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Modeling of electrodialysis of metal ion removal from pulp and paper mill process stream
Chemical Engineering Science 55 (2000) 4629}4641
Modeling of electrodialysis of metal ion removal from pulp and paper mill process stream Zhiqiang Yu, Wudneh Admassu* Department of Chemical Engineering, University of Idaho, Moscow, ID 83844-1021, USA Received 10 March 1999; accepted 22 March 2000
Abstract A theoretical model of an electrodialysis process for the removal of metal ions in process stream of the pulp and paper industry has been developed in this article. Based on a constant applied electric potential gradient as a power supply, the model starts from a di!erential equation of mass balance, and presents an analytical solution describing concentration pro"les of cations as electrolytes in electrodialysis channels. The relationships among metal ion concentration, electrical current density, and removal e$ciency are also discussed. 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Electrodialysis; Metal ion; Concentration pro"les; Current density; Removal e$ciency
1. Introduction Metal ions in a pulp and paper mill process stream include sodium (Na), potassium (K), calcium (Ca), magnesium (Mg), manganese (Mn), aluminium (Al), chromium (Cr), copper (Cu), nickel (Ni), titanium (Ti), iron (Fe), mercury (Hg), lead (Pb), silver (Ag) and zinc (Zn). Many of these metals are essential nutrients at low levels but toxic at higher levels. Metal ions in process streams originate from one or more of four sources: wood chips, chemicals used in the pulping process, additives used in paper making, or products of corrosion (Springer, 1986; Bryant, 1993). Under speci"c operating conditions, some metal ions are added to improve the process e$ciency, but have to be removed for other processes. For example, Na> and K> should be removed from the streams in the pulping process before entering into the recovery boiler to prevent fouling and corrosion. In general, the concentrations of metal ions in pulp and paper mill process streams are far below toxicity levels (Springer, 1986). However, in zero discharge process mills metal ions, such as Na> and K>, will accumulate in process stream during recycling, so the removal of metal ions from the recycling streams of pulp and paper
* Corresponding author. Tel.: #1-208-885-8918; fax: #1-208-8887462. E-mail address: [email protected] (W. Admassu).
mill becomes very important. To address the metal ion accumulation problem in zero discharge pulp and paper mills, the e!ective removal of the excess metal ions from the process water is critical. Removal of metal ions from aqueous solutions has traditionally meant precipitation (Janson, Kenson & Tucker, 1982; Beszedits & Netzer, 1986). Often, this technique is now unpopular because it produces a large amount of sludge for disposal (La Grega, Buckingham & Evans, 1994; Kim & Weininger, 1982; Totura, 1996). Another alternative, electrochemical metal recovery, is a feasible process, but process streams of pulp and paper mills are often dilute and therefore the electrochemical recovery has low e$ciency (Larson & Wiencek, 1994). Solvent extraction of metals has been extensively used in hydrometallurgical operations. The major disadvantage of this technique is that the extraction is limited by equilibrium consideration (Raghuraman, Tirmizi, Neena & Wiencek, 1994). Currently, the application of membrane separation processes (MSP) for metal ion removal has been increasing rapidly (Larson & Wiencek, 1994; Sternberg, 1987; Grinstead & Paalman, 1989; Zhu & Elimelech, 1995). The advantages of MSP are the high-quality "ltrate that is suitable for discharge or reuse, little or no chemical addition and its compactness. Electrodialysis (ED) is a membrane process, in which an applied electrical potential di!erence causes ion transport through a membrane. The ED unit normally consists of anion- and cation-selective membranes arranged alternately in parallel between an anode and a cathode.
0009-2509/00/$ - see front matter 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 0 1 - 9
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Fig. 1. Schematic diagram of the ED process.
As a consequence of the driving force (the electric "eld), anions in a solution permeate through the anion-transfer membrane and cations permeate through the cationtransfer membranes, resulting in concentration and dilution in adjoining compartments. The principle of the ED process is illustrated in Fig. 1, which shows a schematic diagram of a typical ED cell arrangement consisting of a series of anion- and cation-exchange membranes arranged in an alternating pattern between an anode and a cathode to form individual cells. Considerable research shows that the primary relationships or factors controlling an ED process are dissociation of salts (or materials to be treated) in water, Faraday's law, Ohm's law, and membrane properties (Bersillon, Anselme, Mallevialle, Aptel & Fiessinger, 1992; Gavach et al., 1994; Kikuchi, Gotoh, Takashashi, Higashino & Drano!, 1995; Kraaijeveld, Sumberova, Kuindersma & Wesselingh, 1995; Andres, Riera, Alvarez & Audinos, 1994; Nakamura 1993, 1995; Shaposhnik, Vasil'eva & Praslov, 1995; Popov, 1996; Koter, 1995; Shapovalov & Tyurin, 1996). The application of ED in the paper and pulp industry focuses on the following three aspects: (1) the recovery of valuable products from spent liquors and kraft liquors; (2) the production of chemicals used in the bleaching process, such as caustic soda; and (3) the disposal of e%uent (Zubets & Lebedinskaya, 1981; Mishra & Bhattacharya, 1984; Filatov, Vishnevskaya, Nikolaeva & Rubtsova, 1988; Kaverzina, Nepenin, Filatov & Vishnevskaya, 1988, 1989; Paleologou et al., 1994; Adachi & Hanada, 1994). However, there are limited studies on controlling of metal ions in pulp and paper industry process stream by ED processes. In principle, the concentration pro"les of metal ions and electrical current density are the two dominant topics being investigated (Rosler, Maletzki
& Staude, 1992; Probstein, 1989; Strathmann, 1992; Tanaka, 1991, MacNeil & McCoy, 1989). These have involved concentration polarization, membrane fouling, removal e$ciency and electrical energy consumption. Most researchers have frequently dealt with di!erent types of di!erential equations that need tedious and/or numerical analyses. Traditionally, the ED process uses galvanostatic condition for charge balance in which a constant electrical current is applied to pass through the electrodes as a power supply. As an alternative approach, this article would like to address modeling of the ED process based on a constant applied electric potential gradient as a power supply. Using this method the governing di!erential equation can be simpli"ed and an analytical solution describing concentration pro"les of cations as electrolytes in electrodialysis channels can be obtained. The relationships among metal ion concentration, electrical current density, and removal e$ciency are further discussed. The model also describes the transport of metal ions in boundary layers at membrane surfaces, the limiting current density as a function of #ow velocity, and the trend of increase of current density vs. increase of removal e$ciency by which an optimum ED process can be designed.
2. Theoretical modeling 2.1. Transport equations The mass balance in terms of molar concentration of metal ions in the direction perpendicular to the #ow stream is given by *C G # ) C u "! ) J #r , G G G *t
(1)
where C is the molar concentration of species i, u the G molar velocity, J the total molar #ux, and r the related G G chemical reaction rate. For no chemical reaction and an incompressible #uid, r "0, ) u "0, which gives G *C G #u ) C "! ) J . (2) G G *t The total molar #ux depends upon the e!ects of di!usion, migration and convection in the ED channels. For su$ciently dilute solutions, because the solute species and their gradients do not interact, the #ux contributions from di!usion, migration, and convection can be linearly superposed (Probstein, 1989). When considering an aqueous solution where only cation and anion involved, the total molar #ux are represented by J "!z v FC
!D C #C u. ! ! ! ! ! ! !
(3)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
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process requires that the induced potentials as resistance should be much less than the external electrical potential as driving force, the total electric potential gradient
is approximately a constant. By Eqs. (2) and (3) there is *C ! #u ) C "z v F ) (C
)#D C . ! ! ! ! ! ! *t Fig. 2. Electrodialysis with a cation exchange membrane.
Here, C are the concentrations of cation or anion; ! z the valence number; v the mobility related to the ! ! di!usivity D by the relation D "v RT (here v and ! ! ! ! D are taken as constants), F the Faraday constant,
! the electric potential gradient, and u the mass #ow velocity of #uid. The molar average velocity u is replaced by the mass average velocity u, since in su$ciently dilute solutions u "u. Eq. (3) is a well-known Nernst}Planck equation, in which the right-hand side represents the contributions of #ux from migration, di!usion, and convection respectively. In the galvanostatic case the electric potential will be a function of location due to a constant current supply. In this modeling, will include the external applied potential and two induced potentials, i.e., the electrostatic potential and polarization potential, that is
" # # , (4) NJ here is the external electrical potential, is the NJ electrical potential due to the concentration polarization, and is the electrostatic potential due to the concentra tion di!erence of ions. It is important to characterize these contributions before the modeling simpli"cation. For example, in a case of ED with a cation exchange membrane, the potential contributions are shown as in Fig. 2 by the transport of cations. Fig. 2 states the following: (1) the external electrical potential, generates an ex ternal electric "eld E (
'0, E (0) as a driv ing force on cation transport; (2) the electrical potential due to the polarization (C !C )(C and C are the concentrations at the A ? A ? membrane surfaces in concentrated and dilute sides),
produces a polarization "eld E (
(0, NJ NJ NJ E '0); NJ (3) the electrostatic potential due to the concentration gradients,
produces an electrostatic "eld E ( C'0,
'0, E (0); (4) the total induced electrical potential ( plus ) will NJ QR be in the opposite direction of , leading to E !E '0, otherwise, the law of energy conservaNJ tion will be violated. In this paper,
is taken as a constant. Since the ED
(5)
Here, the "rst term on the left-hand side is the rate of change of mass per unit volume, and the second term is the contribution due to convection; on the right-hand side the "rst term is due to the migration e!ect under the electrical "eld, and the second term is due to the di!usive e!ect. Because of the motion of charged species, there will be an electrical current. The current density is de"ned as i"FRzJ. Using Eq. (3) the current density in dilute aqueous solution can be written as i "!F
Rz v C !FRz D C ! ! ! ! ! ! ! #FuRz C . ! !
