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Ind. Eng. Chem. Res. 2005, 44, 5380-5387
Expression for the Film Mass-Transfer Coefficient of Charged Solutes in a Liquid Stream Flowing in Packed Beds of Charged Particles and Charged Porous Monoliths Athanasios I. Liapis* Department of Chemical and Biological Engineering and Biochemical Processing Institute, University of MissourisRolla, Rolla, Missouri 65409-1230
Numerous biochemical and chemical separation and reaction systems have low Reynolds and high Peclet numbers and involve mass transfer of charged solutes between a pressure-driven flowing liquid stream and packed beds of charged particles or charged porous monoliths. For such systems, an expression for determining the film mass-transfer coefficient of a charged solute in a pore (channel) was derived from fundamental expressions of physics. In the derivation of the expression for the film mass-transfer coefficient, mass transport by the mechanisms of convection, diffusion, and electrophoretic migration was taken into account. By considering geometrical (physical) similarity between all pores in a packed bed of charged particles or in a charged porous monolith and the existence of a macroscopic pressure field with uniform gradient, the film mass-transfer coefficient is found to be the same over all pores regardless of size. The values of the parameters in the derived expression for the film mass-transfer coefficient depend on the value of the size of the electrical double layer (Debye length), the magnitude of the zeta potential on the surface of the pores, the relative concentrations of the cations and anions of the supporting electrolyte and of the charged solute, the interaction (adsorption) isotherm of the charged analyte with the charged pore surface, and the values of the charge and Peclet numbers of the charged analyte and the cations and anions of the background/buffer electrolyte. The expression for the film mass-transfer coefficient presented in this work could be used to analyze and correlate experimental data on the rate of mass transfer between charged porous monoliths or packed beds of particles having charged pore surfaces and a flowing liquid stream containing charged species. 1. Introduction The rate of mass transfer between beds of particles or porous monoliths and a flowing fluid stream is needed in the analysis and design of the many systems used for adsorption, desorption, ion exchange, chromatography, heterogeneous catalysis, and catalyst regeneration. Film mass-transfer coefficients are needed to determine the rate of mass transfer between beds of particles or porous monoliths and a flowing fluid stream, and for this purpose, numerous studies have been carried out with the object of measuring mass-transfer coefficients in packed beds and correlating the results.1-4 In systems with low Reynolds, Re, and high Peclet, Pe, numbers (many practical systems in bioseparations and biochemical reaction engineering have low Reynolds and high Peclet numbers) it has been shown5-8 that for a single pore (channel) or a porous medium9 whose pores have similar geometry the dimensionless Sherwood number, Sh, for a neutral (uncharged) solute is given by the following expression:
Sh )
( )
Kfλ vmλ ) δ1 Dmf Dmf
1/3
) δ1Pe1/3
(1)
In eq 1, Kf denotes the film mass-transfer coefficient of the solute, Dmf is the free molecular diffusion coefficient of the solute, λ represents the linear characteristic * Telephone: (573) 341-4414. Fax: (314) 965-9329. E-mail:
[email protected].
dimension of the pore (e.g., radius), vm denotes the mean velocity of the fluid in the pore, and δ1 is a proportionality constant that depends only on the geometry of the pore.8,9 The mean velocity, vm, of the mobile liquid phase in eq 1 could represent the mean velocity of a pressuredriven flow9 or of an electroosmotic10,11 flow. In a pressure-driven flow, vm in eq 1 is proportional to ∆p(λ/ µ), where ∆p represents the pressure drop across a pore of length l1 and µ is the viscosity of the flowing fluid. But the pressure drop across a pore, ∆p, could be determined from l1 (∆P/L), where ∆P is the total pressure over the whole length L of the packed bed or porous monolith. By considering that the length of the pore, l1, is proportional to the linear characteristic dimension, λ, of the pore (l1 ) β1λ, where β1 is the proportionality constant), it can be shown that the mean velocity, vm, of the fluid becomes proportional to λ2(∆P/ L)(1/µ) and when the expression (vmλ)/Dmf is substituted in eq 1 for the Peclet number, Pe, it was shown9 that, for pressure-driven flows in packed beds or porous monoliths whose pores (channels) have similar geometry, the film mass-transfer coefficient, Kf, of a neutral solute could be determined from the expression
Kf ) δ2(Dmf)2/3
(µ1) (∆PL) 1/3
1/3
(2)
where µ denotes the viscosity of the fluid, ∆P represents the total pressure drop in the packed bed or porous monolith, L denotes the length of the packed bed or porous monolith, and δ2 is a proportionality constant
10.1021/ie049120w CCC: $30.25 © 2005 American Chemical Society Published on Web 01/26/2005
