Control of the Energy Flux in a Model of Complex Heat Transfer

Andrey Kovtanyuk*, Alexander Chebotarev, Nikolai Botkin and Karl-Heinz Hoffmann

The steady-state diffusion model (P$_1$ approximation) describing radiative, conductive, and convective heat transfer processes in a bounded domain $G\subset \mathbb{R}^3$ is considered. The governing equations have the following normalized form: \begin{equation}\tag{$1$} -a \Delta \theta + \textbf{v} \cdot \nabla \theta + b \kappa_a \theta^4= b \kappa_a \varphi,\;\; - \alpha \Delta \varphi + \kappa_a \varphi = \kappa_a \theta^4. \end{equation} Here $\theta$ is the normalized temperature, $\varphi$ the normalized radiation intensity averaged over all directions, $\textbf{v}$ a prescribed velocity field of the medium, and $\kappa_a$ the absorption coefficient. The constants $a$, $b$, and $\alpha$ are defined as follows: $$a=\frac{k}{\rho c_v},\quad b = \frac{4\sigma n^2 T_{max}^3}{\rho c_v}, \quad \alpha=\frac{1}{3\kappa - A \kappa_s},$$ where $k$ is the thermal conductivity, $c_v$ the specific heat capacity, $\rho$ the density, $\sigma$ the Stefan-Boltzmann constant, $n$ the refraction index, $T_{max}$ the maximal temperature in the non-scaled model, $\kappa := \kappa_s + \kappa_a$ the extinction coefficient (the total attenuation factor), $\kappa_s$ the scattering coefficient. The coefficient $A \in [-1,1]$ describes the anisotropy of scattering, the case $A=0$ corresponds to the isotropic scattering. It is assumed that the boundary $\Gamma: =\partial G$ consists of three open parts $\Gamma_{1,2,3}$ such that $\Gamma_i\cap\Gamma_j =\emptyset, \;\; i\not= j$, and $\Gamma=\overline{\Gamma}_1\cup \overline{\Gamma}_2\cup\overline{\Gamma}_3$. The parts $\Gamma_2$ and $\Gamma_3$ are considered as inflow and outflow regions, respectively, so that $\textbf{v}\cdot \textbf{n}|_{\Gamma_2}\leq 0$ and $\textbf{v}\cdot \textbf{n}|_{\Gamma_3}\geq 0$, where $\textbf{n}$ denotes the outer boundary normal. The part $\Gamma_1$ is assumed to be flow impermeable. Moreover, the functions $\theta$ and $\varphi$ satisfy the following boundary conditions: \begin{equation}\tag{$2$} \theta|_{\Gamma_1 \cup \Gamma_2}=\Theta_0, \;\; \frac{\partial \theta}{\partial \textbf{n}}|_{\Gamma_3}=0,\;\; \alpha \frac{\partial \varphi}{\partial \textbf{n}} + u(\varphi -\Theta_{0}^4)|_{\Gamma_1}=0,\;\;\alpha \frac{\partial \varphi}{\partial \textbf{n}} + \frac{1}{2}\varphi |_{\Gamma_2\cup \Gamma_3}=0. \end{equation} Here, the notation $\partial/\partial \textbf{n}$ is used for the normal derivative; $\Theta_{0}$ is a prescribed nonnegative temperature distribution on $\Gamma_1 \cup \Gamma_2$; and the function $u$, describing the reflective properties of the part $\Gamma_1$, is considered as a control input. To introduce an objective functional for the control system $(1)$ and $(2)$, notice that the vector field $\textbf{q}=-a\nabla \theta+\theta \textbf{v}-\alpha b\nabla \varphi$ describes the density of the energy flux, and the quantity \begin{equation}\tag{$3$} J=\int_{\Gamma_3}(\textbf{q}\cdot\textbf{n})\,d\Gamma = \int_{\Gamma_3}\left((\textbf{v}\cdot \textbf{n})\theta+\frac{1}{2}b\varphi\right)d\Gamma \end{equation} is proportional to the energy escaping through $\Gamma_3$ per time unit. Thus, the optimization of the heat removal assumes finding a control input $u(\cdot)$ that yields solutions $\theta$ and $\varphi$ of the problem $(1)$ and $(2)$ under the restrictions \begin{equation}\tag{$4$} u_1(\xi)\leq u(\xi)\leq u_2(\xi)\mbox{ on }\Gamma_1,\;\; 0\leq \theta(x)\leq M,\;\; 0\leq \varphi(x)\leq M^4\mbox{ in }G, \end{equation} and maximizes the objective functional $J$. Here, $u_1(\xi)$ and $u_2(\xi)$ are some prescribed functions, and $M=\max \Theta_{0}$. The paper presents new apriori estimates on solution of the control system $(1)$ and $(2)$ that ensure the solvability of the control problem and provide optimality conditions of the first order. Moreover, conditions on the model parameters, the geometry of the domain $G$, and the velocity field $\textbf{v}$ are formulated to provide the unique solvability of the adjoint system, which ensures a regularity of the control problem $(1)$-$(4)$. This theoretical analysis provides backgrounds for the development of numerical procedures for finding optimal control inputs in problems of complex heat transfer.

Mathematics Subject Classification: 35Q93

Keywords: Complex heat transfer; Optimal control

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