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Effects of product condensation on reaction-enhanced vaporization rates and on the transition from homogeneous to heterogeneous reaction in high-temperature metal oxidation

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Abstract

In order to predict the vaporization rate of fume-forming condensed phases into reactant-rich gases it is necessary to account for the strong augmentation of vapor transport rates caused by bulk flow into the reaction-condensation zone. For this purpose a simple film theory is developed based on the idealization of a thin reaction-condensation “front” supplied by convection-diffusion processes normal to the surface, but with condensate removal by convection processes parallel to the vaporizing surface. The steady-state position or “stand-off distance” of the front and, hence, its effect on the observed mass transfer rate, are determined by the demands of local reactant stoichiometry at the front and species conservation throughout the two-part film, subject to the imposed vapor and oxidizer boundary concentrations. As applied to the problem of metal vaporization into reactive atmospheres, it is shown that while we recover the results of Turkdogan, Grieveson, and Darken in highly dilute isothermal systems (i.e., metals far from their boiling points vaporizing into dilute oxygen-inert gas mixtures) appreciable additional effects due to the condensation process are predicted in the limit of oxidizer-rich ambient gases. Combining this extended theory with the notion that there is an upper limit to the attainable magnitude of the vaporization rate enhancement at which the reaction regime must change from homogeneous to heterogeneous, leads to predicted critical (transition) oxidizer pressures valid even approaching the practically important limit of pure gaseous ozidizer. On the assumption that this transition will occur when the metal vaporization rate approaches that expected in a vacuum (as has been experimentally observed by Turkdogan, Grieveson, and Darken), illustrative calculations are presented for the metals Na, Mg, Ca, Al, Be, Fe, and Zr vaporizing into oxygen-inert gas mixtures, and conclusions are drawn concerning the impossibility of the vapor phase reaction mode at surface temperatures too far below the metal boiling (or sublimation) point. It is also shown that the formalism can be extended to incorporate an alternative critical condition, perhaps more realistic under some experimental conditions, viz., oxygen penetration to the surface due to the finite thickness of the reaction zone. In either event, the model can be readily used to predict the homogeneous-heterogeneous transition locus and its dependence on the occurrence or nonoccurrence of product condensation, and physical, chemical, and aerodynamic parameters.

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Abbreviations

B :

Mass transfer driving force parameter; Eq. (32)

c P :

Specific heat (capacity)

D i−j :

Fick binary diffusion coefficient for chemical species i and j

D :

Oxidizer-to-metalDρ-ratio; Eq. (24)

d :

Diameter; Eq. (26)

E :

Activation energy of the metal oxidation homogeneous rate constant; Eq. (30)

F RXN :

Reaction factor;m "/m "0

G 1 :

Dimensionless temperature gradient function defined by Eq. (40)

G 2 :

Dimensionless temperature gradient function defined by Eq. (41)

h i :

Specific enthalpy of chemical species i

I :

Inert (non-participating) gas

k :

bimolecular reaction rate constant; Eq. (30)

k :

Boltzmann constant

Le :

Dimensionless transport group (Lewis number) defined by Eq. (43)

ℒ :

Lewis number ratio defined by Eq. (42)

m i :

Mass of a molecule of chemical species i

ℳ i :

Molecular weight of chemical species i (gm/gm-mole)

m "i :

Total mass flux of chemical species i (relative to observer fixed at y = 0).

M :

Metal species

Nu :

Nusselt number defining conditions of external heat/mass transport

n i :

Number density of chemical species i

O:

Oxidizer species

p i :

Partial pressure of chemical species i

p :

Total pressure (expressed in atmospheres)

P :

Reaction product

Q :

Heat of gas phase reaction (1) per unit mass of metal vapor; Eq. (38)

r :

Mass of oxidizer that reacts with unit mass of metal vapor

R :

Universal gas constant

T :

Absolute temperature

w ‴i :

Source strength for chemical species i due to homogeneous chemical reaction [(mass) (volume)−1 (time)−1]

x :

Stoichiometric ratio of oxidizer to metal atoms in the resulting metal oxide

Y i :

Local mass fraction of chemical species i

y :

Distance measured normal to the vaporizing surface

α :

Evaporation coefficient

δ :

Effective film (boundary layer) thickness

δ chem :

Effective thickness of the homogeneous metal-oxygen vapor reaction zone

ε/k :

Energy well-depth parameter for the Lennard-Jones 6:12 potential

Λ :

Molar heat of metal vaporization

λ :

Thermal conductivity of gas mixture

ρ :

Total mass density of gas mixture

σ :

Molecular size parameter in the Lennard-Jones 6:12 potential

bp:

At the boiling point corresponding to the prevailing total pressure

B = 0:

In the absence of appreciable net interfacial mass transfer

chem:

Pertaining to the reaction zone

eq:

At local equilibrium

e:

At the outer edge of the film (boundary layer)

eff:

Effective value

f:

At the reaction “front” (cf. Fig. 1)

M:

Pertaining to the metal

MO:

Pertaining to the metal oxide

mp:

At the equilibrium melting point

RXN:

In the presence of homogeneous chemical reaction

vac:

Into a vacuum (absence of “background” gas)

w:

At the gas-metal interface (cf. Fig. 1)

1:

In zone 1 (0 <y <y f); cf. Fig. 1

2:

In zone 2 (y f <y < Δ); cf. Fig. 1

*:

Pertaining to the critical (transition) condition from homogeneous to heterogeneous chemical reaction

o:

In the absence of homogeneous chemical reaction

\(\overline {\left( {} \right)} \) :

Average value

+:

On + side

−:

On − side

(s):

Solid phase

(c):

Condensed phase

(g):

Gaseous (vapor) phase

Δ():

Change in

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Authors and Affiliations

  1. Department of Engineering and Applied Science, Chemical Engineering Group, Mason Laboratory, Yale University, New Haven, Connecticut

    Daniel E. Rosner (Associate Professor)

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  1. Daniel E. Rosner
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Supported by the U.S. Air Force Office of Scientific Research, Propulsion Division, under contract F44 620-70-C-0026.

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Rosner, D.E. Effects of product condensation on reaction-enhanced vaporization rates and on the transition from homogeneous to heterogeneous reaction in high-temperature metal oxidation. Oxid Met 4, 1–25 (1972). https://doi.org/10.1007/BF00612504

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