llqtud-llquld mass transfer and Its retardation by macromolecular effect on the retardmg mechanism by a steady adsorption at the periphery and a steady desorption m the centre[32] Some of these dtierences suggest that theu boundary layer flow was essentially different, probably due to both the higher stirring speeds and the apphcatlon of small-mesh wue-netting m their vessels Davies et al [26,43-47] have mainly studied turbulently stirred hquld-hquld and gas-liquid systems, so that not all of their results are comparable to ours Davies et al [44] found, e g an increase m retardation with stu-rmg rate from a spread protein (at low N no effective eddies were present which could be damped) However, Davies and Mayers[43] determined for their system an QNmaX comparable to ours and Davies and Khan[45] also found that not the absolute value of y IS relevant but only Its gradient From this comparison it IS clear that differences in hydrodynamic and geometnc condltlons may affect the results considerably and may even reverse a certain trend Therefore It remains risky to draw general conclusions from one particular series of measurements, although the underlymg pnnclples agree closely CONCLUSIONS
1 The mterfaclal resistance plays no part m the KC1 transfer from water to I-butanol only the butanol phase resistance 1s rate determining 2 The retardation m mass transfer due to spread copolymers can be accounted for quantltatlvely by consldermg the mteractlons between the hydrodynamic boundary layer and the y gradlent induced by the compressed copolymerlc surface excess 3 The surface activity of polyvinyl alcohol-acetate copolymers IS, for the water-butanol interface, strongly dependent on the acetate content Acknowledgemenr-The who skdfully
performed
authors are Indebted to Mr A Korteweg, most of the expenments
NOTATION
A a ; L ml2 N R QN r
interfacial area, m2 ratio of Vb and VW molar concentration, mol m-’ dlffuslon coefficient, mz s-’ characterlstlc length of the system, m dlstnbutlon coefficient ( = c,/c,) rotation speed, mm-’ overall mass transfer resistance, s m-’ retardation at rotation speed N partial mass transfer resrstance s m-‘, usually placed between brackets and always occumng with either superscnpt A or subscnpt A, 6, w or u radial distance m, often occurring with subscnpt 0, 1, 2 or 3 and never with a superscnpt amount of (co)polymer spread, mg m-* time, s volume, m3 velocity, m s-’ distance coordinate, m
669
adsorptlon
Greek svmbols < positive proportionahty constant, positive proportionahty constant, interfacial tension, N m-’ boundary layer thickness, m experimental constant vlscoslty, kg m-’ s-’ specdic conductance, R-’ m-’ density, kg rn-’ shear stress, kg m-’ s? molar flux, mol m-’ s-’ dlmenslonless z-coordinate
/
Bracket
()
m* s-’ m s-’
average
value over the cross
sectlon
Superscripts and subscrIpts In general, these refer to the quantity or substance mentioned b refers to the butanol phase ex experimental max maximum N refers to the rotation speed in the butanol phase r radial, at distance r th theoretical w refers to the aqueous phase CT mterfacial property Dtmenslonless groups Re Reynolds number = vpL1q SC Schmidt number = v/(pD)
REFERENCES
[l] Davies J T and Rldeal E K , Znierfacral Phenomena, Chap 7 Academic Press, New York 1961 [2] Davies J T , Adu Chem Engng 1963 4 1 [3] Davies J T , Turbulence Phenomena Academic Press, New York 1972 J,AZChEJ 1%410671 [4] KmgC I51 Szekelv J . Chem Erwtn SCI 1%5 20 141 Scholtkns’B J R , gh-D Thesis, Agrtcultural Umverslty, Wagenmgen. The Netherlands, Commumc Agnc Umv Wagenmgen 1977 77-7, avmlable upon request from B H BlJsterbosch 171 Davies J T ,J Phys Chem 1950 54 185 PI Tung L H and Drlckamer H G , J Chem Phys 1952 20 6 191 &felt J H and Dnckamer H G , I Chem Phys 1955 23 1095 [lOI McManamey W J , Chem Engng Scr I%1 15 210 1111 Vlgnes A, J Chrm Pkys 1960 57 980 I121 Rosano H L , Duby P and Schulman J H , J Phys Chem 1961 65 1704 L Y , Ostrovskn M V and [I31 Kremnev L Y , Skvtklrskn Abramzon A A, Zhurn Pnkl Khrm 1%5 38 24% I141 Tmg H P , Bertrand G L and Sears D F, Bmpkys / 1966 6 813 PJI Schulman J H , Ann New York Acad SCI 1%6 137 860 C T and Bllsterbosch B H , Berlckte [I61 De Jonge-Vleugel vom VZ Intern Kongr fur grenzflachenaktrve Stoffe, Band 2, 469-82 Carl Hanser Verlag, Munchen 1973 [I71 Albery W J , Couper A M , Hadgraft J and Ryan C , J C S Furaday Z 1974 70 1124 [18] Chandrasekhar S and Hoelscher H E , A Z Ch E J 1975 21 103 [19] Albery W J , Burke J F , Leffier E B and Hadgraft J , JCS Faraday Z 1976 72 1618
B J R
670
SCHOLTENS
[20] Thomas W J and Ismad S I A, Chem Engng Commun 1976287 [21] Garner F H and Hale A R , Ckem Engng .%I 1953 2 157 [22] Lmdland K P and TerJesen S G , Chem Engng Scr 1956 5 [23] hdge L K and Heldeger W J , A ZCh E I 1970 16 602 [24] Burnett J C and Hlmmelblau D h4 , A ZCh EJ 1970 16 185 [25] MacRltchae F , J Collard Interface SCI 1976 57 393 [26] Davies J T , Proc R SIX Lundon 1966 A290 515 [27] Scrlven L E and Sternlmg C V , Nature 1960 187 186 [28] Sawlstowskl H , m Recent Advances m Lcqurd-Llqutd Extraction (Edrted by Hanson C ), pp 293-366 Pergamon Press, Oxford 1971 1291 Sawlstowskl H , bd Chem Engng 1973 15 35 [30] Scholtens B J R and Bljsterbosch B H , FEBS Lett 1976 62 233 [31] Nltsch W and Hdlekamp K ,Chem Ztg 1972 96 254 [32] Kmep P , Thesis T U Munich 1974 E N , Zianspon [33] Bard R B , Stewart W E and LIghtfoot J Wdey. New York 1960 Phenomena [34] Abramowltz M and Stegun A , Handbook of Mathemabcal Fanctrons Dover, New York 1%8 [35] Schbchtmg H , Boundan, Layer Theory McGraw-Hdl. New York 1968 [36] Lock R C , Q JMech Appl Math 1951 4 42 [37] Beck W J and Bakker C A P , Appl Scr Res 1961 A10 241 [38] Scholtens B J R and Bljsterbosch B H , to be pubhshed [39] Nitsch W , Raab M and Kruep P , Benchte vom VI Intern Kongr furgrenzflachenaktrve Stoffe, Band 2, 147-55 Carl Hanser Verlag, Munchen 1973 [40] Nltsch W Raab M and Wkedholz R , Chem Ing Tech 1973 45 1026 [41] y$;i 5”;’ and Heck K D, Warme- und Stofiabertragung [42] Nttsch W [43] Davies J 16 55 [44] Davies J Scz 1964
