  We model a microchannel cooling system and consider the optimization of its
shape by means of shape calculus. A three-dimensional model covering all
relevant physical effects and three reduced models are introduced. The latter
are derived via a homogenization of the geometry in 3D and a transformation of
the three-dimensional models to two dimensions. A shape optimization problem
based on the tracking of heat absorption by the cooler and the uniform
distribution of the flow through the microchannels is formulated and adapted to
all models. We present the corresponding shape derivatives and adjoint systems,
which we derived with a material derivative free adjoint approach. To
demonstrate the feasibility of the reduced models, the optimization problems
are solved numerically with a gradient descent method. A comparison of the
results shows that the reduced models perform similarly to the original one
while using significantly less computational resources.
