This post discusses recent advances in the collaborative project “OptiMag“, carried out between Magnotherm Solutions GmbH and TU Darmstadt. It was addressed recently in the ECMI annual report as well as in blog posts in 2023 and 2024. It investigates magnetocaloric refrigeration devices that use solid magnetocaloric materials (MCMs) as the working medium and changing magnetic fields as forcing. They have the potential for higher energy efficiency and also eliminate direct greenhouse gas emissions through refrigerant leakage.
The two central components of the refrigeration device are the Active Magnetic Regenerator (AMR) and a permanent magnet (PM) assembly. The former is an elongated porous structure made of magnetocaloric material and functions as the thermally active element. The MCM reacts to the application and removal of a magnetic field by heating up or cooling down. By cyclically pumping fluid through the AMR, a temperature difference is built up between its ends, and heat is pumped from cold side to hot side. The periodic magnetization is achieved by the movement of the rotating permanent magnet assembly (see figure, left). The aim of the project is the optimization of the PM assembly considering its effect on the performance of the AMR directly.
In a recent contribution, Wiesheu et al. applied topology and shape optimization to the PM assembly, optimizing for a desired shape of the produced temporal magnetic field profile (see figure, center; rotation angle is linked to time by the constant rotation speed). This was successful; however, this specific objective function is heuristic in nature: the actual objective is the performance of the AMR, and the objective function based on the field profile does not predict this directly. Extending this work, the simulation of the PM assembly is coupled to a numerical model of the AMR (see figure): a series of magnetostatic simulations is conducted for different rotation angles. The magnetic field at the center of the AMR (marked by the crosses) is sampled for each angle and translated to a temporal field profile. This is then passed as an input to a 1D simulation of the AMR, which is modelled by a transient convection-diffusion equation with magnetocaloric heat capacity and heat source terms. In the periodic steady state of this model, the cooling power as main quantity of interest is computed, which is then passed to the optimizer as the objective value.
The simulation setup is made suitable for optimization by emphasizing computational efficiency and by providing parameter sensitivities for the objectives, so that a gradient-based optimizer can be used: The magnetic part is based on isogeometric analysis (IGA) and thus allows for spline-based geometry description and freeform optimization. The simulation of the rotation is sped up by the use of harmonic mortaring and parameter gradients of the sampled magnetic field are efficiently calculated using an adjoint method. To speed up the convergence to the periodic steady state solution, the shooting method is employed in the AMR simulation. The sensitivity of the cooling power against the given H-profile is computed using the adjoint method for ODEs with an adaptation to be able to handle the steady state problem. Care must be taken to ensure sufficient smoothness of the magnetocaloric material data.
With the coupled optimization setup, it is now possible to optimize directly with respect to the AMR performance. A proof-of-concept experiment showed a 10% improvement in simulated cooling power when using the coupled optimization compared to optimizing against a reasonable heuristic objective. However, it remains to be investigated whether this is a general result and whether a heuristic could be found which is on par with the performance of the coupled optimization.
The team at TU Darmstadt consists of Prof. Sebastian Schöps, Prof. Oliver Weeger, Boian Balouchev, Yusuf Elbadry, Melina Merkel and Michael Wiesheu. Magnotherm is represented by Dr. Maximilian Fries, Dimitri Benke and Tim Sittig. The research is funded by the LOEWE Project 1450/23-04 via the Hessian Ministry of Science and Art and the Hessen-Agentur.