Non-newtonian carriers in a batch electrophoretic cell with joule heating: hydrodynamic considerations and mathematical aspects

NON-NEWTONIAN CARRIERS IN A BATCH ELECTROPHORETIC CELL WITH JOULE HEATING: HYDRODYNAMIC CONSIDERATIONS AND MATHEMATICAL ASPECTS

M.A.Bosse¹, P.Arce² and A.Vasquez³

¹Department of Chemical Engineering, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile

Phone: [phone], Fax: [phone], E-mail: [e-mail]

²Department of Chemical Engineering and Geophysical Fluid Dynamics Institute, GFDI. Florida State University,

Tallahassee, Florida 32308, USA. E-mail: [e-mail]

³Department of Mathematics, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile

E-mail: [e-mail]

(Received: August 30, 2000 ; Accepted: November 22, 2000)

Abstract - The Joule heating effects on hydrodynamics are examined for the case of a non-Newtonian fluid flowing in a batch electrophoretic cell. The potential benefits of using non-Newtonian fluids in electrophoretic separations could lead, for example, to an increase in the efficiency of separations and to a better scale-up of the operation for the processing of large quantities on a pilot or even an industrial scale. The analysis conducted in this research is based on the formulation of simplified mathematical models that can capture the most relevant aspects of the physics of the free convection that takes place in a batch electrophoretic cell. The first aspect is associated with the carrier fluid when no solute sample (of particles or macromolecules) is added to the device. This is, indeed, a carrier fluid completely free of solute problem. This aspect of the research is referred to as the carrier fluid problem. The second aspect involved in the investigation is the analysis of the mass transport (i.e., convective-diffusive) process associated with the motion of a solute sample added to the carrier fluid. This particular aspect of the research is referred to as the solute problem of the free-convection electrophoretic cell. The two problems are sequentially coupled and the solution of the carrier fluid problem must be performed first in order to have the velocity profile to study the solute problem. Details about hydrodynamic considerations and the solution approach are discussed in this contribution.

Keywords: electrophoresis, non-Newtonian fluids, Joule heating generation.

INTRODUCTION

The idea of using electrical fields to separate biomacromolecules has been around for some time (Bier, 1960). There are a number of applications related to the use of electrical fields to separate, for example, proteins, DNA, cells and organelles (Hunter, 1995; Bosse, 1998), and they are included in the general area of electrophoresis. However, considerably more work needs to be carried for elucidating problems related to the loss of separation efficiency due to the free convection flows that are generated by the Joule heating effect. Joule heating generation is due to the resistance of the buffer to the passage of an electrical current inside the electrophoresis device.

In past applications, several approaches were introduced to limit the effect of Joule heating generation. Among them, one can cite the adjustment of the design of the cell in order to take advantage of better heat transfer conditions (Clifton, 1993); also, the concentration of buffer has been changed parametrically to discover optimal conditions for eliminating counterflows (Probstein, 1991) and material (such as gels) has been used to increase separation while keeping heat generation at a low level. One of the alternatives that was suggested (Dobry and Finn, 1958 a & b) is the modification of the rheology of the fluid carrier in order to reduce adverse effects originating from free convection flows. This aspect, though, has not been systematically studied by using the hydrodynamic aspects and the convective-diffusive transport taking place inside the electrophoresis cell. In this contribution, the power-law model has been used to study the effect of non-Newtonian fluids on minimizing the impact of Joule heating generation on the free convection flows and, indirectly, on attenuating such an effect on the separation efficiency of the electrophoresis device.

This contribution follows the strategy recently introduced by Bosse and Arce (2000 a&b) to study the effect of Joule heating generation in batch electrophoresis cells with Newtonian fluids. This approach is based on a sequential coupling of the energy equation and the momentum conservation equation. The various aspects involved in the technique will be discussed in the sections below.

TEMPERATURE PROFILE

As mentioned in Bosse and Arce (2000a), the carrier fluid problem consists of two sub-problems. The first one is focused on the temperature profile and the second one, on the hydrodynamic velocity profile. This section will focus on the analysis of the heat transfer problem, while section three will analyze the hydrodynamic aspects.