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that depends only on the geometry of the pore (through δ1) and on the geometric ratio (through β1). The result in eq 2 indicates that in pressure-driven flows the film mass-transfer coefficient, Kf, does not depend in an explicit manner on λ. Liapis and Grimes10 derived an expression for the film mass-transfer coefficient, Kf, of a neutral solute when the flow of the liquid is electrically driven; electroosmotic flows are employed in microchannel devices that have been and are being developed and increasingly employed in the separation (e.g., capillary electrochromatography), identification, and synthesis of chemicals, pharmaceuticals, and biotechnology products.10,12-14 In electroosmotic flows, the film masstransfer coefficient, Kf, is also a function of the ratio R/θ, where R is the channel radius and θ is the size of the electrical double layer (Debye length10). But the correlating expressions for the film masstransfer coefficient of a solute1-4 as well as those equations for the film mass-transfer coefficient derived from fundamental expressions of physics,5-10 could be employed to determine the value of the film masstransfer coefficient only if the solute of interest is a neutral (uncharged) species.9,10,13,14 Unfortunately, numerous researchers and designers have been using over the years these inappropriate for charged solutes film mass-transfer expressions to determine the values of the film mass-transfer coefficients of charged solutes, and of course, the magnitude of the value of the film masstransfer coefficient of a charged solute determined by the currently available expressions should be highly questionable.15 In the field of bioseparations, most solutes of interest in the liquid phase are charged and their separation often involves the interaction of the charged solute of interest with a charged solid surface (e.g., ion-exchange chromatography); in such systems, the values for the height equivalent to a theoretical plate (HETP), determined from the van Deemter,16 the Knox,17 or the Horvath and Lin18 expressions,3 could be erroneous not only because these expressions were constructed by using very limiting physically assumptions about the mathematical form of the concentration gradient of the solute and do not account for the electrophoretic migration of the charged solute and the effects of the transport of the cations and anions of the background/buffer electrolyte on the net mass flux of the charged solute,15,19-25 but also because the value of the film mass-transfer coefficient of the charged solute required in the van Deemter, the Knox, or the Horvath and Lin expressions for evaluating3 the HETP cannot be determined by the currently available equations for calculating the value of the film mass-transfer coefficient since these equations could be applicable, if their very restrictive assumptions could be satisfied, only to neutral (uncharged) solutes. In this work, an expression is constructed that could be used to determine the film mass- transfer coefficient of a charged solute in a pore (channel) or in a porous medium whose pores could have different sizes but similar geometry, when the flow of the liquid stream is pressure driven and the solid surface of the pore(s) is charged. 2. System Formulation and Analysis A charged analyte in a flowing liquid solution in a pore (channel) is considered as shown in Figure 1, and the charged analyte interacts with the charged surface (wall) of the pore. The cations and anions of the
Figure 1. Concentration boundary layer in the vicinity of the charged surface of a pore or channel in which a liquid stream containing charged species flows.
background/buffer electrolyte are taken to be components 1 and 2, respectively, while the charged adsorbate (analyte) is taken to be component 3. An electrical double layer whose thickness (Debye length) θ is determined by the ionic strength of the solution7,15 is formed close to the charged surface of the channel (pore), and there is no charge neutrality within the electrical double layer because the number of counterions will be large compared with the number of coions. In a packed bed or porous monolith,26-28 the interstitial channels (pores) for bulk flow will be forming a porous medium whose pores will be of differing sizes λ (λ denotes the linear characteristic dimension of a pore (e.g., radius)). In the construction of the theoretical approach, the following considerations are made: (i) The porous medium has many pores and the pores could have different sizes. (ii) The pores of the porous medium have similar geometry.9,26,27 (iii) The porous medium could be described by the pore-network model.9,26-31 The pore-network model approach has been found to provide useful insights and results, especially with respect to the understanding of the physics of mass transfer in porous media. (iv) Systems having low Reynolds and high Peclet numbers are considered. There are many practical systems that have low Reynolds and high Peclet numbers, especially in bioseparation and biochemical reaction engineering systems involving macromolecules; the free molecular diffusion coefficients of the cations and anions of the background/buffer electrolyte could be of the order of 10-9m2/s and the free molecular diffusion coefficient of a macromolecule could be of the order of 10-10-10-13 m2/s while the kinematic viscosity, ν (ν ) µ/F), of water (in many bioseparation and biochemical reaction engineering systems the solvent or carrier liquid is water) is of the order of 10-6 m2/s. The Reynolds and Peclet numbers are defined as follows:
