Diffusion coefficients of binary solutions can be measured accurately by observation at long times of isothermal diffusion in a vertical cell closed at the ends (restricted diffusion). The present analysis, which accounts for the effects of solvent flux and variable solution properties, demonstrates that the experiment yields a well-defined, differential diffusion coefficient, even in concentrated solutions.Observation is effected by Rayleigh interference optics, and results for aqueous potassium chloride solutions confirm the accuracy of the method. JOHN SCOPEAccurate values of diffusion coefficients of binary mixtures, including their dependence on composition and temperature, are required to provide complete transportproperty data (somewhat analogous to thermodynamic data like activity coefficients), to permit quantitative design of separation processes like distillation and electrochemical deposition, and to permit detailed comparison with theoretical results. Among theoretical results we have in mind, on the one hand, the treatment of simple systems where variable physical properties have been included in the analysis and, on the other hand, the theoretical prediction or correlation of transport properties on a molecular basis such as the Debye-Htickel theory.For binary solutions of electrolytes, three methods are used to measure the diffusion coefficient: the magnetically stirred diaphragm cell, optical observation of diffusion from an initially sharp boundary, and (restricted) diffusion in a box of finite height with observation of the concentration distribution by measurement of the electrical conductivity at two positions in the box. The last method, developed by Hamed beginning in 1949, appeals to the intuition because the maximum concentration difference in the system vanishes as the experiment progresses, and an absolute determination of the diffusion coefficient requires only the height of the box.One question we raise is just what diffusion coefficient is measured and is this complicated by the reference velocity for the diffusion flux, the precise definition of transport properties for concentrated solutions, or the variation of physical properties with composition?Harned's method is restricted to very dilute solutions of electrolytes because adequate resistance measurement with solutions of higher conductivity requires larger cell dimensions where free convection becomes a problem. The second question is how can the method be extended to concentrated solutions? CONCLUSIONS AND SIGNIFICANCEAn analysis of the restricted diffusion process shows that the measurement yields the differential diffusion coefficient D at the final average concentration. This is the diffusion coefficient which appears in the flux Equations (13) experiment due to volume changes on mixing (or diffusion).
A new formulation and algorithm is described for computing the solution to an overdetermined linear system, Ax b, with possible errors in .both A and b. This approach preserves any affine structure of A or [AIb], such as Toeplitz or sparse structure, and minimizes a measure of error in the discrete Lp norm, where p 1, 2, or x. It can be considered as a generalization of total least squares and we call it structured total least norm (STLN). The STLN problem is formulated, the algorithm for its solution is presented and analyzed, and computational results that illustrate the algorithm convergence and performance on a variety of structured problems are summarized. For each test problem, the solutions obtained by least squares, total least squares, and STLN with p-1, 2, and were compared. These results confirm that the STLN algorithm is an effective method for solving problems where A or b has a special structure or where errors can occur in only some of the elements of A and b.
The Total Least Squares (TLS) method is a generalization of the least squares (LS) method for solving overdetermined sets of linear equations Ax b. The TLS method minimizes jj Ej?r]jj F where r = b?(A+E)x, so that (b?r) 2 Range(A+E), given A 2 C m n , with m n and b 2 C m 1. The most common TLS algorithm is based on the singular value decomposition (SVD) of A j b]. However, the SVD based methods may not be appropriate when the matrix A has a special structure, since they do not preserve the structure. Recently, a new problem formulation, called Structured Total Least Norm (STLN), and algorithm for computing the STLN solution have been developed. The STLN method preserves the special structure of A or A j b], and can minimize the error in the discrete L p norm, where p = 1; 2 or 1. In this paper, the STLN problem formulation is generalized for computing the solution of STLN problems with multiple right-hand sides AX B. It is shown that these problems can be converted to ordinary STLN problems with one right-hand side. In addition, the method is shown to converge to the optimal solution in certain model reduction problems. Furthermore, the application of the STLN method to various parameter estimation problems is studied in which the computed correction matrix applied to A or A j B] keeps the same Toeplitz structure as the data matrix A or A j B], respectively. In particular, the L 2 norm STLN method is compared with the LS and TLS methods in deconvolution, transfer function modeling and linear prediction problems, and shown to improve the accuracy of the parameter estimates by a factor of 2 to 40 at any signal-to-noise ratio.
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