Parabolic scaling rate constants

The implication of Wagner s theory was that the parabolic scaling rate constants, memy of which were known for certain gas-metal combinations, should be quantitatively related to two basic types of information. 

After the most severe oxidation exposure of the current study (w), the oxidation layers were rare and only about 100 nm thick for A and 500 nm for B. For material B an estimate for the parabolic oxidation rate constant less than 0.0005 nf/h was calculated from the oxidation scale thickness of 500 nm (Fig.lc). The value is low when compared to parabolic rate constants for SiC in pure atmospheric oxygen at 900°C. However, direct comparison is misleading since in hot gas filters SiC is covered by the binder. This means that water needs time for diffusion through the binder before the oxidation occurs. In addition an initial oxidation layer of unknown thickness may be present. 

Table 3.2 Relationships between variously defined parabolic oxidation rate constants (for the scale stoichiometry expressed as M X).

Figure 5-2. Parabolic oxidation rate constant, p, for the growth of various oxide scales as a function of temperature (Arrhenius plot).

Where L is the oxide scale thickness (in cm), kp is the parabohc growth rate constant referring to as the oxygen mass uptake (in g cm s ), t is time (in s), p is the density of the scale (in g cm ), and 0 is the weight fraction of oxygen in the scale. A plot of the scale thickness as a function of the oxidation time for various parabolic growth rate constants is shown in Fig. 5. 

Fig. 6 Measured oxidation rate constants for selected uncoated and coated FSS [9, 10]. The grey square referred to as acceptable values set for the oxide scale thickness and parabolic oxidation rate constants after 10 kh operation. Reproduced here with kind permission from The Electrochemical Society 2009...

Based on these experimental data, setting an oxide scale around 1 pm after 10 kh operation for an alloy with a parabolic oxidation rate constant around 10 g cm " s requires coated alloys for temperature above 750 °C. There are a few alloy candidates fulfilling all requirements to operate the cell at temperature below 700 °C, but it is anticipated that continuous progress in material design will enable to improve oxidation resistance of alloys. 

This parabolic law, which indicates that diffusion is rate-limiting, is of overwhelming importance for scale formation. Wagner (1933) showed that the parabolic scale constant (and hence, rate of oxidation) can be calculated using the enthalpy of formation of the corrosion product, the electrical conductivity of the protective film and the transport number of the ions and electrons in the film. 

In addition to the laboratory-scale reactors described here, there are numerous more specialized reactors in use. However, as mentioned previously, the performance of these reactors must lie somewhere between the mixing limits of the PFR and the CSTR. Additionally, when using small laboratory reactors, it is often difficult to maintain ideal mixing conditions, and the state of mixing should always be verified (see Chapter 8 for more details) prior to use. A common problem is that flow rates sufficiently large to achieve PFR behavior cannot be obtained in a small laboratory system, and the flow is laminar rather than turbulent (necessary for PFR behavior). If such is the case, the velocity profile across the reactor diameter is parabolic rather than constant. 

To relate the permeation flux to the parabolic rate constant, Wagner further assumed quasi-steady-state growth conditions. This assumption implies that the flux into the reaction layer is equal to the flux out of it and that there is no accumulation of material in the film. In other words, at any time, the flux was not a function of. r, but was only a function of time. This condition is shown schematically in Fig. 7.18c for various times during scale growth. Mathematically it implies that the flux is inversely proportional to Ax, and hence dx in Eq. (7.73) can be replaced by Ax. Make that substitution, and note that the rate at which the oxide layer is growing is given by... 

Since the scale formed on iron above 570 °C is predominantly wustite, growth of this layer controls the overall rate of oxidation. However, since the defect concentrations in wustite at the iron-wustite and wustite-magnetite interfaces are fixed by the equilibria achieved there, for any given temperature, the parabolic rate constant will be relatively unaffected by the external oxygen partial pressure. Increasing the oxygen partial pressure in the gas phase should, theoretically, lead to an increase in the relative thickness of the haematite layer. However, since this layer only accounts for about 1% of the metal-scale thickness, any variation in rate constant with oxygen partial pressure will be difficult to detect. 

On the other hand, refractory-metal sulphides are both very stable and slow growing. This has been addressed by Douglass and coworkers, who demonstrated that the addition of Mo and Mo plus A1 to nickel substantially reduced the parabolic rate constant for sulphidation, by about five orders of magnitude for the composition Ni-30 wt% Mo-8 wt% Al and, in the case of iron alloys, by six orders of magnitude for the composition Fe-30 wt% Mo- 5 wt% Al. In these cases the scales formed were AI0.5M02S4, which gave excellent resistance to sulphidation, but would form molybdenum oxides in oxidizing atmospheres. ... 

Fig. 11 Arrhenius-plot relating the parabolic rate constant for the growth ofAl203 scales and data for diffusion in AI2O3, (the scale for the grain boundary diffusion data at the right has been adjusted, so that if 5 = 1 nm, the grain boundary diffusion coefficient corresponds to the bulk diffusion scale) [51].

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