Knudsen void gas heat transport in fibrous media

2005

Ind. Eng. Chem. Res.

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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.

Section: η(R)x′i‚η(r′)] (42)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2005

Ind. Eng. Chem. Res.

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“…The cylinders with orientation ω appear as random overlapping circles (Figure a) of radius a in the ω plane. The probability P that no circle center lies within the area A in the ω plane can be written in terms of the density n , per unit area of circle centers in the ω plane: The void fraction Φ can be interpreted as the probability that a random chosen point in the slab falls in the void or the probability that in the ω plane no circle has its center within a distance a of the random point. Then from eq 14, we have The surface area ζ, overlapped or not, of those cylinders with orientation ω within a unit total volume of slab is From its product with the void fraction (eq 15), we obtain the exposed surface per unit total bed volume We will consider two fiber beds made up of freely overlapping cylinders all of the same radius a with central axes aligned in the same direction ω .…”

Section: Scattering Integral Gmentioning

confidence: 99%

Fiber Bed Effective Emissivities from Multiple-Scattering Calculations

2004

Ind. Eng. Chem. Res.

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The effective emissivity of a random fiber bed is a thermodynamic coefficient of importance for high-temperature radiation heat transfer. A multiple-scattering theory is developed that provides complementary (i.e., same scattering order) upper and lower bounds for the fiber bed emissivity. An attractive feature of this method is that this pair of bounds comes from the evaluation of one integral. Results for single-scattering bounds include analytical estimates with rigorous error bounds for all porosities, local fiber surface emissivities, and several representative bed edge random fiber arrangements, i.e., all fibers perpendicular and all fibers parallel to the bed edge. The estimates are tight for fiber surface emissivities above 0.5 at any porosity or for all fiber surface emissivities at low bed porosities. The analytical, single-scattering results provide a parametric study of the fiber bed effective emissivity enhancement above the local fiber surface emissivity for all fiber surface emissivities, porosities, and two limiting fiber bed structures. The largest enhancement (33.4%) is found at the midrange of the fiber emissivity values and in the limit of unit void fraction for fibers oriented perpendicular to the bed edge surface.

“…5,28 It suffices to consider those cylinders with axial orientation vector j and the j plane perpendicular to this direction. Those cylinders with orientation j appear as randomly overlapping circles of radius a in the j plane with centers randomly placed.…”

Section: Prager Upper Bound On K Dmentioning

confidence: 99%

Diffusion and surface reaction in random fibrous media

The Journal of Chemical Physics

2000

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Alternative solution for diffusion to two spheres with first-order surface reactionIn the classic ''trapping'' problem, the reactant is uniformly generated at a constant rate across a matrix phase, and diffuses to the interface of a dispersed cylinder phase, where it instantly reacts. Prager's upper bound on the effective reaction rate constant k ͑a ''best'' bound based on void-point nearest-neighbor-surface statistics͒ is derived for randomly placed, freely overlapping, infinitely long cylinders of radius a, where each cylinder has an arbitrary orientation with respect to the others. To compare with simulation data for an isotropic bed of overlapping spheres of radius a, the variational upper bound is considered for the case of the three-dimensional, isotropic overlapping cylinder bed with totally random mutual cylinder orientations. Once a correction factor of 3/2 for the surface area ratio of the overlapping sphere to overlapping cylinder beds is applied, the analytical variational cylinder bed bound is nearly coincident with the sphere dispersion simulation curve. The small differences observed at lower dispersion densities are consistent with the expected diffusion flux differences in the sphere and cylinder geometry. By comparing the variational upper bound with other simulation results for randomly overlapping, aligned, elongated, prolate spheroids, a maximum range for the k change due to mutual rotation between neighboring cylindrical reaction sites ͑29% and 32%, respectively, at solid volume fractions of 0.30 and 0.50͒ is obtained. The necessary distribution moments for the corresponding ''relaxation time'' lower bound are also given. The Doi lower bound on the effective reaction rate ͑a ''best'' bound based on two-point void-void F vv , void-surface F vs , and surface-surface F ss correlations͒ is addressed. The twopoint correlations for an isotropic bed of overlapping cylinders with random mutual orientation in three dimensions are presented. Their behavior is discussed, and a logarithmic singularity in F ss at the two-point distance of 2a is pointed out, that renders the Doi bound indeterminate.