  A model based on microcanonical Monte Carlo method is used to study the
application of the temperature gradient along a two-dimensional (2D) Ising
system. We estimate the system size effects on thermal conductivity, $K$, for a
nano-scale Ising layer with variable size. It is shown that $K$ scales with
size as $ K=cL^\alpha$ where $\alpha$ varies with temperature. Both the
Metropolis and Cruetz algorithms have been used to establish the temperature
gradient. Further results show that the average demon energy in the presence of
an external magnetic field is zero for low temperatures.
