  The goal of this note is to provide most of the technical details involved in
the application of the Chapman-Enskog method to solve the revised Enskog
equation to Navier-Stokes order. Explicit expressions for the transport
coefficients and the cooling rate are obtained in terms of the coefficient of
restitution and the solid volume fraction by using a new Sonine approach. This
new approach consists of replacing, where appropriate in the Chapman-Enskog
procedure, the local equilibrium distribution (used in the standard first
Sonine approximation) by the homogeneous cooling state distribution. The
calculations are performed in an arbitrary number of dimensions.
