Mass fraction, defined

At this point we have 17 equations in 17 unknowns mo, mi, mo.me.mi, mg, m,t to mn, Ta, ft, ft, and the four mass fractions defined by Equations 7 through 10. The system of equations can be solved by an equation-solving program such as E-Z Solve. 

Since mass is a more practical property to measure than moles, flowrates are often given as mass flowrates rather than molar flowrates. When this occurs, it is convenient to express concentrations in terms of mass fractions defined similarly to mole fractions. 

Throughout this discussion we have used the numerical fraction of molecules in a class as the weighting factor for that portion of the population. This restriction is not necessary some other weighting factor could be used equally well. As a matter of fact, one important type of average encountered in polymer chemistry is the case where the mass fraction of the ith component is used as the weighting factor. Defining the mass of material in the ith class as mj, we write... 

The particle mass retained by each sieve is determined by weighing after drying when necessary, and each fraction is designated by the sieve size it passed and the size on which it was retained. The sieve diameter of a particle is therefore defined as the size of the sieve aperture through which the particle in question just passes through. Mass fractions of the particles are then presented in tabular or graphical form. 

This quantity, which is often referred to as magma density or soHds concentration (mass of crystals per unit system volume), is often an important process variable. A cumulative mass fraction of crystals having a size less than U can also be defined as... 

Ziebold (1967) has defined the detection limit of any analytical technique in terms of the minimum mass fraction (MMF) detectable according to the expression ... 

The reaction rates Rt will be functions of the state variables defining the chemical system. While several choices are available, the most common choice of state variables is the set of species mass fractions Yp and the temperature T. In the literature on reacting flows, the set of state variables is referred to as the composition vector [Pg.267]

Note that if the mass fractions are used to define the composition vector, then by definition... 

Let Yrj denote the mass fractions of the K chemical species describing the reacting flow. By definition, KYa—. Assuming that the chemical species are numbered such that the major species (e.g., reactants) appear first,2 followed by the minor species (e.g., products), we can define a linear transformation by... 

In the computer code, a sorting algorithm can be used to put the mean mass fractions (Ya) in descending order before defining Xp. By keeping track of the order of the indices, one can easily define the inverse transformation needed to compute Ya from Xp. 

If the mass fractions are sorted in descending order, then all of the null mass fractions will be grouped together at the end of the array. The procedure can thus be simplified by using only the nonzero mass fractions to define Xp. In practice, one can define a lower limit for Yrl and set the correction term to zero for mass fractions below this limit. 

Flashover occurs in a room causing 200 g/s of fuel to be generated. It is empirically known that the yield of CO is 0.08. Yield is defined as the mass of product generated per mass of fuel supplied. The room has a volume of 30 m3 and is connected to a closed corridor that has a volume of 200 m3. The temperatures of the room and corridor are assumed uniform and constant at 800 and 80 °C respectively. An equation of state for these gases can be taken as pT = 360 kg K/m3. Steady mass flow rates prevail at the window of the room (to the outside air) and at the doorway to the corridor. These flow rates are 600 and 900 g/s respectively. Derive expressions for the mass fraction of CO in the room and corridor as a function of time assuming uniform concentrations in each region. 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

It is convenient to work in terms of the void fraction a and the mass fraction of gas w, which is also known as the quality. The void fraction is defined as the time average fraction of the cross-sectional area through which the gas flows ... 

Following the description given in the previous section, the detailed chemical system is mapped on two controlling variables the mixture fraction (Z) and a reaction progress variable (c). The mixture fraction describes the mixing of the species and enthalpy, while the progress variable follows the advance of the chemical reaction. Hence, the species mass fractions, temperature, density and species chemical source terms become functions of Z and c. The mixture fraction is defined according to [19] as ... 

It is relatively straightforward to develop the controlling parameters of a stirred reactor process. If ip is defined as the unbumed mass fraction, it must follow that the fuel-air mass rate of burning RB is... 

Defining mA as the mass fraction of A, one obtains the following proper form for the diffusion of species A in terms of mass fraction ... 

Under the inherent assumption that the mass fractions of the reactants are not changing, further interesting insights can be obtained by rearranging Eq. (7.22). If the reaction proceeds at a constant rate corresponding to T(), a characteristic reaction time rr can be defined as... 

We define the mass bias factor g = e and finally obtain the expression for the so-called mass-fractionation power-law as ... 

Figure 10. On this unusually fractionated block of 40 cycles, the effect of the noise-induced correlation observed on Zn isotopes can be separated from the mass fractionation effects. The solid line corresponds to mass-dependent fractionation, while the dashed line is defined by counting statistics. When the larger Zn/ "Zn and Zn/ "Zn ratios are plotted against each other (top), counting statistics tend to pull the results away from the mass fractionation line. When the smaller "Zn/"Zn is considered (bottom), counting statistics has essentially no effect. Data acquired using the VG Plasma 54 of Lyon.

Let us define the intercept of the mass fractionation line with thcT-axis as ... 

Early studies of Mg isotope ratios in geological materials used the notation A Mg to mean per mil deviations from a standard as expressed in Equation (1) above, a convention that persists today (e.g., Elsu et al. 2000). The values assigned to A "Mg in those studies represent the level of mass-dependent isotopic fractionation relative to the standard. The same convention defined fi Mg as the per mil deviation from the standard after correction for the mass fractionation evidenced by A "Mg. In this system of nomenclature, A values refer to mass dependentfractionations while 5 values refer to deviations from mass-dependent fractionation (i.e., the S Mg defines excesses in Mg relative to mass fractionation attributable to decay of the extinct nuclide Al). In some cases A "Mg has been replaced by the symbol Fn,g (Kennedy et al. 1997) where the F refers to fractionation. ... 

There is a regularly returning confusion, why the generated X-ray intensity is proportional to the mass fraction of the elements present, in contrast to the atomic fraction. This fact is contradictory to our first physical anticipation, since the ionization cross section is defined for one atom, so it is the number of atoms that should be important (irrespective of their masses). It is a correct assumption and we recall it below, how the mass fraction results from the starting number of atoms. 

Having obtained a set of mineral components that satisfactorily reproduces Che data, we have defined Che matrix of Equation 5. Given jC and C, Equation 5 Is Chen used Co solve for For Che alblte cluster with 107 samples and 4 mineral components, F Is 107 X 4 matrix containing the mass fractions of each mineral component In each sample. Because these mass.fractions sum to 1.0 for each sample, assuming we are accounting for all the mineral matter, we solve an overdetermined set of simultaneous equations of Che form... 

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