Describing blow-ups locally

Question

Let $X$ be a scheme and $U$ an open subscheme of $X$. Let $C \subseteq X$ be a closed subscheme of $X$ that is properly contained in $U$. I would like to know if we have $$\text{Bl}_{C}(X) \cong \text{Bl}_{C}(U) \cup (X \setminus U)$$ or even equality? The notation $\text{Bl}_A(B)$ means the blow-up of a scheme $A$ along its closed subscheme $B \subseteq A$. We may assume all of our schemes are affine + of finite type over some field $K$ if that is needed. Any answer/reference is appreciated.