Re ) vmλ/ν Pei )
vmλ , Dmf,i
for i ) 1, 2, 3
(3) (4)
In eqs 3 and 4, vm represents the mean velocity of the fluid. The total mass flux of each species i is equal to the sum of the diffusional, electrophoretic, and convective mass fluxes of each component i.7,15 The mechanism of diffusion is characterized by the diffusion coefficient
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Dmf,i (i ) 1, 2, 3) while the mechanism of electrophoretic migration is characterized32 by the mobility mi (mi ) (Dmf,izie)/kT, for i ) 1, 2, 3). Because we consider systems having low Reynolds and high Peclet numbers, the mechanism of viscous shearing will be considered to be uncoupled7-9 from the mechanisms of diffusion and electrophoretic migration. (v) The accumulation terms of the continuity equations of components 1, 2, and 3 will be taken to be equal to zero (quasi-steady-state condition). Furthermore, the continuity equations of components 1, 2, and 3 reduce to a boundary layer problem as systems with high Peclet numbers are considered in this work. (vi) Dilute mixtures are considered, and the crossdiffusion coefficients are taken to be very much smaller than the diagonal terms of the diffusivity tensor and were set equal to zero in the study reported in this work for the reasons presented in refs 7 and 33. Furthermore, the rate of production or consumption of each species i (i ) 1, 2, 3) per unit volume of the fluid is equal to zero. (vii) The mass transfer by diffusion and electrophoretic migration in the x1 direction (Figure 1) could be neglected because the magnitudes of the diffusional and electrophoretic mass fluxes in the x1 direction could be significantly smaller than the magnitude of the convective mass flux in the x1 direction.7,9,14,15,34,35 (viii) The effect of inertia on fluid flow is considered to be negligible; systems with low Reynolds, Re, numbers are considered in this work. (ix) The radius of curvature of the wall of the pore is considered to be significantly smaller than the thickness of the boundary layer. This consideration allows the analysis of the continuity equations for components 1, 2, and 3 to be made in a system of rectangular coordinates. (x) The linear characteristic size, λ (e.g., radius), of each pore (channel) is considered to be much larger than the size of the electrical double layer, θ, so that overlapping of the electrical double layers does not occur; the dimensions of the interstitial channels for bulk flow in packed beds or porous monoliths satisfy this consideration.14,26,27 (xi) From the application of the Π (Pi) theorem,36 the following expression is obtained for the mean velocity, vm, of the fluid:
vm ∝ ∆p(λ/µ)
(5)
In expression 5, ∆p represents the pressure drop across a pore. Since expression 5 would apply to the components of the velocity at any point in the pore (channel), then one could consider that the velocity profile would be scaled in relation to the applied pressure and would be fixed. From the above considerations, the continuity equations for components 1, 2, and 3 for an isothermal system of constant density, F, and constant diffusion coefficients Dmf,1, Dmf,2, and Dmf,3, are as follows:
( ) ( ) [( )( ) ( )] 2
vx1
∂Ci ∂ Ci ) Dmf,i + ∂x1 ∂x22
z ie ∂Ci ∂Φ ∂2Φ Dmf,i + Ci , kT ∂x2 ∂x2 ∂x22
for i ) 1, 2, 3 (6)
In eq 6, zi (i ) 1, 2, 3) represents the charge number of component i, e is the charge of an electron, k represents
the Boltzmann constant, T denotes the absolute temperature of the system, Ci (i ) 1, 2, 3) is the concentration of species i in the boundary layer, Φ represents the electrical potential in the boundary layer, and vx1 is the velocity component along direction x1. Furthermore, in eq 6 the velocity component vx1 is taken to be a function of x2 (as shown below in eq 8) and the cross section of the pore does not vary along the x1 direction. The electrical potential, Φ, can be obtained through Poisson’s eq 7 that relates the spatial variation in the electrical field to the charge distribution, which for a medium of uniform dielectric constant is
( ) 3
( )
ziCi ∑ i)1
F
∂2 Φ
)-
∂x22
(7)
The permittivity of the medium is equal to the permittivity of a vacuum, o ) 8.854 × 10-12 C2 N-1 m2 (C V-1 m-1) multiplied by the relative dielectric constant (also known as the relative permittivity) r, which is dimensionless;7 in eq 7, F represents the Faraday constant whose units are C mol-1. Equations 6 and 7 are coupled, and the velocity profile in the vicinity of the wall of the pore could be linear, and thus, the velocity component vx1, could be given by the expression in eq 8
vx1 ) x2(∂vx1/∂x2)
(8)
The concentrations Ci∞ (i ) 1, 2, 3) represent the concentrations Ci (i ) 1, 2, 3) of the three components at (a) x1 ) 0 for 0 e x2 e ∞ and (b) x2 ) ∞ for 0 e x1e l1; the parameter l1 represents the length of the pore along direction x1 and is a constant. It is important to mention at this point that (i) the inlet boundary condition will be considered at x1 ) 0 for 0 e x2 e ∞; (ii) the pore wall boundary condition will be considered at x2 ) 0 for 0 e x1 e l1; and (iii) the free-stream boundary condition will be considered at x2 ) ∞ for 0 e x1e l1. The boundary conditions for eqs 6 and 7 are as follows:
