  We study an analogue of the Calabi flow in the non-K\"ahler setting for
compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a
priori estimates for the evolving metric along the flow given a uniform bound
on the Chern scalar curvature. If the Chern scalar curvature remains uniformly
bounded for all time, we show that the flow converges smoothly to the unique
Chern-Ricci-flat metric in the $\partial\bar{\partial}$-class of the initial
metric.
