The physical significance of formulae for the thermal conductivity and viscosity of gaseous mixtures

1963 ◽  
Vol 276 (1364) ◽  
pp. 69-82 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Binary Mixture ◽  
Linear Equations ◽  
Physical Significance ◽  
Simultaneous Equations ◽  
Total Thermal Conductivity ◽  
Rigorous Analysis ◽  
Gaseous Mixtures ◽  
Polyatomic Gases ◽  
Simultaneous Linear Equations

A physical interpretation is made of various complicated formulae which have been given for the thermal conductivity A and viscosity n of mixtures of gases. The interpretation is based on the recognition of two principal effects operating in the transport of heat or momentum through gaseous mixtures. The first (and larger) effect is that molecules of one species impede transport of heat or momentum by other species. The second effect is a transfer of the transport of heat (or momentum) from one species to another. When the transfer of transport is neglected, equations of the form proposed by Sutherland (1895) and Wassiljewa (1904) follow immediately. For the thermal conductivity (symbols used are defined in the main text): A = E{ n i /( n i A i -1 +Ea ij n j )] There is an identical expression for the viscosity, though the values of a ij are different from those for the thermal conductivity. The significance to be given to each term of the sum over i is that of a quotient of (i) a force driving conduction, proportional to n i , and (ii) a resistance due to species i(n i A i -1 ) and to all other species (En j a ij ). Here A i -1 is the resistance offered by species i to its own transport, a ij and the resistance offered by the species j to transport by i . When the transfer of transport is taken into account, two simultaneous linear equations have to be solved for a binary mixture, and the solution has the form of the quotient of quadratics familiar in the theoretical analysis of mixtures of monatomic gases. For a mixture of N constituents, the resulting expression for A appears as a quotient of determinants. Application of the same principles also throws light on the transport of internal energy in mixtures of polyatomic gases. The total thermal conductivity may be divided into contributions from each species, and each such contribution may be further subdivided into internal and translational parts: thus for a binary mixture it is necessary to replace the pair of simultaneous equations by four. Such linear equations represent a direct generalization of the equations of Mason & Monchick (1962) for a simple gas. In an appendix, the physically significant parameters of the generalized approach employed here are compared explicitly with the predictions of rigorous analysis for mixtures of monatomic gases.

On the Solution of the Numerical Simultaneous Equations Arising in the Analysis of Redundant Structures

1945 ◽  
Vol 49 (411) ◽  
pp. 104-111
Author(s):  
F. J. Turton
Keyword(s):  
Strain Energy ◽  
Linear Equations ◽  
Simultaneous Equations ◽  
Distribution Method ◽  
Energy Applications ◽  
Moment Distribution ◽  
Simultaneous Linear Equations ◽  
Moment Distribution Method ◽  
Mathematical Accuracy ◽  
The Moment

The application of strain energy or slope-deflection methods in the analysis of redundant structures leads to a number of simultaneous linear equations with numerical coefficients; the equations may be obtained in such order that each successive equation contains one new unknown, until all the unknowns are so included. This is the only condition essential for the method to be described in the present paper, but the labour is much reduced in slope-deflection and strain energy applications by the fact that most (or all) of the equations contain very few of the unknowns. The method to be given reduces the solving of these equations to a column of successive evaluations, followed by the solution, by algebraic methods, of a small number of simultaneous equations; and a final column of evaluations. In the remaining paragraphs a number of problems are examined to show how the equations may be obtained in suitable sequence for the method to apply. Following an application to the determination of secondary stresses, the operations involved in the moment-distribution method and in this method are compared. A numerical example is worked out in the simple case of §2, and it is shown how any order of mathematical accuracy in the roots may be ensured, provided that sufficient figures have been retained to permit that accuracy.

scholarly journals On the Evaluation of Determinants, the Formation of their Adjugates, and the Practical Solution of Simultaneous Linear Equations

1933 ◽  
Vol 3 (3) ◽  
pp. 207-219 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Linear Equations ◽  
Statistical Correlation ◽  
Simultaneous Equations ◽  
Normal Equations ◽  
Practical Solution ◽  
Simultaneous Linear Equations ◽  
Reciprocal Matrix ◽  
Image Position

There are various methods in existence for the practical solution of a set of simultaneous equationsSome of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

On a Simple Method for Solving Simultaneous Linear Equations by a Successive Approximation Process

1935 ◽  
Vol 39 (292) ◽  
pp. 349-351 ◽  
Author(s):  
J. Morris
Keyword(s):  
Linear Equations ◽  
Iteration Process ◽  
Simultaneous Equations ◽  
Engineering Science ◽  
Slide Rule ◽  
Approximation Process ◽  
Simple Method ◽  
Moment Distribution ◽  
Hardy Cross ◽  
Simultaneous Linear Equations

Simultaneous equations of three or more variables are notoriously trouble some to solve numerically and, moreover, are frequently sensitive for certain relative values of the variables. This usually precludes the use of the slide rule and in consequence resort has to be had to accurate working to an increasing number of decimal places involving much labour even with a calculating machine.A method is here given which is applicable to certain classes of equations of frequent occurrence in mathematical physics and engineering science, which method is a combination of what is knovwi as the Iteration process and the means of applying this principle to end moment distribution devised by Professor Hardy Cross.

