A Fractional Derivative Modeling of Heating and Cooling of LED Luminaires

Eduardo Balvís,Ricardo Bendaña,Sonia Zaragoza,Iván Area,Humberto Michinel,Angel Paredes,Alicia V Carpentier

doi:10.3390/math8030362

Abstract

In the context of energy efficient lighting, we present a mathematical study of the heating and cooling processes of a common type of luminaires, consisting of a single light-emitting diode source in thermal contact with an aluminum passive heat sink. First, we study stationary temperature distributions by addressing the appropriate system of partial differential equations with a commercial finite element solver. Then, we study the temporal evolution of the temperature of the chip and find that it is well approximated with a fractional derivative generalization of Newton’s cooling law. The mathematical results are compared and shown to largely agree with our laboratory measurements.

Highlights

The rapid development of the technology of high-power light-emitting diode (HP-LED)devices has had a great impact on the lighting industry [1]
The simplest formalism is Newton’s law of cooling, which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. This has very limited applicability [18] since, for instance, it cannot account for the thermal dynamics of systems where radiation or natural convection play a relevant role
We show that the steady-state temperature distribution can be accurately described by the solution of the system of equations that describes heat transfer in the aluminum passive heat sink and the surrounding air, proving that the maximal temperature of the system, which is that reached by the chip itself, can be kept well below its threshold of damage

Summary

Introduction

The rapid development of the technology of high-power light-emitting diode (HP-LED). devices has had a great impact on the lighting industry [1]. Heat dissipation is an interesting mathematical problem and an important technical issue in many industries [4] It plays an important role in the context of HP-LED cooling and different types of solutions have been explored in the literature. The second question (Section 3) is the variation in time of the temperature of the chip itself during the heating and cooling processes Rather than analyzing this complex problem from physical first principles, we look for a simple equation that can properly describe the evolution. We find that a fractional ordinary differential equation (FODE), which is the generalization of the simplest form of Newton’s cooling law, fits the experimental data quite satisfactorily Encouraged by this result, we speculate on the possible applications of FODEs in similar setups of mathematical engineering

Stationary Spatial Temperature Profile of the Luminaire

Conclusions and Outlook