How to insulate a pipe?

Fabien Caubet, Carlos Conca, Marc Dambrine, Rodrigo Zelada.
How to insulate a pipe?
HAL hal-04772321

Problem statement.

We are interested in the optimal insulation of a pipe where hot water runs.
As the pipe may have a complex geometry, the flow of the hot fluid is described by the stationary incompressible Navier-Stokes equations. \[\displaystyle{\left \{ \begin{array}{rll} -\nu \Delta \boldsymbol{u}^\epsilon + (\nabla \boldsymbol{u}^\epsilon )\boldsymbol{u}^\epsilon + {\dfrac{1}{\rho}}\nabla p^\epsilon & = 0 & \text{ in } \Omega_{1}^\epsilon, \\ \text{div}(\boldsymbol{u}^\epsilon) & = 0 & \text{ in } \Omega_{1}^\epsilon, \\ \boldsymbol{u}^\epsilon & = \boldsymbol{u}_{\mathrm{D}} & \text{ on } \Gamma_{\mathrm{D}} ^\epsilon, \\ \sigma(\boldsymbol{u}^\epsilon,p^\epsilon)\boldsymbol{n} & = 0 & \text{ on } \Gamma_{\mathrm{N}} ^\epsilon, \\ \boldsymbol{u}^\epsilon & = 0 & \text{ on } \Gamma_1 ^\epsilon, \end{array} \right.}\]

The temperature solves: \[\left \{ \begin{array}{rcll} -\text{div}(\kappa_1 \nabla \mathsf{T}_{1} ^\epsilon) + \boldsymbol{u}^\epsilon \cdot \nabla \mathsf{T}_{1} ^\epsilon & = & 0 & \text{ in } \Omega_{1}^\epsilon, \\[4pt] -\text{div}(\kappa_{\mathrm{m}} \nabla \mathsf{T}_{\mathrm{m}} ^\epsilon) & = & 0 & \text{ in } \Omega_{\mathrm{m}} ^\epsilon, \\[4pt] -\text{div}(\kappa_2 \nabla \mathsf{T}_2 ^\epsilon) & = & 0 & \text{ in } \Omega_2 ^\epsilon, \\[4pt] \mathsf{T}_1 ^\epsilon & = & \mathsf{T}_{\mathrm{D}} & \text{ on } \Gamma_{\mathrm{D}} ^\epsilon, \\[4pt] \displaystyle{\kappa_1 \frac{\partial \mathsf{T}_1 ^\epsilon}{\partial \boldsymbol{n}} } & = & 0 & { \text{ on } \Gamma_{\mathrm{N}}^\epsilon,} \\[6pt] %\Gamma_N, \\[4pt] { \displaystyle{\kappa_{\rm m} \frac{\partial \mathsf{T}_{\rm m} ^\epsilon}{\partial \boldsymbol{n}} }} & = & 0 & { \text{ on } \Gamma_{\mathrm{m, N}} ^\epsilon,} \\[4pt] %\Gamma_N, \\[4pt] % \color{magenta}\displaystyle{\kappa_2 \frac{\partial \mathsf{T}_2^\epsilon}{\partial \boldsymbol{n}} } & \color{magenta} = & \color{magenta}0 & \color{magenta} \text{ on } \Gamma_{\mathrm{e, N}} ^\epsilon, \\[4pt] %\Gamma_N, \\[4pt] \displaystyle{\kappa_2 \frac{\partial \mathsf{T}_2^\epsilon}{\partial \boldsymbol{n}} + \alpha \mathsf{T}_2^\epsilon} & = & \alpha \mathsf{T}_{\rm ext} & \text{ on } \Gamma_{\mathrm{R}} ^\epsilon, \\[4pt] %\left [\mathsf{T}^\epsilon \right] & = 0 & \text{ on } \Gamma_1 ^\epsilon \cup \Gamma_2 ^\epsilon , \\[4pt] \mathsf{T}_1 ^\epsilon & = & \mathsf{T}_{\mathrm{m}} ^\epsilon & \text{ on } \Gamma_1 ^\epsilon , \\[4pt] \mathsf{T}_2 ^\epsilon & = & \mathsf{T}_{\mathrm{m}} ^\epsilon & \text{ on } \Gamma_2 ^\epsilon , \\[4pt] %\displaystyle{\left [ \kappa \frac{\partial \mathsf{T}^\epsilon}{\partial \boldsymbol{n}} \right]} & = 0 & \text{ on } \Gamma_1 ^\epsilon \cup \Gamma_2 ^\epsilon , \displaystyle{\kappa_1 \frac{\partial \mathsf{T}_1^\epsilon}{\partial \boldsymbol{n}} } & = & \displaystyle{\kappa_{\mathrm{m}} \frac{\partial \mathsf{T}_{\mathrm{m}} ^\epsilon}{\partial \boldsymbol{n}} } & \text{ on } \Gamma_1 ^\epsilon , \\[6pt] \displaystyle{\kappa_2 \frac{\partial \mathsf{T}_2^\epsilon}{\partial \boldsymbol{n}} } & = & \displaystyle{\kappa_{\mathrm{m}} \frac{\partial \mathsf{T}_{\mathrm{m}} ^\epsilon}{\partial \boldsymbol{n}}} & \text{ on } \Gamma_2 ^\epsilon , \end{array} \right.\]

