Computation of the Fluxes

Only in these limiting cases is the computation of the fluxes from an exact solution of the Maxwell-Stefan equations so straightforward. In most cases of practical importance we must make use of the full matrix solution (Eq. 8.3.24). 

The computation of the fluxes from either of Eqs. 8.3.24 necessarily involves an iterative procedure (except for the special cases discussed above), partly because the themselves are needed for the evaluation of the matrix of correction factors and also because an explicit relation for the matrix [0] cannot be derived as a generalization of Eq. 8.2.16 for binary mass transfer there is no requirement in matrix algebra for the matrices [FFq] be equal to each other even though the fluxes calculated from both parts of these equations must be equal. Indeed, these two matrices will be equal only in the case of vanishingly small mole fraction differences (yg Tg) and vanishingly small mass transfer rates. In almost all cases of interest these two matrices are quite different. An explicit solution was possible for binary systems only because all matrices reduce to scalar quantities. 

The method of successive substitution can be a very effective way of computing the from Eqs. 8.3.24 when the mole fractions at both ends of the diffusion path y-g and y g, are known. In practice, we start from an initial guess of the fluxes and compute the rate factor matrix [ ]. The correction factor matrix [a] may be calculated from an application of Sylvester s expansion formula (Eq. A.5.20) 

The fluxes ( /) can then be calculated from either of Eqs. 8.3.24 (having previously calculated [/3] and [A ]). The new estimates of the are used to recalculate [0] and the procedure is repeated until convergence is obtained. An initial estimate of the can be computed from Eqs. 8.3.24 with the correction factor matrices set equal to the identity matrix [/]. This procedure is summarized in Algorithm 8.1 and illustrated in Example 8.3.1. 

Before continuing we note that this procedure will not always converge and, if it does, it may not do so particularly efficiently. In Section 8.3.4 we discuss some of the computational subtleties of these equations and provide a more efficient and robust algorithm for computing the fluxes. 

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Equations 8.3.52-8.3.57 were presented as an exact solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures by Burghardt (1984). Equations 8.3.52-8.3.57 are somewhat less useful than Eqs. 8.3.15-8.3.24 because we need to know the composition profiles in order to evaluate the matrizant. Even if the profiles are known, the computation of the fluxes from either of Eqs. 8.3.62 or 8.3.63 is not straightforward and not recommended. It is with the development in Section 8.4 in mind that we have included these results here. 

For computation of the fluxes themselves. Algorithms 8.1, 8.4, and 8.5 may be used more or less as written simply replace all quantities in molar units by the corresponding quantities in the turbulent eddy diffusivity model. 

A. CALHET Direct computation of the flux distribution heterogeneity effect by the CALHET code. 

With the complete concentration profile c,(x,y) at hand, it is straightforward to compute the magnitude of the flux /, towards the plane ... 

The concept of elementary flux modes has resulted in a vast number of applications to analyze and predict the functionality of metabolic networks [64, 65, 138, 241 243]. Software resources that allow for the computation of elementary flux modes are fisted in Table V [178, 224], It should be noted that, due to their definition as an exhaustive enumeration of possible flux distributions,... 

Given the inherent limits of a purely computational approach to obtain an estimate of the flux distribution of a metabolic system, an experimental determination of metabolic fluxes is paramount to the construction and validation... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

The influence of any metabolite is described by a single bme-dependent variable, the concentration. The temporal evolution of the variables is then computed from ordinary differenbal equations (ODEs) that predict the immediate change of a variable according to the size of the fluxes that produce or degrade the particular metabolite. In turn, the size of each flux depends on a well defined set of variables, i.e. the instantaneous concentrations of substrates, products, cofactors and effectors, and a set of parameters. The state of the model, i.e. the values of all metabolite concentrations, evolves by updating the variables over subsequent time intervals which are repeated for different external conditions to yield theoretical time courses of concentrations and fluxes. 

Recently, a 10 days continuous time series measurements of N2 at 5 m depth off Massachusetts, provided the possibihty to estimate the air-sea transfer coefficient of N2 and the subsequent computation of the air—sea fluxes of the reactive gases CO2 and O2 (McNeil et al, 2006). 

Stigebrandt, A. (1991). Computations of oxygen fluxes through the sea surface and the net production of organic matter with application to the Baltic and adjacent seas. Limnol. Oceanogr. 36, 444—454. 

Figure 1 Computer-generated curves of the flux vs. increasing concentrations of substrate transported by either a simple diffusion process fit to the equation J = 5(S) or by a facilitated diffusion process fit to the equation J = 100(S)/(0.4 + S).

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

In practical applications of the linear-response formalism, it is often more convenient to express the response in terms of a displacement V rather than a flux J, and we will therefore focus on this case here. The response of a gel to an applied stress is, for instance, conveniently expressed as a strain, and likewise, the response of a dielectric material to an applied electric field is often expressed as a polarization [which is closely related to the so-called electric displacement field (89)]. As a mechanical analogue would indicate, a flux is generally proportional to a velocity, and the displacement is therefore computed as the time integral of the flux. 

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