  Internally heated convection involves the transfer of heat by fluid motion
between a distribution of sources and sinks. Focusing on the balanced case
where the total heat added by the sources matches the heat taken away by the
sinks, we obtain \emph{a priori} bounds on the minimum mean thermal dissipation
$\langle |\nabla T|^2\rangle$ as a measure of the inefficiency of transport. In
the advective limit, our bounds scale with the inverse mean kinetic energy of
the flow. The constant in this scaling law depends on the source--sink
distribution, as we explain both in a pair of examples involving oscillatory or
concentrated heating and cooling, and via a general asymptotic variational
principle for optimizing transport. Key to our analysis is the solution of a
pure advection equation, which we do to find examples of extreme heat transfer
by cellular and `pinching' flows. When the flow obeys a momentum equation, our
bound is re-expressed in terms of a flux-based Rayleigh number $Ra$ yielding
$\langle |\nabla T|^2\rangle\geq CRa^{-\alpha}$. The power $\alpha$ is $0, 2/3$
or $1$ depending on the arrangement of the sources and sinks relative to
gravity.
