Difference equations, first-order, solution

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

The solution for Ya is simple, even elegant, but what is the value of F It is equal to the mass holdup divided by the mass throughput. Equation (1.41), but there is no simple formula for the holdup when the density is variable. The same gas-phase reactor will give different conversions for A when the reactions are A 2B and A —> B, even though it is operated at the same temperature and pressure and the first-order rate constants are identical. 

The best solution to such numerical difficulties is to change methods. Integration in the reverse direction eliminates most of the difficulty. Go back to Equation (9.15). Continue to use a second-order, central difference approximation for d a/d, but now use a first-order, forward... 

Solution estimation for difference bormdary-value problems by the elimination method. In tackling the first boundary-value problem difference equation (21) has the tridiagonal matrix of order TV — 1... 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

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. 

The parameter y reflects the sensitivity of the chemical reaction rate to temperature variations. The parameter represents the ratio of the maximum temperature difference that can exist within the particle (equation 12.3.99) to the external surface temperature. For isothermal pellets, / may be regarded as zero (keff = oo). Weisz and Hicks (61) have summarized their numerical solutions for first-order irreversible... 

Analytical solution is possible only for first or zero order. Otherwise a numerical solution by finite differences, method of lines or finite elements is required. The analytical solution proceeds by the method of separation of variables which converts the PDE into one ODE with variables separable and the other a Bessel equation. The final solution is an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

The parameter [3 is related to the contrast. If (3A> > 1, equation 1 reduces to that of a simple first order reaction (such as CEL materials are usually assumed to follow (6)). If 3A< < 1, the reaction becomes second order in A In a similar manner, the sensitized reaction varies between zero order and first order. For the anthracene loadings required by the PIE process (13,15), A is close to 1M, so (3 > > 1 is required for first order unsensitized kinetics. Although in solution, 3 for DMA is -500, and -25 for DPA (20), we have found [3 =3 for DMA/PEMA, and (3=1 for DPA/PBMA. Thus although the chemical trends are in the same direction in the polymer as in solution, the numbers are quite different, indicating a substantial... 

The effective singles-and-doubles equation for the Cauchy vectors, equation (25), has the same structure as the effective singles-and-douhles equation for the first-order CC3 amplitudes (see equation (26) of Ref. [21]), the main difference being that now the right-hand side depends on the solution vector of lower order. Equation (25) may thus be implemented following the scheme described in Ref. [21] with an extra external loop over n (i.e., the Cauchy vectors order). 

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

The equations used in these models are primarily those described above. Mainly, the diffusion equation with reaction is used (e.g., eq 56). For the flooded-agglomerate models, diffusion across the electrolyte film is included, along with the use of equilibrium for the dissolved gas concentration in the electrolyte. These models were able to match the experimental findings such as the doubling of the Tafel slope due to mass-transport limitations. The equations are amenable to analytic solution mainly because of the assumption of first-order reaction with Tafel kinetics, which means that eq 13 and not eq 15 must be used for the kinetic expression. The different equations and limiting cases are described in the literature models as well as elsewhere. 

We now have to solve this equation for different values of Pe/,. Solutions are plotted in Figure 8-9 for p(t) with D i = 0 and the conversion versus position with a first-order reaction. 

There are two main ingredients that go into the semiclassical tiunover theory, which differ from the classical limit. In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a trarrsmission probabihty T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a qrrantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

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