Concentration polarization and other boundary layer effects in the pervaporation of chlorinated hydrocarbons

91

Desalination, 95 (1994) 91-113 Elsevier Science B.V. Amsterdam - Printed in The Netherlands

Concentration polarization and other boundary layer effects in the pervaporation of chlorinated hydrocarbons C. Dotremont, S. Van den Ende, H. Vandommele and C. Vandecasteele Department of Chemical Engineering, Katholiek B-3tWi Leuven (Belgium)

Universiteit Leuven, De Croylaan 46,

(Received Allgust 25, 1993; in revised form November

15, 1993)

suMrviARY

The effect of concentration polarization on the pervaporation performance was studied for several chlorinated hydrocarbons (Cl-HC’s). The feed flow rate was varied between l-5 l/min in a test cell unit, provided with a flat sheet membrane (plate-and-frame module). For this configuration and for all Cl-HC’s studied, the flux decreased below a critical flow rate of 3 l/min. Cl-HC’s with a high permeability showed a considerable flux decline, while for organic compounds with a low permeability like propanol, the decrease in flux was negligible. A model was derived in order to estimate the yield which gave good agreement between the calculated and experimental results. The addition of salts caused an important flux decline for trichloroethylene. The effect was reversible, could be reduced by an increase of the Re number and could not be explained by pore-blocking of the zeolite pores. Different salts caused the same decrease of trichloroethylene flux but did not affect the water flux. Fluxes were not influenced by the pH of the feed mixture, but the viscosity of the feed mixture had a considerable effect on both the trichloroethylene and water flux. Keywords: Pervaporation, boundary layer resistance,

OOll-9164/94/$07&I

chlorinated hydrocarbons, mass transfer

concentration

0 1994 Elsevier Science B.V. All rights reserved.

SSDI OOll-9164(94)00008C

polarization,

92 SYMBOLS

ci

D

d(r) d(R)

d/l Ji

k.I,L ki,M ki,OV 1

L P

Q R Re R i,L R i,M R i,OV s SC

Sh wi Y

-

-

-

concentration of component i, m3/m3 diffusion coefficient, m2/s channel height (m) at position r (radius of module) channel height (m) at r=R; d(R)=0401 m hydraulic diameter, m pervaporation flux of component i, m3/(m2 s) boundary layer mass transfer coefficient, m/s membrane mass transfer coefficient, m/s overall mass transfer coefficient, m/s membrane thickness, m characteristic measure of the module, m permeability, m2/s flow rate, m3/s module radius (0.1524 m) Reynolds number boundary layer resistance, s/m membrane resistance, s/m overall resistance, s/m flow velocity, m/s Schmidt number Sherwood number weight fraction of component i, g/g yield l

direction z direction r -

coordinate perpendicular to the membrane surface radial coordinate

Greek 6 1

-

V

-

thickness of the liquid boundary layer, m dynamic viscosity, kg/m s kinematic viscosity, m2/s l

Subscripts e i j s

-

inlet component i (preferentially permeating component) component j (for binary mixtures: component j=water) exit

Superscripts b m

-

bulk

membrane

93 INTRODUCTION

The removal of chlorinated hydrocarbons (Cl-HC’s) from water by pervaporation has already been intensively studied [l-4]. Good results were obtained with organophilic composite membranes having a (silica&-filled) PDMS top layer which combines high Cl-HC fluxes with high selectivities towards the organic compounds. However, in some cases, depending on the flow conditions and the type of waste water, the pervaporation performance can decrease, and a flux decline can be observed [5]. According to the resistance-in-series model [4,6], the pervaporation process implies the mass transfer of the permeating component from the bulk of the feed towards the membrane and subsequently the transport of this component through the membrane. Transport through the membrane is mainly determined by the membrane characteristics on the one hand and the properties of the permeating component on the other. In order to obtain a high mass transfer towards the membrane (corresponding with a low boundary layer resistance), appropriate flow conditions (Re number) for the considered module are required. In conditions of too low feed velocity, an insufficient supply of the preferentially permeating component(s) from the bulk to the membrane causes a depletion layer near the membrane with a flux decline as a consequence. In literature, this phenomenon is described as concentration polarization. In comparison with other membrane processes, in pervaporation concentration polarization is generally assumed to be of minor importance. Nevertheless, Cod and Lipski [6] and Psaume et al. [7] showed that when the permeability of the considered component is high, as in the removal of trichloroethylene from water, the effect can become very severe. In a recent study, Karlsson et al. [8] studied the effect of concentration polarization during the pervaporation of aroma compounds through PDMS membranes without zeolite, and they showed that in some cases the diffusive mass transfer in the feed becomes rate determining for the pervaporation process rather than the diffusive mass transfer in the membrane. However, the effect of concentration polarization was examined on a multicomponent aroma mixture. We have shown earlier [9] that coupling phenomena can generally not be ignored in this case: the transport through the membrane of one component is influenced by the presence of the other component(s), and flux predictions, even in the absence af concentration polarization, are difficult to make. Both phenomena, concentration polarization and coupling, may cause a decrease in partial fluxes; and in order to make a clear distinction between them, experiments should be carried out in conditions where only one effect can play a part.

