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Application of an exact integral transform formulation for temperature estimation in solid-state electronics

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Abstract

This work presents an application of an exact analytical approach to estimate the temperature field of solid-state electronics (SSE). A partial lumped approach in the chip’s height was performed, obtaining a two-dimensional mathematical model. The internal heat generation (HG) regions caused by Joule effect due to internal components were modeled as Gaussian functions, and the classical integral transform technique (CITT) was used to solve the problem. The developed formulation was applied in three illustrative HG layouts for chips. In addition, the finite difference method was also implemented for verification and comparison purposes. The CITT provided fast computing and accurate results with small truncation orders in problems with multiple non-uniform heated regions. Furthermore, the formulation presented in this work may be applied to any SSE internal HGs layout, being a useful tool for thermal assessment of SSEs.

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Abbreviations

T :

Temperature (\(^{\circ}\)C)

k :

Thermal conductivity (W/(m K))

x, y :

Cartesian coordinates (m)

L, H :

SSE x and y dimensions, respectively (m)

\(T_{f}\) :

Temperature of the surrounding air (\(^{\circ }\)C)

\(T_{o}\) :

Reference temperature (\(^{\circ }\)C)

h :

Convection heat transfer coefficient (W/(m\(^2\)K))

\(N_{n}\) :

Normalization integrals

\(\dot{g}'''\) :

Overall heat generation (HG) (W/m\(^3\))

G :

Overall dimensionless HG

\(G_{e}\) :

Total dimensionless intensity of each individual HG region

erf:

Error function

\(\xi ,\,\eta\) :

Dimensionless Cartesian coordinates

\(\xi _{0},\,\eta _{0}\) :

HG center coordinate

\(\delta\) :

SSE z-dimension (m)

\(\beta\), \(\gamma\) :

Aspect ratios

\(\varPsi\), \(\lambda\) :

Eigenfunction and eigenvalue, respectively

\(\sigma\) :

HG dispersion parameter

\(\varTheta\) :

Dimensionless temperature

RE:

Relative error

\(\epsilon\)-RMS:

Root mean square error

\(\Delta \xi\), \(\Delta \eta\) :

FDM distance between nodes in \(\xi\) and \(\eta\) directions, respectively

\({\mathrm{Bi}}_L\) :

Biot number based on L-dimension

\(\mathrm{Bi}_{\delta }\) :

Biot number based on \(\delta\)-dimension

\(\bar{\, }\) :

Integral transform

n :

CITT index

\(n_{\max }\) :

Truncation order for CITT

i, j :

FDM node indexes

\(i_{\max }\), \(j_{\max }\) :

FDM number of nodes in \(\xi\) and \(\eta\) direction, respectively

k :

Index of each individual HG

\(k_{\max }\) :

Total number of individual HGs

CITT:

Related to the CITT

FDM:

Related to the FDM

CITT-Conv.:

Fully converged CITT result

References

  1. Abdelmlek KB, Araoud Z, Ghnay R, Abderrazak K, Charrada K, Zissis G (2016) Effect of thermal conduction path deficiency on thermal properties of leds package. Appl Therm Eng 102:251–260. https://doi.org/10.1016/j.applthermaleng.2016.03.100

    Article  Google Scholar 

  2. Bar-Cohen A, Wang P (2009) Thermal management of on-chip hot spot. In: ASME 2009 Second international conference on micro/nanoscale heat and mass transfer. American Society of Mechanical Engineers, pp 553–567. https://doi.org/10.1115/MNHMT2009-18548

  3. Chalhub DJNM, Sphaier LA (2011) Analysis of integral transform solutions for thermally developing flow in microchannels with isothermal wall. In: International conference on nanochannels, microchannels, and minichannels, vol 1. Edmonton, Alberta, Canada, ASME, pp 523–530. https://doi.org/10.1115/ICNMM2011-58175

  4. Chalhub DJNM, Sphaier LA (2011) Comparisons between integral transform, finite volumes and combined solution strategies for convective heat transfer problems. In: ASME/JSME 2011 8th thermal engineering joint conference. ASME/JSME 2011 8th Thermal Engineering Joint Conference. Honolulu, Hawaii, USA, ASME, pp T10058. https://doi.org/10.1115/AJTEC2011-44547

