8
 
The wall-jet electrode
 

The hydrodynamics of the wall-jet electrode (WJE) are considerably more complex than in the case of the channel or rotating disc electrodes. Fortunately most experiments are conducted at one of two limits. The first is formally referred to as a 'wall-jet' where the jet is very small compared to the electrode radius. Analytical expressions (given in section 8.1.1) have been derived for the velocity profile near the electrode surface in this limit. The second is referred to as a 'wall-tube' or 'micro-jet' where the electrode radius is much smaller than the jet radius. Analytical expressions (given in section 8.2), based on two empirically determined parameters, have been derived for the velocity profile close to the electrode surface. For an intermediate case, no analytical solutions appear to have been attempted, though such a system could be simulated numerically using computational fluid dynamics (usually finite element) software to solve the Navier-Stokes equations.

As will be shown in the conclusions chapter, both of the limiting geometries offer experimental advantages - the former spans a very wide range of timescales and the latter can reach very high mass-transport rates. Unfortunately, the lack of theory available has limited their experimental adoption. This chapter attempts to redress this situation, acknowledging that some modelling has already been done for the wall-jet using the BI method, ignoring radial diffusion (see below). The importance of radial diffusion is therefore assessed and efficient modelling methods are introduced utilising conformal mappings. Finally, working curves and surfaces are computed for common electrochemical mechanisms allowing mechanistic analysis of steady-state voltammetric data.

8.1 Wall-jet simulations in conformal space

Previous attempts1 at a two-dimensional WJE simulation incorporating radial diffusion, using a rectangular mesh in cylindrical polar co-ordinates, failed to give acceptable convergence due to the highly non-uniform concentration distribution in the vicinity of the electrode.

In order to overcome this problem, the WJE simulations by Compton et al.2, relied on the frontal nature of the BI method (section 2.3) to incorporate an expanding grid via an interpolation technique. The frontal nature of the simulation, however, meant that radial diffusion could not be included. Given the decrease in radial velocity towards the disc edge, one might expect this to be significant and this was postulated as a possible explanation for discrepancies between simulations and experimental data at low flow rates (a simulation study was also conducted using the BI method using a displaced radial diffusion term)3.


Figure 8.1: Expanding grid used by Compton et al., based on interpolation.


Figure 8.2: Wall-jet velocity profile.

A more flexible and accurate method to account for the non-uniformity than interpolating between frontal solution vectors is to perform the simulations in a conformal space based on the hydrodynamics. This was done by Laevers et al.4, using a frontal application of the Crank-Nicolson method (see section 2.1.3) to simulate the chronoamperometric response without radial diffusion. The transformation is generalised in this work and applied to the full mass transport equation (including radial diffusion). The following hydrodynamic equations form the basis of the transformation.

8.1.1 Velocity field

Glauert5 studied the velocity distribution in a wall-jet, shown in Figure 8.2, and found an exact solution of the boundary layer equation. The radial component of the convection is given by2,5:

(8.1)

and the normal component by:

(8.2)

where the curvilinear co-ordinate h, is related to r and z by:

where , and and n=5/4.(8.3)

The functions f'(h) and h(h) are related though the following equations:

(8.4)
(8.5)
(8.6)

in terms of the quantity g, a function of h, defined implicitly by:

(8.7)

Figure 8.3: Functional dependence of h on g. The 1st and 3rd order series expansions are also shown.

This is plotted in Figure 8.3 together with one and three term series expansions of g:

(8.8)

where a = 3.086x10-3 and b = 4.8991x10-5. Thus 2-term expressions for vr and vz may be found:

and (8.9)

which, as h(r)0, collapse to the 1-term expressions:

and (8.10)

The one and two term expressions are plotted in Figure 8.4. Compton et al.2 found that the diffusion layer was within h=0.4, which would imply that a 1st-order approximation of the convection would be adequate. It was found in the simulations reported here that the difference in the current, using 1st or 2nd order approximations of the convection, was indistinguishable (to 6 or 7 s.f.) over the range of Peclet numbers studied.

Figure 8.4: (a) normal and (b) radial velocity components (at r=0.2) as a function of h.

