  We present the development of a two-stage PrNi$_5$ continuous demagnetization
refrigerator at the Institut N\'{e}el/CNRS and numerical simulations of its
performance. The thermal model used in the simulations is discussed in detail
including the likely sources of heating. We demonstrate the effects of the
critical thermal links including superconducting heat switches as well as the
heat conductivity of the PrNi$_5$, accounting for the dependence of cooling
power on the PrNi$_5$ rod diameter. Our simulations show that if extreme care
is taken to minimize the thermal resistance between the nuclear stages, a
sample temperature of 1~mK can be maintained under a 20~nW heat load.
