Sample space variables velocity

A semi-colon is used in the argument list to remind us that V is an independent (sample space) variable, while x and t are fixed parameters. Some authors refer to fyx (Vj x, f ) as the one-point, one-time velocity PDF. Here we use point to refer to a space-time point in the four-dimensional space (x, t). 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

The variables that govern turbulent reacting flows have large, random fluctuations. This suggests a statistical treatment of the variables which leads to the use of multivariable joint probability density functions (PDF) t). i) denotes the probability that a system at location x and time r is in a state between and + d, where denotes a particular state in the underlying sample space Q. In this notation, for instance, = (p,p, T, u, Ti,..., Yn) denotes a vector whose components are flow and thermo-chemical random variables. More precisely, p denotes the density, p the pressure, T the temperature, u the velocity, and Ti,..., Tw the N species mass fractions. 

The formal FDF is defined by Fl v, ip,, rj, x t) where, v, tp, and 77 are the velocity vector, the scalar array, the sensible internal energy and pressure in the sample space, respectively. The function F has all of the properties of a probability density function in that the filtered value of any function of the velocity and/or scalar variables is obtained by its integration over the sample spaces ... 

The first step in E-cat testing is to bum the carbon off the sample. The sample is then placed in a MAT unit (Figure 3-13), the heart of which is a fixed bed reactor. A certain amount of a standard gas oil feedstock is injected into the hot bed of catalyst. The activity i.s reported as the conversion to 430°F (221°C) material. The feedstock s quality, reactor temperature, catalyst-to-oil ratio, and space velocity are four variables affecting MAT results. Each catalyst vendor uses slightly different operating variables to conduct micro activity testing, as indicated in Table 3-2. 

The key variables in the steam reforming experiments were the steara/carbon and tl /carbori ratio In the reactor feed and the gas hourly space velocity (GHSV) calculated for normal temperature and pressure (NTP), The H20/carbon and H2/carbon levels were varied according to the incidence of carbon deposition on each catalyst sample. The GHSV was adjusted at the beginning of each experiment to approximate differential conditions (<26 mol% conversion) of the hydrocarbon feedstcick into the product. Normal laboratory SNG production conditions wore as follows ... 

The data harmonization was performed on several hydrocarbon species and CO oxidation as well as on NO, -CO reaction. A small review of some of the data treated in this study is presented in Table 1. The first four columns in Table 1 indicate the experimental variables such as the catalyst type, the sample weight, the gas volumetric flow rate and inlet gas mixture composition. The fifth column reports the calculated space velocity based on the given data, and in the last column the reference from which the data were collected is given. There are several factors to note in the information presented in Table 1. 

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