8.1.2 Simulation in a general curvilinear space

In order to formulate a simulation method (which incorporates radial diffusion effects) in a conformal space similar to that of Laevers et al., we seek a curvilinear transformation:

a = a(r,z) (r) a = a(r,y) where y = y(r,z)(8.11)

As shown in Appendix 3, under this transformation, the mass-transport equation becomes:

(8.12)

(symbols are defined in the Appendix). The mixed second derivatives present an added complication.


Figure 8.5: Stencils on which the finite difference representation of mixed second derivatives may be formulated. On a 9=point stencil the four corners may be used. On a 7 point stencil, the average may be taken between two sets of four points.

These may be represented as finite differences at the corners of a 9-point stencil:

(8.13)

or the average of the top-left and bottom-right squares of a 7-point stencil:

(8.14)

Note that although the two partial derivatives are not necessarily equal, they reduce to the same finite difference formula. Introducing the coefficients (for brevity):




(8.15)

The finite difference form of the mass-transport equation is therefore:

(8.16)

This may be represented on a 7-point stencil (such as that for MGD1, introduced in section 4.1.1) as:

A1j,k(kyd - kys + kx - kvy) uj-1,k
A2j,k(-kx) uj-1,k+1
A3j,k(krd - krs) uj,k-1
A4j,k(-2kyd - 2krd - 2kx) uj,k
A5j,k(krd + krs) uj,k+1
A6j,k(-kx) uj+1,k-1
A7j,k(kyd + kys + kx + kvy) uj+1,k

Since the curvilinear transformation physically corresponds to a stretching of the z-co-ordinate moving along the electrode, the simulation space (depicted in Figure 8.6) and the boundary conditions (shown in Table 8.1) remain as they would be in cylindrical polar co-ordinates.


Figure 8.6: Simulation space for the WJE.

Table 8.1: Boundary conditions for WJE simulations in conformal space.

Zone

Boundary condition

r = 0, all y(symmetry)
r = rmax, all y(waste fluid)
y = 0, r < rea = 0 (electrode)
y = 0, r > re(insulator)
y = ymax, all ra = 1 (bulk solution)

The implementation of the boundary conditions, shown in Table 8.2, is complicated by the cross-derivatives as the 'extra' boundary nodes (2 and 6) must reinforce the boundary conditions. Note that at r=0 there is a singularity in the Laplacian operator (since 1/r = infinity), but since the simulation space stops one node short of the boundary, this problem is avoided (if the boundary were to be simulated, a Maclaurin expansion could be used to avoid the problem).

Table 8.2: Finite difference implementation of boundary conditions (in MGD1 notation).
Disc centrek=1, j=1...NGYA>4j,k = A>4j,k + A3j,k; A3j,k = 0
k=1, j=1...NGYA>7j,k = A>7j,k + A6j,k; A6j,k = 0
Wastek=NGX, j=1...NGYA>4j,k = A4j,k + A>5j,k; A5j,k = 0
k=NGX, j=2...NGYA1j,k = A1j,k + A2j,k; A2j,k = 0
Bulk solutionj=NGY, k=1...NGXA7j,k = 1
j=NGY, k=2...NGXA>6j,k = 1
Electrodej=1, k=1...kEA1j,k = 0
j=1, k=2...kE-1A>2j,k = 0
Insulatorj=1, k=kE...NGXA4j,k = A4j,k + A1j,k; A1j,k = 0
j=1, k=kE-1...NGXA5j,k = A5j,k + A2j,k; A2j,k = 0


Figure 8.7: Curvilinear 'boundary layer' co-ordinate


Figure 8.8: Convergence plot comparing trapezoidal and rectangular integration.

So far, all the formulation has been for a general function, y. This approach has the advantage that functions can readily be interchanged and thus optimised without reworking the algebra. We now introduce a specific function for y which allows the full 2-D steady-state problem to be solved efficiently.

8.1.3 A conformal space based on the boundary layer

As Leavers et al. suggested, the hydrodynamic co-ordinate, h, (plotted in Figure 8.7) may be used as the basis of the transformation:

(8.17)

(where n=5/4). The partial derivatives of this variable are


(8.18)

These may be substituted directly into the finite-difference equation (8.16). The current may be evaluated from:

(8.19)

or as a first-order finite difference form using rectangular quadrature:

(8.20)

As in the case of the channel electrode (see section 3.9), significant error cancellation occurs if rectangles are used to perform the integration rather than trapezia. This is illustrated in Figure 8.8.

