886
I n d . E n g . C h e m . Res. 1995,34, 886-894
Effects of Axial and Orthogonal Applied Electric Fields on Solute Transport in Poiseuille Flows. An Area Averaging Approach S. G. Sauer, B. R. Locke,* and P.Arce* Department of Chemical Engineering, Florida Agricultural and Mechanical University and Florida State University College of Engineering, Tallahassee, Florida 3231 6-2175
The effects of axial and normal applied electrical fields on the transport of solutes in Poiseuille flows is investigated using the area average approach to the microscopic conservation equations. The averaged solute species continuity equation features an effective dispersion coefficient and a n effective velocity that arise due to the applied electric fields, and that are functions of the electric field parameters. The three specific cases considered in this paper include (1)a single field applied parallel to the fluid flow, (2) a single field applied perpendicular to the flow, and (3) one field applied parallel to the flow and a n additional field applied perpendicular to the flow. This analysis can be used to describe the transport of dilute noninteracting solutes in such separations as capillary electrophoresis with secondary fields and field flow fractionation with coaxial and perpendicular electric fields, and in electrostatic aerosol classifiers.
Introduction The application of two orthogonal electric fields (i.e., one field in the direction of the flow and the other perpendicular to it) to enhance electrophoretic separations is of current interest for such processes as capillary electrophoresis (Lee et al., 1991)and gel electrophoresis (Chu et al., 1989). An orthogonal field superimposed on laminar flow in a rectangular channel may also be of interest for enhancing separations in field flow fractionation (Giddings, 1993). Lee et al. (1991) have developed a method to directly control the electroosmotic flow in capillary electrophoresis by using an external electric field to induce a radial potential gradient. In addition, orthogonal fields in cylindrical annular channels have been used in aerosol studies for particle sizing (Wang and Flagan, 1990). Most of the previous work devoted to modeling electrokinetic phenomena in laminar flow systems has primarily considered the effects of an electric field applied parallel to the flow axis (Overbeek, 1950; Overbeek and Wiersema, 1967; Dukhin and Derjaguin, 1974; Hunter, 1981, 1989; Hiemenz, 1986). Melcher (Melcher, 1981; Eringen and Maguin, 1990; Woodson and Melcher, 1968) considered the steady flow of a positively charged gas ion between two charged screens placed in a long cylinder with insulated walls in both plug flow and laminar flow. Levine et al. (1975) considered electrokinetic flow in a narrow parallel-plate channel and in a cylindrical pore of aqueous solutions with a uniform electric field applied parallel to the direction of fluid flow. Gajdos and Brenner (1978) analyzed field flow fractionation with an orthogonal field (primarily gravitational) of large nonspherical particles. Cwirko and Carbonell (1989) used the method of spatial averaging for analyzing electrolyte transport across charged cylindrical pores. The pore surface was assumed to have a constant charge per unit area and the ion transport was considered steady. Analytical expressions for the radial potential profile were developed and were shown to be valid when the radial deviations of the electric potential from the areaaveraged potential are small. In the present study, the method of spatial averaging (Slattery, 1981;Whitaker, 1985; Zanotti and Carbonell,
* To whom correspondence should be addressed.
@( y=B)=K, /-
Figure 1. Schematic of process.
1984a-c) is used to determine the effect of electric fields on solute transport where the fields are applied in the direction of the flow (i.e., axial direction),in the direction perpendicular to the flow (i.e., normal direction), or in both directions. This study focuses on a rectangular channel as shown in Figure 1; however, the methodology can be readily extended to cylindrical geometry. The fluid in the channel is assumed to be Newtonian, incompressible, and under steady state conditions. The flow is laminar, and in addition, it is assumed that the velocity and solute concentration fields have no effect on the electric field. This allows a sequential coupling between the electrical field equation and the molar species continuity equation. Furthermore, the electrical field equation can be averaged and its solution derived independently of the other balance equations in the model. Upon substitution of the average potential and the average velocity into the molar species continuity equation, effectivefield dependent parameters including the dispersion coefficient,the velocity, and a source term arise. The resulting averaged species continuity equation is then solved by the method of moments to show the effects of electric fields on the behavior of pulses of solutes injected in the system.
