Vector mass fraction

Here, Z is the mass fraction vector, Z=[z, Zg], and is analogous to a concentration vector for each component in the system. G is the total mass flow rate flowing through the CSTR (G is therefore always constant). It is also assumed that the rate vector is now expressed in terms of mass fractions and catalyst mass W, so that the units of r(Z) are (mass of component reacted/(mass of catalyst X time)). 

The mass fraction vector, z, is then defined in a similar manner to the concentration vector C as follows ... 

That is, for any mixture containing n species, we may associate the mixture with a mass fraction vector z. From a geometric viewpoint, z is the vector in K" mass fraction space associated with a unique magnitude and direction. 

The system of equations may be combined into mass fraction vectors... 

Hence, the geometric interpretation of this is that a mixture resulting from combinations of mixtures Zj and produces a mass fraction vector z that lies on a straight line between Zj and in z -Zb space. Linear mixing is therefore maintained in mass fraction space. 

Next we consider how the three fundamental reactor types found in AR theory may be expressed in terms of the mass fraction vector z. These expressions are derived directly from the molar versions of the same expression, and converted using the mass fraction formulae given in Sections 9.2.1-9.2.4. 

The feed mass fraction vector Zf may be computed using W and nj. The feed vector written in terms of mass flow rates is given by... 

The total mass flow is then determined as the sum of the elements in mj, G = 16 +18 + 28 = 62kg/s. The feed mass fraction vector is then... 

To compare the regions obtained in Chapter 8 to those given here, mass fraction vectors belonging to the stoichiometric subspace must be converted to equivalent concentration vectors. Since the components are assumed to behave as ideal gases, the conversion from species mass fractions to species molar fractions is straightforward. Hence, species concentration c, for an ideal gas mixture is given by... 

Consider Equations (6-10) that represent the CVD reactor problem. This is a boundary value problem in which the dependent variables are velocities (u,V,W), temperature T, and mass fractions Y. The mathematical software is a stand-alone boundary value solver whose first application was to compute the structure of premixed flames.Subsequently, we have applied it to the simulation of well stirred reactors,and now chemical vapor deposition reactors. The user interface to the mathematical software requires that, given an estimate of the dependent variable vector, the user can return the residuals of the governing equations. That is, for arbitrary values of velocity, temperature, and mass fraction, by how much do the left hand sides of Equations (6-10) differ from zero ... 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

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... 

Figure 9.1 Inverse partial melting problem in the three-dimensional space of elements 1, 2, 3 when the source is known. Projection of the source onto the sample subspace provides the mass-fraction of each phase of the molten source. If one phase is at the origin (sterile phase), every representative point can be shifted by a constant vector.

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

In these one-dimensional equations, the independent variable is the spatial coordinate z, and the dependent variables are the temperature T and the species mass fractions Yk. The continuity equation is satisfied exactly by m" = pu, which is a constant. Other variables are the z component of the mass-flux vector jkz, the molar production rate of species by chemical reaction 6)k, the thermal conductivity A, the species enthaplies hk, and the molecular weights Wk. The diffusion fluxes are determined as... 

Since there is not a continuously differentiable relationship between the inlet and outlet flows, the Gauss divergence theorem (i.e., the V- operation) has no practical application. Recall that, by definition, the surface unit vector n is directed outward. The sign of the mass-fraction difference in Eq. 16.68 is set by recognizing that the inlet flow velocity is opposite the direction of n, and vice versa for the exit. The overall mass-continuity equation,... 

The feed vector F0 is similar to F, and contains mass fractions, which are added to the circuit after eveiy transition. In the case shown in Fig. 2a, it contains only the first non-zero vector of the three, which is the fraction size distribution /o in the feed to the circuit. The rest two vectors are the zero vectors of the same size. 

Here G(a) is the intensity scattered by the particles with diameters between a and art-da, kfi is Boltzmann s constant, T the temperature, k the scattering vector (dependent on scattering angle and wavelength), t the correlation time and n the viscosity. To find the mass fraction distribution, F(a), we use... 

We consider a thin, horizontal, fused-silica plate of area. I and thickness H. Initially, both horizontal surfaces of the plate are in contact with air (Figure 2.3c). We assume that the air is completely soluble in silica. At time t 0. the air below the plate is replaced by pure helium, which is appreciably soluble in silica. The helium slowly penetrates into the plate by molecular motion, and eventually appears in the air above the plate. This molecular transport of one substance relative to another is known as diffusion. The air above the plate is removed rapidly, so that there is no measurable helium concentration there. In this system, the index / shows helium and j shows silica, and the concentrations are given by the mass fractions w, and i. respectively. Eventually, the concentration profile tends toward a straight line with increasing t, and we have w, = r(U at the bottom surface, and w, = 0 at the top surface of the plate. At steady-state diffusion, the molar flow vector that is the flow rate of helium per unit area j, is proportional to the concentration gradient in terms of the mass fraction V r,... 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

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