Example using forward-backward

In summary, using work collected from forward and backward paths greatly improves the accuracy of the estimates, and for the symmetric system studied here eliminates the bias. In our particular example, the cumulant estimators using forward and backward work data produce the most precise free energy estimates, followed by Bennett s optimal estimator. However, this somewhat poorer performance of the optimal estimator is caused in part by the high degree of symmetry of the system studied. 

In principle, we can use any combination of forward, backward and center finite differences to replace the first two differential terms of Eq. 10.72. However, our choice is limited for two reasons. First, we need to achieve numerical stability of the solution (see Section 10.3.5.1 below). For example, the stability analysis shows that the finite difference scheme which replaces the first term of Eq. 10.72 by a... 

For example, let us consider a two-component problem and use the forward-backward scheme to calculate numerical solutions, with a Courant number equal to fli for the first component. In linear chromatography, the contribution of the numerical dispersion for this first component would be exactly equivalent to the true HETP, Hi, if the space and time increments are chosen according to Eqs. 10.86 and 10.96... 

As previously discussed, some statisticians prefer to start with a larger model (backward elimination) and from that model, eliminate x, predictor variables that do not contribute significantly to the increase in SSr or decrease in SSg. Others prefer to build a model using forward selection. The strategy is up to the researcher. A general rule is that the lower-order exponents appear first in the model. This ensures that the higher-order variables are removed first if they do not contribute. For example,... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

In example 2 a simpler approach is used to correctly handle backward cycles (co-products). The difference to the forward cycle is that the co-product quant B created by a quant A cannot be used as predecessor of A, because cycles in the quant network are not allowed (violates the cause effect principle). A model can avoid this cycles using aggregation in such a way that cycles are within these quants A and B (see Figure 4.15). 

Note that in the mechanistic schemes presented, the dissolution steps of reactant and products have been omitted for the sake of brevity. These include, for example CO (g) <-> CO (1), C02 (1) <-> C02 (g), and H2 (1) <-> H2 (g). From the standpoint of thermodynamics, when the equilibrium lies far to the right, reactions are deemed to be irreversible and may be denoted with a forward arrow - symbol. In cases where the reaction is considered to be reversible (i.e., equilibrium lies somewhere in the middle), the forward and backward arrows (e.g., <-> ) are employed. In some cases, however, we do not specify reversible/irreversible steps, and therefore arrows (e.g., or <-> ) might be used in a general sense. From a kinetic standpoint, in some cases a step will be defined that is considerably slower than the others (i.e., the rate determining step) in those cases, the remaining steps may be considered to be pseudo-equilibrated. The reader must therefore use discretion in interpreting the mechanistic schemes. The context of the discussion should clue the reader into how to interpret the arrows. 

As shown in this chapter for the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaced by their finite-differenced equivalents. These finite-differ-enced forms of the model equations are shown to evolve as a natural consequence of the balance equations, according to Franks (1967), and as derived for the various examples in this book. The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end-sections can be better approximated by the forward and backward differences as derived in the previous examples. The various forms of approximation based on the use of central, forward and backward differences have been listed by Chu (1969). 

In the unit considered in Example 14.2A, the weak liquor is fed to effect 0 and flows on to 0 and then to . The steam is also fed to 0, and the process is known as forward-feed since the feed is to the same unit as the steam and travels down the unit in the same direction as the steam or vapour. It is possible, however, to introduce the weak liquor to effect and cause it to travel from to to , whilst the steam and vapour still travel in the direction of 0 to to . This system, shown in Figure 14.6, is known as backward-feed. A further arrangement for the feed is known as parallel-feed, which is shown in Figure 14.7. In this case, the liquor is fed to each of the three effects in parallel although the steam is fed only to the first effect. This arrangement is commonly used in the concentration of salt solutions, where the deposition of crystals makes it difficult to use... 

If one would be able to derive from the experimental data an accurate rate equation like (12) the number of terms in the denominator gives us the number of reactions involved in forward and backward direction that should be included in the scheme of reactions, including the reagents involved. The use of analytical expressions is limited to schemes of only two reaction steps. In a catalytic sequence usually more than two reactions occur. We can represent the kinetics by an analytical expression only, if a series of fast pre-equilibria occurs (as in the hydroformylation reaction, Chapter 9, or as in the Wacker reaction, Chapter 15) or else if the rate determining step occurs after the resting state of the catalyst, either immediately, or as the second one as shown in Figure 3.1. In the examples above we have seen that often the rate equation takes a simpler form and does not even show all substrates participating in the reaction. 

The lesser and greater functions are not time-ordered and arguments of the operators are not affected by time-ordering operator. Nevertheless we can write such functions in the same form (283). The trick is to use one time argument from the forward contour and the other from the backward contour, for example... 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

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