The continuous stirred-tank reactor (CSTR), also known as vat- or backmix reactor, mixed flow reactor (MFR), or a continuous-flow stirred-tank reactor (CFSTR), is a common model for a chemical reactor in chemical engineering and environmental engineering. A CSTR often refers to a model used to estimate the key unit operation variables when using a continuous agitated-tank reactor to reach a specified output. The mathematical model works for all fluids: liquids, gases, and slurries.

The behavior of a CSTR is often approximated or modeled by that of an ideal CSTR, which assumes perfect mixing. In a perfectly mixed reactor, reagent is instantaneously and uniformly mixed throughout the reactor upon entry. Consequently, the output composition is identical to composition of the material inside the reactor, which is a function of residence time and reaction rate. The CSTR is the ideal limit of complete mixing in reactor design, which is the complete opposite of a plug flow reactor (PFR). In practice, no reactors behave ideally but instead fall somewhere in between the mixing limits of an ideal CSTR and PFR.

A continuous fluid flow containing non-conservative chemical reactant A enters an ideal CSTR of volume V.

Integral mass balance on number of moles N_A of species A in a reactor of volume V:

1. ${\displaystyle [{\text{Net accumulation of}}~A]=[A~{\text{in}}]-[A~{\text{out}}]+[{\text{Net generation of}}~A]}$

2. ${\displaystyle {\frac {dN_{A}}{dt}}=F_{Ao}-F_{A}+V\nu _{A}r_{A}}$

where

Applying the assumptions of steady state and ν_A = −1, Equation 2 simplifies to: