Design of Heating Coils Based on Space-Filling Fractal Curves for Highly Uniform Temperature Distribution

Heating coils utilize the concept of resistive heating to convert electrical energy into thermal energy. Uniform heating of the target area is the key performance indicator for heating coil design. Highly uniform distribution of temperature can be achieved by using a dense metal distribution in the area under consideration, however, this increases the cost of production significantly. A low-cost and efficient heating coil should have excellent temperature uniformity while having minimum metal consumption. In this work, space-filling fractal curves, such as Peano curve, Hilbert curve and Moore curve of various orders, have been studied as geometries for heating coils. In order to compare them in an effective way, the area of the geometries has been held constant at 30 mm × 30 mm and a constant power of 2 W has been maintained across all the geometries. Further, the thickness of the metal coils and their widths have been kept constant for all geometries. Finite Element Analysis (FEA) results show Hilbert and Moore curves of order-4, and Peano curve of order-3 outperform the typical double-spiral heater in terms of temperature uniformity and metal coil length.

Arithmetic-Analytic Representation of Peano Curve

In this work, we obtained a nonmatrix analytic expression for the generator of the Peano curve. Applying the iteration method of fractal, we established a simple arithmetic-analytic representation of the Peano curve as a function of ternary numbers. We proved that the curve passes each point in a unit square and that the coordinate functions satisfy a Hölder inequality with index α=1/2, which implies that the curve is everywhere continuous and nowhere differentiable.

A construction of a Peano curve

PEANO, G.:

USING PEANO CURVES TO CONSTRUCT LAPLACIANS ON FRACTALS

We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.