Ideal distribution function

Equation (1) describes the idealized distribution function, obtained from an infinite number of sample measurements, the so-called parent population distribution. In practice we are limited to some finite number, n, of samples taken from the population being examined and the statistics, or estimates, of mean, variance, and standard deviation are denoted then by x, and s respectively. The mathematical definitions for these parameters are given by equations (2H4),... 

Table 8-1 gives tlie relationships between tlie age distribution functions and Figure 8-6 shows the age distribution functions of ideal reactors. 

An Excel spreadsheet (Example8-7.xls) was used to determine the various RTD functions and the computer program PROGS 1 was used to simulate the model response curve with the experimental data. The results show the equivalent number of ideally mixed stages (nCSTRs) for the RTD is 13.2. The Gamma distribution function from Equation 8-143 is ... 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Once the wavelength dependence of the molecular form factor F(nqo) is known from the reasonable model of layer organization, the ratios r /ti may be calculated. The value of these ratios (for example, T2/T1, T3/T1) give a good guide to the sharpness of the distribution function f(z) - for an ideal crystal f(z) would be an array of delta-functions and T2 = Ti = = = 1. From the... 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

Except for the case of an ideal plug flow reactor, different fluid elements will take different lengths of time to flow through a chemical reactor. In order to be able to predict the behavior of a given piece of equipment as a chemical reactor, one must be able to determine how long different fluid elements remain in the reactor. One does this by measuring the response of the effluent stream to changes in the concentration of inert species in the feed stream—the so-called stimulus-response technique. In this section we will discuss the analytical form in which the distribution of residence times is cast, derive relationships of this type for various reactor models, and illustrate how experimental data are treated in order to determine the distribution function. 

For a few highly idealized systems, the residence time distribution function can be determined a priori without the need for experimental work. These systems include our two idealized flow reactors—the plug flow reactor and the continuous stirred tank reactor—and the tubular laminar flow reactor. The F(t) and response curves for each of these three types of well-characterized flow patterns will be developed in turn. 

Note that in this case the right side of equation 11.1.68 is zero for t = 0 and unity for t = 00. Figure 11.9 contains several F(t) curves for various values of n. As n increases, the spread in residence time decreases. In the limit, as n approaches infinity the F(t) curve approaches that for an ideal plug flow reactor. If the residence time distribution function given by 11.1.69 is differentiated, one obtains an... 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

Examination of the Fermi-Dirac distribution function Eq. (2.41) shows that the condition for applicability of the ideal-gas distribution to electron velocities is... 

The distribution function (24) for an ideal gas, shown in figure 6 is known as the Maxwell-Boltzmann distribution and is specified more commonly [118] in terms of molecular speed, as... 

Abstract. A theory of relativistic ideal gas (RIG), fluxons and electrons is presented. A distribution function of velocities (FRS) and the equation of state of the RIG are found, together with the distribution function of the observed frequencies. 

This paper is organized as follows. Section 2 presents non-trivial properties of the velocity distribution functions for RIG for quasi and ordinary particles in one dimensions. In section 3 we find the state equation for relativistic ideal gas of both types. Section 4 presents the distribution function for the observed frequency radiation generated for quasi and ordinary particles of the relativistic ideal gas, for fluxons under transfer radiation and radiative atoms of the relativistic ideal gas. Section 5 presents a generalization of the theory of the relativistic ideal gas in three dimensions and the distribution function for particles... 

Earlier the velocity distribution function of quasi particles of a relativistic ideal gas for a one dimensional system, for example, fluxons in thermalized Josephson systems and electrons in a high temperature plasma was found. 

Let us introduce non-trivial properties of the distribution function of velocities of a relativistic ideal gas for both cases. For this we remind that the probability to find a particle with velocity between vx and... 

In order to define the statistical characteristics of a many particle system, for instance an ideal gas, their distribution function with some defined physical parameters (for example, velocity, momentum, energy, etc) should be fully determined. In particular it is physically important to define the velocity of particles corresponding to the most probable state, which is the maximum of the distribution function. 

Figure 1. (la) Distribution function of the velocity for the relativistic ideal gas of gluxions. (2a) Distribution functions of the observable frequencies. (3a) Most probable values of the observable frequencies as function of a. (4a) Absolute minimal realization of most probable states of system. 

The chapter begins with a reiteration and extension of terms used, and the types of ideal flow considered. It continues with the characterization of flow in general by age-distribution functions, of which residence-time distributions are one type, and with derivations of these distribution functions for the three types of ideal flow introduced in Chapter 2. It concludes with the development of the segregated-flow model for use in subsequent chapters. 

The characterization of flow by statistical age-distribution functions applies whether the flow is ideal or nonideal. Thus, the discussion in this section applies both in Section 13.4 below for ideal flow, and in Chapter 19 for nonideal flow. 

In this section, we derive expressions for the age-distribution functions E and F for three types of ideal flow BMF, PF, and LF, in that order. Expressions for the quantities W, I, and H are left to problems at the end of this chapter. The results are collected in Table 13.2 (Section 13.4.4). 

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