8.1.4 Further refinements to the space transformation for efficient 2-D modelling


Figure 8.9: Slice through Figure 8.11, at one node above the electrode surface.

Figure 8.10: Slice through Figure 8.11 in the normal co-ordinate at the disc centre.


Figure 8.11: Simulated concentration distribution in (r,h) space at log10 Ps = 5.45. Parameters are: NGX=121, NGY=121, ld=1, hmax=1, n=0.0089cm2s-1, D=2x10-5cm2s-1, [A]bulk=1x10-6 molcm-3, vf=0.05cm3s-1, re=0.2cm, rjet=0.0345cm, 2nd order convection, upwind differences in r, central in h.

At high Peclet numbers, the concentration distribution in the conformal space resembles a rotating disc electrode (RDE). The flux is uniform across the electrode surface, shown in Figure 8.9, until the electrode edge where it rises sharply. The flux in the normal co-ordinate, shown in Figure 8.10, is linear near the electrode surface, then curving towards its bulk value away from the electrode. At lower Peclet numbers, radial diffusion becomes more significant and the concentration distribution tends to that of a microdisc. This suggests that secondary transformations may be applied based on those used to improve the efficiency of microdisc or RDE simulations.

One possibility would be to use a second conformal transformation such as Amatore and Fosset's6 to transform the (r,h) space into a closed form. There are a number of disadvantages with this, however:

Therefore, based on the work of Taylor et al.8, expanding grids were used to refine the mesh in the diffusion layer and around the boundary discontinuity at the electrode edge.

8.1.4.1 Sigmoidal expansion of the radial co-ordinate


Figure 8.12: Sigmoidal co-ordinate transformation to concentrate finite difference nodes at the edge of a disc. Several values of g are shown.

In the case of the microdisc electrode, the singularity at the electrode edge is the major impediment to convergence. The effect is less pronounced in the wall-jet as the radial convection increases with the distance from the jet centre - this effectively 'smears out' the concentration gradient at the electrode edge. Even so, the concentration profile is still steepest at the electrode edge and also (due to the 2pr integration factor) this is the zone of the electrode that contributes the largest fraction of the current, hence any error in this zone will be amplified. This suggests, particularly at low Peclet numbers, that the simulation accuracy could be improved by transforming the radial co-ordinate via a sigmoidal function to concentrate nodes at the electrode edge, as shown in Figure 8.12. Taylor et al.8 used a Fermi-Dirac function (sigmoid) for this purpose:

(8.21)

As with the channel electrode (Chapter 6) a dimensionless expansion parameter, gd:

gd = re/g(8.22)

was used in the simulations, which is independent of the electrode radius.

In order to perform the transformation (which is presented in Appendix 3):

a(r,y) => a(r,y) where r=r(r)(8.23)

we require the first and second derivatives of the transformed variable with respect to r, since the mass-transport equation is now given by:

(8.24)

The first derivative is:

(8.25)

and the second derivative:

(8.26)

A simulated concentration profile is shown in Figure 8.13, showing the contraction of the mesh at the electrode edge resulting in better definition of the rapidly changing concentration gradient in this region.


Figure 8.13: Simulated concentration profiles in (r,h) space and (r,h) space with an expansion parameter of 10. The concentrations at all h values are shown on the plot. Parameters are: NGX=24, NGY=25, ld=1,hmax=1, n=0.0089, D=2x10-5, [A]bulk=1x10-6, vf=0.05cm3s-1, re=0.2cm, rjet=0.0345cm, 2nd order convection, upwind differences in r, central in h.

Whilst conducting simulations in this space, it became apparent that at low Peclet numbers the large values of R (dimensionless) required to accommodate radial diffusion were resulting in values of r very close to 1 which were being rounded up due to finite machine precision. As was found with ChE simulations in section 6.3, this resulted in a singular system of equations.

If e is the machine precision, the point where the co-ordinate transformation breaks down is:

1-rmax < e(8.27)

where rmax is the value at the downstream boundary, given by:

(8.28)

Thus the co-ordinate transformation may be used only when:

ld.gd < ln(e-1-1) » ln(e-1)(8.29)

Since ld is fixed by the Peclet number, this effectively imposes an upper limit on the amount of grid expansion which decreases as the amount of radial diffusion increases:

gd < ln(e-1)/ld(8.30)

The value of ln(e-1) was found to be approximately 86 using 8 byte double precision floating-point variables on an SGI Indigo2 (tm). This is inconvenient, although it is partially offset by the concentration profile being more spread out at low Peclet numbers, thus a lower grid expansion factor is required. It should be noted that similar problems should occur with any other transformation function which is used to expand the grid nodes 'exponentially' towards infinity, but is fixed at a finite point (other than the origin) such as the electrode edge. This is one argument for adopting the approach of Pastore et al. 9, fixing the grid at infinity and allowing the electrode size to vary with the mesh size. The main disadvantage of such a method is that a transformation to an equivalent dimensionless problem is required to simulate a given electrode length (for example in the ChE simulations in section 6.3 this was achieved by compensating the cell height). This is especially important if one wishes to use 2nd or higher-order convection approximations in WJE simulations where the current is no longer a sole function of the Peclet number.

A second disadvantage with this transformation (both for the WJE and microdisc) becomes apparent when one considers a simulation involving homogeneous kinetics, such as an ECE process. Although a sigmoidal transformation might improve the definition of species A, the concentration distribution of kinetically unstable species falls away rapidly along the radial co-ordinate, thus a transformation which shifts points from the centre to the edge of the disc would impede the definition of these species. Since Taylor et al.8 only considered heterogeneous kinetics, they did not encounter this problem. In Chapter 9, the conformal mapping of Amatore and Fosset6 is shown to be well-suited to the simulation of homogeneous processes at a microdisc electrode.

8.1.4.2 Exponential expansion of the normal co-ordinate

Returning to the WJE, the major obstacle to convergence of the (r,h) simulations was the accurate description of the concentration gradient near the electrode surface, hence rather high values of NGY were required for satisfactory accuracy. An exponentially expanding grid in the normal co-ordinate mimics the wall-jet concentration profile in (r,h) space reasonably well and has been successfully applied to the RDE10. This may be incorporated into the generic WJE transformation by transforming h through Feldberg's ln(1+ax) expanding grid function:

where and (8.31)

The first derivatives are:

(8.32)

and

(8.33)

The second derivatives are:

(8.34)

and

(8.35)

Again, these can be substituted directly into the finite difference form of the transformed mass transport equation. A simulated concentration profile is shown in Figure 8.14. The transformation packs more nodes close to the electrode surface, so that the curvature of the diffusion layer is reduced in the transformed co-ordinate.


Figure 8.14: Concentration profile in (r,y) space with an expansion parameter of 10. Parameters are: NGX=121, NGY=121, ld=1, hmax=1, n=0.0089cm2s-1, D=2x10-5cm2s-1, [A]bulk=1x10-6 molcm-3, vf=0.05cm3s-1, re=0.2cm, rjet=0.0345cm, 2nd order convection, upwind differences in r, central in y.


Figure 8.15: Convergence plot for (r,h) mapping and the (r,y) mapping with roughly optimal and greater than optimal expansion parameters. Of course as the expansion parameter tends to zero, the mapping tends to the (r,h) case.

Optimisation of the expansion parameter is made awkward by the fact that the current continually drops with increasing grid expansion, thus unlike the case with an exponentially expanding normal co-ordinate in the ChE (see section 6.1), there is no turning point to signify the optimal value. For this example system, by examining the rate of convergence in the r-co-ordinate, optimal values of the expansion parameter were found to be around 2.5 (using hmax=1), as shown in Figure 8.15, and since hmax does not need to be varied, this value will suffice for most simulations. The rate of convergence is considerably accelerated - the number of nodes required is reduced by a factor of 6 in this case. The main advantage of this transformation, however, is when fast kinetics are simulated, since it concentrates nodes in the reaction layer. Of course the exponential function only roughly approximates the shape of the concentration distribution (consider the linearity near the electrode surface in Figure 8.10). It might be possible to construct a more efficient (at least for an electron transfer) closed-space transformation for the normal co-ordinate based on the Hale transformation for the RDE (see section 7.2).

