Rectangular electrodes

Because of its advantages (high sensitivity and selectivity, low cost and miniaturization) amperometric detection has been frequently used in flow injection analysis (FIA) and RP-HPLC. However, it has been established that the peak area (detector response) considerably depends on the flow rate. A general approach has been proposed to predict the effect of flow rate on the peak area in FIA and RP-HPLC. The general form of the correlation describing the flow in a parallel plate cell with short rectangular electrodes is... 

The same reactor concept would be valid in a hot wall system, if the entire reactor were placed in a furnace. In this way, temperature uniformity would be excellent by definition. Obviously, this would be awkard and not economically attractive. However, if our electrode geometry were of two parallel, narrow rectangular electrodes, the structure could conveniently fit into a hot tube (much like a diffusion furnace). In this way, a large batch load could be run in a relatively hot furnace tube. Such a hot wall system is shown in Figure 15. 

Fig. 10.5(a). Transfer function, H, relating modulated current response to modulated flow rate for tube electrode (Reference [20]), rectangular electrode embedded in a wall (Reference [12]) and modulated RDE (amplitude only). The dimensionless modulation frequency see text) is the ratio of the time scale for diffusion across the concentration boundary layer to the timescale for modulation of the hydrodynamics. 

Using an alternative geometry Evans et al. [16] developed the channel stopped flow method (CSFM). This technique, to date, has been used to measure solution diffusion coefficients (independent of knowledge of the concentration of the electroactive species) and crystal dissolution kinetics. The channel flow cell consisted of a rectangular electrode, typical dimensions 2.5 mm long and 6.25 mm wide, situated in a rectangular duct, 10 mm wide and 0.25-1.0 mm high. The electrode was placed a suitable distance... 

Fig. 10.15. Experimental (—) and theoretical (—) chronoamperometric response for the diffusion-limited oxidation of 2 x 10-3 mol dm-3 Fe(CN)S in 0.1 mol dm-3 KC1 at a rectangular electrode, 2.5 mm long and 6.25 mm wide, in a 0.5 mm high channel flow cell under channel stopped flow conditions. The initial volume flow rate of the solution was 0.197 cm3 s-1, which gave a limiting current at the channel electrode, defined as / . At time, f.top, solution flow was retarded (Evans et al., in preparation). The theoretical data has been simulated assuming Df (cn)2 = 6.5 x 10-6 cm2 s l.

For a rectangular electrode of length L and of width W, the expression of the impedance is given by... 

Figure 13.4 Normalized global convective-diffusion impedance for a small rectangular electrode, The solid line represents the low-frequency solution (equation (13.37)), and the dashed line represents the high-frequency solution (equation (13.40)). Overlap is obtained for 6 < K < 13, with the dimensionless frequency K, given by equation (13.34). (Taken from Delouis et al. and reproduced with permission of The Electrochemical Society.)...

The previous equations allow determination of the local flux. To obtain the response of the electrode itself, it is necessary to integrate the local flux over the electrode surface. For a rectangular electrode. 

Square or rectangular electrodes placed either horizontally. or vertically... 

In the case of channel electrodes, the solution containing the electroactive species flows in a channel such as that shown in Figure 8.3 where a rectangular electrode of length Xe and width w is placed on the channel floor. The mass transport by convection can be controlled through the channel design, the electrode size and the flow rate. Moreover, this setup enables the incorporation of electrochemical measurements to flow systems as well as its use in spectroelectrochemistry and photoelectrochemistry [4]. 

Cells with rectangular electrodes are very similar to the battery in Fig. 10.13 except that the pins are replaced by a thin membrane, thus reducing the inter-plate gap and aiding the rapid discharge of the cell. Figures 10.14 and 10.15 show the designs of a spirally wound cylindrical cell and a button cell. 

Silver oxide/zinc can be manufactured in rechargeable cells with free potassium hydroxide electrolyte and pasted rectangular electrodes. Such cells are expensive but have a good energy density compared with Pb/acid or Ni/Cd (70 Wh kg ) and are also capable of high discharge rates for short periods. Other alkaline secondary batteries which have been manufactured include NiO(OH)/Zn, NiO(OH)/Fe and Ag2 0/Cd. 

THE PROBLEM Estimate the electrolyte voltage requirement for an electrolyzer with a single gas evolution reaction with a current efficiency of 100%. The rectangular electrodes are 0.1m wide and 0.25 m long the electrolyte gap is 0.01 m. 

In FAMS, the gas-phase mobility separation of ions in an electric field is achieved at atmospheric pressure [145-147], In its simplest design, the FAIMS device comprises two parallel rectangular electrodes at a uniform distance (Fig. 4.9). One of the electrodes is grounded, while at the other an asymmetrical waveform is applied. The asymmetric waveform is characterized by a significant difference in voltage in the positive and negative polarities of the waveform. FAMS nti-... 

The location of a planar rectangular electrode of width w and length / in a Cartesian coordinate system is shown in Fig. 12.11a. The origin of the coordinate system is at the centre of the electrode. For normalisation we make use of the same equations as for the UMBE. 

Fig. 12.11 (a) Cartesian coordinates for a rectangular electrode of length / and width w. (b) Computational domain for the rectangular electrode in normalised space coordinates. 

Figure 12.11b presents the rectangular electrode in dimensionless space. Because of the symmetry of the rectangular UME it is sufficient to consider only a quarter of the rectangle. X,, Ymax and Zmax mark the maximum values of the space variables taken into account for the computation. Analogous to Eqs. (12.37) and (12.38) for the UMDE, the limits are defined as... 

Fig. 12.12 The unequally spaced grid for the three-dimensional simulation of a rectangular electrode with L = 2. (a) XY plane (b) XZ plane...

To calculate the current at the rectangular electrode, Eq. (12.97) has to be solved numerically using new concentrations C. This can be done by repeated application of the trapezoidal rule in two dimensions... 

Examples of transient simulations for different values of L, starting with a square shaped electrode (L = 1) are shown in [277] and [276]. The percentage deviation between transients calculated for two-dimensional diffusion from the analytical equation of Aoki et al. [12, 288] and three-dimensional simulation with decreasing value of L has been presented [277]. An Aoki type equation where the coefficients were evaluated by fitting to three-dimensional numerical simulations to describe transients at square and rectangular electrodes has been published [276]. 

Another popular method (almost always with a transformation) is ADI, as this allows solution by a tridiagonal Thomas algorithm separately for the rows and columns. An ADI variant was also the choice for the three-dimensional problem of a rectangular electrode [81]. See Chap. 12 for more details. 

Conventional electtowinning EW), as well as electioiefining, utilizes rectangular electrodes (planar starter sheets) having surface areas that occupy the space Imxlm approximately. The anodic and cathodic electrodes are suspended vertically and alternatively at a distance 3 cm < x < 10 cm in the cells. The electrodes in one cell are connected in parallel and the cells are connected in series as shown in Figure 7.3. This is a classical electrode-ceU arrangement for reducing the net cell potential drop and ohmic resistance [27,34]. 

Recessing the electrodes into insulating wells. In the case of disk electrodes a recession of 0.1-0.4 times the disk radius has already major positive effects, the more the recession depth the better. The required depth in the case of rectangular electrodes is not mentioned in the literature, but the positive influence of recession on current distribution is certain. 

In 2004, the Cooks group at Purdue built a rectilinear-LQIT (RLQIT) with rectangular electrodes, and they previously built cylindrical ion traps (CIT). Both of these traps are attractive alternatives to XQITs that use hyperbolic electrodes because they are easier to build. 

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