  We consider the accurate investigation of the energy current and its
components, heat and work, in some boundary driven quantum spin systems. The
expressions for the currents, as well as the associated Lindblad master
equation, are obtained via a repeated interaction scheme. We consider small
systems in order to analytically compute the steady distribution to study the
current in the steady state. Asymmetrical XXZ and quantum Ising models are
detailed analyzed. For the XXZ chain we present cases in which different
compositions of heat and work currents, obtained via the repeated interaction
protocol, lead to the same energy current, which may be obtained via the
Lindblad master equation. For the quantum Ising chain, we describe a case of
zero energy current and novanishing heat and work currents. Our findings make
clear that to talk about heat in these boundary driven spin quantum systems we
must go beyond an investigation involving only the Lindblad master equation.