The general energy equation for the system yields (Bird et al., 1960)

The main assumptions used for the problem under consideration are steady state, T=T(y), constant properties (m , k, C_p), motion only in the z direction, no end effects, Boussinesq fluids, and negligible viscous effects for the energy equation.

Equation (1) can now be reduced considerably due to the kinematics of the free convection cell and the assumptions stated above. The kinematics for the system sketched in Figure 1 is given by since a one-dimensional flow has been assumed.

Since the system is examined under Boussinesq conditions, density is not a function of either time or space for mass conservation. The velocity in the x and y directions is assumed to be negligible, compared to the velocity in the z direction, i.e., the 1-D flow assumption is used here. Therefore, equation (1) becomes

Equation (3) also assumes that the conduction term for temperature is only a function of y, as stated by the second assumption above. In addition, convective transport of heat is neglected in the axial direction and the conductive terms are essentially those that drive the heat flow across the system (Batchelor, 1954). The physical meaning of equation (3) is that conduction in the y direction is balanced by Joule heat generation.

The boundary conditions for the system sketched in Figure 1 are

Although other (more general) boundary conditions could be adopted (Boland et al., 2000), the simplest possible situation to describe the temperature field has been selected. After integration, applying the above boundary conditions to equation (3), one arrives at

Now, introducing a new non-dimensional parameter, the Joule heating number, f, equation (6) can be written as follows

As mentioned in Bosse and Arce (2000a), the temperature profile given by equation (7) is required for the study of the hydrodynamic velocity profile. Analysis of this profile is the subject matter off the section below.

HYDRODYNAMIC PROFILE OF NON-NEWTONIAN FLUIDS

The general equation of motion for a non-Newtonian fluid in the cell sketched in Figure 1 is

The same assumptions as those used in the heat transfer model (see the section above) are adopted here. Thus, the same kinematics and simplified continuity equation are valid here. In addition, the pressure terms are only a function of the z direction and density is a function of temperature only. This last result is due to the use of the Boussenesq approximation, i.e., the only term in the Navier-Stokes equation affected by the variation in density is the buoyancy force term (Gebhardt et al., 1988).

The continuity equation features (in general) a spatial dependence of density on position. However, for this particular case, since the Boussenesq approximation will be used, the incompressible assumption is helpful in order to obtain a relatively simple, yet accurate, hydrodynamic model

for the description of the velocity profile inside the cell. Under these conditions, equation (9) becomes

Function r(T) must be determined in order to solve equation (10). In deriving the density function, r(T), it will be expanded in a Taylor series in the T variable around some unspecified temperature value (Bird et al., 1960) as follows:

where the coefficient of volume expansion is

For this particular system, the pressure variation is solely appreciable in the z direction. Furthermore, it is assumed that the pressure gradient in the cell is compensated for the weight of the fluid (see Bird et. al., 1960). Under these assumptions equation (10) becomes

The physical meaning of this equation is that the viscous forces are balanced out by the buoyancy forces. Substituting equation (6) into equation (13), and by integrating equation (13) once, the following equation is obtained:

Now the yz component of the stress tensor for non-Newtonian fluids, using the power-law and CEF models, can be written as

Upon substitution of equation (15) into equation (14), one obtains the following differential equation for the velocity profile of power-law fluids flowing inside the free convection cell sketched in Figure 1

For simplicity, one can express equation (16) in the following way:

where one can define the following expression:

where F(y,f ) corresponds to

and where the following group of parameters has been identified:

Thus, the general (formal) velocity profile for the fluid can be computed as

where C₁ and C₂ are constants of integration.

If one substitutes equation (19) into equation (21), the general velocity profile of the fluid (power- law model) can be computed as

As used in the analysis by Bosse (1998), one effective way to solve the integral in equation (22) is by doing a series expansion of the polynomial indicated in the integrand of this equation.

