Jose Lage


In this study, geometrically complex planar heat exchangers, designed in line with the Constructal Law and operating at steady-state, are investigated numerically. The work is divided into two parts, one focusing on diffusion heat transfer in a rectangular plane and another on conjugate diffusion-convection heat transfer in a circular plane heat exchanger.

In the first part, a heat generating rectangular solid volume made of a low conductivity material is cooled through a small, isothermal side-section of the domain. The diffusion cooling process is improved by distributing within the heat generating material a fixed amount of a high conductivity material. The question of how to best distribute the high conductivity material to cool the domain and at the same time optimize the decrease in its maximum temperature is answered via geometric optimization based on Constructal Law. The result is a T-shaped network type distribution for the high conductivity material embedded in the heat generating volume. However, this embedded approach, called here the “in-plane” distribution, is of very limited practical use for being too intrusive to the domain. An alternative proposed and investigated here is the “out-of-plane” distribution, in which the high conductivity material network is placed on top of the heat generating plane. Three different network distributions with increased complexity and same specifications (i.e., same uniform heat generation rate, planar aspect ratio and thickness of generating volume, and same volumes of base material and high conductivity material) are investigated numerically for both, in-plane and out-of-plane configurations. The main objective is to compare the heat transfer effectiveness achieved by each configuration. This aspect is very important because, if the effectiveness are comparable, the option of using the out-of-plane distribution could alleviate the practical limitations of the in-plane (embedded) configuration. An additional effort in this first part of the project is to extend the analysis to a non-dimensional parametric study, where the number of different networks is increased to six and different amounts of low and high thermal conductivity materials are investigated. This part if not only to establish if in-plane and out-of-plane yield similar results, but also to understand how the high/low thermal conductivity volume ratio influences the removal of heat out of the heat generating volume.

The results, obtained numerically, show the two networks, i.e., in-plane and out-of-plane, yield nearly identical temperature distributions and heat transfer effectiveness (to within the numerical uncertainty achieved by the simulations) for all configurations tested. Hence, one can confidently use the out-of-plane networks instead of the in-plane networks in practical applications of T-shaped network cooling cold plates. The results also confirm the robustness of the thermal design and extend previous work, showing an increase in network complexity (e.g., by increasing the number of assemblies in each network) indeed yields better cooling performance, even when some of the stringent assumptions imposed in the analysis of building the networks are not fully satisfied.

In the second part of the work, a heat generating material in a circular planar domain (a disk) is now cooled via convection through circular channels from the center to the periphery of the domain. The optimum distribution of the fixed volume convection channels in the domain, again following Constructal Law, yields a tree-shaped network starting with one inlet at the center of the disk, and flowing through bifurcating channels toward a set number of outlets distributed uniformly at the periphery of the disk. Here the thermal-hydraulic performance of the resulting tree-shaped flow networks is obtained numerically and compared to the simpler, radial flow networks (feeding the same number of peripheric outlets) for cooling the disc. The flow entering at the center of the disc is assumed isothermal and fully developed. Two tree-shaped flow networks are considered, having either six (one bifurcation level) or twelve (two bifurcation levels) outlets, conceived for maximizing the cooling and minimizing the flow resistance. Both, tree-shaped and radial flow configurations are set with the same disc solid and fluid (channels) volumes, and same uniformly distributed volumetric heat generation rate in the solid region.

The results show the best performance flow distribution depends on factors that go beyond the channels being either radial or bifurcating, highlighting the very complex heat transfer interaction in this conjugated convection-diffusion system. Moreover, the flow separation effect intrinsic to the channel bifurcations and neglected in previous studies, is essential to the thermal design as it affect not only the overall pressure loss but also the overall heat transfer performance, particularly when the flow channel Reynolds number is high, as one would expect.

Degree Date

Summer 8-4-2021

Document Type


Degree Name



Lyle School of Engineering


Jose Lage

Second Advisor

Prof. David Willis

Third Advisor

Prof. Ali Beskok

Fourth Advisor

Prof. Volkan Otugen

Fifth Advisor

Dr. Antohe Bogdan

Subject Area

Engineering, General/Other

Number of Pages




Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License