scholarly journals Transport Coefficients of Hyperonic Neutron Star Cores

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 203
Author(s):  
Peter Shternin ◽  
Isaac Vidaña
Keyword(s):  
Thermal Conductivity ◽  
Neutron Star ◽  
Shear Viscosity ◽  
Transport Coefficients ◽  
Dense Matter ◽  
Baryon Number ◽  
Nucleon Nucleon ◽  
Total Thermal Conductivity ◽  
Nucleon Force ◽  
Density Region

We consider transport properties of the hypernuclear matter in neutron star cores. In particular, we calculate the thermal conductivity, the shear viscosity, and the momentum transfer rates for npΣ−Λeμ composition of dense matter in β–equilibrium for baryon number densities in the range 0.1–1 fm−3. The calculations are based on baryon interactions treated within the framework of the non-relativistic Brueckner-Hartree-Fock theory. Bare nucleon-nucleon (NN) interactions are described by the Argonne v18 phenomenological potential supplemented with the Urbana IX three-nucleon force. Nucleon-hyperon (NY) and hyperon-hyperon (YY) interactions are based on the NSC97e and NSC97a models of the Nijmegen group. We find that the baryon contribution to transport coefficients is dominated by the neutron one as in the case of neutron star cores containing only nucleons. In particular, we find that neutrons dominate the total thermal conductivity over the whole range of densities explored and that, due to the onset of Σ− which leads to the deleptonization of the neutron star core, they dominate also the shear viscosity in the high density region, in contrast with the pure nucleonic case where the lepton contribution is always the dominant one.

On predicting thermal conductivity of a binary mixture of solids.

1966 ◽  
Vol 3 (2) ◽  
pp. 287-288 ◽  
Author(s):  
J. M. FOSTER ◽  
C. B. SMITH ◽  
R. I. VACHON
Keyword(s):  
Thermal Conductivity ◽  
Binary Mixture

scholarly journals Effective Thermal Conductivity of Open Cell Polyurethane Foam Based on the Fractal Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Kan Ankang ◽  
Han Houde
Keyword(s):  
Thermal Conductivity ◽  
Fractal Dimension ◽  
Effective Thermal Conductivity ◽  
Polyurethane Foam ◽  
Solid Phase ◽  
Fractal Theory ◽  
Heat Radiation ◽  
Total Thermal Conductivity ◽  
Equivalent Thermal Conductivity ◽  
Open Cell

Based on the fractal theory, the geometric structure inside an open cell polyurethane foam, which is widely used as adiabatic material, is illustrated. A simplified cell fractal model is created. In the model, the method of calculating the equivalent thermal conductivity of the porous foam is described and the fractal dimension is calculated. The mathematical formulas for the fractal equivalent thermal conductivity combined with gas and solid phase, for heat radiation equivalent thermal conductivity and for the total thermal conductivity, are deduced. However, the total effective heat flux is the summation of the heat conduction by the solid phase and the gas in pores, the radiation, and the convection between gas and solid phase. Fractal mathematical equation of effective thermal conductivity is derived with fractal dimension and vacancy porosity in the cell body. The calculated results have good agreement with the experimental data, and the difference is less than 5%. The main influencing factors are summarized. The research work is useful for the enhancement of adiabatic performance of foam materials and development of new materials.

An efficient approach to the seismogram synthesis for a basin structure using propagation invariants

1996 ◽  
Vol 86 (2) ◽  
pp. 379-388 ◽  
Author(s):  
H. Takenaka ◽  
M. Ohori ◽  
K. Koketsu ◽  
B. L. N. Kennett
Keyword(s):  
Integral Equation ◽  
Inverse Fourier Transform ◽  
Linear Equations ◽  
Computation Time ◽  
Reduction Technique ◽  
Final Solution ◽  
Space Domain ◽  
New Approach ◽  
Simultaneous Linear Equations ◽  
Single Integral

Abstract The Aki-Larner method is one of the cheapest methods for synthetic seismograms in irregularly layered media. In this article, we propose a new approach for a two-dimensional SH problem, solved originally by Aki and Larner (1970). This new approach is not only based on the Rayleigh ansatz used in the original Aki-Larner method but also uses further information on wave fields, i.e., the propagation invariants. We reduce two coupled integral equations formulated in the original Aki-Larner method to a single integral equation. Applying the trapezoidal rule for numerical integration and collocation matching, this integral equation is discretized to yield a set of simultaneous linear equations. Throughout the derivation of these linear equations, we do not assume the periodicity of the interface, unlike the original Aki-Larner method. But the final solution in the space domain implicitly includes it due to use of the same discretization of the horizontal wavenumber as the discrete wavenumber technique for the inverse Fourier transform from the wavenumber domain to the space domain. The scheme presented in this article is more efficient than the original Aki-Larner method. The computation time and memory required for our scheme are nearly half and one-fourth of those for the original Aki-Larner method. We demonstrate that the band-reduction technique, approximation by considering only coupling between nearby wavenumbers, can accelerate the efficiency of our scheme, although it may degrade the accuracy.

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