A Unit-Cell Shape Optimization Approach for Maximizing Heat Transfer in Periodic Fin Arrays at Constant Solid Temperature

TL;DR

The paper presents a unit-cell–based optimization framework to maximize heat transfer in periodic fin arrays at constant solid temperature, leveraging macro-scale isothermal-solid models to reduce full-array simulations to a single unit cell. It combines an adjoint-free-shape optimization with an augmented Lagrangian approach and geometric constraints to obtain optimal fin shapes under either fixed pressure drop or fixed flow rate, while quantifying the impact of entrance effects. Results show heat-transfer gains up to 104% in a fictitious case, but only about 8% when both flow rate and pressure drop are constrained; the accuracy of unit-cell predictions is strongly affected by how temperature scaling is formulated and by entrance-region effects. The work highlights the potential and limits of unit-cell models for rapid design of extended heat transfer surfaces and points to future improvements in accounting for entrance effects and broader operating conditions. Overall, the method offers a computationally efficient pathway to optimize periodic fin arrays with meaningful gains, while identifying critical sources of modeling error to address in future work.

Abstract

Periodic fin structures are often employed to enhance heat transfer in compact cooling solutions and heat exchangers. Adjoint-based optimization methods are able to further increase the heat transfer by optimizing the fin geometry. However, obtaining optimal geometries remains challenging in general because of the high computational cost of full array simulations. In this paper, a unit cell optimization approach is presented that starts from recently developed macro-scale models for isothermal solid structures. The models exploit the periodicity of the problem to reduce the computational cost of evaluating the array heat transfer to that of a single periodic unit cell. By combining these models with a geometrically-constrained free-shape optimization approach, optimal fin geometries are obtained for the periodic fin array that maintain a minimal fin distance. Moreover, using an augmented Lagrangian approach, also the average pressure gradient and barycenter of the fin can be fixed. On a fictitious use-case, heat transfer increases up to 104 \% are obtained. When also flow rate is constrained in addition to maintain a high effectiveness, only up to 8 \% heat transfer increase is observed. Finally, the errors of the unit-cell optimization approach are investigated, indicating that with a good choice of cost functional formulation, errors of the approach as low as 1-2 \% can be obtained for the periodically developed part of the array. Finally, the entrance effect to the heat transfer is found to be non-negligible with a contribution of 10-15 \% for the considered fin array. This advocates for further research to extend the unit-cell models towards improved modeling of entrance effects.

Figures (7)

• Figure 1: This figure illustrates a periodic fin array, highlighting a unit cell and its fluid domain ${\Omega}_{}$ and fin boundary $\Sigma_f$.
• Figure 2: Figure illustrating the components of the shape optimization subproblem, including the geometrical box to which the fin is constrained and the unit-cell domain boundaries. The colors illustrate the magnitude of the initial domain deformation field $\boldsymbol{\mathcal{V}}_{}$.
• Figure 3: Geometry of cylinder array and unit cell.
• Figure 4: Initial (a) and optimized fin arrays obtained by optimizing the normalized (b) and scaled (c) cost function applied at unit-cell level, with the color representing the temperature $T$.
• Figure 5: Heat transfer increase achieved by optimization as a function of pressure drop, for the cost function without $\theta$-normalization. The shapes corresponding to the optima are illustrated, with the color scale representing the periodic temperature $\theta$.