On a theory for system-level cooling of close electronics enclosures by PCM-filled heat sinks—Exact solution and second law performance limits C. Naaktgeboren a,⇑ , A.T. Franco b , S.L.M. Junqueira b a Federal University of Technology, Paraná, Guarapuava Campus, Research Center for Thermal Sciences, Av. Profa. Laura Pacheco Bastos, 800, Industrial, CEP 85053-525 Guarapuava, PR, Brazil b Federal University of Technology, Paraná, Curitiba Campus, Ecoville Headquarters, CERNN – Research Center for Rheology and Non-Newtonian Fluids, Rua Dep. Heitor Alencar Furtado, 5000, Ecoville CEP 81280-340 Curitiba, PR, Brazil article info Article history: Received 15 May 2018 Received in revised form 3 July 2018 Accepted 4 July 2018 Keywords: System-level electronics cooling Stefan problem Zero-phase modeling Analytical solution Second law of thermodynamics abstract This paper takes a fundamental approach on the simple, however widespread, application of closed elec- tronics enclosures that are passively cooled by internally mounted heat sinks filled with phase change material (PCM) and exposed to periodic temperature environments. The regime of interest is that of peri- odic steady state (PSS). A model that allows for an analytical solution of the Stefan-like problem in the finite PCM domain and of the energy balance coupling of the closed enclosure is devised in dimensionless terms by using a lumped formulation for the enclosure and for the PCM domain, the later being known as zero-phase modeling (Naaktgeboren, 2007). The model is one that allows for internally reversible heat transfers in the PCM and in the enclosure domains, while retaining the finiteness (irreversibilities) of the heat interactions between the environment, the enclosure, and the PCM-filled heat sinks, as well as the heat sources irreversibilities, so as to allow for meaningful outcomes. Results include melting and freezing interface location history—conversely, melting and freezing times—and a design condition for perpetual occurence of phase change; hence, latent heat interactions. Moreover, a figure of merit for passive temperature regulation of real (irreversible) electronics enclosures is proposed as a temper- ature attenuation effectiveness. The limit imposed by the second law of thermodynamics on the temper- ature attenuation effectiveness is analytically derived and shown to depend on only two (out of the four) dimensionless parameters of the model. The original expression for the enclosure temperature attenua- tion effectiveness in the reversible limit, obtained herein, is of theoretical and practical significance. Ó 2018 Elsevier Ltd. All rights reserved. 1. Introduction The problem of a closed enclosure with internal work-into-heat dissipations (heat sources) that exchanges heat with a periodic temperature environment finds application in system-level elec- tronics cooling of outdoor systems [2,3] and on some types of buildings [4,5]. Electronics can be harmed by either internal tem- perature peaks or excessive temperature variations along the day-and-night cycle [6]—factors that also impact comfort on buildings. The issue of passive temperature control of electronics by employing phase change materials (PCM)—as part of Latent Heat Thermal Storage (LHTS) systems—has been mainly addressed by means of local temperature management, i.e., on an electronic component level [7–10], on electronic or solar module level [11,6,12], and on portable device [13,14] treatments. Such local and mesoscale [14,15] thermal management approaches deal mostly with the mitigation of local temperature peaks. In order to address the issue of temperature fluctuation along the day-and-night cycle of larger systems, a system-level tempera- ture control may be used [3,5,16]. In periodic steady state (PSS) regime, the temperature of the enclosure (system) is expected to oscillate around a mean enclosure temperature. Therefore, reduc- ing the enclosure thermal amplitude also means reducing its peaks of temperature, thus aiding all other local and mesoscale tempera- ture control systems in place. Electronics enclosure with PCM-filled heat sinks are complex systems from the point of view of transport phenomena. New tech- nologies and specific systems are often studied both experimen- tally and numerically. Pertinent numerical studies include solution to the Stefan problem, recirculating flows in enclosures, phase change heat transfer, conjugate (coupled) heat transfer prob- lems, and electronics cooling. An extensive compilation on such studies can be found on this [17] review. https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.012 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: NaaktgeborenC.PhD@gmail.com (C. Naaktgeboren). International Journal of Heat and Mass Transfer 127 (2018) 535–543 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt