We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive the generalized diffusion and heat systems on the evolving surface. Moreover, we investigate the boundary conditions for the two systems to study their conservation and energy laws. As an application, we make a mathematical model for a diffusion process on an evolving double bubble. Specifically, this paper is devoted to deriving the representation formula for the unit outer co-normal vector to the boundary of a surface.