  We investigate the macroscopic behavior of asymmetric attractive zero-range
processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate
of order $N^\beta$, where $\beta \in \mathbb{R}$ and $N\in\mathbb{N}$ is the
scaling parameter. We prove that the hydrodynamic limit of this particle system
is described by the unique entropy solution of a hyperbolic conservation law,
supplemented by a boundary condition depending on the range of $\beta$. Namely,
if $\beta \geqslant 0$, then the boundary condition prescribes the particle
current through the origin, whereas if $\beta<0$, the destruction of particles
at the origin has no macroscopic effect on the system and no boundary condition
is imposed at the hydrodynamic limit.
