Transfer rates

However, a note of caution should be added. In many multiphase reaction systems, rates of mass transfer between different phases can be just as important or more important than reaction kinetics in determining the reactor volume. Mass transfer rates are generally higher in gas-phase than liquid-phase systems. In such situations, it is not so easy to judge whether gas or liquid phase is preferred. 

In addition to the advantage of high heat transfer rates, fluidized beds are also useful in situations where catalyst particles need frequent regeneration. Under these circumstances, particles can be removed continuously from the bed, regenerated, and recycled back to the bed. In exothermic reactions, the recycling of catalyst can be... 

At low currents, the rate of change of die electrode potential with current is associated with the limiting rate of electron transfer across the phase boundary between the electronically conducting electrode and the ionically conducting solution, and is temied the electron transfer overpotential. The electron transfer rate at a given overpotential has been found to depend on the nature of the species participating in the reaction, and the properties of the electrolyte and the electrode itself (such as, for example, the chemical nature of the metal). 

At higher current densities, the primary electron transfer rate is usually no longer limiting instead, limitations arise tluough the slow transport of reactants from the solution to the electrode surface or, conversely, the slow transport of the product away from the electrode (diffusion overpotential) or tluough the inability of chemical reactions coupled to the electron transfer step to keep pace (reaction overpotential). 

Ladanyi B M and Hynes J T 1986 Transition state solvent effects on atom transfer rates in solution J. Am. Ohem. Soc. 108 585-93... 

After some straightforward manipulations of A3.8.22. the PI-QTST estimate of the proton transfer rate constant can be shown to be given by 48... 

Chemical reactions can be studied at the single-molecule level by measuring the fluorescence lifetime of an excited state that can undergo reaction in competition with fluorescence. Reactions involving electron transfer (section C3.2) are among the most accessible via such teclmiques, and are particularly attractive candidates for study as a means of testing relationships between charge-transfer optical spectra and electron-transfer rates. If the physical parameters that detennine the reaction probability, such as overlap between the donor and acceptor orbitals. 

Figure C1.5.12.(A) Fluorescence decay of a single molecule of cresyl violet on an indium tin oxide (ITO) surface measured by time-correlated single photon counting. The solid line is tire fitted decay, a single exponential of 480 5 ps convolved witli tire instmment response function of 160 ps fwiim. The decay, which is considerably faster tlian tire natural fluorescence lifetime of cresyl violet, is due to electron transfer from tire excited cresyl violet (D ) to tire conduction band or energetically accessible surface electronic states of ITO. (B) Distribution of lifetimes for 40 different single molecules showing a broad distribution of electron transfer rates. Reprinted witli pennission from Lu andXie [1381. Copyright 1997 American Chemical Society.

Figure C3.2.10.(a) Dependence of electron transfer rate upon reaction free energy for ET between biphenyl radical anions and various organic acceptors. Experiments were perfonned with the donors and acceptors frozen into...

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

Early studies showed tliat tire rates of ET are limited by solvation rates for certain barrierless electron transfer reactions. However, more recent studies showed tliat electron-transfer rates can far exceed tire rates of diffusional solvation, which indicate critical roles for intramolecular (high frequency) vibrational mode couplings and inertial solvation. The interiDlay between inter- and intramolecular degrees of freedom is particularly significant in tire Marcus inverted regime [45] (figure C3.2.12)). 

Beratan D N, Betts J N and Onuchic J N 1991 Protein electron transfer rates set by the bridging secondary and tertiary structure Science 252 1285-8... 

Spears K G, Wen X and Zhang R 1996 Electron transfer rates from vibrational quantum states J. Phys. Chem. 100 10 206-9... 

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

A transfer rate constant can be obtained by applying a Boltzmann distribution, and by writing the concentration of reactant present as... 

Mavri, J., Berendsen, H.J.C. Calculation of the proton transfer rate using density matrix evolution and molecular dynamics simulations Inclusion of the proton excited states. J. Phys. Chem. 99 (1995) 12711-12717. 

Van der Spoel,D., Berendsen, H.J.C. Determination of proton transfer rate constants using ab initio, molecular dynamics and density matrix evolution calculations. Pacific Symposium on Biocomputing, World Scientific, Singapore (1996) 1-14. 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

These are the flux relations associated with the dusty gas model. As explained above, they would be expected to predict only the diffusive contributions to the flux vectors, so they should be compared with equations (2.25) obtained from simple momentum transfer arguments. Equations (3,16) are then seen to be just the obvious vector generalization of the scalar equations (2.25), so the dusty gas model provides justification for the simple procedure of adding momentum transfer rates. 

Oxygen solubility Oxygen tents Oxygen transfer Oxygen transfer rate Oxygen transport Oxyhalide Oxyhemoglobin... 

Mass transfer rates may also be expressed in terms of an overall gas-phase driving force by defining a hypothetical equiHbrium mole fractionjy as the concentration which would be in equiHbrium with the bulk Hquid concentration = rax ) ... 

Design Procedure. The packed height of the tower required to reduce the concentration of the solute in the gas stream from to acceptable residual level ofjy 2 may be calculated by combining point values of the mass transfer rate and a differential material balance for the absorbed component. Referring to a sHce dh of the absorber (Fig. 5),... 

The term dqljdt represents the overall rate of mass transfer for component / (at time t and distance averaged over a particle. This is governed by a mass transfer rate expression which may be thought of as a general functional relationship of the form... 

This rate equation must satisfy the boundary conditions imposed by the equiUbrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to 2ero at equiUbrium. It maybe a linear driving force expression of the form... 

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

Design Methods. Improvements ia the ability to predict multicomponent equilibrium and mass-transfer rate performance will allow significant improvements ia the design of new adsorption systems and ia the energy efficiency of existing systems. 

The search for a suitable adsorbent is generally the first step in the development of an adsorption process. A practical adsorbent has four primary requirements selectivity, capacity, mass transfer rate, and long-term stabiUty. The requirement for adequate adsorptive capacity restricts the choice of adsorbents to microporous soUds with pore diameters ranging from a few tenths to a few tens of nanometers. 

In addition to the fundamental parameters of selectivity, capacity, and mass-transfer rate, other more practical factors, namely, pressure drop characteristics and adsorbent life, play an important part in the commercial viabiUty of a practical adsorbent. 

Temperature and pressure are not considered as primary operating variables temperature is set sufficiendy high to achieve rapid mass-transfer rates, and pressure is sufficiendy high to avoid vaporization. In Hquid-phase operation, as contrasted to vapor-phase operation, the required bed temperature bears no relation to the boiling range of the feed, an advantage when heat-sensitive stocks are being treated. 

From equation 23, it can be seen that the higher the power input per unit volume, the lower the oxygen transfer efficiency. Therefore, devices should be compared at equal transfer rates. AH devices become less energy efficient as rates of transfer increase (3). 

Aerator type Materials of constmction Oxygen transfer efficiency, OTE, % Oxygen transfer rate, OTR, g/(W-h)... 

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