Ind. Eng. Chem. Res. 2007, 46, 7627-7636
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Role of Soil Porosity and Electrical and Thermal Resistivities on Free-Convective Transport. Heat Transfer and Hydrodynamic Aspects in an Annular Geometry Mario A. Oyanader† and Pedro E. Arce*,‡ Chemical Engineering Department, UniVersidad Cato´ lica del Norte, AVenida Angamos 0610, Antofagasta, Chile, and Department of Chemical Engineering, Tennessee Technological UniVersity, Prescott Hall 214, CookeVille, Tennessee 38505
The electrokinetic process of soil remediation is affected by different transport driving forces that are responsible for the motion of the bulk fluid and solute species. In particular, the electro-mechanisms, that is, electroosmosis, electrophoresis, and electromigration, may compete with buoyancy and advection promoting distinct flow regimes. The earlier applications of electrokinetic phenomena, that is, electro-assisted drug delivery, microelectrophoretic separations, and material processing, just to name a few, mainly in the area of electrophoresis, neglected this competition and therefore the hydrodynamics of electrokinetic systems was considered simpler. Field test results demonstrate that this is not the case with soil cleanup processes. The unique characteristics of soil porous media call for a different approach and are in need of further analysis. In this contribution, the basic aspects of the behavior of such a system are captured by using an annular capillary model. Under the proposed geometry, a differential model is formulated using simplifying assumptions to maintain the mathematical aspects to a minimum level, and a solution is presented for the different fields, that is, the temperature and the velocity. Several numerical examples are shown to portray the flow situations found in the system for a selection of values of the parameter space. From the analysis of these graphic representations, a qualitative and semiquantitative description of the different flow regimes inside the annular channel is obtained. Particularly interesting in this study is the inclusion of the resistive heating effect in the core of the annular capillary channel as a force term. Temperature developments are explained and analyzed under different scenarios. This information is useful to identify further aspects for the investigation and delineate a systematic approach for a more rigorous description. Implications for design of devices and cleaning strategies are also included. The results obtained in this study are useful to promote a deeper understanding of the behavior of the system, to have a better idea about the experimental effort needed for validation of the different trends, and to lead to important guidelines for improving the separation or cleaning efficiency in a given application. 1. Introduction and Motivation It is an enormous pleasure and honor to contribute to the Dr. Alberto E. Cassano Festschrift by reporting part of our work on electrokinetic soil remediation. (Within Dr. Cassano’s legacy, his constant belief that the use of fundamental principles is one effective way to advance technological applications takes an important role. We hope to illustrate this aspect of his vast influence on many of us by presenting the analysis described here.) The use of electrical fields in soil decontamination has generated different versions of the intended application, that is, cation selective membrane, ceramic casting, Lasagna, electrochemical ion exchange, electrokinetic bioremediation, electrochemical geooxidation, and electrosorb, to name a few.1,2 All these versions of technology constitute the so-called “electrokinetic remediation” in which electrical fields are applied to the impacted area to induce controlled contaminant movement. The inability of most technologies to succeed in heavy metals soil remediation has invoked other approaches for soil cleanup involving electrical fields. The test results of such approaches indicate that this promising technique is effective not only with heavy metals but with radioactive species, dense nonaqueous phase liquids, petroleum hydrocarbons, and several organic compounds as well2-5 and that it can be applied in unsaturated and saturated contaminated soils.2 Nevertheless and * To whom all correspondence should be addressed. E-mail: parce@ tntech.edu. † Universidad Cato´lica del Norte. Interchange Visitor at Tennessee Technological University. ‡ Tennessee Technological University.
despite these promising results, uncertainty, lack of protocols, inability to extrapolate results, and scaling problems are some of the technology’s drawbacks in in situ implementations.6,7 From a practical point of view, it seams to be that the missing link to success is the understanding of the fundamentals that applied to electrokinetic processes in the soil matrix. Understanding the principles of electrokinetics is useful for design and operation purposes in the processes involved. In the area of electrophoresis, these principles are fairly well understood for most of the earlier applications of electrokinetic phenomena.8 For instance, it is very well-known that a combination of different transport driving forces is responsible for the motion of solute through a capillary channel. Although the movement of the solute species is the most important result, it can be caused by different mechanisms and not necessarily by electrophoresis only which is one key characteristic of separation processes.9,10 On the contrary, environmental applications in porous media are mostly characterized by electroosmosis and electromigration as driving mechanisms. For simplicity, the way in which these mechanisms collaborate with other driving forces, that is, buoyancy and hydrodynamics, has been either ignored or neglected as it is a usual rule of thumb in other chemical/environmental processes. The characteristics of soil, that is, heterogeneous, anisotropic, low-permeability, and so forth, call for a different approach where the competition among buoyancy, hydrodynamics, and electroosmosis/electromigration cannot be overlooked because it may promote different types of flow regimes and, therefore, affect solute transport. The authors in previous works have demonstrated and
10.1021/ie070236c CCC: $37.00 © 2007 American Chemical Society Published on Web 06/16/2007
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Figure 1. Geometrical sketch of the annular capillary channel and coordinate system used in the analysis.
reported these hydrodynamics aspects using rectangular, cylindrical, and annular geometries.10-12 Although the basic equations for the transport of solutes under an applied electrical field are known in the literature,3,13,14 it is the systematic analysis of the different effects and parameter ranges that has not been completed, and it could be still be missing items such as to know how to avoid the pitfalls of electrokinetic-based cleaning methods in field testing. To the best of our knowledge, the three contributions made by Oyanader et al.10-12 are the first and latest efforts to provide design criteria for the different electrokinetic applications involving the three previously mentioned main driving forces. Initially, these authors conducted their systematic analysis using a rectangular geometry and then a cylindrical geometry, and recently they concentrated their study in an annular capillary system.12,15 Although the objectives of these works have been rather modest, these geometrical approaches have revealed interesting hydrodynamics aspects to be considered in the design and operation of electrokinetic-based processes in soil remediation. These findings justify the continuation of the study in an annular representation of a porous medium. The authors are perfectly aware that no single geometry can completely describe field conditions; however, there are further aspects not yet considered that can be studied by adopting this proposed geometry. In fact, the annular approximation is more realistic for a porous medium than for a rectangular or cylindrical geometry in that it provides an additional richer avenue for the analysis of the effects of porosity and resistive heating of the core on the hydrodynamics of the system. Consequently, this work concentrates on developing the respective annular model and seeks the understanding of the different flow regimes that may be possible in an annular channel. 2. Model Formulation Consider the system being analyzed of an annular channel of length L, inner radius R0, outer radius R, and inclination of an angle γ with respect to the horizontal line (Figure 1). The annular channel is exposed to a constant electrical field B E. The outer wall surface presents uniform temperature as defined by it interaction with the temperature of the surroundings T∞. The axes (r and x) have been placed coincidently with the lower
end of the capillary channel, x, and the origin of the r-axis at the center of channel. This choice of the coordinate axis is the normal convention in cylindrical coordinates. The system being modeled requires at least two main aspects of transport phenomena for the proposed analysis to be considered, that is, heat transfer and hydrodynamics. From the respective transport equations, profiles of temperature and axial velocity will be obtained for the study of the system behavior. In the sections below, the description of these different aspects is included. 2.1. Model for the Heat Transfer Process. By following the strategy introduced by Bosse and Arce16 the first step in the analysis of the system shown in Figure 1 is the identification of the heat transfer model that will lead to the temperature profile inside the core and annular regions of the cylindrical capillary. The main assumptions involved in the derivation of the heat transfer model include the following: (a) The capillary has a length H with an outer radius denoted by R and an internal radius denoted by RR, (b) the capillary holds the ratio between the width (RR - R) and the high H that assures that end effects are negligible, and (c) the outer surface, located at the radius R, is in contact with an environment that is at a temperature value T∞ while the core of the annulus is associated with two different possibilities, that is, adiabatic and non-adiabatic core. In this contribution, the situation that leads to a non-adiabatic core is studied; that is, the fluid inside the annular channel and the core show a nonzero resistance to the electrical current and, therefore, heat generation takes place. In this case, the generated heat is assumed constant with time and uniform across the annular channel. In addition, the capillary is assumed to be inclined an angle γ with respect to the horizontal line. This is a typical situation found in separation media or soil. The reality is, however, that a distribution of capillary orientations is found in an actual medium. In this contribution, we focus on only one of such possibilities to maintain simplicity. As it was anticipated, the situation described above yields two cases of heat transfer that need to be analyzed separately; this is the heat generation in the core and in the fluid. The analysis of the core presents similarities with the electric heating of a wire for which Bird et al.17 demonstrated its modeling. In this case, the wire is the core of the annular channel. By applying Fourier’s law the following equation is obtained:
-KC
dT Se ) r dr 2
(1)
where KC is the thermal conductivity of the core of the annular channel, T is the temperature of the system in the domain of the core, r is the radial coordinate, and Se is the heat generation rate per volume unit. Now, the following non-dimensional variable for temperature is defined
SeR2 T - T∞ r 2 ; φe ≡ r′ ≡ ; θ ≡ R T∞ 2KCT∞
(2a)
where r′ is the dimensionless radial coordinate, θ is the dimensionless temperature of the system, and φe2 the resistive heating number, with T∞ being the ambient temperature beyond r ) R (r′ ) 1). Then, it is possible to render the system to a non-dimensional form. There are different options to select a non-dimensional radius, and these methods depend upon the type of scaling that is used. Here, we borrow the one used by Wu and Papadopoulos18 because it is useful for comparison purposes. Furthermore, the prime of the radius will be dropped for simplicity.
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By using the non-dimensional variables identified above, eq 1 reduces to
-
∂θ ) φe2r ∂r
(2b)
Equation 2b can be easily solved assuming that KC is constant and using the boundary condition
θCore ) θFluid at r ) R
(3)
In the fluid part of the system, following the ideas of Bosse and Arce,19 the analysis is centered on situations where the assumption of low Reynolds numbers applies and, therefore, the system is dominated by conduction-type regimes.20 Because the ratio of (R - RR)/H is small, the conduction can be safely assumed in the radial direction of the annulus. The fluid is assumed Newtonian with constant thermal properties (see section 3 below). The nonzero resistance to the electrical current, under the assumptions described above, is known as the Joule heating effect, and in such a case the energy equation17 reduces to
-
1 d dθ r ≡ φ2 r dr dr
( )
(4)
In eq 4, the right-hand side, the Joule heating number φ2 (please note that the core and the fluid have been assumed to have different electrical properties), has been identified as
QR2 φ ≡ KFT∞ 2
-
dθ ) βφe2r at r ) R dr
(6a)
-
dθ ) Nuθ at r ) +1 dr
(6b)
In the previous expression eq 6a, β, the thermal core-fluid ratio, is defined as
KC KF
(7)
with φe2 being the resistive heating number previously defined in eq 2a above, and in eq 6b, the Nusselt number, Nu, is defined as
Nu ≡
hR KF
( )
∂Vx 1 ∂ ∂p rµ ) - F(T)gx r ∂r ∂r ∂x
(8)
where h is the heat transfer coefficient, and all the other parameters have been defined above. 2.2. Model for the Hydrodynamics. The fluid in the annular channel, as described above, is assumed to be Newtonian and incompressible for the mass conservation aspects and under steady-state conditions. This fluid is also assumed to have constant properties everywhere except for the density in the buoyancy force term. This is, in fact, the assumption suggested by Boussinesq.23 All the assumptions described in section 2
(9)
where the function F(T) is computed by a first-order Taylor approximation around a mean temperature Tm of the system17
F(T) ) F(Tm) - βmFm(T - Tm)
(10)
and where βm is the volumetric compressibility of the fluid at a mean temperature Tm. Now, the parameter Tm is determined by the total mass conservation condition that may be stated as
2π
∫RRRF(Tm) Vx(r, Tm)r dr ) 0
(11)
To have a convenient way of analyzing the different aspects related to the velocity profile Vx(r), the following dimensionless variables and numbers are proposed.
V′x )
(5)
where, Q is the Joule heating generation and KF is the thermal conductivity of the fluid of the annular channel. The conservation of energy eq 4 needs boundary conditions at both the inner wall and the outer wall surface of the annular channel; although different types of boundary conditions are possible21,22 in this analysis we are interested in a simple case. Therefore, the flux or Robin boundary conditions21 are selected, and they are given by the following equations:
β≡
above are assumed valid for the hydrodynamic flow problem as well. In particular, the “no end effects” and the conductiondominated regime (i.e., small magnitude of velocity field) are invoked here. Moreover, a pressure gradient is assumed to be present, but its magnitude must be relatively small to comply with the assumption of a small velocity field. Under these assumptions, the axial or x component of the Navier-Stokes equation17 is given by
Gr )
VxFmR µ
βmFm2R3T∞g µ2
(13a)
Gr βm T∞
(13b)
VxRFm µ
(14)
∂hP Gr* Gr* + sin(γ) ∂x Re Re
(15)
Gr* ) Re ) Pm )
(12)
The Grashoff numbers, Gr and Gr*, represent the buoyancy to viscous forces due to changes in temperature and density, respectively. The Reynolds number, Re, defined in eq 14 represents the inertia to viscous forces. In this analysis Re ) 1 can be easily demonstrated given the characteristic velocity used to reduced Vx to a dimensionless variable. Additionally, a convenient combination of Grashoff and Reynolds numbers has been mathematically applied to dimensionally reduce the total hydraulic head gradient, which yields the dimensionless number Pm, eq 15. By using these numbers and variables in the NavierStokes component, eq 10, the following dimensionless differential equation is obtained
( )
Gr 1 d dVx sin(γ) (θ - θm) r ) Pm r dr dr Re
(16)
The prime in the non-dimensional variables V′z(r) and r′ is dropped (see eq 16) to maintain simplicity of the notation. This equation shows several terms that account for the different forces present in the system. The left side of the equation is the viscous term, the first term on the right side represents the pressure driven force, and the second term is related to the buoyancy effects.
