Warped time

The uniformity lies in the fact that K is not a function of x and this allows the problem to be reduced to linearity by suitably warping time. Astarita12 shows that in this way any order of reaction may be achieved for the lump. Indeed Astarita and Aris13 showed that any kinetics could be imitated by choosing F and K adroitly. We shall return to uniform kinetics later, but, for the moment, will stick with the linear case. 

For representing reaction and diffusion in stretching lamellar structures, Ottino and Ranz (101) introduced a warped time tw defined by ... 

The assumption of uniformity is in fact justified for some realistic kinetic schemes, such as Langmuir isotherm catalyzed reactions, Michaelis-Menten kinetics, and others (Aris, 1989 Cicarelli et al, 1992). The assumption bears a more than superficial analogy with those systems termed pseudo-monomolecular by Wei and Prater (1962). Mathematically, it is a very powerful assumption By crossing out the dependence of F[ ] on x, its value has been reduced from an infinite-dimensional vector (a function of x) to a scalar. This simplification makes Eq. (102) a quasilinear one, and it can be integrated explicitly by introducing a warped time scale t(0. (0) = 0. The solution, as can be verified by inspection, is... 

Again, it is useful to define a warped time r(t), t(0) = 0, which is delivered by the following differential equation ... 

An important point about Eq. (128) is that it shows that, in the warped time scale T, the behavior is entirely independent of the form of the functional F[ ] the latter influences only the relationship between the warped time r and the actual time t. The case of first-order intrinsic kinetics (Aris and Gavalas, 1966) is recovered by simply setting t = r. 

Cicarelli et al. (1992) have developed the solution of Eq. (128) by a perturbation expansion, with e the perturbation parameter. They consider the special case of Langmuir isotherm kinetics, where F[ ] = 1/(1 -I- ). At the zero-order level, C = exp(r). This result simply reflects the fact that, in the distorted time scale r where the kinetics are linear, two species are formed from one at every reaction step, and hence the total concentration grows exponentially. This, however, does not include the fact that end products are being formed, and thus disappear from the spectrum of concentrations (at the zero-order level, e = 0 and no end products are formed). The critical warped time tq at which the zero-order approximation breaks down is estimated as —In e that is, it is well in excess of unity. Even for linear kinetics, there is an induction time significantly longer than the inverse of the kinetic constant during which very few end products are formed (this is even more true for nonlinear kinetics of the type considered). The solution can be obtained formally at all levels of perturbation the first-order level is of particular relevance because it yields (to within order e) the total amount of end products formed up to the critical time. 

A warped time technique is again useful. Let the warped dimensionless residence time W be defined as... 

The whole problem has thus been reduced to the solution of Eq. (134), which is a functional equation for the single scalar W. It is, however, a nasty functional equation, not so much for the possible nonlinearity of the functional F[ ] itself, but because the argument function is nonlinear in the unknown W. The warped time technique is again useful, but, contrary to what happens in the single-component case, the solution for the CSTR is more difficult than the one for the plug flow reactor or batch reactor. 

Another description of the dynamics of (2.87) can be extracted from the observation that there is a change of variables that eliminates the advection term in (2.87). This is done (Ranz, 1979), even for the case of a time-dependent A = A(t), by replacing x and t by new spatial and temporal coordinates and r (a stretched coordinate and a warped time) given by... 

Equation 4.2 describes the change in composition with time in the still. This equation can also be further modified by dividing it by DIH to obtain a differential equation in terms of a warped time variable dt = plH)d and letting (V/b) = r + 1, where r is the reflux ratio, resulting in Equation 4.3 ... 

Equation 8.8 may be further simplified in terms of a warped time variable, d = V/H)dt, such that... 

When a mixture of nc-components undergoes rirx simultaneous equilibrium chemical reactions, the RCM expression may be described in terms of transformed molar compositions and a reaction-warped time (Ung and Doherty, 1995a),... 

Another example is that of reaction and diffusion taking place in stretching aggregates. 6(a) being the striation thickness, decreasing with age, some authors [32] have introduced a warped" time t defined by... 

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