for 0 e x2 e ∞,
at x1 ) 0,
Ci ) Ci∞ ,
at x2 ) 0,
Ci ) Ci,equil ,
at x2 ) ∞,
Ci ) Ci∞,
at x2 ) 0,
Φ|x2)0 ) ζw,equil,
for 0 e x1 e l1, i ) 1, 2, 3 (10)
for 0 e x1 e l1,
∂Φ
|
∂x2
x2)∞
i ) 1, 2, 3 (11)
[ ]
for 0 e x1 e l1
(12)
( )
ziDmf,i ∑ |x )∞ ∂x i)1 3
at x2 ) ∞,
i ) 1, 2, 3 (9)
∂Ci
2
)-
2
, zi2e Dmf,iCi|x2)∞ i)1 kT for 0 e x1 e l1 (13) 3
∑
In eq 10, Ci,equil (i ) 1, 2, 3) represents the equilibrium concentration of component i in the liquid layer adjacent to the surface of the pore. In eq 12, ζw,equil represents the value of the zeta potential on the surface of the pore
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when equilibrium conditions for the concentrations of the three components in the liquid layer adjacent to the pore surface have been attained; for the adsorbate (component 3), this would mean that its concentration, C3,equil, in the liquid layer adjacent to the pore surface is in equilibrium with its concentration, CS3,equil, in the adsorbed phase on the surface of the pore. Equation 13 is obtained from the condition that the component of the current density vector along direction x2 is equal14 to zero at x2 ) ∞; this condition, along with the continuity of charge equation,7,15 maintains electroneutrality in the liquid layers located at positions along x2 that are relatively far away from the pore surface. The dimensionless variables γ1, γ2, v′x1,C′i (i )1, 2, 3), and Φ′ are defined as follows:
γ1 ) C′i )
()
λ x1, θ2
Ci Ci∞
γ2 )
x2 , θ
v′x1 )
for i ) 1, 2, 3,
to rescale again without altering the structure of the boundary conditions given in eqs 17-21. By letting
R1Ψ
( )( ) ( ) [( )( ) ( )]
z ie ∂C′i ∂Φ′ ∂2Φ′ ζw + C′i , kT ∂γ2 ∂γ2 ∂γ22
for i ) 1, 2, 3 (15)
( ) ( )[ ]
( ) ( )
[( )( ) ( )] ( ) ( ) [( )(
for i ) 1, 2 (24)
( )]
z3e ∂C′3 ∂2C′3 ∂C′3 ∂Φ′ ∂2Φ′ ) + + C′ ζ w 3 ∂γ1 kT ∂Ψ ∂Ψ ∂Ψ2 ∂Ψ2 (25)
)
( ) () [ ]
In eq 14, ζw represents the zeta potential on the pore surface at time t ) 0 (for time t e 0, no adsorption of adsorbate onto the pore surface has taken place) and the value of ζwcan be measured experimentally.7,12,37 Equations 6, 7, and 9-13 in terms of dimensionless variables have the following forms:
∂v′x1 ∂C′i ∂2C′i ) + Peiγ2 ∂γ2 ∂γ1 ∂γ22
(23)
∂C′i ∂2C′i ) + ∂γ1 ∂Ψ2 ∂C′i ∂Φ′ ∂2Φ′ + C′i , ∂Ψ ∂Ψ ∂Ψ2
PeiPe3-1R1Ψ
vm (14)
R1 ) ∂v′x1/∂γ2 eqs 15-21 become as follows:
z ie ζ kT w
Φ ζw
(22)
and
vx1
Φ′ )
Ψ ) γ2Pe31/3
3
∂2Φ′ ∂Ψ
)-
2
θ2
ζw
F
Pe-2/3 3
ziCi∞C′i ∑ i)1
for 0 e Ψ e ∞,
at γ1 ) 0,
C′i ) 1 ,
at Ψ ) 0,
C′i ) C′i,equil,
at Ψ ) ∞,
C′i ) 1 ,
at Ψ ) 0,
Φ′|Ψ)0 )
(26)
i ) 1, 2, 3 (27)
for 0 e γ1e 1, i ) 1, 2, 3 (28)
3
∂2Φ′
)-
∂γ22
θ2
at γ2 ) 0,
C′i ) C′i,equil,
at γ2 ) ∞,
C′i ) 1,
(16)
i ) 1, 2, 3 (17)
for 0 e γ1 e l2, i ) 1, 2, 3 (18)
for 0 e γ1 e l2, ζw,equil , ζw
[
()
1 ∂Φ′ |γ2)∞ ) at γ2 ) ∞, ∂γ2 ζw
for 0 e γ1 e 1,
i ) 1, 2, 3 (29)
for 0 e γ2 e ∞,
C′i ) 1,
Φ′|γ2)0 )
ziCi∞C′i ∑ i)1
ζw
at γ1 ) 0,
at γ2 ) 0,
F
i ) 1, 2, 3 (19)
for 0 e γ1 e l2 3
ziDmf,iCi∞ ∑ i)1
(
|
∂C′i ∂γ2
γ2)∞
]
(20)
)
, zi2e Dmf,iCi∞(C′i|γ2)∞) i)1 kT for 0 e γ1 e l2 (21) 3
∑
In eqs 18-21, l2 ) (λ/θ2)l1, and thus, l2 is a dimensionless constant. The free-stream boundary condition given by eqs 19 and 21 is in effect at infinity with respect to the boundary layer; this condition at infinity permits one
ζw,equil , ζw
[
()
1 ∂Φ′ at Ψ ) ∞, |Ψ)∞ ) ∂γ2 ζw
for 0 e γ1 e 1 3
ziDmf,iCi∞ ∑ i)1
(
(30)
|
∂C′i ∂Ψ
Ψ)∞
)
]
, zi2e Dmf,iCi∞(C′i|Ψ)∞) i)1 kT for 0 e γ1 e 1 (31) 3
∑
The solution of eqs 24-31 depends on the values of θ, ζw, z1, z2, z3, and C′i,equil(i ) 1, 2, 3) and on the values of the Peclet numbers of components 1, 2, and 3, but it does not depend on the size λ of the pore. The depth of the stream channel or tube that contains the boundary layer is represented by ∆β; the depth of the stream tube is along the axis x3, which is normal to x1 and x2 (Figure 1). If the depth ∆β is normalized with respect to the parameter λ2/θ, and if the normalized depth is denoted by ∆γ, then the amount of adsorbate (component 3) per unit time that is transferred to the surface of the pore along the whole stream channel or tube, would be given by
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∫A(N3|x )0) dA ) ∫A 2
[
( | ) ( | )] (∫ [ | ( | )]
Dmf,3
∂C3 ∂x2
z3e ∂Φ D C| kT mf,3 3 x2)0 ∂x2 (C3∞Pe1/3 3 Dmf,3λ)
x2)0
Sh3 ) Γ8Pe1/3 3
dA )
∂C′3 ∂Ψ
+
Ψ)0
)
z3e ∂Φ′ ζ C′ | ∆γ dγ1 (32) kT w 3 Ψ)0 ∂Ψ Ψ)0 But
∫01
( | ) ( | ) ∂C′3 ∂Ψ
Ψ)0
∆γ dγ1 ) Γ1 ≡ constant
∆γ dγ1 ) Γ2 ≡ constant ∫01C′3|Ψ)0 ∂Φ′ ∂Ψ Ψ)0
(33) (34)
By introducing expressions 33 and 34 into eq 32, the following expression is obtained:
∫AN3|x )0 dA ) (C3∞Pe1/3 3 Dmf,3λ) 2
[
Γ1 + Γ2
( )] z3e ζ kT w
(35)
Thus, the right-hand side of eq 35 provides the amount of component 3 (adsorbate) per unit time that is transferred to the surface of the pore along the whole stream tube. Also, the amount of adsorbate per unit time that is transferred to the surface of the pore along the whole stream channel is equal to Γ3Kf,3(C3∞ C3,equil)λ2, where Γ3 is a constant characteristic of the geometry of the pore (Γ3 > 0). By equating the term Γ3Kf,3(C3∞ - C3,equil)λ2 with the right-hand-side of eq 35 and letting
1 - C′3,equil ) Γ4 ≡ constant, Γ1 ) Γ6 ≡ constant, Γ5
