The phonon Boltzmann transport equation is used to calculate the thermal conductivity in clamped silicon nanowires and study both boundary and confining effects of phonon scattering. The solution method includes partly diffuse and specular phonon reflections at the boundaries and introduces a spectral extinction coefficient. This parameter is derived from the Umklapp relaxation time accounting for thermal resistive processes. A numerical scheme based on the discrete ordinates method has been developped. Results are presented for several wire lengthes and extrapolated to infinte wires. Fourier's law is retrieved both theoretically and numerically for acoustically thick media when only specular reflections occur.