  We consider the symmetric simple exclusion process in the interval $[-N,N]$
with additional birth and death processes respectively on $(N-K,N]$, $K>0$, and
$[-N,-N+K)$. The exclusion is speeded up by a factor $N^2$, births and deaths
by a factor $N$. Assuming propagation of chaos (a property proved in a
companion paper "Truncated correlations in the stirring process with births and
deaths") we prove convergence in the limit $N\to \infty$ to the linear heat
equation with Dirichlet condition on the boundaries; the boundary conditions
however are not known a priori, they are obtained by solving a non linear
equation. The model simulates mass transport with current reservoirs at the
boundaries and the Fourier law is proved to hold.