and Weber G T and Mayers
T , Kllner 19 583
, Chem Zng Tech G R A,
Chem
A A and Ratchff
G A,
Scr
Chem
[45] Davies J T and [46] McManamey W R , Chem Engng [47] McManamey W Engng Scr 1975
Assummg Ideal overall reststance formulated as
Khan W , J , Davies Scr 1973 J , Multam 30 1536
(Rb) = where
the average
1%1 Engng
Cbem Engng Scr 1%5 20 713 J T , Woolen J M and Coe J 28 1061 S K S and Davies J T , Chem
APPENDIX mlxmg m both stlrred phases, the with respect to the 1 butanol phase
molar
(Cdnbw
(4”) = (V,,/A) V, denotes facud area
-
average can be
Cb)t(4”)
flux (4”) (mot
(Al)
mm2 s-‘) IS defined
dcddt
the volume of the I-butanol Usmg the trlvlal formulae
by (A2)
phase
and A the Inter
c, = c,’ - acb
(A3)
C, ’ = cbe(a + m b:)
(A4)
and
where I respectively e refer to the uutlal and a = V,/V,, we can rewrite (Al) as
and equlhbrmm
(Rt,) = A(1 + ambw)(cbe - c,+)/(Vb dct,/dt) which
value
(AS)
yields (R ) =
1976 48 715
Engng
eta1
b
AU +
ambv)
V, d In (cb’ -
provided A, V, and am& are constant, ally found to hold for the system studled
dt Cb)
which [6]
(‘46)
was experiment
LIQUID-LIQUID MASS TRANSFER BY MACROMOLECULAR B J R SCH0LTENS.t Laboratory
S BRUIN*
w/o
AND ITS RETARDATION ADSORPTION and B H BIJSTERBOSCH
for Physical and Collard Chemistry, Agricultural The Netherlands
Umverslty,
De Dre.l]en 6, Wagenmgen,
(Recewed 13 June 1978, accepted 18 September 1978) Abstract-The transfer of KCI from water to I-butanol has been measured in a Lewis cell, under lammar flow con&Ions This system appears to extublt no slgmficant mterfaclal resistance The effect of spread macromolecules on the mass transfer 1s shown to be prlmardy of a hydrodynarmc ongm On basis of the boundary layer theory and the convective diffusion equation a simple quantltatlve estimate can be made of the mfluence of a (partial) rlgichfication of the tnterface on the transfer process INTRODUCTION
has been studled extensively durmg the last few decades, confllctmg views have been presented on the existence of an appreciable mterfacml resistance and on the mechamsm by which surfactants retard mass transfer Usually[l4] the resistance to transfer of a substance A from phase 1 to phase 2 IS described by a series of partial resistances (two film theory) Although
solvent
extractIon
(RzA) = (rIAm2,* + r_*m*,*
+ r+*)
(1)
where (IZzA)(s m-‘) 1s the average overall resistance with respect to phase 2, rIA, r2A and I-,^ are, respectively, the two hquld phase resistances and the mterfaclal the dlstnbutlon reastance, while mZIA represents coefficient of A between phases 2 and 1 Actually, the resistances equal the ratlo of the relevant concentration difference and the resultmg mass flux Whereas the dlffuslonal hquld phase resistances can be predicted for a number of systems with simple geometry and boundary layer flow, the value of r,^ IS unknown Q pnon and IS simply neglected m many cases [l-3] In pnnclple, Its value can be obtamed from comparison of with the calculated values of r,* and r2* In @2%x practice, however, this procedure 1s only successful when the expertmental accuracy IS extremely high (static systems) or when the boundary layer flow IS exactly known These prerequlsltes are almost never met l61 Yet, regularly studies appear m which. whether correctly or assumed to play a, someor not, r,^ IS determmed times even dommant, part (e g [7-201) Several different explanations have been proposed for the retardmg influence surfactants may have on the transfer process These are based on possible changes in either the hydrodynamics near the Interface, affecting
*Present address Central Laboratory DSM, Fundamental Polymer Research Department, P0 Box 18, Geleen, The Netherlands *Department of Food Process Engmeenng, Agncultural Unlverslty, De DrelJen 12, Wagenlngen, The Netherlands $In pnnclple, an excess InterfacIal shear v~scoslty may also alter the flow, but tins effect IS always of neghglble magmtude [2] 641
r,* and rzA (e g [l-3,6]), or m r,“, which may be caused by several effects (e g [ll. 16.21-25]), or m both The hydrodynamics near the Interface always play a very important role m the Interphase transfer process Therefore It IS worthwhde to mvestlgate tf and how these may be changed by adsorption The mere presence of a clean hqmd-hquld interface has a pronounced influence on the boundary condltlons of both hquld flows at the interface the tangential velocity components are equal m both phases, whtle the normal velocity components vanish, m ad&Ion, the momentum flux perpendicular to the interface IS contmuous Adsorption may influence the hydrodynamics by Its effect (pursued by a decrease m mterfaclal tenslon, y) on these boundary condltlons I The mterfacml tension can only influence the boundary layer flow when the interface IS curved (capillary force), (by an or when Its value vanes along the interface uneven dlstnbutlon of surfactant molecules) This uneven dlstnbutlon may be caused by a boundary layer flow and m this case the induced gradient m y will oppose Its own orlpn Only for the most mterestmg case from a technological pomt of view, VIZ turbulent flow, a quantitative theory has been proposed[3,26] to explam the retarding effect of monolayers Although the complex hydrodynamics were slmphfied considerably m thts at least quahtafive predlctlons have been analysls, confirmed by expenments On the other hand it has been suggested that the rnterfacral resistance may be increased by adsorption by (partly) blockmg the mterface (sieve effect) or by a chemical mteraction of A with the surfactant or the Interface Itself (free energy barrier) [ 11, 16,2 l-251 The precise mechamsm of retardation determmes which of the three terms at the right hand side of (1) changes most as compared to the surfactant-free case This Implies that any observed retardation can only be explamed exclusively by, e g a change m r,^, provided the second term of (1) 1s at least comparable in magnitude to the two other terms (bottleneck principle) The controversial views on the importance of r,^ and on the mechamsm of retardation by adsorptlon can be studled under deliberately chosen expenmental conditions such that the hquld phase resistances are relatlvely small and still amenable to calculations To meet