Model Formulation The molar species continuity equation of a nonreacting macromolecular component in the dilute solution limit, with the molar flux given by the Nernst-Planck
0888-5885/95/2634-0886$09.00/0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 887 constitutive theory (Newman, 1967,i 1973; Slattery, 1981) is
+
+
ac - PVC = VDVC VWCV@ at
(1)
where c is the concentration of the macromolecular species, D is the dilute solution diffusion coefficient of the macromolecular species, v is the mass average velocity of the fluid, @ is the electric potential, and u is the electrophoretic mobility of the macromolecular species given by
where z is the charge on the macromolecular species, ;R is the gas constant, T i s the absolute temperature, and 9%Faraday's constant. It is implicity assumed that this species does n9t contribute significantly to the electrical field. Under the assumptions previously stated, the velocity field can be obtained upon solution of the Navier-Stokes equation
Dv Dt
Q -= -VP
+ u v 2 v - Qf
which includes the electric body force term f, and the electric potential can be determined via the Poisson equation
applied electrical fields which may lead to new insights for the design and operation of electrophoretic separators. The Navier-Stokes equation with only viscous and pressure terms, and with the no-slip boundary conditions a t the walls of a rectangular channel, leads to the usual Haggen-Poiseuille flow solution given by (Whitaker, 1981)
(4) where L is the channel length, p is the viscosity of the fluid, B is the channel width, and AP is the pressure drop in the channel. The fact that only one component of the velocity vector, vr, is nonzero comes from the kinematic assumption of fully developed flow in the axial direction of the channel. Equation 1 for solute transport in a rectangular channel as shown in Figure 1 reduces to
The walls of the channel are assumed to be impermeable to solute transport, and thus the no-flux boundary conditions are given by
Dacl
ay y=o
(3) where 6 is the dielectric permittivity of the fluid and zi and ci represent the charge and concentration of the small current carrying species. Poisson's equation reduces t o Laplace's equation when it is assumed that the solute of interest does not affect the potential profile, i.e., in the dilute solution limit when the solute has a mobility much smaller than the current carrying electrolytes, when the current is not limited by mass transfer of reactants to the electrodes, when the transport medium has a constant electrical conductivity, and when Joulean heating is neglected (Newman, 1973; Henry, 1931; Dukhin and Derjaguin, 1974; Smith and White, 1993). In the general case, eqs 1-3 are coupled and must be solved simultaneously for all charged species. However, the present study shall consider the simplified case where the body forces can be neglected in the equation of motion (i.e., eq 21, where the species of interest does not contribute to the electrical field and where the Laplace equation can be used for the electric field. These simplifying assumptions allow for the solution of the velocity and electric potential fields independently of the solute transport given by eq 1. This is a key aspect for the methodology used in this paper and is one of the differences between this work and that of Cwirko and Carbonell (1989) where the Poisson equation was included for a case with charged walls. The work of Carbonell and Cwirko considered transport in very small charged cylindrical pores where electroneutrality could not be assumed. The present study considers the case where the electrical double layers are very thin and this approximation can be made. Another important difference between this work and that of Cwirko and Carbonell (1989) is the consideration of two
+ ' Oacl y=B
?/
+uc(B)-
y =E
= 0 (6)
where both diffusive and electroconvectivefluxes must be considered. At the initial state no solute is assumed to be flowing in the system, i.e.
t = 0,
c ( x y )= 0
(7)
Yx,y
The Laplace equation for the electric potential in a two-dimensional system such as the one in Figure 1 is given by
a2@ -a2@ +
n
&'
n
ay"
A constant or zero applied electric field at the wall will lead to the Dirichlet boundary conditions given by
@(x=O) = 0, @(x=L)= Kl
@a)
@(y=O) = 0, @(y=B)= K2
6%)
where K1 and K2 represent the applied potentials in the x and y directions, respectively.
The major objective of the present study is to consider the effects of a secondary applied field on the solute transport under the influence of a primary field colinear with the fluid flow. Two limiting cases will first, however, be considered as a preliminary analysis of the system. These include the application of a single field parallel to the flow, i.e., K2 = 0 in eq 9b, and the application of a single field perpendicular to the flow, i.e., Kl = 0 in eq 9a. Both of these cases of course give rise to only one-dimensionalelectric fields and represent simplified limiting cases. The remaining case will consider the two-dimensionalLaplace equation with one
888 Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995
field applied parallel to the flow, i.e., eq 9a an orthogonal field through eq 9b.
