# Estimation of the heat conducted by a cluster of small cavities and characterization of the equivalent heat conduction

## TL;DR

This paper develops a method to estimate heat conduction in a cluster of small cavities using time domain integral equations, deriving effective heat conductivity with explicit error estimates.

## Contribution

It introduces a novel approach to model heat conduction in cavity clusters via time domain integral equations and derives explicit error bounds for the effective conductivity.

## Key findings

- Heat conduction dominated by sum over individual cavity heats after interactions.
- Derived explicit error estimates for the effective heat conductivity.
- Presented a method favoring space variables using Laplace operator-based potentials.

## Abstract

We estimate the heat conducted by a cluster of many small cavities. We show that the dominating heat is a sum, over the number of the cavities, of the heats generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a close, and invertible, system of time domain integral equations of a second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a 3D bounded domain, with explicit error estimates in terms of that cluster. At the analysis level, we use time domain integral equations. Doing that, we have two choices. First, we can favor the space variable by reducing the heat potentials to the ones related to the Laplace operator (avoiding Laplace transform). Second, we can favor the time variable by reducing the representation to the Abel integral operator. As the model under investigation has time-independent parameters, we follow here the first approach.