  5. Chalhub DJNM, Sphaier LA, Alves LS de B (2014) Integral transform analysis of Poisson problems that occur in discrete solutions of the incompressible Navier-Stokes equations. J Phys Conf Ser 547(1):012040. https://doi.org/10.1088/1742-6596/547/1/012040

    Article  Google Scholar 

  6. Chalhub DJNM, Sphaier LA, Alves LS de B (2014) Semi-analytical method for the solution of the Poisson equation derived from the Navier–Stokes using integral transform. In: ASME 2014 12th International conference on nanochannels, microchannels and minichannels collocated with the ASME 2014 4th joint US-European fluids engineering division summer meeting, Chicago, Illinois, USA, V001T12A016, ASME. https://doi.org/10.1115/ICNMM2014-22238

  7. Chalhub DJNM, Sphaier LA, Alves LS de B (2016) Integral transform solution for thermally developing slip-flow within isothermal parallel plates. Comput Therm Sci 8(2):147–161. https://doi.org/10.1615/ComputThermalScien.2016015459

    Article  Google Scholar 

  8. Cotta RM (1993) Integral transforms in computational heat and fluid flow. CRC Press, Cleveland

    MATH  Google Scholar 

  9. Cotta RM, Knupp DC, Quaresma JN, Lisboa KM, Naveira-Cotta CP, Zotin JLZ, Miyagawa HK (2020) Integral transform benchmarks of diffusion, convection–diffusion, and conjugated problems in complex domains. In: 50 Years of CFD in engineering sciences. Springer, pp 719–750. https://doi.org/10.1007/978-981-15-2670-1_20

  10. de Almeida AP, Naveira-Cotta CP, Cotta RM (2020) Transient three-dimensional heat conduction in heterogeneous media: integral transforms and single domain formulation. Int Commun Heat Mass Transf 117:104792. https://doi.org/10.1016/j.icheatmasstransfer.2020.104792

    Article  Google Scholar 

  11. de Barros L, Sphaier L (2019) Improved lumped analysis of Graetz problems with axial diffusion. J Braz Soc Mech Sci Eng. https://doi.org/10.1007/s40430-019-2061-8

    Article  Google Scholar 

  12. de Souza JRB, Lisbôa KM, Allahyarzadeh AB, de Andrade GJA, Loureiro JBR, Naveira-Cotta CP, Freire APS, Orlande HRB, Silva GAL, Cotta RM (2016) Thermal analysis of anti-icing systems in aeronautical velocity sensors and structures. J Braz Soc Mech Sci Eng 38(5):1489–1509. https://doi.org/10.1007/s40430-015-0449-7

  13. Feuillet V, Scudeller Y, Jarny Y (2009) The discrete boundary resistance method for thermal analysis of solid-state circuits and devices. Int J Therm Sci 48(2):372–382. https://doi.org/10.1016/j.ijthermalsci.2008.05.011

    Article  Google Scholar 

  14. Formalev VF, Kolesnik SA, Kuznetsova EL (2017) Time-dependent heat transfer in a plate with anisotropy of general form under the action of pulsed heat sources. High Temp 55(5):761–766. https://doi.org/10.1134/S0018151X17050066

    Article  Google Scholar 

  15. Hahn D, Osizik M (2012) Heat conduction. Willy, Hoboken

    Book  Google Scholar 

  16. Huang MJ, Shaw YR, Chien HC (2019) Thermal spreading resistance of heat sources on rectangular flux channel under non-uniform convective cooling. Int J Therm Sci 145:105979. https://doi.org/10.1016/j.ijthermalsci.2019.105979

    Article  Google Scholar 

  17. Iyengar M, Schmidt R (2006) Analytical modeling for prediction of hot spot chip junction temperature for electronics cooling applications. In: Thermal and thermomechanical proceedings 10th intersociety conference on phenomena in electronics systems. ITHERM, IEEE, pp 87–95. https://doi.org/10.1109/ITHERM.2006.1645326

  18. Kartashov É (2017) Integral transformations for the generalized equation of nonstationary heat conduction in a partially bounded region. J Eng Phys Thermophys 90(6):1279–1287. https://doi.org/10.1007/s10891-017-1684-9

    Article  Google Scholar 

  19. Kartashov E (2020) Analytical approaches to the analysis of unsteady heat conduction for partially bounded regions. High Temp 58(3):377–385. https://doi.org/10.1134/S0018151X20030086

