Thermal Computations for Electronics

Gordon N. Ellison has a BA in Physics from the University of California at Los Angeles (UCLA) and an MA in Physics from the University of Southern California (USC). His career in thermal engineering includes eight years as a Technical Specialist at NCR and 18 years at Tektronix, Inc., retiring from the latter as a Tektronix Fellow. Over the last 15 years Mr. Ellison has been an independent consultant and has also taught the course, Thermal Analysis for Electronics, at Portland State University, Oregon. He has also designed and written several thermal analysis computer codes.

Introduction


Primary mechanisms of heat flow


Conduction


Application example: Silicon chip resistance calculation


Convection


Application example: Chassis panel cooled by natural convection


Radiation


Application example: Chassis panel cooled only by radiation 7


Illustrative example: Simple thermal network model for a heat sinked power transistor


Illustrative example: Thermal network circuit for a printed circuit board


Compact component models


Illustrative example: Pressure and thermal circuits for a forced air cooled enclosure


Illustrative example: A single chip package on a printed circuit board—the problem


Illustrative example: A single chip package on a printed circuit board—Fourier series solution


Illustrative example: A single chip package on a printed circuit board—thermal network solution


Illustrative example: A single chip package on a printed circuit board—finite element solution


Illustrative example: A single chip package on a printed circuit board—methods compared


Thermodynamics of airflow


The first law of thermodynamics


Heat capacity at constant volume


Heat capacity at constant pressure


Steady gas flow as an open, steady, single stream


Air temperature rise: Temperature dependence


Air temperature rise: T identified using differential forms of ΔT,ΔQ


Air temperature rise: T identified as average bulk temperature


Airflow I: Forced flow in systems


Preliminaries


Bernoulli’s equation


Bernoulli’s equation with losses


Fan testing


Estimate of fan test error accrued by measurement of downstream static pressure


Derivation of the "one velocity" head formula


Fan and system matching


Adding fans in series and parallel


Airflow resistance: Common elements


Airflow resistance: True circuit boards


Modeled circuit board elements


Combining airflow resistances


Application example: Forced air cooled enclosure


Airflow II: Forced flow in ducts, extrusions, and pin fin arrays


The airflow problem for channels with a rectangular cross-section


Entrance and exit effects for laminar and turbulent flow


Friction coefficient for channel flow


Application example: Two-sided extruded heat sink


A pin fin correlation


Application example: Pin fin problem from Khan, et al.


Flow bypass effects according to Lee


Application example: Flow bypass method using Muzychka and Yovanovich correlation


Application example: Flow bypass method using HBT friction factor correlation


Flow bypass effects according to Jonsson and Moshfegh


Application example: Pin fin problem using Jonsson and Moshfegh correlation


Airflow III: Buoyancy driven draft


Derivation of buoyancy driven head


Matching buoyancy head to system


Application example: Buoyancy-draft cooled enclosure


System models with buoyant airflow


Forced convective heat transfer I: Components


Forced convection from a surface


The Nusselt and Prandtl numbers


The Reynold’s number


Classical flat plate forced convection correlation: Uniform surface temperature, laminar flow


Empirical correction to classical flat plate forced convection


correlation, laminar flow


Application example: Winged aluminum heat sink


Classical flat plate forced convection correlation: Uniform heat rate per unit area, laminar flow


Classical flat plate (laminar) forced convection correlation extended to small Reynold’s number


Circuit boards: Adiabatic heat transfer coefficients and adiabatic temperatures


Adiabatic heat transfer coefficient and temperature according to M. Faghri, et al.


Adiabatic heat transfer coefficient and temperature according to R. Wirtz


Application example: Circuit board with 1.5 in. / 1.5 in. / 0.6 in. convecting modules


Application example: Circuit board with 0.82 in./ 0.24 in. /0.123 in. convecting modules


Forced convective heat transfer II: Ducts, extrusions, and pin fin arrays


Boundary layer considerations


A convection/conduction model for ducts and heat sinks


Conversion of an isothermal heat transfer coefficient referenced to inlet to referenced to local air


Nusselt number for fully developed laminar duct flow corrected for entry length effects


A newer Nusselt number for laminar flow in rectangular (cross-section) ducts


Nusselt number for turbulent duct flow


Application example: Two-sided extruded heat sink


Flow bypass effects according to Jonsson and Moshfegh


Application example: Heat sink in a circuit board channel using the flow bypass method of Lee


