We consider a fluid flowing within a microchannel embedded in a long bar or strip. If the thermal conductivity of this bar is varied spatially in exactly the right way, the moving fluid can be subjected to arbitrary heating and/or cooling, simply by holding two opposing edges of the bar at different temperatures. With this concept in mind, the objective of this study was to develop a rigorous procedure to identify the pattern of thermal conductivity needed to achieve any particular heating/cooling profile. It is relatively straightforward to fabricate such a bar using laminated composite materials, whose precisely graded composition is comprised of comb-like metal structures embedded within polymeric layers. Within certain practical limits, one can thereby obtain relatively steep (>100C/mm) heating and cooling gradients, which makes it possible to cycle over a relatively wide range of temperatures (>100C) over a relatively short length scale (<10mm). When the desired temperature profile is known, the task at hand is the determination of the required model geometry. This inverse problem can be solved by embedding an inner loop of thermofluidic analysis within an outer loop of numerical optimization. Given a reasonable starting guess, one can move iteratively towards a comb geometry that can better match the desired temperature profile. Of practical value, the starting guess can have far fewer degrees-of-freedom than the optimal solution. In this particular study, the inner loop was performed using finite element methods (i.e., COMSOL Multiphysics 4.2), and the outer loop was implemented using SQP constrained optimization (i.e., MATLAB R2011).

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