(6)
2.2. Simplixcations To simplify the theoretical model, one can assume a case in which cations and anions are completely dissociated by the external electrical "eld. For example, potassium chloride (KCl) is a chief form of chemical compound present as K> and Cl\ in the pulp and paper process streams. Both K> and Cl\ are known to accelerate the plugging of the #ue gas passages and increase the rate of the corrosion of the superheat tubes in the recovery boiler system by lowering the melting points of the dust (Shenassa, Reeve, Dick & Costa, 1995; Hupa, 1993). Under the condition of complete dissociation of cation and anion, K> and Cl\, only the transport behavior of cations in aqueous solution is considered. This is because the anion transport is assumed to be symmetrical. In a symmetric multicell ED system, if both cation and anion exchange membranes have the same transport properties, such as the transport numbers, resistance, etc., the concentration distributions of cations and anions in the separation region will be symmetric (except in the cells adjacent to the electrodes). A schematic of the #ows in dialysate and concentrate channels separated by a cation exchange membrane is shown in Fig. 3, where the cation exchange membrane is set at x"0 and its thickness is negligible compared to the channel width. For a quasi-steady state, the time variation of concentration *C /*t"0. Also, the following assumptions are > used: (1) for the whole channel the neutrality of charge is satis"ed, that is, "0 (see the appendix); (2) the di!usion term along the #ow direction D*C /*y is negligible compared to the convection >
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 4. Schematic of boundary layers.
Due to the electroneutrality within the #uid (Deen, 1998) Fig. 3. Schematic of #ows in ED channels.
term u *C /*y, where u is the velocity component W > W in the #ow direction; (3) assume that the in#uence of #ow pro"le of #uid is negligible in our model in both dialysate and concentrate channels. Therefore, a plug #ow of the #uid is assumed. Because of the dilute nature of the solution, the viscosity of the solution is approximately that of water at the given temperature. Under this assumption, the #ow velocity u is now constant in the channels everywhere along the #ow direction, i.e., u"uy( , y( is a unit vector in the y-direction. Therefore, according to schematic of #ows in Fig. 3 it follows that *C u ) C "u > > *y
(7)
*C d
) (C
)" C )
#C " > ) , > > > *x dx
(8)
D C "D > > >
*C *C *C ># > +D >. > *x *x *y
(9)
By inserting Eqs. (7)}(9) into Eq. (5), will result in u
*C d *C *C > "D > #z v F >. > *x > > d *x *y
(10)
z C #z C "0. > > \ \ Using Eq. (11)}(13) will result in Fz C d
dC >" > > . dx R¹ dx
(13)
(14)
Here, C "C , and z "!z are considered for > \ \ > K> and ClU in aqueous solution. 2.3. Concentration distributions By de"ning parameters A"u/D , B"z v F/D ; > > > > d /dx, and a dimensionless reduced concentration CH"C /C , Eq. (10) becomes 5 *CH *CH *CH " #B . (15) A *x *x *y Here, C is the initial concentration of the solution, and C is de"ned as the wall concentration in the boundary U layers near the interface (0)"x")x ). As shown in ? Fig. 4, x is a thickness of the boundary layer between ? either membrane and dialysate or membrane and concentrate (here x ;h, h is the half-width of ED channel). ? This modeling assumes that signi"cant cation transport occurs only in these boundary layers, that is, in the region of 0)"x")x , or 0)"x"/x )1. Since x ;h, when ? ? ? x'x and xPh, it could be taken as xPR. ? The boundary conditions for Eq. (15) are
Eq. (10) shows that the concentration of cations in the dialysate channel is a function of both coordinates x and y. For current density, in this modeling i O0 but > i "0 (because only a cation exchange membrane is \ considered). Since the electrical current of interest here is in the direction of the electric "eld that is perpendicular to the #ow of #uid, the e!ect of convection contribution on current uC in Eq. (6) is negligible. Then Eq. (6) ! becomes
x"0, CH"C (CH" ), Q V xPh(x'x ), CH"C (CH" ), ? F VF y"0, CH"1 for x'x . (16) ? Eq. (15) indicates that in either dialysate or concentrate channel the variations of concentration along the y-axis (#ow direction) and in the x-axis (channel width) are independent. Therefore, the general solution can be written as
dC Fz D C d
>! > > > i "!D Fz . > > > dx R¹ dx
CH(x, y)"f (y)g(x). (11)
dC Fz D C d
\! \ \ \ 0"!D Fz . \ \ dx R¹ dx
(12)
(17)
Now Eq. (15) can be written as
A df 1 dg dg " #B . f dy g dx dx
(18)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
For the dialysate channel, the cation concentration decreases along the #ow direction. Hence, Eq. (18) can be de"ned to be equal to a separation constant, !j, which is related to the boundary conditions. Therefore, df j # f"0, dy A
(19)
dg dg #B #jg"0. dx dx
(20)
Eq. (20) is a second-order homogeneous ordinary di!erential equation. Its characteristic equation (or auxiliary equation) is K#BK#j"0.
(21)
There are two roots for Eq. (21), B K "! +1G(1!4j/B) ,. 2
(22)
Then, a general solution for Eq. (20) is g(x)"N exp(K x)#N exp(K x), (23) with the integration constants N and N . By combining Eq. (23) and a general solution of Eq. (19) the reduced concentration in the dialysate channel will be CH(x, y)"f (y)g(x)"+m exp(K x)
j #m exp(K x),exp ! y . A (24)
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Substituting for m , Eq. (26) is
j j h exp ! x B B
CH(x, y)" exp
j #m exp(!Bx) exp ! y . A
(28)
Both Eqs. (26) and (28) are satis"ed with the boundary conditions in Eq. (16). But the condition at x"0, CH"C (CH" ) could not be used conveniently to deQ V termine m because the surface value C (CH" ) cannot Q V be measured experimentally. So the continuity condition of electrical current at x"0 is introduced here to determine m . For a "xed location of y, at the interface between membrane and solution, the expressions of current density i(x) in terms of concentration gradient and potential gradient are given by Eqs. (11) and (14). Here, surface "eld strength (d /dx) is used to replace (d /dx) K B indicating a di!erent potential gradient at the interface between dialysate and membrane to that between concentrate and membrane, i.e., (d /dx) . The wall concenK A tration C is used to replace C emphasizing the mass U > transfer at the boundary layers. Using the continuity of current both in the boundary layer (0(x(x ) and the ? interface between the membrane surface and solution (x"0), Eq. (14) becomes dC U dx
V
z FC d
" > Q R¹ dx
.
(29)
K B
Here, the integration constants m and m need to be determined using the boundary conditions. For the purpose of simpli"cation, by assuming 4j/B;1 (see Section 3), Eq. (22) becomes
Here, the surface concentration C is de"ned as Q C "C " (Eq. (28)). Using Eqs. (28) and (29), Q U V at x"0,
j K "! , B
dC U dx
j K "!B# . B
(25)
The assumption of 4j/B;1 indicates that j/B;B, so that K !B. Then, Eq. (24) is j CH(x, y)"f (y)g(x)"+m exp ! x B
This is only a numerical approximation where only symbol substitutions were made and it does not change the form of Eq. (24) which satis"es Eq. (15) exactly. At y"0, CH"1 for x'x . As x becomes large, ? exp(!Bx)P0 faster than exp(!j/Bx) because of j/B;B. This yields
V
z FC d
> Q R¹ dx
(27)
K B
j h B
j j ! !Bm exp ! y , B A (30)
z F d
" > R¹ dx
exp
K B
j h B
j #m exp ! y . A
j #m exp(!Bx),exp ! y . A (26)
j m "exp h . B
" exp
(31)
Since B"z v F/D (d /dx), then > > > m "
j exp h B !
d
R¹ # dx z F K B > d
d
1# dx dx K B
d
dx
j B
d
dx
.
(32)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Therefore, under similar assumptions 4m/B;1 and m/B;1 there is
And "nally,
C j U "exp ! y C A
j j h exp ! x B B
exp
d
dx
!
d
R¹ j # dx z F B K B > d
d
1# dx dx K B
;exp
j h exp(!Bx) . B
d
dx
(33)
d
d
C dx dx U " 1! K B C d
d
1# dx dx K B
j ;exp(!Bx) exp ! y . A
m #n exp(Bx ) exp y . A
Eq. (33) is a complete form of solution for Eq. (15). Now if we use the approximations that j/B&0, exp((j/B)h)+1 and exp(!(j/B)x)P1, Eq. (33) will be
C m m CH" U " exp ! h exp ! x C B B
(35)
In order to express Eq. (35) in the same form as Eq. (34), the variable x shall be transferred back to its original coordinate, x "!"x". Thus, Eq. (35) becomes
m C m CH" U " exp ! h exp "x" B C B
m #n exp(!B"x") exp y . A
(36)
Using the same approach as Eqs. (29)}(34), the concentration distribution for the concentrate channel is given by
(34)
Eq. (34) represents a wall concentration distribution of cation in the dialysate channel where x+0, For the concentrate channel, there are two conditions to be considered: (1) Physically, (a) the steady-state assumption ensures that there is no retention of cation on membranae surfaces and in the membrane. Mass conservation shows the bulk concentration of cation will increase along the #ow direction while in the dialysate channel it decreases; (b) as shown in Fig. 4, the mass transfer only occurs near interfaces and the concentration should tend to be constant as xP!h. Because of this consideration, a new separation constant m for Eq. (15) should be used for the concentrate channel instead of !j (dialysate channel). The relation between m and j can be determined using conservation of mass. (2) Mathematically, Fig. 3 shows that along the x-axis with reference to the interface (x"0) the concentration distribution in the concentrate channel is an o!set mirror image of that in the dialysate channel. Using a transformation of coordinates, x"!x (x referring to a mirror image system of x while m is used instead of !j for Eq. (18).
d
d
C dx dx U " 1# K A C d
d
1! dx dx K A
;exp(!B"x") exp
m y . A
(37)
Eqs. (34) and (37) present a complete cation concentration distribution in ED channels. These concentration distributions show the following characteristics: (1) When x+0, Eqs. (34) and (37) show the variations of concentration in the boundary layers. When x'x ? and xPh, the concentrations are basically independent of x. This indicates that the separation mainly occurs at the membrane surfaces. (2) As functions of y, the concentration in the dialysate channel decreases along the y-direction while it increases in the concentrate channel. (3) In general, because of concentration polarization, the "eld strengths at the membrane surfaces (d /dx) K B and (d /dx) are di!erent. These parameters should be K A determined experimentally, but their ratio can be estimated by the continuity condition of electrical current at speci"ed locations.
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
2.4. Current density
C "C " , the result is Q U V
Using Eqs. (11), (14) and (34), the current density can be represented as
i > C !C "! x U Q ? 2D Fz > >
d
d
dx dx K B i "!2D Fz C B U B > > d
d
1# dx dx K B
j ;exp(!Bx) exp ! y A
2D Fz i " > > C " . U V x ?
