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

Enskog kinetic theory for $d$-dimensional dense granular gases