Let $X$ be a scheme, let $i : Z \to X$ be a closed subscheme, let $Y := \mathrm{Bl}_{Z}(X)$ be the blowup of $X$ at $Z$ with projection $\pi : Y \to X$. Suppose $U \supseteq X \setminus Z$ is an open subscheme of $X$ for which $\pi^{-1}(U) \to U$ is an isomorphism. Does there exist a closed subscheme $Z' \to X$ with support contained in $X \setminus U$ and for which the blowup $Y' := \mathrm{Bl}_{Z'}(X)$ is $X$-isomorphic to $Y$?

Motivation: If we blow up a subscheme of a normal scheme $X$, the maximal open subscheme $U$ for which $\pi^{-1}(U) \to U$ is an isomorphism contains all the codimension 1 points of $X$, so I was naively wondering whether all blow ups of a normal scheme are obtained by blowing up a closed subscheme of codimension at least 2.

Thoughts: If $A$ is a ring and $I$ is an ideal of $A$ and $a \in A$ is a nonzerodivisor, then the Rees algebras $\bigoplus_{n \ge 0} I^{n}$ and $\bigoplus_{n \ge 0} (aI)^{n}$ are isomorphic so if the center of the blowup is of the form $\operatorname{Spec} A/(aI)$, I can replace it with $\operatorname{Spec} A/I$, but I don't know how to recognize an ideal as being of the form $aI$.