This paper describes the conceptual design and performance of balanced two-stream counterflow heat exchangers, in which each stream flows as a tree network through its allotted space. The two trees in counterflow are like two palms pressed against each other. The paper develops the relationships between effectiveness and number of heat transfer units for several tree-counterflow configurations: (i) constructal dichotomous trees covering uniformly a rectangular area, (ii) trees on a disk-shaped area, and (iii) trees on a square-shaped area. In configurations (ii) and (iii) each stream flows between the center and the periphery of the area. Configurations (i) and (ii) are trees with minimal resistance to fluid flow. Configuration (iii) is designed by minimizing the length of each duct in the network. The paper reports the formula for the number of heat transfer units in each configuration. Unlike in counterflows formed by two parallel streams, in which the longitudinal temperature gradient is constant, in the counterflow formed by two trees the longitudinal temperature gradient is steeper as one approaches the periphery of the tree canopy. The application of dendritic heat exchangers to devices with maximal transport density is discussed, e.g., electronics cooling, fuel cell architectures, etc.

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