662
B J R
SCHOLTENS et
these demands we developed a transport vessel wrth a flat hquld-hqmd Interface and centro-symmetic lammar tangential flow Smce model measurements of this type may be severely dlsturbed by spontaneous (Marangom) mstablhtles [U-29], theu absence must be checked thoroughly dunng the transfer process In a recent commumcatlon [30] It was tentatively suggested that vmation of the hydrodynamics may enable the overall resistance to be separated mto the sum (T,%~,~ + rzA) quantitative and r,^rn2, * In this paper we present arguments m favour of this approach, whzch provides indeed a key to a better understanding of the mterfaclal phenomena
The overall resistance was measured for the KC1 transfer (starting at a concentration of 0 1 mol 1-I) from with I-butanol) to I-butanol water (always saturated (saturated with water) The transport vessel (Fig l), rather similar to that of Nltsch et 01 [31,32], contamed large propellers to obtwn relatively high overall flow rates with as little turbulence as possible These were counterrotattBecause KCI was the only electrolyte under conslderatlon, we refram from using the mdex KCI when mentlonmg (I&), cb or
al
mg, pumping the llquld m inward, mainly radial. duection along the interface which was flat and without slgmficant waves Draft tubes with (SIX) baffles guaranteed a systemattc fiow near the interface and optimum partlclpatlon of all hquld m the transfer process In order to achieve comparable hydrodynamics at both sides of the interface, we always chose equal values of Re for both phases (&/AL = ub/vy = 2 78) Reproducible results were obtained by always venfymg (with a cathetometer) the uutlal posrtlons of all parts that could Influence the hqtnd flow and adlusting d necessary For that same reason RO wire netting was used, m contrast to[31,32] In order to suppress any mterfenng vlbratlons, the supporting construction, carrying two identical vessels, was mounted on a heavy concrete slab that rested on rubber plugs The glass jackets were thermostatted at 25 OO+ 0 01°C (&)t was determined from the time-dependent KC1 concentration m the butanol phase, which was measured conductometncally In order to convert the conductance m-l) to the KC1 concentration cb, the expenKb (i-i-’ mental relation cb = 1 201(#&)’ 1304mol I-’ was used, which held true for the concentration range 0 I16 mmol 1-l (mb, = 172 X lo-*) The value of (&) was obtained usmg eqn (A6) (see the Appendix) By applying moderate sturmg speeds (40 < & C 90 mm-‘) a lammar
SupportIng construction
I
Fig 1 Vertical cross section of one transport vessel and the supportmg construction, 1, conductance cell, 2, upper stu-rmg shaft (#J = 0 60 cm) with a three-blade. nght-hand propeller ($ = 5 1 cm, 2 7 cm above the interface). both stainless steel, 3, stamless steel shaft connected to a baffle of the draft tube, 5, thermostat& double-walled Pyrex glass vessel (mslde 9 = 8 08 c 0 02 5 90 cm, outside 4 = 6 36 cm, he@ = 2 50 cm) with SIX bafRes (2 60 x 0 75 mterface, 7, ldentlcal lower draft tube. reshng with three baffles m holes m stainless steel plug (4 = 5 8 and 1 5 cm, height = 0 6 and 3 4 cm) with a hutyl
4, PVC cover with butyl rubber O-nng, cm), 6, upper draft tube (mslde $ = cm’), the lower edge 1 2 cm above the the plug 8, butyl rubber gasket ring, 9, rubber 0-ring, fixed by a PVC nut (20),
10, lower stvrmg shaft, ldentlcai to 2, 11, toothed belt (Synchroflex), tooth = 0 5 cm, 12, toothed pulley (& = 3 cm), 13, Teflon shde rmg, 14, brass shde bearmg, fixed on I5 with msulatmg material. IS, upper ;tlummmm plate
(thickness 0 6cm), 16, lockmg ring, 17, brass column (9 = 1 Ocm, he& = 20cm). 18, shaft with nut to fix the cover (length = IS cm). 19, lower alummmm plate (33 x 22 x 0 3 cm’) with two insulating rubber mats. 20, PVC nut, 21, msulatmg PVC columns, 22, alumuuum support (33 x 20 x 0 3 cm’) Both propellers and draft tubes are smutted symmetncally with regard to the Interface, whde VW = 225 and V, = 210 cm3
Llqtud-bquld
mass transfer and Its retardation
velocity profile along the interface was guaranteed with 500 > Re, = Reb c 1200 Although theoretically Marangem mstabrhtres are possible for the present transfer dIrectIon, we checked very thoroughly that they were absent at the chosen concentration dfierenceE61 In addition we studied the effect of three, only water soluble, polymenc substances on the KCI transfer PVAH, B2 and B4 The first 1s a completely hydrolysed polyvmyl alcohol (vrscoslmetnc mean molar mass = 118 kg mol-‘), while the other two are p-ally reacetylated copolymers denved from PVA-H, contammg 6 8 and 21 3 mole% vinyl acetate respectively These substances were spread at the interface from a concentrated solution, usmg a micrometer syringe Further detmls of the expenmental procedures, the completely automated determination of (&) and all materials are given extensively eIsewhere [6]
by macromolecular
v, = -a/r
N, = constant
I
‘1
v,=-f3z/r
W w il
r
0
@
(2)
Since u, pomts to r= 0 m the experimental set-up, we only consider negative constants, which may stil depend on z The followmg two cases ~111 be examined (see Fig 2) v,=
-fflr
(3)
and (4)
v, = -@IT
where a(m’ s-‘) and @(m s-l) are stants For the diffusion of A out of the lammar boundary layer, the equation (e g [33], p 559) can be approximated
ah ar
tl,-=D,4-7.