(40)
Eq. (40) has a form of
d
dx 2D Fz C > K B "! > R¹ d
d
1# dx dx K B j ;exp(!Bx) exp ! y . A
(39)
Here, i is in the opposite direction of x. So, > C !C '0, which satis"es the case of concentration in U Q the dialysate cell. When C P0, i "i . If only the Q > magnitude of i is considered, then
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i JC exp
(38)
Eq. (38) presents the electrical current density in the boundary layer for the dialysate channel. It indicates the following properties for the current density. (1) At the entrance (y"0), There is a maximum value of current density in the y-direction towards the exit (y"¸), the current density decreases continuously as a function of exp(!(j/A)y). (2) At the membrane surface (x"0), the current density also reaches a maximum value in the x-direction. In the boundary layer, the current density decreases as xPx . Beyond the boundary layer, as ? xPh, iP0. This strongly shows that the current density depends upon the concentration gradient. (3) The current density is directly proportional to the strength of electrical "eld represented by the surface "eld (d /dx) that K B shows a direct relationship between the ion transport and the electrical "eld. (4) A direct proportional relation between the current density and the initial concentration shows that the strength of ion transport depends upon the initial condition. (5) The current density is an indirect function of #uid velocity, which is included in the parameter de"ned in Eq. (15). The limiting current density i is another important physical quantity in ED process, which is de"ned as the current density at the membrane surface when the surface concentration tends to zero in the dialysate channel. So, i is the maximum current that may pass through a given membrane area without creating adverse e!ects, i.e., higher electrical resistance or lower current utilization. The limiting current density determines the minimum membrane area required achieving a certain removal e!ect. In this modelling, by combining Eqs. (11) and (14) and integrating both sides with the boundary condition
!uH , u
(41)
with an e!ective #ow velocity uH"D jy. For a large > value of #ow velocity u, the limiting current density, i , becomes only sensitive to the initial concentration C , because of the large #ow velocity could provide almost a uniform ion concentration everywhere in the dialysate channel. Eq. (41) presents that the limiting current density, i , is a function of both initial concentration C and #ow velocity u. This is consistent with the expres sions of limiting current density by Strathmann (1992) and Tanaka (1991). In the latter, the convection #ow velocity is perpendicular to the membrane surface, which may be limited by the permeability of the membrane.
3. Results and discussion 3.1. Parameter estimation One way to estimate the parameter j is by introducing the expected removal e$ciency E(%). In the dialysate channel the removal e$ciency E(%) is de"ned as C !C E(%)" ;100%, C
(42)
C is the average bulk concentration at any location in the y-axis, which is de"ned as
FC dx 1 F C 5 " C dx. 5 Fdx 2h
(43)
Here, h is the half-width of the dialysate channel shown in Fig. 2. When 2hB<1, C is approximately equal to C exp(!jy/A). Therefore, for whole channel
C !C j * ;100%" 1!exp ! ¸ E(%)" C A ;100%,
(44)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
which shows
A E(%) j"! ln 1! . ¸ 100
(45)
For 0(E(%)(100, Eq. (46) con"rms that j'0 is satis"ed. For the concentrate channel, the bulk concentration increases along the #ow direction. To compare with the dialysate channel, the separation constant m is related to the gain e$ciency E (%), which can be de"ned as
C !C m ;100%" exp ¸ !1 E (%)" * C A ;100%,
(46)
and then
E (%) A . m" ln 1# 100 ¸
(47)
When there is no retention of cations at the membrane surfaces and in the membrane, the amount of cations removed from the dialysate channel should equal to that gained in the concentrate channel. This indicates that numerically E (%)"E(%).
Geometric sizes Flow distance Flow width Half-channel width Average "lm thickness Number of channel pair Characteristics of -uid Average #ow speed Valence (K>) Initial concentration Di!usion coe$cient Density Viscosity Applied condition Temperature Constant electrical "eld Removal e$ciency
Symbol ¸ (m) = (m) h (mm) x (mm) ?
u (cm/s) z C (ppm) D (cm/s) o (g/cm) k (kg/m s)
0.01&1 1 1000 (or 2.56;10\ mol/cm) 10\ 1 (as water) 993;10\ (as water at 300 K)
¹ (K) 300 (&703F) d /dx(vol/m) 100 (or 100 kg m/sC)
Physical constant Faraday constant Gas constant
E(%)
50
F (C/mol) R (J/K mol)
96480 8.413
(48)
Eqs. (46)}(48) provide a method to determine the separation constants j and m by only one parameter E(%). To estimate the magnitude of j/B, the parameters in either dialysate or concentrate channel are taken from Table 1. An initial concentration of KCl in aqueous solution, C , is taken as 1000 ppm, as a minor chemical component in the recycling stream of dust rinsing by Venturi scrubber, which is connected to the electrostatic precipitator of recovery boiler in the pulp and paper processing plant. Usually, a single pass through an ED cell removes only 30}60% of metal ions (MacNeil & McCoy, 1989). Using E(%)"50 as an example in the modeling, it follows that j 1 E(%) " ln 1! B ¸ 100
Table 1 Typical values in ED process (used in parameter estimation for the model)
uR¹ "0.018/mm, z D F(d /dx) > >
z v F d z F d
B" > > " > "3.823/mm, D dx R¹ dx > m "0.011/mm B
(49) (50) (51)
Typically, the distance between membrane is in millimetres (Strathmann, 1992). Here, h"5 mm is taken. Hence, the factor assumed in Eqs. (43) and (44) is 2hB"(2)(5 mm)(3.82 mm)"38, satisfying 2hB<1. Also these con"rm that the assumptions 4j/B;1 and 4m/B;1 are valid.
3.2. Concentration proxles To plot concentration pro"les in both dialysate and concentrate channels, the surface "eld strengths at the interfaces between membrane surfaces and solution (d /dx) and (d /dx) or their ratio shall also be K B K A estimated at di!erent location along y-direction. For a speci"ed location y"yH, at x"0, Eqs. (11), (14), (34) and (37) with i " "i " give > B V > A V d
d
j dx dx d
K B exp ! yH 1! A d
d
dx K B 1# dx dx K B
m "exp yH A
d
d
dx dx K A 1# d
d
1! dx dx K A
d
dx
.
K A (52)
Let c "(d /dx) / (d /dx), c "(d /dx) / (d /dx), K B K A then Eq. (52) becomes
c c c 1! "c 1# 1#c 1!c yH (1#E (%)/100) ;exp ln ¸ (1!E(%)/100)
. (53)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
4637
Fig. 5. Plots of c vs. c at various locations yH/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
for E(%)"E (%)"50, at the entrance (yH"0) Eq. (53) shows
c c c 1! "c 1# , 1#c 1!c and at the exit (yH"¸) Eq. (53) gives
Fig. 6. (a) Concentration pro"les at various locations y/¸ (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}); (b) an enlargement of concentration pro"les at y/¸"0.
(54)
c c c 1! "3c 1# . (55) 1#c 1!c When both c and c are small, for example, c "0.01 as a constant, Eqs. (54) and (55) show approximately yH"0, c +c , (56) yH"¸, c +c /3. (57) These indicate that (1) at the entrance, the surface "eld strengths (d /dx) and (d /dx) are the same due to K B K A the starting point at the location of separation; (2) along the channel, (d /dx) '(d /dx) , because of cations K B K A crossing the membrane from low concentration to high concentration; (3) when (d /dx) +constant in the diK B alysate channel, the ratio of c /c decreases from 1 to 1/3, which may be related to decreasing ion concentration along the channel. The plots of c vs. c at some speci"ed locations yH/¸"0.25, 0.50, 0.75 and 1.0 are shown in Fig. 5. In the range 0)c )0.1, the plots of c vs. c show approxim ately linear relations which decrease from 1 to 1/3. Based on these estimations, the relative cation concentration pro"les are plotted in Fig. 6 by taking c "0.01 and related values of c from Fig. 5 at yH/¸"0, 0.25, 0.50, 0.75 and 1.0. The concentration pro"les presented in Fig. 6 indicate the following. (1) At a "xed location in the
channel, the concentrations vary only at the vicinity of interface; as xPh, i.e., in the middle region of cells, the concentrations tend to be constant. This indicates that in the middle region of channel, the #uid with #ow velocity u acts as a reservoir of cations to provide a constant local concentration of charged particles. (2) In the y-axis the concentration in the dialysate channel goes down while that in the concentrate channel goes up, which presents the e!ects of ion removal. Especially, at the entrance (y"0), the concentrations in both dialysate and concentrate channels are the same. (3) The bulk concentrations in both dialysate and concentrate channels are essentially determined by C exp(!jy/A). Eq. (3) could be used to estimate the electrical potential due to the concentration polarization . Therefore, NJ zvFC
"D C.
(58)
With the membrane thickness *l, and v"D/R¹, * " "R¹*C/zFC NJ
(59)
Taking C"(C #C )/2, *C"(C !C ), where A B A B C and C are the concentrations at membrane surfaces A B (x"0) in concentrate side and dilute side. Then, at the exit (y"¸) and with removal e$ciency E of 50%, C "1.50C , C "0.5C . A B
(60)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 7. E!ect of electrical "eld on concentration pro"les at d /dx"10 (}*}), 20 (}䉫}), 50 (}䉭}), 70 (}*}) and 100 V/m (};}).
Fig. 8. Current density in the boundary layer at various locations y/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
Therefore,
"R¹*C/zFC"0.026 V. (61) NJ For the applied external electric "eld"100 V/m, with the channel width of 2h"1 cm,
"1 V. (62) So the assumption < is reasonable. NJ The e!ect of the electrical "eld d /dx on the concentration pro"les mainly comes from parameter B. When d /dx is taken as a numerical variable, Fig. 7 shows a variation of concentration at the entrance (y"0) for the electrical "eld d /dx of 10, 20, 50, 70 and 100 V/m. The response of the concentration pro"le to the variation in the electrical "eld appears to be signi"cant in the boundary layers. The variation of concentration gradient indicates that the current density strongly depends upon the strength of the electrical "eld.
