Surface Species Governing Equations

The time-rate-of-change of surface species k due to heterogeneous reaction is given by Eq. 11.102. As discussed above, the effects of surface chemistry must be accounted for as boundary conditions on gas-phase species through flux-matching conditions such as Eq. 11.123. For a transient simulation, a differential equation for the site fraction Zk of surface species k can be written 

Equation 11.131 assumes that the total surface site density T is constant. Section 17.7.1 uses the equation above in the formulation of a transient stagnation flow problem, for example. 

In a steady-state calculation, it must be the case that the surface species concentrations (or site fractions) are not changing with time, that is, 

At steady state the surface species concentations have to adjust themselves consistent with the adjacent gas-phase species concentrations such that the condition sk = 0 is satisfied. In a steady-state reacting flow simulation, such as discussed in Sections 6.2 and 7.7, the surface-species governing equations are taken to be 

Note that Eq. 11.133 is imposed for one fewer than the total number of surface species (i.e., for Ks - 1 species). A normalization condition, Eq. 11.134, is used for one of the surface species (arbitrarily the last surface species, k = Ks) to make the system of equations well posed. 

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Effectiveness factor approach (rj-approach) accounts for diffusion limitations in the washcoat. rj-approach is based on the assumption that one target species determines overall reactivity (Deutschmann, 2008). An effectiveness factor for a first-order reaction is calculated for the chosen species based on the dimensionless Thiele modulus (< ) (Hayes et al., 2007, 2012), and all reaction rates are multipfied by this factor at the species governing equation at the gas-surface interface. is calculated as... 

If the reaction is pseudo-first-order with respect to the gaseous reactant (i.e., if there is liquid reactant taking part in the reaction, its concentration is in excess and uniform) and if the reaction occurs only at the catalyst surface, the governing dimensionless material-balance equations for the reacting species in three phases are given as10... 

For an ideal-gas mixture with a constant average molecular weight, the term (dh/dYi)p p y. j i appearing in equation (55) is simply the (total) specific enthalpy of species i [that is, the quantity hi given in equation (1 -11)], and the dimensionless parameter Gi becomes Gi = hi/(Cp fT) thus the term Yj=x is a measure of the rate of liberation of heat by the chemical reaction. Equation (54) is a useful replacement for equation (45) in studies of the characteristic surfaces of the governing equations. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

While the gas concentration distribution in the gas charmel is obtained by the solution of the Navier-Stokes equation along with the governing equation for mass species transport, the overall resistance for convective mass transport and convection mass transfer rate from bulk gas stream to the adjacent electrode surface is often given by the convection mass transfer coefficient. For such a case, the convective mass transfer rate equation over a surface of area A can be written as... 

In this review, wherever electrochemistry is concerned, the reversibility of a reaction refers firstly to the chemical reversibility. It also requires that the electron transfer reaction occurs at such a rate that the rate of the whole electrodic process, which is measured by the output current of the electrode, is controlled by the diffusion of the redox species towards the electrode surface. Furthermore, the surface concentrations of O and R at a given potential should be governed by the Nemst equation. 

In order to derive mathematical equations able to describe the movement of a species towards or from an electrode surface it would be necessary to know the physical laws which govern the three modes of the mass transport. 

These equations are applicable for a reaction proceeding under pseudo-first-order conditions, i.e. when the concentration of the solute species is constant right up to the gas/liquid interface. It is thus possible to examine the possibility that reaction may occur in a film for the catalyst reoxidation and reduction reactions separately, if the two-stage redox mechanism is appropriate. The penetration theory leads to a series of coupled nonlinear partial differential equations which have to be solved numerically with appropriate boundary conditions. For example, if y is the distance from the melt surface, the equation governing the concentration of species B in time and space is given in (15). 

The dependence of the rate of elementary electron transfer reactions on the applied potential, , is governed by the Butler-Volmer equation. For irreversibly adsorbed redox active species, this rate can be expressed, without loss of generality, in terms of the surface concentration of the reduced form of the species, Tred, as follows ... 

K inches of surface chemical reactions are governed by the elementary reactions llial constitute the reaction mechanism. In accordance with the principle of mass in lion, rates are proportional to surface concentrations (or surface densities) of piulicipating species surface sites, specifically adsorbed solutes, and non-npiTilically adsorbed solutes. The site stoichiometry model presented above is nih il used to express rate equations for surface chemical reactions. 

Equation (2.3-55) is in the form of a rate being governed by two resistances in series—diffusion and chemical reaction. If I k SIOAB (fast surface reaction), die rale is governed by diffusion, while if Ilk 6/Dar (slow reaction rate), the rate is governed by cheraical kinetics. This additivity of resistances is only obtained when linear expressions relate rates and driving forces and wonld not be obtained, for example, if Ihe surface reaction kinetics were second order. More complex kinatic situations can be analyzed in a similar fashion where reaction stoichiometry at the surface provides information on (be flux ratio of various species. 

The equations governing the voltammetric method (e.g., assuming only species O is present initially) include the same ones as used previously, namely the mass-transfer equations [such as (5.4.2)] and the initial and semi-infinite conditions (5.4.3) and (5.4.4). However, the flux condition at the electrode surface is different, because the net reaction involves the electrolysis of diffusing O as well as O adsorbed on the electrode, to produce R that diffuses away and R that remains adsorbed. The general flux equation is then... 

Once a model has been selected, an additional conservation or continuity equation Is required to describe the transient species of Interest. The eddy coefficients that govern the transport of the transient have been taken to be the same as those for heat transport [see e.g., ( ]. Thus, the temporal development of the transient can be modeled along with the development and diurnal variation of the stability of the water column. In this manner, the effects of wlnd-stlrrlng and surface heat flux as well as of photochemical processes on the vertical profile of a transient can be Investigated. 

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