thus
posltlve
con-
interface into the of continuity of A by
a2cA az
provided a pseudo-steady state approxlmatlon can be made (&,/at = 0) and the velocity profile IS not affected by the dlffuslon process The boundary conditions for the solution of (5) ~th (3) or (4) read
I,
@
,2--Bd-A r0
663
In the followmg we shall solve a slmphfied form of the contuuuty equation for the transfemng component which IS applicable for the two ultunate cases (I e the completely moblle resp ngdified interface) From these solutions the concentration profiles can be evaluated and from these, m turn, the hqmd phase resistances In addition, for all intermediate cases, VIZ a partly covered Interface, the solution will be shown to be a combmatlon of the two ultimate situations Consider an axially symmetic, flat liquid-hqutd Interface at z = 0, extending between r. and rl (rO > r,), wth a lammar boundary layer for z Z=0 If we assume that the flow 1s stationary (au/at = 0), that the density remams constant and that only a radial velocity component (u,) 1s present along the interface, the continuity equation for the flowing hquld (e g [33]. p 83) can be readily mtegrated to
ANALYSIS
One fundamental requirement to analyse the mass transfer process and to understand the effect of adsorption quantitatively, 1s the theoretical estimate of both hqurd resistances m the absence and presence of adsorbed molecules Thus, the innermost part of the boundary layer flow should be known precisely for these situations Flow visuahzation showed a rather systemauc, mamly radial flow pattern m both phases near the mterface[6] For the clean hquld-hquld Interface we shall assume a radial plug flow (Ag 2a), since It cannot resist any tangential shear stress Later this approximation ~111 be shown to mtroduce no serious error m the estimate of both liquid phase resistances In the presence of su&zctunt the radial flows will drive the adsorbed molecules to the centre of the interface, until the induced mterfacld tension gradient 1s able to oppose the two shear stresses Since adsorptlon of the macromolecular substances used was shown to be u-reverslble[6], they could not desorb from the centre into the aqueous solution Ultimately, the interface may be completely covered by an adsorptlon layer, the surface excess mcreasmg towards the centre, so that the interfacial velocity is zero everywhere, Just as for a hquld-sohd interface For the mnermost part of the boundary layer in this situation. a radially symmetnc velocity 1s assumed, mcreasmg linearly with the distance from the interface (Fig 2b)
I
adsorptIon
CA = CA,
at
z>O
and
r=ro
c.4 = CA,
at
z=O
and
rlsrsro
CA = CA,
at
z=m
and
r,srcr,
(6) 1
In order to solve (5) with (3) and (6) we introduce dlmenslonless vanabIe
the
z I
4
r
I_ 0
Frg 2 Schemattc representatton of the two hydrodynannc boundary layers along the Interface whrch extends between rO and r,
Inserting
into (5) leads to d*c, CdcA _ dk+2dC-O
(8)
664
B J R SCHOLTENS
while the boundary
conditions
can now be formulated
as
eta1
for r, s r s r2 and mobile for r2 < r c r. (Fig 3a) In the mobile, region the flow may be approximated by (3) and in the inner region by (4) For the outer region the calculations are identical to those for the clean interface, but now with ri = r2 However, for the inner region the solution of the completely rigid interface does not apply since the boundary conditions at r = r2 are now given by (10) and not by the appropriate form of (9a) However, we can still use solution (16) by applying a mathematical trick consider another, hypothetical interface completely immobilized for r, s r c r3 (with r+ < r3 < r& as drawn in Pig 3b We choose r3 m such a way that at r = r2 the concentration penetration depth for this case IS equal to that calculated with (10) for the outer region Obviously, (rA) for the partly covered interface is then given by outer,
CA = CA,
at
(9a)
CA = CA0
at
i=O
(9b)
The solution
of (8) satisfying
these boundary
conditions
IS
cAo-cA = CA.7- CA.=
I
c
0
exp (- 6’14) dr = PW2,
I)
(10)
_ exp (- t’l4) dS I II
where P(1/2,5) is an incomplete I-function (e g [34], p 260) From this concentration profile, the local molar flux 42, and the average mass transfer resistance (rA) can be evaluated with standard methods
(18) and
h) =
(CA- - c.&&r(ro2 - r,a) = I(1/2)(rclZ - r**)“* 2(2aD,)‘” (ro’ - rtZ)“2 = 1 5%(aD,)“’
(12)
Since the determination of r3 as a function of r,,, rz, (I, p and DA IS tedious-the concentration profiles both contain incomplete P-functions-the approximate method of Von Karman and Pohlhausen (e g [U]. p 187) was used As a trial solution for the concentration profile we used (for z s S (r)) CA
Analogously, the solution of (5) with (4) and (6) can be obtained by introduction of
’=‘(o,($Inserting
r2))“’
(13)
(C&T-- CA&i - z/s(r))’ +
=
CA,
(19)
