[membrane science and technology] ion exchange membranes - fundamentals and applications volume 12...

Overall Mass Transport

Chapter 6

6.1. OVERALL MEMBRANE PAIR CHARACTERISTICS AND MASS

TRANSPORT ACROSS A MEMBRANE PAIR

When an electrolyte solution is supplied to an electrodialyzer and anelectric current is passed through it, ions and a solution in a desalting cell aretransported to a concentrating cell across a cation and an anion exchangemembrane. The quantity of ions JS and a solution JV transported across a pairof membranes per unit area and per unit time at current density i are expressedby the overall mass transport equation, Eqs. (6.1) and (6.2) (Tanaka, 2006).

JS ¼ Zi

F

¼ C00JV ¼ li mðC00 C0Þ ¼ li mDC (6.1)

JV ¼ fi þ rðC00 C0Þ ¼ fi þ rDC (6.2)

Z is current efficiency, F the Faraday constant, C0 and C00 electrolyte concen-trations in a desalting and a concentrating cell, respectively, l the overall trans-port number, m the overall solute permeability, f the overall electro-osmoticpermeability and r the overall hydraulic conductivity, and these parameters aretermed the overall membrane pair characteristics altogether. The term ‘‘overall’’means that the parameters are the sum of the contributions of a cation and ananion exchange membrane. It also means that the parameters are the sum of thecontributions of many kinds of ions dissolving in an electrolyte solution. Pa-rameters li and mDC in Eq. (6.1) stand for the electro-migration and solutediffusion, respectively, and parameters fi and rDC in Eq. (6.2) correspond tothe electro-osmosis and concentration–osmosis. JS/i and JV/i against DC/i yieldstraight lines, so that l, m, f and r are obtained from the intercepts and thegradients of the lines based on the electrodialysis experiment as explained inSection 2.10.

JS/i and JV/i vs. DC/i plots are obtained by repeating the electrodialysis bychanging current density. The plots are not influenced by the concentrationpolarization and are obtained by the electrodialysis of seawater as shown inFig. 6.1. On the basis of many electrodialysis experiments of seawater describedabove, the regularity in ion exchange membrane characteristics is found fromthe plot of l(eq C1), m(cm s1) and f(cm3C1) against r(cm4 eq1 s1) asshown in Fig. 6.2. The plots for a 0.5M NaCl solution electrodialysis aremarked by asterisks in Fig. 6.2, indicating that the plotting is done on the samelines for seawater electrodialysis.

DOI: 10.1016/S0927-5193(07)12006-4

dx.doi.org/10.1016/S0927-5193(07)12006-4.3d

0.00 0.05 0.10 0.15 0.20 0.25 0.300

1

2

3

4

5

JS

/i(1

0-5

eq

C-1

) J

V/i

(10

-3c

m3C

-1)

C /i (eq A-1cm-1)

=1.213 × 10-2cm4eq-1s-1=1.406 × 10-3cm3 C-1=1.429 × 10-6cm s-1=9.724 × 10-6eq C-1

J V/i

JS/i

Figure 6.1 JS/i vs. DC/i plot and JV/i vs. DC/i plot (Tanaka, 2006).

Ion Exchange Membranes: Fundamentals and Applications82

Mizutani and Nishimura (1970) investigated microstructure of cation ex-change membranes by converting them into porous inert membranes having no ionexchange component on treatment with hydrogen peroxide, and determined ap-parent pore size, tortuosity factor, number of pores and pore size distribution in theporous membranes. Fig. 6.3 gives the experimental result indicating the effects ofporosity e on water contentW and specific resistance v taking tortuosity factor s asa parameter, showing W and v to be zero and infinite, respectively, at e-0. Thephenomena described here means that the mass transport is increased with theporosity e, and the membrane loses electric conductivity at e ¼ 0 and becomes aninsulator. Based on this experimental result, it is concluded that m, f and r aredecreased with the decrease of e and approach zero at e-0, and that the membranepair electrical resistance R is increased with the decrease of e and becomes infinityat e-0, and further that the water content of the membrane W is decreased withthe decrease of e and becomes zero at e-0, so we have the following equations:

lim!0

m ¼ lim!0

f ¼ lim!0

r ¼ lim!0

1

R

¼ lim

!0W ¼ 0 (6.3)

Accordingly,

limr!0

m ¼ limr!0

f ¼ limr!0

1

R

¼ lim

r!0W ¼ 0 (6.4)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(10-2cm4eq-1s-1)

(10-5

eq C

-1)

(

10-4

cm s

-1)

(

10-3

cm3 C

-1)

Figure 6.2 r vs. l, m and f (Tanaka, 2006).

