Stochastic modelling of axial dispersion in external ... - Springer Link

Bioprocess Engineering 23 (2000) 543±549 Ó Springer-Verlag 2000

Stochastic modelling of axial dispersion in external-loop airlift bioreactors M. Gavrilescu, O. Muntean, R. Z. Tudose

Abstract The paper presents a model of the motion of a particle subjected to several transport processes in connection with mixing in two phase ¯ow. A residence time distribution technique coupled with a one-dimensional dispersion model was used to obtain the axial dispersion coef®cient in the liquid phase, Dax . The proposed model of Dax for an external-loop airlift bioreactor is based on the stochastic analysis of the two-phase ¯ow in a cocurrent bubble column and modi®ed for the speci®c ¯ow in the airlift reactor. The model takes into account the riser gas super®cial velocity, the riser liquid super®cial velocity, the Sauter bubble diameter, the riser gas hold-up, the downcomer-to-riser cross sectional area ratio. The proposed model can be applied with an average error of 20%.

Hd HL HR L l1 Neq ni

pij QG QF Re t tC List of symbols tM AD ui downcomer cross sectional area, m2 VL AR riser cross sectional area, m2 aij intensity of passage from vi to vj vi vLD Bo Bodenstein number, Eq. (43) C0 vLR initial tracer concentration, kg/m3 vM C…t† actual concentration of tracer, kg/m3 vSG Cr dimensionless concentration of tracer (Cr ˆ C…t†=C0 ) vSGR Dax axial dispersion coef®cient in the overall bioreac- vSL tor, m2 /s vSLR DD w downcomer diameter, m x DR riser diameter, m z dB bubble diameter, m dS (d32 ) Sauter diameter, m HD downcomer liquid height, m Greek eG Received: 6 July 1999 t s M. Gavrilescu (&) sr Department of Environmental Engineering, Faculty of Industrial Chemistry, Technical University of Iasi, Mangeron Str. 71, 6600 ± Iasi, Romania

O. Muntean Department of Chemical Engineering, Faculty of Industrial Chemistry, University ``Politehnica'' Bucharest, Romania R. Z. Tudose Department of Transfer Phenomena and Chemical Engineering, Faculty of Industrial Chemistry, Technical University of Iasi, Mangeron Str. 71, 6600 ± Iasi, Romania

dispersion height, m unaerated liquid height, m riser liquid height, m length of region under investigation, m average length followed by the ¯uid element with the velocity v1 , m number of perfectly mixed tank-in-series number of bubbles belonging to i dimensional class probability of velocity change from vj to vi gas ¯ow rate, m3 /s liquid phase feed ¯ow rate, m3 /s Reynolds number, Eq. (21) current time, s circulation time, s mixing time, s density of probability liquid nominal volume in the reactor, m3 velocity of a ¯uid element, m/s linear liquid velocity in downcomer, m/s linear liquid velocity in riser, m/s velocity of gas±liquid mixture, m/s gas super®cial velocity, m/s riser gas super®cial velocity (vSGR ˆ QG =AR ), m/s liquid super®cial velocity, m/s riser liquid super®cial velocity, m/s convective ¯uid velocity, m/s dimensionless axial coordinate (x ˆ z=L) axial coordinate, m symbols gas holdup kinematic viscosity, m2 /s dimensionless time (s ˆ t=tC ) dimensionless mixing time (sr ˆ tM =tC )

1 Introduction Airlift bioreactors are gas±liquid contacting devices characterized by ¯uid circulation in a clear and de®ned cyclic pattern, through channels designed speci®cally for this purpose. The ¯ow patterns in bioreactors is often very different from that speci®cally to the ideal reactors (the perfectly mixed batch, the plug-¯ow and the perfectly mixed continuous tank reactors). The two-phase ¯ow in airlift reactors is still beyond the overall knowledge. The mixing phenomena is important in the design of bioreactors, since the time spent in each of

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Bioprocess Engineering 23 (2000)

the environments that the bioreactor offers, age of other elements that an element of ¯uid does contact, are of crucial importance in evaluating the bioreactor performances. The mixing phenomenon is of particular importance for design, modelling and operation, as well as for scale-up from laboratory to industrial scale. Two fundamentally mixing effects superimpose each other in the airlift bioreactors [1]:

544

± axial mixing at each circulation; ± backmixing due to the recycling. Most of the work dealing with the study of liquid phase dispersion coef®cients in airlift bioreactors had attempted to characterize the liquid mixing in terms of an overall axial dispersion for the whole reactor. Axial dispersion in the circulating ¯ow is caused, as in real tubular ¯ow, by stream pro®le, turbulence, dead space and molecular diffusion. The resulting axial mixing can be mathematically formulated according to two models [2, 3]:

ui …x; t ‡ Dt† ˆ

n X

pji uj …x

vi Dt; t†; i ˆ 1; 2; . . . ; n ;

jˆ1

…1†

± the particles having the velocity vi which arrive in the point x at the moment t ‡ Dt, originated from the group of those with the velocity vj , which have been found at the distance vi Dt from x at the moment t; these particles pass from vj to vi in the interval Dt; ± the model P takes into account all the velocities possible, that is j pji ˆ 1, so that:

ui …x; t ‡ Dt† ˆ

n X

pji uj …x

vi Dt; t†;

i ˆ 1; 2; . . . ; n :

jˆ1

From Eqs. (1, 2), it follows that:

…2†

uj …x; t ‡ Dt†

ui …x vi Dt; t† Dt 20 1 n X 1 ˆ 4@ pij Aui …x vi Dt; t† Dt jˆ1;i6ˆj 3 n X pij uj …x vi Dt; t†5; i; j ˆ 1; 2; . . . ; n ;

± the tanks-in-series model, where the real circulation loop is replaced by a series of consecutive, equal volume ideally stirred tank reactors, Neq , resulting in the same longitudinal mixing effect; ± the diffusion model, where the disturbances in plug ¯ow, although physically different, but all of the essentially statistical nature, are considered according to the jˆ1;i6ˆj molecular diffusion laws, by summing them all up in the effective diffusion coef®cient, Dax , which is comprised in …3† the Bodenstein or PeÂclet numbers. ± the velocity change is dependent on the previous velocity at previous moments and the probability of this This work was undertaken in order to investigate the change is proportional to the considered time interval liquid phase axial dispersion in external-loop airlift bioronly, that is: eactors and to develop mathematical models based on a stochastic approach.

2 Formulation of the model The proposed model starts from the hypothesis that the ¯ow and the turbulent mixing are diffusional processes. Also, the phase dispersion is the consequence of the fact that there are several zones which move with different velocities in the continuous phase and an exchange of aleator particles occurs between these zones and the dispersed phase. In this situation, to characterize the dispersion intensity, it is necessary to know the phase velocity spectra as well as their change frequencies. To characterize the mixing in these conditions, a stochastic simpli®ed model was used, because in biphasic ¯ow and, in particular, in vertical gassed columns, the velocity ®eld is very large. The model would imply a ®nite number of processes which can describe the essence of the phenomenon, based on the following hypothesis:

pij ˆ aij Dt;

i; j ˆ 1; 2; . . . ; n :

…4†

Using the power series in the ®rst term of Eq. (3):

Dui ‡


: …5† Dx For Dt ! 0 and Dx ! 0 the following model results: 0 1 n n X oui oui @ X ˆ vi aij Aui ‡ aji uj ; ot ox jˆ1;i6ˆj jˆ1;i6ˆj ui …x

vi Dt; t† ˆ ui …x; t†

vi Dt

i ˆ 1; 2; . . . ; n :

…6†

For aij ˆ const., the model (6) corresponds to a connection through a Markonian process of some elementary transport process with the velocity vi , i ˆ 1; 2; . . . ; n. The hyperbolic model (6) may tend to the classic dispersion model (Eq. 7) [4]:

oui oui o2 ui ‡w ˆ Dax 2 ; …7† ox ox ± the ¯uid elements which belong to the continuous phase ot can have one of the velocities vi , i ˆ 1; 2; . . . ; n; where w is the velocity of convective ¯uid ¯ow, and D is ± one particle belonging to the dispersed phase having the the axial dispersion coef®cient. velocity vi reaches the point x at the moment t, with the The system (6) can be written in a matrical form as density of probability ui …x; t†; follows: ± in the interval of time Dt, the velocity of a particle ou changes from vj to vi with the probability pji ; in this ˆ Vu ‡ Qu ; …8† ot situation it follows that:

M. Gavrilescu et al.: Stochastic modelling of axial dispersion in airlift bioreactors

where

v1 o ox 0
Vˆ

0

u1 u 2
u ˆ ;

u n P a j6ˆ1 1j a12
Qˆ


a1n

a21 P j6ˆ2

a2j

a2n

0







o v2 ox









0

a31









a32









a3n








0 0
;

o vn ox an1 an2 :…9† P anj j6ˆn

The system (6) is also equivalent with the Eq. (10):

 o det ot

V

 Q uˆ0 :

…10†

P qi vi w ˆ Pi ; i qi P i6ˆj qij vi vj ; Dax ˆ P i qi

…11† …12†

where qi is the determinant of the principal minor of Q with the ligne and the column i eliminated, and qij is the determinant of the principal minor with the ligne and the column i eliminated, respectively. When Q has only two columns, qij ˆ 1. It was considered that other terms from the model (10) are negligible. In this manner, it appears that the dispersion models are approximations of the polystochastic models, considering that the in¯uence of the convective and diffusional terms are dominant. The conditions necessary to use the model (7) with the coef®cients (11) and (12) were established by Pinski [5]. Particularized for bubble columns in gas±liquid ¯ow, the model (6) may become as follows:

a12 u1 ‡ a21 u2 ;

…13†

a21 u2 ‡ a12 u1 ; a21 a21

 ;

v1 a21 ‡ v2 a12 ˆ 0 :

A similar condition can be written using gas voidage in the form:

v1 eG ‡ …1

eG †v2 ˆ 0 ;

…17†

resulting that:

a21 eG ˆ : a12 1 eG From Eqs. (15) and (18) it follows that: Dax ˆ

v21 eG : eG †…a12 ‡ a21 †

…1

…18†

…19†

To determine the passage intensities (a12 ; a21 ) it is necessary to start from the mechanism of the velocity changes. The particles carried by the gas bubble which moves with the velocity v1 can change their velocity from v1 to v2 owing to the two eddies formed in the bubble wake. The term a12 which can be considered as the number of passages from v1 to v2 per unity of time is calculated with the following relationship:

v1 ; …20† l1 where l1 is the average length followed by the ¯uid element with the velocity v1 . The coef®cient 2 refers to the two possibilities of velocity change. The length l1 can be computed using the Kolmogoroff isotropic turbulence theory [7] from the condition that the Reynolds number is 1 (Eq. 21): v2 † : …21† t The value Re ˆ 1 corresponds to the hypothesis that the ®rst eddies behind the gas bubble appeared when the inertial and viscous forces are equal [7]. For a bubble with the diameter dB , the following approximation should be used: Re ˆ

…v1

dB ; …22† 4 because each of the two eddies had the size which can be approximated with dB =2. Also, it can be considered that every ¯uid element ¯ows over a half of the eddy diameter before leaving it, and pass from v1 to v2 . In these conditions, the following equation can be written: l1 ˆ

a12 ˆ q1 ˆ

…16†

a12 ˆ

From Eq. (10) it follows that:

ou1 ou1 ˆ v1 ot ox ou2 ou2 ˆ v2 ot ox so that  a12 Qˆ a12

v1 v2 : …15† a12 ‡ a21 For stagnant liquids in the column, w ˆ 0, so that [6]: Dax ˆ

a21 ;

q2 ˆ

a12 :

l21

eG †

;

…23†

and, respectively for axial dispersion coef®cient:

From Eqs. (11), (12) the global velocity can be obtained:

v1 a21 ‡ v2 a12 wˆ ; a12 ‡ a21 as well as the axial dispersion coef®cient:

2t…1

…14†

Dax ˆ

v2SG dB2 ; 32eG …1 eG †

…24†

where the super®cial gas velocity, vSG is represented by:

vSG ˆ ev1 :

…25†

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Bioprocess Engineering 23 (2000)

For circulating liquid columns such as airlift reactors, the gaseous phase ¯ow and its different holdup in the special reactor zones (riser, downcomer) generate a net liquid ¯ow and the overall velocity represent the sum of the two phases super®cial velocities [8]:

vM ˆ vSGR ‡ vSLR ;

…26†

or

vM ˆ vGR eGR ‡ vLR …1 546

eGR † ;