Fig. 9. Limiting current density vs. #ow velocity at various locations y/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
3.3. Current density proxle Fig. 8 shows a current distribution i in the boundary U layer (Eq. (38)). At the membrane surface (x"0), i reaches its maximum for all locations along the #ow U direction. Because i directly depends upon the local U bulk concentration, it has the highest value at the entrance (y"0), and then decreases along the #ow direction. Also, as xPh, the current density decreases exponentially for all locations in the y-axis. From the interface (x"0) to our assumed boundary layer thickness x "0.5 mm, the current density drops by more ? than 85% of its initial value at the interface; and at x"1.0 mm, i is only about 2% of its magnitude at the U
surface. So, the cation transport could be considered as occurring in the boundary layer. The functional behaviours of limiting current density vs. #ow velocity as predicted by Eqs. (40) and (41) are shown in Fig. 9. This model shows the following distinguishable features. (1) At the entrance, for all values of u, the limiting current density is the same. This is because of a uniform concentration in the aqueous solution. (2) When the #ow velocity of #uid is low (u(0.1 cm/s), i increases as u increases. This indicates that the #uid functions as a reservoir of cations, increasing #ow
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 10. Average surface current density in #ow direction vs. removal e$ciency E(%).
velocity, increases the rate of replacement of charged particles in #uid, and then increases the value of limiting current density. (3) In the region 0.1(u(1 cm/s, the increasing of i tends to be slow. (4) When u*1 cm/s, i tends to be saturated. This indicates at larger #ow velocity, the fast replacement of charged particles in #uid reaches a critical point after which the concentration distribution along the #ow direction almost is a constant. At this time, only the variation of initial concentration can change the limiting current density. 3.4. Removal ezciency An average surface current density along the y-axis, i is calculated and plotted as a function of removal U W e$ciency E(%) in Fig. 10. With the same initial concentration C , for a high assumed removal e$ciency E(%), there will be a lower bulk concentration at exit; and a smaller current density. To estimate the trend of operating e$ciency, the increase of current density *i and U W the average removal e$ciency E (%) are de"ned as *i "iJ>!iJ , (63) U W U W U W E (%)"[E (%)#E (%)]/2. (64) J> J The increase of current density *i vs. the average U W removal e$ciency E (%) is plotted in Fig. 11, in which the ratio of *i /E (%) vs. E (%) is also presented. U W The trend *i /E (%) shows that: (1) in the region of U W low E (%), such as 0(E (%)(30, although there is a continuous decreasing of *i /E (%), it has relatively U W large values; (2) in the intermediate region, 40(E (%)(80, there is approximately #at trend of
4639
Fig. 11. Plots of the increase of current *i vs. the average removal U W e$ciency E (%) (}䉫}), and the ratio of *i /E (%) vs. E (%) U W (}*}).
*i /E (%). This indicates in this region the change of U W *i is almost proportional to E (%); and (3) when U W E (%)'80, the ratio of *i /E (%) shows a slight U W increase. These characteristics may provide an optimum in design and operation processes. The whole trend of *i /E (%) indicates that to meet a certain requireU W ment of concentration reduction, it would be more economical to use multiple ED unit, rather than to increase the removal e$ciency directly by increasing the power consumption, the strength of electrical "eld, and the length of ED unit. The plots in Fig. 12 are with various initial concentration C "1000, 750, 500, 375 and 300 ppm. All curves in Fig. 12 have similar curvature providing optimum region. For example, in the pulp and paper process unit, when the content of K> in recovery boiler dust is constant, the initial concentration C will be a function of the ratio of volumetric liquid #ow rate Q (m/min) to gas #ow rate Q (m/min) in the Venturi * % scrubber. Usually, the ratio Q /Q is about 10\ m/m * % (Nevers, 1995; Cooper & Alley, 1994). Small value of Q /Q will provide high initial concentration that is * % good for ED operation, but it may cause corrosion in the Venturi scrubber system. So these considerations should be balanced in design.
4. Conclusions Under the condition of a constant applied electrical potential gradient, the analytical expressions of metal ion concentration and electrical current density have been
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
¹ u u uH x ? z
temperature mass #ow velocity molar #ow velocity e!ective #ow velocity thickness of boundary layer valance
K m/s m/s m/s m
electrical potential the rate of (d /dx) K B and (d /dx) to d /dx K A separation constants in Eq. (10) mobility, D"vR¹
J/C dimensionless
dimensionless
Greek letters
c ,c j, m v
dimensionless mmol/s/J
Subscripts and superscripts Fig. 12. *i /E (%) vs. E (%) at various initial concentration U W C "1000 ppm (}*}), 750 ppm (}䉫}), 500 ppm (}䉭}), 375 ppm (}*}) and 300 ppm (};}).
given by the simpli"cations of the second order partial di!erential equation derived from an ED channel mass balance. The relationships among the ion concentration, electrical current density, location in the channel, electrical "eld, #ow velocity, and removal e$ciency have been simulated. The modeling results show that: (1) the concentration distributions indicate that the separation process mainly occurs at the membrane surfaces; (2) the current density depends upon the concentration gradient and is directly proportional to the strength of electrical "eld represented by the surface "eld; (3) the limiting current density increases with the increase of #ow velocity, but will be saturated after a critical value; and (4) the trend of increase of current density vs. increase of removal e$ciency allows ED process design optimization.
Notation A, B C CH D E(%), E (%) F h i J ¸ m, n r R
parameter in Eq. (15) m\ molar concentration mol/m C /C , dimensiondimensionless U less concentration di!usivity m/s removal or gain e$ciency dimensionless Faraday constant C/mol half-width of channel m electrical current density A/m molar #ux mol/s/m #ow length in channel m integration constants in dimensionless Eqs. (21) and (26) chemical reaction rate mol/s/m gas constant J/K/mol
0 i m, c & m, d $ w
initial condition species i membrane surface at concentrate or dialysate cation or anion in the boundary layer
Appendix A In this paper, is constant, but and may not NJ be constant (but could be constant in steady-state case). Then, "0, O0, and O0. But the op NJ eration condition requires < , and < , and NJ QR also the law of energy conservation indicates E 'E . NJ Therefore,
" # # 0. (A.1) NJ When O0, the problem of polarization decreases the NJ removal e$ciency. When approaches (in magnitude) NJ
, transport process stops. So for electrodialysis pro cess to be functional < . Otherwise, the depolariz NJ ation process (membrane clean up) is necessary. The value of cannot be measured directly, but can NJ be estimated by monitoring the total electrical potential or the removal e$ciency. In Poisson's equation ( "!o /e), "0 repC resents no net charge can be created in the #ow channel (for o ,0). That is, under electric "eld, when one cation C reaches cathode, there must be one anion reaching anode.
References Adachi, H., & Hanada, F. (1994). Production acids and alkalies from pulping black liquors. Canadian Pattern Application, 22 February, 1994, CA 2, 104,53 (Cl.B09B3/00).
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641 Andres, L. J., Riera, F. A., Alvarez, R., & Audinos, R. (1994). Separation of strong acids by electrodialysis with membranes selective to monovalent ions. An approach to modeling the process. Canadian Journal of Chemical Engineering, 72(5), 848}853 (in English). Bersillon, J. L., Anselme, C., Mallevialle, J., Aptel, P., & Fiessinger, F. (1992). Membranes. New York: International Science Publisher. Beszedits, S., & Netzer, A. (1986). Removal of heavy metals from wastewaters. Toronto, Canada: B&L Information Services. Bryant, P. S. (1993). Transition metal measurement and control on closed kraft mills with hydrogen peroxide bleach lines. Ph.D. dissertation, in Chemical Engineering Department, University of Idaho, Idaho, USA. Cooper, C. D., & Alley, F. C. (1994). Air pollution control (p. 231). Prospect Heights, IL: Waveland Press, Inc. Deen, W. M. (1998). Analysis of transport phenomena (pp. 453}463). New York: Oxford University Press. Filatov, B. N., Vishnevskaya, N. S., Nikolaeva, V. P., & Rubtsova, S. V. (1988). Ultra"ltration and electrodialysis of liquor from hot re"ning. Bumazhnaya Promyshlennost, 10, 13}14 (in Russian). Gavach, C., Bribes, J. L., Chapotot, Maillols, J., Pourcelly, G., Sandeaux, J., Sandeaux, R., & Tugas, I. (1994). Improvements of the selectivity of ionic transport through electrodialysis membranes in relation with the performances of separation electromembrane processes. Journal of Physics IV, 4, 233}243 (in English). Grinstead, R. R., & Paalman, H. H. (1989). Metal ion scavenging from water with "ne mesh ion exchangers and microporous membranes. Environmental Progress, 8(1), 139}144. Hupa, M. (1993). Recovery boiler chemistry * the picture becomes sharper. Paperi ja Puu * Paper and Timer, 75(5), 310}319. Janson, C. E., Kenson, R. E., & Tucker, L. H. (1982). Treatment of heavy metals in wastewaters. Environmental Progress, 1(3), 212}216. Kaverzina, T. P., Nepenin, Yu. N., Filatov, B. N., & Vishnevskaya, N. S. (1988). Recovery of chemicals and organic substances from liquors of hot re"ning. Khimiya Drevesing, 6, 59}61 (in Russian). Kaverzina, T. P., Nepenin, Yu. N., Filatov, B. N., & Vishnevskaya, N. S. (1989). Recovery of chemicals and organic substances from liquors of hot re"ning. Otkrytica. Izobret, 17, 103 (in Russian). Kikuchi, K., Gotoh, T., Takashashi, H., Higashino, S., & Drano!, J. S. (1995). Separation of amino acids by electrodialysis with ion-exchange membranes. Journal of Chemical Engineering of Japan, 28(1), 103}109 (in English). Kim, B. M., & Weininger, V. (1982). Electrolytic removal of heavy metals from wastewaters. Environmental Progress, 1(2), 121}125. Koter, S. (1995). In#uence of the layer "xed charge distribution on the performance of an ion-exchange membranes. Journal of Membrane Science, 108(1}2), 177}183 (in English). Kraaijeveld, G., Sumberova, V., Kuindersma, S., & Wesselingh, H. (1995). Modeling electrodialysis using the Maxwell}Stefan description. Chemical Engineering Journal (Lausanne), 57(2), 163}175 (in English). La Grega, M. D., Buckingham, P. L., & Evans, J. C. (1994). Hazardous waste management (pp. 408}409). New York: McGraw-Hill. Larson, K. A., & Wiencek, J. M. (1994). Mercury removal from aqueous streams utilizing microemulson liquid membrances. Environmental Progress, 13(4), 253}262.
