# A multi-objective optimization problem in mixed and natural convection for a vertical channel asymmetrically heated

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## Abstract

This paper deals with a multi-objective topology optimization problem in an asymmetrically heated channel, based on both pressure drop minimization and heat transfer maximization. The problem is modeled by assuming steady-state laminar natural convection flow. The incompressible Navier-Stokes equations coupled with the convection-diffusion equation, under the Boussinesq approximation, are employed and are solved with the finite volume method. In this paper, we discuss some limits of classical pressure drop cost function for buoyancy-driven flow and, we then propose two new expressions of objective functions: the first one takes into account work of pressure forces and contributes to the loss of mechanical power while the second one is related to thermal power and is linked to the maximization of heat exchanges. We use the adjoint method to compute the gradient of the cost functions. The topology optimization problem is first solved for a Richardson (Ri) number and Reynolds number (Re) set respectively to Ri ∈{100,200,400} and Re = 400. All these configurations are investigated next in order to demonstrate the efficiency of the new expressions of cost functions. We compare two types of interpolation functions for both the design variable field and the effective diffusivity. Both interpolation techniques have pros and cons and give slightly the same results. We notice that we obtain less isolated solid elements with the sigmoid-type interpolation functions. Then, we choose to work with the sigmoid and solve the topology optimization problem in case of pure natural convection, by setting Rayleigh number to {3 × 10^{3},4 × 10^{4},5 × 10^{5}}. In all considered cases, our algorithm succeeds to enhance one of the phenomenon modeled by the proposed cost functions without deteriorating the other one. The optimized design obtained suppresses any reversal flow at the exit of the channel. We also show that the thermal exchanges are improved by computing the Nusselt numbers and bulk temperature. We conclude that the new expressions of objective functions are well suited to deal with natural convection optimization problem in a vertical channel.

## Keywords

Natural convection Vertical channel Topology optimization Objective functions Adjoint sensitivity analysis Sigmoid function## Notes

### Acknowledgments

Computations have been performed on the University of Reunion Island supercomputer facility.

### Compliance with Ethical Standards

### **Conflict of interest**

The authors declare that they have no conflict of interest.

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