**THE ROLE OF THE RHEOLOGICAL PROPERTIES OF NON-NEWTONIAN FLUIDS IN CONTROLLING DISPERSIVE MIXING IN A BATCH ELECTROPHORETIC CELL WITH JOULE HEATING**

M.A.Bosse^{1}, P.Arce^{2}, S.A.Troncoso^{1} and A.Vasquez^{3}

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

Phone: 56-55-355193, Fax: 56-55-355917, E-mail: mbosse@ucn.cl

^{2}Department of Chemical Engineering, Geophysical Fluid Dynamics Institute, GFDI,

Florida State University, Tallahassee, FI 32308, USA, E-mail: arce@eng.fsu.edu

^{3}Department of Mathematics, Universidad Católica del Norte, Casilla 1280,

Antofagasta, Chile, E-mail: avasquez@ucn.cl

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

Abstract -The problem of the effect of Joule heating generation on the hydrodynamic profile and the solute transport found in electrophoretic devices is addressed in this article. The research is focused on the following two problems: The first one is centered around the effect of Joule heating on the hydrodynamic velocity profile and it is referred to as "the carrier fluid problem." The other one is related to the effect of Joule heating on the solute transport inside electrophoretic cells and it is referred to as "the solute problem". The hydrodynamic aspects were studied first to yield the velocity profiles required for analysis of the solute transport problem. The velocity profile obtained in this study is analytical and the results are valid for non-Newtonian fluids carriers. To this end, the power-law model was used to study the effect of the rheology of the material in conjunction with the effect of Joule heating generation inside batch electrophoretic devices. This aspect of the research was then effectively used to study the effect of Joule heating generation on the motion of solutes (such as macromolecules) under the influence of non-Newtonian carriers. This aspect of the study was performed using an area-averaging approach that yielded analytical results for the effective diffusivity of the device.

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

INTRODUCTION

In general, the effect of Joule heating on the free convection process of electrophoretic devices has not been studied systematically using basic principles (Hunter, 1995). Therefore, the general goal of this research, which follows the strategy recently introduced by Bosse and Arce (2000 a&b), is to promote a fundamental understanding of the role of Joule heating on such a process, with the purpose of helping in the future design of devices where electrophoretic transport plays a role.

]]> The specific objectives related to the analysis of the solute-free carrier fluid problem are:*(a)* The analysis of the effect of Joule heating on the hydrodynamic velocity profile of the electrophoretic cell sketched in Figure1 for non-Newtonian types of carriers.

*(b)* The study of the sensitivity of the rheology of the carrier fluid in free convection currents inside the cell. Two rheological models, i.e., the power-law model and the CEF (Criminale-Ericksen-Fibley) model for viscoelastic fluids, will be used in the study.

The objectives associated with the solute problem are centered on the magnitude of the effect of Joule heating on the transport parameters, such as dispersive mixing, inside the free-convection electrophoretic cell.

In this particular situation, a method of area averaging (Whitaker, 1985) has been very helpful for *a priori* determination of the effect of Joule heating on the effective diffusion and effective convection in the device.

The system shown in Figure 1 depicts the general purpose of the research presented here of studying the role played by Joule heating in a batch electrophoretic cell. For example, a basic question to ask would be, what would happen if an electrical field of a relatively high intensity is applied to a free convection system, such as the one sketched in Figure 1? Figure 1 shows a free convection cell where "Q," or the heat generation term, is equal to zero. The schematic diagram shows that the cell walls are maintained at two temperatures, T_{2} (hot side) and T_{1} (cool side). The cell has a length, L, and a thickness, w. The width of the cell is given by d. L is assumed to be L >> d or w. The coordinate system is placed at one of the ends of the cell (at the hot side). The velocity profile shown in Figure 1 is for a typical case of a non-Newtonian fluid placed within the system, with no heat generation.

**]]>**

PROBLEM ONE: THE CARRIER FLUID PROBLEM

As introduced by Bosse and Arce (2000a), the carrier fluid problem has two related sub-problems. The first one is centered around the heat transfer aspects and the second one, on the hydrodynamic aspects. The analysis of these two problems is presented (sequentially) in the section below. Many mathematical details were reported in Bosse et al. (2001, this issue), and only a presentation of the problem with an abbreviated discussion will be included here.

Temperature Profile

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-type fluids, and negligible viscous effects (i.e., the energy dissipation will be assumed to be negligible).

Equation (1) can now be reduced considerably due to the kinematics of the free convection cell under the assumptions stated above. The kinematics for the system sketched in Figure 1 are

since a one-dimensional flow has been assumed.

]]> In addition, the system is examined under incompressible flow conditions, from a mass conservation point of view, and therefore, density is not a function of either space or time. The velocity in the x and y directions is assumed to be negligible, compared to the velocity in the z direction. Therefore, equation (1) becomesEquation (3) also assumes that conduction is only important in the y direction since the ratio d/L<<1. 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. This is, in fact, the so-called Batchelor assumption (1954). The physical meaning of equation (3) is that conduction in the y direction is balanced by Joule heat generation. In other words, the heat generated by the Joule effect is released (from the cell) mainly by conduction.