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MacNeil, J., & McCoy, D. E. (1989). In H. M. Freeman, Standard handbook of hazardous waste treatment and disposal (pp. 6}103). New York: McGraw-Hill. Mishra, A. K., & Bhattacharya, P. K. (1984). Alkaline black liquor treatment by batch electrodialysis. Canadian Journal of Chemical Engineering, 62(5), 723}727. Nakamura, K., (1993, 1995). Method and devices for metal ion removal in recycling spent solutions, Jpn. Kokai Tokkyo Koho JP 07, 108, 276 [95,108,282] (Cl.C02F1/461), 25 April 1995, Appl. 93/253,446, 08 Oct 1993; 9pp. Nevers, N. de. (1995). Air pollution control engineering (p. 258). New York: McGraw-Hill. Paleologou, M., Cloutier, J. -N., Ramamurthy, P., Berry, R. M., Azarnionch, M. K., & Dorica, J. (1994). Pulp and Paper of Canada, 95(10), 36}40. Popov, A. (1996). Electrodialysis through liquid ion-exchange membrances and the oil-water interface. In A. G. Volkov, D. W. Deamer, Liquid-Liquid Interfaces 1996 (pp. 333}361). (in English). Probstein. R. F. (1989). Physicochemical hydrodynamics (p. 42). Boston: Butterworths. Raghuraman, B., Tirmizi, V., Neena, V., & Wiencek, J. (1994). Emulsion liquid membrances for wastewater treatment: Equilibrium models for some typical metal-extraction systems. Environmental Science and Technology, 28(6), 1090}1098. Rosler, H. -W., Maletzki, F., & Staude, E. (1992). Ion transfer across electrodialysis membrances in the overlimiting current range: chronopotentiometric studies. Journal of Membrane Science, 72, 171}179. Shaposhnik, V. A., Vasil'eva, V. I., & Praslov, D. B. (1995). Concentration "elds of solutions under electrodialysis with ion-exchange membranes. Journal of Membrane Science, 101(1}2), 20}30 (in English). Shapovalov, S. V., & Tyurin, V. I. (1996). Mathematical model for #ow and mass transfer in electromembrane cells with macrovortical #uid #ow. Russian Journal of Electrochemistry, 32(2), 214}219 (in English). Shenassa, R., Reeve, D. W., Dick, P. D., & Costa, M. L. (1995). Chloride and potassium control in closed kraft mill liquor cycles, 1995 International Chemical Recovery Conference, Co-sponsorred by the Technical Section, CPPA and TAPPI, Toronto, Canada, B177. Springer, A. M. (1986). Industrial environmental control * pulp and paper industry (pp. 15}16). New York: Wiley. Sternberg, S. (1987). Membrane utilization in hazardous metal removal from wastewater in the electronic industry. Environmental Progress, 6(3), 139}144. Strathmann, H. (1992). Membrane handbook (pp. 217}262). W. S. Winston Ho, K. K. Sirkar, New York: Van Nostrand Reinhold. Tanaka, Y. (1991). Concentration polarization in ion exchange membrane. Journal of Membrane Science, 57, 217}235. Totura, G. (1996). Innovative uses of specialty ion exchange resins provide new cost-e!ective options for metal removal. Environmental Progress, 15(3), 208}212. Zhu, X., & Elimelech, M. (1995). Fouling of reverse osmosis membranes by aluminium oxide colloids. Journal of Environmental Engineering, 884}892. Zubets, N. N., & Lebedinskaya, G. A. (1981). Use of electrodialysis for the preparation of alkali from anion-exchange "lter regeneration liquors. Teoriya i Praktika Sorbtsionnykh Protsessov, 14, 117}119 (in Russian).
Modeling of electrodialysis of metal ion removal from pulp and paper mill process stream Zhiqiang Yu, Wudneh Admassu* Department of Chemical Engineering, University of Idaho, Moscow, ID 83844-1021, USA Received 10 March 1999; accepted 22 March 2000
Abstract A theoretical model of an electrodialysis process for the removal of metal ions in process stream of the pulp and paper industry has been developed in this article. Based on a constant applied electric potential gradient as a power supply, the model starts from a di!erential equation of mass balance, and presents an analytical solution describing concentration pro"les of cations as electrolytes in electrodialysis channels. The relationships among metal ion concentration, electrical current density, and removal e$ciency are also discussed. 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Electrodialysis; Metal ion; Concentration pro"les; Current density; Removal e$ciency
1. Introduction Metal ions in a pulp and paper mill process stream include sodium (Na), potassium (K), calcium (Ca), magnesium (Mg), manganese (Mn), aluminium (Al), chromium (Cr), copper (Cu), nickel (Ni), titanium (Ti), iron (Fe), mercury (Hg), lead (Pb), silver (Ag) and zinc (Zn). Many of these metals are essential nutrients at low levels but toxic at higher levels. Metal ions in process streams originate from one or more of four sources: wood chips, chemicals used in the pulping process, additives used in paper making, or products of corrosion (Springer, 1986; Bryant, 1993). Under speci"c operating conditions, some metal ions are added to improve the process e$ciency, but have to be removed for other processes. For example, Na> and K> should be removed from the streams in the pulping process before entering into the recovery boiler to prevent fouling and corrosion. In general, the concentrations of metal ions in pulp and paper mill process streams are far below toxicity levels (Springer, 1986). However, in zero discharge process mills metal ions, such as Na> and K>, will accumulate in process stream during recycling, so the removal of metal ions from the recycling streams of pulp and paper
* Corresponding author. Tel.: #1-208-885-8918; fax: #1-208-8887462. E-mail address: [email protected] (W. Admassu).
mill becomes very important. To address the metal ion accumulation problem in zero discharge pulp and paper mills, the e!ective removal of the excess metal ions from the process water is critical. Removal of metal ions from aqueous solutions has traditionally meant precipitation (Janson, Kenson & Tucker, 1982; Beszedits & Netzer, 1986). Often, this technique is now unpopular because it produces a large amount of sludge for disposal (La Grega, Buckingham & Evans, 1994; Kim & Weininger, 1982; Totura, 1996). Another alternative, electrochemical metal recovery, is a feasible process, but process streams of pulp and paper mills are often dilute and therefore the electrochemical recovery has low e$ciency (Larson & Wiencek, 1994). Solvent extraction of metals has been extensively used in hydrometallurgical operations. The major disadvantage of this technique is that the extraction is limited by equilibrium consideration (Raghuraman, Tirmizi, Neena & Wiencek, 1994). Currently, the application of membrane separation processes (MSP) for metal ion removal has been increasing rapidly (Larson & Wiencek, 1994; Sternberg, 1987; Grinstead & Paalman, 1989; Zhu & Elimelech, 1995). The advantages of MSP are the high-quality "ltrate that is suitable for discharge or reuse, little or no chemical addition and its compactness. Electrodialysis (ED) is a membrane process, in which an applied electrical potential di!erence causes ion transport through a membrane. The ED unit normally consists of anion- and cation-selective membranes arranged alternately in parallel between an anode and a cathode.
0009-2509/00/$ - see front matter 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 0 1 - 9
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 1. Schematic diagram of the ED process.
As a consequence of the driving force (the electric "eld), anions in a solution permeate through the anion-transfer membrane and cations permeate through the cationtransfer membranes, resulting in concentration and dilution in adjoining compartments. The principle of the ED process is illustrated in Fig. 1, which shows a schematic diagram of a typical ED cell arrangement consisting of a series of anion- and cation-exchange membranes arranged in an alternating pattern between an anode and a cathode to form individual cells. Considerable research shows that the primary relationships or factors controlling an ED process are dissociation of salts (or materials to be treated) in water, Faraday's law, Ohm's law, and membrane properties (Bersillon, Anselme, Mallevialle, Aptel & Fiessinger, 1992; Gavach et al., 1994; Kikuchi, Gotoh, Takashashi, Higashino & Drano!, 1995; Kraaijeveld, Sumberova, Kuindersma & Wesselingh, 1995; Andres, Riera, Alvarez & Audinos, 1994; Nakamura 1993, 1995; Shaposhnik, Vasil'eva & Praslov, 1995; Popov, 1996; Koter, 1995; Shapovalov & Tyurin, 1996). The application of ED in the paper and pulp industry focuses on the following three aspects: (1) the recovery of valuable products from spent liquors and kraft liquors; (2) the production of chemicals used in the bleaching process, such as caustic soda; and (3) the disposal of e%uent (Zubets & Lebedinskaya, 1981; Mishra & Bhattacharya, 1984; Filatov, Vishnevskaya, Nikolaeva & Rubtsova, 1988; Kaverzina, Nepenin, Filatov & Vishnevskaya, 1988, 1989; Paleologou et al., 1994; Adachi & Hanada, 1994). However, there are limited studies on controlling of metal ions in pulp and paper industry process stream by ED processes. In principle, the concentration pro"les of metal ions and electrical current density are the two dominant topics being investigated (Rosler, Maletzki
& Staude, 1992; Probstein, 1989; Strathmann, 1992; Tanaka, 1991, MacNeil & McCoy, 1989). These have involved concentration polarization, membrane fouling, removal e$ciency and electrical energy consumption. Most researchers have frequently dealt with di!erent types of di!erential equations that need tedious and/or numerical analyses. Traditionally, the ED process uses galvanostatic condition for charge balance in which a constant electrical current is applied to pass through the electrodes as a power supply. As an alternative approach, this article would like to address modeling of the ED process based on a constant applied electric potential gradient as a power supply. Using this method the governing di!erential equation can be simpli"ed and an analytical solution describing concentration pro"les of cations as electrolytes in electrodialysis channels can be obtained. The relationships among metal ion concentration, electrical current density, and removal e$ciency are further discussed. The model also describes the transport of metal ions in boundary layers at membrane surfaces, the limiting current density as a function of #ow velocity, and the trend of increase of current density vs. increase of removal e$ciency by which an optimum ED process can be designed.
