Modelling of carbon-carbon composite ablation in rocket nozzles (3)

The products the propellant combustion contain numerous molecules. Due to its very high temperatures, the homogeneous reactions occurring in the gas phase are very fast and the gas mixture is always in a local chemical euilibrium. Water being the most abundant among them, it will be the only modeled species. This molecule is sufficiently diluted in typical propergol combustion products to the assumed to have no impact on the flow characteristics. Moreover, in the case of laminar boundary layers, the transverse pressure gradient is zero. On the other hand, there is a temperature gradient through the boundary layer, and, consequently, the equilibrium concentration of water is a function of the distance to the wall.

Mass transfer of water is governed by the chemical potential gradients for this species. In contract with species which are not consumed or produced by the wall, there is a strong transverse gradient of the difference between the equilibrium chemical potential and its actual value. In response to this, there is a water diffusive flux oriented towards the wall, which has to be combined to a convective flux oriented tangentially. The magnitude of the convective flux decreases when getting close the the wall; some distance away, it is so strong that it ensures completion of chemical equilibrium. In order to describe as simply as possible this transfer, we choose to display a model in which we substitute the chemical potential difference by the more familiar quantity concentration, which will be considered constant on the boundary layer limit.

Heterogeneous reactions:

The composite surface is subjected to several heterogeneous reacting leading to the gasification of the solid wall. Numerous studies have been done in order to identify the chemical mechanisms and kinetics of carbon gasification at the composite scale under laboratory conditions. Some studies were focused on the mass loss in nozzle throat applications. These studies agreed on the fact that mass loss in the nozzle throat application is mainly due to carbon oxidation by H2O and CO2 present in the gaseous products of propergols combustion

C(s) + H2O(g) → CO(g) + H2(g)

C(s) + CO2(g) → 2CO(g)

Libby et al assume that the kinetic rates of the two oxidation mechanisms are identical. Under this asumption, water vapor being ten times more concentrated than carbon dioxide in the nozzle, the mass loss is mainly caused by H2O.

The data from these studies can be collected into a global formalism with H2O as a single oxidizing species, and an apparent first-order rate law with constant k:

k=KsTse-Ea/RTs

In this expression, the values of the pre-exponential coefficient Ks lie bwtween 1 and 50 m/s/K and the activation energy Ea is about 40 kcal/mol. The Ks variation is linked to the nanotexture and nanostructure of carbon and will depend on the considered solid phase.

Model set-up:

The global ablation phenomenon is the result of the competition between several process. In order to compare the influence of each of these processes, several dimensionless numbers have to be considered in addition to the previously defined Reynolds number:

l The mass Peclet number Peμ=uolμ/D, where uo is the magnitude of the reference fluid velocity in the global frame, compares the velocity of advective to diffusive transfer mechanisms.

l The Damkihler number Daμ=klμ/D compares the surface reaction to the diffusion process. In the absence of advection, large values of Da lead to a diffusion-limited process for which the wall concentration is close to zero and the mass loss does not depend anymore on the reactivity k. On the contrary, when Da<1, reaction is the limiting factor and the concentraction field tends to be uniform at the value imposed at the upper part of the boundary layer.

l The condensation ratio W=co/cs, which is the ratio between the recession and reaction velocity, is also the ratio between the solid volume concentration Cs and the gas concentration in the bulk phase, close to the top of the boundary layer.

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