A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 86
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Indeed the DME is actually such a system, but wastreated as a stationary electrode by the approximate method described in Chapter 7.Methods involving convective mass transport of reactants and products are sometimescalled hydrodynamic methods; for example, the techniques involving measurement oflimiting currents or i-E curves are called hydrodynamic amperometry and voltammetry,respectively.The advantage of hydrodynamic methods is that a steady state is attained ratherquickly, and measurements can be made with high precision (e.g., with digital voltmeters), often without the need for recorders or oscilloscopes. Moreover, at steadystate, double-layer charging does not enter the measurement.
Also, the rates of masstransfer at the electrode surface in these methods are typically larger than the rates ofdiffusion alone, so that the relative contribution of mass transfer to electron-transferkinetics is often smaller.1 Even though it might first appear that the valuable timevariable is lost in steady-state convective methods, this is not so, because time entersthe experiment as the rotation rate of the electrode or the solution velocity with respect to the electrode. Dual-electrode techniques can be employed to provide thesame kind of information that reversal methods do in stationary electrode techniques.These methods are also of interest in the continuous monitoring of flowing liquids,and in treatments of large-scale reactors, such as those employed for electrosynthesis(see Section 11.6).Construction of hydrodynamic electrodes that provide known, reproducible masstransfer conditions is more difficult than for stationary electrodes.
The theoretical treatments involved in these methods are also more difficult and involve solving aAlthough diffusional fluxes at ultramicroelectrodes can often exceed those available by convection at largerelectrodes.331332Chapter 9. Methods Involving Forced Convection—Hydrodynamic Methodshydrodynamic problem (e.g., determining solution flow velocity profiles as functions ofrotation rates, solution viscosities, and densities) before the electrochemical one can betackled.
Rarely can closed-form or exact solutions be obtained. Even though the numberof possible electrode configurations and flow patterns in these methods is limited only bythe imagination and resources of the experimenter, the most convenient and widely usedsystem is the rotating disk electrode. This electrode is amenable to rigorous theoreticaltreatment and is easy to construct with a variety of electrode materials.
Most of what follows deals with it and its variations.9.2 THEORETICAL TREATMENTOF CONVECTIVE SYSTEMSThe simplest treatments of convective systems are based on a diffusion layer approach. Inthis model, it is assumed that convection maintains the concentrations of all species uniform and equal to the bulk values beyond a certain distance from the electrode, 8. Withinthe layer 0 < x < 8, no solution movement occurs, and mass transfer takes place by diffusion. Thus, the convection problem is converted to a diffusional one, in which the adjustable parameter 8 is introduced. This is basically the approach that was used in Chapter1 to deal with the steady-state mass transport problem.
However, it does not yield equations that show how currents are related to flow rates, rotation rates, solution viscosity,and electrode dimensions. Nor can it be employed for dual-electrode techniques or forpredicting relative mass-transfer rates of different substances. A more rigorous approachbegins with the convective-diffusion equation and the velocity profiles in the solution.They are solved either analytically or, more frequently, numerically.
In most cases, onlythe steady-state solution is desired.9.2.1 The Convective-Diffusion EquationThe general equation for the flux of species j , Jj, is (equation 4.1.9)Z F\Jj = -DjVCj - jfDf^+ CjV(9.2.1)where on the right-hand side, the first term represents diffusion, the second, migration,and the last, convection. For solutions containing an excess of supporting electrolyte, theionic migration term can be neglected; we will assume this to be the case for most of thischapter (see, however, Section 9.3.5). The velocity vector, v, represents the motion of thesolution and is given in rectilinear coordinates byx, y, z) = ivx + ju y + kvzwhere i, j , and к are unit vectors, and vx, vy, and vz are the magnitudes of the solutionvelocities in the x, v, and z directions at point (x, y, z). Similarly, in rectilinear coordinates,da^ i ^ +jdC;dCs^+k ^(9.2.3)The variation of Cj with time is given bydC,= - V • Jj = div Jj(9.2.4)9.2 Theoretical Treatment of Convective Systems333By combining (9.2.1) and (9.2.4), assuming that migration is absent and that Z)j is not afunction of x, y, and z, we obtain the general convective-diffusion equation:-A = D j V 2 C j - v • VCj(9.2.5)The forms for the Laplacian operator, V2, are given in Table 4.4.2.
