Elimination of time as an independent

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. 

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

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. 

The elimination of diffusion effects or their treatment as an independent correction factor will be described in Section IV, and it will therefore be sufficient to use the expression (4) as a starting point. For those reactions involving pressure change, this expression must be translatable in terms of a variation of gas pressure with time. For this purpose the reaction to be expected needs to be known, for then dn/dt will be expressible... 

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 ... 

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. 

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. 

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). 

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