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To complete the problem for the electro-hydrodynamic velocity profile, the non-slip boundary conditions, at the inner and outer wall surfaces, will be assumed for the annular channel
Vx ) 0 at r ) R
(17a)
Vx ) 0 at r ) 1
(17b)
Equation 16 can be solved by simple substitution of the temperature and electrostatic potential profile equations followed by mathematical integration and constants’ evaluation using the boundary conditions of eq 17a,b. The mathematical analysis is described in the next section. 3. Solution to Model Equations This section focuses on the development of the solution to the differential models for heat transfer, electrostatic field, and hydrodynamic velocity profile. These models, described in section 2, are sequentially coupled, and it is possible to obtain their solutions taking advantage of this characteristic. The sources of coupling between the energy equation and the equation of motion involved in this contribution lie on the energy-convective term and in the potential variation of the physical properties such as viscosity and thermal conductivity. The Batchelor assumption20 of using a conductive-dominated heat transfer regime with a heat source has been validated for variety of experimental conditions.24 Furthermore, the Boussinesq approximation is the standard assumption for many problems with the presence of buoyancy driven flows.23 Under this set of assumptions, the sequential solution approach used in this study (see below) is the most effective approach to obtaining the results herein reported. First of all, the heat transfer and electrostatic models must be solved to obtain the temperature and electrostatic potential profile equations. Second, the velocity profile must be computed using the hydrodynamic model. After an expression for the velocity profile has been determined it is introduced in the mass conservation equation to isolate the characteristic mean temperature of the system, Tm. This last step constitutes the closing loop of the solution procedure for the entire system. The detailed results of the described strategy are discussed in the next sub-sections. 3.1. Heat Transfer Model Solution. As describe in section 2.1 above, in terms of heat transfer there are two differential models to solve. The first one is related to the core of the annular channel represented by eq 1 with boundary condition eq 3. The integrated solution is computed as
{ ( )}
φe R r 2 1; 0ereR 2 R
θ(r) )
{
θMIN )
θMAX )
{
}
{
}
Equation 18b is an analytical function of the position of the annular channel (please note that if φe2 is equal to zero, the case reduces to the adiabatic case), across the radial direction, and it is very useful in the computation of the hydrodynamic velocity profile to be described in a section below. However, some interesting limiting cases can be derived from eq 18b when
{
1 φ2 φ2 2 - R + βφe2R2 Nu 2 2
}
(19b)
{
}
φ2 φ2 2 (1 - R2) + R - βφe2R2 ln |R| + 4 2 1 φ2 φ2 2 - R + φe2R2 Nu 2 2
{
}
(19c)
These expressions produce the lowest and highest temperature in the fluid system for any value of the Joule heating parameter, φ2, and resistive heating number, φe2. Also, the temperature difference between any value and the lowest value is readily given by
∆θ(r) )
{
}
φ2 2 φ 2 φ2 2 - r + R - φe2R2 ln |r| 4 4 2
(19d)
This equation becomes useful to predict temperature differences at any location of the annular channel and the surface of such domain. 3.2. Hydrodynamic Model Solution. The dimensionless Navier-Stokes equation, eq 16, can be integrated after the temperature function θ(r) has been replaced by eq 18b, valid for the fluid part. The solution, after applications of the boundary conditions eq 17 have been used, yields
Vx(r) )
A0 2 A1 (r - 1) + (r4 - 1) 4 16 A2 2 (2r ln |r| - r2 + 1) + A3 ln |r| (20) 8
where the following parameters and expressions have been identified in the function above:
(18a)
φ2 2 φ 2 φ2 2 - r + R - βφe2R2 ln |r| + 4 4 2 1 φ2 φ2 2 - R + βφe2R2 ; R e r e 1 (18b) Nu 2 2
(19a)
Also, the situation of maximum temperature in the fluid can be derived from eq 18 by substituting r ) R. This is
[
A1 ) A2 ) A3 )
(
]
2
φe φ2 Gr sin(γ) (1 - R2) + β R + θMIN - θm Re 4 2 (21a)
The second solution, related to the fluid and described by eq 4 with boundary conditions eq 6a,b, is readily computed as
θ(r) )
}
φ 2 φ2 2 φ2 2 - r + R - βφe2R2 ln |r| 4 4 2
The situation of minimum temperature, located at the wall surface (location r ) 1), for any Nusselt number value can be computed from
A0 ) Pm -
2 2
θ(r) ) θ(R) +
examined. For example, the situation of a high convective cooling system, high Nusselt number values, leads to
φ2 Gr sin(γ) Re 4
(
(21b)
)
Gr φ2 sin(γ) R2 - βφe2R2 Re 2
(21c)
A1 A2 A0 (1 - R2) + (1 - R4) + (2R2 ln |R| - R2 + 4 16 8 1 1) (21d) ln |R|
)
The parameters A0 and A2 are closely related to the buoyancy driven term, affected by the Joule and resistive heating effect, and are porosity dependent. Only A0 is related to the pressure
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Figure 2. Dimensionless temperature profiles (inside the annular capillary channel and the core) for various values of the heat generation parameter, Nusselt number, geometrical factor, R, and the thermal core-fluid ratio, β. The cases depicted are as follows: Nusselt number Nu ) 5, with no resistive heating φe2 ) 0 and R ) 0.4 (a); Joule heating number, φ2 ) 0.5, with no resistive heating φe2 ) 0 and R ) 0.4 (b); Nusselt number Nu ) 10 (cooling system), with Joule heating number, φ2 ) 0.1 and R ) 0.2 (c); Nusselt number Nu ) 10 (cooling system), with Joule heating number, φ2 ) 0.1 and R ) 0.4 (d); Nusselt number Nu ) 1 (less cooling system), with no resistive heating φe2 ) 0 and Joule heating number, φ2 ) 0.1 (e); and Nusselt number Nu ) 5, Joule heating number, φ2 ) 0.3, with no resistive heating φe2 ) 0.3 and R ) 0.3 (f).
