Exponential distribution, continuous distributions

For systems following invariant growth the crystal population density in each size range decays exponentially with the inverse of the product of growth rate and residence time. For a continuous distribution, the population densities of the classified fines and the product crystals must be the same at size Accordingly, the population density for a crystallizer operating with classified-fines removal is given by... 

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

To answer the question as to whether the fluorescence decay consists of a few distinct exponentials or should be interpreted in terms of a continuous distribution, it is advantageous to use an approach without a priori assumption of the shape of the distribution. In particular, the maximum entropy method (MEM) is capable of handling both continuous and discrete lifetime distributions in a single analysis of data obtained from pulse fluorometry or phase-modulation fluorometry (Brochon, 1994) (see Box 6.1). 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Another approach is known as the Distributed Activation Energy Model (DAEM). This model recognizes that devolatilization occurs through many simultaneous reactions. To express this process in a mathematically tractable manner, these reactions are all presumed to be first order and to be describable by a continuous distribution of kinetic rates with a common pre-exponential and a defined distribution function of activation energy [43],... 

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

For a given type of x-values distribution, the size of the x-values array (number of blocks ra, ) plays approximately the same role as the number of scans N. Theoretically, the relative precision of any relaxation rate estimate is proportional to the square root of both and N. This, of course, presumes that Ub is anyway large enough to carry out the analysis. For example, values as small as 4 may be sufficient in mono-exponential cases, while continuous distributions spreading over several orders of magnitude require a logarithmic distribution of x-values and Ub values of over 100. 

The polystyrene data were collected from a steady state, continuous, well-mixed reactor. The initiator was n-butylli-thlum for data of Figure 2 and was azobisisobutylnitrile for data of Figure 3. Toluene was used as a solvent. The former polymerizatl n y ields an exponential population density distribution ( ), M /M = 1.5 the latter yields a molar distribution defined as th product of degree of polymerization and an exponential ( ), M /M = 2.0. Standards utilized in calibration of both instrumen s ftere polystyrene supplied by Pressure Chemical Company. 

Equation (8) can be generalized by considering more than two kinds of centers. If there are many different kinds of active centers, their relative proportions may be represented approximately by a continuous distribution function, as has been suggested by Constable (2). In particular, it may be assumed that their relative numbers decrease exponentially with decreasing activation energy A . Thus the number of active centers dn involving activation energies between A and AE -+ dAE can be assumed to be... 

In the detection of the autocorrelation functions in self-beat spectroscopy, solution polydispersity can lead to a non-exponential form. If we assume that there are no contributions to the autocorrelation function except those from translational diffusion for the different types of molecules, we can consider two simple cases a continuous distribution of solute particle sizes and several distinct components in a solution. We shall approach the two cases by determining their effect on the observed correlation function. 

In studies of low-mobility insulators, two types of continuous trap distributions are commonly used the exponential distribution of traps [364] and the Gaussian distribution of traps [365]. 

Due to the functional form of Eq. (185), which, except for the constant coefficient, is identical to the SCL j-U characteristics for a continuous exponential distribution of the form (175) or (176) derived in the case of the infinitely sharp point traps, the experimental data arising from the discrete macrotrap background can, without scrutiny, be mistakenly attributed to the continuous exponential distribution of point traps. The transition from the low-to high-field regions of the current occurs at a voltage (see Fig. 71)... 

The age of a fluid element is defined as the time it has resided within the reactor. The concept of a fluid element being a small volume relative to the size of the reactor yet sufficiently large to exhibit continuous properties such as density and concentration was first put forth by Danckwerts in 1953. Consider the following experiment a tracer (could be a particular chemical or radioactive species) is injected into a reactor, and the outlet stream is monitored as a function of time. The results of these experiments for an ideal PFR and CSTR are illustrated in Figure 8.2.1. If an impulse is injected into a PFR, an impulse will appear in the outlet because there is no fluid mixing. The pulse will appear at a time ti = to + t, where t is the space time (r = V/v). However, with the CSTR, the pulse emerges as an exponential decay in tracer concentration, since there is an exponential distribution in residence times [see Equation (3.3.11)]. For all nonideal reactors, the results must lie between these two limiting cases. 

Thus, the crossover from a strictly exponential to an anomalous relaxation pattern can be associated with the change of a continuous distribution of relaxation times (a = 1) into a fractal-like one (0 < a = df < 1). 

Hence, we can generate an exponential random variable by generating a random variable U and then use Eq. (33.6) to draw a sample from an exponential distribution with mean X. This method of simulating continuous variables is called the inverse transformation method. Although the method can be applied to any distribution, either continuous or discrete, the problem with the inverse transformation approach is that it is often difficult to invert the CDF, if it even exists, to an analytical solution. 

The continuous phase is well mixed but there is a dispersed phase. The particles in the dispersed phase behave as PFRs. They are in contact with the continuous phase but are isolated from each other and have an exponential distribution of residence times. This case is treated in Examples 11.17 and 15.12. 

A perfect mixer has an exponential distribution of residence times W t) = exp(—r/7). Can any other continuous flow system have this distribution Perhaps, surprisingly, the answer to this question is a definite yes. To construct an example, suppose the feed to a reactor is encapsulated. The size of the capsules is not critical. They must be large enough to contain many molecules but must remain small compared to the dimensions of the reactor. Imagine them as small ping-pong balls as in Figure 15.1 la. 

In those cases where it is not practical to mix discrete classes of narrowly sized particles, an approximation yields a continuous particle size distribution. Mathematically, it is described by the Fuller distribution [B.42], an exponential distribution in which the exponent must be between 1/3 and 2/3. 

Gordiets Vibrational Distribution Function in Non-Thermal Plasma. Compare the discrete Gordiets vibrational distribution with continuous distribution function (3-141). Pay special attention to the exponential decrease of the vibrational distribution functions at high vibrational energies in the case of low translational temperatures (To < ftoS). 

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