Overall Mass Transport 83

Referring to Eq. (6.4), the plots in Fig. 6.2 are expressed by the following empiricalequations (Tanaka, 2006).

l ¼ l1 þ l2r l1 ¼ 9:208 106 l2 ¼ 1:914 105 (6.5)

m ¼ mr m ¼ 2:005 104 (6.6)

f ¼ n1r0:2 þ n2r n1 ¼ 3:768 103 n2 ¼ 1:019 102 (6.7)

in which Eq. (6.4) is realized in Eqs. (6.6) and (6.7).r vs. membrane pair electric resistance R ( ¼ RK+RA, O cm2) and mem-

brane pair water content W ( ¼ (WK+WA)/2, g H2O/g dry membrane) is in-dicated, respectively, in Figs. 6.4 and 6.5, expressed by the following empiricalequations, in which Eq. (6.4) is satisfied. Subscripts K and A refer to a cationexchange membrane and an anion exchange membrane, respectively.

R ¼ pr1 p ¼ 5:107 102 (6.8)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.850

100

200

300

400

500

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

(Ω c

m)

W (

%)

2.1

=6.9

3.8

2.4

2.4

Figure 6.3 Effect of porosity on water content and specific electric resistance of ionexchange membranes (Mizutani and Nishimura, 1970).


W ¼ q1r0:5 þ q2r q1 ¼ 3:785 q2 ¼ 6:375 (6.9)

The electrolyte concentration in a concentrating cell C00 is introduced from Eqs.(6.1) and (6.2) as follows:

C00 ¼1

2r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ 4rB

q A

(6.10)

A ¼ fi þ m rC0 (6.11)

B ¼ li þ mC0 (6.12)

Equations (6.5)–(6.7) indicate that r is a leading parameter and represents all ofthe overall membrane pair characteristics. l, m and f (and R, W) are computedby substituting r in Eqs. (6.5)–(6.7) (and (6.8), (6.9)). C00 is computed by sub-stituting i and C0 in Eqs. (4.10)–(4.12). Accordingly, the electro-migration li, thesolute–diffusion mDC, the electro-osmosis fi and the concentration–osmosis

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

8

9

10

R(

Ω c

m2 )

(10-2cm4eq-1s-1)

Figure 6.4 Relationship between overall water permeability and membrane pair elec-trical resistance (Tanaka, 2006).


rDC are determined using Eqs. (6.5)–(6.7) and (6.10)–(6.12) by setting r, i andC0 as parameters.

Here, we calculate the mass transport across a membrane pair JS, li, mDC,JV, fi and rDC based on the computation described above by settingr ¼ 1.00 102 cm4 eq1 s1 and C0 ¼ 6 104 eq cm3 and they are plottedagainst i. The results are presented in Fig. 6.6, indicating that mDC is negligibleas compared to li, and that rDC is predominant at lower i and fi is predom-inant at larger i.

JS, JV, C00 and Z are computed using Eqs. (6.1), (6.2), (6.5)–(6.7) and (6.10)–

(6.12), i ¼ 3Adm2 and C0 ¼ 0.6 eq dm3. They are plotted against r and areshown in Fig. 6.7, which means that the mass transport can be analyzed using r.

6.2. THE OVERALL MASS TRANSPORT EQUATION AND THE

PHENOMENOLOGICAL EQUATION

The phenomenological equation, Eqs. (5.59) and (5.60) and the overallmass transport equation, Eqs. (6.1) and (6.2) are substantially identical, so we

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

W (

gH2O

/g d

ry m

em

br.

)

(10-2cm4eq-1s-1)

Figure 6.5 Relationship between overall water permeability and water content (Tanaka,2006).


have the following equations:

JS;K þ JS;A ¼ JS (6.13)

JV;K þ JV;A ¼ JV (6.14)

From Eqs. (6.13) and (6.14), the overall membrane pair characteristics are rep-resented by the following equations.

l ¼tK þ tA 1

F¼

JS

i

DC¼0

(6.15)

m ¼ RT ½ðoK þ oAÞ fLP;KsKð1 sKÞ þ LP;AsAð1 sAÞgCS

¼ RT ½ðoK þ oAÞ sð1 sÞðLP;K þ LP;AÞCS ¼

JS

DC

i¼0

ð6:16Þ

f ¼ bK þ bA ¼JV

i

DC¼0

(6.17)

0 2 4 6 8 10 120

4

8

12

16

20 00

2

4

6

8

10

(2)

(1)

JV

i J S

i(A/dm2)

(10-7

eq c

m-2

s-1)

Figure 6.6 Electro-migration, solute diffusion, electro-osmosis and concentration os-mosis in ion exchange membrane electrodialysis.