2. Theoretical modeling 2.1. Transport equations The mass balance in terms of molar concentration of metal ions in the direction perpendicular to the #ow stream is given by *C G # ) C u "! ) J #r , G G G *t
(1)
where C is the molar concentration of species i, u the G molar velocity, J the total molar #ux, and r the related G G chemical reaction rate. For no chemical reaction and an incompressible #uid, r "0, ) u "0, which gives G *C G #u ) C "! ) J . (2) G G *t The total molar #ux depends upon the e!ects of di!usion, migration and convection in the ED channels. For su$ciently dilute solutions, because the solute species and their gradients do not interact, the #ux contributions from di!usion, migration, and convection can be linearly superposed (Probstein, 1989). When considering an aqueous solution where only cation and anion involved, the total molar #ux are represented by J "!z v FC
!D C #C u. ! ! ! ! ! ! !
(3)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
4631
process requires that the induced potentials as resistance should be much less than the external electrical potential as driving force, the total electric potential gradient
is approximately a constant. By Eqs. (2) and (3) there is *C ! #u ) C "z v F ) (C
)#D C . ! ! ! ! ! ! *t Fig. 2. Electrodialysis with a cation exchange membrane.
Here, C are the concentrations of cation or anion; ! z the valence number; v the mobility related to the ! ! di!usivity D by the relation D "v RT (here v and ! ! ! ! D are taken as constants), F the Faraday constant,
! the electric potential gradient, and u the mass #ow velocity of #uid. The molar average velocity u is replaced by the mass average velocity u, since in su$ciently dilute solutions u "u. Eq. (3) is a well-known Nernst}Planck equation, in which the right-hand side represents the contributions of #ux from migration, di!usion, and convection respectively. In the galvanostatic case the electric potential will be a function of location due to a constant current supply. In this modeling, will include the external applied potential and two induced potentials, i.e., the electrostatic potential and polarization potential, that is
" # # , (4) NJ here is the external electrical potential, is the NJ electrical potential due to the concentration polarization, and is the electrostatic potential due to the concentra tion di!erence of ions. It is important to characterize these contributions before the modeling simpli"cation. For example, in a case of ED with a cation exchange membrane, the potential contributions are shown as in Fig. 2 by the transport of cations. Fig. 2 states the following: (1) the external electrical potential, generates an ex ternal electric "eld E (
'0, E (0) as a driv ing force on cation transport; (2) the electrical potential due to the polarization (C !C )(C and C are the concentrations at the A ? A ? membrane surfaces in concentrated and dilute sides),
produces a polarization "eld E (
(0, NJ NJ NJ E '0); NJ (3) the electrostatic potential due to the concentration gradients,
produces an electrostatic "eld E ( C'0,
'0, E (0); (4) the total induced electrical potential ( plus ) will NJ QR be in the opposite direction of , leading to E !E '0, otherwise, the law of energy conservaNJ tion will be violated. In this paper,
is taken as a constant. Since the ED
(5)
Here, the "rst term on the left-hand side is the rate of change of mass per unit volume, and the second term is the contribution due to convection; on the right-hand side the "rst term is due to the migration e!ect under the electrical "eld, and the second term is due to the di!usive e!ect. Because of the motion of charged species, there will be an electrical current. The current density is de"ned as i"FRzJ. Using Eq. (3) the current density in dilute aqueous solution can be written as i "!F
Rz v C !FRz D C ! ! ! ! ! ! ! #FuRz C . ! !
(6)
2.2. Simplixcations To simplify the theoretical model, one can assume a case in which cations and anions are completely dissociated by the external electrical "eld. For example, potassium chloride (KCl) is a chief form of chemical compound present as K> and Cl\ in the pulp and paper process streams. Both K> and Cl\ are known to accelerate the plugging of the #ue gas passages and increase the rate of the corrosion of the superheat tubes in the recovery boiler system by lowering the melting points of the dust (Shenassa, Reeve, Dick & Costa, 1995; Hupa, 1993). Under the condition of complete dissociation of cation and anion, K> and Cl\, only the transport behavior of cations in aqueous solution is considered. This is because the anion transport is assumed to be symmetrical. In a symmetric multicell ED system, if both cation and anion exchange membranes have the same transport properties, such as the transport numbers, resistance, etc., the concentration distributions of cations and anions in the separation region will be symmetric (except in the cells adjacent to the electrodes). A schematic of the #ows in dialysate and concentrate channels separated by a cation exchange membrane is shown in Fig. 3, where the cation exchange membrane is set at x"0 and its thickness is negligible compared to the channel width. For a quasi-steady state, the time variation of concentration *C /*t"0. Also, the following assumptions are > used: (1) for the whole channel the neutrality of charge is satis"ed, that is, "0 (see the appendix); (2) the di!usion term along the #ow direction D*C /*y is negligible compared to the convection >
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 4. Schematic of boundary layers.
Due to the electroneutrality within the #uid (Deen, 1998) Fig. 3. Schematic of #ows in ED channels.
term u *C /*y, where u is the velocity component W > W in the #ow direction; (3) assume that the in#uence of #ow pro"le of #uid is negligible in our model in both dialysate and concentrate channels. Therefore, a plug #ow of the #uid is assumed. Because of the dilute nature of the solution, the viscosity of the solution is approximately that of water at the given temperature. Under this assumption, the #ow velocity u is now constant in the channels everywhere along the #ow direction, i.e., u"uy( , y( is a unit vector in the y-direction. Therefore, according to schematic of #ows in Fig. 3 it follows that *C u ) C "u > > *y
(7)
*C d
) (C
)" C )
#C " > ) , > > > *x dx
(8)
D C "D > > >
*C *C *C ># > +D >. > *x *x *y
(9)
By inserting Eqs. (7)}(9) into Eq. (5), will result in u
*C d *C *C > "D > #z v F >. > *x > > d *x *y
(10)
z C #z C "0. > > \ \ Using Eq. (11)}(13) will result in Fz C d
dC >" > > . dx R¹ dx
(13)
(14)
Here, C "C , and z "!z are considered for > \ \ > K> and ClU in aqueous solution. 2.3. Concentration distributions By de"ning parameters A"u/D , B"z v F/D ; > > > > d /dx, and a dimensionless reduced concentration CH"C /C , Eq. (10) becomes 5 *CH *CH *CH " #B . (15) A *x *x *y Here, C is the initial concentration of the solution, and C is de"ned as the wall concentration in the boundary U layers near the interface (0)"x")x ). As shown in ? Fig. 4, x is a thickness of the boundary layer between ? either membrane and dialysate or membrane and concentrate (here x ;h, h is the half-width of ED channel). ? This modeling assumes that signi"cant cation transport occurs only in these boundary layers, that is, in the region of 0)"x")x , or 0)"x"/x )1. Since x ;h, when ? ? ? x'x and xPh, it could be taken as xPR. ? The boundary conditions for Eq. (15) are
Eq. (10) shows that the concentration of cations in the dialysate channel is a function of both coordinates x and y. For current density, in this modeling i O0 but > i "0 (because only a cation exchange membrane is \ considered). Since the electrical current of interest here is in the direction of the electric "eld that is perpendicular to the #ow of #uid, the e!ect of convection contribution on current uC in Eq. (6) is negligible. Then Eq. (6) ! becomes
x"0, CH"C (CH" ), Q V xPh(x'x ), CH"C (CH" ), ? F VF y"0, CH"1 for x'x . (16) ? Eq. (15) indicates that in either dialysate or concentrate channel the variations of concentration along the y-axis (#ow direction) and in the x-axis (channel width) are independent. Therefore, the general solution can be written as
dC Fz D C d
>! > > > i "!D Fz . > > > dx R¹ dx
CH(x, y)"f (y)g(x). (11)
dC Fz D C d
\! \ \ \ 0"!D Fz . \ \ dx R¹ dx
(12)
(17)
Now Eq. (15) can be written as
A df 1 dg dg " #B . f dy g dx dx
(18)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
For the dialysate channel, the cation concentration decreases along the #ow direction. Hence, Eq. (18) can be de"ned to be equal to a separation constant, !j, which is related to the boundary conditions. Therefore, df j # f"0, dy A
(19)
dg dg #B #jg"0. dx dx
(20)
Eq. (20) is a second-order homogeneous ordinary di!erential equation. Its characteristic equation (or auxiliary equation) is K#BK#j"0.
(21)
There are two roots for Eq. (21), B K "! +1G(1!4j/B) ,. 2
(22)
Then, a general solution for Eq. (20) is g(x)"N exp(K x)#N exp(K x), (23) with the integration constants N and N . By combining Eq. (23) and a general solution of Eq. (19) the reduced concentration in the dialysate channel will be CH(x, y)"f (y)g(x)"+m exp(K x)
j #m exp(K x),exp ! y . A (24)
4633
Substituting for m , Eq. (26) is
j j h exp ! x B B
CH(x, y)" exp
j #m exp(!Bx) exp ! y . A
(28)
Both Eqs. (26) and (28) are satis"ed with the boundary conditions in Eq. (16). But the condition at x"0, CH"C (CH" ) could not be used conveniently to deQ V termine m because the surface value C (CH" ) cannot Q V be measured experimentally. So the continuity condition of electrical current at x"0 is introduced here to determine m . For a "xed location of y, at the interface between membrane and solution, the expressions of current density i(x) in terms of concentration gradient and potential gradient are given by Eqs. (11) and (14). Here, surface "eld strength (d /dx) is used to replace (d /dx) K B indicating a di!erent potential gradient at the interface between dialysate and membrane to that between concentrate and membrane, i.e., (d /dx) . The wall concenK A tration C is used to replace C emphasizing the mass U > transfer at the boundary layers. Using the continuity of current both in the boundary layer (0(x(x ) and the ? interface between the membrane surface and solution (x"0), Eq. (14) becomes dC U dx
V
z FC d
" > Q R¹ dx
.
(29)
K B
Here, the integration constants m and m need to be determined using the boundary conditions. For the purpose of simpli"cation, by assuming 4j/B;1 (see Section 3), Eq. (22) becomes
Here, the surface concentration C is de"ned as Q C "C " (Eq. (28)). Using Eqs. (28) and (29), Q U V at x"0,
j K "! , B
dC U dx
j K "!B# . B
(25)
The assumption of 4j/B;1 indicates that j/B;B, so that K !B. Then, Eq. (24) is j CH(x, y)"f (y)g(x)"+m exp ! x B
This is only a numerical approximation where only symbol substitutions were made and it does not change the form of Eq. (24) which satis"es Eq. (15) exactly. At y"0, CH"1 for x'x . As x becomes large, ? exp(!Bx)P0 faster than exp(!j/Bx) because of j/B;B. This yields
V
z FC d
> Q R¹ dx
(27)
K B
j h B
j j ! !Bm exp ! y , B A (30)
z F d
" > R¹ dx
exp
K B
j h B
j #m exp ! y . A
j #m exp(!Bx),exp ! y . A (26)
j m "exp h . B
" exp
(31)
Since B"z v F/D (d /dx), then > > > m "
j exp h B !
d
R¹ # dx z F K B > d
d
1# dx dx K B
d
dx
j B
d
dx
.
