Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient

Abstract

SUMMARYThis paper is motivated by an optimal boundary control problem for the cooling process of molten and already formed glass down to room temperature. The high temperatures at which glass is processed demand to include radiative heat transfer in the computational model. Since the complete radiative heat transfer equations are too complex for optimization purposes, we use simplified approximations of spherical harmonics coupled with a practically relevant frequency bands model. The optimal control problem is considered as a partial differential algebraic equation (PDAE)-constrained optimization problem with box constraints on the control. In this paper, we augment the objective by a functional depending on the state gradient, which forces a minimization of thermal stress inside the glass. To guarantee consistent and grid-independent values of the reduced objective gradient at the end of the cooling process, we pursue two approaches. The first includes the temperature gradient with a time-dependent linearly decreasing weight. In the second approach, we augment the objective functional by the final state tracking and final state gradient optimization. To determine an optimal boundary control, we apply a projected gradient method with the Armijo step size rule. The reduced objective gradient is computed by the continuous adjoint approach. The arising time-dependent PDAEs are numerically solved by variable step size one-step methods of Rosenbrock type in time and adaptive multilevel finite elements in space. We present two-dimensional numerical results for an infinitely long glass block and compare the two different approaches derived to ensure consistency at the end of the cooling process.

Apparatus for measuring the emittance of materials from far infrared to visible wavelengths in extreme conditions of temperature

“…In the manufacturing sector such as the glass industry, the need to master thermal heat transfer in the plants is mandatory to achieve efficient energy use and optimization of the costs [1,2]. The quality of products necessitating high temperature treatments is also often related to the accuracy of the control of temperature.…”

Combination of an adaptive multilevel SQP method and a space-time adaptive PDAE solver for optimal control problems

“…To evaluate the reduced Hessian applied to a given direction s u (13), first, the linearized state system (14) has to be solved to achieve the linearized state s y . Next, the second adjoint system (15) has to be solve with respect to the current state y, the adjoint state ξ, the linearized state s y and the control update s u .…”

“…Therefore, we compare the multilevel SQP method to a multilevel gradient method with Armijo line search [13,14]. Even though one optimization iteration of the gradient method is significantly faster than one of the SQP method, there are also significantly more iterations necessary to reduce the scaled gradient under the predefined tolerance which all in all leads to a multiplication of computing time by a factor 5.…”

Optimal control of complex heat transfer

“…A considerable number of works is devoted to problems of control of evolutionary systems describing the radiative heat transfer (see, e.g. [8], [9], [10], [11]). In the mentioned works, the transfer of radiation is described by the integraldifferential equation of radiative transfer or its approximations.…”