IEM model for a stirred reactor

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as 

The mean concentration vector is found by assuming that the CSTR is homogeneous on large scales (i.e., well macromixed).101 Using the fact that the age distribution in a well macromixed CSTR is exponential, 

100 The IEM model is easily extended to treat multiple feed streams and time-dependent feed rates (Fox and Villermaux 1990b). 

102 Although this definition can be employed to find the mean concentrations, it is numerically unstable for small values of fm. In this case, an equivalent integro-differential equation is preferable (Fox 1989 Fox and Villermaux 1990b). 

---