Closed form solutions

What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visuaHzable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

The presence of D g 26 governing differential equation and the boundary conditions renders a closed-form solution impossible. That is, in analogy to both bending and buckling of a symmetric angle-ply (or anisotropic) plate, the variation in lateral displacement, 5vy, cannot be separated into a function of x alone times a function of y alone. Again, however, the Rayleigh-Ritz approach is quite useful. The expression... 

Although a closed-form solution can thus be obtained by this method for any system of first-order equations, the result is often too cumbersome to lead to estimates of the rate constants from concentration-time data. However, the reverse calculation is always possible that is, with numerical values of the rate constants, the concentration—time curve can be calculated. This provides the basis for a curve-... 

Li s algorithm for generating closed form solutions for the power spectra of CA attractors consists essentially of five main parts ([li87], [li89b]) ... 

While a valid and useful answer to the first question can often be found, there is at least one significant drawback to this approach so many simplifying assumptions must usually be made about the real system, in order to render the top-level problem a soluble one, that other natural, follow-up questions such as "Why do specific behaviors arise or How would the behavior change if the system were defined a bit differently cannot be meaningfully addressed without first altering the set of assumptions. An analytical, closed-form solution may describe a behavior, however, it does not necessarily provide an explanation for that behavior. Indeed, subsequent questions about the behavior of the system must usually be treated as separate problems. 

Many reaction schemes with one or more intermediates have no closed-form solution for concentrations as a function of time. The best approach is to solve these differential equations numerically. The user specifies the reaction scheme, the initial concentrations, and the rate constants. The output consists of concentration-time values. The values calculated for a given model can be compared with the experimental data, and the rate constants or the model revised as needed. Methods to obtain numerical solutions will be given in the last section of this chapter. 

In earlier treatments, we have dealt with kinetic schemes with closed-form solutions that is, with solutions that could be realized by approximations such as steady state or prior equilibrium. 

Schemes II and III can be solved only if [I] can be approximated at the steady-state value. If that approximation is not valid, then neither [A], nor [P], has a closed-form solution. Schemes II and III have a fixed stoichiometry, this being 2A = P for Scheme II and A + B = P + Q for Scheme III. Scheme I, on the other hand, has a variable stoichiometry, intermediate between the extremes A = P (when it fe tB]) and A + B = Q (when k k2[B]).

Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

The modern discipline of Materials Science and Engineering can be described as a search for experimental and theoretical relations between a material s processing, its resulting microstructure, and the properties arising from that microstructure. These relations are often complicated, and it is usually difficult to obtain closed-form solutions for them. For that reason, it is often attractive to supplement experimental work in this area with numerical simulations. During the past several years, we have developed a general finite element computer model which is able to capture the essential aspects of a variety of nonisothermal and reactive polymer processing operations. This "flow code" has been Implemented on a number of computer systems of various sizes, and a PC-compatible version is available on request. This paper is intended to outline the fundamentals which underlie this code, and to present some simple but illustrative examples of its use. 

Later, an analytical closed-form solution for was derived [26] by treating the density change as a small perturbation and assuming parabolic wing shape. Numerical studies with detailed reaction mechanisms [33,34] demonstrated that the enhancement of can be primarily attributed to the flow redirection effect, and the contributions of the preferential diffusion and/or strain were <15%. 

Magnetic dipole interaction Hm (4.47) and electric quadmpole interaction //q (4.29) both depend on the magnetic quantum numbers of the nuclear spin. Therefore, their combined Hamiltonian may be difficult to evaluate. There are closed-form solutions of the problem [64], but relatively simple expressions exist only for a few special cases [65]. In Sect. 4.5.1 it will be shown which kind of information can be obtained from a perturbation treatment if one interaction of the two is much weaker than the other and will be shown below. In general, however, if the interactions are of the same order of magnitude, eQV Jl, and... 

It is a common experience in science when we try to solve a new problem to find that an exact solution proves elusive. If no closed-form solution to the problem seems to be available, some process of iterative improvement may be needed to move from the best approximate solution to one of acceptable quality. With persistence and perhaps a little luck, a suitable solution will emerge through this process, although many cycles of refinement may be required (Figure 5.1). 

The form of the solutions that one obtains for reactions of this type is dictated by the relative values of the four reaction orders. For some choices of these orders it is not possible to obtain simple closed form solutions to these rate equations unless one places additional restrictions on the initial composition of the reaction mixture. The three possible-cases are discussed below. 