In-line and staggered pin fin heat sinks


Application example: Thermal resistance of a pin fin heat sink


Natural convection heat transfer I: Plates


Nusselt and Grashof numbers


Classical flat plate correlations


Small device flat plate correlations


Application example: Vertical convecting plate


Application example: Vertical convecting and radiating plate


Vertical parallel plate correlations applicable to circuit board channels


Application example: Vertical card assembly


Recommended use of vertical channel models in sealed and vented enclosures


Conversion of heat transfer coefficients referenced-to-inlet air to referenced-to-local air


Application example: Enclosure with circuit boards - enclosure temperatures only


Application example: Enclosure with circuit boards - circuit board temperatures only


Application example: Enclosure with circuit boards, comparison with CFD


Application example: Single circuit board enclosure with negligible circuit board radiation


Illustrative example: Single circuit board enclosure with radiation exchange between interior enclosure walls and circuit board, results compared with experiment


Illustrative example: Metal walled enclosure with ten circuit boards


Illustrative example: Metal walled enclosure with heat dissipation provided


Natural convection heat transfer II: Heat sinks


Heat sink geometry and some nomenclature


A rectangular U-channel correlation from Van de Pol and Tierney


Design plots representing the Van de Pol & Tierney correlation


A few useful formulae


Application example: Natural convection cooled, vertically oriented heat sink


Application example: Natural convection cooled, nine fin heat sink compared to test data


Thermal radiation heat transfer


Blackbody radiation


Spacial effects and the view factor


Application example: View factors for finite parallel plates


Non-black surfaces


The radiation heat transfer coefficient


Application example: Radiation and natural convection cooled enclosure with circuit boards


Radiation for multiple gray-body surfaces


Hottel script F (F) method for gray-body radiation exchange


Application example: Gray-body circuit boards analyzed as infinite parallel plates


Application example: Gray-body circuit boards analyzed as finite parallel plates


Thermal radiation networks


Thermal radiation shielding for rectangular U-channels (fins)


Application example: Natural convection and radiation cooled heat sink


Application example: Nine fin heat sink, compared with test data


Application example: Natural convection and radiation cooled nine fin heat sink


Illustrative example: Natural convection and radiation cooled heat sink


Conduction I: Some basics


Fourier’s law of heat conduction


Application example: Mica insulator with thermal paste


Thermal conduction resistance of some simple structures


The one-dimensional differential equation for heat conduction


Application example: Aluminum core board with negligible air cooling


Application example: Aluminum core board with forced air cooling


Application example: Simple heat sink


Fin efficiency


Differential equations for more than one dimension


Physics of thermal conductivity of solids


Thermal conductivity of circuit boards (epoxy-glass laminates)


Application example: Epoxy-glass circuit board with copper


Thermal interface resistance


Application example: Contact resistance for an aluminum joint


Conduction II: Spreading resistance


The spreading problem


Fixed spreading angle theories


Circular-source, semi-infinite media solution by Carslaw and Jaeger (1986)


Rectangular-source, time dependent, semi-infinite media solution by Joy & Schlig (1970)


Other circular source solutions


Rectangular source on rectangular, finite-media with one convecting surface: Theory


Rectangular source on rectangular, finite-media: Design curves


Application example: Heat source centered on a heat sink (Ellison, 2003)


Application example: IC chip on an alumina substrate


Rectangular source on rectangular, finite-media with two convecting surfaces: Theory


Exploring the difference between one-sided and two-sided Newtonian cooling


Including the effect of two different ambients to the two-sided spreading theory


Application example: Heat sink with two convecting sides, one finned and one flat


Square source on square, finite-media with one convecting surface - time dependent (Rhee and Bhatt, 2007)


Additional mathematical methods


Thermal networks: Steady-state theory


Illustrative example: A simple steady-state, thermal network problem, solutions compared


Thermal networks: Time-dependent theory


Illustrative example: A simple time-dependent, thermal network problem


Finite difference theory for conduction with Newtonian cooling


Programming the pressure/airflow network problem


Finite element theory - the concept of the calculus of variations


Finite element theory - derivation of the one-dimensional Euler-Lagrange equation


Finite element theory - application of the one-dimensional Euler-Lagrange equation


Finite element theory - derivation of the two-dimensional Euler-Lagrange equation


Finite element theory - application of the Euler-Lagrange equation to two dimensions


Appendices





Bibliography





Index