Approximation techniques reaction curve

IV.44 Consider the feedback loop shown in Figure PIV.3a. Select the settings of the PI controller using the Cohen-Coon tuning technique. In a graph paper display the actual process reaction curve and its first-order plus dead time approximation. 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

The results discussed in this article were mostly obtained with ultrahigh vacuum systems at total pressures not exceeding 10"4 Torr, whereas real catalysis is performed in the atmospheric pressure regime. This general pressure gap raises the serious question to which extent experiments of the type described using the spectroscopic techniques of "surface science are relevant at all for real-life catalysis. A general answer to this problem can certainly not yet be offered. However, a rather favorable situation is found in the present case, as long as the discussion is confined to temperatures below 7 ax at which the reaction rate reaches is maximum rmax (cf., for example, Fig. 35). This situation has been discussed in detail in Section IV for palladium and holds as well for the other platinum metals since the shape of the r(T) curve is always quite similar. It has been shown that the kinetics may then approximately be described by... 

The equilibrium constants for a particular donor were determined by stirring appropriate quantities of the donor and BF3 for several hours in a suitable reaction vessel (24). Replicate aliquots of BF3 before and after equilibration were analyzed for boron-10 by means of a 6-inch, 60°-sector ratio mass spectrometer. In our experiments the amount of boron trifluoride in the gas phase was deliberately kept small, compared with the amount of boron trifluoride in the liquid phase. For this condition, the ratio of boron-10 to boron-11 in the gas before and after equilibration approximated the true single-stage fractionation factor, B/ B(liquid)/ B/ B(gas). When corrected for the BF3 present in the liquid phase in excess of the 1 1 mole ratio required by the molecular addition compound (Table II), these single-stage fractionation factors represented the isotopic equilibrium constants for Reaction 2. Equilibrium constants for the exchange of boron between BF3 gas and fifteen addition compounds of BF3 are shown in Table III. Curves of the form, log Keq = (b/T) — a, were fitted to the data by means of the least squares technique. From the slopes of these curves and the values of the isotopic... 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

---