Skip to main content

Thermal Processes in Electronic Equipment at Uncertainty


At present, electronic systems are thermally designed on the basis of the assumption that all the parameters and factors that determine the thermal processes are fully known and unambiguously determined, id est, that they are determinate. However, the practice of creation and operation of real electronic systems shows that the real values of determining parameters and factors, as well as the thermal processes and temperature distributions, are uncertain and can take any values within some intervals of their variation with an equal probability. The disregard for the interval stochastic character of the thermal processes leads to design errors and development of uncompetitive electronic systems. This article elaborates a method that permits modeling non-stationary interval stochastic thermal processes in an electronic system at interval uncertainty of input factors and parameters. The method is based on obtaining equations for non-stationary statistical measures (mathematical expectations, variances, mean square deviations, and covariances) of thermal processes at specified statistical measures of input data. The article gives an example of applying the elaborated method to thermal processes in a real electronic system that consists of electronic modules with printed circuit boards, as well as integrated microcircuits, resistors, and other electronic components installed on them.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1

    Adomian, G., Stochastic Systems, New York: Academic Press, 1983.

  2. 2

    Wang, C., Qiu, Z., and Chen, X., Uncertainty Analysis for Heat Convection-Diffusion Problem with Large Uncertain-but-Bounded Parameters, Acta Mech., 2015, vol. 226, no. 11, pp. 3831–3844.

  3. 3

    Wang, C. and Qiu, Z., Hybrid Uncertain Analysis for Steady-State Heat Conduction with Random and Interval Parameters, Int. J. Heat Mass Transfer, 2015, vol. 80, pp. 319–328.

  4. 4

    Claudio, R., Avila, da S.Jr., Azikri de Deus, H.P., Kozlik, A.Jr., and Garcia, O.S., Application of the Method of Galerkin to Non Linear Problem Stochastic Heat Conduction One-Dimensional,Appl. Mech. Mater., 2015, vol. 751, pp. 325–330.

  5. 5

    Ellison, G.N., Thermal Computations for Electronics. Conductive, Radiative, and Convective Air Cooling, New York: CRC Press, 2011.

  6. 6

    Feller, W., An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed., New York: Wiley, 1970.

  7. 7

    Keller, C.J. and Antonetti, V.W., Statistical Thermal Design for Computer Electronics, Electronic Pack. Prod., 1979, vol. 19, no. 3, pp. 55–62.

  8. 8

    Madera, A.G., Simulation of Stochastic Heat Conduction Processes, Int. J. Heat Mass Transfer, 1994, vol. 37, no. 16, pp. 2671–2677.

  9. 9

    Madera, A.G., Modelirovanie teploobmena v tekhnicheskikh sistemakh (Modeling of Heat Transfer in Technical Systems), Moscow: Akad. V.A. Mel’nikov Sci. Found., 2005.

  10. 10

    Madera, A.G., Hierarchical Method for Mathematical Modeling of Stochastic Thermal Processes in Complex Electronic Systems,Comput. Res. Model., 2019, vol. 11, no. 4, pp. 613–630.

  11. 11

    Madera, A.G., Interval-Stochastic Thermal Processes in Electronic Systems: Analysis and Modeling, J. Eng. Thermophys., 2017, vol. 26, no. 1, pp. 17–28.

  12. 12

    Madera, A.G., Interval-Stochastic Thermal Processes in Electronic Systems: Modeling in Practice, J. Eng. Thermophys., 2017, vol. 26, no. 1, pp. 29–38.

  13. 13

    Pugachev, V.S., Probability Theory and Mathematical Statistics for Engineers, Oxford: Pergamon Press, 1984.

Download references


This work was performed as part of the budget project of Scientific Research Institute for System Analysis RAS (Program of Basic Scientific Research SP 14, theme no. 0065-2019-0001, AAAA-A19-119011790077-1).

Author information



Corresponding author

Correspondence to A. G. Madera.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Madera, A.G., Kandalov, P.I. Thermal Processes in Electronic Equipment at Uncertainty. J. Engin. Thermophys. 29, 170–180 (2020).

Download citation