Rigorous Solution

This is a more difficult equation to solve than that for the solubility of Pb(I03)2 in distilled water, and its solution is not immediately obvious. A rigorous solution to equation 6.34 can be found using available computer software packages and spreadsheets. 

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an-... 

Rapid Approximate Design Procedure. Several simplified approximations to the rigorous solutions have been developed over the years (57—60), but they aU. remain too compHcated for practical use. A simple method proposed in 1989 (61,62) uses a correction factor accounting for the effect of axial dispersion, which is defined as (57)... 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

V and L, are found from Equations 9 and 10. The improved procedure is better for rigorous solution of complicated absorber designs. 

The solutions to a problem of this magnitude can be found in references [3, 7] and others. Figures 9-16 and 9-17 are torsional mode shape diagrams of some typical systems. While the rigorous solution to the multimass damped system is not within the scope of this book, several interesting points should be made. 

Obviously, the assumptions involved in the foregoing derivation are not entirely consistent. A transverse strain mismatch exists at the boundary between the fiber and the matrix by virtue of Equation (3.8). Moreover, the transverse stresses in the fiber and in the matrix are not likely to be the same because v, is not equal to Instead, a complete match of displacements across the boundary between the fiber and the matrix would constitute a rigorous solution for the apparent transverse Young s modulus. Such a solution can be found only by use of the theory of elasticity. The seriousness of such inconsistencies can be determined only by comparison with experimental results. 

Torres-Marchal [110] and [111] present a detailed graphical solution for multicomponent ternary systems that can be useful to establish the important parameters prior to undertaking a more rigorous solution with a computer program. This technique can be used for azeotropic mixtures, close-boiling mixtures and similar situations. 

This system for evaluating multicomponent adjacent key systems, assuming constant relative volatility and constant molai overflow, has proven generally satisfactory for many chemical and hydrocarbon applications. It gives a rigorous solution for constant molai overflow and volatility, and acceptable results for most cases which deviate from these limitations. 

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

This particular scheme has been the subject of many, many publications. Some deal with the rigorous solutions, but more treat various approximate solutions such as the steady-state and prior equilibrium approximations. Several assumptions, valid much of the time, convert the full expressions into more tractable forms. They are the subject of the next two sections. 

In this case Eq. (1.71) reduces to a differential equation. Its rigorous solution with initial conditions Kj(0) = rd, Kj(0) = 0 gives... 

With time-dependent computer simulation and visualization we can give the novices to QM a direct mind s eye view of many elementary processes. The simulations can include interactive modes where the students can apply forces and radiation to control and manipulate atoms and molecules. They can be posed challenges like trapping atoms in laser beams. These simulations are the inside story of real experiments that have been done, but without the complexity of macroscopic devices. The simulations should preferably be based on rigorous solutions of the time dependent Schrddinger equation, but they could also use proven approximate methods to broaden the range of phenomena to be made accessible to the students. Stationary states and the dynamical transitions between them can be presented as special cases of the full dynamics. All these experiences will create a sense of familiarity with the QM realm. The experiences will nurture accurate intuition that can then be made systematic by the formal axioms and concepts of QM. 

A more rigorous solution based on marching ahead in enthalpy according to Equation 7.42 is given in the next 16 lines of code. The temperature is found from the enthalpy using a binary search. The code is specific to the initial conditions of this problem. Results are very similar to those for marching temperature directly. 

Particle Size Laser Refractometiy is based upon Mie scattering of particles in a liquid medium. Up until about 1985, the power of computers supplied with laser diffraction instruments was not sufficient to utilize the rigorous solution for homogeneous spherical particles formulated by Gustave Mie in 1908. Laser particle instrument manufacturers therefore used approximations conceived by Fraunhofer. 

Thus, the rigorous solution of kinetic equation describing the change in electric conductivity of a semiconductor during adsorption of radicals enables one to deduce that information on concentration of radicals in ambient volume can be obtained measuring both the stationary values of electric conductivity attained over a certain period of time after activation of the radical source and from the measurements of initial rates in change of electric conductivity during desactivation or activation of the radical flux incident on the surface of adsorbent, i.e. 

MULTICOMPONENT SYSTEMS RIGOROUS SOLUTION PROCEDURES (COMPUTER METHODS)... 

The basic steps in any rigorous solution procedure will be ... 

If a rigorous solution to this problem is at all possible, it consists of two parts ... 

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

To solve Equation 9.51, it is necessary to know the values of not only a ,-j and 9 but also x, d. The values of xitD for each component in the distillate in Equation 9.51 are the values at the minimum reflux and are unknown. Rigorous solution of the Underwood Equations, without assumptions of component distribution, thus requires Equation 9.50 to be solved for (NC — 1) values of 9 lying between the values of atj of the different components. Equation 9.51 is then written (NC -1) times to give a set of equations in which the unknowns are Rmin and (NC -2) values of xi D for the nonkey components. These equations can then be solved simultaneously. In this way, in addition to the calculation of Rmi , the Underwood Equations can also be used to estimate the distribution of nonkey components at minimum reflux conditions from a specification of the key component separation. This is analogous to the use of the Fenske Equation to determine the distribution at total reflux. Although there is often not too much difference between the estimates at total and minimum reflux, the true distribution is more likely to be between the two estimates. 

Intuitively [46], the dSS approximation is more likely to hold for times greater than the time to practically reach SS ( ss 8 /ttDm)- Thus, the approximation agrees with the rigorous solution (transient diffusion with the bulk value restored at r = ro + <5m) for most of the range seen in Figure 15, but not for... 

Figure 15. Comparison of curves Jm (a) and /u (b) versus t predicted by different submodels for a system with linear adsorption. Continuous line rigorous solution of transientwithboundaryconditioncM(/o + 0 = V7 dashedline rigorous(transient)...

---