BOOKCOMP, Inc. — John Wiley & Sons / Page 381 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 381 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [381], (121) Lines: 4609 to 4609 ——— 0.91072pt PgVar ——— Normal Page PgEnds: T E X [381], (121) N number of sides in a polygon, dimensionless number of discrete sources, dimensionless number of microcontacts, dimensionless N(k ) numerator function, dimensionless n counter, dimensionless Hertz elastic parameter, dimensionless combination parameter, dimensionless contact spot density, 1/m 2 P perimeter, m pressure, N/m 2 or Pa P g,∞ reference gas pressure, N/m 2 or Pa P m mean contact area pressure, N/m 2 or Pa Pr Prandtl number, dimensionless Q heat transfer rate, W Q g gap heat transfer rate, W q heat flux, W/m 2 R thermal resistance, K/W R g thermal resistance of gap, K/W R mac macroscopic thermal resistance of gap, K/W R mic microscopic thermal resistance of gap, K/W R r radiation resistance, K/W R ∗ c combination of terms, dimensionless R ∗ g combination of terms, dimensionless R ∗ j combination of terms, dimensionless R ∗ r combination of terms, dimensionless r radial coordinate, m S f material yield or flow stress, N/m 2 s side dimension, m T temperature, K T g gas molecule temperature, K T g,∞ reference temperature, K ∆T temperature drop or difference, K ∆T j joint temperature drop, K T area-averaged temperature, K T 1 temperature, K T 2 temperature, K t layer thickness, m thickness of elastic layer, m time, s t 1 layer 1 thickness, m thickness of isotropic plate, m t 2 layer 2 (substrate) thickness, m u local gap thickness, dimensionless position, dimensionless w(x,y) total local displacement, m BOOKCOMP, Inc. — John Wiley & Sons / Page 382 / 2nd Proofs / Heat Transfer Handbook / Bejan 382 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [382], (122) Lines: 4609 to 4696 ——— 0.77428pt PgVar ——— Normal Page PgEnds: T E X [382], (122) w o approach of contacting bodies due to loading, m X c coordinate of center of eccentric rectangular area, m x length coordinate, m Y mean plane separation, m Y c coordinate of center of eccentric rectangular area, m y distance, m length coordinate, m z length coordinate, m Greek Letter Symbols α ratio of semimajor axes, dimensionless thermal diffusivity, m 2 /s thermal conductivity ratio, dimensionless accommodation parameter or coefficient, dimensionless α 1 accommodation coefficient, dimensionless α 2 accommodation coefficient, dimensionless β combination of terms, dimensionless fluid property parameter, dimensionless β m,n eigenvalue, dimensionless Γ(x) gamma function of argument x, dimensionless γ aspect ratio parameter, dimensionless ratio of specific heats, dimensionless γ T combination of terms, dimensionless ∆ change in, dimensionless physical parameter, m 2 /N δ local gap thickness, m δ 0 local gap thickness under zero-load conditions, m δ m eigenvalue, dimensionless δ n eigenvalues of J n (x), dimensionless radius ratio, dimensionless ellipse aspect ratio, dimensionless emissivity of hemisphere, dimensionless emissivity of disk, dimensionless relative contact spot size, dimensionless c contact strain, dimensionless ζ ellipsoidal coordinate, m dummy variable, dimensionless θ temperature excess, K θ area averaged temperature rise, K θ(r,z) temperature excess field, K θ(τ) ellipsoidal temperature rise, K θ o centroid temperature rise, K θ s temperature rise due to spreading, K κ parameter, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 383 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 383 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [383], (123) Lines: 4696 to 4756 ——— 0.93477pt PgVar ——— Normal Page PgEnds: T E X [383], (123) thermal conductivity ratio, dimensionless Λ mean free path length of gas molecules, m Λ g molecular mean free path length at reference temperature, m Λ o reference value of mean free path length, m λ dummy variable, dimensions vary relative mean free path length, dimensionless λ 1 combination of terms, dimensionless λ 2 combination of terms, dimensionless λ n eigenvalue, dimensionless µ dynamic viscosity, N · s 2 /m positive root of an equation, dimensionless molecular weight ratio, dimensionless ν arbitrary order of Bessel function, dimensionless Poisson’s ratio, dimensionless ξ length ratio, dimensionless ρ radius of curvature, m radius of elastic hemisphere, m ρ n,e boundary condition parameter, dimensionless rectangular aspect ratio, dimensionless combination of terms, dimensionless σ Stefan–Boltzmann constant, 5.67 ×10 −8 W/m 2 · K 4 effective surface roughness, m or microns τ thickness, dimensionless τ 1 thickness, dimensionless τ 2 thickness, dimensionless τ ∗ combination of terms, dimensionless φ angle, rad combination of terms, dimensionless φ n combination of terms, dimensionless ϕ combination of terms, dimensionless ϕ + layer parameter, dimensionless ϕ − layer parameter, dimensionless ψ combination of terms, dimensionless spreading resistance, dimensionless spreading–constriction parameter, dimensionless amplitude angle, rad ψ mac macroscopic spreading–constriction parameter, dimensionless ψ mic microscopic spreading–constriction parameter, dimensionless ψ o combination of terms, dimensionless ψ n combination of terms, dimensionless ψ e,i combination of terms, dimensionless ψ ∗ thermal elasto constriction parameter, dimensionless ψ 12 dimensional spreading resistance in layer–substrate ω angle, rad ∇ 2 Laplacian operator, 1/m 2 BOOKCOMP, Inc. — John Wiley & Sons / Page 384 / 2nd Proofs / Heat Transfer Handbook / Bejan 384 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [384], (124) Lines: 4756 to 4756 ——— 0.20874pt PgVar ——— Normal Page PgEnds: T E X [384], (124) Subscripts a nominal value ave average B Brinnell c flux tube area active area contact c, 1 contact 1 c, 2 contact 2 circle circle e elastic contact radius ei layer thickness parameter ep elastic–plastic radius ellipse ellipse g gap g, 1 gap 1 g, 2 gap 2 g, ∞ gas conductivity under continuum conditions j joint layer layer 1 layer one layer 2 layer two layers L thick layer m counter mean ma macrogap mi microcontact mac macroscopic max maximum metal metal mic microscopic n normal component counter n, e combination of terms new new value o outer p plastic contact radius polymer q layer thickness parameter r radiation or radiative s spreading thin layer source area source source sink sink BOOKCOMP, Inc. — John Wiley & Sons / Page 385 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 385 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 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