Boundary conditions heterogeneous kinetics

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

The quantitative requirements as in the onedimensional case, are determined by the reaction kinetics, the physical state, and the equations of stat e of the material or of its components if the chge is heterogeneous. The shock- terminating rarefaction is here provided by the three- dimensional geometry and does not need a pressure-relieving rear boundary condition as in the one-dimensional case. If the shock wave is inadequate for detonation initiation, a deflagration frequently occurs instead. In Section VI,B of Ref 66, it was shown that for the correct boundary conditions a deflgrn can create a shock wave which can initiate a deton... 

The Surface Chemkin formalism [73] was developed to provide a general, flexible framework for describing complex reactions between gas-phase, surface, and bulk phase species. The range of kinetic and transport processes that can take place at a reactive surface are shown schematically in Fig. 11.1. Heterogeneous reactions are fundamental in describing mass and energy balances that form boundary conditions in reacting flow calculations. 

Wilson and Liu showed that both location and travel time probabilities can be calculated directly, using a backward-in-time version of traditional continuum advection-dispersion modeling. In addition, they claimed that by choosing the boundary conditions properly, the method can be readily generalized to include linear adsorption with kinetic effects and 1st order decay. An extension of their study for a 2D heterogeneous aquifer was reported in Liu and Wilson [39]. The results for travel time probability are in very close agreement with the simulation results from traditional forward-in-time methods. 

D Me-S surface alloy and/or 3D Me-S bulk alloy formation and dissolution (eq. (3.83)) is considered as either a heterogeneous chemical reaction (site exchange) or a mass transport process (solid state mutual diffusion of Me and S). In site exchange models, the usual rate equations for the kinetics of heterogeneous reactions of first order (with respect to the species Me in Meads and Me t-S>>) are applied. In solid state diffusion models, Pick s second law and defined boundary conditions must be solved using Laplace transformation. 

The value of iT,a gives the normalizing factor for currents I = i/iT,The general solution of equations involving heterogeneous kinetics with respect to a tip above a conducting substrate can be obtained under reasonable boundary conditions in the form of two-dimensional integral equations (see Chapter 5). 

For homogeneous reactions, the reaction kinetics enter into die balance equation while for heterogeneous inactions, die kinetics appear in die boundary conditions. The solution of Eq. (2.3-36) with the boundary conditions CA = CA. at z = 0 and CA CA2 at z = 5 provides an expression for the flux of A into ihe film at z = 5 ... 

This is a mathematical expression for the steady-state mass balance of component i at the boundary of the control volume (i.e., the catalytic surface) which states that the net rate of mass transfer away from the catalytic surface via diffusion (i.e., in the direction of n) is balanced by the net rate of production of component i due to multiple heterogeneous surface-catalyzed chemical reactions. The kinetic rate laws are typically written in terms of Hougen-Watson models based on Langmuir-Hinshelwood mechanisms. Hence, iR ,Hw is the Hougen-Watson rate law for the jth chemical reaction on the catalytic surface. Examples of Hougen-Watson models are discussed in Chapter 14. Both rate processes in the boundary conditions represent surface-related phenomena with units of moles per area per time. The dimensional scaling factor for diffusion in the boundary conditions is... 

The kinetic rate constant kj corresponds to the kinetics of heterogeneous surface-catalyzed chemical reactions in the boundary conditions, whereas the rate law is written on a pseudo-volumetric basis when chemical reaction terms are included in the mass transfer equation. 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Step 11. Write all the boundary conditions that are required to solve this boundary layer problem. It is important to remember that the rate of reactant transport by concentration difhision toward the catalytic surface is balanced by the rate of disappearance of A via first-order irreversible chemical kinetics (i.e., ksCpJ, where is the reaction velocity constant for the heterogeneous surface-catalyzed reaction. At very small distances from the inlet, the concentration of A is not very different from Cao at z = 0. If the mass transfer equation were written in terms of Ca, then the solution is trivial if the boundary conditions state that the molar density of reactant A is Cao at the inlet, the wall, and far from the wall if z is not too large. However, when the mass transfer equation is written in terms of Jas, the boundary condition at the catalytic surface can be characterized by constant flux at = 0 instead of, simply, constant composition. Furthermore, the constant flux boundary condition at the catalytic surface for small z is different from the values of Jas at the reactor inlet, and far from the wall. Hence, it is advantageous to rewrite the mass transfer equation in terms of diffusional flux away from the catalytic surface, Jas. 

Singular perturbation analysis was employed to study the velocity of pulled fronts, and it was shown that the solvability integrals diverge [103, 104, 448]. Therefore we will use this method only for non-KPP kinetics. We assume 5 = 0(e), weak heterogeneities, i.e., S = as in (6.51) with a = 0(1). Equation (6.51), together with the corresponding boundary conditions, becomes... 

Before we proceed further, we need to differentiate a homogeneous and a heterogeneous reaction system. We will do so based on the relative rates of mass transfer and reaction kinetics. A fast reaction can only take place at the interface therefore, the rate expression is expressed in the boundary condition. On the contrary, a slow reaction at rates comparable to that of diffusion resides mathematically in the continuity equation. In vector form, we express the continuity equations as follows ... 

To simulate positive feedback situation, one has to replace the no flux condition for boundary 3 representing the substrate surface with c=1. Minor modifications in the input file allow the simulation of the SECM responses for recessed or protruding tips, and finite heterogeneous kinetics at the tip and/or substrate. 

In order to consider chromatographic processes in a more universal manner as processes in which film control of heterogeneous mass-exchange is also possible, dimensionless criteria for the conditions of formation of sharp zone boundaries may be represented by the parameter A [124,125]. The evaluation of this parameter is carried out on the basis of dynamic (chromatographic) and kinetic experiments ... 

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