Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyCompact Thermal Networks for Conjugate HeatTransfer by Moment MatchingLorenzo Codecasa, Dario D’Amore, Paolo MaffezzoniPolitecnico di Milano, Milan, Italye-mail: {codecasa, damore, pmaffezz}@elet.polimi.itAbstract— The problem of constructing compact thermal modelsfor conjugate heat transfer problems is faced. A novelnotion of thermal multi-port modeling conjugate heat transferis given and a novel moment matching method is introduced forconstructing compact models of such thermal multi-ports. Theresulting compact models preserve the passivity and reciprocityproperties of the original thermal problem and can exhibit highlevels of accuracy and compactness.I. INTRODUCTIONThe question of constructing compact models of conjugateheat transfer problems is crucial in many situations, such asin internally forced convection problems. Various techniqueshave been proposed in literature for constructing compactthermal models of heat conduction problems in electronicsystems. However conjugate heat transfer problems cannotin general be accurately modeled by conduction problems inwhich boundary conditions are represented by heat exchangecoefficients [1]. Thus ad hoc techniques for constructing compactmodels of conjugate heat transfer problems are needed.The firast attempt reported in literature for the construction ofcompact thermal models for conjugate heat transfer problems,as far as the authors know, is due to M. N. Sabry [1].In this paper the problem of constructing compact thermalmodels for conjugate heat transfer problems is faced. A novelapproach for constructing such compact models is obtainedby generalizing previous results by the authors for conductionheat transfer problems. Precisely the notion of thermal multiportmodeling conjugate heat transfer is given, by properlydefining its port variables. It is shown that the proposeddefinitions of port variables lead to a thermal multi-port whichpreserves the main thermodynamic properties of the conjugateheat transfer problem. In this way the second principle ofthermodynamics for the conjugate heat transfer problem ispreserved in the form of the passivity property of the thermalmulti-port. This property ensures the stability of dynamicthermal networks obtained by connecting such thermal multiports.Besides it is shown that a reciprocity relation holdsfor conjugate heat transfer problems, which generalizes thereciprocity relation for conduction heat transfer problems [2],and that this relation is preserved in the form of a reciprocityrelation for the thermal multi-ports. As with definition of portvariables for the thermal multi-ports modeling conduction heattransfer [3], also the definitions of port variables for thermalmulti-ports modeling conjugate heat transfer can be used forconstructing boundary condition independent (BCI) models.The moment matching method developed by the author forthe model reduction of heat conduction problems [4], is thenextended to construct compact models of the thermal multiportsmodeling conjugate heat transfer problems. The resultingcompact models preserve the passivity and reciprocity propertiesof the original thermal problem and can exhibit highlevels of accuracy and compactness.The resulting algorithm can be used to construct both staticand dynamic compact models and implies only the solution ofthe conjugate heat transfer problem at steady state or at mostin the frequency domain. Moreover, as a post-processing, fromthe solution of the compact model of the thermal multi-port,the whole spatio-temporal distribution of temperature and heatflux in the conjugate heat transfer problem can be recovered.As a reference problem, a prototype of a simple electronicsystem composed of two dinstinct heat sources and a microchannelhas been considered. A compact model of the thermalmulti-port modeling the corresponding conjugate heat transferproblem have been constructed. A high level of accuracyand compactness has been observed. The differences of theresponses of such compact model have also be evidenced withrespect to the responses of the compact models obtained bymodeling the problem as a conduction heat transfer problem.The remaining of this paper is organized as follows. Insection II conjugate heat transfer problems are introduced andtheir advection-diffusion equation model is discussed. Thermalnetworks for conjugate heat transfer problems are introducedin section III. A novel method for generating compact modelsof such thermal networks, based on Galerkin’s projection andMoment Matching, is proposed in sections IV, V. Numericalresults are presented in section VI.II. CONJUGATE HEAT TRANSFER PROBLEMLet Ω be a bounded region in which an incompressible fluidwith uniform properties can flow with a stationary velocityfield v(r), r being the position vector. The temperature risex(r,t) in Ω can then be assumed to be ruled by the followingadvection-diffusion equation in Ωc(r) ∂x∂t (r,t)+∇·(−k(r)∇x(r,t))++ c(r)v(r) ·∇x(r,t)=g(r,t) (1)in which c(r) is the volumetric heat capacity, k(r) is the thermalconductivity and g(r,t) is the generated power density.Eq. (1) has to be completed with conditions on ∂Ω, boundaryof Ω. The following Robin-like boundary conditions are©EDA Publishing/THERMINIC 2008 52ISBN: 978-2-35500-008-9