Heat transfer in a medium in which many small particles are embedded

TL;DR

This paper derives a limiting heat transfer equation for a medium embedded with many small particles, considering non-periodic distributions and the asymptotic behavior as particle size diminishes.

Contribution

It introduces a new model for heat transfer in media with numerous small particles without assuming periodicity, extending previous theories.

Findings

Derived the limiting heat equation for complex particle systems.

Established conditions for particle size, number, and spacing.

Provided a framework for non-periodic particle distributions.

Abstract

The heat equation is considered in the complex system consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies a Newton-type boundary condition is imposed. An equation for the limiting field is derived when the characteristic size $a$ of the small bodies tends to zero, their total number $\mathcal{N}(a)$ tends to infinity at a suitable rate, and the distance $d = d(a)$ between neighboring small bodies tends to zero $a << d$. No periodicity is assumed about the distribution of the small bodies.