Γ3Γ4 ) Γ5 ≡ constant
Γ2 ) Γ7 ≡ constant (36) Γ5
the following expression for the dimensionless Sherwood number, Kfλ/Dmf,3, of component 3 is obtained:
Sh3 )
[
(38)
x2)0
1
0
the pore surface is uncharged and, for this case, the value of ζw is equal to zero, then eq 37 gives
+
( )]
Kf,3λ z3e ζ Pe1/3 ) Γ6 + Γ7 3 Dmf,3 kT w
(37)
It should be mentioned here that the charged species have been treated as point charges. Detailed results on surface and volume exclusion effects recently obtained by Zhang et al.25 from molecular dynamics simulation studies indicate that, when the charged adsorbate is a macromolecule, in the near proximity of the charged solid surface where the length scale is much smaller than the Debye length, the Poisson-Boltzmann equation7 may have to be replaced by a modified PoissonBoltzmann expression.24,25,38,39 In this derivation, the surface and volume exclusion effects of the charged species are now being accounted through the influence they could have in the values of the parameters Γ1, Γ2, and Γ4, and therefore, the influence of these effects on the value of the Sherwood number, Sh3, will finally be included in the values that will be determined from experimental data for the parameters Γ6 and Γ7 in eq 37. If the adsorbate is an uncharged (neutral) species with z3 ) 0 or if the adsorbate is a charged species but
where Γ8 is a proportionality constant whose role could be similar to that of the parameter δ1 in eq 1. It is important to indicate here that the magnitude of the first term (Γ6Pe1/3 3 ) in the right-hand side of eq 37 for a charged solute interacting with a charged surface could be different than the magnitude of this term if the solute was neutral (uncharged) or if the solute was charged but the surface was uncharged, because the transport of a charged adsorbate is influenced by the competitive interplay of the diffusional and electrophoretic mass transfer mechanisms of the charged adsorbate and of the cations and anions of the supporting electrolyte;15,20-23 for this reason, the magnitudes of the parameters Γ6 (eq 37) and Γ8 (eq 38) could be different. It is very important to note here that the results in expressions 37 and 38 are quite general not only for any distortion to the geometry of the pore (channel) shown in Figure 1 but also for a “channel” of more complex geometry with several inlets and outlets, considering, of course, that for each surface section the relation in eq 18 or eq 28 could be inferred. Therefore, the definition of “channel” or “pore” can be allowed to become less rigid to some repeated geometrical unit of the packed bed or porous monolith for which similar inlet boundary conditions occur. This relaxation of the definition of “pore” appears to get around, to some extent, the criticism of the pore-network model that the distinction between nodes and pores is rather vague. The magnitude of the term (z3e/kT)ζw in eq 37 depends on the values of z3, T, and ζw. Typical values of ζw for charged adsorbent particles37 over an operational range of pH values (the magnitude of ζw depends on the value of the pH of the solution15,37) vary between -50 and -100 mV for negatively charged particles and between 50 and 100 mV for positively charged particles. Therefore, if T ) 293.15 K and z3 ) +1, then the range of values of the term (z3e/kT)ζw could vary between -1.987 and -3.974 considering that the positively charged adsorbate interacts with a negatively charged surface; of course, if T ) 293.15 K and z3 ) - 1, then again the range of values of the term (z3e/{kT)ζw could vary between -1.987 and -3.974 considering now that the negatively charged adsorbate interacts with a positively charged surface. Furthermore, the gradient of the dimensionless electrical potential, ((∂Φ′/∂Ψ)|Ψ)0), evaluated at the surface of the pore (eq 34) will be negative (when either a positively charged adsorbate interacts with a negatively charged surface or a negatively charged adsorbate interacts with a positively charged surface) and since the value of the dimensionless concentration of the adsorbate, C′3|Ψ)0, evaluated at the surface of the channel is always positive, then the constant Γ2 in eqs 34 and 36 will be a negative number. It is worth mentioning here that the magnitude of (∂Φ′/∂Ψ)|Ψ)0 could be large.15,20 It is also important to indicate here that, for the same background/buffer electrolyte, changes in the values of the ionic strength and pH of the solution will affect the size of the Debye length, θ, and the value of the zeta potential, ζw, respectively (and if such changes affect the values of the diffusion coefficients, then the values of the Peclet numbers, Pe1, Pe2, and Pe3 in eqs 24-26 will be affected), as well as the values of