For this purpose and for algebraic convenience, let’s redefine the integrand of equation (22) as

If one now expands function f(y) in a Taylor series, then the following expression is obtained:

By expanding around y_o = 0, i.e., the origin of coordinates, where v_z (y)=0, one obtains the MacLaurin series:

By evaluating and substituting all the derivatives into equation (25) and by truncating the series in the third term, one arrives at the following function.

or, simply,

where the following quantities have been defined:

Now, by substituting equation (27) into equation (22), the following integral equation for the velocity profile may be written:

where the group of parameters

has been identified.

Now, equation (32) is easily integrated to obtain the velocity profile v_z(y) as

By using the non slip boundary conditions at the wall (see Figure 1), i.e.,

and by substituting q from equation (29), the integration constants indicated in equation (34) may be calculated as

By substituting c₁ and c₂ into equation (34) and by expressing v_z as a function of h, which is defined by , one arrives at

This velocity profile is still a function of the mean temperature, T, of the system but equation (38) must satisfy the mass conservation principle

This equation leads to an additional constraint. By substituting the expression of the velocity profile given by equation (38) into (39) and by integrating, the following algebraic equation is derived:

By dividing equation (40) by p and by substituting (l/p) and (h/p), whose expressions are



and



respectively, one obtains a second-order polynomial for c₁:

The solution of equation (43) is simply

Parameters m₁, m₂, and m₃ have been defined as



Additionally, the following parameters have been identified:




Constants A, B, and C were defined as in equation (20). By rearranging equation (44), it is possible to obtain a simpler expression:



where




and



From the analysis written above, one observes that there is only one equation for c₁ as a function of C, which implies two unknowns, and only one equation. It is, therefore, necessary to have one additional equation to solve for C and c₁. Such an equation can be obtained from the boundary condition for c₁ ,given by equation (36) above.

Thus,

By dividing equation (52) by p and substituting the expressions for (l/p) and (h/p), which are given by equations (41) and (42) above, one obtains a polynomial of the third degree in respect to C:



where

r₁, r₂, r₃, and r₄ are



and parameters r₂₁, r₃₁, r₃₂ and r₄₁ are defined as



By substituting the expressions for r₁, r₂, r₃ and r₄ into equation (53), the following algebraic equation is derived:



and by substituting c₁, given by equation (52), and by multiplying by D³ /C², one obtains



The procedure used to obtain C involves the following aspects: First, obtain an expression that allows for the separation of the terms that contain R (the square root), defined by equation (50), from the rest of the terms. This expression can then be squared with the objective of dealing with a simpler mathematical expression. It is observed from equation (57) that by using step one above, one can use the negative or the positive value of R to accomplish this analysis. Second, solve the resulting equation of eight degrees with the purpose of obtaining C.

The procedure involves the following steps:

Step 1: By using, for example, the negative value for R in equation (57), one obtains

By defining the following groups of parameters:




and by substituting the expressions for a, R and D given by equations (49), (50) and (51) into equations (59) and (60) one is able to rewrite functions F and G as




Equation (58) is now simply



Step 2: Now, in order to obtain C, one needs to square expression (63) with the objective of eliminating the square root; thus,

By using equations (61) and (62), one may write

and

Finally, by rearranging equation (64), one obtains an eight-degree polynomial in the unknown C:



Polynomial equation (67) is very useful for obtaining C. For example, by using the Mathematica software one can find the eight roots or the eight values of C for every power-law fluid. In other words, for every value of n, one can calculate the value of C for the different values of the Joule heating parameter, f. Some of the values of the roots are imaginary and some are real numbers. The selection of the best values of C was based on the values that satisfy equation (56) with a physical meaning.

To obtain a dimensionless velocity, it is necessary to multiply by the expression of the velocity of power-law fluids given by equation (38).