The shape optimization problem

We consider the following shape optimization problem:
given a prescribed volume~\(V_0 >0\) of insulating material, minimize the criterion \(J\) by \[ J(\Omega_2) = \int_{\Gamma_{\mathrm{R}}} \left(\kappa_2 \frac{\partial \mathsf{T}_2}{\partial \boldsymbol{n}} \right)^2 \, \mathrm{d}s =\int_{\Gamma_{\mathrm{R}} } \alpha^2 (\mathsf{T}_2 - {\mathsf{T}_{\rm ext}})^2 \, \mathrm{d}s , \] where the temperature \(\mathsf{T}\) solves the approximate convection-diffusion problem and the fluid speed solves the Navier-Stokes system.

Contributions.

Use of an asymptotic model for the temperature.

The pipe wall is assummed thick. In order to avoid a mesh at the size of this small parameter, the pipe's wall are modelled thanks to a non standard interface condition involving second order tangential derivatives: the original problem set in three materials is replaced by the two domains problem \[ \left\{\begin{array}{rcll} -\text{div}(\kappa_1 \nabla \mathsf{T}_1) + \boldsymbol{u} \cdot \nabla \mathsf{T}_1 & = & 0 & \text{ in } \Omega_{1}, \\[4pt] -\text{div}(\kappa_2 \nabla \mathsf{T}_2 ) & = & 0 & \text{ in } \Omega_2, \\[4pt] \mathsf{T}_1 & = & \mathsf{T}_{\mathrm{D}} & \text{ on } \Gamma_{\mathrm{D}}, \\[4pt] \displaystyle{\kappa_1 \frac{\partial \mathsf{T}_1}{\partial \boldsymbol{n}} } & = & 0 & \text{ on } \Gamma_{\mathrm{N}}, \\[4pt] % \displaystyle{\kappa_2 \frac{\partial \mathsf{T}_2}{\partial \boldsymbol{n}} } & = & \color{magenta}0 & \color{magenta}\text{ on } \Gamma_{\mathrm{e,N}}, \\[4pt] \displaystyle{\kappa_2 \frac{\partial \mathsf{T}_2}{\partial \boldsymbol{n}} + \alpha \mathsf{T}_2} & = & \alpha {\mathsf{T}_{\rm ext}} & \text{ on } \Gamma_{\mathrm{R}}, \\[6pt] \displaystyle{\left < \kappa \frac{\partial \mathsf{T}}{\partial \boldsymbol{n}}\right >} & = & -\kappa_{\mathrm{m}} \epsilon^{-1} \left [\mathsf{T} \right] & \text{ on } \Gamma, \\[8pt] \displaystyle{\left [ \kappa \frac{\partial \mathsf{T}}{\partial \boldsymbol{n}} \right]} & = & \epsilon \text{div}_\tau(\kappa_{\mathrm{m}} \nabla_\tau \langle \mathsf{T} \rangle) - \kappa_{\mathrm{m}} H [\mathsf{T}] & \text{ on } \Gamma,\\[8pt] \displaystyle{\kappa_i \frac{\partial \mathsf{T}_i}{\partial \boldsymbol{n}}} & = & 0 & \text{ on } \partial \Gamma, i=1,2 , \end{array} \right. \] where \( [u] \) is the jump of \(u\) and \(< u>\) its mean across the interface.

The computational domain for the temperature.

Derivation of shape derivative.

See the paper for the expressions.