8.1.5 Effects of radial diffusion

It is shown in Appendix 4 that under a 1st order convection approximation, the dimensionless current is a sole function of the shear-rate Peclet number, Ps. In this section WJE simulations in conformal (r,h) space are used to quantify this relationship. At high Peclet numbers, this should limit to Levich behaviour. The departure from Levich behaviour at lower Peclet numbers allows a quantitative assessment of the importance of radial diffusion. Simulations were run for the parameters shown in Table 8.3.

Table 8.3: parameters used in WJE simulations
Kinematic viscosityn8.9x10-3 cm2s-1
Diffusion coefficientD3.2x10-6 cm2s-1
Electrode Radiusre4.025x10-1 cm
Nozzle radiusrjet3.45x10-2 cm
Bulk concentration[A]bulk1x10-6 mol cm-3
Space downstreamld2
number of nodes in RNGX641
Number of nodes in hNGY641


Figure 8.16: Working curve for a simple electron transfer showing the departure from Levich behaviour at low Peclet numbers due to radial diffusion. The response of the channel electrode (transformed through an equivalent diffusion layer thickness - see section 10.3) is also shown for comparison.

The simulation data is summarised in Figure 8.16, a log-log working curve of dimensionless current vs. Peclet number. The linear Levich region is evident at high Peclet numbers as is the departure from this behaviour due to radial diffusion effects at low Peclet numbers. From the simulation data one may conclude that radial diffusion effects become significant (i.e. augment the current by >1%) below log10Ps = 3.5. The experimentally accessible11 range of Peclet numbers span from ca. 1.4 (where radial diffusion augments the current by 11%) up to 9.0 (where the Levich equation is obeyed). Note that the amount of axial diffusion is greater at the equivalent channel microband than radial diffusion at the wall-jet.

8.1.6 Working curves and surfaces for analysis of steady-state voltammetry

Since radial diffusion effects are negligible over most of the experimentally-accessible range of Peclet numbers, a working curve of the experimental observable vs. dimensionless rate constant can be used for the analysis of most kinetic data. This also allows a solver such as the BI or frontal PKS method (see section 4.5) to be used to conduct these simulations, saving substantially in CPU time. One problem with using a frontal solver is the choice of the initial concentration vector corresponding to the jet impinging on the electrode surface. Both Laevers et al.4 and Compton et al.2 used the bulk concentration in their BI simulations, but Figure 8.10 shows that the concentration is uniform in h across the disc surface at steady-state. This means that in a steady-state simulation, if the bulk concentration is used for the initial concentration vector, the current is artificially augmented and a very high number of nodes in the x co-ordinate are required for accurate simulations. This is illustrated in Figure 8.17. Since the simulations of Laevers et al.4 were time-dependent (chronoamperometry), the error from the incoming concentration boundary would increase as the current approached steady-state.

In such a simulation, the concentrations relax to the true value by the end of the electrode, and this should also, in fact, be the same concentration vector as at the disc centre. Therefore the incoming concentration vector may be 'pre-relaxed' before the simulation commences.


Figure 8.17: Electrode surface concentrations. Errors are evident in the BI simulations due to the jet boundary condition. The 'true' value was simulated using NGX=50 and 20 pre-relaxations.


Figure 8.18: Convergence plot of frontal WJE simulations showing the importance of pre-relaxing the incoming concentration vector. 20 cycles of the Thomas Algorithm (on the mass-transport coefficients at k=1) were used to relax the initial concentration vector.

This can be done by sweeping the concentration vector of the incoming species until it reaches a steady value (since only 10-20 iterations are required, this is computationally inexpensive). The actual BI or Frontal PKS simulation may then be conducted (adding any kinetic terms) using this vector as the starting approximation. This dramatically improves the convergence of the simulations, as shown in Figure 8.18.

Working curves were simulated using the BI and frontal PKS methods for EC, EC2, ECE, EC2E, DISP1 and DISP2 mechanisms using the parameters shown in Table 8.4 (though since the working curves are dimensionless, many of these are arbitrary). An exponentially expanding grid in the y co-ordinate with an expansion parameter of 2.5 was used to improve accuracy. A uniform mesh spacing was used in the radial co-ordinate.