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the C′i,equil and ζw,equil in eqs 18 and 20, respectively (or in eqs 28 and 30, respectively), and thus, the values of the constants Γ1, Γ2, and Γ4 in eqs 33, 34, and 36 will be affected and through them the values of the parameters Γ6 and Γ7 in eq 37 will change, and thus, the magnitude of the Sherwood number, Sh3, will be affected. The values of the parameters Γ6 and Γ7 in eq 37 have not yet been determined because at this time no experimental data from controlled packed bed or porous monolith adsorption experiments involving a charged solute in an electrolytic solution in contact with the charged particles of the bed or the charged surfaces of the pores of the monolith where the zeta potential had been measured, could be found in the literature. It could be considered that a major reason for the lack of such experimental data may be the fact that the mechanism of electrophoretic migration of charged solutes had not been taken under account in the development of expressions1-8 for the determination of the film masstransfer coefficient, and therefore, the contribution of the electrophoretic migration of charged solutes to the total mass flux of the charged species in the liquid film surrounding the charged packed particles had not been accounted for and, thus, had been neglected. But it has been shown15,24,25 that the contribution of the mass flux due to the electrophoretic migration of charged solutes to the total mass flux of charged species in the liquid film surrounding charged particles can be significant. Thus, the present work clearly indicates that experimental data are needed to be measured under controlled conditions in packed beds of charged particles or charged porous monoliths involving liquid solutions with charged solutes interacting with the charged particles of the packed bed or the charged surfaces of the pores of the monoliths. The present work provides a theoretical expression that could be used to analyze and correlate the data obtained from such controlled experiments. Detailed simulations15,20-24 employing complex models that require tedious and time-consuming calculations have shown that the effect of the electrical double layer on the mass transfer of the charged solute depends on a number of factors that include (i) the size of the electrical double layer, (ii) the relative concentrations of the cations and anions of the supporting electrolyte and of the charged adsorbate, (iii) the relative values of the diffusion coefficients of the cations and anions of the supporting electrolyte and of the charged adsorbate, (iv) the relative values of the charge numbers z1, z2, and z3, (v) the value of the electrostatic potential, Φ, at the charged pore surface, and (vi) the adsorption mechanism of the charged adsorbate. A special case7,20 in electrochemical systems is encountered when the concentrations of the cations and anions of the electrolyte are significantly high (this makes the size of the electrical double layer to be very small) and at the same time these concentrations are very much higher in magnitude than the magnitude of the concentration of the charged adsorbate, and furthermore, the adsorption of the adsorbate onto the charged pore surface is negligible. For this special case, the experimentally determined mass-transfer characteristics of the charged solute appear to be similar7,20 to those of a neutral solute, and in this case, one could expect that the magnitute of the second term (Γ7(z3e/kT)ζwPe1/3 3 ) in the right-hand side of eq 37 may be negligible. In this special case, the value of C′3|Ψ)0 in eq 34 could be very, very small (negligible)
and lengthy in computational time calculations using the complex and computationally tedious model of Grimes and Liapis15 indicate that the magnitude of the integral in eq 34 would be negligible, and thus, the value of the parameter Γ2 would be negligible, which would then imply from eq 36 (for such a system, Γ4 = 1, and Γ5 = Γ3) that the value of the parameter Γ7 would be negligible and this would mean that the contribution of the second term (the charge term) in the right-hand side of eq 37 could be negligible. This result for the special case supports the structure of the expression derived and given in eq 37, since such an experimentally determined result could be ascertained from the derived in this work eq 37. To evaluate the performance of a packed bed (e.g., packed chromatographic column) or porous monolith, one has to have an expression for the film mass-transfer coefficient, Kf,3. If ∆P represents the total pressure drop over the whole length L of the packed bed or porous monolith, then the pressure drop ∆p across a pore (see expression 5) is related5 to ∆P through the following expression:
∆p ∝ λ (∆P/L)
(39)
It is worth mentioning here that in eq 39 it is considered that the pore length is taken to be proportional to the linear characteristic dimension λ (e.g., radius) of the pore, as dicussed above in the Introduction. The combination of expressions 5, 37, and 39 provides the following expression for Kf,3:
Kf,3 ∝ (Dmf,3)2/3
() ( ) [ 1 µ
1/3
∆P L
1/3
( )] z3e ζ kT w
(40)
( )]
(41)
Γ6 + Γ7
Expression 40 could also be written as
Kf,3 ) δ3(Dmf,3)2/3
() ( ) [ 1 µ
1/3
∆P L
1/3
Γ6 + Γ7
z3 e ζ kT w
