Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$ (i.e. a closed, reduced subscheme). Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with center $Z$.
My question is: Under which conditions on $X$ and $Z$ does $E$ have the same number of irreducible components as $Z$?
For example, if $X$ is smooth and the irreducible components of $Z$ are smooth and disconnected, then the statement holds. There are examples of surfaces $X$ which are singlar at a single point $Z$ and the exceptional divisor is not irreducible, so I know that it is not always true.
I suppose that I will have to assume $X$ smooth, but I am wondering if the assumption on $Z$ can be weakened. I would be most happy if it was always true for smooth $X$ because $Z$ is a reduced subscheme, but I do not really know whether that is to be expected.