Table 8.4: Parameters used in BI simulations for generation of working curves
Number of nodes in rNGX1000
Number of nodes in yNGY1000*
Diffusion coefficientD1x10-5 cm2s-1
Kinematic viscosityn0.0089 cm2s-1
Electrode radiusre0.2 cm
Nozzle radiusrjet0.0345 cm
Volume flow ratevf0.01cm3s-1
Bulk concentration[A]bulk1x10-6 mol cm-3
† increased at high rate constant to keep simulation accurate (and stable in the case of the BI method). * increased at high rate constants to account for the thinning of the reaction layer.
The working curves, given in terms of the dimensionless rate constant:
(8.36)

are summarised in Figure 8.19. At lower Peclet numbers, a working surface (of Neff vs. log K vs. log Ps) is necessary for the analysis of kinetic data. Surfaces were simulated for ECE and DISP1 mechanisms using ILU(1) preconditioned, multigrid accelerated BiCGStab(4) (introduced in section 4.3) on a mesh of 513x513 nodes.

Figure 8.19: Working curves for common electrochemical mechanisms in the limit of negligible radial diffusion (a) Effective number of electrons for ECE, EC2E, DISP1 and DISP2 processes, (b) shift in dimensionless half-wave potential for EC and EC2 processes.

The simulations were run in strips of fixed Ps, increasing the rate constant to traverse a working curve in log K. The concentration distribution at the previous rate constant was used as the starting approximation for the next, considerably reducing the number of V-cycles after the first rate constant was simulated. The tolerance had to be tightened to 1x10-14 to ensure satisfactory convergence (possibly due to the close starting approximation). Again, an exponentially expanding grid in the y co-ordinate with an expansion parameter of 2.5 was used to improve accuracy. Figure 8.20 shows how, moving across each working surface, the working curves change with Peclet number. Comparing the response at Ps = 10 with that in the limit of no radial diffusion, the working curve is distorted at most by 2.5%. Comparing this with the augmentation in current for a simple E process - approximately 18% at Ps = 10, shows that the radial diffusion effect is largely cancelled out in Neff.





Figure 8.20: Change in working curve at low Peclet numbers (plotted as a ratio to the high Peclet number response).

8.2 Simulations of the micro-jet electrode

Homann12 and then Frössling13 derived an asymptotic solution to Navier-Stokes equations, valid close to the electrode surface as a series expansion:

(8.37)

and

(8.38)

where a is an undetermined flow rate-dependent parameter with units of s-1,

(8.39)

and the coefficients (according to Frössling) are:

a2 = 0.656, a3 = -0.16667, a6 = 0.0036444 and a7 = 0.00039682(8.40)

Albery and Bruckenstein14 proposed use of the just the first term in the series expansion:

and (8.41)

where

(8.42)

Chin and Tsang15 found empirically that for 0.1 < rjet/zjet < 2.5:

(8.43)

and this was also confirmed experimentally by Macpherson et al.16,17. Note however that Chin and Tsang's determination was based on only four experimental data points. In order to assess the validity of Frössling's boundary layer approximation and Chin & Tsang's empirical result for the hydrodynamics independently, Coles18 is currently simulating the MJE numerically using the commercial fluid dynamics package, FIDAP19. For now, it will be assumed that the velocity field predicted by the above equations is a reasonable approximation within the diffusion layer and use this as a basis to investigate over what range of Peclet numbers the MJE is uniformly accessible.

Since the wall-tube geometry is at least approximately uniformly accessible, there is little to be gained from using a hydrodynamic conformal transformation as was applied to the wall-jet. Therefore simulations were conducted on a simple rectangular mesh in cylindrical polar co-ordinates. The simulation space and boundary conditions are exactly as in Figure 8.6 and Table 8.1 for the WJE except that the normal co-ordinate is now z rather than y. The central finite difference form of the steady-state mass transport equation is:

(8.44)

which has no cross derivatives hence the finite difference implementation boundary conditions, shown in Table 8.5, is simpler than in the WJE. Since the mass transport equation may be represented by a 5-point stencil in this case, a wider range of solvers could be applied. It was found in practice that MGD1 was unstable at both low and high Peclet numbers even if diagonal dominance of the coefficient matrix was maintained by upwind convection terms, so Stone's SIP (see section 2.6.2.4) was used as the solver for these simulations.

A simulated concentration profile is shown in Figure 8.21, clearly conforming to the hypothesis of uniform accessibility - above the electrode surface the concentration contours are linear.