    Article  Google Scholar 

  20. Knupp DC, Abreu LA (2016) Explicit boundary heat flux reconstruction employing temperature measurements regularized via truncated eigenfunction expansions. Int Commun Heat Mass Transf 78:241–252. https://doi.org/10.1016/j.icheatmasstransfer.2016.09.012

    Article  Google Scholar 

  21. Knupp DC, Naveira-Cotta CP, Cotta RM (2013) Conjugated convection-conduction analysis in microchannels with axial diffusion effects and a single domain formulation. J Heat Transf 135(9):091401. https://doi.org/10.1115/1.4024425

    Article  Google Scholar 

  22. Knupp DC, Naveira-Cotta CP, Cotta RM (2014) Theoretical-experimental analysis of conjugated heat transfer in nanocomposite heat spreaders with multiple microchannels. Int J Heat Mass Transf 74:306–318. https://doi.org/10.1016/j.ijheatmasstransfer.2014.03.005

    Article  Google Scholar 

  23. Knupp DC, Cotta RM, Naveira-Cotta CP, Kakaç S (2015) Transient conjugated heat transfer in microchannels: integral transforms with single domain formulation. Int J Therm Sci 88:248–257. https://doi.org/10.1016/j.ijthermalsci.2014.04.017

    Article  Google Scholar 

  24. Knupp DC, Naveira-Cotta CP, Renfer A, Tiwari MK, Cotta RM, Poulikakos D (2015) Analysis of conjugated heat transfer in micro-heat exchangers via integral transforms and non-intrusive optical techniques. Int J Numer Methods Heat Fluid Flow 25(6):1445–1462. https://doi.org/10.1108/HFF-08-2014-0259

    Article  MATH  Google Scholar 

  25. Kumar T (2004) A serial solution for the 2-d inverse heat conduction problem for estimating multiple heat flux components. Numer Heat Transf Part B Fundam 45(6):541–563. https://doi.org/10.1080/10407790490277940

    Article  Google Scholar 

  26. Lee S, Early M, Pellilo M (1997) Thermal interface material performance in microelectronic packaging applications. Microelectron J 28:1–2. https://doi.org/10.1016/S0026-2692(97)87871-8

    Article  Google Scholar 

  27. Li Q, Zhu X, Xuan Y (2015) Modeling heat generation in high power density nanometer scale GaAs/InGaAs/AlGaAs PHEMT. Int J Heat Mass Transf 81:130–136. https://doi.org/10.1016/j.ijheatmasstransfer.2014.10.020

    Article  Google Scholar 

  28. Lu X, Shi T, Xia Q, Liao G (2012) Thermal conduction analysis and characterization of solder bumps in flip chip package. Appl Therm Eng 36:181–187. https://doi.org/10.1016/j.applthermaleng.2011.12.028

    Article  Google Scholar 

  29. Mikhailov MD, Ozisik MN (1984) Unified analysis and solutions of heat and mass diffusion. Wiley, New York

    Google Scholar 

  30. Moore G (1965) Moore’s law. Electron Mag 38(8):114

    Google Scholar 

  31. Moosaie A (2008) Non-Fourier heat conduction in a finite medium subjected to arbitrary non-periodic surface disturbance. Int Commun Heat Mass Transf 35(3):376–383. https://doi.org/10.1016/j.icheatmasstransfer.2007.08.007

    Article  MathSciNet  Google Scholar 

  32. Muzychka YS (2006) Influence coefficient method for calculating discrete heat source temperature on finite convectively cooled substrates. IEEE Trans Compon Packag Technol 29(3):636–643. https://doi.org/10.1109/TCAPT.2006.880477

    Article  Google Scholar 

  33. Naveira-Cotta CP, Cotta RM, Orlande HRB, Fudym O (2009) Eigenfunctions expansions for transient diffusion in heterogeneous media. Int J Heat Mass Transf 52(121–122):5029–5039. https://doi.org/10.1016/j.ijheatmasstransfer.2009.04.014

    Article  MATH  Google Scholar 

  34. Naveira-Cotta CP, Cotta RM, Orlande HRB (2011) Inverse analysis with integral transformed temperature fields: identification of thermophysical properties in heterogeneous media. Int J Heat Mass Transf 54(7–8):1506–1519. https://doi.org/10.1016/j.ijheatmasstransfer.2010.11.042