r ¼ RTðsKLP;K þ sALP;AÞ ¼ RTsðLP;K þ LP;AÞ ¼JV

DC

i¼0

(6.18)

Here, s is the membrane pair reflection coefficient defined for simplification ofthe expression as follows:

s ¼LP;Kf1 sKð1 sKÞg þ LP;Af1 sAð1 sAÞg

LP;K þ LP;A(6.19)

6.3. REFLECTION COEFFICIENT r, HYDRAULIC CONDUCTIVITY LP

AND SOLUTE PERMEABILITY x

Yamauchi and Tanaka (1993) measured sK, LP,K and oK of commerciallyavailable cation exchange membrane Neocepta CL-25T by means of pressure-driven dialysis of KCl supplied sucrose for generating pressure gradient. Theexperimental works were performed under the circumstances in which the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0

2

4

6

8

10

12

C"

JV

JS

J S(1

0-7eq

cm

-2s-1

) J

V(1

0-5cm

s-1

) C

"(eq

dm

-3)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

(10-2cm4eq-1s-1)

Figure 6.7 Effect of overall hydraulic conductivity on ionic flux, volume flux, electrolyteconcentration in a concentration cell and current efficiency (Tanaka, 2006).


following equations hold:

sK ¼1

LP;K

JV;K

DpS

DPDpi¼0

(6.20)

LP;K ¼JV;K

DP Dpi

DpS¼0

(6.21)

oi ¼JS;K

DpS

JV¼0

(6.22)

Here, DP is mechanical pressure difference, Dpi and DpS are osmotic pressuredue to impermeable and permeable solutes, respectively. The results are pre-sented in Table 6.1 indicating LP,K ¼ 1.62 1013 cm2 smol g1 eq1,oK ¼ 0 smol cm1 g1 and s ¼ 1.

In order to analyze LP,K and oK in Table 6.1, the following equations areintroduced from Eqs. (6.16) and (6.18):

LP;K ¼r

2RTsK(6.23)

Table 6.1 Hydraulic conductivity, solute permeability and reflection coefficient of acation exchange membrane measured in the pressure dialysis of a KCl and a sucrosesolution

LP,K 1.62 1013 cm2smol g1eq1 0.5M sucrose/watersystem

oK 0 Smol cm1 g1 0.01M KCl/0.1M KClsystem

sK 1 0.01M KCl/0.1MKCl+0.5M sucrosesystem

Note: Neocepta CL-25T 251C.Source: Yamauchi and Tanaka (1993).


oK ¼m

2RTþ LP;Kð1 sKÞC

S (6.24)

Here, m and r are divided by 2 because these are the membrane pair charac-teristics and the characteristics of a cation exchange membrane and that of ananion exchange membrane are assumed to be equivalent in this situation. Thelogarithmic mean concentration C

S in Eq. (6.24) is calculated using Eqs. (5.61)and (6.10)–(6.12). Substituting m and r of Neocepta CL-25T/AVS-4T membraneevaluated in the electrodialysis experiment and sK ¼ 1 (Table 6.1) into Eqs(6.23) and (6.24), LP,K and oK are computed as shown in Table 6.2. LP,K inTable 6.2 is less to some extent than LP,K ¼ 1.62 1013 cm2 smol g1 eq1 inTable 6.1, however, both give fairly good agreement. oK in Table 6.1 is eval-uated to be zero, however, extremely small oK values are detected in Table 6.2.The argumentation described here demonstrates the overall mass transportequation to be in agreement with the phenomenological equations based on theirreversible thermodynamics.

6.4. PRESSURE REFLECTION COEFFICIENT AND CONCENTRATION

REFLECTION COEFFICIENT:ELECTRIC CURRENT SWITCHING OFF

CONCEPT

Equations (6.15)–(6.18) show that sK and sA appear in m and r and do notappear in l and f. These events and Eqs. (6.1) and (6.2) mean that sK and sA donot exert an influence on mass transport with electric current passing. Accord-ingly, in order to understand the behavior of sK and sA in an electrodialysisprocess, it is necessary to create the image of zero current density. In otherwords, it is reasonable to image the electric current interruption (switch off)for a moment in the electrodialysis process operating under a constant electriccurrent, and to assume the disappearance of the electro-migration and electro-osmosis in this moment. Here, we assume further that DC and resulting solute

Table 6.2 Hydraulic conductivity and solute permeability of a cation exchange membrane estimated from the overall membrane paircharacteristics m and r

C0 (103 eq cm3) m (106 cm s1) r (103 cm4 eq1 s1) LP,K (1013 cm2 s2mol g1 eq1) oK (1017 smol cm1 g1)

0.294 1.458 6.354 1.27 2.900.577 3.208 6.788 1.35 6.381.132 0.908 7.566 1.51 1.811.920 3.575 7.691 1.53 7.11

Note: Neocepta CL-25T/AVS-4T 291C, seawater electrodialysis.Source: Tanaka (2006).

