Equation limiting conditions

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

This is the gas-phase mass-transfer limited condition, which can be substituted into Eq. (14-71) to obtain the following equation for calculating the height of packing for a dilute system ... 

For a dilute system in which the liquid-phase mass-transfer limited condition is valid, in which a veiy fast second-order reaction is involved, and for which Nna E veiy large, the equation... 

In general, the sonic or critical velocity is attained for an outlet or downstream pressure equal to or less than one half the upstream or inlet absolute pressure condition of a system. The discharge through an orifice or nozzle is usually a limiting condition for the flow through the end of a pipe. The usual pressure drop equations do not hold at the sonic velocity, as in an orifice. Conditions or systems exhausting to atmosphere (or vacuum) from medium to high pressures should be examined for critical flow, otherwise the calculated pressure drop may be in error. 

Ideally the design engineer requires an equation which condenses all this information and from which he can calculate the effect of a particular chemical upon a range of materials, and the limiting conditions of say temperature, concentration and velocity. To achieve this objective he needs to know which of the properties of the chemical and the material are the most important in determining the interaction leading to corrosion. 

Equation (6) is then discussed in terms of two limiting conditions. The first is... 

Two flow models are used to estimate the mean condensation coefficient in horizontal tubes stratified flow, Figure 12.45a, and annular flow, Figure 12.45. The stratified flow model represents the limiting condition at low condensate and vapour rates, and the annular model the condition at high vapour and low condensate rates. For the stratified flow model, the condensate film coefficient can be estimated from the Nusselt equation, applying a suitable correction for the reduction in the coefficient caused by... 

Equation (23) implies that the current density is uniformly distributed at all times. In reality, when the entire electrode has reached the limiting condition, the distribution of current is not uniform this distribution will be determined by the relative thickness of the developing concentration boundary layer along the electrode. To apply the superposition theorem to mass transfer at electrodes with a nonuniform limiting-current distribution, the local current density throughout the approach to the limiting current should be known. 

There are several interesting forms of equation 8.3.20 that correspond to various limiting conditions. For example, if both VR and are time invariant, we have the situation corresponding to a shift from one steady-state operating condition to a second, and the quantity... 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

Low IQ, quasi-reversible kinetics. Under these circumstances, we can use the approximation represented by equation (1,35) in which we assume that [O]0 a [O ]. The equation is generally applied under two limiting conditions ... 

Equations (2.14) and (2.15) are important under sewer conditions, not just because nongrowth limited conditions may exist. The equations are also the basis for description of the kinetics when the growth of the biomass takes place under substrate or other external environmental conditions limiting the growth rate. 

Equation (5.15) may be extended to describe transformations under substrate-limited conditions by including a Monod expression (cf. Section 2.2.1). The information from Section 5.6.1 indicates that the saturation constant for nitrate, KS0, is about 2-3 gN03 nr. ... 

For limiting nutrients, cellular concentrations are constant under conditions of steady-state growth. To ensure that the limiting nutrient is not diluted in the microbial population, kmt must be greater than the maximal growth rate, /imax. This limiting condition sets a minimum for the value of the Monod constant, Kmd = / max /[7]- Note that while Monod kinetics are more applicable than first-order kinetics for many ecological uptake processes, solutions of the above equations require considerably more a priori information [48]. 

When biological uptake does not perturb the external medium, then /int can be given by equation (35). As discussed above, this limiting condition is assumed to occur in both the free-ion activity and biotic ligand models. When Ka[M] < 1, then (cf. equation (7)) ... 

Under steady-state conditions, the internalisation flux equals the rate of supply by diffusive transport and chemical reactions. As was shown earlier (cf. equations (12) and (13)), the maximum flux (rate) of solute internalisation by a microscopic cell under diffusion-limited conditions can be given by ... 

