Model hierarchy for the shape optimization of a microchannel cooling system

Abstract

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.

Optimal control of the Sabatier process in microchannel reactors

“…4 and 5. To remedy this, we now use ideas similar to [19] to derive a reduced one-dimensional model of the reactor which is significantly easier to solve numerically.…”

“…While the boundary Γ wall is still present from this point of view due to the geometrical constraints, it now only acts as a mathematical boundary of the domain and does not interact physically with the fluid. To model the fluid velocity in such a porous medium, we use a Brinkman equation with slip boundary conditions, in analogy to [19] and the references therein, given by the system…”

“…Additionally, we have to modify the temperature equation as we can no longer incorporate the effect of the wall temperature on the gas mixture via a Dirichlet boundary condition due to the previously mentioned reasons. Hence, we proceed analogously to [19] and use the following equation: where h f,s is the so-called interfacial heat transfer coefficient which models the heat transfer between the gas mixture and the wall. We computed the value of h f,s numerically, as described in [19] and the references therein, and the result is shown in Table 1.…”

“…The purpose of this paper is to investigate the optimization of such a microchannel reactor for the Sabatier reaction using techniques from PDE-constrained optimization. Such methods are widely used in various physical applications, e.g., for the optimization of chemical reactions in conventional sized reactors [12][13][14], the optimization of semiconductors [15,16], glass cooling processes [17,18], or for the optimal shape design of microchannel cooling systems [19,20], aircrafts [21,22], and polymer spin packs [23,24]. The optimization of microchannel reactors has also been investigated previously using derivative-free approaches only, e.g., in [25][26][27][28].…”

“…First, we present a three-dimensional model that contains all important physical and chemical effects of the reactor. Second, we derive a one-dimensional model from the first one using a homogenization procedure similar to [19]. As both models are given by strongly coupled and highly nonlinear systems of PDEs, we use our software package cashocs [30] for the numerical solution of the subsequent optimization problems.…”