Derivation of the Average Differential Model for Effective Solute Transport The use of effective (i.e., averaged or macro) transport equations is useful for investigating the behavior of a given physical system from both basic and design points of view. These averaged equations feature effective diffusive and convective transport coefficients that are direct functions of important parameters such as the molecular diffusivity of the solute, the convective (hydrodynamic) velocity of the fluid, and electrical parameters. The study of these coefficients allows for a-priori characterization of the behavior of the system, and thus it gives useful insights into the design strategy of, for example, an electrophoretic separation process. In this section of the paper, the spatial averaging approach (Slattery, 1981;Whitaker, 1985)will be applied to derive average differential models whose effective transport coefficients will be studied in detail. The methodology requires a closure scheme (Whitaker, 1985) that is derived following the general procedure by Gray (1975). In this procedure, physical and geometrical assumptions are made in order to neglect second order terms of the deviation field variables. In summary, area averaging will be defined and then applied to the electrical field equation, hydrodynamic velocity profile and the molar species continuity equation. Area Averages. The area average of a general spatial function h(xy,z)can be defined as
As shown in eq 10, the net effect of applying an area average to a function such as h(xy,z) is to reduce the number of independent geometrical variables. Application of this average to the hydrodynamic velocity profile given by eq 4 gives
@(XY)= (@)
+ 6,(XJy>
(13)
To obtain the deviation field, 6(xy), the definition of the deviation, eq 13,is inserted in the Laplace equation, eq 8, and the averaged expression, eq 12, is subtracted from the result to give
Following the approach of Carbonell and Whitaker (Paine et al., 1983; Zanotti and Carbonell, 1984a-c; Whitaker, 1986a,b),the long channel approximationwill allow neglecting the second term of the left hand side of eq 14. This assumption will be used to determine the deviation of the electric potential and the average potential for the three cases under consideration. Case I: For the case of a one-dimensionalelectric field applied in the direction parallel to the flow the Laplace equation, eq 8, can be solved to give
Case 2: For the case of a one-dimensional electrical field applied perpendicularly to the flow field, the solution to eq 8 leads to
Case 3: When two fields are applied simultaneously as given in eqs 9a and 9b, it will be necessary to solve eq 14 t o determine the deviation field. Using a long channel approximation, it is reasonable to neglect the axial variation of the deviation of the potential with respect to the variation orthogonal to the flow axis to give
a% ay2
a&lB
=O
(17)
Bay0
Using eq 13, the boundary conditions, eq 9b, reduce to
(11) which is independent of the axial coordinate x . Area Average Potential Fields. In order to determine the average electric potential, eq 10 can be applied to the Laplace equation, eq 8, t o give
9
+--
=o
(12)
The value of the gradient of the potential at the walls of the channel (y = 0, y = B ) is not known for the case of a known applied potential a t those walls (i.e., Dirichlet boundary conditions, eq 9b). In order to relate the average values with the point values of the electrical potential for the cases under analysis, it will be necessary to develop a closure scheme which invokes the use of a deviation variable of the field that can be determined using geometrical or physical assumptions. This deviation variable may be related to the average of the and to its point values, W x y ) as proposed potential, (a), by
(@)
+ 6(0) = 0
(a)+ 6 ( B )= K2
(18)
The solution of eq 17 for 6 will allow for the determination of 6 in terms of the-normal variable y and the average potential ((a). @, as will be shown in a subsequent section, is required in order to determine the effective diffusion coefficient and the effective velocity. In order to solve eq 17 subject to boundary conditions eq 18, it will be assumed, following Paine et al. (1993),that all the axial variation of @ can be accounted for through a linear dependence on ((a) and all the orthogonal variation can be accounted for through two unknown functions f i )andg(y). A solution of the form
6 = fer) + g(Y>(a)
(19)
can therefore be proposed. In order to obtain closure, the f and g fields must be obtained. Substitution of eq 19 into eqs 17 and 18 and combining like terms with ((a) leads to f”
-
2’1;
=0
fC0) = 0
fCB) = K2
(20)
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 889
-%I:=
g”
0
1 +g(O)=O
1 +g(B)= 0 (21)
with the constraints @ = 0 and (g) = 0. The solutions of eqs 20 and 21 lead to
0.8
0.6 1
IGsertion of this result into eq 12, noting that a@/ay = a@/8y, yields the differential equation for the axial variation of the average potential
0.4
II/
-
equation (24) Oequation (25)
i
0.2 ,
(23) The solution of this equation with boundary conditions, eq 9a, is
b 0.058 0.050
~
{;Ai [UC, sinh(A$)
“ 4 =
t
J
P
Equation 24 is the average potential derived by using an area averaging approach that involves a closure scheme where assumptions were made in order to eliminate certain terms. To test the validity of these assumptions in the case of the specific problem under analysis here, the exact solution to the Laplace equation (eq 8) can be determined and the area average defined in eq 11can be applied; the result can then be compared with the average potential given by eq 24. This procedure also provides further understanding of the behavior of the average potential with respect to the axial coordinate (0 < x -= L). The solution to the Laplace equation (eq 8) may be obtained by following standard procedures (see, for example, Carlaw and Jaeger (1981)) involving separation of variables. The application of the averaging operation given by eq 11leads to the following result. ((a)
1
Oo4_: 0.033
$ L
’ -0 equation (24) equation (25) 0.2
0.8
0.6
1.o
x/L
0.060
0.050 t
I
$
1
0 0.030 .040:i
‘I
0.020 L
i
0.010
-
0.4
I
i=I
‘i
oto
0.000
-
equation (24) Oequation (25)
I J
0.2
0.4
0.6
0.8
1.o
XlL
sinh(&)]]
/sinh($)}
(25)
where Ai = n(2i - 1). Figure 2 shows the comparison of the average potential given by eq 24 and the average potential given by the exact solution of the Laplace equation given by eq 25. It should be noted that for the cases considered here, i.e., with the long column approximation, the results are very similar. Furthermore, the average potential is approximately constant throughout the x domain (0 x < L)except very close to the end points at x = 0 and x = L. This will allow for the assumption of a constant averaged electrical potential, (a), in the deviation of the potential function given by eq 22.
