The Pulse Input Tracer Experiment and Analysis

Chemical engineers also use this kind of experiment. It can be utilized to great advantage in chemical reactors to find the residence time distribution of the reactor, a crucial piece of information which links microscopic flow behavior, that is, fluid dynamics, to measurables of the system, such as chemical conversion and selectivity. For vessels that are not used for reaction processes, but are used for other operations that are also critically dependent upon mixing, this tracer experiment provides a great deal of insight into how the system behaves. We can analyze how a pulse of injected tracer would behave in the well-stirred vessel we have been analyzing here. 

Imagine that an injection is made as a pulse of tracer, the concentration of which can be measured in the tank and in the exit stream as a function of time. For a laboratory vessel, the injection may be done by hand with a syringe full of tracer such as a dye or a radioactively tagged molecule. For larger vessels at pilot and production scale ingenious methods have been invented for putting a pulse of tracer into the unit. Ideally, the pulse should be added instantaneously, which means in as short a time period as possible. In other words, the time to add the tracer must be much shorter than the time required to wash it out of the unit. 

How can we model such a problem To do the analysis we need to introduce and become comfortable with two new functions the Dirac-Delta function and the UnitStep (or Heaviside) function. The Dirac-Delta function is infinitely intense and infinitesimally narrow—like a pulse of laser light. We can imagine it arising in the following way. We begin by considering a pulse that is quite broad, such as the function that is plotted here  

We can sharpen this function in time by decreasing the value of the time constant 6 , as follows  

As we decrease the time constant the function becomes more intense in and around the f = 0. Doing this in the limit of 6 - 0 transforms this into the infinitely intense pulse of infinitely short time duration. We can use this Dirac-Delta function, once we know more about its properties and how it is implemented in Mathematica. 

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