94

The purpose of the present study was to obtain more information on the effect of concentration polarization in the pervaporation of Cl-HC’s from water. To make sure that coupling effects did not affect the partial fluxes, all experiments were carried out with binary mixtures of one Cl-HC and water. The Cl-HC’s selected for this study had a different permeability in the considered membrane. Of course, concentration polarization is expected to be most pronounced for Cl-HC’s with a high permeability and negligible for components with a low permeability. Nevertheless, the pervaporation results obtained for some real waste waters at optimum flow conditions deduced from previous experiments with synthetic binary mixtures, still showed a lower flux. In general, real waste waters contain a mixture of contaminants, including salts and viscous components, which can cause an additional resistance to the mass transfer of the organics towards the membrane. This, of course, implies that the flow conditions which were adjusted for pure synthetic binary mixtures are no longer adequate for real waste waters, and further optimization must be carried out. Additional pervaporation experiments were therefore performed with synthetic binary mixtures (Cl-HCYwater) containing different salts or a viscous component or at different pH in order to gain more information on the influence of these contaminants on the pervaporation performance. Moreover, from the resistance-in-series model, the boundary layer resistance was calculated to describe the concentration polarization phenomenon.

RESISTANCE-IN-SERIES

MODEL

According to the resistance-in-series model [4], the flux of the permeating component i through the membrane Ji can be described as

Eqn. (1) implies a negligible permeate concentration and can be applied here since the concentration of the permeating components is very low due to their low partial pressure in the permeate ( I 10 mbar). The reciprocal overall mass transfer coefficient (l/k,,) is defined as the sum of a liquid boundary layer resistance (Ri.3 and a membrane resistance

95

1 k i,OV

= Ri,*” = Ri fL + Ri,M = $

i,L

+1 k i,M

(2)

The overall mass transfer coeffkient ki,ov can be deduced from experimental measurements of the feed concentration and flux Ji of the permeating component i. The pervaporation test cell used in the present study (Lab Test Cell Unit from GFT-Le Carbone) is equipped with a plate-and-frame module as shown in Figs. la and lb. If it is assumed that the membrane shows a high selectivity for component i, a depletion layer on the feed side of the membrane is formed (Fig. 2).

Fig. la. Schematic cross-section plate-and-frame module.

wi,r(0) =

Wi,rm

of the

Fig. lb. Top view of the membrane in the module.

I

1Wi,r

membrane

L 2

Ji

Wi,r(Z)

Fig. 2. Concentration profile at the feed side of the membrane.

96

For the plate-and-frame module, we derived the following equations for the mass transfer coefficient of the boundary layer (l/k,+ l/k, is related to the Sherwood number (Sk): d kLdh

Sh = -

D

(3)

= aRebScC

where Re=(d,J/v, Sc=v/D, and a, b, c and d are constants. These constants depend on the geometry of the module and for a plate-and-frame module, Eqn. (3) becomes [5] Sh = 1.86 Reli3 SC”~ (dh/L)*‘3

(4)

This equation was theoretically derived for ideal channels with a fully developed stream pattern. As will be shown below, it can be used to fit experimental yield curves. The plate-and-frame configuration was approximated by a flow between two plates. It must be taken into account that channel height d is not constant and is a function of r, so that L = d(R) dh = -4 (2 a r) d(r) = 2&.)