Figure 2. Comparison of average potential with average of exact solution: case 3. (a) K1 = K2 = 1,LIB = 10;(b) KI= 0.01,KZ= 0.1,L/B = 10; (c) K1 = 0.05, K2 = 0.10,LIB = 10.
Area Average of Concentration. Applying the definition of the average to the molar species continuity equation (eq 5) results in the following expression
In order to resolve the terms containing the concentration, velocity, and electrical potential deviations, a closure scheme analogous to that used for the applied
890 Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995
potential must be developed. The application of the approach suggested by Gray (1975) for the case of concentration variables starts with c ( x y ) = (e)
+ E(xy)
shown in Figure 2 eq 26 reduces to at
(27)
The equation for the concentration deviation is obtained by subtracting the averaged equation (eq 26) from the microscopic differential equation of transport (eq 5 ) with the definition of the deviation (eq 27) inserted t o give
+
+
-=De,a 2 w - v,, - u - (e)a(@.>
ax2
ax
(;
ax
and (37)
The quasi-steady-state approximation for the deviation field and the long channel approximation are two assumptions that were made previously, and they will be used again here. In addition, it will be assumed that the average concentration is much larger than the deviation of the concentration to give
+ u((c)+ E ) ?Y
=B-
ax
(29)
where the velocity deviation is given by - (u)
= -6(~)(($)~ -
k)+ i)
(30)
The gradient of the deviation of the potential that appears in eq 29 can be calculated from the expressions derived previously for the three cases. Equation 29 can be readily solved in a manner analogous to Cwirko and Carbonell (1989) and Paine et al. (1983) to give (31)
where expj - u g )
ab)=
(35)
where
ax
B =u
1
-1
(32)
(expj -Ug))
The last term in eq 35 can be expanded to give
where
veff= v,, = v,,
- a(@)
ax
(39)
and the last term in eq 39 may be viewed as an effective source term. Table 1gives the D,ffand peff for the one dimensional field cases (i.e., case 1 and case 2). For the case where the field is applied parallel to the bulk flow, the effective dispersion term reduces to the Taylor-Aris limit and the effectivevelocity Veffis unchanged. Note that in this case Veff = Veff- uKl/L indicating that the effective hydrodynamic velocity is modified by the electric field. This characteristic leads to the conclusion that the effective convective transport depends upon both the hydrodynamic and electrical fields. This physically implies that the field is large enough to reverse the net convective transp2rt. It is intere5ting t o note that from the equation for Veff the sign of Veff reverses when the values of the parameters are such that uKd(u)L > 1. When the field is applied perpendicularly to the hydrodynamic flow, the effective dispersion and velocity terms are now functions of a new dimensionless group Q K2u/D which represents the ratio of electric fieldinduced convection normal to the flow to diffusive transport. In the limit as Q goes to zero the dispersion term reduces to the Taylor-his limit of 1/210 and the effective velocity goes to the average velocity. Figure 3 shows the dependence of the effective velocity and dispersion coefficient on Q for case 2. For purposes of clarity eq 36 can be rearranged to yield
(33) (34)
In order to determine the effective transport parameters, it is necessary to use eq 31, eq 30, and the appropriate potential deviation in eq 26. Effective Transport Parameters. In the case where 6 is independent of (@) as in case 1 and case 2, and where it is dependent on (@)but(@) is independent of x , as was assumed for case 3 on the basis of the comparison with the solution of the point equations as
where Pe = (u)B/D. Figure 3a shows V,d(u) as a function of Q, and Figure 3b shows 8 as a function of Q. Npte that in this case, since a(@)/& is independent of x , Veff= Veg. In both cases the qualitative natures of the two functions are similar since a sigmoidal shape can be seen where the dispersion and velocity go to zero as the electric field (52) increases. This is due to the higher orthogonal electric field impeding the rate of passage of a solute through the channel. From the quantitative point of view the numbers involved in Ved ( u ) are fairly different from those associated with 8.The
Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995 891 Table 1. Comparison of Effective CoefAcients for One-Dimensional Fields veff 44
Des JD
applied potential parallel
1+-Pe2 210
perpendicular
1 + Pe
+
Q(1 + exp(-QN]] 1 - exp(-Q)
Q(l exp(-Q)) 1 - exp(-Q)
Q = K 2-U0
0.8
\
I
L
1.0
k
0.5
n=-0.9 ----
p'
m . 3
;!I
--- MM .. 03 -_-m.9
/'
./'
0.0
0
10
1
n
100
1000
no "." -10.0
10000
10.0
20.0
30.0
