Steady-State Limit

We consider the single load case in the steady-state limit. Let the indentor have been moving in a negative x direction, at speed V, for a long time. If transient effects have died away then 

It is convenient to choose length units and origin so that oo,b = [-1.1] and time units such that F = 1. This means that any quantity of dimension length, which has value c in our initial units, becomes 

The subsidiary conditions (3.3.27, 30) become, in the steady-state limit and for this coordinate system. 

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In many cases the concentration of a substance can be determined by measuring its steady-state limiting diffusion current. This method can be used when the concentration of the substance being examined is not very low, and other substances able to react in the working potential range are not present in the solution. 

The properties of the voltammetric ultramicroelectrode (UME) were discussed in Sections 2.5.1 and 5.5.1 (Fig. 5.19). The steady-state limiting diffusion current to a spherical UME is... 

Unsteady-state effects and transition times are also significant in forced convection. Whenever the mass-transfer boundary layer is large anywhere on the working electrode surface, the commonly employed experimental time range of a few minutes may not be adequate to reach the steady-state limiting current. 

Experimental data relative to unsteady-state mass transfer as a result of a concentration step at the electrode surface are not available. However, for a linear increase of the current to parallel-plate electrodes under laminar flow, Hickman (H3) found that steady-state limiting-current readings were obtained only if the time to reach the limiting current at the trailing edge of the plate (see Section IV,E), expressed in the dimensionless form of Eq. (18), is... 

The minimum time necessary to obtain steady-state limiting currents by a current ramp was found to be... 

Fig. 9. Logarithmic plot of apparent limiting-current density as a function of current increase rate at a rotating-disk electrode i — apparent limiting current density i, = true steady-state limiting current density di/dt = current increase rate (A cm-2 sec-1) (u = rotation rate (rad sec-1). [From Selman and Tobias (S10).]...

In many cases mass transfer is not the sole cause of unsteady-state limiting currents, observed when a fast current ramp is imposed on an elongated electrode. In copper deposition, in particular, as a result of the appreciable surface overpotential (see Section III,C) and the ohmic potential drop between electrodes, the current distribution below the limiting current is very different from that at the true steady-state limiting current. 

Lichtner, P. C., E. H. Oelkers and H. C. Helgeson, 1986, Interdiffusion with multiple precipitation/dissolution reactions transient model and the steady-state limit. Geochimica et Cosmochimica Acta 50, 1951-1966. 

THE STEADY-STATE LIMIT FOR TWO PARALLEL INTERNALISATION ROUTES... 

On taking the Laplace transform, and noting that the steady-state limit (f->°° where dp/dt 0) is equivalent to the limit s -+ 0, and using the initial condition (131), this reduces to the spherically symmetric... 

The kinetics of the A + B - 0 bimolecular reaction between charged particles (reactants) is treated traditionally in terms of the law of mass action, Section 2.2. In the transient period the reaction rate K(t) depends on the initial particle distribution, but as f -> oo, it reaches the steady-state limit K(oo) = K() = 47rD/ieff, where D — Da + >b is a sum of diffusion coefficients, and /4fr is an effective reaction radius. In terms of the black sphere approximation (when AB pairs approaching to within certain critical distance ro instantly recombine) this radius is [74]... 

The well-known equation i = 4nFDCr (i limiting current, n number of electrons implied in the electrochemical process, F Faraday constant, D diffusion coefficient, C electroactive specie concentration and r radius of the disk) describes the theoretical steady-state limiting currents of the disk UMEs. This equation is useful to determine the effective radius of a disk UME and to estimate diffusion coefficients. In this sense, the above-mentioned polished carbon disk UMEs have been characterised through the limiting currents obtained in solution with known parameters, i.e. ferrocyanide aqueous solutions (0.05 M and 2M KC1) [118]. The experimental limiting currents were fairly accurately described by this equation ( + 10%). When the effective radius is determined, this equation can be employed to obtain unknown diffusion coefficients. In this way, we have estimated the diffusion coefficients for /i-carotene in several aprotic solvents with different electrolytic concentrations [123]. 

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

The time evolution of the cathodic limiting current (Eq. 2.147) has been plotted in Fig. 2.15 together with that obtained for a planar electrode (Eq. 2.28) and the constant steady-state limiting current for a spherical electrode given by... 

The overall (effective) reaction rate of a bimolecular reaction is, in general, determined by both the diffusion rate and the chemical reaction rate. A steady-state limit for the effective reaction rate is approached at long times and the rate constant takes a simple form. When the chemical reaction is very fast, the overall rate is determined by the diffusion rate, which is proportional to the diffusion constant. In the opposite limit where the chemical reaction is very slow, the overall rate is equal to the intrinsic rate of the chemical reaction. 

Thus, for t —> oo we obtain the steady-state limit for the effective rate constant ... 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

In adiabatically operated industrial hydrogenation reactors temperature hot spots have been observed under steady-state conditions. They are attributed to the formation of areas with different fluid residence time due to obstructions in the packed bed. It is shown that in addition to these steady-state effects dynamic instabilities may arise which lead to the temporary formation of excess temperatures well above the steady-state limit if a sudden local reduction of the flow rate occurs. An example of such a runaway in an industrial hydrogenation reactor is presented together with model calculations which reveal details of the onset and course of the reaction runaway. 

In the steady-state limit, J is independent of x. In accordance with eqn. (97), D is likewise a constant. Thus we can write... 

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