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III. Modifications of the model for metal flame erosion and nozzle ablation

Although in our basic model we introduced provides deep insight into the dynamics of the case breach fault for a given propellant geometry and given design curve Sb = f(Rp), experimental results obtained during the ground firing tests suggest that more details have to be added to the model to reproduce experimentally observed time-traces. In particular, the experimental results demonstrate some deviations in the nozzle ablation and metal flame erosion rates from that given by the Eqs (4), (5), (8). These deviations are related mainly to the complex geometry of the actual fault in the forward closure to the heating of the metal case during flame erosion of the metal walls, and to the erosion of the nozzle wall surface during ablation process that changes the rate of ablation. Some other physical effects such as thermal expansion of the materials and wave formations on the walls of the nozzle throat will be considered in more detail elsewhere.

Model of the metal flame erosion in complex geometry

To take into account real geometry of the fault (see Figure 8) and the heating of the metal we have to modified equation (5) of the model by substituting it with the following set of equations

image059

Here, heat of combustion and velocity of burning or flame erosion of metal surface are reacting with oxidizing agents of the flow of combustion products (hot gas). Moreover, we take into account the geometry of the fault and the heating of the metal surface due to the convective flow and radiation with the values of and given by equations

image073

/image075

Here, the emissivity of the hot gas, %AL, is the percentage of aluminum in a solid propellant, and is the Stefan-Boltzmann constant:

image077


rmp33 rmp34

Figure 8 (left) Sketch of the fault geometry. (right) Synthetic data. Dynamics of the hole growth that take into account radiation, and flame erosion (blue solid line) as compared to the growth dynamics that neglects erosion (black dashed line) and growth dynamics neglecting both erosion and radiation (red dotted line). Note the nonlinear regime of the fault dynamics.

Model of the nozzle ablation with surface flame erosion

To take into account the effect of the nozzle surface roughness on the ablation velocity we introduce factor nr. With this correction the rate of ablation has the form

image083

where Rnto = Rnt(0) and

image087
image089

image091, image093, image095 are fitting coefficients. It is clear from Eqs (17)-(19) that the ablation rate is increased approximately twice as the characteristic roughness length lr becomes larger than rmp_ch3_sp02

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