40.0
U
b
0'0050 0.0040
0.0
I 1 i i I
O.O1'
I
0.005
/,
0.0030
m 0.0020 L
\
0.0010
0.0000 1
0
'
'
" " "
'
'
"""'
10
100
1000
10000
-0.010 I -10.0
0.0
10.0
20.0
30.0
I 40.0
n Figure 3. Effective velocity (a)and dispersion (b) coefficients for one-dimensional field orthogonal to flow: case 2.
Figure 4. Effective velocity (a) and dispersion (b) coefficients for two-dimensional field with orthogonal applied field: case 3.
ratio Vefp/(u) is never greater than 1,indicating that the perpendicular field by itself does not enhance the convective transport of the solute. Both the velocity and dispersion terms are symmetric with respect to a change in the sign of Q. This, follows from the zero flux at the walls. For both case 1and case 2 with one-dimensional fields the last term in eq 39 is zero. Case 2 has some similarities to that solved by Gajdos and Brenner (1978) where a, generalized Taylor-his methodology was applied for field flow fractionation with an orthogonal body force. Their analysis focused on nonspherical particles and wall effect, whereas the present paper treats macromolecules as points imbedded in a continuum and does not consider wall effects. For the cases of electric field applied simultaneously in the parallel and perpendicular direction of the flow the effective dispersion and velocity terms must be computed numerically using eq 36. Figure 4 shows the effective velocity and dispersion terms for case 3 as a function of the average potential through the dimen-
sionless group U = ((a) (ulD)and for various values of 9,the orthogonal electric field. It should be noted that in this case ((a) is considered to be independent of x , the channel length, and this will allow the use of eq 22 with ((a) constant. The validity of this assumption is illustrated in Figure 2 for several values of the applied fields, K1 and K2. The long channel approximation would also be consistent with this constant field approximation. Figure 4a shows the main characteristics of the effective velocity when two electrical fields are present. A general trend that the results show is that the major effect of the orthogonal electrical field, 9, appears in the region of small mobilities (i.e., U FZ 0). The effect of 9 seems to disappear asymptotically for large (positive and negative) values of the average potential indicated by the parameter U. Also, the function of the effective velocity for this case with respect to the average potential, U , shows a nonsymmetrical shape which indicates that the sign of U yields opposite effects on the average velocity for this case. As
U
892 Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995
U goes to large negative values, VeR/(U)becomes very small for all values of S2. A negative U indicates that the (a) is inducing solute transport opposite to the hydrodynamics of the system thus resulting in a slower movement of the solute. The opposite is true as U becomes large and positive. Figure 4b presents the effect of the two applied fields on the effective dispersion. This figure illustrates the behavior of the effective dispersion with respect to the average potential U as a function of the orthogonal field, S2. The region to the right of the value U = 0 shows the largest effect of Q near the values of small mobility (i.e., U RZ 01, but this effect decreases asymptotically for large (positive) values of U. In this limit, the effective dispersion coefficient approaches the molecular diffusion coefficient, which indicates a significant reduction in radial dispersion. As U approaches zero, 8 approaches a minimum (the Taylor-Aris limit for S2 = 0) and dispersion effects are a t a maximum due t o the way in which 8 is defined. The other region (to the left of the value U = 0)shows that the effect of the parameter S2 increases for large (negative) values of the average potential U. Also, as U becomes large and negative, 8 increases above zero indicating that the effective dispersion is less than the molecular diffusion. When comparing the two regions mentioned above, we see that it is in the second region (i.e., the region to the left of U = 0) that the most significant effect of the secondary field is observed, with the larger positive field increasing 8 the most. We also observe that the large positive orthogonal potential (i.e., S2 = 0.9)yields the biggest increase in 8. An opposite effect can be seen in the region t o the right of the value U = 0. An inversion of the behavior occurs in the region to the left of U = 0 but around values close t o U = 0. The results of this figure could be useful in selecting the different regions of operation of, for example, a separator where two simultaneous electrical fields are applied.