IonExchangeMem

branes:

FundamentalsandApplica

tions

90


diffusion and solution concentration–osmosis exist as it is just after the inter-ruption of an electric current.

In order to discuss the behavior of the reflection coefficient in an elect-rodialysis process, we express the volume flow JV and exchange flow JD in an ionexchange membrane pair by the following equation introduced by Schlogel(1964).

JV ¼ ðLP;K þ LP;AÞDPþ ðLPD;K þ LPD;AÞRTDC ¼JS

CS

þJW

CW

(6.25)

JD ¼ ðLDP;K þ LDP;AÞDPþ ðLD;K þ LD;AÞRTDC ¼JS

CS

JW

CW

(6.26)

DC and DP are pressure difference and concentration difference across themembrane, respectively. LP is the hydraulic conductivity and LD is the exchangeflow parameter. LPD is the osmotic volume flow coefficient and LDP is theultrafiltration coefficient, between which the Onsager reciprocal relationLPD ¼ LDP (Onsager, 1931) is satisfied. JW is the flux of water molecules, C

S

and CW are, respectively, logarithmic average concentration (Eq. (5.61)) of sol-

utes and water (solvent) between a desalting cell and a concentrating cell. Itshould be noticed that Eqs. (6.25) and (6.26) are originally defined in the pres-sure driven transport (pressure dialysis) of neutral species on the promise of noelectric current.

In a pressure driven process, putting DC ¼ 0 in Eqs. (6.25) and (6.26) getto the following equations being applicable in pressure dialysis.

ðJVÞDC¼0 ¼ ðLP;K þ LP;AÞDP (6.27)

ðJDÞDC¼0 ¼ ðLDP;K þ LDP;AÞDP (6.28)

Staverman (1951) and Kedem–Katchlsky (Kedem and Katchalsky, 1958) definethe reflection coefficient s by Eq. (6.29) introduced from Eqs. (6.27) and (6.28).

s ¼ JD

JV

DC¼0

¼ LDP;K þ LDP;A

LP;K þ LP;A(6.29)

s defined by Eq. (6.29) is identical with the parameter included in Eqs (5.57) and(5.58). Here, we term s presented by Eq. (6.29) as ‘‘pressure reflection coeffi-cient’’, taking note that s exhibits a pressure difference DP driven phenomenon.

In the electrodialysis process, DP is relatively low and possible to neglect,and at just after an electric current interruption situation, DC exists as it is.Putting DP ¼ 0 in Eqs. (6.25) and (6.26), we have the following equations beingapplicable to diffusion dialysis.

ðJVÞDP¼0 ¼ ðLPD;K þ LPD;AÞRTDC (6.30)

ðJDÞDP¼0 ¼ ðLD;K þ LD;AÞRTDC (6.31)


Here, we define another reflection coefficient s0 by Eq. (6.32) introduced from Eqs.(6.30) and (6.31) which are defined at just after an electric current interruption.

s0 ¼ JD

JV

DP¼0

¼ LD;K þ LD;A

LPD;K þ LPD;A(6.32)

We term s0 presented by Eq. (6.32) as ‘‘concentration reflection coefficient’’, pay-ing attention that s0 expresses a concentration difference DC driven phenomenon.

Regarding the Onsager reciprocal relation LPD ¼ LDP, in Eqs. (6.29) and(6.32):

ss0 ¼LD;K þ LD;A

LP;K þ LP;A(6.33)

Canceling JW=CW in Eqs. (6.25) and (6.26):

JD ¼ 2JS

CS

JV (6.34)

From Eqs. (6.25), (6.32) and (6.34), s0 is introduced as follows:

s0 ¼ 1 21

CS

JS

JV

¼ 1 2

JS

CS

JS

CS

þJW

CW

(6.35)

s0 defined in Eq. (6.35) means the permselectivity between ions and water mole-cules at just after an electric current interruption. JS and JV are expressed as thefollowing equations at just after an electric current interruption by putting i ¼ 0 inEqs. (6.1) and (6.2), and canceling the minus sign of the second term in Eq. (6.1).