Only in the case where A0 = B0 is it possible to obtain a closed form solution to this... 

In this case + nx differs from m2 + n2 and there are a variety of possible forms that the rate expression may take. We will consider only some of the more interesting forms. In this case elimination of time as an independent variable leads to the same general result as in the previous case (equation 5.2.50). As before, in order to obtain a closed form solution to this equation, it is convenient to restrict our consideration to a system in which A0 = B0. In this specific case equation 5.2.50 becomes... 

A closed form solution to this equation exists for all values of + nf) — (m2 + n2)]. However, the resultant function will depend on this difference. One case that occurs often is that in which the difference is unity. 

In this subsection we have treated a variety of higher-order simple parallel reactions. Only by the proper choice of initial conditions is it possible to obtain closed form solutions for some of the types of reaction rate expressions one is likely to encounter in engineering practice. Consequently, in efforts to determine the kinetic parameters characteristic of such systems, one should carefully choose the experimental conditions so as to ensure that potential simplifications will actually occur. These simplifications may arise from the use of stoichiometric ratios of reactants or from the degeneration of reaction orders arising from the use of a vast excess of one reactant. Such planning is particularly important in the early stages of the research when one has minimum knowledge of the system under study. 

For first-order irreversible reactions and identical space times it is possible to obtain closed form solutions to differential equations of the form of 8.3.61. In other cases it is usually necessary to solve the corresponding difference equations numerically. 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

In this case a closed form solution is possible with... 

For isothermal systems, it is occasionally possible to eliminate the external surface concentrations between equations 12.6.1 and 12.6.2 and arrive at a global rate expression involving only bulk fluid compositions (e.g., equation 12.4.28 was derived in this manlier). In general, however, closed form solutions cannot be achieved and an iterative trial and error procedure must be employed to determine thq global rate. One possible approach is summarized below. 

G.A. Swartz, Closed-Form Solution of I-V Characteristic for a-Si H Solar Cells Isamu Shimizu, Electrophotography Sachio Ishioka, Image Pickup Tubes... 

We consider two cases, one with a higher Peclet number than the other. Disper-sivity tt[, in the first case is set to 0.03 m in the second, it is 3 m. In both cases, the diffusion coefficient D is 10-6 cm2 s-1. Since Pe L/oti., the two cases on the scale of the aquifer correspond to Peclet numbers of 33 000 and 330. We could evaluate the model numerically, but Javandel el al. (1984) provide a closed form solution to Equation 20.25 that lets us calculate the solute distribution in the aquifer... 

Closed form solutions are not possible or are quite complex algebraically in most other cases. 

For multiple reactions, material balances must be made for each stoichiometry. An example is the consecutive reactions, A = B = C, for which problem P4.04.52 develops a closed form solution. Other cases of sets of first order reactions are solvable by Laplace Transform, and of course numerically. 

In the classical literature analysis the system equations were manipulated to eliminate V y, the velocity of the solid bed consumption in the thickness direction (y direction), from the analysis by using the assumption that the solid bed reorganizes. This allowed a straightforward differential analysis and a closed form solution in the cross-channel x direction for solids melting. In this analysis, the y-direction velocity V y is retained as a variable because this facilitates the calculation of the change in bed thickness, which was found to be very important in the reevaluation of the literature data, as shown in Figs. 6.9 and 6.10. 

The Gaussian plume foimulations, however, use closed-form solutions of the turbulent version of Equation 5-1 subject to simplifying assumptions. Although these are not treated further here, their description is included for comparative purposes. The assumptions are reflection of species off the ground (that is, zero flux at the ground), constant value of vertical diffusion coefficient, and large distance from the source compared with lateral dimensions. This Gaussian solution to Equation 5-1 is obtained under the assumption that chemical transformation source and sink terms are all zero. In some cases, an exponential decay factor is applied for reactions that obey first-order kinetics. A typical solution (with the time-decay factor) is ... 

Due to the complex mixed-mode nature of composite delamination, no closed form solutions have been developed yet to express the influence of governing parameters that control the edge delamination behavior. Under tensile loading, delamination is normally preceded by a number of transverse cracks, particularly in the 90° plies. Because of the presence of these cracks, the location of delamination is not unique as in the case of compressive loading, which invariably results in gross buckling of the laminate. The path of delamination along the axial direction varies... 

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