Effect of the Applied Fields on the Separation of Two Solutes
a lo8 ,
-U,=Z.XIO ’cm‘iv-sec - u2=3.x1O”cm‘N-sec
--_-u2=4.x1O+cm’N-sec
---u2=5.x10.’cm2N-sec
1oo
lOl0
b
10’
looo
lola
I \ 1
10‘
1
1 oz0
io’’
K, ( V I
,
1o6
1
- K,=l .O
---- K,=10. --- K,=100. 1
I
lo., 1
1o-6
1oo us (cm2/V-sec)
1o.2
1o2
II
1o4
Figure 5. Time to achieve a resolution of 2 for one-dimensional cm2/s, B = field orthogonal to flow: case 2. D1 = DZ= 1 x 0.01 cm, R = 2, (v) = 1 c d s . (a) t vs &(u); (b) t vs UB.
A and species B. The length of time required to separate the two species for a given resolution is
In order t o consider the effects of the applied field(s) on the separation of two components, the method of moments will be applied. Following the approach of Aris (1956) the nth moment can be defined as (41)
Defining the following groups
where (c) and all its derivatives are zero at fw. For eq 38 with the last term set to zero (i.e., in the case where the average potential is independent of the axial solution) the mean position of a band of solute in the channel at any time t is given by p,’
= M,/Mo = vent
MJM,, - CUI’) = 2De&
= A , B (46)
allows eq 45 to be written as (42)
and the band spreading is given by p2’
z
(43)
For Gaussian peaks (Carbonell and McCoy, 1974) the resolution can be given as (44)
which should be 1 2 for good separation between species
.=[
I’
-k J[(1 - G(B)Pe,2](PeA/Pe,)
H ( B ) - H(A)
(47)
where species A is considered the reference species. Figure 5a shows plots of z versus the applied field K2 for the case of a field applied normal to the fluid motion. It is interesting to note that a minimum occurs. At very low applied fields both solutes travel with the mean hydrodynamic velocity and little separation is possible; however, as the field increases the time for a given separation decreases because the electric field has a
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 893
a iooooo 10000
a larger negative field reduces the time for separation to the least as in Figure 6 close to Vug = 0. When the axial field is large or the mobility difference is large and UB < 0, the larger negative values of Rug produce the shortest times for separation due to the attraction of the charged walls for the solute.
,i;\\
1,\ i.
1001
Summary and Conclusions 101
! 1
0.0
.-
10.0
20.0
30.0
us
b
'Ooo0
I
1000
r
100
10
i
.10.0
-8.0
-6.0
-4.0
-2.0
"B
Figure 6. Time to achieve a resolution of 2 for case 3 as a function of a dimensionless orthogonal field. (a) Positive values of UB;(b) negative values of UB,PeA = Pes = 1.