driven term. A1 is not affected by resistive heating but by buoyancy forces. Finally, A3 is a combination of the previous parameters and consequently inherits their characteristic driven by porosity. Some of the qualitative and semi-qualitative information about the flow given by the hydrodynamic velocity profile is analyzed in the section below. 3.4. Solution of the Mass Conservation Condition. The main objective in the solution of eq 11 is to obtain an expression for the system mean temperature, Tm, or its equivalent dimensionless form, θm. The expression in this case corresponds to
θm ) θMIN - φ2f1(R) + βφe2f2(R)
(22)
where the following functions of the parameter R have been identified:
f1(R) ) [29R6 - 87R4 - 3R2 + 31 + 6R2 ln(R) 102R6 ln(R) + 30R7 + 72R4 ln(R) + 48R6 ln(R)2][48(R4 3R2 + 2)]-1 (23a) f2(R) ) [-16R6 ln(R) + 5R7 - 16R4 + 7R2 + 12R4 ln(R) + 4R6 + 8R6 ln(R)2][4(R4 - 3R2 +2)]-1 (23b) In eqs 22 and 23, it is observed that θm is a linear function of the parameter β, the Joule heating generation, φ2, and resistive
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Figure 3. Dimensionless 3D temperature profiles (inside the annular capillary channel and the core) for thermal core-fluid ratio β ) 1 and four limiting cases associated with Joule heating, φ2, and resistive heating generation, φe2: φ2 ) 0.05 and φe2 ) 0.01 (a); φ2 ) 0.6 and φe2 ) 0.01 (b); φ2 ) 0.05 and φe2 ) 0.6 (c); and φ2 ) 0.6 and φe2 ) 0.6 (d).
heating generation, φe2. In contrast, θm is a nonlinear function of the dimensionless core radius, R. In addition, θm presents an inverse relationship with the Nusselt number, Nu, implicit in the θMIN term. 4. Illustrative Results and Discussion To compute meaningful numerical values for the temperature and velocity profiles, a dimensional analysis must be performed on key system parameters. From the analysis, a valid range of values of these parameters are determined to study their influence on the temperature and velocity field. This is first accomplished by restricting the range of feasible Joule heating generation number, φ2, and resistive heating number, φe2, to those values that in combination do not imply a change in the fluid phase. Under ambient conditions, change of phase will occur at temperature values of approximately θ g 0.28 yielding a range of the Joule heating generation number of 0 e φ2 e 0.9 and resistive heating number of 0 e φe2 e 1.0. Another physical constraint taken into consideration is that the hydraulic head gradient is balanced by gravitational forces, and therefore its value can be neglected. Another parameter that needs further analysis is β, the thermal core-fluid ratio. Typical environmental applications include water as the main carrier fluid and different types of sandy and clayed soils as porous media.
The described criteria are implicit in all the calculations herein presented. Figure 2a-f shows the temperature profile along the radial coordinate, r, for different combinations of key parameters such as the Joule heating generation number, φ2, resistive heating number, φe2, Nusselt number, Nu, geometrical factor, R, and thermal core-fluid ratio, β. The following main observations are noticed. Increments of temperature values, θ(r), are yielded as higher values of the Joule heating generation number, φ2, influence the system. A flat temperature profile is observed at the core location when the resistive heating number, φe2, is kept to its zero value (see Figure 2a). This type of behavior is suppressed when larger values of the Nusselt number, Nu, are affecting the system (Figure 2b). On the contrary, temperature increments are observed, as a swelling-like effect, at the core region when higher values of the resistive heating number, φe2, are present (see Figure 2c). This phenomenon is amplified by increments of the geometrical factor, R, values (see Figure 2d), the influence of the Joule heating effect, at a given Nusselt and resistive heating number values, on system temperature increases with smaller values of the geometrical factor, R (see Figure 2e). Finally, as the core conductivity becomes more important (higher values of the thermal core-fluid ratio, β) the system temperature increases with a stiff slope profile at the fluid region and a semiflat profile across the core (see Figure 2f).
