Vector of species

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

General Considerations. The continuity equations for a column vector of species taking place in a reaction-diffusion wave phenomenon take the form... 

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 ... 

Mass flux vector of species i due to molecular transport, kg/(m s)... 

The column vector of species fluxes along a pore segment is then... 

It is unknown for several species that how they were introduced into the Baltic Sea, although shipping is assumed to be the most dominating vector of species introduction, followed by aquaculture activities. In order to explain why so many species invaded the Baltic Sea, following arguments come into consideration ... 

Description In the same manner that one might define a mixture concentration compactly as a vector of species concentrations, it is also possible to form a vector, of equal dimension to the concentration vector, associated with the rate of reaction. If benzene and toluene are the components of interest to the problem, the concentration vector is C = [Cb, c-p]. It is natural to express the corresponding rates of reaction for benzene and toluene as a two-component column vector as well. The rate vector, r(C), in Ca-Cp space is hence defined as follows ... 

An extended Stefan-Maxwell diffusion equation can be constructed by setting up a homogeneous, linear relationship between the vector of diffusion driving forces and the vector of species velocity differences. 

We now consider these separators to be fixed in space and focus attention on a particulcU surface area Sj of any separator shown in Figure 2.1.1. Let Vij be the local average velocity vector of species i with respect to coordinate axes fixed on the surface area Sj. If the mass concentration of species i at this location on such a surfiice area is indicated by pg, with p,j being the local value of the total mass concentration, we define the local mass average velocity vector Vg by... 

Quite often the molar rate of inflow or outflow of mass with respect to the surface area Sj of a separator is of greater interest. In such a case, the molar concentration Cy is used. The quantities corresponding to riy, wy and Wy are then Ny, Wy and Wy, the local molar flux vector of species i, the molar rate of inflow or outflow of species i and the total molar rate of inflow or outflow of aU species, respectively ... 

Tables 3.1.3A, 3.1.3B value of /, in region k diffusive molar flux vector of species 1 (4.2.63) temperature gradient driven molar flux vector of species 1 (4.2.62) z-components of flux vectors /, and /s, y-component of flux vector /J z-components of flux vectors /, /J, / and... 

For a dynamic experiment [32,33] where the vector of species is n = [glucose ethanol biomass CO acetate] the change in mole number at the end of the experiment (in moles) was ... 

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