where δ3 is a proportionality constant. The parameter δ3 in eq 41 depends only on the geometry of the pore and also incorporates the effect of the geometric ratio, as discussed in the Introduction for the parameter δ2 of eq 2. The result shown in eq 41 indicates that the film mass-transfer coefficient, Kf,3, does not depend on the linear characteristic size, λ, of the pore in an explicit manner, and thus, for a packed bed or porous monolith containing similar pores of differing size, Kf,3 is the same for every channel (pore) of the packed bed or porous monolith, depending on (a) the pressure gradient, ∆P/ L, applied, (b) the pH of the solution that could affect the value of the zeta potential, ζw, on the charged surface of the pores as well as the value of the charge number, z3, of the solute, and (c) the ionic strength of the solution that affects the size of the electric double layer which in turn could affect the values of the parameters Γ6 and Γ7. Conclusions For separation as well as chemical and biochemical reaction systems with low Reynolds and high Peclet numbers and involving the interaction of a charged solute with a charged surface, an expression for the film mass-transfer coefficient, Kf,3 of the charged solute in a pore (channel) was derived from fundamental expressions of physics when the flow is pressure driven. The
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derived expression indicates that the film mass-transfer coefficient, Kf,3, does not depend on the linear characteristic size, λ, of the pore in an explicit manner, and thus, for a packed bed or porous monolith containing similar pores of differing size, Kf,3 is the same for every pore (channel) of the packed bed or porous monolith. The values of the parameters that characterize the expression that could be used to determine the value of Kf,3, depend on the value of the size of the electrical double layer (Debye length) θ, the magnitude of the zeta potential, ζw, on the surface of the pores, the relative concentrations of the cations and anions of the supporting electrolyte and of the charged solute, the interaction (adsorption) isotherm of the charged analyte with the charged pore surface (the interaction isotherm plays a very important role in establishing the value of C′3,equil in eq 18 or eq 28), and the values of the charge and Peclet numbers of the charged analyte and the cations and anions of the background/buffer electrolyte. The expression for the film mass-transfer coefficient, Kf,3, presented in this work can be used to analyze and correlate experimental data on the rate of mass transfer between charged porous monoliths or packed beds of particles having charged pore surfaces and a flowing liquid stream containing charged species. Acknowledgment The author thanks Dr. B. A. Grimes at Johannes Gutenberg-Universita¨t Mainz, Mainz, Germany, for a discussion on electrophoretic migration. Literature Cited (1) Wilson, E. J.; Geankoplis, C. J. Liquid Mass Transfer at Very Low Reynolds Numbers in Packed Beds. Ind. Eng. Chem. Fundamentals 1966, 5, 9. (2) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. (3) Wankat, P. C. Rate-Controlled Separations; Elsevier Science Publishers Ltd.: Essex, England, 1990. (4) Geankoplis, C. J. Transport Processes and Separation Process Principles; Prentice Hall: Upper Saddle River, NJ, 2003. (5) Levich, V. G. Physicochemical Hydrodynamics; PrenticeHall: Englewood Cliffs, NJ, 1962. (6) Pfeffer, R.; Happel, J. An Analytical Study of Heat and Mass Transfer in Multiparticle Systems at Low Reynolds Numbers. AIChE J. 1964, 10, 605. (7) Probstein, R. F. Physicochemical Hydrodynamics; Wiley: New York, 1994. (8) Deen, W. M. Analysis of Transport Phenomena; Oxford University Press: New York, 1998. (9) Liapis, A. I. Affinity Adsorption Separations in High Performance Liquid Chromatography and Perfusion Chromatography Systems: The Effects of Pore-Size Distribution and Fractal Pores on Column Performance. Math. Model. Sci. Comput. 1993, 1, 397. (10) Liapis, A. I.; Grimes, B. A. Film Mass Transfer Coefficient Expressions for Electroosmotic Flows. J. Colloid Interface Sci. 2000, 229, 540. (11) Grimes, B. A.; Liapis, A. I. Expressions for Evaluating the Possibility of Slip at the Liquid-Solid Interface in Open Tube Capillary Electrochromatography. J. Colloid Interface Sci. 2003, 263, 113. (12) Liapis, A. I.; Grimes, B. A. Modeling the Velocity Field of the Electroosmotic Flow in Charged Capillaries and in Capillary Columns Packed with Charged Particles: Interstitial and Intraparticle Velocities in Capillary Electrochromatography Systems. J. Chromatogr. A 2000, 877, 181. (13) Grimes, B. A.; Liapis, A. I. The Evolution and Implications of the Concentration Profiles of an Analyte in Porous Adsorbent Particles Packed in a Capillary Electrochromatography Column Operated in the Analytical Mode. J. Sep. Sci. 2002, 25, 1202.
(14) Grimes, B. A.; Lu¨dtke, S.; Unger, K. K.; Liapis A. I. Novel General Expressions that Describe the Bevahior of the Height Equivalent of a Theoretical Plate in Chromatographic Systems Involving Electrically-Driven and Pressure-Driven Flows. J. Chromatogr. A 2002, 979, 447. (15) Grimes, B. A.; Liapis, A. I. The Interplay of Diffusional and Electrophoretic Transport Mechanisms of Charged Solutes in the Liquid Film Surrounding Charged Nonporous Adsorbent Particles Employed in Finite Bath Adsorption Systems. J. Colloid Interface Sci. 2002, 248, 504. (16) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Longitudinal Diffusion and Resistance to Mass Transfer as Causes of Nonideality in Chromatography. Chem. Eng. Sci. 1956, 5, 271. (17) Knox, J. H. Practical Aspects of LC Theory. J. Chromatogr. Sci. 1977, 15, 352. (18) Horvath, Cs.; Lin, H.