Solving the equation

We now have expressions for the two derivatives, which can be substituted into (11.15) to give 

The two undetermined constants c and k may now be chosen so that the equality holds for all values of-0. Equation (11.21) can be rewritten as 

In dimensionless terms, the wave travels with a steady velocity 

Returning to the original rate constants, diffusion coefficient, etc., this gives the speed as 

We may compare the speed of this wave, which will take approximately 6 s to move 1 cm, with the corresponding diffusion of B in the absence of reaction. In this latter case, the diffusion time is simply of the order r2/D, so for r = 1 cm we have tdiff ss 10s s. The diffusion-only wave would propagate more than four orders of magnitude slower than the reaction front. 

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The Fresnel equations predict that reflexion changes the polarization of light, measurement of which fonns the basis of ellipsometry [128]. Although more sensitive than SAR, it is not possible to solve the equations linking the measured parameters with n and d. in closed fonn, and hence they cannot be solved unambiguously, although their product yielding v (equation C2.14.48) appears to be robust. 

For a given potential energy function, one may take a variety of approaches to study the dynamics of macromolecules. The most exact and detailed information is provided by MD simulations in which one solves the equations of motion for the atoms constituting the macromolecule and any surrounding environment. With currently available techniques and methods it is possible... 

One may attempt to correct an integrator by an energy projection, i.e., in the general case of integrating some Hamiltonian H q,p), after a specified number I of steps, we would solve the equation... 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

We have used the shorthand notation for the integrals in the final expression. Note that the two-electron integrals may involve up to four different basis functions /x, v, A, a), which may in turn be located at four different centres. This has important consequences for the way in which we try to solve the equations. 

Gaussian elimination and test it by solving the equation set in Exercise 2-13. 

Some systems can give quantitative results from known pieces of data complete with proper units. For example, these systems can take all the starting information and then determine a set of equations from the available list that can yield the desired result. The program could subsequently convert units or algebraically solve the equations if necessary. 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Linear Equations A hnear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or X = —B/A. Linear equations in n variables have the form... 

Galerldn Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerldn finite element method an additional idea is introduced the Galerldn method is used to solve the equation. The Galerldn method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. 

Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix and its LU fac tors are... 

Various procedures for solving Eqs. (13-149) to (13-161), ranging from a complete tearing method to solve the equations one at a time, as shown by Distefano, to a complete simultaneous method, have been studied. Regardless of the method used, the following considerations generally apply ... 

Multistage CSTR This model has a particular importance because its RTD curve is beU-shaped hke those of many experimental RTDs of packed beds and some empty tubes. The RTD is found by induction by solving the equations of one stage, two stages, and so on, with the result. 

Note that Eq. (40d), in fact, is redundant, because the other three equations form a closed set. Nonetheless, if we solve the equations of motion for s as well, we can use the following, as a diagnostic tool, because this quantity has to be conserved during the simulation even though //nos6 is no longer a Hamiltonian ... 

A variety of techniques have been introduced to increase the time step in molecular dynamics simulations in an attempt to surmount the strict time step limits in MD simulations so that long time scale simulations can be routinely undertaken. One such technique is to solve the equations of motion in the internal degree of freedom, so that bond stretching and angle bending can be treated as rigid. This technique is discussed in Chapter 6 of this book. Herein, a brief overview is presented of two approaches, constrained dynamics and multiple time step dynamics. 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

In order to solve the equation for cr, it is only neeessary to find the partial derivative of the funetion with respeet to eaeh variable. This may be simple for some funetions, but is more diffieult the more eomplex the funetion beeomes and other teehniques may be more suitable. 

The data on blades in an axial-flow eompressor are from various types of easeades, sinee tlieoretieal solutions are very eomplex, and their aeeuraey is in question beeause of the many assumptions required to solve the equations. The most thorough and systematie easeade testing has been eondueted by NACA staff at the Lewis Researeh Center. The bulk of the easeade testing was earried out at low maeh numbers and at low turbulenee levels. 

Various theoretical and empirical models have been derived expressing either charge density or charging current in terms of flow characteristics such as pipe diameter d (m) and flow velocity v (m/s). Liquid dielectric and physical properties appear in more complex models. The application of theoretical models is often limited by the nonavailability or inaccuracy of parameters needed to solve the equations. Empirical models are adequate in most cases. For turbulent flow of nonconductive liquid through a given pipe under conditions where the residence time is long compared with the relaxation time, it is found that the volumetric charge density Qy attains a steady-state value which is directly proportional to flow velocity... 

The equations of state will not be further described or presented in more detail as tliey are unfortunately somewhat difficult to solve without the use of a computer. Full details are available in the referenced material for those wishing to pursue this subject further. In the past, these equations required the use of a mainframe computer not only to solve the equations themselves, but to store the great number of constants required. This has been true particularly if the gas mixture contains numerous components. With the power and storage capacity of personal computers increasing, the equations have the potential of becoming more readily available for general use... 

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

Step 8 Solve the Equations. Many material balances can be stated in terms of simple algebraic expressions. For complex processes, matrix-theory techniques and extensive computer calculations will be needed, especially if there are a large number of equations and parameters, and/or chemical reactions and phase changes involved. 

Selecting the most suitable numerical techniques to solve the equations. 

If in the analysis of a problem there is a set of simultaneous equations then the use of matrices can be a very convenient shorthand way of expressing and solving the equations. For example, consider the following set of equations ... 

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