A Scale Analysis Approach to Thermal Contact Resistance

A new analytical model is developed for predicting thermal contact resistance (TCR) of non-conforming rough contacts of bare solids in a vacuum. Instead of using probability relationships to model the size and number of microcontacts of Gaussian surfaces, a novel approach by employing the “scale analysis methods” is taken. It is shown that the mean size of the microcontacts is proportional to the surface roughness and inversely proportional to the surface asperity slope. A general relationship for determining TCR is derived by superposition of the macro and the effective micro thermal resistances. The present model allows TCR to be predicted over the entire range of nonconforming rough contacts from conforming rough to smooth Hertzian contacts. It is demonstrated that the geometry of heat sources on a half-space for microcontacts is justifiable and that the effective micro thermal resistance is not a function of surface curvature. A comparison of the present model with 604 experimental data points, collected by many researchers during the last forty years, shows good agreement for the entire range of TCR. The data covers a wide range of materials, mechanical and thermophysical properties, micro and macro contact geometries, and similar and dissimilar metal contacts. Nomenclature A = area, m a = radius of contact, m b = flux tube radius, m c = scale analysis constant c1, c2 = Vickers microhardness coefficients, GPa, − CS = carbon steel 1Ph.D. Candidate, Department of Mechanical Engineering. 2Associate Professor, Director, Microelectronics Heat Transfer Laboratory. Member ASME . 3Distinguished Professor Emeritus, Department of Mechanical Engineering. Fellow ASME. dv = Vickers indentation diagonal, μm dr = increment in radial direction, m E = Young’s modulus, GPa E0 = effective elastic modulus, GPa F = external force, N h = contact conductance, W/mK Hmic = microhardness, GPa H 0 = c1 (σ0/m) c2 , GPa k = thermal conductivity, W/mK L = length scale ≡ bL/ (σ/m), m m = effective mean absolute surface slope ns = number of microcontacts P = pressure, Pa P ∗ = non-dimensional pressure ≡ F/ ¡πH 0b2L¢ Q = heat flow rate, W R = thermal resistance, K/W R∗ = non-dimensional thermal resistance T = temperature, K Y = mean surface plane separation, m Greek α = non-dimensional parameter ≡ σρ/aH δ = max surface out-of-flatness, m ε = flux tube relative radius, a/b θ = angle of the surface asperities, rad ψ = spreading resistance factor ρ = radius of curvature, m σ = RMS surface roughness, μm σ0 = roughness reference value =1 μm τ = non-dimensional parameter ≡ ρ/aH υ = Poisson’s ratio, − 1 Copyright c ° 2003 by ASME Subscripts 0 = value at origin 1, 2 = surface 1,2 a = apparent EC = elastoconstriction H = Hertz j = joint L = large mic = microcontact P = plastic deformation r = real s = small, solids v = Vickers