    Article  MATH  Google Scholar 

  35. Pei C, Fu G, Zhao Y, Wan B, Cheng Y (2017) Simplified approach for steady thermal analysis of chips with variable power based on spatial autocorrelation. Appl Therm Eng 120:347–357. https://doi.org/10.1016/j.applthermaleng.2017.04.010

    Article  Google Scholar 

  36. Peterson G, Ortega A (1990) Thermal control of electronic equipment and devices. In: Advances in heat transfer, vol 20. Elsevier, pp 181–314. https://doi.org/10.1016/S0065-2717(08)70028-5

  37. Rinaldi N (2000) Thermal analysis of solid-base devices and circuits: an analytical approach. Solid State Electron 44(10):1789–1798. https://doi.org/10.1016/S0038-1101(00)00120-9

    Article  Google Scholar 

  38. Rogié B, Monier-Vinard E, Nguyen MN, Bissuel V, Laraqi N (2018) Practical analytical modeling of 3d multi-layer printed wired board with buried volumetric heating sources. Int J Therm Sci 129:404–415. https://doi.org/10.1016/j.ijthermalsci.2018.03.016

    Article  Google Scholar 

  39. Sharma CS, Schlottig G, Brunschwiler T, Tiwari MK, Michel B, Poulikakos D (2015) A novel method of energy efficient hotspot-targeted embedded liquid cooling for electronics: an experimental study. Int J Heat Mass Transf 88:684–694. https://doi.org/10.1016/j.ijheatmasstransfer.2015.04.047

    Article  Google Scholar 

  40. Silva GR, Knupp DC, Naveira-Cotta CP, Cotta RM, Neto AJS (2020) Estimation of slip flow parameters in microscale conjugated heat transfer problems. J Braz Soc Mech Sci Eng 42(263):1–16. https://doi.org/10.1007/s40430-020-02328-z

    Article  Google Scholar 

  41. Sphaier LA, Cotta RM, Naveira-Cotta CP, Quaresma JNN (2011) The unit algorithm for solving one-dimensional convection–diffusion problems via integral transforms. Int Commun Heat Mass Transf 38(5):565–571. https://doi.org/10.1016/j.icheatmasstransfer.2010.12.036

    Article  Google Scholar 

  42. Streetman BG, Banerjee S et al (1995) Solid state electronic devices. Prentice Hall, Englewood Cliffs, NJ

    Google Scholar 

  43. Tavakkoli F, Ebrahimi S, Wang S, Vafai K (2016) Thermophysical and geometrical effects on the thermal performance and optimization of a three dimensional integration circuit. J Heat Transf ASME 138(8):082101

    Article  Google Scholar 

  44. Wang P, Yang B, Bar-Cohen A (2009) Mini-contact enhanced thermoelectric coolers for on-chip hot spot cooling. Heat Transf Eng 30(9):736–743. https://doi.org/10.1080/01457630802678391

    Article  Google Scholar 

  45. Wolfram S (2003) The mathematica book. Cambridge University Press, New York/Champaign, IL

    MATH  Google Scholar 

  46. Xiao C, He H, Li J, Cao S, Zhu W (2017) An effective and efficient numerical method for thermal management in 3d stacked integrated circuits. Appl Therm Eng 121:200–209. https://doi.org/10.1016/j.applthermaleng.2017.04.080

    Article  Google Scholar 

  47. Yang XJ (2017) New integral transforms for solving a steady heat transfer problem. Therm Sci 21(1):79–87. https://doi.org/10.2298/TSCI17S1079Y

    Article  Google Scholar 

  48. Zhang J, Yan H, Li Y, Niu P, Yang Q (2019) An analytical model of thermal performance for an eccentric heat source on a rectangular plate with double-sided convective cooling. AIP Adv 9(2):025002. https://doi.org/10.1063/1.5080771

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support provided by CNPq (Grant 423158/2018-0), FAPERJ (Grant: E-26/202.637/2018) and CAPES, Brazilian agencies for the fostering of science.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Lívia M. Corrêa & Daniel J. N. M. Chalhub

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  1. Lívia M. Corrêa
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Correspondence to Daniel J. N. M. Chalhub.

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Corrêa, L.M., Chalhub, D.J.N.M. Application of an exact integral transform formulation for temperature estimation in solid-state electronics. J Braz. Soc. Mech. Sci. Eng. 43, 203 (2021). https://doi.org/10.1007/s40430-021-02912-x

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