where the concentration boundary layer 6(r) is only a function of r With this relation eqn (5) was solved again with (3) or (4) and (6) This resulted in S(r) = (6DA(ro2 - r’)/a)“’
into (5) gives
(20)
and 42, = (&$+)j’m while the same boundary conditions (9) now hold when 5 IS substituted for 5 The solution of (14) then reads CA0 -
CA
CA, - CA,
= P(1/3,5)
for the completely
mobile
(c*, interface,
- c,,)
(21)
and in
(15)
so that
(16) and (rA)
=
2U1/3)(ro+- r,*)“’ = (ro’- r,Y 3(6/3D,,‘)“’
Now we consider only partly covered
1 017(BDA2)‘”
(17)
the same interface, but this time by surfactant molecules ngidified
Fig 3 SchematIc representanon of the hydrodynamrc dlffus#on (8) boundary layer for a hqmd-hquld mterface r, and rlr llgrdlfied for r, G r < r2
(A) and between
Llquld-llquld mass transfer and Its retardanon by macromolecular adsorptIon S(r) = 18D*(r**
- rZ)I/3p3
were found 9~ 10m3 m Smce no slgmficant differences within this volume, these measurements provided us with the average radial bulk velocities at r = 2 5 x lop2 and /zJ = 6 x 10e3 m A curve fitting analysis ylelded
(22)
and
dir =
(fj$ppJ”(c,,
-
CA_)
(23)
for the completely covered Interface Notice the close agreement with the exact solutions From (20) and (22) r3 can now easdy be derived as an lmphclt function of the other variables 2(D/+(ro2 - r*+)/a)’ = 3(D,4(r,2 - r2+)/j3)2
(24)
Making use of experimental values (see results), it can be shown that insertion of r2 instead of the evaluated r3 in (18) for this system results m a value for (rA) that IS maximally 1% too low-as exemplified rn Table 1 for an mtermedlate Nb Therefore It 1s Justified to replace r3 by r2 in (18), resulting in
which reduces to (12) and (17) for the two ultimate cases, Equation (25) VlZ r2 = r, and r2 = ro. respectively enables us, with a simple Iterative method, to determine rz and so the fractional coverage of the Interface, provided all other expenmental values are known Therefore it may provide us with a method to study the dynamic behavlour of adsorbed macromolecular layers in a vanable shear field varlatlon of hydrodynamics (a and /3) will induce a different (rA) from which the new r2 can be evaluated RESULTS In both phases hquld velocity measurements were performed by photographing suspended polystyrene partlcles[6] At four different stlmng speeds their average velocity was determined m the volume confined 2x10-*
665
vbr
=
-63x
lo-‘(N,
mm)14m
VW = - 2 3 x lo-’ (iVb mm)’ 4 m
K?r,t
(m)
35 0 300 2s 0 20 0 15 0 100 50
35 1 30 2 25 4 20 7 16 1 11 8 82
10%
+ MM)*
/3 = 5 7 x 10e3 (vrp”/(6
(27)
x 10-4~)“z m
s-’
(29)
where the numerical constant In the numerator has the dlmensron m1’2 In Table 2 we have collected all relevant hydrodynamic parameters for the experimental conditions the hydroSince mbw IS very low (1 72 X lo-‘), dynamics in both phases are comparable (Re, = Reb) and Db < D, (q, = 2 87X 10p3, vW = 1 2 x lop3 Pa s), It wrll be clear that for the present system the sum of the hquld phase resistances 1s completely controlled by rb The values of r. and rl are given by the geometry of the set-up (they amount to 4 04 x lo-’ and 0 m, respectively), so that the only unknown quantity whtch thwarts the
108L(d&
l@&(M
(m’ s-l)
(m’ s-‘)
(m’ s-‘)
34 45 53 59 63 66 67
I 12 0 89 0 67 048 0 30 0 15 004
I 14 093 0 73 0 54 0 37 021 008
A%” 06 09 11 11 11 09 06
Wetermmed from (24) with rO= 0 0404(m), Q, = 3 7x 10m4(m2 s-l), 0 34 (m s-l), DbKc’= 3 x IO-” (m* s-‘) SThe sum of the integrals m the denommator of (18). with r, = 0 3The second rntegral evaluated with the relevant rS value lIThe second mtegral evaluated with the relevant r2 value *he relative increase In the sum of the two mtegrals by usmg r2 Instead of rS CEsVd34NoS-E
s-’
For the completely ngd Interface p,, and & were estimated from the analogy of the flows with that m the vlcuuty of a stagnatlon pomt (e g [353, p 88), smce at the penphery of the mterface the two flows approach lt at nght angles The actual flow IS radtally away from the “stagnation cucle” in inward direction. so that the solution for plane flow seems reasonably appropriate From the solution of this veloczty profile we obtamed, using (4) at the afore mentioned coordmates
addition the Integrals of (18) are evaluated, the effect of using rz Instead of ra can be Inferred from the last four columns
(m)
(26)
The radial mterfaclal velocity, z),,, of a completely mobde Interface was estimated with relations derived by Lock[36] Assuming the water and butanol streams to be parallel and lammar, thrs resulted in v,, = 0 77 x v,, = -4 8 x lop5 (Nb mln)’ a and so, using (3) at r = 2 5 X lo-’ m
Table 1 The value of r, as a function of r2 at N* = 60 nun-‘, as calculated from (24) In
lO%,
s-’
&, =
B I R SCHOLTENS et al
666 Table 2 Approximate Nil (mm-‘) 40 50 60 70 80 90
values for tbe hydrodynamic
10&t cm s-‘)
- 1oLt (m SK’)
1I
04
15 19 24 29 34
05 07 09 11 13
Nb (mm-‘) 40 50 60 70 80 90
(rt.)