(32)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Therefore, under similar assumptions 4m/B;1 and m/B;1 there is
And "nally,
C j U "exp ! y C A
j j h exp ! x B B
exp
d
dx
!
d
R¹ j # dx z F B K B > d
d
1# dx dx K B
;exp
j h exp(!Bx) . B
d
dx
(33)
d
d
C dx dx U " 1! K B C d
d
1# dx dx K B
j ;exp(!Bx) exp ! y . A
m #n exp(Bx ) exp y . A
Eq. (33) is a complete form of solution for Eq. (15). Now if we use the approximations that j/B&0, exp((j/B)h)+1 and exp(!(j/B)x)P1, Eq. (33) will be
C m m CH" U " exp ! h exp ! x C B B
(35)
In order to express Eq. (35) in the same form as Eq. (34), the variable x shall be transferred back to its original coordinate, x "!"x". Thus, Eq. (35) becomes
m C m CH" U " exp ! h exp "x" B C B
m #n exp(!B"x") exp y . A
(36)
Using the same approach as Eqs. (29)}(34), the concentration distribution for the concentrate channel is given by
(34)
Eq. (34) represents a wall concentration distribution of cation in the dialysate channel where x+0, For the concentrate channel, there are two conditions to be considered: (1) Physically, (a) the steady-state assumption ensures that there is no retention of cation on membranae surfaces and in the membrane. Mass conservation shows the bulk concentration of cation will increase along the #ow direction while in the dialysate channel it decreases; (b) as shown in Fig. 4, the mass transfer only occurs near interfaces and the concentration should tend to be constant as xP!h. Because of this consideration, a new separation constant m for Eq. (15) should be used for the concentrate channel instead of !j (dialysate channel). The relation between m and j can be determined using conservation of mass. (2) Mathematically, Fig. 3 shows that along the x-axis with reference to the interface (x"0) the concentration distribution in the concentrate channel is an o!set mirror image of that in the dialysate channel. Using a transformation of coordinates, x"!x (x referring to a mirror image system of x while m is used instead of !j for Eq. (18).
d
d
C dx dx U " 1# K A C d
d
1! dx dx K A
;exp(!B"x") exp
m y . A
(37)
Eqs. (34) and (37) present a complete cation concentration distribution in ED channels. These concentration distributions show the following characteristics: (1) When x+0, Eqs. (34) and (37) show the variations of concentration in the boundary layers. When x'x ? and xPh, the concentrations are basically independent of x. This indicates that the separation mainly occurs at the membrane surfaces. (2) As functions of y, the concentration in the dialysate channel decreases along the y-direction while it increases in the concentrate channel. (3) In general, because of concentration polarization, the "eld strengths at the membrane surfaces (d /dx) K B and (d /dx) are di!erent. These parameters should be K A determined experimentally, but their ratio can be estimated by the continuity condition of electrical current at speci"ed locations.
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
2.4. Current density
C "C " , the result is Q U V
Using Eqs. (11), (14) and (34), the current density can be represented as
i > C !C "! x U Q ? 2D Fz > >
d
d
dx dx K B i "!2D Fz C B U B > > d
d
1# dx dx K B
j ;exp(!Bx) exp ! y A
2D Fz i " > > C " . U V x ?
(40)
Eq. (40) has a form of
d
dx 2D Fz C > K B "! > R¹ d
d
1# dx dx K B j ;exp(!Bx) exp ! y . A
(39)
Here, i is in the opposite direction of x. So, > C !C '0, which satis"es the case of concentration in U Q the dialysate cell. When C P0, i "i . If only the Q > magnitude of i is considered, then
4635
i JC exp
(38)
Eq. (38) presents the electrical current density in the boundary layer for the dialysate channel. It indicates the following properties for the current density. (1) At the entrance (y"0), There is a maximum value of current density in the y-direction towards the exit (y"¸), the current density decreases continuously as a function of exp(!(j/A)y). (2) At the membrane surface (x"0), the current density also reaches a maximum value in the x-direction. In the boundary layer, the current density decreases as xPx . Beyond the boundary layer, as ? xPh, iP0. This strongly shows that the current density depends upon the concentration gradient. (3) The current density is directly proportional to the strength of electrical "eld represented by the surface "eld (d /dx) that K B shows a direct relationship between the ion transport and the electrical "eld. (4) A direct proportional relation between the current density and the initial concentration shows that the strength of ion transport depends upon the initial condition. (5) The current density is an indirect function of #uid velocity, which is included in the parameter de"ned in Eq. (15). The limiting current density i is another important physical quantity in ED process, which is de"ned as the current density at the membrane surface when the surface concentration tends to zero in the dialysate channel. So, i is the maximum current that may pass through a given membrane area without creating adverse e!ects, i.e., higher electrical resistance or lower current utilization. The limiting current density determines the minimum membrane area required achieving a certain removal e!ect. In this modelling, by combining Eqs. (11) and (14) and integrating both sides with the boundary condition
!uH , u
(41)
with an e!ective #ow velocity uH"D jy. For a large > value of #ow velocity u, the limiting current density, i , becomes only sensitive to the initial concentration C , because of the large #ow velocity could provide almost a uniform ion concentration everywhere in the dialysate channel. Eq. (41) presents that the limiting current density, i , is a function of both initial concentration C and #ow velocity u. This is consistent with the expres sions of limiting current density by Strathmann (1992) and Tanaka (1991). In the latter, the convection #ow velocity is perpendicular to the membrane surface, which may be limited by the permeability of the membrane.
3. Results and discussion 3.1. Parameter estimation One way to estimate the parameter j is by introducing the expected removal e$ciency E(%). In the dialysate channel the removal e$ciency E(%) is de"ned as C !C E(%)" ;100%, C
(42)
C is the average bulk concentration at any location in the y-axis, which is de"ned as
FC dx 1 F C 5 " C dx. 5 Fdx 2h
(43)
Here, h is the half-width of the dialysate channel shown in Fig. 2. When 2hB<1, C is approximately equal to C exp(!jy/A). Therefore, for whole channel
C !C j * ;100%" 1!exp ! ¸ E(%)" C A ;100%,
(44)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
which shows
A E(%) j"! ln 1! . ¸ 100
(45)
For 0(E(%)(100, Eq. (46) con"rms that j'0 is satis"ed. For the concentrate channel, the bulk concentration increases along the #ow direction. To compare with the dialysate channel, the separation constant m is related to the gain e$ciency E (%), which can be de"ned as
C !C m ;100%" exp ¸ !1 E (%)" * C A ;100%,
(46)
and then
E (%) A . m" ln 1# 100 ¸
(47)
When there is no retention of cations at the membrane surfaces and in the membrane, the amount of cations removed from the dialysate channel should equal to that gained in the concentrate channel. This indicates that numerically E (%)"E(%).
Geometric sizes Flow distance Flow width Half-channel width Average "lm thickness Number of channel pair Characteristics of -uid Average #ow speed Valence (K>) Initial concentration Di!usion coe$cient Density Viscosity Applied condition Temperature Constant electrical "eld Removal e$ciency
Symbol ¸ (m) = (m) h (mm) x (mm) ?
u (cm/s) z C (ppm) D (cm/s) o (g/cm) k (kg/m s)
0.01&1 1 1000 (or 2.56;10\ mol/cm) 10\ 1 (as water) 993;10\ (as water at 300 K)
¹ (K) 300 (&703F) d /dx(vol/m) 100 (or 100 kg m/sC)
Physical constant Faraday constant Gas constant
E(%)
50
F (C/mol) R (J/K mol)
96480 8.413
(48)
Eqs. (46)}(48) provide a method to determine the separation constants j and m by only one parameter E(%). To estimate the magnitude of j/B, the parameters in either dialysate or concentrate channel are taken from Table 1. An initial concentration of KCl in aqueous solution, C , is taken as 1000 ppm, as a minor chemical component in the recycling stream of dust rinsing by Venturi scrubber, which is connected to the electrostatic precipitator of recovery boiler in the pulp and paper processing plant. Usually, a single pass through an ED cell removes only 30}60% of metal ions (MacNeil & McCoy, 1989). Using E(%)"50 as an example in the modeling, it follows that j 1 E(%) " ln 1! B ¸ 100
Table 1 Typical values in ED process (used in parameter estimation for the model)
uR¹ "0.018/mm, z D F(d /dx) > >
z v F d z F d
B" > > " > "3.823/mm, D dx R¹ dx > m "0.011/mm B
(49) (50) (51)
Typically, the distance between membrane is in millimetres (Strathmann, 1992). Here, h"5 mm is taken. Hence, the factor assumed in Eqs. (43) and (44) is 2hB"(2)(5 mm)(3.82 mm)"38, satisfying 2hB<1. Also these con"rm that the assumptions 4j/B;1 and 4m/B;1 are valid.
3.2. Concentration proxles To plot concentration pro"les in both dialysate and concentrate channels, the surface "eld strengths at the interfaces between membrane surfaces and solution (d /dx) and (d /dx) or their ratio shall also be K B K A estimated at di!erent location along y-direction. For a speci"ed location y"yH, at x"0, Eqs. (11), (14), (34) and (37) with i " "i " give > B V > A V d
d
j dx dx d
K B exp ! yH 1! A d
d
dx K B 1# dx dx K B
m "exp yH A
d
d
dx dx K A 1# d
d
1! dx dx K A
d
dx
.
K A (52)
Let c "(d /dx) / (d /dx), c "(d /dx) / (d /dx), K B K A then Eq. (52) becomes
c c c 1! "c 1# 1#c 1!c yH (1#E (%)/100) ;exp ln ¸ (1!E(%)/100)
. (53)
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
4637
Fig. 5. Plots of c vs. c at various locations yH/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
for E(%)"E (%)"50, at the entrance (yH"0) Eq. (53) shows
c c c 1! "c 1# , 1#c 1!c and at the exit (yH"¸) Eq. (53) gives
Fig. 6. (a) Concentration pro"les at various locations y/¸ (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}); (b) an enlargement of concentration pro"les at y/¸"0.
