# Analytical properties of Graetz modes in parallel and concentric configurations

## TL;DR

This paper develops an analytical framework for solving the generalized Graetz problem in layered cylindrical or parallel domains, enabling explicit, mesh-less computation of heat transfer modes for heat exchanger applications.

## Contribution

It introduces a new analytical approach to compute Graetz modes explicitly in layered configurations, simplifying the spectral problem and facilitating mesh-less solutions.

## Key findings

- Explicit formulas for eigenvalues via zeros of an analytical series
- Recursive computation of eigenfunctions (Graetz modes)
- Mesh-less, software-compatible solution method

## Abstract

The generalized Graetz problem refers to stationary convection-diffusion in uni-directional flows. In this contribution we demonstrate the ana-lyticity of generalized Graetz solutions associated with layered domains: either cylindrical (possibly concentric) or parallel. Such configurations are considered as prototypes for heat exchanger devices and appear in numerous applications involving heat or mass transfer. The established framework of Graetz modes allows to recast the 3D resolution of the heat transfer into a 2D or even 1D spectral problem. The associated eigefunctions (called Graetz modes) are obtained with the help of a sequence of closure functions that are recursively computed. The spectrum is given by the zeros of an explicit analytical serie, the truncation of which allows to approximate the eigenvalues by solving a polynomial equation. Graetz mode computation is henceforth made explicit and can be performed using standard softwares of formal calculus. It permits a direct and mesh-less computation of the resulting solutions for a broad range of configurations. Some solutions are illustrated to showcase the interest of mesh-less analytical derivation of the Graetz solutions, useful to validate other numerical approaches.