1 17 1 12 109 108 1 05 104
100 8 55 7 58 6 76 6 17 5 68
116920 14 9552009 8 242007 732?005 646kOO4 5 88kOO3
-r 1
(m2 s-‘)
08 12 15 19 22 26
(&Lx/(&l
lo-‘(G)th (s m-l)
to-‘UW,, (s m-‘)
s-‘)
10
m the vlcmlty
101lx,
- lo%,, (m
theoretical estimation of the butanol resistance with (12) or (17) IS 0, Its value was denved from the hmltmg molar conductance of KC1 m the butanol phase, using the Nernst equation for completely dlssoclated electrolytes (which was shown to be apphcable[6]) Making allowance for the low actlvlty coefficient m the butanol phase[6] and assummg equal amontc and catiomc calculated equivalent conductances, we La= 3 x lo-” m2 s-l for the relevant concentration range Experimental values for (I&.) and theoretical values for (rb) (calculated with 12 and 28), both In the absence of any polymenc mater&, are collected in Table 3 as a function of Nb ( = 2 78 NW) The effect of the apphed (co)polymers was diverse, depending on the total amount spread, S (which vmed between 0 2 and 2 x ld mg me2), the acetate content and Nb Even at the highest Se”*’ no slgnficant mfluence was observed at any h$, On the other hand, the two copolymers caused a defimte retardation of the Table 3 Companson of expenmental (RI,) and theoretlcal values as a function of the sturmg speed
parameters
21 29 37 47 56 66
(m
of the Interface
Bb s-‘)
0 15 0 24 0 34 047 063 0 80
(m
BW s-‘)
006 009 0 13 0 17 0 23 0 29
Fig 4 The expenmental retardation &a* for different Nb as a functton of the amount of B2 spread at the mterface, 0, Nb = 40, & Nb = 50, Cl, Nb = 60, and 7, Nb = 70,80 and 90 mu-’
transport rate, that can be expressed as the quotient of the overall resistances wtth and wlthout macromolecular matenal present and IS mdrcated by QN = Experunental (Ra) ,,.,~co~~Ymer/(R~) at N,, = N mm-’ values of QN vs S can be found m Fig 4 for B2 and m Fig 5 for B4 These values are the average of at least four different experunents (whde the (Rb) values are based on approximately 45 runs) For both co-polymers the retardation at constant S Increases with decreasmg stimng speed Whrle for B2 the maximum retardation seems not yet to be reached, at the highest SH4 It IS attamed for most to 6 1 *O 2) Since several QN values Nb (amountmg could be determmed dunng one experlmental run, we also studled the effect of the hlstory of the spread layers on QN a senes of measurements with Na successively 40, 90, 40, 90 and 4Onun-’ showed QW and QW to be essentially constant m all cases
102
lo3 rng m-2
Fig 5 The expenmental retardation QN” for dtierent Nb as a function of the amount of B4 spread at the interface, 0, Nb = 40, & Nb = 50, [7. Nb = 60. a, Nb = 70, A. Nb = 80. and n, Nb = 90 nun-‘, V, represents a comcldmg point for at least two of the four lowest rotation speeds and v for at least two of the four highest rotation speeds
Llqutd-hquld
mass transfer and Its retardation
by macromolecular
adsorptron
667
DISCUSSION
the expertments m the absence of any macromolecufar material, a comparison of expenmental overall with theoretical liquid phase resistances 1s mven in column 4 of Table 3 Although on first sight these results might indicate a small r, a closer mspection of eqns (I), (12) and (28) reveals that the relative conmbutlon of r, must grow with Nb, while column 4 shows the opposite trend Insertion of (28) m (12) shows that (rblfh should be proportIona to Nb-“’ If an mterfaclal resatance were operative, (Rb)ex would be approximately proportional to N to a less negative power However, linear regression analysis of (Rb)cx and log Nb yielded -0 84 as the power, mdlcatmg the absence of an mterfacial resistance We think that the observed devtatlons from theoretical maccuracy m predictions are not due to any (systematic) CY_or & [6] Rather these should be found m differences between expenmental and theoretlcal flow profiles, such as end effects Indeed, flow vlsuahzatlon studles[6] showed the radlaf flow along the mterface to be less effective for ra 3 5 x lo-* m and particularly for r =S 2 X lo-’ m than for the regton in between, especially at low Nb The effects became less distinct with mcreasmg Ns, consistent with the findings in Table 3 and also with results of transfer expenments with a very thm plate at the centre or rmg at the penphery of the interface161 In the calculation of (rb)* plug flow (Fig 2a) was assumed whale m fact a small velocity gradient au,/a_z exists at z = 0 (as depicted at the left of Fig 6) Using results of Beek and Bakker[37& It can easdy be shown, however, that this approxlmatlon does not mtroduce errors larger than 2% Having considered all aspects thoroughly, we conclude that the mterfacml resistance for the present system IS very small d at all present, so that the butanol phase resistance LS the only rate determining step m the transfer process Because of the neghpble tnterfactal resistance, the eflect of the spread copolymers on the transfer process must obviously be an increase m the butanol phase resistance Elsewhere[6] It was shown that there IS a one-to-one correspondence between adsorption (mterfacial excess) and retardation for this system, that Db IS unaffected by macromolecular layers at the interface and that no excess interfacial shear vlscosrty can be attnbuted to these layers Therefore, any increase m (Rb) can only be caused by dynamic utteractions between the Bow along the Interface and the compressed macromolecular layer, resultmg m (partly) nsddicatlon Actually, the flow along the ngd interface is not of the structure indicated m Fig 2, but rather hke that at the nght of Fig 6 However, since Scb = 11 x lo’, only the very innermost part of the hydrodynanuc boundary layer plays a part m the mass transfer process, for whrch the former profile IS a good approxlmatron The effect that complete immobdlzatton of the interface can have on the butanol phase resistance was calculated from (12) and (17) Insertron of (28) and (29) yldds that QNnv” should be independent of Nbr amountmg to 4 9, which IS close to our expenmental results for For
Fig
6
Boundary layer flow along an Interface that extends between r, and r, and IS covered for r, s r s r,
