Elimination of time as an independent variable

Elimination of Time as an Independent Variable. Let us assume that we have the following scheme  

Note that the third equation is not independent of the first two it is related by the stoichiometry of the over-all reaction. We shall thus work with the first two equations. If we differentiate equation 2 of the set (III.7A.2) again and combine with equations 1 and 2 of the same set to eliminate A, we obtain a nonlinear, differential equation 

By using the condition that B = 0 when A = A this is reduced to 

The right-hand side of Eq. (III.7A.8) is always negative because 

1 — A/Ao, which represents the fraction of A used up, is always positive. Thus the left-hand side is negative also, which implies that B/X is always less than 1. Of the terms on the left-hand side, the logarithmic term is always negative and always the larger of the two terms in absolute magnitude. Further, we can show that, when A is used up (that is, A = 0), there will always be some B left. The amount of B left will depend both on the values of K and Ao. 

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Elimination of time as an independent variable between equations 5.2.19 and 5.2.21 gives... 

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... 

One general technique that is often useful in efforts to analyze the behavior of these systems is the elimination of time as an independent variable. 

Citral has three sites of hydrogenation and as shown in (Fig. 1) the mechanism for its reduction is complex. However we have shown previously (ref. 3) that, after the elimination of time as an independent variable, the LANGMUIR-HINSELWOOD rate equation for each component in the liquid phase can be integrated. The integrated equations B = f( A ), C = g( 8 ), i01 = h ( B ), E = i ( B ) and F = j( 8 ) depend on six ratios k g/k, k /k, kgb g/k b. ... k bp/k b which have been computed simultaneously. The agrement between the calculated product compositions as a function of hydrogen consumed and the experimental data is excellent as illustrated in (Fig. 2). Therefore the computed ratio k o /kjb for each step has been used as selectivity criteria. 

Case I The Orders with Respect to Each of the Reactants Are Equal (/M2 = / i and 2 = i). In this case elimination of time as an independent variable gives... 

This method is always applicable when the original equations are of such form that eliminating time as an independent variable leads to a differential equation in which the variables are separable. The following represent two further examples of its use and results. 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

The proper design of commercial pyrolysis reactors requires a suitable expression for the intrinsic rate of the reactions. As intrinsic rate equations cannot yet be predicted, especially for the ultrapyrolysis regime, experimental data is required. This data is best obtained from bench-scale laboratory reactors, rather than from pilot plants or commercial-scale units. In laboratory scale pyrolysis reactors, the design and operating conditions can be chosen to reduce or eliminate the effects of mass and heat transfer, contaminants and catalytic surfaces from the observed measurements, thus allowing for the development of accurate expressions. It is most advantageous if the laboratory reactor is operated isothermally (in space and time), so that the temperature can be considered as an independent variable. Also, the pressure should be ideally kept constant. 

The second type of problem does not involve reaction time or space time. The question here is given a system of reactions with known kinetics and given the concentration of one component at some (unspecified) time (or space time), what are the concentrations of the other species at that time (or space time) This type of problem is referred to as a time-independent problem. Time-independent problems can be solved by forming the ratio of various reaction rates to eliminate time (or space time) as an explicit variable. However, the solution to such problems provides no information about the time or reactor volume required to obtain a given composition. 

For purposes of reactor design, the distinction between a single reaction and multiple reactions is made in terms of the number of extents of reaction necessary to describe the kinetic behavior of the system, the former requiring only one reaction progress variable. Because the presence of multiple reactions makes it impossible to characterize the product distribution in terms of a unique fraction conversion, we will find it most convenient to work in terms of species concentrations. Division of one rate expression by another will permit us to eliminate time as an explicit independent variable, thereby obtaining expressions that are convenient for examining the effects of changes in process variables on the product distribution. 

What is the most meaningful way to express the controllable or independent variables For example, should current density and time be taken as the experimental variables, or are time and the product of current density and time the real variables affecting response Judicious selection of the independent variables often reduces or eliminates interactions between variables, thereby leading to a simpler experiment and analysis. Also inter-relationships among variables need be recognized. For example, in an atomic absorption analysis, there are four possible variables air-flow rate, fuel-flow rate, gas-flow rate, and air/fuel ratio, but there are really only two independent variables. 

Pseudo-first-order rate constants for carbonylation of [MeIr(CO)2l3]" were obtained from the exponential decay of its high frequency y(CO) band. In PhCl, the reaction rate was found to be independent of CO pressure above a threshold of ca. 3.5 bar. Variable temperature kinetic data (80-122 °C) gave activation parameters AH 152 (+6) kj mol and AS 82 (+17) J mol K The acceleration on addition of methanol is dramatic (e. g. by an estimated factor of 10 at 33 °C for 1% MeOH) and the activation parameters (AH 33 ( 2) kJ mol" and AS -197 (+8) J mol" K at 25% MeOH) are very different. Added iodide salts cause substantial inhibition and the results are interpreted in terms of the mechanism shown in Scheme 3.6 where the alcohol aids dissociation of iodide from [MeIr(CO)2l3] . This enables coordination of CO to give the tricarbonyl, [MeIr(CO)3l2] which undergoes more facile methyl migration (see below). The behavior of the model reaction closely resembles the kinetics of the catalytic carbonylation system. Similar promotion by methanol has also been observed by HP IR for carbonylation of [MeIr(CO)2Cl3] [99]. In the same study it was reported that [MeIr(CO)2Cl3]" reductively eliminates MeCl ca. 30 times slower than elimination of Mel from [MeIr(CO)2l3] (at 93-132 °C in PhCl). 

In this brief chapter we hope we have been able to establish in the reader s mind that the coloring of plastics materials is not a simple process. However, we would like the reader to know that it is also not an impossible problem. If one takes a sound scientific approach to variables analysis as it relates to color, for the most part the difficulties can be eliminated. As you have seen, there are many variables that must be contended with and these variables do not always act independent of each other. This means we need to define, understand, and control as many variables as possible. We suggest you start with the simplistic first theorem, which states The most likely reason that your new computer is not working is you don t have it plugged in (actual data from computer support companies). Start with the simple and work to the complex it save lots of time and is good, sound scientific thinking. Below are some simple questions to help you remember the basic variables that most often cause color problems. It is by no means all inclusive, for there are times when the solutions are complicated, but this is usually the exception and not the rule. 

As seen above, laser assisted and controlled photofragmentation dynamics can conceptually be viewed in two different ways. The time-dependent viewpoint offers a realistic time-resolved dynamical picture of the basic processes that are driven by an intense, short laser pulse. For pulses characterized by a long duration (as compared to the timescales of the dynamics), the laser field can be considered periodic, allowing the (quasi-) complete elimination of the time variable through the Floquet formalism, giving rise to a time-independent viewpoint. This formalism not only offers a useful and important interpretative tool in terms of the stationary field... 

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