Model hierarchy for the shape optimization of a microchannel cooling system

Sebastian Blauth,René Pinnau,Christian Leithäuser

doi:10.1002/zamm.202000166

Abstract

AbstractWe model a microchannel cooling system and consider the optimization of its shape by means of shape calculus. A three‐dimensional model covering all relevant physical effects and three reduced models are introduced. The latter are derived via a homogenization of the geometry in 3D and a transformation of the three‐dimensional models to two dimensions. A shape optimization problem based on the tracking of heat absorption by the cooler and the uniform distribution of the flow through the microchannels is formulated and adapted to all models. We present the corresponding shape derivatives and adjoint systems, which we derived with a material derivative free adjoint approach. To demonstrate the feasibility of the reduced models, the optimization problems are solved numerically with a gradient descent method. A comparison of the results shows that the reduced models perform similarly to the original one while using significantly less computational resources.

Highlights

For small devices, such as chemical microreactors and electronic equipment, it is critical to have a cooling system that is able to absorb a lot of heat over a small surface area since the performance of these devices is directly related to their operating temperature
These techniques are based on the so-called shape derivative, which measures the sensitivity of a shape due to infinitesimal deformations, and the topological derivative, which measures the sensitivity of a geometry with respect to the insertion of an infinitesimally small hole, see, e.g., [14,61] for shape calculus and [45] for topological sensitivity analysis
We describe the numerical solution of shape optimization problems using (2.6) as an example and explain the details and modifications we use for the reduced models

Summary

Introduction

For small devices, such as chemical microreactors and electronic equipment, it is critical to have a cooling system that is able to absorb a lot of heat over a small surface area since the performance of these devices is directly related to their operating temperature For this purpose, cooling systems based on microchannels have been used (see, e.g., [24, 32, 42] and the references therein). A more general approach for design optimization consists of using shape or topological sensitivity analysis These techniques are based on the so-called shape derivative, which measures the sensitivity of a shape due to infinitesimal deformations, and the topological derivative, which measures the sensitivity of a geometry with respect to the insertion of an infinitesimally small hole, see, e.g., [14,61] for shape calculus and [45] for topological sensitivity analysis. To the best of our knowledge, the optimization of a microchannel cooling system by means of shape calculus has only been investigated in our earlier work [6], where we rigorously analyzed the theoretical aspects of this problem

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Results

Conclusion