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following equation, now called the Noyes-Nernst equation, represents the flux density across the solute interface:
(2.1)
where: |
|
|
V |
= |
volume of solution |
C∞ |
= concentration in the bulk |
|
Csat |
= |
saturation concentration |
A |
= |
area of interface |
D |
= |
diffusion coefficient |
δ* |
= |
boundary layer thickness |
t |
= |
time |
By including the surface reaction rate constant, K, Berthoud [2.6] derived the following equation:
(2.2)
which indicated that the driving force for dissolution was a combination of both the interface chemical reaction and the inter diffusion of the products and reactants. The derivation from first principles of the empirical constant, δ*, came after the development of boundary layer theory by Prandtl [2.7] and Levich [2.8]. The most important consequence of these theories for the experimentalist was that the effective boundary layer thickness, δ*, of a rotating disk was independent of its radius and proportional to the square root of the angular velocity [2.9].
2.2.2 Crystalline Materials
Attack by Molten Glasses
The use of a single diffusion coefficient, as was done in Eqs. (2.1) and (2.2), even in multicomponent systems was verified by Cooper and Kingery [2.10]. They, along with Samaddar et al. [2.11] and Oishi et al. [2.12], described in detail the theory of corrosion by liquids in ceramic systems (i.e., alumina, mullite, fused silica, and anorthite in Al–Ca–silicate liquid). Diffusion through the boundary
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layer was determined to be the rate-limiting step during dissolution. The composition of the boundary layer may vary depending upon whether diffusion is more or less rapid than the boundary reaction. The basic equation describing the rate of solution under free convection with density being the driving force is:
(2.3)
where: |
|
|
g |
= acceleration due to gravity |
|
∆ρ |
= (ρi-ρ∞)/ρ∞(ρi=saturated liquid density and |
|
ρ∞ |
= |
original) |
v |
= |
kinematic viscosity |
x |
= distance from surface of liquid |
|
Di |
= |
interface diffusion coefficient |
C* |
= |
a concentration parameter |
δ* |
= effective boundary layer thickness |
|
R |
= |
solute radius |
The exponential term was introduced as a correction for cylindrical surfaces. Since experimental tests often involve cylindrical specimens, these equations have been developed for that geometry. In practical applications, the condition relating to the corrosion of slabs is most predominant. However, if the sample diameter is large compared to the boundary layer thickness, the two geometries give almost identical results.
After a short induction period (which is of no consequence in practical applications) in which molecular diffusion predominates, the rate of corrosion becomes nearly independent of time. As a surface corrodes, the interface, if denser than the corroding medium, will be eroded away due to free convection caused by density variation. Use of this equation implies that one has at his disposal data relating to the variation of density and viscosity with temperature. In cases where these data are not available, the investigator will need to determine them prior to any calculation of corrosion rates.
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Fundamentals |
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Hrma [2.13] has used the work of Cooper and Kingery to discuss further the rates of corrosion of refractories in contact with glass. The following equation given by Hrma describes the corrosion under the condition of free convection due to density difference:
(2.4)
where: |
|
|
jc |
= |
rate of corrosion |
c |
= |
solubility of material in liquid |
D |
= |
coefficient of binary diffusion |
g |
= |
acceleration due to gravity |
v |
= |
kinematic viscosity |
L |
= |
distance from surface of liquid |
ρ |
= |
relative variation of density |
k |
= |
constant=0.482 |
This is essentially the same equation as that of Cooper and Kingery, without the exponential term.
Many corrosive environments associated with ceramic materials involve diffusion into the corroding medium, and thus increased velocity of the medium increases corrosion. Thus, if transport in the liquid were important, the corrosion rate must be evaluated under forced convection conditions. In such cases, the rate depended upon the velocity of forced convection:
(2.5)
The terms D* and v* were introduced since diffusivity and viscosity may be composition-dependent. The important point of this equation was that the rate of corrosion depended upon the square root of the angular velocity ω.
In the majority of practical cases, the solubility of the material in the liquid and the density of the liquid change much more slowly than the viscosity of the liquid. Under isothermal
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Chapter 2 |
conditions, the viscosity change is due to compositional changes. Thus, the predominant factor in the corrosion of a material by a liquid is the viscosity of the liquid [2.14,2.15]. This, however, does not hold for every case since liquid composition does affect the solubility of the solid [2.16]. These relationships hold quite well for the corrosion of a solid below the liquid surface. At the surface, where three states of matter are present, the corrosion mechanism is different and much more severe.