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Figure 4. Typical hydrodynamic behavior of the physical system for Grashoff number Gr ) 1, Joule heating number, φ2 ) 0.3, and resistive heating effect φe2 ) 0. Dimensionless-axial velocity profiles (inside the annular capillary channel) are shown for different values of the geometrical factor, R ) 0.1 (a); R ) 0.3 (b); R ) 0.5 (c); and R ) 0.8 (d).
Figure 3a-d presents three-dimensional (3D) temperature profiles predicted for the core-annular problem. In particular, Figure 3 illustrates temperature development for different combinations of low and high values of the Joule heating number, φ2, and resistive heating number, φe2. It seems that, when the core presents no resistive heating, φe2 ) 0, the stress of Joule heating produces very low impact in the core temperature (see Figure 3a,b). In practical terms, low resistive annular cores do not affect significantly the temperature profiles. Finally, comparing Figure 3c,d, the effect of resistive heating generation dominates the system over the Joule heating effect. This implies that stronger buoyancy forces may develop in the fluid part of the systems if resistive heating on the core occurs. Figure 4a-d is a 3D representation of the velocity profile function obtained in section 3.3 above. In particular, Figure 4 illustrates different cases of velocity profile with the geometrical factor, R, as the parameter and given values of Joule heating number, φ2 ) 0.3, and resistive heating number, φe2 ) 0. A generalized flow reversal condition is observed for all values of the parameter R; however, the magnitude of the velocity field, Vx(r), decreases as R decreases. For example, for the value of R ) 0.8, the magnitude of the velocity vector becomes very small compared with the velocity vector yielded by the value of R ) 0.1. In other words, as the cross sectional area of the annular region decreases, the heat generation in the fluid also
decreases, and, therefore, smaller values of buoyancy driven flow profiles are yielded. This phenomenon is in agreement with the temperature behavior under the same conditions. Figure 5a-d presents similar illustrations as in Figure 4 but introduces a limiting case of resistive heating number, φe2 ) 1, a considerably high value. If Figure 5a-d is compared with Figure 4a-d the influence of an additional source of generation amplified the magnitude of the velocity vector in every case presented. The electrical resistivity of the core introduces a new source of the heat generation that modified the temperature profile in the fluid, and, therefore, higher buoyancy effects are observed in the annular region. This fact indicates that a cylindrical capillary model will underpredict values of the driving force for free-convective flows with respect to the annular capillary model. Moreover, the assumption of an adiabatic core on the annular geometry will fail to identify the role of the core and the annulus cross-sectional areas in the heat transfer phenomenon and in the hydrodynamic of the system taken place in the annular section. Comparatively, Figure 5a-d demonstrates that the hydrodynamics of the system is more affected when the value of the ratio “area of the core/area of the annulus” is in the neighborhood of one. This fact suggests that porosity may play a very important role in electrokinetic applications in fibrous and/or porous media in controlling the effect of the Joule and resistive heating.
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Figure 5. Typical hydrodynamic behavior of the physical system for Grashoff number Gr ) 1, Joule heating number, φ2 ) 0.3, resistive heating effect φe2 ) 1 and thermal core-fluid ratio β ) 1. Dimensionless-axial velocity profiles (inside the annular capillary channel) are shown for different values of the geometrical factor, R ) 0.1 (a); R ) 0.3 (b); R ) 0.5 (c); and R ) 0.8 (d).
