Unit impulse function

The (unit) impulse function is called the Dirac (or simply delta) function in mathematics.1 If we suddenly dump a bucket of water into a bigger tank, the impulse function is how we describe the action mathematically. We can consider the impulse function as the unit rectangular function in Eq. (2-20) as T shrinks to zero while the height 1/T goes to infinity ... 

To answer this question, away from the context of PF, consider a characteristic function / ( ) that, at t = b, is suddenly increased from 0 to 1/C, where C is a relatively small, but nonzero, interval of time, and is then suddenly reduced to 0 at t = b + C, as illustrated in Figure 13.6. The shaded area of C(l/C) represents a unit amount of a pulse disturbance of a constant value (1/C) for a short period of time (C). As C - 0 for unit pulse, the height of the pulse increases, and its width decreases. The limit of this behavior is indicated by the vertical line with an arrow (meaning goes to infinity ) and defines a mathematical expression for an instantaneous (C - 0) unit pulse, called the Dirac delta function (or unit impulse function) ... 

F. UNIT IMPULSE FUNCTION. By definition, the z transformation of an impulse-sampled function is... 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

In addition, plasma concentrations from an IV dose or from administration of IR formulation such as an oral solution or rapidly dissolving oral formulation are needed to estimate the unit impulse function. 

The mathematical statement of the above type of experiment is to inject an impulse of tracer into the vessel inlet at time zero. This is represented by the Dirac delta function or perfect unit impulse function ... 

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... 

Transform of an impulse function the Dirac delta or unit impulse function is given as... 

Thus, F(9) is a unit step function shifted along the time axis by 0 and E(0) is a unit impulse function shifted along time axis by 0. Sketches of F(0) and E(0) for ideal PFR are shown in Figures 3.52 and 3.53, respectively. 

RTD methods are based on the concept of age distribution functions and make use of the experimentally measured or calculated residence time distribution of fluid elements in a reactor vessel (Figure 12.3-1, C and D). A Lagrangian perspective is taken and the age of a fluid element is defined as the time elapsed since it entered the reactor. In what follows, steady state operation of a vessel fed with a volumetric flow rate F is considered. A residence time distribution (RTD) experiment can be performed with inert tracers, such that at an instant of time all fluid elements entering a reactor or process vessel are marked. The injection of an impulse of tracer into the vessel at time zero can be mathematically represented by means of the Dirac delta function or perfect unit impulse function ... 

Impulse Input. The unit impulse function discussed in Chapter 3 has the simplest Laplace transform, U s) = 1 (Eq. 3-24). However, true impulse functions are not encountered in normal plant operations. To obtain an impulse input, it is necessary to inject a finite amount of energy or material into a process in an infinitesimal length of time, which is not possible. However, this type of input can be approximated through the injection of a concentrated dye or other tracer into the process (see Example 3.7). 

The collocation method chooses the weighting functions to be the Dirac delta (unit impulse) function ... 

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