(54)
c c c 1! "3c 1# . (55) 1#c 1!c When both c and c are small, for example, c "0.01 as a constant, Eqs. (54) and (55) show approximately yH"0, c +c , (56) yH"¸, c +c /3. (57) These indicate that (1) at the entrance, the surface "eld strengths (d /dx) and (d /dx) are the same due to K B K A the starting point at the location of separation; (2) along the channel, (d /dx) '(d /dx) , because of cations K B K A crossing the membrane from low concentration to high concentration; (3) when (d /dx) +constant in the diK B alysate channel, the ratio of c /c decreases from 1 to 1/3, which may be related to decreasing ion concentration along the channel. The plots of c vs. c at some speci"ed locations yH/¸"0.25, 0.50, 0.75 and 1.0 are shown in Fig. 5. In the range 0)c )0.1, the plots of c vs. c show approxim ately linear relations which decrease from 1 to 1/3. Based on these estimations, the relative cation concentration pro"les are plotted in Fig. 6 by taking c "0.01 and related values of c from Fig. 5 at yH/¸"0, 0.25, 0.50, 0.75 and 1.0. The concentration pro"les presented in Fig. 6 indicate the following. (1) At a "xed location in the
channel, the concentrations vary only at the vicinity of interface; as xPh, i.e., in the middle region of cells, the concentrations tend to be constant. This indicates that in the middle region of channel, the #uid with #ow velocity u acts as a reservoir of cations to provide a constant local concentration of charged particles. (2) In the y-axis the concentration in the dialysate channel goes down while that in the concentrate channel goes up, which presents the e!ects of ion removal. Especially, at the entrance (y"0), the concentrations in both dialysate and concentrate channels are the same. (3) The bulk concentrations in both dialysate and concentrate channels are essentially determined by C exp(!jy/A). Eq. (3) could be used to estimate the electrical potential due to the concentration polarization . Therefore, NJ zvFC
"D C.
(58)
With the membrane thickness *l, and v"D/R¹, * " "R¹*C/zFC NJ
(59)
Taking C"(C #C )/2, *C"(C !C ), where A B A B C and C are the concentrations at membrane surfaces A B (x"0) in concentrate side and dilute side. Then, at the exit (y"¸) and with removal e$ciency E of 50%, C "1.50C , C "0.5C . A B
(60)
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 7. E!ect of electrical "eld on concentration pro"les at d /dx"10 (}*}), 20 (}䉫}), 50 (}䉭}), 70 (}*}) and 100 V/m (};}).
Fig. 8. Current density in the boundary layer at various locations y/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
Therefore,
"R¹*C/zFC"0.026 V. (61) NJ For the applied external electric "eld"100 V/m, with the channel width of 2h"1 cm,
"1 V. (62) So the assumption < is reasonable. NJ The e!ect of the electrical "eld d /dx on the concentration pro"les mainly comes from parameter B. When d /dx is taken as a numerical variable, Fig. 7 shows a variation of concentration at the entrance (y"0) for the electrical "eld d /dx of 10, 20, 50, 70 and 100 V/m. The response of the concentration pro"le to the variation in the electrical "eld appears to be signi"cant in the boundary layers. The variation of concentration gradient indicates that the current density strongly depends upon the strength of the electrical "eld.
Fig. 9. Limiting current density vs. #ow velocity at various locations y/¸"0 (}*}), 0.25 (}䉫}), 0.50 (}䉭}), 0.75 (}*}) and 1.00 (};}).
3.3. Current density proxle Fig. 8 shows a current distribution i in the boundary U layer (Eq. (38)). At the membrane surface (x"0), i reaches its maximum for all locations along the #ow U direction. Because i directly depends upon the local U bulk concentration, it has the highest value at the entrance (y"0), and then decreases along the #ow direction. Also, as xPh, the current density decreases exponentially for all locations in the y-axis. From the interface (x"0) to our assumed boundary layer thickness x "0.5 mm, the current density drops by more ? than 85% of its initial value at the interface; and at x"1.0 mm, i is only about 2% of its magnitude at the U
surface. So, the cation transport could be considered as occurring in the boundary layer. The functional behaviours of limiting current density vs. #ow velocity as predicted by Eqs. (40) and (41) are shown in Fig. 9. This model shows the following distinguishable features. (1) At the entrance, for all values of u, the limiting current density is the same. This is because of a uniform concentration in the aqueous solution. (2) When the #ow velocity of #uid is low (u(0.1 cm/s), i increases as u increases. This indicates that the #uid functions as a reservoir of cations, increasing #ow
Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
Fig. 10. Average surface current density in #ow direction vs. removal e$ciency E(%).
velocity, increases the rate of replacement of charged particles in #uid, and then increases the value of limiting current density. (3) In the region 0.1(u(1 cm/s, the increasing of i tends to be slow. (4) When u*1 cm/s, i tends to be saturated. This indicates at larger #ow velocity, the fast replacement of charged particles in #uid reaches a critical point after which the concentration distribution along the #ow direction almost is a constant. At this time, only the variation of initial concentration can change the limiting current density. 3.4. Removal ezciency An average surface current density along the y-axis, i is calculated and plotted as a function of removal U W e$ciency E(%) in Fig. 10. With the same initial concentration C , for a high assumed removal e$ciency E(%), there will be a lower bulk concentration at exit; and a smaller current density. To estimate the trend of operating e$ciency, the increase of current density *i and U W the average removal e$ciency E (%) are de"ned as *i "iJ>!iJ , (63) U W U W U W E (%)"[E (%)#E (%)]/2. (64) J> J The increase of current density *i vs. the average U W removal e$ciency E (%) is plotted in Fig. 11, in which the ratio of *i /E (%) vs. E (%) is also presented. U W The trend *i /E (%) shows that: (1) in the region of U W low E (%), such as 0(E (%)(30, although there is a continuous decreasing of *i /E (%), it has relatively U W large values; (2) in the intermediate region, 40(E (%)(80, there is approximately #at trend of
4639
Fig. 11. Plots of the increase of current *i vs. the average removal U W e$ciency E (%) (}䉫}), and the ratio of *i /E (%) vs. E (%) U W (}*}).
*i /E (%). This indicates in this region the change of U W *i is almost proportional to E (%); and (3) when U W E (%)'80, the ratio of *i /E (%) shows a slight U W increase. These characteristics may provide an optimum in design and operation processes. The whole trend of *i /E (%) indicates that to meet a certain requireU W ment of concentration reduction, it would be more economical to use multiple ED unit, rather than to increase the removal e$ciency directly by increasing the power consumption, the strength of electrical "eld, and the length of ED unit. The plots in Fig. 12 are with various initial concentration C "1000, 750, 500, 375 and 300 ppm. All curves in Fig. 12 have similar curvature providing optimum region. For example, in the pulp and paper process unit, when the content of K> in recovery boiler dust is constant, the initial concentration C will be a function of the ratio of volumetric liquid #ow rate Q (m/min) to gas #ow rate Q (m/min) in the Venturi * % scrubber. Usually, the ratio Q /Q is about 10\ m/m * % (Nevers, 1995; Cooper & Alley, 1994). Small value of Q /Q will provide high initial concentration that is * % good for ED operation, but it may cause corrosion in the Venturi scrubber system. So these considerations should be balanced in design.
4. Conclusions Under the condition of a constant applied electrical potential gradient, the analytical expressions of metal ion concentration and electrical current density have been
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Z. Yu, W. Admassu / Chemical Engineering Science 55 (2000) 4629}4641
¹ u u uH x ? z
temperature mass #ow velocity molar #ow velocity e!ective #ow velocity thickness of boundary layer valance
K m/s m/s m/s m
electrical potential the rate of (d /dx) K B and (d /dx) to d /dx K A separation constants in Eq. (10) mobility, D"vR¹
J/C dimensionless
dimensionless
Greek letters
c ,c j, m v
dimensionless mmol/s/J
Subscripts and superscripts Fig. 12. *i /E (%) vs. E (%) at various initial concentration U W C "1000 ppm (}*}), 750 ppm (}䉫}), 500 ppm (}䉭}), 375 ppm (}*}) and 300 ppm (};}).
given by the simpli"cations of the second order partial di!erential equation derived from an ED channel mass balance. The relationships among the ion concentration, electrical current density, location in the channel, electrical "eld, #ow velocity, and removal e$ciency have been simulated. The modeling results show that: (1) the concentration distributions indicate that the separation process mainly occurs at the membrane surfaces; (2) the current density depends upon the concentration gradient and is directly proportional to the strength of electrical "eld represented by the surface "eld; (3) the limiting current density increases with the increase of #ow velocity, but will be saturated after a critical value; and (4) the trend of increase of current density vs. increase of removal e$ciency allows ED process design optimization.
Notation A, B C CH D E(%), E (%) F h i J ¸ m, n r R
parameter in Eq. (15) m\ molar concentration mol/m C /C , dimensiondimensionless U less concentration di!usivity m/s removal or gain e$ciency dimensionless Faraday constant C/mol half-width of channel m electrical current density A/m molar #ux mol/s/m #ow length in channel m integration constants in dimensionless Eqs. (21) and (26) chemical reaction rate mol/s/m gas constant J/K/mol
0 i m, c & m, d $ w
initial condition species i membrane surface at concentrate or dialysate cation or anion in the boundary layer
Appendix A In this paper, is constant, but and may not NJ be constant (but could be constant in steady-state case). Then, "0, O0, and O0. But the op NJ eration condition requires < , and < , and NJ QR also the law of energy conservation indicates E 'E . NJ Therefore,
" # # 0. (A.1) NJ When O0, the problem of polarization decreases the NJ removal e$ciency. When approaches (in magnitude) NJ
, transport process stops. So for electrodialysis pro cess to be functional < . Otherwise, the depolariz NJ ation process (membrane clean up) is necessary. The value of cannot be measured directly, but can NJ be estimated by monitoring the total electrical potential or the removal e$ciency. In Poisson's equation ( "!o /e), "0 repC resents no net charge can be created in the #ow channel (for o ,0). That is, under electric "eld, when one cation C reaches cathode, there must be one anion reaching anode.
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