Figure 6a-d shows different cases of the 3D velocity profile with the geometrical factor, R, and the thermal core-fluid ratio, β, as parameters for given values of Joule heating number, φ2 ) 0.3, and resistive heating number, φe2 ) 1. The transition between parts a to b and parts c to d of Figure 6 demonstrates that as the thermal conductivity of the core increases the magnitude of the velocity field also increases. In addition, this effect is larger when there is an even balance of sectional areas between the core and the annulus. This phenomenon was already discussed in the previous paragraph. 5. Summary and Concluding Remarks The effect of a combination of three different driving forces has been analyzed, that is, pressure driven, electroosmosis, and buoyancy, to determine the different types of flow regimes that may occur in an annular capillary channel. This particular geometry is useful to closely describe porous media in a wide variety of situations. Moreover, soil temperature development under field conditions can be associated with resistive heating of the annular core and the Joule heating effect. On the basis of the geometry selected, a few interesting results have been obtained, and these will be highlighted briefly below. First, a family of flow regimes has been identified for characteristic parameter values of the system under study. For
example, competing buoyancy forces with electroosmotic forces change the flow direction and yield flow reVersal regimes. As any of these two forces becomes the most important driving force, the flow reversal effect vanishes and yields pressure driven types of flows. Also, when the buoyancy force is negligible electroosmosis competes with pressure driven forces; however, electroosmosis seams to be stronger, and, therefore, no flow reVersal regimes are observed. In addition, temperature development due to resistive heating induces buoyancy driven flows near the walls. A similar effect is observed when the electrical potential at the inner wall is at its maximum. For the analysis of porosity, higher values of the dimensionless core radius (lower porosity values) favor electroosmotic driven flows, diminishing flow reVersal regimes and pressure driven flows. The generation of different flow regimes due to the control of different driving forces has also been described in other important applications; for example, Cerro and Scriven25 presented an elegant analysis in terms of the integral method proposed by Von Karman for the case of coating flows. Second, few design criteria that constrain the values of the parameter space are reported based on the model equations developed in this study. For example, expressions for predicting limiting cases have been obtained for temperature and total hydraulic head gradient. As the mixing, caused by flow
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Figure 6. Typical hydrodynamic behavior of the physical system for Grashof number Gr ) 1, Joule heating number, φ2 ) 0.3, and resistive heating effect φe2 ) 1. Dimensionless-axial velocity profiles (inside the annular capillary channel) are shown for different values of thermal core-fluid ratio and geometrical factor: β ) 0.2, R ) 0.1 (a); β ) 3.0, R ) 0.1 (b); β ) 0.2, R ) 0.5 (c); and β ) 3.0, R ) 0.5 (d).
reversals, reduces the efficiency of the electrokinetic operation, these criteria may be useful to prepare efficient cleaning protocols in soil remediation. In addition, the interpretation of experimental results related to electro-remediation processes may be carried out more accurately using these criteria as simulation tools. From a practical point of view, the analysis reported here has implications that should be considered in a scale up operation. The dimensionless analysis of the typical conditions of an electro-remediation process has revealed that, for certain parameter ranges, different types of flow reVersals regimes may occur and, therefore, mixing could be a serious detriment to soil cleaning efficiency. Also, porosity seems to play an important role changing the hydrodynamics of the system in direct relationship with electroosmosis. The usual uncertainties of the electrokinetic remediation processes may be avoided if these findings are used as guidelines. Chances of flow reversals increase with the development of high Joule heating and resistive parameters. This phenomenon can be observed by temperature development during field operations. The net result is that flow reversals promote mixing rather than separation which is the main purpose of the electrokinetic remediation technique.
Finally, and accordingly with the goal of this work, a manageable description of the system from an analytical point of view has been obtained. The formal hydrodynamic velocity profile is calculated analytically and, therefore, avoiding complex numerical simulations. This profile is quite useful to analyze the solute/contaminant concentration inside the soil. This analysis will be reported elsewhere. Acknowledgment The support given to M.A.O. by Universidad Cato´lica del Norte, Antofagasta, Chile, is gratefully acknowledged. In addition, the support given to M.A.O. by the Commission Fulbright through out the LASPAU program for faculty enhancement for Latin America has made this research possible, and the authors wish to thank Mrs. Sonia Wallenger (Fulbright-LASPAU, Harvard University) for her continued support and encouragement during the course of this research. Discussions with Ryan O’Hara and Jennifer Pascal are gratefully acknowledged. Nomenclature Ai ) parameters of the axial velocity model with i ) 0, ..., 3 fi ) parameters of the system mean temperature
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Gr ) the Grashoff number Gr* ≡ Gr/βmT∞ B E ) constant electrical field applied on the annular channel H ) capillary length h ) the convective coefficient Kc ) the thermal conductivity of the core of the annular channel KF ) the thermal conductivity of the fluid of the annular channel L ) annular channel of length Nu ) the Nusselt number Pm ) the total hydraulic head gradient Q ) the Joule heating generation Re ) the Reynolds number Se ) the heat generation rate per volume unit. T ) the temperature of the system T∞ ) temperature of the surroundings Vx ) the axial velocity V′x ) the dimensionless axial velocity x ) the axial coordinate r ) the radial coordinate r′ ) the dimensionless radial coordinate R0 ) annular channel inner radius R ) annular channel outer radius Greek Letters R ) the radius of the annular core β ) the thermal core-fluid ratio βm ) the volumetric compressibility of the fluid at a mean temperature γ ) annular channel inclination angle with respect to the horizontal line φ2 ) the Joule heating parameter φe2 ) the resistive heating number ξ ) transversal or radial coordinate θ ) the temperature of the system θm ) the temperature of the system at a mean temperature F ) the fluid density Fm ) the fluid density at a mean temperature Literature Cited (1) Childress, V. W. Resources In Technology: Electrokinetic Remediation. The Technology Teacher 2002, 61 (4), 15-19. (2) Virkutyte, J.; Sillanpaa, M.; Latostenmaa, P. Electrokinetic Soil Remediation - Critical Overview. Sci. Total EnViron. 2002, 289, 97-121. (3) Probstein, R. An Introduction to Physicochemical Hydrodynamics; J. Wiley: New York, 1991. (4) Sharma, A.; Locke, B. R.; Arce, P.; Finney, W. Preliminary Study of Pulsed Corona Discharge for the Degradation of Organic Waste in Aqueous Solutions. Hazard. Waste Hazard. Mater. 1993, 10, 209. (5) Chilingar G. V.; Loo, W. W.; Khilyuk, L. F.; Katz, S. A. Electrobioremediation of Soils Contaminated With Hydrocarbons And Metal: Progress Report. Energy Sources 1997, 19, 129-146.