2 (2ar)

Re = [s(r) (2d(r))] / v (local Re number) Sh = [ki,L(r) (2d(r))]/Di

(local Sh number)

Substituting these local parameters in Eqn. (4) gives an expression for the local mass transfer coefficient ki,L(r):

ki,L(rl = 1.86 Re ‘I3 SC ‘I3 [(2d(r)) ld(R)]1’3 [Di/ (2d (r))]

(5)

This local mass transfer coefficient can be rewritten and rearranged (see Appendix A) in order to find a general expression for ki,L: ki,L = 552.2 (ReR)1’3SC ‘I3 Di

(6)

97

In this model the assumption was made that fir is negligible in comparison with w$, which is certainly the case in conditions of low feed velocity. Therefore, the determination of &,L will be most accurate at a low feed velocity.

Membranes

In the experiments two types of organophilic membranes were used: zeolite-filled and non-filled membranes. The former type consists of a dense PDMS top layer (30 pm) filled with hydrophobic zeolite (silicalite, 60% filling degree), fixed on a support layer of PAN on polyester fabric; the latter has a uniform PDMS top layer (20 pm). The membranes were supplied by GFT-Le Carbone (Neunkirchen-Heinitz, Germany). Apparatus The pervaporation experiments were performed with the equipment as described previously (module with a flat sheet membrane) [lo]. The feed temperature was 50°C and the downstream pressure 10 mbar. The permeate was condensed by liquid N,.

RESULTS AND DISCUSSION

EJgcectofthe feed flow

rate

The effect of concentration polarization on flux and selectivity has been studied by varying the flow rate between l-5 Urnin, corresponding to a Reynolds number ranging from approximately 100-700 as derived from Eqn. (B5) (see Appendix B). All experiments were carried out with zeolitefilled membranes. In a previous study [lo] we already showed that for the membranes considered the permeability decreases from trichloroethylene (P= 1.233 x 10s9 m2/s) over l-chloropropane @‘=7.146x lo-” m2/s) and chloroform (P=5.583 x lo-” m2/s), to 2chloropropane (P = 2.083 x lo-” m2/s). n-Propanol is rather polar and hardly shows any permeability through zeolite-filled PDMS membranes. These organic compounds were selected for this study because of their significant difference in permeability.

98

Fig. 3 presents the flux of trichloroethylene as a function of the flow rate for three different feed concentrations. No effect of concentration polarization could be observed above a flow rate of 3 I/m (Re,r 380), whereas a significant flux decrease occurs at lower flow rates. The water flux on the other hand is independent of the flow rate (Fig. 4): ail fluxes measured at six different flow rates coincide within the experimental uncertainty. This, of course, was to be expected since water

8or 70 60 -

flux

[g/m2.h]

b JO0 ppm LI 300 ppm 0 200 ppm

50 40 -

flow rate

01

0

[I/min]

1

I

,

I

/

I

1

2

3

4

5

6

Fig. 3. Flux of trichloroethylene

as a function of the flow rate.

flux HZ0 [g/m2.h]

120 r

100

80 0 1 I/min v 2 I/min 0 2.5 l/n-kin 0 3 l/min l 4 l/min v 5 l/mm

60

40

20 0 -

1

50

100

I 150

I 200

I 250

I 300

retentate cone. [ppm] I I I 350

400

Fig. 4. Water flux as a function of the retentate concentration

450

for six different feed flows.

99 X of the

maximum

flux

100

80

60

40

v

20

OL

1

0.5

trwhloroethyiene 1-chloropropane chloroform 2-chloropropane n-propanol -

- -

-

rate [I/min]

flow I

I

/

1

I

I

1.0

1.5

2.0

2 5

3.0

3.5

Fig. 5. Flux :as a percent of the maximum flux (measured at 3 Vmin) as a function of the flow rate for five different organic compounds.

0.000

1 0

I

/

I

I

I

I

1

100

200

300

400

500

600

700

Fig. 6. Yield (I’) as a function of the Reynolds number (Re).

