Kinetics Rosen model

In order to develop an intuition for the theory of flames it is helpful to be able to obtain analytical solutions to the flame equations. With such solutions, it is possible to show trends in the behavior of flame velocity and the profiles when activation energy, flame temperature, diffusion coefficients, or other parameters are varied. This is possible if one simplifies the kinetics so that an exact solution of the equation is obtained or if an approximate solution to the complete equations is determined. In recent years Boys and Corner (B4), Adams (Al), Wilde (W5), von K rman and Penner (V3), Spalding (S4), Hirschfelder (H2), de Sendagorta (Dl), and Rosen (Rl) have developed methods for approximating the solution to a single reaction flame. The approximations are usually based on the simplification of the set of two equations [(4) and (5)] into one equation by setting all of the diffusion coefficients equal to X/cpp. In this model, Xi becomes a linear function of temperature (the constant enthalpy case), and the following equation is obtained ... 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

Pellet [35] and Rasmuson and Neretnieks [36] extended the solution of Rosen by including axial dispersion, but still assuming that the kinetics of adsorption-desorption is infinitely fast. Later, Rasmuson [37] extended the earUer solution and calculated the profile of a breakthrough curve (step bormdary condition, or frontal analysis) in the framework of the general rate model (Eqs. 6.58 to 6.64a), which includes axial dispersion, the film mass transfer resistance, the pore diffusion, and a first-order slow kinetics of adsorption-desorption. 

AR Walmsley, T Zhou, MI Borges-Wahnsley, BP Rosen. A kinetic model for the action of a resistance efflux pump. J Biol Chem 276 6378-6391, 2001. 

We turn to the more complicated but important problem of ionic surfactant adsorption, and start with the salt-free case where strong electrostatic interactions are present. In Fig. 3 we have reproduced experimental results published by Bonfillon-Colin et al. for SDS solutions with (open circles) and without (full circles) added salt [13]. The salt-free ionic case exhibits a much longer process with a peculiar intermediate plateau. Similar results were presented by Hua and Rosen for DESS solutions [21]. A few theoretical models were suggested for the problem of ionic surfactant adsorption [22-24], yet none of them could produce such dynamic surface tension curves. It is also rather clear that a theoretical scheme such as the one discussed in the previous section cannot fit these experimental results. On the other hand, addition of salt to the solution screens the electrostatic interactions and leads to a behavior very similar to the non-ionic one. We shall return to this issue in Section 4. We thus infer that strong electrostatic interactions affect drastically the adsorption kinetics. Let us now study this effect in more detail. We follow the same lines presented in the previous section while adding appropriate terms to account for the additional interactions. 

Concerning eventual interfacial processes, there is an abundance of literature. Various techniques have been used to characterize interfaces/ interphases (Schradder and Block, 1971 Di Benedetto and Scola, 1980 Ishida and Koenig, 1980 Rosen and Goddard, 1980 Ishida, 1984 Di Benedetto and Lex, 1989 Thomason, 1990 Hoh et al, 1990 Schutte et al, 1994). Round-robin tests showed that no analytical method is able to provide unquestionable results (Pitkethly et al, 1993). Even in cases where the interface response to humid ageing has been unambiguously identified from studies on model systems (Kaelble et al., 1975, 1976 Salmon et al, 1997), it seems difficult, at this stage, to build a non-empirical kinetic model of the water effects on interfaces/interphases in composites. 

Rosen [40] is a pioneer in the field of two-phase emulsion polymerization mechanisms and kinetics. He studied the influence of the polymer phase separation on the efficiency of grafting reactions and demonstrated the physical limits of these grafting reactions. This was followed by the work of Chiu [41] with an attempt to develop a mechanistic model that can be used to predict the experimental kinetic data obtained from two-phase emulsion polymerizations that was not successful. 

Chelatable lead is widely accepted as representing removal from soft tissue (e.g., Chisolm and Barltrop, 1979), but some mobilizable compartment for lead storage in bone must also be providing a sizeable contribution. Evidence for the bone source includes (1) the age dependency of chelatable lead in non-occupationally exposed subjects, whereas lead in soft tissue is rather invariant with age (Araki, 1973 Araki and Ushio, 1982) (2) experimental animal (Hammond, 1971, 1973) and in vitro bone culture data (Rosen and Markowitz, 1980) showing removal of lead from bone and (3) the tracer modelling data of Rabinowitz et al (1977), which define a bone compartment for lead which is kinetically well mixed with those for blood and soft tissue. 

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