In closed form

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE. 

Many distributions can be represented in closed form except for the Normal and Lognormal types. The CDF for these distributions can only be determined numerically. For example, the 3-parameter Weibull distribution s CDF is in closed form, where ... 

The benefit using the approach over other methods is that it can be used to solve the general reliability equation for combination of distribution, including when there are multiple load applications. This is only possible when the loading stress is described in closed form. 

Flanged Rectangular Opening TTie flow field for an infinitely flanged rectangular opening was first solved in closed form by Tyaglo and Shepelev. 

Furthermore, for any two-mirror telescope in which each mirror is individually corrected for spherical aberration (i,e. composed of two conics) one can solve for the third order aberrations in closed form. 

Differentiation and setting dbout/dT = 0 gives a transcendental equation in Toptimai that cannot be solved in closed form. The optimal temperature must be found numerically. 

For the case of a laser pulse having a square time profile with width T and constant power F, at the center of the Gaussian beam, the above integral can be evaluated in closed form. For r = 0 (the center of the laser beam) the temperature increase is given by ... 

Starting with an arbitrary set of poles, Hild used Brogan s method ( ) to determine the matrices K and P of Figure 6. The integrations of Equation 19 were performed in closed form on the linearized equations and a gradient search was conducted in "pole space" to minimize P. All poles were restricted to negative, real, and distinct values. 

From any real input, this provides an output in the range 0,1. The Tanh function (Figure 2.23) has a similar shape, but an output that covers the range -1, +1. Both functions are differentiable in closed form, which confers a speed advantage during training, a topic we turn to now. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

Here again, the functions Auv and Buv have to be computed numerically however this form is interesting because if we use the approximation (A.30), these integrals may be obtained in closed form and, for equal masses mv = ma, we easily get ... 

Because die outlet concentrations will not depend on it, micromixing between duid particles can be neglected. The reader can verify this statement by showing that die micromixing term in the poorly micromixed CSTR and the poorly micromixed PFR falls out when die mean outlet concentration is computed for a first-order chemical reaction. More generally, one can show that die chemical source term appears in closed form in die transport equation for die scalar means. 

Note that in order to close (1.16), the micromixing time must be related to the underlying flow field. Nevertheless, because the IEM model is formulated in aLagrangian framework, the chemical source term in (1.16) appears in closed form. This is not the case for the chemical source term in (1.17). 

The two terms on the right-hand side of this expression appear in closed form. However, the molecular transport term vV2 (Ut) is of order Re 1, and thus will be negligible at high Reynolds numbers. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

For the Kratzer potential, which is quadratic in the variable (r - re)/r, the eigenvalues for / 0 can be obtained in closed form. This is also the case for the potential quadratic in (r-r2/r) (Gol dman et al., 1960). This potential does not, however, tend to a finite value as r —> oo. 

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

The matrix elements of the operator (2.140) must, in general, be evaluated numerically.9 However, when N is large (a situation that is almost always encountered in actual spectra), the matrix elements of the exponential operator can be evaluated in closed form. Since the operator ft is a scalar, its matrix elements do not depend on J and one has... 

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