larger effect on the species with a larger electrophoretic mobility. As the field gets very large, however, the time for the given separation again increases because both species are retained in the channel. Figure 5b shows the same data plotted as t versus Vug for specific values of K2. This figure shows a decrease in time as the mobility increases at given applied fields. Figure 6 shows the time required to achieve a resolution of 2 for case 3 as a function of Vug = (@)$$Dug). In this plot, p$Dg is assumed equal to ~ A I Dat A pug = 1 x as the reference value. As Vug increases relative t o this reference value, in Figure 6a, the time required for the given separation decreases and the magnitude of this decrease is significantly affected by the secondary field. The greatest decrease in separation time is obtained with the larger positive secondary field. Figure 6b shows the effect of the secondary field on the separation time as UB becomes negative. The opposite effect of the secondary field is observed in this region with the largest reduction in separation time occurring for the larger negative secondary field. An interesting change in behavior can be observed in this figure. At UBclose to zero the longest time for separation occurs for QB = -2; however as UBgets larger negative this value of Qug has the shortest time for the given separation. When Vug is negative, the average axial field (@) is of opposite sign to Vug, the electrophoretic mobility. This would imply that the axial electrophoretic transport is opposite t o the hydrodynamic convective transport. When the axial field is small, the primary effect on the separation is the orthogonal field, and therefore,
In the present study, the method of spatial averaging has been applied to three specific cases of applied electrical fields in laminar flow in a channel. For the case of a single field applied parallel t o the flow axis, the effect of the field simply alters the effective velocity in the direction of the flow (see Table 1). In this case, there is no effect of the electric field on the dispersion coefficient and the dispersion coefficient will simply be given by the Taylor-Aris result. When the single electric field is applied normal to the flow axis, both the effective velocity and the effective dispersion coefficients will be affected by the magnitude of the electric field, given by the dimensionless parameter, R. Both of these terms are sigmoidal functions of the logarithm of R, and they both reduce to the Taylor-Aris result at zero applied electric field and to zero at very large electric fields. The limit at very large fields reflects the fact that the normal field is so large that no solute can pass through the channel. The effect of a one-dimensional orthogonal field on the separation of two components shows an optimal value of the secondary field strength that will give a minimum time for a given resolution. This optimum reflects the fact that at very low fields the separation is reduced since both species travel with the parabolic hydrodynamic flow and a t very high fields both species are retained in the channel. When two orthogonal fields are applied simultaneously to the system, the effective dispersion term is reduced for any value of the secondary perpendicular field, with a maximum occurring near U = 0, a dimensionless parameter utilized t o demonstrate the effects of the solute mobility, and given by the Taylor-Aris limit. The effective velocity is significantly affected by the magnitude of the secondary field in the region for small mobilities. For any value of the secondary field, the effective velocity approaches zero at large negative mobilities and a limiting value greater than 1 at large positive mobilities, due to the dominance of the axial field in these regions. The presence of the secondary field reduces the time required to achieve a given resolution between two noninteracting solutes when U and R are of the same sign.
Acknowledgment This work is based on a senior undergraduate honors thesis in chemical engineering written by S.G.S. In addition, we would like to thank Shell Oil Company for summer support to S.G.S. to work as a Shell Scholar in Chemical Engineering at the FAMU/FSU College of Engineering during the summer of 1992. Preliminary versions of this paper were presented at the 1993 Annual Meeting of the AIChE in St. Louis, MO.
Nomenclature B = height of the channel, cm c = the molar species concentration, mol/cm3 D = diffusion coefficient of species i, cm2/s 9=Faraday's constant, 96 487 C/mol F = velocity deviation function given after eq 36
894 Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995
G(i)= (J - D,fi/Di)/Pei2
H ( i ) = Veff/(U) KI = potential applied at L , V KZ = potential applied at B , V L = length of t h e channel, cm N = molar flux of t h e ion, mol/(cm2 s)
M,,= n t h
moment
P = fluid pressure, J/cm3 Pe = Peclet number, dimensionless = vB/D R = universal gas constant, 8.314 C V/(mol K) R, = resolution u = ion mobility in molar species continuity equation, cm2/ (V s)
U = (@)u/D= dimensionless mobility u = mass average velocity, c d s V = aluB/D x = axial coordinate z = valence of t h e ion ( ) = averaged quantity
Greek Letters 6
= dielectric permittivity of a fluid, C/(V cm)
;1 = eigenvalue
p = fluid viscosity, g/(cm s) p i = n t h moment = density, g/cm3 ef = body force term, dyn g/cm3 z = dimensionless time @ = electric potential, V R = KZdD = dimensionless orthogonal field e = (1 - Deff/D)/Pe2
e
Subscripts
A, B = species A or B eff = effective transport t e r m
Literature Cited Aris, R. On the Dispersion of a Solute in a Fluid Flowing Through a Tube. Proc. R . SOC.A 1956,235,67. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Carbonell, R. G.; McCoy, B. J. Moment Theory of Chromatographic Separations: Resolution and Optimization. Chem. Eng. J. 1974,9,115. Carslaw, H. S.; Jaeger, J. C. Heat Conduction, 2nd ed.; Oxford University Press: Oxford, UK, 1981. Chu, G.; Vollrath, D.; Davis, R. W. Separation of Large DNA Molecules by Contour-Clamped Homogeneous Electric Fields. Science 1986,234,1582. Cwirko, E. H.; Carbonell, R. G. Transport of Electrolytes in Charged Pores: Analysis Using the Method of Spatial Averaging. J . Colloid Interface Sci. 1989,129,513. Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. Eringen, C., Maugin, G. A., Eds. Electrohydrodynamics of Continuum ZI: Fluids and Complex Media; Springer-Verlag: Berlin, 1990. Gadjos, L. J.; Brenner, H. Field Flow Fractionation: Extensions to Nonspherical Particles and Wall Effects. Sep. Sci. Technol. 1978,13,215. Giddings, J. C. Field Flow Fractionation: Analysis of Macromolecules, Colloidal, and Particulate Materials. Science 1993,260, 1456. Gray, W. G. A. Derivation of the Equations for Multi-phase Transport. Chem. Eng. Sci. 1975,30,229.
