A Closed Form Solution

A priori bounds for the state variables can be derived by choosing ij such that ViVj 0 and writing Eq.(2.2.16) as 

With the exception of linear equations, closed form solutions can be obtained only in the case of a single reaction in a planar region. An example is provided by the irreversible reaction 

By eliminating / from the above equations and integrating, there is obtained 

Amundson and Raymond [2] have investigated the number and stability of solutions of the problem (2.3.11)—(2.3.13). Following their procedure we set il/ rj)=drj/dC and use ly as a new independent variable and as a new dependent variable and obtain from Eq. (2.3.11) 

The above substitution is valid under the condition that 0( ) has a single zero, therefore applies to irreversible as well as reversible reactions. 

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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-... 

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]).

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 case a closed form solution is possible with... 

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... 

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. 

For problems with relatively simple boundary and initial conditions, solutions can probably be found in a library. However, it can be difficult to find a closed-form solution for problems with highly specific and complicated boundary conditions. In such cases, numerical methods could be employed. For simple boundary conditions, solutions to the diffusion equation in the form of Eq. 4.18 have a few standard forms, which may be summarized briefly. 

For arbitrary surface heat fluxes the problem must be solved numerically. However, if the desired quantity is the surface heat flux necessary to produce a specified free-boundary motion, Eqs. (120)-(124) become linear. By a judicious choice of X a closed-form solution can be obtained. This procedure was previously followed by Baer and Ambrosio (Bl) for the semi-infinite slab. With the choice... 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

The development of an insensitive controller can of course be accomplished by repetitive simulations, but this by itself is an inefficient and usually impractical approach. The design of such a controller using standard linear optimal control methods has not proven to be fruitful as yet, since inclusion of sensitivity measures in the performance index does not yield to a closed form solution (57), (58). There is a need for improved methods which can realize desired sensitivity characteristics as well as high performance without resorting to extensive interactive calculations Davison (59) has recently suggested one such approach. 

We can obtain a closed-form solution for the bound part of the problem by substitut-, ing Eq. (10.6) into Eq. (10.3a), giving a first-order integro-differential equation for... 

The reader is asked to confirm that all degrees of freedom are fulfilled and the model can be solved. The steady-state model has a closed-form solution. Here we only present the Da - zAi2 dependence in Eq. (4.32) and Figure 4.6(b) ... 

Fick s second law is a partial differential equation that defines the change in concentration within a phase due to the process of molecular diffusion. Fick s second law can be solved numerically, or it can be directly solved to obtain a closed form solution for simplified boundary and initial conditions. 

The model of polarizable dipolar chromophores suggests that the 3D nuclear reaction field of the solvent serves as a driving force for electronic transitions. Even in the case of an isotropic solute polarizability, two projections of the reaction field should be included the longitudinal (parallel to the difference solute dipole) component and the transverse (perpendicular to the difference dipole) component. The 8 function in Eq. [18] eliminates integration over only one of these two field component. The integral still can be taken analytically resulting in a closed-form solution for the Franck-Condon factor... 

In the first two cases, setting up and solving the differential equation was straightforward. In case 2 a closed form solution could not be obtained, so the equation was solved numerically. 

In Eq (7), the bridging force, Pi, is a function of the flexural displacement z(xi) which is given by the bridging law. Thus, it is mathematically difficult to obtain a closed-form solution of Eq. (7). Instead, an iteration method is used to obtain a numerical solution. In the iterative calculation, we add the displacement, 5, and the crack length, h, step by step by a tiny increment. In the first step, we give a tiny increase in the crack length. 

As was the case with the free-draining model, the relaxation spectrum of the first, collective modes is flat, whereas it is unchanged with respect to the unperturbed state for more localized modes (see Figure 6). With the open chain, analytical difficulties prevent us from obtaining a closed-form solution. Numerical calculations show that the same results hold for the open chain... 

Lapidus and Amundson [85] showed that, in the case of a linear isotherm, it is possible to derive a closed-form solution to the system of partial differential equations combining the mass balance equation and a first-order mass transfer kinetic equation. This solution is valid only for analytical applications of chromatography and carmot be extended to nonlinear isotherms. 

The second consequence of the assmnption of a linear isotherm is to make simple the mathematics of describing the migration of these independent, individual bands and of calculating their retention times and profiles. As we show later in this chapter, an analytical solution or, at least, a closed-form solution in the Laplace domain can be obtained with any model of linear chromatography. This is certainly not the case in nonlinear chromatography. 

A closed-form solution of Eq. 6.24 has been derived by Lapidus and Amundson [3], Levenspiel and Smith [17], Carberry and Bretton [18], Reilley et al. [19], and Wicke [20]. All these authors used an "open-open" boundary condition, i.e., conditions assuming that the column stretches to infinity in both directions (z —00, dC/dz = 0 z oo, dC/dz = 0), and that a Dirac 6 z) pulse of solute is injected at z = 0. With these boundary conditions, the solution of Eq. 6.24 is given by... 

Consider the series reaction scheme modeled in example 2.2.1. Redo this problem if the first reaction is first order and the second reaction is second order (Rice and Do, 1995).[1] Obtain a closed form solution and plot the concentration profiles for typical values of rate constants. 

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