The boundary conditions selected for the system sketched in Figure 1 are:

As mentioned in Bosse et al. (2001, this issue), other boundary conditions could be selected but here the emphasis was on choosing the simplest physical situation possible.

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

]]>where

The temperature profile given by equation (7) is required for analysis of the hydrodynamics of the cell in Figure 1. This study is presented below.

**Hydrodynamic Velocity Profile**

The general equation of motion for a non-Newtonian fluid in the cell sketched in Figure 1 is (Bird et. al., 1960)

The same assumptions as those used in the heat transfer model will be adopted here. Thus, the same kinematics and simplified continuity equation will be valid here. Furthermore, the Boussenesq approximation is valid for the study, 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 a spatial dependence of density on position. However, since for this particular case, the Boussenesq approximation is valid, then the incompressible condition helps to simplify the hydrodynamic velocity profile. 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), a Taylor series in the T variable is used 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. Under this set of assumptions equation (10) becomes

The physical meaning of this equation is that the viscous forces are just balanced by the buoyancy forces generated by temperature differences. 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 (Bird et al., 1960) and CEF models (Syrjälä, 1998), 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 and CEF fluids flowing in the free convection cell sketched in Figure 1:For simplicity, one can express equation (16) in the following way:

where one can identify the following expression:

Function F(y,f) corresponds to

and the coefficients to

]]> Thus, the general (formal) velocity profile for the fluid can be computed aswhere C_{1} (see equation (19)) and C_{2} are constants of integration.

If one substitutes equation (19) into equation (21), the following expression is obtained:

One effective way to solve the integral form in equation (22) is by performing a series expansion of the polynomial indicated in the integrand of this equation. For this purpose let’s redefine, for algebraic convenience, the following function:

If one now expands f(y) in a Taylor series around y_{0}=0, the origin of the coordinates, where v_{z}(y)=0, one obtains the following MacLaurin series:

Next, by evaluating and by substituting the expression for f(0), f¢(0), f¢¢(0) and f¢¢¢(0) into equation (24), one obtains

]]>where

The boundary condition for the problem under analysis comes from the nonslip boundary condition at the wall, i.e.,

]]> Applying boundary conditions given by equation (31) and equation (32),

Substituting C_{1} and C_{2},

This function requires the value of (see equation (20)). However, equation (35) must satisfy the mass conservation equation, i.e.,

This equation introduces an additional restriction, and now by substituting the expression of the velocity profile given by equation (35) into equation (36) and integrating, one obtains

Next, using equation (33) and equation (37), one can obtain the values for the integration constant, C_{1}, and parameter C. For this purpose, the Newton-Raphson procedure for nonlinear equations, has been used. Once one has the values for C and C_{1}, it is possible to evaluate the velocity profile for power-law and CEF fluids. The non-dimensional expression for the velocity profile can be obtained simply by multiplying equation (35) by

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

and the expressions for the values k_{1}, k_{2} and k_{3}, dimensionless values independent of the temperature difference between the walls, are:

Finally, by dividing equation (38) by the Grashof number, it is possible to obtain a "universal" or "reduced" velocity profile for power-law and CEF fluids:

where U_{z} is defined as

Further details about the solution procedure can be found in Bosse et al. (2001, this issue).

**]]>**

PROBLEM TWO: THE SOLUTE PROBLEM

Once the temperature and velocity profiles have been determined, the strategy introduced by Bosse and Arce (2000a) suggests the analysis of the solute problem. This task is performed in this section of the article. Since the electrophoretic cell described in Figure 1 is a rectangular cell, the system may be modeled by using Cartesian coordinates under the assumption that proteins in the buffer behave as a homogeneous solution. This assumption is valid only for the protein concentration in the dilute limit. In addition, the model assumes constant density (see section two above) and diffusivity. Furthermore, it will be assumed that the solute (i.e., proteins) moves at the local fluid velocity, in this case, the buffer velocity.

The analysis of solute motion inside the cell will consider the transport of, for example, proteins due to convection as well as diffusive transport in the y direction (i.e., transversal direction of the electrophoretic device). The presence of a DC electrical field in the transversal direction will add a solute migration in this direction. The equation for this case is (Newman, 1991)

Equation (43) needs boundary conditions for the system that is shown in Figure 1. These are

The strategy to be used in the analysis of the model, based on equation (43) and boundary conditions, will be centered on the area-averaged approach. For this reason, additional boundary conditions are not required in this study. The method of area averaging was used to obtain effective transport coefficients that will help to understand the effect of Joule heating on the separation efficiency of solutes in an electrophoretic device. The most interesting feature of this method is the derivation of analytical constitutive equations for these transport parameters involved in the effective transport of solute in the cell.

