Transform of Distribution Functions

Certain operations in chemical engineering are modelled by elementary distribution functions. Such operations as the instantaneous closing of a valve can be modelled by the so-called step function. The quick injection of solute into a flowing stream, such as that done in gas chromatography, can be modelled by the so-called impulse function. Moreover, these distribution-type functions can be shifted in time. 

Consider first the unit step function, illustrated schematically in Fig. 9.5. It is, of course, impossible to cause real physical systems to follow exactly this square-wave behavior, but it is a useful simulation of reality when the process is much slower than the action of closing a valve, for instance. The Laplace transform of u(t) is identical to the transform of a constant 

In fact, it can be proved that all delayed functions are multiplied by exp(-Tj), if T represents the delay time. For instance, suppose a ramp function, r(t) = t 

We note that as the time delay is reduced, so t 0, then the original step and ramp functions at time zero are recovered. 

The unit impulse function. Sit), is often called the Dirac delta function. It behaves in the manner shown in Fig. 9.8. The area under the curve is always unity, and as 6 becomes small, the height of the pulse tends to infinity. We can define this distribution in terms of unit step functions 

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