100

is the major component in the binary mixture and has a low permeability through the membrane. The concentration of water at the feed side of the membrane may therefore for practical applications be considered constant during the whole experiment, even at low feed velocity. Concentration polarization was observed for all compounds studied, from a flow rate lower than 3 Urnin, even for n-propanol. As expected, the relative decrease in flux was more pronounced for the compounds with the highest permeability, as shown in Fig. 5. As proposed by Psaume et al. [7], we defined the yield Y as Y = 1 - [Wi,,/Wi,~]

(7)

and derived the following equation for Y for the plate-and-frame module (Appendix B): Y = 1 - exp -552.2 (ReR)1’3 1

sc”3

q

(rRZ,QJ]

The value of Dj, the diffusion coefficient at infinite dilution, can be derived from the Wilke-Chang equation @=0.9x 10s9 m2/s for trichloroethylene). Moreover, Q, =Re, u R v (see Appendix B, B5), so that for trichloroethylene l

l

l

Yd = 1 - exp [-0.585(Re)-2’3]

(9)

The experimental yield (Y,,) as derived from the pervaporation results is plotted as a function of the Reynolds number in Fig. 6. The best fit to the experimental values is given by the solid line; the calculated yield (Y&) is given by a dotted line. There is good agreement between both curves, proving that this model provides a rather adequate approximation of the flow pattern in the module used in this study (see Resistance-in-Series Model). Similar curves generated for 1-chloropropane and 2-chloropropane also showed good agreement with the experimental results. Effect of salts Beside organic components, most waste waters also contain inorganic components (e.g., salts). So far the effects of salts on the pervaporation performance have only been studied occasionally [ll]. Therefore, in this study pervaporation experiments were carried out with the binary mixture

101

80

IIUX

[ihQ.hl -

3 --

I/mm

5.28

60

I/min .

L 400

ppm

0

.

ppm

300

40

20

salt

0

I

I

I

0

50

100

!

150

[g/l]

cone 1

200

Fig. 7. Flux of trichloroethyleneas a functionof the NaCl concentration.

waterkrichloroethylene containing different amounts of NaCl using zeolitefilled membranes. The feed flow rate was 3 Vmin in a first set of experiments and 5.28 l/min in a second set. The trichloroethylene flux in relation to the salt concentration is given in Fig. 7. In the presence of NaCl a strong flux decline can be observed, indicating that the resistance to mass transfer is considerably higher in the presence of NaCl than in the absence of the salt. However, when the flow rate is increased from 3 l/min to 5.28 l/min, the flux decrease is less important. This experiment shows that the flow conditions which were adjusted for pure binary mixtures in order to prevent concentration polarization (a minimum flow rate of 3 l/min) are insufficient in the presence of salts. The zeolite-filled PDMS membrane used in the previously described experiment for a NaCl concentration of 150 g/l was reused without rinsing of the surface and without any addition of salt. Fig. 8 gives the retentate concentration vs. time for a pure mixture of trichloroethylene/water (reused membrane=dotted line) in comparison with that for a membrane not used previously (unused membrane=solid line). Both curves do not differ significantly, indicating that in pervaporation the influence of salts on the mass transfer is a completely reversible phenomenon. Figs. 9a and 9b are SEM photographs of the zeolite-filled PDMS composite membrane which was used in the pervaporation of a trichloroethylene/water mixture to which 150 g/l of KC1 was added. An X-ray

102 retentate

600

cont.

[PPml

r --

nonused membranen used membrane

300 200 100

0

-

' 0

I 50

Fig. 8. Trichloroethylene

0

I

I

100

150

retentate concentration

pervaporatmntime

[mm]

I 200

vs. time for a used and unused membrane.