The application of the method of area-averaging consists of several steps that can be summarized as follows. The first step involves the definition of the proper area averaging for a general spatial function f(x,y,z) (Whitaker, 1985):

]]>Applying this property, equation (43) becomes

Then by using step 2 (Use of Gray’s Decomposition), step 3 (Deviation Field Model), step 4 (Closure Problem), step 5 (Solution) and step 6 (details of the procedure can be found in Bosse, 1998), one is able to arrive at the equation for the effective coefficients.

The application of this sequence of steps results in the expression for the effective diffusion coefficient:

where

]]> Equation (48) is an analytical expression that is useful to obtain information about the magnitude of the dispersive mixing in the cell. This analysis is given in the next section.

RESULTS

The results obtained here for the temperature and the velocity profiles are analytical. Therefore, numerical calculations will be presented in this section only to illustrate the behavior of the system under certain conditions. Temperature results will be calculated first, and then, hydrodynamic calculations will follow.

Temperature Profile

As is known for the case of f=0, which represents the case with no heat generation, the temperature profile is linear. This type of behavior has already been studied and a parabolic profile, whose maximum value depends on the value of f is now observed. In addition, it is observed that this value, i.e., the maximum temperature, moves towards higher values of h as the values of f increase. For values of f greater than 200, the maximum is attained at h = 0.5 (the center of the cell), as is shown in Figure 2 (D T = 1 K).

]]> An analytical demonstration of this fact may be found in Bosse and Arce (2000a).

Velocity Profile

Several calculations have been made to illustrate the effect of Joule heating generation on the velocity profiles for power-law fluids. In Figure 3 the behavior of velocity with respect to the Joule heating parameter, f, is observed. For example, it is noted that there are two regions within the cell. In the first one, corresponding to the left-hand side of the cell, the velocity increases until it reaches a maximum and then decreases. In the second region (corresponding to the right-hand side of the cell), the opposite behavior is observed. Furthermore, it is observed that the effect of Joule heating generation increases the velocity in both directions, but this effect is less important when the value of the index parameters n, is lower. Another important aspect observed in these figures is that higher velocity values may be obtained when the index parameter, n, decreases.

Effective Diffusion Coefficient

One of the most important aspects to be considered in this study is the effect of rheology (of the fluid) on the evaluation of the effect of Joule heating on the effective diffusion coefficient. It is known that this coefficient measures the magnitude of dispersive mixing in the cell. An increase in the Joule heating parameter, f, causes an increase in the effective diffusion coefficient, and this effect is more important when the value of the index of the fluid, n, increases. Table 1 shows this effect for three different fluids with n = 0.171, 0.229 and 0.52.

]]>

The calculation shows the sensitivity of the effective diffusivity to the change in the parameter n. For example, a change from n=0.171 to n=0.229 (for f =6) does not change the values of D_{eff}/D. However, a change from n=0.229 to n=0.520 implies a change from D_{eff}/D = 1 to D_{eff}/D = 9.06. An immediate conclusion is that low values for parameter n do not significantly change the dispersive-mixing conditions in the cell. Larger values, however, tend to increase dramatically such conditions in the cell. A similar analysis can be made for the case of the effects of parameter f. On the other hand, larger values of either n or f tend to increase dramatically the values of D_{eff}/D. The "worst" case shown in the table is for f=18 and n=0.520, where the dispersive mixing inside the cell probably reaches conditions close to those in the well-mixed situation.

SUMMARY AND CONCLUSIONS

A study of the effect of Joule heating generation on the hydrodynamic velocity profile of non-Newtonian carriers, specifically, those fluids that can be described by a power-law model, and are considered to be either pseudoplastic or dilatant, and by viscoelastic (CEF) rheology, has been conducted in this paper. The results of this study suggest that the Joule heating effect can be attenuated a great deal by changing and selecting the rheology of the non-Newtonian carrier. This observation suggests mixing (by free convection) could potentially be controlled by carefully selecting the operating parameters as well as the nature of the rheology of the carrier fluid flowing through the electrophoretic cell. Further work to validate the predictions reported in this article is in progress and the results will be communicated in future contributions.

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 COBEQ-2000, Aguas de São Pedro, Brazil, September 24-27, 2000. This is contribution number 420 of the Geophysical Fluid Dynamics Institute.

NOMENCLATURE

Variables

C | concentration of solute i (kg/m^{3}) |

mean concentration of solute i (kg/m^{3}) | |

deviation concentration of solute i (kg/m^{3}) | |

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/s^{2}) |

k | thermal conductivity |

k_{e} | electric conductivity (1/ohm cm) |

L | cell length (m) |

m | fluid consistency (J s^{n}/m^{2}) |

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) | |

v | deviation carrier velocity (m/s) |

w | cell thickness (m) |

Greek letters

r | density |

b | coefficient of volume expansion |

m | viscosity |

mi | 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 | effective coefficient |

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