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Numerical solution of the heat equation with non-linear, time derivative-dependent source term
(2010)
The mathematical modeling of heat conduction with adsorption effects in coated metal structures yields the heat equation with piecewise smooth coefficients and a new kind of source term. This term is special, because it is non-linear and furthermore depends on a time derivative. In our approach we reformulated this as a new problem for the usual heat equation, without source term but with a new non-linear coefficient. We gave an existence and uniqueness proof for the weak solution of the reformulated problem. To obtain a numerical solution, we developed a semi-implicit and a fully implicit finite volume method. We compared these two methods theoretically as well as numerically. Finally, as practical application, we simulated the heat conduction in coated aluminum fibers with adsorption in the zeolite coating. Copyright © 2010 John Wiley & Sons, Ltd.
Often, detailed simulations of heat conduction in complicated, porous media have large runtimes. Then homogenization is a powerful tool to speed up the calculations by preserving accurate solutions at the same time. Unfortunately real structures are generally non-periodic, which requires unpractical, complicated homogenization techniques. We demonstrate in this paper, that the application of simple, periodic techniques to realistic media, that are just close to periodic, gives accurate, approximative solutions. In order to obtain effective parameters for the homogenized heat equation, we have to solve a so called “cell problem”. In contrast to periodic structures it is not trivial to determine a suitable unit cell, which represents a non-periodic media. To overcome this problem, we give a rule of thumb on how to choose a good cell. Finally we demonstrate the efficiency of our method for virtually generated foams as well as real foams and compare these results to periodic structures.
Heat production in the windings of the stators of electric machines under stationary condition
(2014)
In electric machines due to high currents and resistive losses (joule heating) heat is produced. To avoid damages by overheating the design of effective cooling systems is required. Therefore the knowledge of heat sources and heat transfer processes is necessary. The purpose of this paper is to illustrate a good and effective calculation method for the temperature analysis based on homogenization techniques. These methods have been applied for the stator windings in a slot of an electric machine consisting of copper wires and resin. The key quantity here is an effective thermal conductivity, which characterizes the heterogeneous wire resin-arrangement inside the stator slot. To illustrate the applicability of the method, the analysis of a simplified, homogenized model is compared with the detailed analysis of temperature behavior inside a slot of an electric machine according to the heat generation. We considered here only the stationary situation. The achieved numerical results are accurate and show that the applied homogenization technique works in practice. Finally the results of simulations for the two cases, the original model of the slot and the homogenized model chosen for the slot (unit cell), are compared to experimental results.
Simulating the electromagnetic‐thermal treatment of thin aluminium layers for adhesion improvement
(2015)
A composite layer material used in packaging industry is made from joining layers of different materials using an adhesive. An important processing step in the production of aluminium-containing composites is the surface treatment and consequent coating of adhesive material on the aluminium surface. To increase adhesion strength between aluminium layer and the adhesive material, the foil is heat treated. For efficient heating, induction heating was considered as state-of-the-art treatment process. Due to the complexity of the heating process and the unpredictable nature of the heating source, the control of the process is not yet optimised. In this work, a finite element analysis of the process was established and various process parameters were studied. The process was simplified and modelled in 3D. The numerical model contains an air domain, an aluminium layer and a copper coil fitted with a magnetic field concentrating material. The effect of changing the speed of the aluminium foil (or rolling speed) was studied with the change of the coil current. Statistical analysis was used for generating a general control equation of coil current with changing rolling speed.
Due to the Renewable Energy Act, in Germany it is planned to increase the amount of renewable energy carriers up to 60%. One of the main problems is the fluctuating supply of wind and solar energy. Here biogas plants provide a solution, because a demand-driven supply is possible. Before running such a plant, it is necessary to simulate and optimize the process. This paper provides a new model of a biogas plant, which is as accurate as the standard ADM1 model. The advantage compared to ADM1 is that it is based on only four parameters compared to 28. Applying this model, an optimization was installed, which allows a demand-driven supply by biogas plants. Finally the results are confirmed by several experiments and measurements with a real test plant.
With proven impact of statistical fracture analysis on fracture classifications, it is desirable to minimize the manual work and to maximize repeatability of this approach. We address this with an algorithm that reduces the manual effort to segmentation, fragment identification and reduction. The fracture edge detection and heat map generation are performed automatically. With the same input, the algorithm always delivers the same output. The tool transforms one intact template consecutively onto each fractured specimen by linear least square optimization, detects the fragment edges in the template and then superimposes them to generate a fracture probability heat map.
We hypothesized that the algorithm runs faster than the manual evaluation and with low (< 5 mm) deviation. We tested the hypothesis in 10 fractured proximal humeri and found that it performs with good accuracy (2.5 mm ± 2.4 mm averaged Euclidean distance) and speed (23 times faster). When applied to a distal humerus, a tibia plateau, and a scaphoid fracture, the run times were low (1–2 min), and the detected edges correct by visual judgement. In the geometrically complex acetabulum, at a run time of 78 min some outliers were considered acceptable. An automatically generated fracture probability heat map based on 50 proximal humerus fractures matches the areas of high risk of fracture reported in medical literature.
Such automation of the fracture analysis method is advantageous and could be extended to reduce the manual effort even further.