Variance-Driven Mean Temperature Reduction in Nonuniformly Heated Radiative-Conductive Systems

Abstract

Radiative-conductive systems are intrinsically nonlinear due to the quartic temperature dependence of thermal radiation. Under fixed total heating power, convexity arguments imply that nonuniform temperature distributions radiate more efficiently and therefore exhibit a lower mean temperature than their isothermal counterparts. However, this conclusion remains qualitative, and an explicit quantitative relation between temperature heterogeneity and mean temperature reduction has been lacking. Here we derive a variance-based analytical expression linking the area-averaged temperature to the corresponding isothermal equilibrium temperature in a nonuniformly heated radiative--conductive system. By integrating the governing equation and performing a systematic second-order expansion about the ambient temperature, we show that the decrease of the mean temperature relative to the isothermal equilibrium value is linearly proportional to the temperature variance, with a proportionality coefficient set solely by the ambient temperature. This result transforms the convexity-based inequality into a quantitative statistical relation within the perturbative regime and provides a physically transparent framework for describing nonlinear radiative averaging in thermally heterogeneous systems.

Figures (5)

• Figure 1: (a) Geometry of the radiative--conductive thin disk model. (b) Representative axisymmetric temperature distribution under localized heating at center.
• Figure 2: Comparison between the full 3D finite-element simulation and the reduced 2D model: (a) mid-plane temperature extracted from the 3D simulation; (b) numerical solution of the reduced governing equation.
• Figure 3: Radial temperature profile $T(r)$ under localized heating. The horizontal lines denote the area-averaged temperature $\bar{T}$ and the corresponding isothermal equilibrium temperature $T_{\rm iso}$. The inset is the zoomed-in comparison between $\bar{T}$ and $T_{\rm iso}$, demonstrating the reduction of the mean temperature in the nonisothermal case.
• Figure 4: Comparison between the exact mean temperature and the variance-based approximation. (a) Radial temperature profile $T(r)$, $T_{\rm iso}$, the numerical mean temperature $\bar{T}_{\mathrm{num}}$, and the analytical result $\bar{T}_{\mathrm{anal}}$. (b) Zoomed-in view highlighting the quantitative agreement between $\bar{T}_{\mathrm{num}}$ and $\bar{T}_{\mathrm{anal}}$, and both lower than $T_{\rm iso}$.
• Figure 5: Absolute error between the numerical mean temperature $\bar{T}_{\rm num}$ and the second-order analytical approximation $\bar{T}_{\rm anal}$ as a function of the maximum temperature nonuniformity $T(0)-T_\textrm{a}$. The error is defined as $|\bar{T}_{\rm num}-\bar{T}_{\rm anal}|$. The logarithmic horizontal axis highlights the broad parameter range over which the second-order approximation remains accurate.