Mohammad Reza Salimpour Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran; Department of Mechanical Engineering, University of California, Riverside, CA 92521 e-mail: salimpour@cc.iut.ac.ir Ahmed T. Al-Sammarraie 1 Mem. ASME Department of Mechanical Engineering, University of California, Riverside, CA 92521 e-mail: aalsammarraie@engr.ucr.edu Azadeh Forouzandeh Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mail: a.foruzande@me.iut.ac.ir Mahsa Farzaneh Department of Mechanical and Energy Engineering, University of North Texas, Denton, TX 76207 e-mail: mahsa.farzaneh@unt.edu Constructal Design of Circular Multilayer Microchannel Heat Sinks Based on the constructal theory concepts, an investigation is carried out to optimize cir- cular multilayer microchannels embedded inside a rectangular heat sink with different numbers of layers and flow configurations. The lower surface of the heat sink is uniformly heated, while both pressure drop and length of the microchannel are fixed. Also, the vol- ume of the heat sink is kept fixed for all studied cases, while the effect of solid volume fraction is examined. All the dimensions of microchannel heat sinks are optimized in a way that the maximum temperature of the microchannel heat sink is minimized. The results emphasize that using triple-layer microchannel heat sink under optimal conditions reduces the maximum temperature about 10.3  C compared to the single-layer arrange- ment. Further, employing counter flow configuration in double-layer microchannel improves its thermal performance, while this effect is less pronounced in the triple-layer architecture. In addition, it is revealed that the optimal design can be achieved when the upper channels of a multilayer microchannel heat sink have bigger diameters than the lower ones. Finally, it is observed while using two layers of microchannels is an effective means for cooling improvement, invoking more layers is far less effective and hence is not recommended. [DOI: 10.1115/1.4041196] 1 Introduction Electronics cooling has dragged a great attention in the few past decades due to the rapid evolution of these components per- formance, and consequently, the need for efficient techniques to keep them working under a proper operating temperature and pre- vent overheating has become substantial [1–7]. Furthermore, com- pactness plays as a crucial aspect of the manufacturing of these devices nowadays; however, this leads to more heat flux intensity which is not desirable. One of the most compact and prominent techniques in this field, compared to other conventional methods, is utilizing the microchannel heat sinks [8–12]. This stems from the fact that they have a large solid–fluid thermal conductive ratio, as well as, the needless for utilizing costly coolants inside their conduits. The concept of the microchannel was proposed for the first time by Tuckerman and Pease [13]. They suggested an effective method for designing microchannels in the laminar and fully developed flow. Wang et al. [14] studied numerically the convec- tive heat transfer in microchannels with an assumption of negligi- ble axial conduction comparing their results to experimental data. Their goal was to investigate the feasibility of using Navier–Stokes equations in microchannels. Farzaneh et al. [15] investigated the effect of using reverting microchannels in the cooling of circular, square, and triangular heat sinks. Mardani and Salimpour [16] invoked constructal theory to optimize the triangu- lar microchannel heat sinks, analytically and numerically. They studied the effect of volume fraction of solid material and pressure drop on the maximum temperature of the microchannel heat sink and observed that increasing the side angle of the triangular microchannel improves its performance. In 1996, Bejan [17] introduced his constructal theory, which is one of the best methods to optimize microchannels. Using this theory, a lot of systems have been optimized during the recent decade [18–21]. For instance, Norouzi and Amidpour [22] utilized constructal theory to optimize the dimensions and flow arrange- ment in heat recovery steam generators. Rocha et al. [23] used constructal theory to design the best architecture of highly con- ductive materials for cooling a disk-shaped body. In another work, Lorenzini et al. [24] designed X-shaped conductive path- ways for cooling a heat generating object. Moreover, Xia et al. [25] studied the flow and heat transfer in complex microchannel heat sinks, numerically and experimentally. Muzychka [26] addressed the constructal design of circular and noncircular ducts in forced convection subjected to finite volume and pressure drop. In another research [27], an increase in heat trans- fer rate per unit volume of the heat exchanger was achieved using additional micropipes in the arrays of circular pipes. Salimpour et al. [28] explored the convective heat transfer in arrays of different microchannels with constraints of constant volume and pressure drop. Further, Salimpour et al. [29] found a definite hydraulic diame- ter at which the overall conductance is maximized and showed that the rectangular and elliptic microchannel heat sinks have the best thermal performance at a specific volume and pressure drop. Vafai and Zhu [30] introduced the concept of double-layer microchannel heat sink. They found that the double-layer micro- channel heat sinks are superior over the conventional single-layer microchannel heat sinks. Later, Lu and Vafai [31] performed a comparative study of innovative microchannel heat sinks like double-layer and multilayer microchannels for the electronic cool- ing purpose. In addition, the effect of multilayers on microchannels was investigated by Saidi and Khiabani [32]. They examined, as well, the effects of length/width ratio, porosity, and fluid properties on the thermal resistance of microchannels. Their main conclusion is that increasing the layers of microchannel can significantly decrease its thermal resistance. Hung et al. [33] analyzed convec- tive heat transfer in a double-layered microchannel with the 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received January 3, 2018; final manuscript received July 24, 2018; published online September 17, 2018. Assoc. Editor: Carey J. Simonson. Journal of Thermal Science and Engineering Applications FEBRUARY 2019, Vol. 11 / 011001-1 Copyright V C 2019 by ASME