The current trend towards miniaturization of electro-mechanical systems (MEMS) has boosted the development of new applications in the field of mini- and micro-heat exchangers. The accompanying growth of the wall-to-channel dimensional ratios is known to increase the role of axial wall conduction, which reduces the effectiveness of counterflow heat exchangers to that of coflow systems in the limit of large axial wall conduction. Understanding the impact of wall conduction on the performance of mini- and micro-heat exchangers is thus critical for the design of high efficient equipment. In this work we investigate the role of axial and transverse wall conduction by consideration of a simple problem, the laminar counterflow parallel-plate heat exchanger, which has been recently used by the authors as a cartoon to represent the heat exchange process in real-life counterflow heat exchangers. Using as starting point the exact series solution of Quintero & Vera [2017], we carry out a thorough parametric study discussing the effect of the two main dimensionless parameters determining the effect of multidimensional wall conduction on heat exchanger performance; namely, the dimensionless wall thermal resistance, k_w~(-1), and the dimensionless wall thickness, Δ_w. Isocontours of heat exchanger effectiveness in the (Δ_w, k_w~(-1)) plane exhibit well-defined asymptotic regimes for limiting values of the parameters corresponding to cases with/withouth significant axial/transverse wall conduction, with smooth transitions between the different regimes ocurrying for values of Δ_w and k_w~(-1) of order unity. The analysis provides accurate conditions for neglecting axial and transverse wall conduction effects, and shows that the optimum wall conductivity, previously discussed in the literature, ceases to exist for moderately thick walls.
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