Boundary conditions surface flux

Neumann boundary condition specifies flux on the surface. The solution to the Laplace equation gives the potential distribution, and differentiation of the potential distribution with respect to the chosen direction (usually orthogonal to the electrode surface) provides the primary current distribution. 

A proper resolution of Che status of Che stoichiometric relations in the theory of steady states of catalyst pellets would be very desirable. Stewart s argument and the other fragmentary results presently available suggest they may always be satisfied for a single reaction when the boundary conditions correspond Co a uniform environment with no mass transfer resistance at the surface, regardless of the number of substances in Che mixture, the shape of the pellet, or the particular flux model used. However, this is no more than informed and perhaps wishful speculation. 

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

In the absence of convective effect, the profiles of > between any two adjacent bubbles exhibits an extremum value midway between the bubbles. Therefore, there exists around each bubble a surface on which d jdr = 3(C )/3r = 0, and hence the fluxes are zero. Using the cell model [Eqs. (158) or (172)] one obtains the following boundary conditions For t > 0... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

This PDE is subject to the initial condition that a = / at t = 0 and boundary conditions that a = a at a = 0 and a = ai at x = oo. The solution is differentiated to calculate the flux as in Equation (11.35). Unlike that result, however, the flux into the surface varies with the exposure time t, being high at first but gradually declining as the concentration gradient at a = 0 decreases. For short exposure times,... 

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

In the presence of surface deposition, the disk boundary condition becomes relatively complex. It states that the gas-phase mass flux of each species to the surface is balanced by the creation or depletion rate of that species by surface... 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

For the radical neutrals, boundary conditions are derived from diffusion theory [237, 238]. One-dimensional particle diffusion is considered in gas close to the surface at which radicals react (Figure 14). The particle fluxes in the two z-directions can be written as... 

The boundary conditions for this early dissolution model included saturated solubility for HA at the solid surface (Cha ) with sink conditions for both HA and A at the outer boundary of a stagnant film (Cha = Ca = 0). Since diffusion is the sole mechanism for mass transfer considered and the process occurs within a hypothesized stagnant film, these types of models are colloquially referred to as film models. Applying the simplifying assumption that the base concentration at the solid surface is negligible relative to the base concentration in the bulk solution (CB CB(o)), it is possible to derive a simplified scaled expression for the relative flux (N/N0) from HPWH s original expressions ... 

A paper by Ozturk, Palsson, and Dressman (OPD), reporting a refinement of the MMSH model, did create some controversy. OPD developed a film model with reaction in spherical coordinates and applied quasi-steady-state assumptions to the boundary conditions at the solid surface [11], They theorized that the flux of all species at the solid surface must be zero, except for HA, or the other species (A-, H+, OH ) would penetrate the solid surface. A debate by correspondence in the Letters to the Editor columns of Pharmaceutical Research ensued [12,13], The reader is invited to evaluate which author s arguments are more convincing. What is difficult to evaluate is whether the OPD model produces dissolution results which are different from those which would be predicted using the MMSH model cast in comparable spherical geometry. Simply, these authors never graphically demonstrate how their model predictions compare to the MMSH model. Algebraically, the solutions to both models appear comparable. 

These authors numerically solved the system of equations with appropriate boundary conditions to derive the time-averaged radiant and conductive heat fluxes between the fluidized bed and the heat transfer surface. Using... 

In the EIPET boundary condition, the electrochemical reduction of electro-active ions adsorbed on the particle provides the essential surface binding interaction which is responsible for particle deposition on the electrode surface. The particle flux is given by... 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

Obviously, many combinations arise different boundary conditions for different species on the same surface, different boundary conditions on different adjacent surfaces (mixed boundary conditions) for the same species, prescribed combinations of flux and concentrations at a certain surface, etc. 

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

Thus N dynamic equations are obtained for each component at each position, within each segment The equations for the first and last segment must be written according to the boundary conditions. The boundary conditions for this case correspond to the following the bulk tank concentration is S0 at the external surface of the biofilm where Z=0 a zero flux at the biofilm on the wall means that dS/dZ=0 at Z = L. 

Using symmetry arguments, the solution for diffusion with no flux at one end can be derived from these equations. Obviously, the concentration profile for zero surface concentration is symmetrical relative to X/2, which means that dC/dx is zero at that point the flux of diffusing substance through this point is zero. Other combinations of boundary conditions can be found in standard textbooks (Carslaw and Jaeger, 1959 Crank, 1976). 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

Figure 2. Description of the initial and boundary conditions for the hydrogen diffusion problem in the pipeline. The parameter / denotes hydrogen flux and C,(P) is normal interstitial lattice site hydrogen concentration at the inner wall-surface of the pipeline in equilibrium with the hydrogen gas pressure P as it increases to 15 MPa in 1 sec. At time zero, the material is hydrogen free,...

As in the full-field formulation, we assigned a zero flux boundary condition, i.e. j = 0 at the outer boundary of the domain as well as on the axis of symmetry ahead of the crack tip (Fig. 5b). Also, along the crack surface, we assumed the NILS hydrogen concentration CL to be in equilibrium... 

The uniformflux of oxygen, S, into the fluid along fhe Pt surface and zero flux along the Au surface provide boundary conditions for the convection-diffusion equation. 

At the electrode snrface, the fluxes of R and O produce current, whereas the flux of Y at the electrode surface is zero, due to its electrochemical inactivity. Hence, the following boundary conditions hold at the electrode surface ... 

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