JS ¼ mðC00 C0Þ (6.36)

JV ¼ rðC00 C0Þ (6.37)

Substituting Eqs. (6.36) and (6.37) into Eq. (6.35):

s0 ¼ 1 2mr

1

CS

(6.38)

s0 is calculated using Eqs. (6.38), (5.61) and (6.10)–(6.12) and substituting l, m,f and rmeasured by electrodialysis. The s0 of commercially available membranesis generally in the range of 0os0o1, which is understandable from Eq. (6.35) thatlinear velocity of water molecules is larger than that of ions just after electriccurrent interruption;

JW

CW

4JS

CS

(6.39)

s0 is plotted against current density i taking C0 as a parameter and as shown inFig. 6.8 indicating that s0 increases with i. This phenomenon is understandable

Figure 6.8 i vs. s0 and LD,K+LD,A.


from Eq. (6.38) assuming C* increases with i and m/r is independent of i as shownin Figs. 6.1 and 6.2 and Eq. (6.6).

6.5. IRREVERSIBLE THERMODYNAMIC MEMBRANE PAIR

CHARACTERISTICS

From the pressure driven dialysis of a KCl solution (Section 6.3),the reflection coefficient s of an ion exchange membrane is generally assumedto be 1. This is presumably owing to dense structure of an ion exchange mem-brane. So, the membrane pair characteristics defined in the phenomenologicalequations are introduced from Eqs. (6.15)–(6.18) putting sK ¼ sA ¼ s ¼ 1 asfollows.

tK þ tA ¼ lF þ 1 (6.40)

oK þ oA ¼mRT

(6.41)

bK þ bA ¼ f (6.42)

LP;K þ LP;A ¼rRT

(6.43)

tK+tA, oK+oA, bK+bA and LP,K+LP,A are computed by substituting l, m, fand r measured by electrodialysis into Eqs. (6.40)–(6.43), and plotted against T,as shown in Fig. 6.9.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

L D,K

+L D

,A (

10−1

3 cm

2 s m

ol g

−1eq

−1)

(LP

,K+

L P,A

)=−(

L PD

,K+

L PD

,A)=

−(L D

P,K

+L D

P,A

)

L D,K+L D,A

(L P,K+L

P,A)=

−(LPD,K

+LPD,A

)=−(L

DP,K+L

DP,A)

(10−2cm4eq−1s−1)

Figure 6.10 Relationship between r and LP, LPD, LDP and LD (Tanaka, 2006).

20 30 40 50 60 700

1

2

3

4

5

6

7

8

9

10

T(°C)

LP,K+LP,A

tK+tA

ω ω

ωω L P

,K+

L P,A

(10-6

mol

cm

4 eq-1

J-1s-1

)(1

0-3cm

3 C-1

)

Figure 6.9 T vs. tK+tA, oK+oA and LP,K+LP,A (Tanaka, 2006).



Putting s ¼ 1 in Eqs. (6.29) and (6.43):

sðLP;K þ LP;AÞ ¼ ðLP;K þ LP;AÞ ¼ ðLPD;K þ LPD;AÞ

¼ ðLDP;K þ LDP;AÞ ¼rRT

ð6:44Þ

LPD,K+LPD,A and LDP,K+LDP,A are calculated using LP,K+LP,A and Eq.(6.44), and are plotted against r in Fig. 6.10.

From Eqs. (6.33) and (6. 43):

LD;K þ LD;A ¼rs0

RT¼ ðLP;K þ LP;AÞs0 (6.45)

LD,K+LD,A is calculated using s0 and Eq. (6.45), and plotted against i inFig. 6.8.

REFERENCES

Kedem, O., Katchalsky, A., 1958, Thermodynamic analysis of the permeability of bio-logical membranes to non-electrolytes, Biochim. Biophys. Acta, 27, 229–246.

Mizutani, Y., Nishimura, M., 1970, Studies on ion-exchange membranes. XXXII.Heterogeneity in ion-exchange membranes, J. Appl. Polym. Sci., 14, 1847–1856.

Onsager, L., 1931, Reciprocal relations in irreversible processes, Phys. Rev., 37, 405–426.Schlogel, R. Z., 1964, Fortschritte der physikalischen Chemie, Band 9.Staverman, A. J., 1951, The theory of measurement of osmotic pressure, Rec. Trav.

Chim., 70, 344–352.Tanaka, Y., 2006, Irreversible thermodynamics and overall mass transport, J. Membr.

Sci., 281, 517–531.Yamauchi, A., Tanaka, Yasuko., 1993, Salt transport phenomena across charged

membrane driven by pressure difference, In: Paterson, R. (Ed.), Effective MembraneProcess – New Prospective, BHR Group Ltd., Information Press Ltd., Oxford,England, pp. 179–185.

[membrane science and technology] ion exchange membranes - fundamentals and applications volume 12...

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