…27†

where vGR is the linear gas velocity in the riser zone, vLR ± the linear liquid velocity in the riser, eGR ± riser gas holdup. To determine the values a12 and a21 respectively, it was also considered the velocity changes from v1 to v2 , as well as from v2 to v1 . The relationship between v1 and v2 is expressed by Eq. (27) and the velocity of the dispersion in the riser is in the following form:

vGR a21 ‡ vLR a12 ˆ vM : …28† a12 ‡ a21 From Eqs. (27, 20, 21, 22, 28) follows in number of passages between the two velocities per unity of time: a12 ˆ

2t…1 l21

eGR †

;

…29†

vM vLR ; …30† vGR vM so that the expression for axial dispersion coef®cient becomes: a21 ˆ a12

Dax ˆ

2 vSLR vSGR d32 ; 32tL eG …1 eG †

Fig. 1. Variation of the axial dispersion coef®cient with the gas super®cial velocity in a bubble column [6] (s experimental data; ÐÐÐ calculated with Eq. (24))

for Newtonian and non-Newtonian liquids as well. The axial dispersion coef®cient of the liquid phase diminishes if the airlift reactor behaviour approaches that of a bubble column, that is AD =AR ! 1, but is dependent on the liquid super®cial velocity. Therefore, the following dependence can be considered [9, 12]:

…31†

where d32 is bubble Sauter diameter, expressed by:

P ni di3 dS ˆ d32 ˆ Pi 2 : i ni di

…32†

Therefore, the speci®c ¯ow behaviour in the airlift reactor obliges us to take into account the liquid super®cial velocity when axial dispersion is investigated. The super®cial liquid velocity in the airlift reactors is dependent on the reactor geometry, expressed in the following form [8±11]:

vSLR  …AD =AR †0:97 ;

…33†

Table 1. Some characteristics of the external-loop airlift reactors used in the investigation Characteristic Unit Scale of operation

AD/AR DR DD HR HD VL á 103

± m m m m m3

Laboratory

Pilot

1.000 0.360 0.030 0.030 0.030 0.018 1.160±1.560 1.100±1.500 1.189±2.165

0.111 0.1225 0.040 0.030 0.200 0.200 0.010 0.070 0.040 4.30±4.70 4.00±4.40 144±170

Fig. 2. Dependence of axial dispersion coef®cient Dax on gas super®cial velocity, vSGR in REL (ÐÐÐ AD =AR ˆ 1:000; б AD =AR ˆ 0:360; ± ± ± AD =AR ˆ 0:111 ; QF = 106 (m3 /s): d 5:00;  8:33;  10.00; D 11.40; ( 12.5)

M. Gavrilescu et al.: Stochastic modelling of axial dispersion in airlift bioreactors

547

Fig. 3. Experimental and calculated (with Eq. (36)) axial dispersion coef®cient values as function of gas super®cial velocity in the laboratory external-loop airlift bioreactor, REL (d experimental values of Dax ; ÐÐÐ calculated values with Eq. (24); ÐÐÐ calculated values with Eq. (36))

1 : …34† …AD =AR †0:97 Also, the axial dispersion coef®cient is inversely proportional to the dimensionless mixing time, sr [1], which essentially represents the number of circulations in the airlift reactor. This corresponds to a given homogeneization degree:

3 Experimental Experiments were performed in two external-loop airlift reactors of laboratory (REL) and pilot scale (REP), respectively [1, 8±12]. Each con®guration investigated consisted of two vertical columns connected at the top and the base by horizontal piping containing values. The most important design characteristics are shown in Table 1. Air was used as gaseous phase with super®cial velocities Dax  1=sr : …35† in the riser ranged between vSGR = 0.016±0.178 m/s in the From experimental studies on ¯ow and mixing resulted laboratory and vSGR = 0.010±0.120 m/s in the pilot reactor, that the minimum number of circulations which correrespectively. Compressed air was introduced at the bottom sponds to an homogeneity degree of 95% was 4 [9, 13, 14]. of the riser by means of a porous plate in REL and a Using these considerations in Eq. (31), the axial discombined ring-multiradial pipes sparger in REP. Tap persion coef®cient can be modelled as follows: water was used as liquid phase. 2 Air ¯ow rate, average gas holdup in the riser and vSLR vSGR d32 Dax ˆ …36† downcomer liquid velocity as well as temperature of the 0:97 : 128t e …1 e †…A =A † Dax 