Henry, D. C. The Cataphoresis of Suspended Particles. Part I-The Equation of Cataphoresis. Proc. R . SOC. London, Ser. A 1931, 133,106. Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Dekker: New York, 1986. Hunter, R. J. Zeta Potential in Colloid Science: PrincipZes and Applications; Academic Press: London, 1981. Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989; Vols. I and 11. Lee, C. S.; Wu, C.-T.; Lopes, T.; Patel, B. Analysis of Separation Efficiency in Capillary Electrophoresis with Direct Control of Electrophoresis by using an External Electric Field. J . Chromatogr. 1991,559,133. Levine, S.;Marriott, J. R.; Neal, G.; Epstien, N. Theory of Electrokinetic Flow in Fine Cylindrical Capillaries at High ZetaPotentials. J . Colloid Interface Sci. 1975a,52,136. Levine, S.; Marriot, J. R.; Robinson, K. Theory of Electrokinetic Flow in a Narrow Parallel-Plate Channel. Faraday Trans. 1975b,11, 7. Melcher, J. R. Continuum Electromechanics; MIT Press: Cambridge, MA, 1981. Newman, J. Transport Processes in Electrolytic Solutions. In Advances in Electrochemistry and Electrochemical Engineering; Tobias, C. W., Ed.; Interscience Publishers: New York, 1967; VOl. 5. Newman, J. Electrochemical Systems; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1973. Overbeek, J. Th. G.; Wiersema, Ph. H. The Interpretation of Electrophoretic Mobilities. In Electrophoresis: Theory, Methods, and Applications; Milan, B., Ed.; Academic Press: New York, 1967. Paine, M. A.; Carbonell, R. G.; Whitaker, S. Dispersion in Pulsed Systems-I. Heterogeneous Reaction and Reversible Adsorption in Capillary Tubes. Chem. Eng. Sci. 1983,38,1781. Slattery, J. C. Momentum, Energy and Mass Transfer in Continua; Krieger, New York, 1981. Smith, C. P.; White, H. S. Theory of the Voltametric Response of Electrodes of Submicron Dimensions. Violation of Electroneutrality in the Presence of Excess Supporting Electrolytes. Anal. Chem. 1993,65,3343. Wang, S. C.; Flagan, R. C. Scanning Electrical Mobility Spectrometer. Aerosol Sci. Technol. 1990,13,230. Whitaker, S.Introduction to Ruid Mechanics; Krieger: Melbourne, 1968. Whitaker, S. The Spatial Averaging Theorem Revisited. Chem. Eng. Sci. 1985,40,1387. Whitaker, S.Flow in Porous Media I: A Theoretical Derivation of Darcy’s Law. Transp. Porous Media l986a,1, 3. Whitaker, S. Flow in Porous Media 11: Governing Equations for Immiscible, Two-Phase Flow. Transp. Porous Media 1986b,1, 105. Woodson, H. H.; Melcher, J. R. Electrochemical Dynamics, Part ZZZ: Elastic and Fluid Media; Wiley: New York, 1968. Zanotti, F.; Carbonell, R. G. Development of Transport Equations for Multiphase Systems-I. Chem. Eng. Sci. 1984a,39,263. Zanotti, F.; Carbonell, R. G. Development of Transport Equations for Multiphase Systems-11. Chem. Eng. Sci. 1984b,39,279. Zanotti, F.; Carbonell, R. G. Development of Transport Equations for Multiphase Systems-111. Chem. Eng. Sci. 1984c,39,299. Received for review January 31, 1994 Revised manuscript received September 19, 1994 Accepted November 3, 1994@
IE9400659 @
Abstract published in Advance ACS Abstracts, February
1, 1995.