-J. Band Spreading in Liquid Chromatography: General Plate Height Equation and a Method for the Evaluation of the Individual Plate Height Contributions. J. Chromatogr. 1978, 149, 43. (19) Liapis, A. I.; Grimes, B. A.; Lacki, K.; Neretnieks, I. Modeling and Analysis of the Dynamic Behavior of Mechanisms that Result in the Development of Inner Radial Humps in the Concentration of a Single Adsorbate in the Adsorbed Phase of Porous Adsorbent Particles Observed in Confocal Scanning Laser Microscopy Experiments: Diffusional Mass Transfer and Adsorption in the Presence of an Electrical Double Layer. J. Chromatogr. A 2001, 921, 135. (20) Grimes, B. A., Ph.D. Dissertation, Department of Chemical and Biological Engineering, University of MissourisRolla, Rolla, MO, 2002. (21) Grimes, B. A.; Zhang, X.; Lacki, K.; Neretnieks, I.; Liapis, A. I. Novel Concepts of Mass Transport and Adsorption of a Charged Analyte onto a Charged Adsorbent Surface Immersed in an Electrolytic Solution: Construction and Analysis of the Mechanisms that Lead to Concentration Rings in the Concentration of the Analyte in the Adsorbed Phase. 22nd International Symposium on the Separation of Proteins, Peptides and Polynucleotides (ISPPP 2002); Heidelberg, Germany, November 10-13, 2002. (22) Zhang, X.; Grimes, B. A.; Wang, J.-C.; Ljunglo¨f, A.; Lacki, K.; Liapis, A. I. Novel Mechanisms and Simulations for Mass Transport and Adsorption of a Charged Analyte in Ion-Exchange Chromatography: Practical Implications. Recovery of Biological Products XI Conference; Banff, Alberta, Canada, September 1419, 2003. (23) Zhang, X.; Grimes, B. A.; Wang, J.-C.; Ljunglo¨f, A.; Lacki, K.; Liapis, A. I. Novel Mechanisms and Simulations for Mass Transport and Adsorption of a Charged Analyte onto a Charged Adsorbent Surface Immersed in an Electrolytic Solution: Practical Implications for Concentration Rings in the Concentration of the Analyte in the Adsorbed Phase of Ion-Exchange Chromatography Systems. 23rd International Symposium on the Separation of Proteins, Peptides and Polynucleotides (ISPPP 2003); Delray Beach, FL, November 9-12, 2003. (24) Zhang, X.; Grimes, B. A.; Wang, J.-C.; Lacki, K. M.; Liapis, A. I. Analysis and Parametric Sensitivity of the Behavior of Overshoots in the Concentration of a Charged Adsorbate in the Adsorbed Phase of Charged Adsorbent Particles: Practical Implications for Separations of Charged Solutes. J. Colloid Interface Sci. 2004, 273, 22. (25) Zhang, X.; Wang, J.-C.; Lacki, K. M.; Liapis, A. I. Molecular Dynamics Simulation Studies of the Transport and Adsorption of a Charged Macromolecule onto a Charged Adsorbent Solid Surface Immersed in an Electrolytic Solution. J. Colloid Interface Sci. 2004, 277, 483. (26) Meyers, J. J.; Liapis, A. I. Network Modeling of the Intraparticle Convection and Diffusion of Molecules in Porous Particles Packed in a Chromatographic Column. J. Chromatogr. A 1998, 827, 197. (27) Meyers, J. J.; Liapis, A. I. Network Modeling of the Convective Flow and Diffusion of Molecules Adsorbing in Monoliths and in Porous Particles Packed in a Chromatographic Column. J. Chromatogr. A 1999, 852, 3. (28) Meyers, J. J.; Crosser, O. K.; Liapis, A. I. Pore Network Modeling of Affinity Chromatography: Determination of the Dynamic Profiles of the Pore Diffusivity of β-Galactosidase and its Effect on Column Performance as the Loading of β-Galactosidase onto Anti-β-Galactosidase Varies with Time. J. Biochem. Biophys. Methods 2001, 49, 123.
Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 5387 (29) Petropoulos, J. H.; Liapis, A. I.; Kolliopoulos, N. P.; Petrou, J. K.; Kanellopoulos, N. K. Restricted Diffusion of Molecules in Porous Affinity Chromatography Adsorbents. Bioseparation 1990, 1, 69. (30) Petropoulos, J. H.; Petrou, J. K.; Liapis, A. I. Network Modeling Investigation of Gas Transport in Bidisperse Porous Adsorbents. Ind. Eng. Chem. Res. 1991, 30, 1281. (31) Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor and Francis: London, England, 1992. (32) Belter, P. A.; Cussler, E. L.; Hu, W.-S. BioseparationsDownstream Processing for Biotechnology; John Wiley and Sons, Inc.: New York, 1988. (33) Liapis, A. I.; Litchfield, R. J. A Note on the Off-Diagonal Terms of the Effective Pore Diffusivity Matrix. Trans. IChemE. 1981, 59, 122. (34) Liapis, A. I. Modeling Affinity Chromatography. Sep. Purif. Methods 1990, 19, 133. (35) Liapis, A. I.; Unger, K. K. The Chemistry and Engineering of Affinity Chromatography. In Highly Selective Separations in
Biotechnology; Street, G., Ed.; Blackie Academic and Professional, an imprint of Chapman & Hall: Glasgow, U.K., 1994; pp 121162. (36) Zlokarnik, M. Dimensional Analysis and Scale-Up in Chemical Engineering; Springer-Verlag: Berlin, Germany, 1991. (37) Lu¨dtke, S., Ph.D. Dissertation, Department of Chemistry and Pharmacy, Johannes Gutenberg-Universita¨t Mainz, Mainz, Germany, 1999. (38) Outhwaite, C. W. A Modified Poisson-Boltzmann Equation for the Ionic Atmosphere Around a Cylindrical Wall. J. Chem. Soc. Faraday Trans. 2 1986, 82, 789. (39) Outhwaite, C. W.; Bhuiyan, L. B. A Modified PoissonBoltzmann Analysis of the Electric Double Layer Around an Isolated Spherical Macroion. Mol. Phys. 1991, 74, 367.
Received for review September 10, 2004 Revised manuscript received November 4, 2004 Accepted November 6, 2004 IE049120W
