  We prove the hydrodynamic limit of a totally asymmetric zero range process on
a torus with two lanes and randomly oriented edges. The asymmetry implies that
the model is non-reversible. The random orientation of the edges is constructed
in a bistochastic fashion which keeps the usual product distribution stationary
for the quenched zero range model. It is also arranged to have no overall drift
along the Z direction, which suggests diffusive scaling despite the asymmetry
present in the dynamics. Indeed, using the relative entropy method, we prove
the quenched hydrodynamic limit to be the heat equation with a diffusion
coefficient depending on ergodic properties of the orientation of the edges.
  The zero range process on this graph turns out to be non-gradient. Our main
novelty is the introduction of a local equilibrium measure which decomposes the
vertices of the graph into components of constant density. A clever choice of
these components eliminates the non-gradient problems that normally arise
during the hydrodynamic limit procedure.
