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The Chapman-Enskog perturbation method for a phonon gas is investigated with the use of Callaway's model for the Boltzmann-Peierls equation. Assuming that the effective relaxation time for normal processes is small and the effective relaxation time for resistive processes is large, this perturbation method proposes to expand the phase density about a displaced Planck distribution and to include the above two relaxation times in the expansion. The main advantage of using the displaced Planck distribution is that the drift velocity of a phonon gas is incorporated into the model in a non-perturbative manner. The result is a system of nonlinear second-order parabolic equations for the energy density and the drift velocity which, unlike the usual set of hydrodynamic equations, does not restrict the magnitude of the individual components of the drift velocity and the heat flux in any way. This system is linearly stable at all wavelengths and is also fully consistent with the second law of thermodynamics in the sense that there exists a macroscopic entropy density which depends locally on the hydrodynamic variables and satisfies the balance equation with a non-negative entropy production due to both resistive and normal processes.