Flux operator

Ultrafiltration equipment suppHers derive K empirically for their equipment on specific process fluids. Flux J is plotted versus log for a set of operation conditions in Figure 6 K is the slope, and is found by extrapolating to zero flux. Operating at different hydrodynamic conditions yields differently sloped curves through C. ... 

A consistent quantal TST (QTST) has been worked out by Miller and coworkers [Miller 1974 Miller et al. 1983 Tromp and Miller 1986 Voth et al. 1989a]. In quantum mechanics the classical flux X is replaced by the symmetrized flux operator... 

It is readily checked that the matrix elements of the flux operator between two states are... 

In these boilers, various interrelated, complex surface chemistry reactions may occur at the metal-water interface, which (apart from the development of a desirable protective magnetite film) can lead to the formation of unwanted deposits. These surface reactions are influenced by the specific heat flux, operating temperatures, and the areas and degree of local metal stress resulting within a particular boiler. 

This use of soda ash as a reactant for FW contaminants is much rarer today than, say, 25 or 30 years ago simply because the steam-water space design and high heat-flux operation of modem boilers cannot tolerate the volumes of carbonate and hydroxide sludges produced. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

This derived expression satisfies conditions a-d mentioned above and based on numerical computatiotf 6-2 seems to bound the exact result from above. It is similar but not identical to Wigner s original guess. The quantum phase space function which appears in Eq. 52 is that of the symmetrized thermal flux operator, instead of the quantum density. 

Using the fact that the symmetrized flux operator commutes with the density matrix, and representing the latter as exp(-/3//) = exp[-(/3 -... 

Figure 7.17 Experiments showing the rate of fouling of 0.22-p.m microfiltration membranes used to treat dilute biomass solutions. The membranes were operated at the fluxes shown, by increasing transmembrane pressure over time to maintain this flux as the membranes fouled [12]. Reprinted from J. Membr. Sci. 209, B.D. Cho and A.G. Fane, Fouling Transients in Nominally Sub-critical Flux Operation of a Membrane Bioreactor, p. 391, Copyright 2002, with permission from Elsevier...

An important recent theoretical development is the direct approaches for calculating rate constants. These approaches express the rate constant in terms of a so-called flux operator and bypass the necessity for calculating the complete state-to-state reaction probabilities or cross-sections prior to the evaluation of the rate constant [1-3]. This is the theme of this chapter. 

After having described the expression for the rate constant within the framework of classical mechanics, we turn now to the quantum mechanical version. We consider first the definition of a flux operator in quantum mechanics.2 To that end, the flux density operator (for a single particle of mass to) is defined by... 

We consider the trace at t = 0, Tr[Fe /2kBT9 P)e l2kBT] = Tr[PA], which defines the operator A. It is noted that both the flux operator F and the A operator are Hermitian operators, which implies that Tr[Tk4] = Tr[(PA)t] = Tr pt] = Tr[AP] = Tr[PA], that is, the trace must always be real valued. Furthermore, if the trace is evaluated in a basis of real-valued functions, then the trace must be equal to zero, since the operators contain the momentum operator, which also contains the imaginary unit i = T. Using this result, Eq. (5.108) can be written in the form... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

---