very high Se4 The quantttative difference m Q”” IS attnbuted to the Imperfectness of the actuaI flow, smce only part of the interface IS flown along effectively due to curvature of the streamlines In the absence of surfactant, this mterfacml flow wdl induce (less effecovely) hqtud movements along the parts that are not reached directly However, for a rtgtd interface this indirect flow mechanism wdl be stall further dimuushed by fmctlonal forces, resulting m a hrgher expertmental Q”” From the expenmental results we may mfer that PVAH desorbs completely after spreadmg, while Bz 1s only weakly surface active usual values lur the surtace excess of macromolecules amount to 0 5-2 mg mm2 Only at high Se4 a plateau value for QN1s found for most Na. but even for this sample apparently a great deal desorbs after spreadmg These findings on the surface activity are in qualitative agreement with interfacial tenslon measurements, as shown m Fig 7 Obviously, the activity increases with acetate content suggesting that the acetate groups are anchor segments at the interface The effects of acetate content and its mtra-molecular dlstrtbutlon have been the subJect of a systematic study on the surface activity of polyvinyl alcohol-acetate copolymers [6,381 While the reduction of v, by an interfacial tension gradient could be vlsuallzed[6] there exists yet another yw,
II
20
mN
1 m’
/ 75
1
If
I-
R
x-
82
o-
Cl
5
lb
Fe
i5
In,”
7 Interfaclal tension between water and I-butanol (= O), and between (co)polymer solutions m water (5g l-‘1 and lbutanoi as a funcuon of tune at 25 O’C Details can be found m Ref [6]. Chap
335
B J R
668
SCHOLTENS
quantitative verlficatlon of this mechamsm To that end we formulate the force balance for an mfimteslmal mterfacial area for which u,~ 1s zero
Insertion
of 7.w
=
-
and eqn (4) and mtegratlon 2t??bPb
+
%@w)/(rz
(31)
r)(avJar) between +
rd
=
rz and r, yields Ayh
-
rd
(32)
In Table 4 the required by values, as deternuned for the area probably flown along directly, are collected as a function of Nb Comparison with Fg 7 shows that the maxunum possible Ay, 1 7 mN m-l, IS of the same order as Ay required for a complete Immobdlzation for all Nb It also explams that PVA-H does not retard the transfer slgmficantly, that B2 cannot reach QMmax and that B4 causes maximum retardation for Nb up to 80 mln-’ For the explanation of the course of QN as a function of Nb and S we can confine ourselves to Rg 5, smce Fig 4 has the same shape as the very left of 5 From (32) Table 4 Estimated values for rnterfacnxl the tenston ddference reqmred to oppose the shear stresses of both flowmg phases
Nb
(mm-‘)
tCakulated
35x10-*and
SCalculated
4 x IO-’ and
Fig 8 The overall
r, = for r,=2xlO-*m r t =fg
mass
2*~ lO-‘
transfer
resistance
etai
we infer that at low Nb (and so at low fib and &.) ay/ar required to oppose both shear stresses wdl also be low Smce the mterfaclal excess was found to be attached n-reversibly at the mterface, It remains constant on varlatlon of Nb Therefore, a given amount of macromolecules 1s able to cover a larger area at lower Nb (rz Increases and consequently aylar decreases) This accounts quahtatlvely for the findmg QN 3 QN+c10at all S With Increase m S the total surface excess grows so that QN Increases, until the Interface IS completely covered for the lowest Na Any further mcrease m S at that Nb will dlmuush the absolute value of y(r) but not of aylar which is the relevant quantity For the higher Nb, however, this increase m S will still have effect until QN maxhas also been attamed for this sltuatlon From eqns (12), (17), (28) and (29) one can Infer that the dependence of (&) on Nb should be the same for a completely mobile and a completely ngd Interface This 1s venfied m Fig 8, where both ultimates show a lmear dependence AU mtermedlates are curved, caused by the change of r2 (and so the area covered) with Nb The deviations from the theoretical flow prevent apphcatlon of (25) to the present results However, modlficatlons of the experlmental set-up are under study m order to arrive at more perfect flow profiles, so that (25) may possibly be apphed to forthcommg results In hterature, Nltsch et al and Davies et al reported on quite slmdar mvestigatlons, so that a mutual comparison of results ESrequued Though Nltsch et al [31,32,39-42] apphed conslderably higher rotation speeds, they also assumed lammar flow and found tr - N” They used the analogy between mass and heat transfer to demonstrate the absence of an mterfacml resistance (for quite another system) Unfortunately, the effect of surfactants on these processes was of different magmtude the retardation of the heat transfer amounted to about a factor 2, the mass transfer was retarded 4-S times The reason of thus discrepancy IS not clear In addltlon they found (R) to be proportlonal to N-’ m most cases, which would imply that the coverage of the Interface is Independent of N, m contrast to eqn (32) and our findmgs Finally, they often used low molecular we&t surfactants, causing a short-crrcultmg
as a funcuon of Nb at Sa4 = O(A), 0 5(O), I( + ), 2(V) and 2 x ld(Cl) mg me2
Llqtud-llquld mass transfer and Its retardation by macromolecular effect on the retardmg mechanism by a steady adsorption at the periphery and a steady desorption m the centre[32] Some of these dtierences suggest that theu boundary layer flow was essentially different, probably due to both the higher stirring speeds and the apphcatlon of small-mesh wue-netting m their vessels Davies et al [26,43-47] have mainly studied turbulently stirred hquld-hquld and gas-liquid systems, so that not all of their results are comparable to ours Davies et al [44] found, e g an increase m retardation with stu-rmg rate from a spread protein (at low N no effective eddies were present which could be damped) However, Davies and Mayers[43] determined for their system an QNmaX comparable to ours and Davies and Khan[45] also found that not the absolute value of y IS relevant but only Its gradient From this comparison it IS clear that differences in hydrodynamic and geometnc condltlons may affect the results considerably and may even reverse a certain trend Therefore It remains risky to draw general conclusions from one particular series of measurements, although the underlymg pnnclples agree closely CONCLUSIONS