This operation gives



or, alternatively,



In the equation above, k₁, k₂, and k3 are dimensionless parameters that are independent of the temperature difference between the walls and they are shown in Table 1. Finally, by rearranging the terms of the equation above, the following function for W_z (h ) is found:





The Grashof number indicated in this equation and for the case of non-Newtonian fluids is defined as



Equation (70) can be expressed simply as



where parametric function F(h,f) with the Joule heating number, f, is defined as



By dividing equation (72) by the Grashof number, one arrives at the "universal" or "reduced" velocity profile for power-law fluids:



In other words, the expression identified in equation (74) is nothing but a "universal" velocity profile that depends upon parameters k_i, where i =1, 2 ,3, and non-dimensional variable h.

In Table 1 a summary of the parameters and variables of the "universal" velocity profile for a power-law fluid is presented.

RESULTS AND DISCUSSION

Several calculations have been made to illustrate the effect of Joule heating generation on the velocity profiles for power-law fluids, both dilatant (n>1) and pseudoplastics (n<1), with values of n ranging between 0.171 and 2.0 (Bosse, 1998). In this contribution and for illustrative purposes, the case of n = 0.575 (Figure 2) will be discussed.

Figure 2



SUMMARY AND CONCLUSIONS

This study has focused on analyzing the behavior of non-Newtonian carrier fluids, i.e., buffers under several conditions of Joule heating generation. The results of the study have shown that the power-law model is very useful in gaining insight into the potential benefits of the use of non-Newtonian fluids in attenuating the fluid mixing effects in an electrophoresis cell. Mixing is due to the free convection flows originated by the Joule heating. In fact, the analysis has made it possible to identify a family of potential fluids that may be highly beneficial for obtaining a very good efficiency with relatively high values of the applied electrical field.

In addition, the study has shown a very effective mathematical approach to deriving the hydrodynamic velocity profile inside the electrophoresis cell. This approach is based on the Taylor expansion of the integrand that appears in the formal velocity profile after integration of the momentum equation. In fact, the actual velocity profile is obtained just by solving an eight-degree polynomial, which is the result of applying the boundary conditions to the system.

Finally, the results of this study suggest that the Joule heating effect can be attenuated or controlled substantially by changing and selecting the rheology of the non-Newtonian carrier. This leads to the observation that mixing (by free-convection) could be potentially diminished not only by a careful selection of operating parameters but also by selecting, i.e., "tuning", the nature of the rheology of the carrier fluid flowing through the electrophoretic cell.

ACKNOWLEDGEMENTS

The financial support given to Maria A. Bosse by Fulbright Laspau and by the Universidad Católica del Norte is gratefully acknowledged. Partial support received by M.A.B. from MARTECH-FSU is also acknowledged. P.A. is grateful for partial support in the form of a "Development Scholar Grant" from the Council of Research and Creativity (FSU). A preliminary version of the article was presented at the COBEQ-2000, Aguas de São Pedro, Brazil, September 24-27, 2000. This is contribution number 419 of the Geophysical Fluid Dynamics Institute.

NOMENCLATURE

Variables

C concentration of solute i (kg/m³) mean concentration of solute i (kg/m³) deviation concentration of solute i (kg/m³) Cp heat capacity of fluid (kJ/kg K) C, c integration constant d cell width (m) D diffusivity E electric field (Volt/cm) g gravitational constant (m/s²) k thernal conductivity k_e electric conductivity (1/ohm cm) L cell length (m) m fluid consistency (J sⁿ/m²) n rheological index, dimensionless p pressure (kpa) Q Joule heating generation (kJ) T temperature (K) average temperature (K) t time (s) v carrier velocity (m/s) mean carrier velocity (m/s) deviation carrier velocity (m/s) w cell thickness (m)

Greek letters

r density b coefficient of volume expansion m viscosity m_i electrophoretic mobility t shear stress tensor dimensionless transversal coordinate Joule heating number, dimensionless temperature difference between the walls

Sub-index

i i component x x direction y y direction z z direction 1 cool side 2 hot side eff Q effective coefficient

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Publication Dates

• Publication in this collection
  25 May 2001
• Date of issue
  Mar 2001

History

• Accepted
  22 Nov 2000
• Received
  30 Aug 2000