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Asymptotes and Asymptotic Analysis for Development of Compact Models for Microelectronics Cooling M

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Asymptotes and Asymptotic Analysis for Development of Compact Models for Microelectronics Cooling M Asymptotes and Asymptotic Analysis for Development of Compact Models for Microelectronics Cooling M. Michael Yovanovich Distiniguished Professor Emeritus Principal ScientificAdvisor Microelectronics Heat Transfer Laboratory University of Waterloo, ON, Canada email: [email protected]. Abstract– In the thermal analysis of microelectronic sys- F (Pr) = Prandtl number function tems various levels of compact models are required to bridge g = gravitational constant, m/s2 thegapbetweentheuseofsimplecorrelationequations G = body gravity function which give quick, partial solutions and the use of costly, time consuming, numerical methods which give complete G = dimensionless reciprocal flux, 1/q∗ solutions with much detail. There are various levels of com- h = heat transfer coefficient, W/m2K pact models: some are based on simple combinations of two 2 HB = Brinell hardness, N/m or more correlation equations; some are based on resistor networks; and some are based on various combinations of k = thermal conductivity, W/mK asymptotic solutions. This paper shows that many different Kn = Knudsen number, Λ/D complex physical problems, when they are nondimensional- L = length of tube, pipe, duct, m ized, exhibit similar trends, i.e., that the complex solution L = optical path length, m varies smoothly between two asymptotic solutions. Rules λ are presented for the method of combining the asymptotes = arbitrary length scale, m and how to calculate the “fitting” parameter. One exam- m,L n = exponents of dimensionless solutions ple shows how to develop a compact model for steady con- M = gas parameter, αβΛ duction across a gas layer for very small and very large Knudsen numbers. A compact model for predicting the Nu = Nusselt number, hL/k complex elastic-plastic contact between a hard sphere and p = fitting parameter a softer substrate is presented. This elastic-plastic contact P = perimeter, m model is required for development of the thermal spreading- constriction model. Another example shows how to model P =gaspressure,Pa the effect of very small and very large Prandtl numbers for Pr = Prandtl number, ν/α forced and natural convection from isothermal plates. Ex- Q = heat transfer rate, W amples are given from forced and natural convection in short QL∗ = dimensionless heat flow, Q /kAθ0 and long ducts and pipes of arbitrary cross section for uni- 2 L form wall temperature and wall heat flux. The simple com- q =heatflux, W/m pact models are found to be accurate when compared with q∗ = dimensionless heat flux numerical results and experimental data, and they are easy r =radius,m to implement. Re = Reynolds number, U /υ Keywords– Asymptotes, asymptotic analysis, compact Ra = Rayleigh number, gβL∆T 3/αν models, steady and transient conduction, radiation, rar- L S∗ = dimensionless shape factor, Q /kAθ0 efaction, contact mechanics, elastic, plastic, elastic-plastic, L L forced and natural convection. t =time,s T =temperature,K Nomenclature U = mean velocity, m/s A = flow area, m2 w = axial velocity, m/s a = contact radius, m a, b = semi major and minor axes of ellipse, m Greek a, b = half dimension of rectangle, m α = thermal diffusivity, m2/s AR = aspect ratio α = accommodation coefficient ci = constants, i =1...5 β = gas parameter cP = specificheat,J/kgK γ = ratio of specificheats,cp/cv D =gapthickness,m γ = symmetry parameters Dh = hydraulic diameter 4A/p, m ∆ = thermal boundary layer thickness, m E1,E2 = Young’s modulus, Pa δ = momentum boundary layer thickness, m f =frictionfactor,2τ/ρv2 ² = aspect ratio, b/a < 1 F =load,N Λ = gas mean free path, m 2 2 F0 = dimensionless time, αt/A µ = dynamic viscosity, Ns/m µ∗ = dimensionless gas parameter, M/D rameters such as Reynolds number or modified Reynolds ν = kinematic viscosity, m2/s for forced convection and Rayleigh and modified Rayleigh ν1, ν2 = Poisson’s ratio number for pipe and channel flows, and the Prandtl num- ρ = fluid density, kg/m3 ber if fluids other than air are used. ρ = radius of curvature, m There is a need for the development simple “compact τ = wall shear stress, Pa models” which incorporate one or more asymptotic solu- θ = temperature excess, T Ti,K tions and which are not limited to particular ranges of the − φ = dimensionless dependent parameter independent parameters. These compact models can there- ξ = dimensionless independent parameter fore be used over broader ranges of the parameters and are Stephan-Boltzman constant, σ = more useful and valuable than the simple correlation equa- 5.67E 8,W/m2K4 tions alone. − This paper will be restricted to several relatively simple, Subscripts but important, compact models which have been applied 0 = reference values to some microelectronics cooling problems. 