Heat conduction in a solid slab embedded with a pipe of general cross-section: Shape Factor and Shape Optimization

We address the problem of two-dimensional heat conduction in a solid slab embedded with an isothermal, symmetric pipe of general cross-section. Similar formulations have applications in continuum mechanics and electricity. The main objective of this work is to develop a Shape Optimization algorithm that will reveal the optimal shapes of the pipe such that the conduction rate is maximized or minimized. This is achieved by optimizing the Shape Factor. To obtain the Shape Factor we transform the pipe into a strip using the generalized Schwarz-Christoffel transformation, and develop an integral equation of the first kind for the temperature gradient using Fourier transform techniques. The integral equation is solved both numerically and analytically/asymptotically. The fact that the Shape Factor is a monotonic function of the length of the strip suggests a Shape Optimization formulation where the objective function is the length of the strip and the variables of the optimization are the parameters of the generalized Schwarz–Christoffel transformation. Optimal shapes for the problem of minimizing the conduction rate are computed numerically and validated with an analytical solution. Numerical results for maximizing the transport rate are also obtained. The versatility and the robustness of the numerical optimization algorithm offers opportunities for improving the design of similar processes with non-linear equality and inequality constraints.