Improved Estimates of High-Temperature Fiber Bed Effective Emissivities from Variational Calculations

Abstract

Rigorous variational upper bounds are presented for the effective bed emissivity εeff of a model semi-infinite fiber bed made up of long, parallel, randomly placed, freely overlapping cylinders for two possible bed edgesfibers perpendicular and fibers parallel to the bed edge. Calculations of bed emissivity variational results for both cases provide significant improvement over first-order scattering upper bounds given in a previous paper. For high void fraction Φ → 1 and fiber surface emissivity values of εs = 0.5, 0.3, and 0.1, respective bed emissivity upper bound reduction improvements of 4.14, 12.2, and 35.8% were generated for the latter bed edge structure, and 4.08, 11.3, and 32.2% were obtained for the former case. Along with the first-order scattering lower bounds on εeff, improved estimates of bed emissivities were calculated for high-temperature, nonmetallic fiber beds with εs ≥ 0.55. Bed emissivity enhancement factors (εeff/εs) of 1.27 for fibers parallel to the semi-infinite bed edge and 1.42 for fibers perpendicular to the bed edge were observed for high fiber bed porosity Φ = 0.9 and εs = 0.55. Results compared favorably with an experimental correlation and Monte Carlo simulations.

Emissivity of High-Temperature Fiber Composites

“…The asymptotic depth where the radiosity has changed to within 1% of the blackbody value is given from variational calculations using the multiplication product of 5 times the decay length, i.e., ( normalδ a ) ¯ = − 5 a λ β ln Φ ( for β = ⊥ , ∥ ) for protruding perpendicular (β = ⊥) or parallel (β = ∥) fibers. The value of λ β obtained in Figure 2 of ref , independent of the scattering calculations of this paper, is insensitive to the porosity Φ, but does change with local fiber−matrix surface emissivity ε s and edge structure (e.g., λ ⊥ = 0.643, λ ∥ = 0.752 for ε s = 0.5, and λ ⊥ = 0.830, λ ∥ = 0.878 for ε s = 0.8. The fact that the values, calculated from eq and listed in Table , agree with the onset of the asymptotes in Figures a−d shows that the large δ/ a asymptotic behavior is caused by the deep bed blackbody radiosity behavior.…”

“…For any surface porosity Φ < 1 in any of the plots of Figures a−d, the effective emissivity curve starts at ε s for the flat surface with no fiber protrusion and increases to an asymptotic maximum as δ/ a becomes large. For either edge structures ⊥ or ∥, it has been shown in an earlier publication on semifinite deep fiber beds that the radiosity decays exponentially into the bed from the external value to an internal blackbody radiosity value. The asymptotic depth where the radiosity has changed to within 1% of the blackbody value is given from variational calculations using the multiplication product of 5 times the decay length, i.e., ( normalδ a ) ¯ = − 5 a λ β ln Φ ( for β = ⊥ , ∥ ) for protruding perpendicular (β = ⊥) or parallel (β = ∥) fibers.…”

Knudsen diffusion through thin fibrous films

“…Numerical simulations require that the lengthy and numerous computer trajectory simulations followed by Monte Carlo averages over sufficient number of replicas of the random geometry be done over and over again for each set of parameters to generate any curves. 4,5 ͑4͒ Variational and multiple scattering bounds can be used to test the merits of a theory or simulation.…”

“…2,3 In both of these cases, Knudsen diffusion within very thin columnar films will play a significant role in the film structural formation and evolution, which in turn will determine the mechanical, thermal and radiative properties. 4,5 For the low pressure-high temperature production by chemical vapor deposition of ceramic matrix composites for high temperature engines, 6 and as well for the chemical vapor deposition of the carbon matrix on the carbon fibers of the preform for the manufacture of aircraft brakes, 7 Knudsen diffusion transports the reactive gas transversely across the axes of the preform fibers oriented, for the most part, parallel to the composite outer edge. The reactive gas deposits the solid matrix material directly on the growing fiber surface, and to model the pathological outside edge pore closure of the fiber preform will require equations for Knudsen diffusion across the fiber axes valid over shorter distances.…”