Optimal boundary control for the heat equation with application to freezing with phase change

Abstract

Abstract-In this paper an approach for optimal boundary control of a parabolic partial differential equation (PDE) is presented. The parabolic PDE is the heat equation for thermal conduction. A technical application for this is the freezing of fish in a vertical plate freezer. As it is a dominant phenomenon in the process of freezing, the latent heat of fusion is included in the model. The aim of the optimization is to freeze the interior of a fish block below −18 o C in a predefined time horizon with an energy consumption that is as low as possible assuming that this corresponds to high freezing temperatures.

The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

“…First, Section 2 provides a brief overview of previous results. Section 3 describes the problem setting of Backi and Gravdahl (2013). Section 4 provides the main stability results, whereupon Section 5 shows a few numerical examples that highlight the results in the previous section.…”

“…An example of a PDE model describing freezing of a specific material (fish species), taking the phenomenon of thermal arrest caused by latent heat of fusion into account, was introduced in Backi and Gravdahl (2013). The parameters have to be state-(i.e.…”

“…This problem was initially introduced in Backi and Gravdahl (2013) for an application that describes the freezing of fish in a vertical plate freezer. Inspired by this problem, the present paper attempts to investigate the stability properties of the Burgers' equation with specific functional forms of ε and its derivatives.…”

“…Strictly speaking the Dirichlet boundary condition is equal to the temperature of the cooling medium, whereas the Neumann boundary condition represents heat flux through the boundary which is proportional to the difference between the temperature at the boundary and the temperature of the cooling medium. The main contribution of the paper is to generalize Backi and Gravdahl (2013) and show that under certain assumptions, the version of Burgers' equation we consider converges to a stationary solution determined by (constant) boundary conditions. The rest of the paper is organized as follows.…”

Optimal boundary control of a contact thawing process for foodstuff

“…As computational power has grown in recent years, complex simulations can provide qualitative and quantitative insights to the system, even in the absence of analytical solutions. An open-loop approach for optimal boundary control of a plate freezing process has been described in Backi and Gravdahl (2013). However, it must be mentioned that thawing is not simply the reversed freezing process, as for example outlined in Olver (2014, Section 4.1).…”