screening on K+ and Cl- ions showed an increased concentration of K+ (Fig. 9a) and Cl- (Fig. 9b) in the zeolite-filled PDMS top layer, while only traces of both ions could be found in the support layers. No evidence could be provided for the formation of a salt layer at the surface of the membrane. In order to check if blocking of the zeolite pores by the deposition of salts may occur and may be responsible for the previously observed flux decline, an additional pervaporation experiment was performed with a nonfilled membrane (without zeolite) for a feed which was a mixture of water, trichloroefhylene and 150 g/l NaCl. A proportional trichloroethylene flux decline of 65% was observed (400 ppm feed concentration) compared to 66% for a filled membrane, which shows that pore-blocking of the zeolite can be precluded. In another set of experiments with zeolite-filled membranes, different salts were successively added to the reference feed mixture of water/ trichloroethylene. The trichloroethylene flux in relation to the salt concentration for different salts, for three feed concentrations of trichloroethylene, and for a feed flow of 3 l/min is given in Fig. 10. No significant difference in flux decrease could be observed, which proves that the increase of resistance to mass transfer is the same for different salts and independent of the ionic strength of the solution (compare single, double and triple charged ions). If some sort of a solid layer were formed by the deposition of salts at the feed side of the membrane, it would cause an additional resistance to the

103

support layer

I

top layer

a support

layer

I

top layer

Fig. 9. SEM and X-ray photographs of the composite membrane.

104

go r flux

k/mz.hl .

80 -

70 -

? 60 -

50 -

40

IXI

0

NaCl

o

Na3P04

b

FeCl3

i) CJCIZ

-

30 -

20 -

10 snl1 cont.

0 ’

I

0.0

I

I

0.5

1.0

Fig. 10. Trichloroethylene

I

1.5

!

1

2.0

2.5

[mole/t]

flux as a function of the salt concentration

J

3.0

for five different salts.

mass transfer across the membrane, resulting in a decrease of the flux. This solid salt layer would be expected to influence both the water flux and the trichloroethylene flux. However, in the previously described experiments, the addition of salts did not affect the water flux, as shown in Fig. 11 where the water flux is plotted against the NaCl concentration for the pervaporation of trichloroethylene/water with the zeolite-filled membrane. Therefore, we assume that instead of the deposition of salt, a layer of somewhat increased salt concentration is formed at the feed side of the membrane. Indeed, trichloroethylene molecules can permeate easily through the membrane and the permeability of the water molecules is rather small but cannot be ignored. On the other hand, salt ions are too polar to sorb and to permeate (only a small fraction was found in the top layer by X-ray analysis). They are completely retained and accumulate at the feed side of the membrane, forming a salt-rich layer, repelling any apolar organic component (like trichloroethylene) and causing a depletion layer of organics at the feed side of the membrane. Working in conditions of higher feed

105

water flux [g/ml.h]

80

20

salt concentration

0

[g/l]

’

I

1

/

1

0

50

100

150

200

Fig. 11. Water flux as a function of NaCl concentration for four different trichloroethylene concentrations.

flow improves the removal of retained molecules and more rapidly provides “new” organic molecules at the feed side of the membrane so that no depletion layer is formed. The influence of pH on the pervaporation performance was examined for the mixture trichloroethylene/water. The pervaporation was performed with the zeolite-filled membrane in a pH range from 1-13. The pH affected neither the trichloroethylene flux nor the water flux. This was expected since trichloroethylene has no functional group(s) which can interact with the H+ or OH - ions. Moreover, the concentration of the H + or OH- ions is too small (maximum 10-l mole/l; compare Fig. 10) to form a layer of increased ion concentration near the membrane. No membrane damage was observed after several hours of pervaporation in rather extreme acid and alkaline conditions. Efect of the viscosity

An increase of the relative viscosity from l-5.4 caused a considerable flux decrease in the pervaporation of the reference mixture trichloroethylene/water. The relative viscosity of the mixture was varied by adding glycerine in different ratios to the reference mixture. The pervaporation experiment was carried out with the zeolite-filled membrane at a feed temperature of 20°C for feed flows of 3 I/m and 5.28 l/min. The partial

106

[e/m2.h] -

3 I:min

1)

0

I 1

0

I 2

I

I

3

4

relative

=

q/qH20

(20°C)

I 5

6

Fig. 12a. Trichloroethyleneflux as a functionof the relative viscosity (T$TQ-,).

flux

50

H20

[g/m2.h] -

45

3 I/rmn -

-

5.28

l/min

40 35 30 25 20 15 10

-

50 0

I 1

/ 2

I 3

I 4

7) relntlve I 5

=

7~jqH20

(20’0

J

6

Fig. 12b. Water flux as a functionof the relative viscosity (qlr)~&.

