Equations elemental mass fractions

The actual volume of each phase in element AV is that of the total volume of the element, multiplied by the respective fractional phase holdup. Hence considering the direction of solute transfer to occur from the aqueous or feed phase into the organic or solvent phase, the mass balance equations become ... 

This form of the partition coefficient, analogous to that used for Fe-Mg fractionation between olivine and melt (see Chapter 1), is necessary only for the rare cases where trace substitution affects Cj and Cp substantially. A number of reviews (O Nions and Powell, 1977 Michard, 1989) describe the various sorts of partition coefficients expressed either in mass-fractions, atom fractions, or normalized to a major element and their respective merits. If the discussion is restricted to a narrow range of chemical compositions (e.g., basaltic systems, Irving, 1978, Irving and Frey, 1984), enough experimental information exists on trace-element partitioning to resort to the wonderfully simple equation (9.1.1). 

Equation (9.3.22) can be used when volume to mass-fraction conversion is needed. In the right-hand side braces, the first term corresponds to the amount of element i entering the molten zone, the last two terms to the amount left behind at z. Assuming... 

The previous equations do not require that the isotopic ratio used for normalization (x-axis) and the ratio to be measured (y-axis) have to be for the same element. It is therefore possible to normalize the isotopic composition of Cu to that of a standard Zn solution without any assumption made on the particular mass-fractionation law. The original formulation of this property by Longerich et al. (1987) calls for identical isotopic fractionation factors for the two elements, but this is not at all a necessary constraint and Albarede et al. (2004) show that, in fact, this very assumption may lead to significant errors. For a Cu sample mixed with a Zn standard, in which the Zn/ Zn ratio of the standard solution is used for normalization, we obtain the expression ... 

When written in full, the numher of unknowns is the sum c + e + r, where c is the number of active cups, e the number of elements for which the value h is needed, and r the number of ratios to measure. We first show how to use a standard solution (or a mixture of standards of different elements) of known isotopic compositions to determine the cup efficiencies. The unknowns are the efficiencies A for each cup and the mass fractionation factors/(or h for other laws). The system of equations is particularly compact since Equation (59) now reduces to ... 

The first three constraints represent orthogonal collocation applied to the differential equations at the collocation points. The next three equations represent mass balances at the separation point. The discretized RTD function and the expression for the mean residence time are given in the final constraints. As la —> 0, this model is equivalent to the original reaction-separation model (PI2). The main difference is that we allow separation only at the end of each element within each element no separation occurs. Although the model appears nonlinear, the nonlinearities are actually reduced when one considers the rates in terms of the mass fractions. The solution to this model then gives us the optimal separation split fractions as a function of time along the reactor. 

The mass fraction A is obtained as the solution to a partial differential equation, which comes from the mass balance. In order to derive this we will consider a coherent region of any size, out of which we will imaginarily cut a body in which diffusion takes place, Fig. 2.57a. The volume of the region is V, and its surface area A. A surface element dA, whose normal n is directed outwards as in Fig. 2.57a, has a mass flow... 

The above derivatives must be evaluated at the interface composition before use in computing the Jacobian elements. This additional complexity in evaluating the derivatives of the vapor-phase mass transfer rate equations arises because we have used mass fluxes and mole fractions as independent variables. If we had used mass fractions in place of mole factions the derivatives of the rate equations would be simpler, but the derivatives of the equilibrium equations would be more complicated. For simplicity, we have ignored the dependence of the mass transfer coefficients themselves on the mixture composition and on the fluxes. 

Despite the uncertainty regarding absolute rates of diffusion for noble gases in a water-filled medium, the relative rates remain a direct function of mass. In principle, for example, the extent of diffusive gas loss for any reservoir can be determined by the magnitude of fractionation of known noble gas elemental ratios using Equation (24) and the appropriate mass fractionation coefficient (Eqn. 27). 

Let and be the mass fractions of glycol in the absorbent solution, respectively, within the element, and inside the layer on the plate and oixo and a i be the corresponding values in the absorbent solution coming to our plate from the previous plate, and at the exit from the previous element. The equation describing the variation of with time is similar to (20.47) and has the form... 

The measurement process for the determination of the mass fraction of an element by INAA in a blank-free environment can be described by the following equation ... 

In the relative method, the mass fraction of the analyte element in the unknown sample is obtained using the mass fraction of the same element in the standard. The equation takes into account the peak areas obtained in the unknown and in the standard the sample weight (mass), W the decay factors S, D, and C (see Eq. (30.26)) differences in neutron flux and differences in detection efficiency due to slightly different counting geometries. 

A similar equation can be obtained for the mixture fraction [2]. Because Z is defined by Eq. (4.7) as the mass fraction of the fuel stream into the mixture, Z is the sum of the mass fractions of the elements contained in the fuel stream. 

Here, p is mass density and yk th mass fraction, t is time and div the divergence operator v is local mass flow velocity (vector) and jk the it-th molecular diffusion flux vector, added to the term pykV representing the convection of particles Ck by the motion of a material element as a whole. So the instantaneous local change (increase) of the Ck-concentration (mass per unit volume) equals minus the amount that escapes from a volume element (the divergence term) plus the amount produced by chemical reactions. Physically, the balance makes sense if we know how the flux jk depends on the gradients (most simply by Pick s law), and how the rates of possible reactions depend on the local state of the element. If also the latter information is available then the balance takes the form of convective diffusion equation, possibly with chemical reactions. [If we have no information on the reaction rates, the w -terms can be eliminated from Eqs. (C.2) by an algebraic transformation in the same manner as in Chapter 4 indeed, it is sufficient to substitute for W, in (4.3.2) and to define the components of column vector n as follows from (C.2).] Observe finally that we have... 

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