(6) U.S. EPA. Environmental Center Report Number SFIM-AEC-ETCR-99022; U.S. Environmental Protection Agency: Washington, DC, 2000. (7) Ho, S. V.; Athmer, C.; Sheridan, P.; Shapiro, A. Scale up Aspect of Lasagna process for In Situ Soil Decontamination. J. Hazard. Mater. 1997, 55, 221-237. (8) Sauer, S. Honors in the Major Thesis, Florida State University, 1993. (9) Oyanader, M. A. Physicochemical Hydrodynamics of Electrokinetics in Soil Remediation. Ph. D. Thesis, Florida State University, 2004. (10) Oyanader, M.; Arce, P.; Dzurik, A. Avoiding Pitfalls In Electrokinetic Remediation: Robust Design And Operation Criteria Based On First Principles For Maximizing Performance In A Rectangular Geometry. Electrophoresis 2003, 24, 3457-3466. (11) Oyanader, M.; Arce, P.; Dzurik, A. Design Criteria For Soil Cleanning Operations In Electrokinetic Remediation. Hydrodynamic Aspects In A Cylindrical Geometry. Electrophoresis 2005, 26, 2878-2887. (12) Oyanader, M.; Arce, P.; Dzurik, A. Design Criteria For Soil Cleaning Operations In Electrokinetic Remediation. Hydrodynamic Aspects In An Annular Geometry. Ind. Eng. Chem. Res. 2005, 44, 6200-6411. (13) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (14) Masliyah, J. Electrokinetic Transport Phenomena; AOSTRA Technical Publication Series No. 12, Alberta, Canada, 1994. (15) Erdmann, L.; Oyanader, M. A.; Arce, P. Effect Of the Joule Heating and Of the Material Voids on Free-Convective Transport In Porous Or Fibrous Media With Applied Electrical Fields. Electrophoresis 2005, 26, 2867-2877. (16) Bosse, M. A.; Arce, P. The Role of Joule Heating in Dispersive Mixing Effects in Electrophoretic Cells: Hydrodynamic Considerations. Electrophoresis 2000, 21, 1018. (17) Bird, R. B.; Stewart, W.; Lightfoot, E. N. Transport Phenomena; J. Wiley: New York, 1960. (18) Wu, R. C.; Papadopoulos, K. D. Electroosmotic flow through porous media: Cylindrical and annular models. Colloids Surf., A 2000, 161 (1), 469-476. (19) Bosse, M. A.; Arce, P. Role of the Joule Heating in DispersiveMixing Effects in a Batch Electrophoretic Cell: Hydrodynamic Aspects. Electrophoresis 2000, 21, 1018-1024. (20) Batchelor, G. K. Q. Appl. Math. 1954, 12, 209. (21) Incropera, F. P.; DeWitt, D. P. Fundamental of Heat and Mass Transfer, 4th ed.; John Wiley & Sons: New York, 1996. (22) Boland, M. A.; Arce, P.; Erdmann, E. Free Convection Flows in Fibrous or Porous Media: A Solution for the Case of Homogeneous Heat Sources. Int. Commun. Heat Mass Transf. 2000, 27 (6), 745. (23) Gebhardt, B.; Jaluria, Y.; Majahan, R. L.; Sommakia, B. BuoyancyInduced Flows and Transport; Hemisphere Publishing Corporation: New York, 1988. (24) Calfa Luna, G. Disen˜o y Construccio´n a Escala Piloto de un Equipo de Electroforesis de Flujo Continuo. Honor Thesis, Universidad Cato´lica del Norte, Antofagasta, Chile, 2001. (25) Cerro, L. R.; Scriven, L. E. Rapid Free Surface Film Flows. An Integral Approach. Ind. Eng. Chem. Fundam. 1980, 19 (1), 40-50.
ReceiVed for reView February 12, 2007 ReVised manuscript receiVed April 16, 2007 Accepted April 18, 2007 IE070236C
