With flow chemistry, material continuously enters and leaves the 'flow reactor'; this is the part of the process where conditions such as reactant concentrations, pH, temperature, pressure and mass transfer between different phases enable a chemical reaction to take place. In batch chemistry, the conditions in the reactor change with time (the concentrations at the start of the reaction will be different than those at the end) whereas in flow chemistry, once steady state has been reached, the conditions within the reactor are constant. A simple example of a flow reactor would be a round-bottomed flask with a constant flow of reactants being pumped in, and a constant flow of product out of it.
The design of the flow reactor can influence the reaction; the simplest flow reactor is a tube into which reactants flow. Here the mixing of material will largely be diffusion driven (so slow), adding internal structures within the tube (static mixers) can improve mixing but introduce additional pressure drop. Tubular reactors are sometimes (and confusingly) referred to as plug flow reactors (PFRs) although the flow within these isn't really plug flow.
An alternative to tubular reactors is the continuous stirred tank reactor (CSTR). Essentially this is a vessel with an agitator, into which fluid enters and leaves. CSTRs are a mainstay in batch manufacture of chemical products due to their flexibility in dealing with a wide range of materials - they span single- and multi-phase chemistry, fermentation requiring gas sparging, crystallisations, viscous materials and many more. CSTRs come in many sizes and process engineers have access to considerable data and a wide range of correlations with which to specify equipment (e.g. mixer type) and operating parameters for specific functions (e.g. suspension of particles, mixing of phases). There is also a lot of know-how in scaling the process (e.g. from small development scale to manufacture scale) with CSTRs.
The fReactor-Classic have been built using many principles of CSTRs. Each separate module is a single CSTR. There are good reasons for using more than one CSTR together - each addition of a module means the processing conditions across different 'elements' of fluid become more unform.
Of course there are other reactor designs, for example for specific multiphasic reactions, exothermic reactions, fast reactions or the use of catalyst beds. A few of these are mentioned briefly under the page covering multiphasic flows, but the focus of these pages is largely around PFRs and CSTRs to support your understanding of flow reactors and to start your exploration of the wider world of flow reactors.