Diffusional steady-state approach

THE DIFFUSIONAL STEADY-STATE (dSS) APPROACH 2.4.1 The Essence of the dSS Approach... 

The following example, taken from Welty et al. ( 1976), illustrates the solution approach to a steady-state, one-dimensional, diffusional or heat conduction problem. 

This problem illustrates the solution approach to a one-dimensional, non-steady-state, diffusional problem, as demonstrated in the simulation examples, DRY and BNZDYN. The system is represented in Fig. 4.2. Water diffuses through a porous solid, to the surface, where it evaporates into the atmosphere. 

Unlike the previously described group of enz5une sensors that rely on catal54ic transformation and that optimally should work under diffusional control, these sensors are obliged to perform under kinetic control. Basically, the measurement of the inhibition can be obtained through steady-state (Fig. 10.13a) or kinetic (Fig. 10.13b) measurements. Figure 10.13 exemplifies the main features of both approaches. The kinetic measurement is obviously preferred from a practical point of view [350] whilst better sensitivity can be achieved when inhibition is calculated firom two separate measurements [351]. 

When the adsorption/desorption kinetics are slow compared to the rate of diffusional mass transfer through the tip/substrate gap, the system responds sluggishly to depletion of the solution component of the adsorbate close to the interface and the current-time characteristics tend towards those predicted for an inert substrate. As the kinetics increase, the response to the perturbation in the interfacial equilibrium is more rapid and, at short to moderate times, the additional source of protons from the induced-desorption process increases the current compared to that for an inert surface. This occurs up to a limit where the interfacial kinetics are sufficiently fast that the adsorption/desorption process is essentially always at equilibrium on the time scale of SECM measurements. For the case shown in Figure 3 this is effectively reached when Ka = Kd= 1000. In the absence of surface diffusion, at times sufficiently long for the system to attain a true steady state, the UME currents for all kinetic cases approach the value for an inert substrate. In this situation, the adsorption/desorption process reaches a new equilibrium (governed by the local solution concentration of the target species adjacent to the substrate/solution interface) and the tip current depends only on the rate of (hindered) diffusion through solution. 

The simplest treatments of convective systems are based on a diffusion layer approach. In this 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. Within the layer 0 x < 5, 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 Chapter 1 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 for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved either analytically or, more frequently, numerically. In most cases, only the steady-state solution is desired. 

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