fluxes of trichloroethylene and water as a function of the relative viscosity are given in Figs. 12a and 12b, respectively. The Reynolds number and the diffusion coefficient of the organic component(s) (Wilke-Chang equation) are inversely proportional to the

107

viscosity. The decrease of the Re number and of Di, due to an increasing viscosity of the feed mixture, leads to a reduction of ki,L (i=organic component); moreover, the decrease of the Re number reduces $,L (j= water) as well. Both the fluxes of trichloroethylene and water are thus affected by the viscosity. The Re number, however, can be increased by increasing the feed flow. In these conditions the effect of the viscosity can be diminished, as is illustrated in Fig. 12. Efect of the flow rate, salts and viscosity on IQ_ In order to describe and to quantify all these phenomena, the liquid boundary layer resistance and the membrane resistance were estimated. The resistances as a function of the feed flow for the pure mixture of trichloroethylene and water are given in Fig. 13. The overall mass transfer coefficient as a function of the feed flow was calculated from experimental measurements using Eqn. (1) while the liquid boundary layer resistance for a specific feed flow was deduced from Eqn. (6). It is assumed that in all these experiments the membrane resistance is constant and can be derived from Eqn. (2). Since Eqn. (6) is most accurate at low feed velocity, the membrane resistance was determined 80000 -

Rev

70000

-

60000

- \

50000

-

40000 30000-

-

20000

-

10000

-

O0

[s/m]

h

h Ri,L

\/

, Ri.$

,

I

1

I

1

2

3

4

5 feed

Fig. 13. Rev, Ri,t and Ri,M as a functionof the feed flow.

flow

[l/min]

108

for a feed flow of 1 l/min, giving ki,L (1 l/min) = 0.000021 m/s

ki,M =

1 ---1 ki,OV

1

= 0.00018 m/s

ki,L.

I = 5464 s/m R i,M = k i.M

As can be seen from Fig. 13, Ri L and Rev increase exponentially below a feed flow rate of 3 Vmin. From this critical point the contribution Of Ri,L to R,, becomes dominant. For a feed flow of 3 l/m and higher, Ri,L and R,, are constant, indicating that concentration polarization effects can be excluded. In the same way the overall mass transfer coefficient and the liquid boundary layer resistance were calculated for the trichloroethylene/water mixtures to which successively NaCl or glycerine was added. The results are shown in Figs. 14 and 15, respectively. Ri L and R,, increase considerably with increasing salt concentration or viscosity. The increase, however, is less pronounced at a higher feed velocity. When the overall resistances are compared, the large value for the viscous mixture is striking, proving that the viscosity of the feed mixture is an important parameter in the pervaporation performance.

CONCLUSIONS

At low feed velocity, concentration polarization becomes important and flow conditions should be optimized in order to keep the flux decrease minimal. However, the flow conditions derived fbr synthetic mixtures were not suitable for real waste waters containing salts and/or viscous compounds. The addition of salts or an increased viscosity of the feed affects the boundary layer resistance and higher flow rates should be provided to avoid a significant flux decrease. It was shown that the effect of the salt was completely reversible, and the flux decrease was independent of the ionic strength of the solution. The overall resistance, consisting of the

109 Rev [s/m]

60000 r

50000

-

40000

-

10000

-

I \I/

0

I 100

, 50

0

\/ 1' Ri,M

I

I

150

200 NaCl cont.

Fig. 14. R,,

[g/l]

Ri,L and Ri,Mas a function of salt concentration. Rev [r/m]

200000

3 I/min

150000

5.28 I/n-h

100000

-P

R1.L

50000

0 0

1

2

‘1’

3

4

5

6

Fig. 15. ROY, Ri,L and Ri,Mas a function of viscosity.

liquid boundary layer resistance and the membrane resistance, was calculated as a function of the flow rate, the salt concentration and relative viscosity. These results indicate that especially an increase in relative viscosity of the feed mixture results in large values for the liquid boundary layer resistance.