Table 8.5: Finite difference implementation of boundary conditions (in MGD1 notation).
Disc centrek=1, j=1...NGYA4j,k = A4j,k + A3j,k; A3j,k = 0
Wastek=NGX, j=1...NGYA4j,k = A4j,k + A5j,k; A5j,k = 0
Bulk solutionj=NGY, k=1...NGXA7j,k = 1
Electrodej=1, k=1...kEA1j,k = 0
Insulatorj=1, k=KE...NGXA4j,k = A4j,k + A1j,k; A1j,k = 0

The current may be evaluated from simple rectangular quadrature:

(8.45)

Given the uniformity of the concentration distribution over the electrode, there is little to be gained from higher order integration formulae.

8.2.1 Effects of radial diffusion


Figure 8.21: Simulated concentration profile at a MJE including radial diffusion effects. Parameters are as in Table 8.6 with vf =0.0242cm3s-1 and zmax=1.7mm . Convection was approximated using 1st order expressions and discretised in both r and z using an upwind scheme.

It is shown in Appendix 5 that, as for the WJE, RDE and ChE, the dimensionless current is a sole function of the shear-rate Peclet number under a 1st order convection approximation. This means that it is possible to quantify the effects of radial diffusion by examining the departure from Levich behaviour at low Peclet numbers. The simulation parameters are shown in Table 8.6.

Table 8.6: parameters used in MJE simulations
Diffusion CoefficientD1x10-5 cm>2s-1
Kinematic viscosityn8.9x10-3> cm2s-1
Electrode Radiusre25 mm
Nozzle Radiusrjet52 mm
Electrode-nozzle separationzjet0.3 cm
Bulk concentration[A]bulk1x10-6 molcm-3
Space downstreamld1.5
Number of nodes in rNGX600
Number of nodes in zNGY600

Figure 8.22: Levich plot for MJE.

The Levich plot shown in Figure 8.22, allows the radial diffusion effect to be assessed. The geometries and flow rates used by Macpherson and Unwin16 spanned Peclet numbers in the range 3.5 < log10Ps < 7. At the lowest, radial diffusion augments the current by approximately 2.5%, and the augmentation drops below 1% at log10 Peclet numbers above 4.5.

It may be desirable to increase the rate of flow in order to achieve higher rates of mass transport, which should be reasonably easy given the relatively modest volume flow rates employed by Macpherson and Unwin (a maximum of 5x10-2 cm3s-1 was used in ref. 16). The unimportance of radial diffusion, together with the assumption that the dominating vz convection term is quadratic in z within the diffusion layer, means that kinetic analysis may be conducted using theory generated for the RDE14. This is discussed in Chapter 10. Until the validity of the hydrodynamic approximations has been confirmed, however, caution is advised.

References

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9 L. Pastore, F. Magno, C.A. Amatore, J. Electroanal. Chem., 301, (1991), 1.
10 S.W. Feldberg, C.I. Goldstein and M. Rudolph, J. Electroanal. Chem., 413, (1996), 25.
11 These values (which are listed in the conclusions chapter) are calculated from values given by C. Brett (e-mail brett@cygnus.ci.uc.pt) in a personal communication from the experimental parameters given in R.G. Compton, A.C. Fisher, G.P. Tyley, J. Appl. Electrochem., 21, (1991), 295.
12 F. Homann, ZAMN, 16, (1936), 153; Forsschg. IngWes, 7, (1936), 1 [both in German]
13 N. Frössling, Lunds Univ. Avd., 2, (1940), 35. [in German]
14 W.J. Albery, S. Bruckenstein, J. Electroanal. Chem., 144, (1983), 105.
15 D.T. Chin, C.H. Tsang, J. Electrochem. Soc., 125, (1978), 1461.
16 J.V.Macpherson, S.Marcar, P.R.Unwin, Anal.Chem., 66, (1994), 2175
17 J.V.Macpherson, M.A. Beaston, P.R.Unwin, J. Chem. Soc. Faraday. Trans., 91, (1995), 899.
18 B.A. Coles, Oxford University, unpublished work. For further information, e-mail bac@physchem.ox.ac.uk.
19 FIDAP is a commercial CFD package (based on a finite element method) made by Fluent Incorporated (http://www.fdi.com).