Multi-mode models

In this section, we present the low-dimensional multi-mode models for different types of non-isothermal homogeneous reactors. Flere, we skip the details of the averaging process and summarize the results. 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

The most important observation that follows from the above analysis of the multi-mode model is that in almost all practical cases, tubular reactor instabilities arise due to mixing/diffusional limitations at the small scales and spread over the reactor. In contrast, pseudohomogeneous models predict (erroneously) that reactor instabilities (ignition, multiple solutions, etc.) arise due to macromixing limitations at the reactor scale. 

The multi-mode model for the above reaction scheme in an isothermal tank reactor is given by... 

In this section, we consider briefly the accuracy and convergence aspects of the multi-mode models derived by the L-S method. We also illustrate the regularization procedure used for the local equation(s) to increase the region of convergence of the multi-mode models. 

Most commercial ovens, whether single-mode or multi-mode models, are protected from potential damage caused by reflected energy by an automatic cut-off mechanism also, some ovens provide additional protection in the form of a circulator that directs reflected energy into a dummy load. A circulator is essentially a device that uses ferrites and static magnetic fields to allow microwaves to pass in the forward direction but diverts reflected waves into a dummy load where the energy is harmlessly dissipated as heat. 

Fig. 4 Power spectrum of light sources expressed by a multi-mode model...

The basic equation used is the same as eq. (3), but with different expression of E(t). The complex amplitude of the light field in a multi-mode model as shown in Fig. 4 is expressed as... 

J. Azaiez, R. Guenette, A. Ait-Kadi, Journal of Non-Newtonian Fluid Mechanics, Entry flow calculations using multi-mode models, 66 (271) (1996)... 

In addition to packed catalyst bed, a fluidized bed irradiated by single and multi-mode microwave field, respectively, was also modeled by Roussy et al. [120]. It was proved that the equality of solid and gas temperatures could be accepted in the stationary state and during cooling in a single-mode system. The single-mode cavity eliminates the influence of particle movements on the electric field distribution. When the bed was irradiated in the multimode cavity, the model has failed. Never-... 

Initial indications are that a multi-mode version of Eqs. (50) and (51) can be useful at a quantitative level in modelling an extensively-investigated LDPE for all geometries of flow [82]. Matching only the linear viscoelastic relaxation spectrum, and adding a segment priority distribution consistent with that derived... 

Fig.20a,b. Transient data in a extension b shear on LDPE from [81] modelled with a multi-mode version of the pom-pom constitutive equation. The linear modes (of weights and relaxation times T, ) have been decorated with the structure of pom-pom elements (adding values of and q to each mode)... 

Another approach consisting of the Liapounov-Schmidt reduction coupled with a perturbation argument is developed in the articles Balakotaiah and Chang (2003), Balakotaiah (2004) and Chakraborty and Balakotaiah (2005). It allows developing multi-mode hyperbolic upscaled models. 

The frequency (o and gate pulse time-delay td dependent fluorescence up-conversion signal F(cojd) have been theoretically modeled by using an overdamped multi-mode Brownian oscillator model [5], The deconvoluted fluorescence spectrum is given by... 

Time-resolved fluorescence of coumarin C522 was determined in water and in host-guest complex with p-cyclodextrin, representing free aqueous and cavity restricted environments, respectively. Experimental fluorescence clearly showed faster dynamics in a case of water. The time parameters of monoexponential fit for water and p-cyclodextrin at 500 nm and 520 nm were determined to be 1.37 ps and 2.02 ps, and 2.97 ps and 7.14 ps, respectively. Multi-mode Brownian oscillator model, as an attempt to simulate the solvation dynamics, supported these fluorescence dynamics results. 

Roussy, G., Jassm, S. and Thiebaut, J.M., Modeling of a fluidized-bed irradiated by a single or a multi-mode electric microwave field distribution, /. Microw. Power Electromagn. Energy, 1995, 30, 178. 

Apart from the heat bath mode, the harmonic potential surface model has been used for the molecular vibrations. It is possible to include the generalized harmonic potential surfaces, i.e., displaced-distorted-rotated surfaces. In this case, the mode coupling can be treated within this model. Beyond the generalized harmonic potential surface model, there is no systematic approach in constructing the generalized (multi-mode coupled) master equation that can be numerically solved. The first step to attack this problem would start with anharmonicity corrections to the harmonic potential surface model. Since anharmonicity has been recognized as an important mechanism in the vibrational dynamics in the electronically excited states, urgent realization of this work is needed. 

As demonstrated by the power spectra in Figs. 12.2a and 12.3b, regulation of the blood flow to the individual nephron involves several oscillatory modes. The two dominating time scales are associated with the period Tsiow 30—40 s of the slow TGF-mediated oscillations and the somewhat shorter time scale Tjast 5—10 s defined by the myogenic oscillations of the afferent arteriolar diameter. The two modes interact because they both involve activation of smooth muscle cells in the arteriolar wall. Our model describes these mechanisms and the coupling between the two modes, and it also reproduces the observed multi-mode dynamics. We can, therefore, use the model to examine some of the phenomena that can be expected to arise from the interaction between the two modes. 

Palese S, Mukamel S, Miller RJD, Lotshaw WT. Interrogation of vibrational structure and line broadening of liquid water by Raman-induced Kerr effect measurements within the multi-mode Brownian oscillator model. I Phys Chem 1996 100 10380-10388. 

B. Multi-mode Hyperbolic Averaged Models for Describing Dispersion Effects in Chromatographs 233... 

IV. Spatially Averaged Multi-mode (Multi-scale) Models for... 

F. Examples Illustrating Use of Multi-mode Homogeneous Reactor Models 260... 

In this section, we present examples to illustrate the usefulness of multi-mode homogeneous reactor models in predicting micromixing effects on yield and selectivity, reactor runaway, etc. 

VI. Accuracy, Convergence and Region of Validity of Multi-mode/ Multi-scale Averaged Models... 

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