Kinetic-transport equations, coupled

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

The goal of this chapter is to describe the application of hydrodynamic electrodes to the study of electrode kinetics and the kinetics of electrode and coupled homogeneous reactions. In order to do this, it is important to describe first the mass transport and how to fulfil experimentally the conditions described by the mass transport equations, i.e. electrode construction and operation. 

Extension of the equilibrium model to column or field conditions requires coupling the ion-exchange equations with the transport equations for the 5 aqueous species (Eq. 1). To accomplish this coupling, we have adopted the split-operator approach (e.g., Miller and Rabideau, 1993), which provides considerable flexibility in adjusting the sorption submodel. In addition to the above conceptual model, we are pursuing more complex formulations that couple cation exchange with pore diffusion, surface diffusion, or combined pore/surface diffusion (e.g., Robinson et al., 1994 DePaoli and Perona, 1996 Ma et al., 1996). However, the currently available data are inadequate to parameterize such models, and the need for a kinetic formulation for the low-flow conditions expected for sorbing barriers has not been established. These issues will be addressed in a future publication. 

The permeate is continuously withdrawn through the membrane from the feed sueam. The fluid velocity, pressure and species concentrations on both sides of the membrane and permeate flux are made complex by the reaction and the suction of the permeate stream and all of them depend on the position, design configurations and operating conditions in the membrane reactor. In other words, the Navier-Stokes equations, the convective diffusion equations of species and the reaction kinetics equations are coupled. The transport equations are usually coupled through the concentration-dependent membrane flux and species concentration gradients at the membrane wall. As shown in Chapter 10, for all the available membrane reactor models, the hydrodynamics is assumed to follow prescribed velocity and sometimes pressure drop equations. This makes the species transport and kinetics equations decoupled and renders the solution of... 

Reactive solute-transport models couple the equations that describe physical transport processes with equations that describe geochemical reactions. These models can be divided into three basic categories (i) equilibrium models, (ii) partial equilibrium models, and (iii) kinetic models. The three are differentiated by the... 

The rate functions in Table 2-2 can be applied to batch kinetic data directly because no transport (flow) processes occur. In flow methods the appropriate transport equation must be coupled to the kinetic rate function to achieve a correct solution. Kinetic-rate functions cannot be applied directly. For the thin-disk method the transport equation (Skopp and McCallister, 1986) is... 

Apart from the coupling of chemical kinetics to the transport equations, the chemical-reaction dynamics itself may pose numerical challenges when a number of different reactions are superposed. In such a case the rate of disappearance of a chemical species i can be written as... 

For reactive multi-component transport, the main problem is the coupling between components through the reactive term. Indeed the reaction kinetics depends on the concentrations of several components, which makes necessary to consider large system of coupled transport equations. For the case of an irreversible reaction, the situation is simplified because the reaction kinetics depends only on reagents and does not depend on the reaction products. Then it is sufficient to formulate the transport equations only for hydrogen and CC>2. 

Determination of the electrochemical performance of the pore requires the relation between the faradaic current density at the interface, y>, and . In order to establish this relation, one needs the following (i) an electric double layer model to relate to OM, (ii) transport equations to relate om to reactant concentrations, and (iii) charge transfer kinetic equations to relate local reactant concentrations and potentials to y>. The continuum model developed by Chan and Eikerling (2011) consists of coupled relations for reactant transport, metal surface charge, and charge transfer kinetics. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

Salt flux across a membrane is due to effects coupled to water transport, usually negligible, and diffusion across the membrane. Eq. (22-60) describes the basic diffusion equation for solute passage. It is independent of pressure, so as AP — AH 0, rejection 0. This important factor is due to the kinetic nature of the separation. Salt passage through the membrane is concentration dependent. Water passage is dependent on P — H. Therefore, when the membrane is operating near the osmotic pressure of the feed, the salt passage is not diluted by much permeate water. 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

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