In the paper of Mukai & Sakai https://link.springer.com/article/10.1007/BF01171494, give a smooth projective curve $C$ and a vector bundle $E$ of rank $r$, they use the Grassmannian bundle $\mathbb{G}(E)$ to get a upper bound of dimension of the Quot scheme $Q_0$, where $Q_0$ parametrizes the locally free quotient of $E$ of rank $\rho$.
I am interesting to the case of a smooth projective surface, i.e., give a smooth projective surface $S$ and a vector bundle of rank $r$, what is the dimension of the Quot scheme $Q_0$, where $Q_0$ parametrizes the torsion free quotient of $E$ of rank $\rho$. But there is an obstruction that we can't get a morphism between $S\times Q_0$ and $\mathbb{G}(E)$. So I think whether there is another method to estimate dim $Q_0$?
More generally, is there a method to estimate the dimension of Quot scheme Quot$_{E/S/k}^{\Phi,H}$ with a fix Hilbert polynomial $\Phi$ and ample line bundle $H$?