L GR

GR

D

R

Bioprocess Engineering 23 (2000)

vSLR ˆ vLR …1

eGR † ;

…38†

vSLD ˆ vLD …1

eGD † ;

…39†

AR …1

eGR †vLR ˆ AD …1

eGD †vLD :

…40†

Axial dispersion parameters were determined using the classical tracer response technique [1, 2, 6, 13]. The axial dispersion model was used to estimate the axial dispersion number: 548

oCr 1 o2 Cr oCr ˆ : …41† os ox Bo ox2 The solution of Eq. (41) for an initial Dirac pulse in the airlift reactors considering the circulation ¯ow is given by [1, 2, 12]: " #  0:5 X 1 Bo …x s†2 Bo : …42† Cr ˆ exp 4ps 4s xˆ1 Fitting the model to an experimental response for an initial Dirac function, the Bodenstein number for the reactor calculated for a minimum standard deviation between the predicted and experimental outlet concentration pro®le, according to the least-square criterion was obtained [1, 12]. For each section of the bioreactor, the predicted outlet concentration pro®le was determined by convoluting the experimental inlet concentration pro®le with the curve of the ¯uid age distribution inside the bioreactor, E, calcuFig. 4. Experimental and calculated (with Eq. (36)) axial disper- lated for an open±open vessel as a function of Bo. Knowing sion coef®cient values as function of gas super®cial velocity in the Bodenstein values, the model parameter, Dax , was calcupilot external-loop airlift bioreactor, REP (d experimental values lated with Eq. (43), where the liquid velocity values were of Dax ; ÐÐÐ calculated values with Eq. (36)) determined experimentally [8±10]:

vSL L

…43† system were measured during experiments, conducted in Bo ˆ Dax ;  batch mode, at 21  1 C and atmospheric pressure. The bubble diameter was evaluated using the chemical Gas holdup was determined using the expansion bed method on the basis of the following relation [10, 15, 16]: method [16±18], based on the determination of the conversion in the gas or liquid phases in the course of a fast Hd …vSGR † HL chemical reaction, that is the sulphite oxidation method : …37† eGR …vSGR ; Hd † ˆ catalyzed by cooper ions. Details on this method were Hd …vSGR † presented in [16]. Before any data was taken, aeration was carried out over a longer period of time to ensure that a steady ¯ow distribution was established in the airlift reactor. On the basis of 4 repeated measurements and uncertainties in the estabResults and discussion lishment of the aerated dispersion column height, an av- Iordache and Muntean [6] applied the model (24) to ®t the erage error of 5% was assigned to gas holdup values. experimental data obtained in a bubble column of Liquid circulation velocity was measured by conducto- 0.0232 m i.d. They found a good correlation between the metric tracer method. It was measured the time between theory and the experimental results, with a maximum the injection of a tracer consisting of 2 ml ± in the labo- deviation of 28% (Fig. 1). ratory, and 10 ml ± in the pilot device, respectively, at the The liquid axial dispersion coef®cient in the airlift bidowncomer entrance under the elbow and the detection oreactors under investigation was found to be strongly point at the downcomer end above the elbow, to avoid the dependent on gas super®cial velocity and geometrical end effects. Knowing the linear distance between the two characteristics of the contactors (Fig. 2). points, the linear liquid velocity in the downcomer, vLD The models expressed by Eqs. (24, 36) were ®tted with was calculated. An average of ®ve repetitions for each run the experimental data obtained in REL and REP. The results are presented in Figs. 3 and 4, for both laboratory was used, the mean deviation being less than 5%. Sepaand pilot external-loop airlift bioreactors, respectively. ration of phases was almost complete at the top of the From Figs. 3, 4 it is evident that the model (24) develreactors. Therefore, only a single phase ¯ow existed in the downcomer. In this way it was possible to determine the oped for bubble columns does not hold for airlift bioreactors. The comparison of the experimental data obtained liquid super®cial velocity in the riser by applying the in the investigated airlift reactors with those calculated continuity equation of liquid ¯ow:

M. Gavrilescu et al.: Stochastic modelling of axial dispersion in airlift bioreactors

using the Eq. (36) reveals that the proposed model ®ts the experimental data with an average error of 20%. The model (36) offers a good concordance between theory and the experimental data and evidences the importance of the consideration of the liquid phase speci®c circulation in the airlift bioreactors when mixing is analyzed. The present investigation clearly evidenced that mixing characteristics in the airlift bioreactors can have a strong in¯uence on performances of these contactors. While itself affected by turbulence and momentum transfer behaviour of phases, mixing in turn affects the driving forces for heat and mass transfer.

