In this talk we consider two interacting particle systems in the finite box with N sites, namely, independent random walks and symmetric exclusion processes. Both systems are in contact with a finite reservoir, where the exit rate is proportional to the inverse of $N^\theta$, where $theta$ is nonnegative. We prove the hydrodynamic limit for these models, which are given by the heat equation with Wentzell boundary conditions at the boundary at the critical parameter $\theta=1$, and exhibit a dynamic phase transition. Moreover, the Wentzell boundary condition is non-linear in the exclusion setting at the critical parameter. Joint work with Patrícia Gonçalves and Matheus Franco.