At the liquid surface, a sharp cut normally develops in the vertical face of the solid material being corroded as shown in Fig. 2.2. This region has been called flux-line, metal-line, or glass-line corrosion (also called the Marangoni effect [2.17]). Pons and Parent [2.18] reported that the flux-line corrosion rate was a nonlinear function of the oxygen potential difference between the surface and the interior of a molten sodium silicate. Cooper and Kingery [2.10] reported that flux-line corrosion was the result of natural convection in the liquid caused by changes in surface forces due to an increase in surface energy of the liquid as solid is dissolved. They also reported that if the surface energy of the liquid were independent of the amount of solid dissolved, no such excessive flux-line corrosion would occur. Hrma
FIGURE 2.2 Corrosion of a vertical face by a liquid.
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Fundamentals |
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[2.13] reported that the additional corrosion at the flux-line depended only upon the variation in surface tension and density, with surface tension being the more important factor. Although this is a well-known phenomenon, no one has investigated it thoroughly to determine a definitive mechanism. In actual practice, quite often, a thermal gradient exists such that the highest temperature exists at the flux line. This temperature difference, however, cannot be the sole driving force for excessive corrosion at the flux line since the same phenomenon is observed in laboratory isothermal studies. This same excessive corrosion occurs at any location where three substantially different materials come in contact with one another. In the above case, it was ceramic, liquid, and atmosphere. It may also occur where two liquids come in contact with a ceramic—a well-known phenomenon in metallurgy. The two liquids in that case are molten metal and an oxide slag.
The temperature dependence of corrosion can be represented by the Arrhenius equation:
(2.6)
Excellent fit of some experimental data to this equation reported by Samaddar et al. [2.11] has indicated that corrosion corresponds to an activated process. Blau and Smith [2.19] have attempted such an interpretation. However, the fact that variations of liquidus compositions, diffusion coefficients, and liquid structure change with temperature suggests that interpreting corrosion as an activated process may be very misleading and at least ineffective. The Arrhenius dependence should be used only for cases where the liquid is far from being saturated with components from the solid, which, according to Woolley [2.20], is the case for practical glassmaking applications.
The corrosion of a flat vertical slab under a thermal gradient is depicted in Fig. 2.3. As the convective flow of the liquid (caused by either forced convection or density changes) removes some of the reaction product interface, the total thickness of the slab decreases and the thermal gradient becomes steeper,
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Chapter 2 |
FIGURE 2.3 Effect of thermal gradient upon corrosion interface: (a) short time and (b) extended time.
assuming that the hot face temperature remains constant, which is very close to actual furnace operations. The actual cold face temperature will rise slightly, but the overall result is a steeper thermal gradient. The thermal gradient through a wall as depicted in Fig. 2.3 is more complex than presented here (actually, it is not linear but a complex 3-D shape), but the overall effect is the same. If the reaction product layer can form between certain temperature limits (2800°F and 2700°F in Fig. 2.3), it is obvious that the layer thickness must become smaller as corrosion proceeds. Thus the corrosion rate decreases with time. It is not uncommon for the flux line of the basin wall of a glass furnace to corrode away approximately onehalf of its thickness in less than 1 year, while the remaining half may take four or five times as long to exhibit the same amount of corrosion.
A downward-facing horizontal surface also exhibits greater corrosion than does a vertical or upward-facing horizontal surface. A downward-facing surface can exhibit excessive
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Fundamentals |
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FIGURE 2.4 A fusion cast alumina-zirconia-silica refractory throat of a TV panel glass furnace exhibiting upward drilling of the throat cover. (Courtesy of Corning, Inc.)
corrosion if bubbles are trapped beneath the horizontal surface.
This is known as upward-drilling since it results in vertically corroded shafts (see Figs. 2.4 and 2.5). Surface tension changes around the bubble cause circulatory currents in the liquid that cause excessive corrosion very similar to fluxline corrosion. Although no scientific comparisons have been made to geological corrosion, examples of something similar to upward drilling can be found. Fig. 2.5 shows a comparison of upward drilling of an AZS* fusion cast refractory paver from a glass furnace and a dolomite boulder (compare also with Fig. 2.4).
* AZS is the common abbreviation used by the industry to represent refractories composed of alumina, zirconia, and silica. Quite often, they are of the fusion cast variety.
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Chapter 2 |
FIGURE 2.5 Comparison of upward drilling between (a) an AZS fusion cast glass furnace paver (top surface towards left) and (b) a dolomite boulder.
Copyright © 2004 by Marcel Dekker, Inc.