  Using vanishing of graded components of local cohomology modules of the Rees
algebra of the normal filtration of an ideal, we give bounds on the normal
reduction number. This helps to get necessary and sufficient conditions in
Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the
normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the
normal reduction number.
