  We consider heat transport in a Luttinger liquid (LL) with weak disorder and
study the Lorenz number for this system. We start at a high-$T$ regime, and
calculate both the electrical and thermal conductivities using a memory
function approach. The resulting Lorenz number $L$ is independent of $T$ but
depends explicitly on the LL exponents. Lowering $T$, however, allows for a
renormalization of the LL exponents from their bare values by disorder, causing
a violation of the Wiedemann-Franz law. Finally, we extend the discussion to
quantum wire systems and study the wire size dependence of the Lorenz number.