110 REFERENCES 1 J. Kaschemekat, J.G. Wijmans, R.W. Baker and I. Blume, Proc. 3rd Int. Conf. on Pervaporation Processes in the Chemical Industry, Bakish Materials Corp., Englewood, NJ, 1988, pp. 405-412. J.M. Watson and P.A. Payne, J. Membr..Sci., 49 (1990) 171-205. R. Abouchar and H. Briischke, Extraction of organics from industrial waste waters by pervaporation. Proc. of Euromembrane 92, Paris. H.H. Nijhuis, Removal of trace organics from water by pervaporation. A technical and economic analysis. Ph.D. Thesis, University of Twente, Enschede, The Netherlands, 1990. 5 M. Mulder, Basic Principles of Membrane Technology, Chap. 7, Kluwer Academic, Dordrecht, The Netherlands, 199 1. 6 P. Cot6 and L. Lipski, Proc., 3rd Int. Conf. on PervaporationProcesses in the Chemical Industry, Bakish Materials Corp., Englewood, NJ, 1988, pp. 449-462. 7 P. Psaume, P. Aptel, Y. Aurelle, J.C. Mora and J.L. Bersillon, J, Membr. Sci., 36 (1988) 373-384. 8 H.O.E. Karlsson and G. Trag&rdh, J. Membrane Sci., 81 (1993) 163-171. 9 S. Goethaert, C. Dotremont, M. Kuijpers, M. Michielsand C. Vandecasteele, J. Membr. Sci., 78 (1993) 135-145. 10 C. Dotremont, S. Goethaert and C. Vandecasteele, Desalination, 91 (1993) 177-186. 11 K.W. Baddeker and G. Bengston, J. Membr. Sci., 53 (1990) 143-158.

APPENDIX

A: DETERMINATION

COEFFICIENT

OF THE BOUNDARY LAYER MASS TRANSFER

(k,,d

The local mass balance of component i is given by

so that

s(r) =

Qt? (2rr) d(r)

(Al)

Re = [s(r) (U(r))] /Y = Q,! [am]

Assuming that the convective flux can be ignored on position r in comparison with the diffusional flux, the local flux of component i can be written as Ji,r dZ = Di P dWi,r(Z)

042)

111

Integration conditions:

across

the boundary

Wi,r (0)

= ST",

at Z,=O

W,,,(S)

= Wipr

at

layer using

the following

boundary

Z=6

gives Ji,r

= ki,Z,

P [wi:

(A3)

- wiT]

with

If the partial flux of the considered component flux expression can be further simplified: J i,r

= ki,L

P

(A4)

w&

The total flux of component

i is then

R

J.r,tot =

is high (wTr4 w&), the

R

(r>(2ar) Pdr Ji,r (2~r)dr = (Wifr,), ki,L

(A%

Substitution of Eqn. (Al) for the Re number in Eqn. (5) (see text) gives an expression for kiL(r) which can be used in Eqn. (A5); moreover, by taking the module dimension into account (d(r)= -0.16 r+0.0132) and after integration across the surface, an expression for Ji,tot is found: J.2,tot = 10 (ReR)1’3 SC

1’3 p (Wan,

Di

In the analogy to the relation between local and global heat transfer, mass transfer coefficient ki,L can be defined as J.r,tot =

ki,L. (,P,jav

P

tTR2)

W) the

(A7)

112

By comparing Eqn. (A6) with Eqn. (A7), an expression for ki,L can be deduced: k

i,L

552.2 (ReR)1’3 SC1’3Di

=

APPENDIX B: DETERMINATION

W)=(6)

OF THE YIELD Y

The local mass balance can also be written as

Q,

P

dwipr = -Ji,r

(2 r r) dr

031)

After substitution of Ji,r by Eqn. (A4) and after integration along the surface with the boundary conditions

W.b 190=

Wie,

at r=O

w.b l,R =

Wis

at r=R

ki,L (1rR2) 1

Eqn. (Bl) becomes ln

wi,c I

W. 1,s

=

Qt!

032)

The yield Y is defined as Y = 1-

[Wi,,/Wi,~]

(B3)

The combination of Eqns. (B2) and (B3) results in

Y = 1 - exp

For r=R, Q, and Re, are given by

(B4)

113

ReR

=

b(R) W (RN / v

so that Q, = aRReR v

@5)

Combining Eqns. (A8) and (B5) with (B4) gives Y = I- exp -552.2 (ReR)1’3 sc 1’3 Di (M,QJ]

036)= (8)