One-parameter distributions

In hydrocarbon liquids other than n-hexane, the procedure for obtaining the thermalization distance distribution could conceivably be the same. However, in practice, a detailed theoretical analysis is rarely done. Instead, the free-ion yield extrapolated to zero external field (see Chapter 9) is fitted to a one-parameter distribution function weighted with the Onsager escape probability, and the mean thermalization length (r ) is extracted therefrom (see Mozumder, 1974 ... 

This canonical ensemble is also a one-parameter distribution function for various values of 9, independent of q and p. 

The exponential distribution was presented in the preceding Section. This distribution is a one-parameter distribution (A). Mathematical statistics uses a large number of distributions, which may serve, for example, to describe empirical data or random processes. Below the probability density functions of several two-parameter distributions are listed, some of which also exist in versions with three parameters. Details are found in [C-l-C-5]. 

For a one-parameter distribution, it suffices to know any moment (higher than the zeroth) to completely define the distribution. The geometric and Poisson distribntions are examples of one-parameter distributions. For a two-parameter distribution, such as the Gaussian, two moments... 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

Another aspect in which Bayesian methods perform better than frequentist methods is in the treatment of nuisance parameters. Quite often there will be more than one parameter in the model but only one of the parameters is of interest. The other parameter is a nuisance parameter. If the parameter of interest is 6 and the nuisance parameter is ( ), then Bayesian inference on 6 alone can be achieved by integrating the posterior distribution over ( ). The marginal probability of 6 is therefore... 

The distribution given in the table should, as for Table 6.21, be used only as a guide, not for designing. There are many cases where one parameter of the building system can have a large influence on the performance of the ventilation system, even when there are low demands on the target levels. 

The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

The line the data supports on a hazard plot determines engineering information relating to the distribution of time to failure. Fan failure data and simulated data are illustrated here to explain how the information is obtained. The methods provide estimates of distribution parameters, percentiles, and probabilities of failure. The methods that give estimates of distribution parameters differ slightly from the hazard paper of one theoretical distribution to another and are given separately for each distribution. The methods that give estimates of distribution percentiles and failure probabilities are the same for all papers and are given first. 

Assuming that our LG is in a local equilibrium, it is reasonable to expect that the one-particle distribution functions should depend only on the macroscopic parameters u x,t) and p x,t) and their derivatives [wolf86c]. While there is no reason to believe that this dependence should be a particularly simple one, it is reasonable to expect that both u and p are slowly varying functions of x and t. Moreover, in the subsonic limit, we can assume that li << 1. 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

Chapter 8 consists of the following in Sect. 8.2 the physical model of the process is described. The governing equations and conditions of the interface surface are considered in Sects. 8.3 and 8.4. In Sect. 8.5 we present the equations transformations. In Sect. 8.6 we display equations for the average parameters. The quasi-one-dimensional model is described in Sect. 8.7. Parameter distribution in characteristic zones of the heated capillary is considered in Sect. 8.8. The results of a parametrical study on flow in a heated capillary are presented in Sect. 8.9. 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

In practice when reservoir parameters such as porosities and permeabilities are estimated by matching reservoir model calculated values to field data, one has some prior information about the parameter values. For example, porosity and permeability values may be available from core data analysis and well test analysis. In addiction, the parameter values are known to be within certain bounds for a particular area. All this information can be incorporated in the estimation method of the simulator by introducing prior parameter distributions and by imposing constraints on the parameters (Tan and Kalogerakis, 1993). 

The discrete Poisson distribution is only characterized by one parameter, the mean Y. The standard deviation is given by sY = Jy and the relative standard deviation by SyreI = 1 / JY. 

State-of-the-art polymeric materials possess property distributions in more than one parameter of molecular heterogeneity. Copolymers, for example, are distributed in molar mass and chemical composition, while telechelics and macromonomers are distributed frequently in molar mass and functionality. It is obvious that n independent properties require n-dimensional analytical methods for accurate (independent) characterization of the different structural parameters. 

VC=C n the yellow solution. In both cases the experimental data are quite well represented considering that there is only one parameter in the fit. The model also adequately describes the Raman excitation profile. The good agreement between the model and the experimental results confirms the basic premise of the model that the individual polymer chains in solution contain a distribution of chain lengths which determine the absorption characteristics of the solutions. 

A one-parameter model, termed the bubbling-bed model, is described by Kunii and Levenspiel (1991, pp. 144-149,156-159). The one parameter is the size of bubbles. This model endeavors to account for different bubble velocities and the different flow patterns of fluid and solid that result. Compared with the two-region model, the Kunii-Levenspiel (KL) model introduces two additional regions. The model establishes expressions for the distribution of the fluidized bed and of the solid particles in the various regions. These, together with expressions for coefficients for the exchange of gas between pairs of regions, form the hydrodynamic + mass transfer basis for a reactor model. 

A quantitative analysis of the structure-retention relationship can be derived by using the relative solubility of solutes in water. One parameter is the partition coefficient, log P, of the analyte measured as the octanol-water partition distribution. In early work, reversed-phase liquid chromatography was used to measure log P values for drug design. Log P values were later used to predict the retention times in reversed-phase liquid chromatography.The calculation of the molecular properties can be performed with the aid of computational chemical calculations. In this chapter, examples of these quantitative structure-retention relationships are described. 

The solution of Eq. (7.35), which represents the stationary temperature distribution, should be of the form 6 = f (q, <5) with one parameter, that is, 6. The condition under which such a stationary temperature distribution ceases to be possible, that is, the critical condition of ignition, is of the form 6 = const = <5crit. The critical value depends upon T(h the geometry (if the vessel is nonspherical), and the pressure through Z. Numerical integration of Eq. (7.35) for various S s determines the critical 6. For a spherical vessel, <5crit = 3.32 for an infinite cylindrical vessel, <5CTit = 2.00 and for infinite parallel plates, X n[ = 0.88, where r becomes the distance between the plates. 

Audisio studied the microtacticity of the 1,4-rrans-polypentadiene [—CH2—CH=CH—CH(CH3)—] in connection with that of the poly-(methyltetramethylene) [—CH2—CH2—CH2—CHfCHs)—] obtained from the preceding compound by reduction (109, 110), and succeeded in evaluating the distribution of the triads mm, mr, and rr. He has proposed an interpretation according to a one-parameter model based on enantiomorphic catalyst sites (111) (see Table 4, column 4, 3/ = 1). 

In practice, when one measures the size distributions of aerosols using techniques discussed in Chapter 11, one normally measures one parameter, for example, number or mass, as a function of size. For example, impactor data usually give the mass of particles by size interval. From such data, one can obtain the geometric mass mean diameter (which applies only to the mass distribution), and crg, which, as discussed, is the same for all types of log-normal distributions for this one sample. Given the geometric mass mean diameter (/) ,) in this case and crg, an important question is whether the other types of mean diameters (i.e., number, surface, and volume) can be determined from these data or if separate experimental measurements are required. The answer is that these other types of mean diameters can indeed be calculated for smooth spheres whose density is independent of diameter. The conversions are carried out using equations developed for fine-particle technology in 1929 by Hatch and Choate. 

The term cross-fractionation (CF) refers to analyses of distributions in differing directions by means of separation processes. Cross-fractionation is a significant tool for the evaluation of the complex distribution which copolymers normally have with respect to molar mass (MMD) and chemical composition (CCD). The idea of CF implies separation by one parameter and subsequent analysis of the fractions obtained for the distribution of the other parameter through another separating process. 

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