6 function delta Dirac distribution

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

This model is directly derived from the Langmuir isotherm. It assumes that the adsorbent surface consists of two different types of independent adsorption sites. Under this assumption, the adsorption energy distribution can be modeled by a bimodal discrete probability density function, where two spikes (delta-Dirac functions) are located at the average adsorption energy of the two kinds of sites, respectively. The equation of the Bilangmuir isotherm is... 

The mathematical derivation of SPH is based on the calculation of a quantity /using a convolution of/-values with the Dirac delta function S. Subsequently, the Dirac distribution is replaced by a kernel functicMi W, which tends to the delta function in the limit case of its characteristic length, the smoothing length h, becoming zero [11,12]... 

Fig. 3-3. Some Important Probability Density Functions and Their Corresponding Distribution Functions. Arrows are used to indicate Dirac delta functions with the height of the arrow indicating the area under the delta function.

When considering the composition inhomogeneity of Markovian copolymers, the finiteness of the chemical size of macromolecules cannot be ignored, because fractional composition distribution W(/ f) in the limit / -> oo turns out to be equal to the Dirac delta function 5(f - X). For macromolecules of finite size f2> 1 the function W(/ f) is the Gaussian distribution whose center and dispersion (Eq. 2) are described by relationships (Eq. 8) and the following one... 

When deriving this expression for the average composition distribution, authors of paper [74] entirely neglected its instantaneous constituent, having taken (as is customary in the quantitative theory of radical copolymerization [3,84]) the Dirac delta-function < ( -X) as the instantaneous composition distribution. Its averaging over conversions, denoted hereinafter by angular brackets, leads to formula (Eq. 101). Note, this formula describes the composition distribution only provided copolymer composition falls in the interval between X(0) and X(p). Otherwise, this distribution function vanishes at all values of composition lying outside the above-mentioned interval. 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

The effects of non-uniform distribution of the catalytic material within the support in the performance of catalyst pellets started receiving attention in the late 60 s (cf 1-4). These, as well as later studies, both theoretical and experimental, demonstrated that non-uniformly distributed catalysts can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniform. Work in this area has been reviewed by Gavriilidis et al. (5). Recently, Wu et al. (6) showed that for any catalyst performance index (i.e. conversion, selectivity or yield) and for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a non-isothermal pellet, with finite external mass and heat transfer resistances, the optimal catalyst distribution remains a Dirac-delta function. 

Dirac delta function (distribution) unit step function, Heaviside function... 

Figure 9.6 Total conversion for inert membrane reactor with catalyst on the feed side (IMRCF), catalytic membrane reactor (CMR) and conventional fixed-bed reactor (FBR) with uniform and Dirac delta catalyst activity distributions as a function of the dimensionless residence time [Yeung et al., 1994]...

Figure 9.7 Effect of location of Dirac delta catalyst activity distribution on toul conversion as a function of dimensionless residence time for (a) IMRCF (b) CMR and (c) FBR [Yeung etal., 1994]...

Catalyst attached to membrane pore surface. The final distribution of the catalyst in the membrane pores can significantly impact the reactor performance. The optimal form of the catalyst distribution for maximizing conversion was studied mathematically by Keller et al. [1984]. They determined that the optimal distribution of the catalyst concentration is of the Dirac delta function. 

The function / is a probability density, indicated by the shading of the lower right part of Fig. 6. To bring about a decomposition of into eigenstates, one must choose the function / as a linear combination of Dirac delta functions (i.e., as an appropriate distribution). 

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