1 The mterfaclal resistance plays no part m the KC1 transfer from water to I-butanol only the butanol phase resistance 1s rate determining 2 The retardation m mass transfer due to spread copolymers can be accounted for quantltatlvely by consldermg the mteractlons between the hydrodynamic boundary layer and the y gradlent induced by the compressed copolymerlc surface excess 3 The surface activity of polyvinyl alcohol-acetate copolymers IS, for the water-butanol interface, strongly dependent on the acetate content Acknowledgemenr-The who skdfully
performed
authors are Indebted to Mr A Korteweg, most of the expenments
NOTATION
A a ; L ml2 N R QN r
interfacial area, m2 ratio of Vb and VW molar concentration, mol m-’ dlffuslon coefficient, mz s-’ characterlstlc length of the system, m dlstnbutlon coefficient ( = c,/c,) rotation speed, mm-’ overall mass transfer resistance, s m-’ retardation at rotation speed N partial mass transfer resrstance s m-‘, usually placed between brackets and always occumng with either superscnpt A or subscnpt A, 6, w or u radial distance m, often occurring with subscnpt 0, 1, 2 or 3 and never with a superscnpt amount of (co)polymer spread, mg m-* time, s volume, m3 velocity, m s-’ distance coordinate, m
669
adsorptlon
Greek svmbols < positive proportionahty constant, positive proportionahty constant, interfacial tension, N m-’ boundary layer thickness, m experimental constant vlscoslty, kg m-’ s-’ specdic conductance, R-’ m-’ density, kg rn-’ shear stress, kg m-’ s? molar flux, mol m-’ s-’ dlmenslonless z-coordinate
/
Bracket
()
m* s-’ m s-’
average
value over the cross
sectlon
Superscripts and subscrIpts In general, these refer to the quantity or substance mentioned b refers to the butanol phase ex experimental max maximum N refers to the rotation speed in the butanol phase r radial, at distance r th theoretical w refers to the aqueous phase CT mterfacial property Dtmenslonless groups Re Reynolds number = vpL1q SC Schmidt number = v/(pD)
REFERENCES
[l] Davies J T and Rldeal E K , Znierfacral Phenomena, Chap 7 Academic Press, New York 1961 [2] Davies J T , Adu Chem Engng 1963 4 1 [3] Davies J T , Turbulence Phenomena Academic Press, New York 1972 J,AZChEJ 1%410671 [4] KmgC I51 Szekelv J . Chem Erwtn SCI 1%5 20 141 Scholtkns’B J R , gh-D Thesis, Agrtcultural Umverslty, Wagenmgen. The Netherlands, Commumc Agnc Umv Wagenmgen 1977 77-7, avmlable upon request from B H BlJsterbosch 171 Davies J T ,J Phys Chem 1950 54 185 PI Tung L H and Drlckamer H G , J Chem Phys 1952 20 6 191 &felt J H and Dnckamer H G , I Chem Phys 1955 23 1095 [lOI McManamey W J , Chem Engng Scr I%1 15 210 1111 Vlgnes A, J Chrm Pkys 1960 57 980 I121 Rosano H L , Duby P and Schulman J H , J Phys Chem 1961 65 1704 L Y , Ostrovskn M V and [I31 Kremnev L Y , Skvtklrskn Abramzon A A, Zhurn Pnkl Khrm 1%5 38 24% I141 Tmg H P , Bertrand G L and Sears D F, Bmpkys / 1966 6 813 PJI Schulman J H , Ann New York Acad SCI 1%6 137 860 C T and Bllsterbosch B H , Berlckte [I61 De Jonge-Vleugel vom VZ Intern Kongr fur grenzflachenaktrve Stoffe, Band 2, 469-82 Carl Hanser Verlag, Munchen 1973 [I71 Albery W J , Couper A M , Hadgraft J and Ryan C , J C S Furaday Z 1974 70 1124 [18] Chandrasekhar S and Hoelscher H E , A Z Ch E J 1975 21 103 [19] Albery W J , Burke J F , Leffier E B and Hadgraft J , JCS Faraday Z 1976 72 1618
B J R
670
SCHOLTENS
[20] Thomas W J and Ismad S I A, Chem Engng Commun 1976287 [21] Garner F H and Hale A R , Ckem Engng .%I 1953 2 157 [22] Lmdland K P and TerJesen S G , Chem Engng Scr 1956 5 [23] hdge L K and Heldeger W J , A ZCh E I 1970 16 602 [24] Burnett J C and Hlmmelblau D h4 , A ZCh EJ 1970 16 185 [25] MacRltchae F , J Collard Interface SCI 1976 57 393 [26] Davies J T , Proc R SIX Lundon 1966 A290 515 [27] Scrlven L E and Sternlmg C V , Nature 1960 187 186 [28] Sawlstowskl H , m Recent Advances m Lcqurd-Llqutd Extraction (Edrted by Hanson C ), pp 293-366 Pergamon Press, Oxford 1971 1291 Sawlstowskl H , bd Chem Engng 1973 15 35 [30] Scholtens B J R and Bljsterbosch B H , FEBS Lett 1976 62 233 [31] Nltsch W and Hdlekamp K ,Chem Ztg 1972 96 254 [32] Kmep P , Thesis T U Munich 1974 E N , Zianspon [33] Bard R B , Stewart W E and LIghtfoot J Wdey. New York 1960 Phenomena [34] Abramowltz M and Stegun A , Handbook of Mathemabcal Fanctrons Dover, New York 1%8 [35] Schbchtmg H , Boundan, Layer Theory McGraw-Hdl. New York 1968 [36] Lock R C , Q JMech Appl Math 1951 4 42 [37] Beck W J and Bakker C A P , Appl Scr Res 1961 A10 241 [38] Scholtens B J R and Bljsterbosch B H , to be pubhshed [39] Nitsch W , Raab M and Kruep P , Benchte vom VI Intern Kongr furgrenzflachenaktrve Stoffe, Band 2, 147-55 Carl Hanser Verlag, Munchen 1973 [40] Nltsch W Raab M and Wkedholz R , Chem Ing Tech 1973 45 1026 [41] y$;i 5”;’ and Heck K D, Warme- und Stofiabertragung [42] Nttsch W [43] Davies J 16 55 [44] Davies J Scz 1964
and Weber G T and Mayers
T , Kllner 19 583
, Chem Zng Tech G R A,
Chem
A A and Ratchff
G A,
Scr
Chem
[45] Davies J T and [46] McManamey W R , Chem Engng [47] McManamey W Engng Scr 1975
Assummg Ideal overall reststance formulated as
Khan W , J , Davies Scr 1973 J , Multam 30 1536
(Rb) = where
the average
1%1 Engng
Cbem Engng Scr 1%5 20 713 J T , Woolen J M and Coe J 28 1061 S K S and Davies J T , Chem
APPENDIX mlxmg m both stlrred phases, the with respect to the 1 butanol phase
molar
(Cdnbw
(4”) = (V,,/A) V, denotes facud area
-
average can be
Cb)t(4”)
flux (4”) (mot
(Al)
mm2 s-‘) IS defined
dcddt
the volume of the I-butanol Usmg the trlvlal formulae
by (A2)
phase
and A the Inter
c, = c,’ - acb
(A3)
C, ’ = cbe(a + m b:)
(A4)
and
where I respectively e refer to the uutlal and a = V,/V,, we can rewrite (Al) as
and equlhbrmm
(Rt,) = A(1 + ambw)(cbe - c,+)/(Vb dct,/dt) which
value
(AS)
yields (R ) =
1976 48 715
Engng
eta1
b
AU +
ambv)
V, d In (cb’ -
provided A, V, and am& are constant, ally found to hold for the system studled
dt Cb)
which [6]
(‘46)
was experiment