For example, for onedimensional diffusion and convection, (9.2.5) is-*r = D*l?-v'-*F(9 2 6)--Note that in the absence of convection (i.e., v = 0 or vy = 0), (9.2.5) and (9.2.6) are reduced to the diffusion equations. Before the convective-diffusion equation can be solvedfor the concentration profiles, Cpc, y, z), and subsequently for the currents from the concentration gradients at the electrode surface, expressions for the velocity profile, \(x, y, z),must be obtained in terms of x, y, z, rotation rate, and so on.9.2.2Determination of the Velocity ProfileAlthough it is beyond the scope of this chapter to treat hydrodynamics in any depth, abrief discussion of some of the concepts, terms, and equations is included to provide somefeeling for the approach and the results that follow.
For an incompressible fluid (i.e., afluid whose density is constant in time and space), the velocity profile is obtained by twodifferential equations with appropriate boundary conditions. The continuity equation,V • v = div v = 0(9.2.7)is a statement of incompressibility, whereas the Navier-Stokes equation,sdtrepresents Newton's first law (F = ma) for a fluid. The left side of (9.2.8) represents ma(per unit volume, since ds is the density), and the right side represents the forces on a volume element (P is the pressure; 17S, the viscosity; and f the force/volume exerted on an element of the fluid by gravity). The term T/SV2V represents frictional forces.
This equationis usually written in the form^j- = ^r- ^P + ^V2v + —dtds(9.2.9)dswhere v = rjs/ds is called the kinematic viscosity and has units of cm2/s. Values for various solutions are given in Table 9.2.1. The term f represents the effect of natural convection arising from the buildup of density gradients in the solution.Two different types of fluid flow are usually considered in hydrodynamic problems(Figure 9.2.1). When the flow is smooth and steady, and occurs as if separate layers (laminae) of the fluid have steady and characteristic velocities, the flow is said to be laminar.For example, the flow of water through a smooth pipe is typically laminar, with the flowvelocity being zero right at the walls (because of friction between the fluid and the wall)and having some maximum value in the middle of the pipe. The velocity profile underthese conditions is typically parabolic.
When the flow involves unsteady and chaotic motion, in which only on the average is there a net flow in a particular direction, it is termed334Chapter 9. Methods Involving Forced Convection—Hydrodynamic MethodsTABLE 9.2.1 Kinematic Viscositiesof 0.1 M TEAP Solutions at 25.0°CaSolutionv, cm2/sH2OH 2 O(0.1MKCl)*MeCNDMSOPyridineDMFAf,/V-DimethylacetamideHMPAD2O0.0091320.0088440.0045360.018960.0095180.0089710.010670.035300.01028aFrom M. Tsushima, K.
Tokuda, and T.Ohsaka, Anal. Chem., 66,4551 (1994).^Contains KC1 instead of TEAP.turbulent. This type of flow might result from a barrier being placed in a pipe to obstructthe flow stream.The solution of the hydrodynamic equations requires modeling the system, writingthe equations in the appropriate coordinate system (linear, cylindrical, etc.), specifying theboundary conditions, and usually, solving the problem numerically. In electrochemicalproblems, only the steady-state velocity profile is of interest; therefore (9.2.9) is solvedfor d\/dt = 0.Often, the equations are rewritten in terms of dimensionless groups of variables.
Onethat occurs in many hydrodynamic problems is the Reynolds number, Re, which relates acharacteristic velocity, i>ch (cm/s), and a characteristic length, / (cm), for the system of interest. In general terms, it is given byRe = vchl/v(9.2.10)The Reynolds number is proportional to fluid velocity, so high values imply high flow orelectrode rotation rates. For flow rates below a level characterized by a certain criticalReynolds number, Recr, the flow remains laminar. When Re > Re cr the flow regime becomes turbulent.General treatments of the formulation and solution of hydrodynamic problems, especially as they relate to problems in electrochemistry, are available (1-6).Figure 9.2.1 Typesof fluid flow. Arrowsrepresent instantaneousLaminar flowTurbulent flowlocal fluid velocities.9.3 Rotating Disk Electrode3359.3 ROTATING DISK ELECTRODEThe rotating disk electrode (RDE) is one of the few convective electrode systems for whichthe hydrodynamic equations and the convective-diffusion equation have been solved rigorously for the steady state.
This electrode is rather simple to construct and consists of a disk ofthe electrode material imbedded in a rod of an insulating material. For example, a commonlyused form involves a platinum wire sealed in glass tubing with the sealed end ground smoothand perpendicularly to the rod axis. More frequently, the metal is imbedded into Teflon, epoxyresin, or another plastic (Figure 9.3.1). Although the literature suggests that the shape of the insulating mantle is critical and that exact alignment of the disk is important (7), these factors areusually not troublesome in practice, except perhaps at high rotation rates, where turbulenceand vortex formation may occur. It is more important that there is no leakage of the solutionbetween the electrode material and the insulator.
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