Problem solving, first-order

Equations (13), (14), and (15) can be used to solve three types of problems involving first-order processes. 

Lax-Wendroff. This is a well known method to solve first-order hyperbolic partial differential equations in boundary value problems. The two step Richtmeyer implementation of the explicit Lax-Wendroff differential scheme is used (8). 

The shifted position procedure discussed in Problem 12.7 is also useful when dealing with problems with radial coordinates, such as problems having cylindrical and spherical geometries. To show this, solve the following problem of first order chemical reaction in a spherical catalyst having the dimensionless form... 

Although the calculation of the Hessian is time consuming, the effort is quickly compensated by the excellent convergence properties of the Newton-Raphson approach [293-295]. This optimization technique solved the convergence problems of first-order MCSCF methods, which optimized orbitals and Cl coefficients in an alternating manner (recall chapter 9). Even perturbative improvements of the four-component CASSCF wave function are feasible and have been implemented and investigated [527]. 

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) ... 

Wlien working with any coordinate system other than Cartesians, it is necessary to transfonn finite displacements between Cartesian and internal coordinates. Transfomiation from Cartesians to internals is seldom a problem as the latter are usually geometrically defined. However, to transfonn a geometry displacement from internal coordinates to Cartesians usually requires the solution of a system of coupled nonlinear equations. These can be solved by iterating the first-order step [47]... 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

System hardware maintenance also presents some challenges. Trial personnel at the Central location first attempt resolution of hardware malfunctions. If they are unable to resolve the problem, personnel at the central location try to resolve the problem, and if that does not solve the problem a repair order is requested from the vendor as long as there is an active warranty. If the problem persists, the participating site is instructed to ship the device to the central location. If the device is the computer, the site coordinator (SC) is instructed to back up the database before shipping the computer. Personnel... 

Equation (11) is the integrated rate expression for a first-order process and can serve as a working equation for solving problems. It is also in the form of the equation of a straight line ... 

Since all the kinetic characteristics of the disappearance of a drug from plasma are the same as those for the pseudo-first-order disappearance of a substance from a solution by hydrolysis, the same working equations [Eqs. (11) and (13)] and the same approach to solving problems can be used. 

In Illustrations 8.3 and 8.6 we considered the reactor size requirements for the Diels-Alder reaction between 1,4-butadiene and methyl acrylate. For the conditions cited the reaction may be considered as a pseudo first-order reaction with 8a = 0. At a fraction conversion of 0.40 the required PFR volume was 33.5 m1 2 3, while the required CSTR volume was 43.7 m3. The ratio of these volumes is 1.30. From Figure 8.8 the ratio is seen to be identical with this value. Thus this figure or equation 8.3.14 can be used in solving a number of problems involving the... 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

In the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

Differential equations of the first order arise with application of the law of mass action under either steady or unsteady conditions, and second order with Fick s or Newton-Fourier laws. A particular problem may be represented by one equation or several that must be solved simultaneously. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

The first-order necessary conditions for this problem, Equations (8.9)-(8.11), consist of three equations in three unknowns (jq, x2, A). Solving (8.9)-(8.10) for jq and x2 gives... 

This problem may be solved in various ways. In particular the first-order change may be obtained easily from general principles (McWeeny, 1955) and yields... 

A third problem was that of fast coke. This coke also exhibited first-order intrinsic burning, but with a rate constant 17 times that of slow coke at 950°F (783 K). Again a fortunate simplification was found so that the kiln equations would not have to be solved for two kinds of coke burning... 

Let me make my own personal preference clear from the outset. I have solved literally hundreds of systems of ODEs for chemical engineering systems over my 30 years of experience, and t have found only one or two situations where the plain old simple-minded first-order Euler algorithm was not the best choice for the problem. We will show some comparisons of different types of algorithms on different problems in this chapter and the next. 

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