Further complications may arise when more concentrated solutions are used, even before reaching the limiting conditions that invalidate equations 13.4 or 13.9. The most noticeable problem is the appearance of multiphoton effects, for which the relation between photoacoustic signal and energy absorbed is no longer linear. As mentioned, this phenomenon is easily detected by a deviation... 

The semi-empirical Pitzer equation for modeling equilibrium in aqueous electrolyte systems has been extended in a thermodynamically consistent manner to allow for molecular as well as ionic solutes. Under limiting conditions, the extended model reduces to the well-known Setschenow equation for the salting out effect of molecular solutes. To test the validity of the model, correlations of vapor-liquid equilibrium data were carried out for three systems the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution studied by van Krevelen, et al. 

Most early theories were concerned with adsorption from the gas phase. Sufficient was known about the behaviour of ideal gases for relatively simple mechanisms to be postulated, and for equations relating concentrations in gaseous and adsorbed phases to be proposed. At very low concentrations the molecules adsorbed are widely spaced over the adsorbent surface so that one molecule has no influence on another. For these limiting conditions it is reasonable to assume that the concentration in one phase is proportional to the concentration in the other, that is ... 

Rate equations, such as equation 17.85, make no attempt to distinguish mechanisms of transfer within a pellet. All such mechanisms are taken into account within the rate constant k. A more fundamental approach is to select the important factors and combine them to form a rate equation, with no regard to the mathematical complexity of the equation. In most cases this approach will lead to the necessity for numerical solutions although for some limiting conditions, useful analytical solutions are possible, particularly that presented by Rosen(41). ... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

In the limiting condition of very low metal ion concentrations, the terms containing [M and [MB] can be neglected in the denominator of Eq. (5.88). This means that [BH] and [H ], in practice, are constant with time, and their initial and equilibrium concentrations are essentially the same. The rate equation then becomes... 

At 273 and s 9.02 the polymerization followed first order kinetics. At 225 (Figure 4) the conversion curve was indistinguishable from a zero-order plot up to 40 percent conversion but if the whole curve was examined the internal order was seen to lie between zero and one. At 250 the internal order also lay between zero and one but the fit was nearer to the first-order plot in monomer. The kinetics are consistent with a Bateup-Yerusalimskii mechanism. At low temperatures the limiting condition of Equation 3 is approached. As the temperature rises the stationary state concentration of the complex decreases and the mechanism shifts to its other limit... 

No mechanistic significance can be attached to the change of order being observed in the present experiments and not in the earlier ones. The kinetic experiments with n-butylmagnesium compounds were carried out at monomer concentrations 10 times higher than permissible in an NMR experiment. This not only favoured the limiting condition of Equation 3, but increasing viscosity restricted measurements to ca. 40 percent conversion. The deviation from zero order kinetics at 225 in the present work was not apparent until this conversion was exceeded. 

Equation (68) shows that 0 increases linearly with an initial slope that equals K. This slope will be larger the farther to the right the equilibrium represented by Equation (61) lies. At higher concentrations, Equation (69) indicates that saturation of the surface with adsorbed solute is achieved. Figure 7.16a shows how these two limiting conditions affect the appearance of the isotherm. 

Spheres. Consider a 5-rich sphere of /3 phase of radius R = R(t) growing in an infinite a matrix under diffusion-limited conditions as shown in Fig. 20.6. This problem can be solved by using the scaling method with r) defined by rj = r/ ADat)1/2. The diffusion equation in the a phase in spherical coordinates in rt-space (see Eq. 5.14) becomes, after transformation into 77-space,... 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

Figure 3.37C shows the profiles that result when the applied potential is sufficiently negative that the concentration of O at the electrode surface is effectively zero. In this case, essentially all of O at the electrode surface must be electrolyzed to R in order to satisfy the Nernst equation. Consequently, O is converted to R as rapidly as it can diffuse to the electrode surface. Since this is the limiting condition, application of even more negative potentials causes no measurable change in the profiles. 

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