Compact Analytical Models for Effective Thermal Conductivity of Rough Spheroid Packed Beds

New compact analytical models for predicting the effective thermal conductivity of regularly packed beds of rough spheres immersed in a stagnant gas are developed. Existing models do not consider either the influence of the spheres roughness or the rarefaction of the interstitial gas on the conductivity of the beds. Contact mechanics and thermal analyses are performed for uniform size spheres packed in SC and FCC arrangements and the results are presented in the form of compact relationships. The present model accounts for the thermophysical properties of spheres and the gas, contact load, spheres diameter, spheres roughness and asperities slope, and temperature and pressure of the gas. The present model is compared with experimental data for SC and FCC packed beds and good agreement is observed. The experimental data cover a wide range of the contact load, surface roughness, interstitial gas type, and gas temperature and pressure. Nomenclature A = area, m aL = radius of macrocontact, m aH = radius of Hertzian contact, m bL = chord of macrogap, m c1 = Vickers microhardness coefficient, Pa c2 = Vickers microhardness coefficient D = sphere diameter, m E = Young’s modulus, Pa 1Ph.D. Candidate, Department of Mechanical Engineering. 2Distinguished Professor Emeritus, Department of Mechanical Engineering. Fellow ASME. 3Associate Professor, Director, Microelectronics Heat Transfer Laboratory. Member ASME. E0 = effective elastic modulus, Pa Fc = normal contact force, N FCC = Face Center Close H∗ = c1 (σ0/m) c2 , Pa Kn = Knudsen number k = thermal conductivity, W/mK L = length, m m = mean absolute surface slope M = gas parameter, m P = pressure, Pa Pr = Prandtl number Q = heat flow rate, W q = heat flux, W/m R = thermal resistance, K/W SC = Simple Cubic T = temperature, K Greek α = non-dimensional parameter, ≡ σρ/aH αT = thermal accommodation coefficient γ = exponent of general pressure distribution γg = ratio of gas specific heats, ≡ cp/cv Λ = mean free path, m υ = Poisson’s ratio ξ = non-dimensional radial position, ≡ r/aL ρ = radius of sphere, m σ = RMS surface roughness, m σ0 = σ/σ0, σ0 = 1 μm τ = non-dimensional parameter, ≡ ρ/aH ω0 = bulk normal deformation at origin, m Subscripts 0 = reference value, value at origin 1, 2 = solid 1, 2 1 Copyright c ° 2004 by ASME a = apparent BR = boundary resistance c = cell g = gas, microgap G = macrogap H = Hertz j = joint L = large, macrocontact r = real s = solid, micro INTRODUCTION Packed beds have a wide variety of applications in thermal systems. One of the significant characteristics of packed beds is the high ratio of solid surface area to volume. This property is useful in applications such as catalytic reactors, heat recovery processes, heat exchangers, heat storage systems, the breeder blanket about fusion reactors [1], and insulators. The insulator packed beds are often immersed in a gas at reduced pressure. The thermal conductivity of packed beds is not isotropic and it is difficult to formulate a model that fully defines the effective thermal conductivity. However, the structure of a packed bed can be modeled assuming regularly packed beds. A regularly packed bed is one in which the same arrangement of spheres, uniform in size, is repeated throughout the bed. Therefore a typical “basic cell” will represent the entire bed. There are three such regular packings: 1) Simple Cubic (SC), 2) Body Center Close (BCC), and 3) Face Center Close (FCC). Tien and Vafai [2] showed that the effective thermal conductivity of a random packed bed filled by a single phase fluid presents two limits. The upper bound can be obtained considering the FCC packing and the lower bound can be represented by the SC packing. Therefore, in this study only the thermal conductivity of the SC and FCC arrangements will be studied. The SC and FCC arrangements for packed beds of uniform diameter spheres are shown in Fig. 1. Each cell is made up of contact regions. A contact region is composed of a contact area between two portions of spheres, surrounded by a gas layer. Many studies have been performed on the prediction of thermal conductivity of packed beds filled with stagnant gas. The existing models can be categorized into two main groups. The first is numerical models, e.g. finite element methods (FEM) which can treat the three dimensional problem by dividing the bed into many cells with temperature and heat flow matched at their boundaries. One should keep in mind that it is a combined thermal and mechanical three dimensional numerical analysis which makes the FEM modeling extremely expensive from the calculative S2 D D FCC