Linear equations, processes governed

There are several reasons for observing differences between the computed results and experimental data. Errors arise from the modeling, discretization and simulation sub-tasks performed to produce numerical solutions. First, approximations are made formulating the governing differential equations. Secondly, approximations are made in the discretization process. Thirdly, the discretized non-linear equations are solved by iterative methods. Fourthly, the limiting machine accuracy and the approximate convergence criteria employed to stop the iterative process also introduce errors in the solution. The solution obtained in a numerical simulation is thus never exact. Hence, in order to validate the models, we have to rely on experimental data. The experimental data used for model validation is representing the reality, but the measurements... 

It is shown that the development of the equations governing THM processes in elastic media with double porosity can be approached in a systematic manner where all the constitutive equations governing deformability, fluid flow and heat transfer are combined with the relevant conservation laws. The double porosity nature of the medium requires the introduction of dependent variables applicable to the deformable solid, and the fluid phases in the two void spaces. The governing partial differential equations are linear in view of the linearized forms of the constitutive assumptions invoked in the formulations. The linearity of these governing equations makes them amenable to solution through conventional mathematical techniques applicable to the study of initial boundary value problems in mathematical physics (Selvadurai, 2000). Such solutions should serve as benchmarks for appropriate computational developments. 

Analysis of Desorption Kinetics. Several features are evident from the kinetics of desorption. Plots of the logarithm of the area are a linear function of the square root of time in agreement with a diffusion controlled process governed by the equation ... 

The operating rules or constraints governing the process (e.g., limited resources) can be expressed as a set of linear equations or inequalities. 

This example elucidates how to systematically and swiftly perform balances of complex plants at steady-state conditions. Of key importance is the linearity of the governing equations describing the process. Due to linearity, which may be found as well for many similar tasks, the governing equations allow for a closed, analytical solution. An analogous problem is presented in Sect. 4.3.3.5 as regards the calculation of a heat exchanger. 

Finite element formulations for linear stress analysis problems are often derived by direct reasoning approaches. Fluid flow and other materials processing problems, however, are often viewed more easily in terms of their governing differential equations, and this is the... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

Some interesting aspects of the interface kinetics appear only when temperature and latent heat are included into the model, if the process of heat conductivity is governed by a classical Fourier law, the entropy balance equation takes the form Ts,= + x w where s = - df dr. Suppose for simplicity that equilibrium stress is cubic in strain and linear in temperature and assume that specific heat at fixed strain is constant. Then in nondimensional variables the system of equations takes the form (see Ngan and Truskinovsky, 1996a)... 

The discussion up to this point has been concerned essentially with metal alloys in which the atoms are necessarily electrically neutral. In ionic systems, an electric diffusion potential builds up during the spinodal decomposition process. The local gradient of this potential provides an additional driving force, which acts upon the diffusing species and this has to be taken into account in the derivation of the equivalents of Eqns. (12.28) and (12.30). The formal treatment of this situation has not yet been carried out satisfactorily [A.V. Virkar, M. R. Plichta (1983)]. We can expect that the spinodal process is governed by the slower cation, for example, in a ternary AX-BX crystal. The electrical part of the driving force is generally nonlinear so that linearized kinetic equations cannot immediately be applied. 

The reactions of poly(styryl)lithium in benzene with an excess of diphenyl-ethylene 272) and bis[4-(l-phenylethenyl)phenyl]ether158) also were found to proceed by a first order process. However, the reactions of poly(styryl)lithium with the double diphenylethylenes l,4-bis(l-phenylethenyl)benzene and 4,4 bis(l-phenyl-ethenyl)l,l biphenyl gave l58) non-linear first order plots with the gradients decreasing with time. This curvature was attributed to departure from a geometric mean relationship between the three dimerization equilibrium constants (Ka, Kb and Kab). The respective concentrations of the various unassociated, self-associated and cross-associated aggregates involved in the systems described by Equations (49) to (51) are dependent upon the relative concentrations of the two active centers and the respective rate constants which govern the association-dissociation events. 

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