Numerical solution of the equations

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations or continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conseiwation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

Inspection of the numerical solutions of the equations shows that, with the exception of Es= 0 kcal/mole, the rate of surface temperature increase with time is very large once the surface temperature reaches approximately 420°K—on the order of 108°K/sec. Because typical autoignition temperatures are of the order of 625°K for composite propellants, the particular value of the ignition temperature does not affect the computed numerical value of the ignition-delay time. 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

A simple example of a computational problem on the U(l) level is the numerical solution of the equation... 

A. Majda and J.A. Sethian. Derivation and Numerical Solution of the Equations of Low Mach Number Combustion. Combust. Sci. Techn., 42 185-205,1984. 

At first glance the appearing equations seem to be very complex. But the numerical solution of the equations is a process which can be done with a computer program. The analytical model offers several advantages compared to simulations. Since such a theoretical ansatz needs only a small amount of computing time, more complex systems can be studied. Moreover our models are not restricted to small lattices which are inavoidably used in computer... 

Note, also, that for any a there exists the trivial solution u/nn = 1 (plane front), which is, however, unstable.] Numerical solutions of the equations are illustrated in Fig. 4. 

A mathematical model of a plant or a section of a plant can be judged only by comparison with actual plant data. The model may be considered as good when the simulated variables can predict with some level of confidence the plant parameters which are important in determining the cost and quality of the finished product. Failures of the model are likely to be a result of (1) oversimplification of the equations that constitute the model, (2) inadequacy of the numerical solution of the equations. 

Numerical micromagnetics, which may be based either on the finite difference or finite element method, resolve the local arrangement of the magnetization which arises from the interaction between intrinsic magnetic properties such as the magnetocrystalline anisotropy and the physical and chemical microstructure of the material. The numerical solution of the equation of motion also provides information on how the magnetization evolves in time. The time and space resolution of numerical micromagnetic simulations is in the order of nanometers and nanoseconds, respectively. 

In order to reduce the complexity of the model two additional simplifying assumptions were made, (a) With typical residence times of 1 second, particle Reynolds numbers of 800 and tube-to-particle diameter ratios of 3, one would expect small values of the wall Biot number thus, a small number of radial finite difference (or collocation) points should be adequate for the numerical solution of the equations (8). (b) It was assumed... 

After the differential equations and boundary conditions have been formulated, there remain the problems of evaluating the various coefficients that appear, and carrying out the numerical solution of the equations. The following quantities are required. 

Outside the limited case of a first-order reaction, a numerical solution of the equation is required, and because this is a split-boundary-value problem, an iterative technique is required. 

The numerical solution of the equations does not in itself introduce a significant modeling error. 

Figure 10.12 Comparison of the numerical solutions of the equations of the equilibrium-dispersive model calculated with the forward-backward algorithm (dashed line) and the method of orthogonal collocation on finite elements (dotted line). 250 iL injection of a 15 g/L solution of (+)-Tr6ger s base. Column length, 25 cm column efficiency, Np = 146 plates F = 0.515 u = 0.076 cm/s. Isotherms in Figure 3.25...

A digital computer will facilitate the numerical solution of the equations derived in parts (a) and (h). 

The order of a method is the integer p such that the difference between the term on the right-hand side of Equation (2.104) and y is proportional to The solution of the difference equation converges to that of the differential equation if the difference between the two solutions vanishes as the step size, h, approaches zero. A difference scheme is said to be zero-stable if the numerical solution of the equation ior fix,y) = 0 is zero. 

A more complete treatment of the classical dynamics of a system containing an ion and a rotating polar molecule involves the numerical solution of the equation of motion (trajectory calculation) by Dugan et al. [64—68]. In their treatment, a capture collision is defined by an ion trajectory that penetrates to within a certain value of r. Their results also show that the locking in of the dipole is not likely to occur because of the conservation of angular momentum. 

The space- and time-dependent generalized Langevin equation (4) is a phenomenological equation. The exact numerical dynamics for any given simulation is not identical to the numerical solution of an STGLE. However, the numerical solution of the equations of motion of a system of hundreds or thousands of particles is in many senses a black box. One may get some numbers, but the dynamics is so complicated that there is very little useful additional information. On the other hand, because so much more is known about the solution of STGLEs it is very useful to try and map the complex dynamics onto an... 

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