Multiscale Approach to Ablation Modeling of Phenolic Impregnated Carbon Ablators

Abstract

A multiscale approach is used to model and analyze the ablation of porous materials. Models are developed for the oxidation of a carbon preform and of the char layer of two phenolic impregnated carbon ablators with the same chemical composition but with different structures. Oxygen diffusion through the pores of the materials and in depth oxidation and mass loss are first modeled at the microscopic scale. The microscopic model is then averaged to yield a set of partial differential equations describing the macroscopic behavior of the material. Microscopic and macroscopic approaches are applied with progressive degrees of complexity to gain a comprehensive understanding of the ablation process. Porous medium ablation is found to occur in a zone of the char layer that we call the ablation zone. The thickness of the ablation zone is a decreasing function of the Thiele number. The studied materials are shown to display different ablation behaviors, a fact not captured by current models that are based on chemical composition only. Applied to Stardust's phenolic impregnated carbon ablator, the models explain and reproduce the unexpected drop in density measured in the char layer during Stardust postflight analyses [Stackpoole, M., Sepka, S., Cozmuta, I., and Kontinos, D., "Post-Flight Evaluation of Stardust Sample Return Capsule Forebody Heat-Shield Material," AIAA Paper 2008-1202, Jan. 2008]. Nomenclature C = oxygen concentration, mol=m 3 D = diffusion coefficient, m 2 =s D dis = dispersion coefficient, m 2 =s d p = mean pore diameter, m f = rarefaction function J = molar ablation rate, mol=m 2 =s Kn = Knudsen number k = reactivity, m=s L D = diffusion length, m L R = reaction length, m L 0 = sample initial length, or thickness, m M = molar mass, kg=mol n = vector normal to the surface P = pressure, Pa Pe = Péclet number P f = local fiber perimeter, m q = local molar flux density, mol=m 2 =s R = ideal gas constant, 8:314 J=mol=K Sx; y; z; t = surface function S p = local horizontal section of a pore, m 2 s = volume surface, m 2 =m 3 T = temperature, K V = averaging volume, m 3 v = velocity, m=s v = molecular agitation mean velocity, m=s x, y, z, t = space (m) and time (s) coordinates = exponential coefficient for tortuosity law of the pore-filling matrix ERSA = effective reactive surface-area coefficient " = porosity = tortuosity = mean free path, m h 2 j i = mean square displacement in direction j, m = density, kg=m 3 = Thiele number = solid molar volume, m 3 =mol Subscripts B = bulk e, eff = effective ERSA = effective reactive surface area f = fiber g = pyrolysis gases i = indicator or index-number K = Knudsen m = matrix ref = reference w = wall

Microstructure and gas-surface interaction studies of a low-density carbon-bonded carbon fiber composite in atmospheric entry plasmas

“…A major outcome from those studies is that the ablation mechanism, and hence the reactivity, strongly depends on the type and density of carbon fibers (themselves relying on the precursor type, and on the processing & graphitization cycles [19]), their connecting interface and defects on the fiber surface. New multi-scale material response models are proposed [20][21][22] in order to take into account the porous micro-structure of the new class of materials. Volume-averaging of this microscopic scale model helped analyzing and better understand the oxidation behavior of carbon fibers, embedded in the charred phenolicpolymer matrix.…”

Analytical modeling of the transient ablation of a 3D C/C composite

“…It is a well know fact that oxidation rates increase with temperature and that diffusion becomes a limiting process [3], even at the fiber length scale [27,38]. In this case, fibers take ogival shapes rather than needle shapes [4], we have shown, as pictured in figure 2-b.…”

Permeability of Ablative Materials Under Rarefied Gas Conditions

“…As has previously been discussed by Lachaud et al [8], if the pore sizes in the internal structure of a heat shield are on the order of micrometres, the gas flow can fall within the transition Knudsen number regime due to the small length scale and the relatively small magnitude of the stagnation pressures encountered during the peak heat flux phase of re-entry. Therefore, the familiar Navier-Stokes-Fourier (NSF) equations are no longer valid because the gas is rarefied and the assumptions of the linear form of the constitutive relations for shear stress and heat flux no longer hold; transport of the fluid must be described by the Boltzmann equation, which for a single species, monatomic gas, has the form:…”

“…This is a very low pressure, but it designed to simulate the very early stages of re-entry, where the external flow is rarefied, and the material can still be considered near to its virgin state. For the pyrolised material, the same outlet pressure was used, which then represents a further point in the re-entry where the vehicle has decelerated significantly and the stagnation pressure has reduced again, as shown in Reference [8].…”

“…The aim of the current work is develop a numerical framework that can be used to promote a better understanding of how the permeability of these materials changes from the virgin to the pyrolised states when gas flow through the materials is rarefied, which is possible both at the start and at the end of the high heat flux phase of an atmospheric entry, as shown in Reference [8]. The improved knowledge of the permeability will allow it to be employed with greater confidence in lower-order ablation models.…”