Summary: We consider the flow of closed convex hypersurfaces in Euclidean space \(\mathbb{R}^{n+1}\) with speed given by a power of the \(k\)-th mean curvature \(E_k\) plus a global term chosen to impose a constraint involving the enclosed volume \(V_{n+1}\) and the mixed volume \(V_{n+1-k}\) of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.