Boundary conditions zero-flux wall

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. 

During the MC simulation, boundary conditions must be applied at the edges of the flow domain. The four most common types are outflow, inflow, symmetry, and a zero-flux wall. At an outflow boundary, the mean velocity vector will point out of the flow domain. Thus, there will be a net motion of particles in adjacent grid cells across the outflow boundary. In the MC simulation, these particles are simply eliminated. By keeping track of the weights... 

The boundary conditions at the wall, on the other hand, influence the performance of the reactor critically, and should be determined as accurately as possible. For the equations of concentration (or conversion) the condition at the wall is that the flux of material normal to the wall is zero, which requires that the directional derivative of concentration normal to the wall be zero. For the tubular reactor, with cylindrical symmetry, the condition is expressed by the equation... 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

The heat flux boundary condition at the wall of the catalyst bed is used when heat is exchanged with an external charmel, where the heat transfer is described using a flux profile. This is the case for an adiabatic catalyst bed as in a prereformer or an autothermal reformer where the heat flux is zero so that the heat transfer equations simplify to a onedimensional model. The heat flux boundary condition is also used when the tubular reformer is coupled with a furnace type model (refer to Section 3.3.6). 

The concentration at the wall, a(7), is found by applying the zero flux boundary condition. Equation (8.14). A simple way is to set a(I) = a(I — 1) since this gives a zero first derivative. However, this approximation to a first derivative converges only 0(Ar) while all the other approximations converge O(Ar ). A better way is to use... 

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. 

The propagation rate is assumed to be second order with respect to the end-group concentration,. p = ka. The boundary conditions are a specified inlet concentration, zero flux at the wall, and symmetry at the centerline. 

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. 

The boundary conditions are otherwise zero flux at the walls and outflow conditions at the outlet(s). 

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,...

Modeling of the packed bed catalytic reactor under adiabatic operation simply involves a slight modification of the boundary conditions for the catalyst and gas energy balances. A zero flux condition is needed at the outer reactor wall and can be obtained by setting the outer wall heat transfer coefficients /iws and /iwg (or corresponding Biot numbers) equal to zero. Simulations under adiabatic operation do not significantly alter any of the conclusions presented throughout this work and are often used for verification... 

Fig. 10.2. A typical non-uniform stationary-state profile. Note the vanishing spatial derivative at the end walls (r = 0 and r = a0) appropriate to zero-flux boundary conditions.

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

The necessity to solve Laplace s equation requires formulating all boundary conditions, and at this point the cell geometry becomes important. Generally, there are two types of boundary conditions that come into play. Any electrically insulating cell wall is mathematically described by zero-flux or von Neumann boundary conditions ... 

Although there might be a granular temperature flux through the wall, little is known about the magnitude of such a flux. Hence, a simple zero gradient boundary condition is normally used ... 

This shows that the forward velocity is a parabolic function of position in the tube varying from zero at the wall where r is equal to Tq, to the maximum value at the center of the cylindrical tube, where is equal to —ro(dP/dx)/(4r ). It is emphasized that equation (6.3.4) is only applicable to laminar or non-turbulent flow in a cylindrical tube. When a liquid flows under different geometrical boundary conditions, the relationship between the flux and the force is not the same. 

The boundary conditions atz = 0 are also unchanged and are given by (6 124b). Finally, because of the presence of the impermeable end walls, the zero-flux condition, (6 125), is preserved, though now in the slightly more general form... 

In general, boundary conditions are difficult to specify and oftentimes difficult to incorporate into the numerical scheme. Typical boundary conditions used are given in [92, 97, 129, 130, 136]. Boundary conditions for the mass continuity Eq. (22) specify a zero electron density at the wall, or an electron flux equal to the local thermal flux multiplied by an electron reflection coefficient. The ion diffusion flux is set to... 

In order to be consistent with the Bohm criterion for ions, the sheath edge is defined as the point where the ions have been accelerated (presumably by the presheath electric field. Fig. 5) to the Bohm velocity, i.e. the presheath is included as part of the bulk plasma. The Bohm flux also provides a boundary condition (applied at the wall because of the thinness of the sheath) for the positive ion continuity equation. The negative ion density is assumed zero at the walls. 

Hong and Bergles [284] have analyzed the thermal entrance solution of heat transfer for a circular segment duct with 20 = 180° (i.e., a semicircular duct). Two kinds of thermal boundary conditions are used (1) a constant wall heat flux along the axial flow direction with a constant wall temperature along the duct circumference, and (2) a constant wall heat flux along the axial flow direction and a constant wall temperature along the semicircular arc, with zero heat... 

Boundary conditions involve the imposition of inlet profiles, the Neumann (3.40) condition of zero gradient on outflows (3.41), the reactive flux on active surfaces (3.43, and the zero flux on inert walls (3.42), as shown in Figure 3.6. 

Since there was no current flow in the refractory walls, the magnetic vector potential. A, was set equal to zero. For heat transfer, a constant heat flux boundary condition, equal to the measured heat loss flux, was specified for this wall. The heat loss fluxes at the side wall for the water-cooled copper panels in Ae bullion and slag were 31.3 kW/m and 1.75 kW/m respectively. The conventional non-slip boundary condition was used for momentum transfer. 

For the mass transfer equation a boundary condition was applied to satisfy the flux continuity at the interface, while zero flux was set at the channel wall. Periodic boundary conditions are used at the front and the back of the computational domain for the velocity, the pressure, and the concentration (see Fig. 7.1). 

The above equation is subject to the zero flux boundary conditions, n.Vn, = 0 at both solid walls and reservoirs. As one can see that the Nemst-Planck equation is coupled with the Poisson equation and the N-S equation, which are very difficult to solve simultaneously. Another difficulty arises when all the species of... 

The principal dimensions of the reactor are shown in Figure 6.1. The numerical values used here are D = 0.48 m //(= 1.02 m H, = 0.34 0 = 0.33 m s = 0.01 m r = 0.044125 m d = 0.0625 m d = 0.04 m and q = 0.07 m. Non-slip boundary conditions are assumed on the vessel wall. Both radial and axial velocities are set to zero on the shaft and impeller disk and the angular velocity is determined by the speed of rotation. On the free surface of the liquid, the axial component of velocity is zero with the other two components of velocity being stress free. Along the central line, below the impeller, the axial component of velocity is stress free and the other two components are zero. The temperature of the jacket at the vessel walls is fixed at 10 °C. Heat is lost by convection and at the free surface and there is an axis of symmetry along the centreline with no flux at the shaft and impeller boundaries. The flow is... 

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