5 Conclusions This work was undertaken in order to investigate the liquid phase axial dispersion in external-loop airlift bioreactors and to develop mathematical models based on a stochastic approach. The mixing phenomena is important in the design of bioreactors, since the time spent in each of the environments that the bioreactor offers, age of other elements that an element of ¯uid does contact, are of crucial importance in evaluating the bioreactor performances. The axial dispersion mixing phenomenon is of particular importance for design, modelling and operation, as well as for scale-up from laboratory to industrial scale, taking into account the special hydrodynamic behaviour of the airlift bioreactors. The model can be applied for the airlift bioreactors with negligible downcomer gas holdup. Although the proposed model is based on several hypothesis and comprises some dependences occuring from an empirical model, it can be applied with an average error of 20%. The paper offers a model for the axial dispersion coef®cient in external-loop airlift bioreactors, based on stochastic analysis of the motion of ¯uid particles subjected to several transport processes during ¯ow, taking into account the fact that the two-phase ¯ow in airlift reactors is still beyond the overall knowledge. References

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2. Levenspiel, O.: Chemical reaction engineering. New York, John Wiley & Sons, 1972 3. Blenke, H.: Loop reactors. In: Ghose, T.K.; Fiechter, A.; Blackebrough N. (eds) Advances in biochemical engineering. Vol. 13, pp. 120±214 Springer, Berlin Heidelberg New York, 1979 4. Levenspiel, O; Bischoff, K.: Patterns of ¯ow in chemical process vessels. Advances in chemical engineering. Vol. 4, John Wiley & Sons, New York, 1963 5. Pinski, M.: Wahrscheinlichkeitsheorie verw. Geb. 9 (1968) 101 6. Iordache, O.; Muntean, O.: Modele stocastice ale anestecaru. I. Dispersia axialaÆ Ãõn curgerea bifazicaÆ gaz-lichid. Rev. Chim. (Bucharest) 31 (1980) 996±999 7. Kafarov, V.: Fundamentals of mass transfer. Moscow, Mir Publishers, 1975, pp. 278, 363 8. Gavrilescu, M.; Tudose, R.Z.: Effects of geometry on hydrodynamics in external-loop airlift reactors. Chem. Eng. Commun. 156 (1996) 89±113 9. Gavrilescu, M.: Study of the ¯ow of heterogeneous systems in airlift reactors. Thesis, Technical University of Iasi, 1997 10. Gavrilescu, M.; Tudose, R.Z.: Study of liquid circulation velocity in external-loop airlift bioreactors. Bioproc. Eng. 14 (1995) 33±39 11. Gavrilescu, M.: Pneumatic bioreactors. Iasi ± Romania, Dosoftei Press, 1997 12. Gavrilescu, M.; Tudose, R.Z.: Residence time distribution of liquid phase in an external-loop airlift bioreactor. Bioproc. Eng. 14 (1996) 183±193 13. Gavrilescu, M.; Tudose, R.Z.: Mixing in airlift reactors. Roumanian Chemical Quarterly Reviews 5 (1997) 53±66. 14. Verlaan, P.: Modelling and characterization of an airlift loop bioreactor. PhD Thesis, University of Wageningen, 1987 15. Gavrilescu, M.; Tudose, R.Z.: Effects of downcomer-to-riser cross sectional area ratio on operation behaviour of externalloop airlift bioreactors. Bioproc. Eng. 15 (1996) 77±85 16. Gavrilescu, M.; Tudose, R.Z.: The speci®c interfacial area in external-loop airlift bioreactors. Bioproc. Eng. 16 (1997) 127±133 17. Schumpe, A.; Deckwer, W.-D.: Analysis of chemical methods for determination of inerfacial areas in gas-in-liquid dispersions with non-uniform bubble size. Chem. Eng. Sci. 35 (1980) 2221±2233 18. Popovic, M.; Robinson, C.W.: The speci®c interfacial area in external-circulation-loop airlifts and a bubble column. I. Aqueous sodium sulphite solution. Chem. Eng. Sci. 42 (1987) 2811±2824

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