0 = corresponding to very small value = corresponding to very large value II. Asymptotes and Simple Compact Model 1∞, 2 = surfaces 1 and 2 development √A = based on square root of area Many complex physical systems exhibit a smooth transi- bl = boundary layer tion between two asymptotic solutions [1,2,3]. The smooth c =critical transition means that there are no discontinuities and no Dh = based on hydraulic diameter sudden changes in slope within the transition region. e =entrance These asymptotes appear in steady and transient inter- fd = fully developed nal and external conduction, steady conduction across a = based on arbitrary length gas layer between two large parallel isothermal plates, radi- RL =radiation ation through a highly porous substance between two large t =thermal isothermal walls, contact between a “rigid” sphere and a x = local value softer substrate, natural and forced internal and external w=wall convection. There are many other examples from conduc- λ =optical tion, fluid flow, and mass transfer; however, they will not be considered in this paper. I. Introduction Since the various problems have different dependent and HE present demands on the thermal analysts who independent parameters and different symbols are used, it Tmust daily deal with microelectronics cooling issues is is necessary to present the asymptotes and the method of daunting. They are required to provide, in very short time, combining them in a general form based on dimensionless reliable estimates of temperatures within devices, packages, dependent and independent parameters. boardsandsystemswhichareoftencooledbyairbyeither The dimensionless dependent parameter is called φ and natural or forced flow. Although experiments and numer- the independent parameter is called ξ. The functional de- ical methods can be employed to find the temperatures pendence φ(ξ) can be relatively simple or quite complex and provide much detail about the temperature distribu- depending on the physical problem. tion throughout the system; these methods are both time The dimensionless parameter φ has two asymptotes cor- consuming and costly. Certainly in the early stages of de- responding to very small and very large values of the di- sign, experiments are not feasible and numerical methods mensionless independent parameter ξ [1,2,3]: may require information that is not readily available. Fig- m φ0 = C0 ξ as ξ 0 ure 1 illustrates that the single correlation equations fall in φ → (1) the lower left corner, the numerical results lie in the upper → ( φ = C ξn as ξ or ξ 1 right corner where much detailed information is available ∞ ∞ →∞ → at some high cost, and the compact models form the rel- The asymptotes φ0 and φ are based on analytical solu- ative large “bridge” between the very low cost and high tions and they consist of a constant∞ which is a positive, real cost regions. There are many different types of compact number. The two constants are denoted as C0 as ξ 0 models ranging from relatively simple models to relatively and as C for ξ or ξ 1. The exponents m and→ n complex models. can have∞ the values→∞ such as→ 0, 1, 1/2, 1/4, 1/3 for example It is apparent that alternative simpler methods are re- [1,2,3]. quired in the first stages of device, package and board devel- It is known from analysis, experiments and numerical opment to give quick and acceptable estimates of temper- work that φ frequently transitions in a smooth manner be- aturelevels.Oneverysimplemethodisbasedontheuse tween the two asymptotes. of correlation equations which have been developed from The plots in Figs. 2 through 5 illustrate four basic, but experimental or numerical data. Often the correlations different, trends. Figure 2 shows both asymptotes φ0 and are limited to the particular ranges of the independent pa- φ increasing with increasing values of ξ, and the solution ∞ 2 Compact Modeling Numerical Models •Highly detailed results •Time consuming sign •Steep learning curve final de tion detail u Complex Models sol Compact Models gn si Simple Models yde r Correlations ina • quick, easy to use m - constrained• constrained to limited to limited rangerange of data of data preli cost Fig. 1. Overview of Simple, Compact and Detailed Models Fig. 2. Asymptotes and Compact Model 1 Fig. 3. Asymptotes and Compact Model 2 φ is concave upwards. For example, external forced and natural convection from single isothermal convex bodies downwards. For example, internal forced and natural con- exhibit this trend. vection within ducts and channels exhibit this trend. Figure 3 shows both asymptotes φ0 and φ decreasing with increasing values of ξ, and the solution∞φ is concave Figure 5 shows the asymptote φ0 to be constant indepen- upwards. For example, steady conduction in spherical wall dent of ξ and the asymptote φ decreases with increasing and external transient conduction from isothermal convex values of ξ, and the solution φ∞is concave downwards.An bodies exhibit this trend. example from tribology is the temperature rise of a heated Figure 4 shows both asymptotes φ0 and φ increasing contact area which moves with constant velocity in the sur- with increasing values of ξ, and the solution ∞φ is concave face of a semi-infinite body [4].
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