Equal flux condition

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... 

Until now we have tacitly assumed that the boundary separates two identical media, for instance, hypolimnetic and epilimnetic water bodies, as in Illustrative Example 19.1. We can intuitively understand that (as stated by Eqs. 19-3 and 19-12) the net flux across the boundary is zero if the concentrations are equal on either side. Yet, how do we treat a boundary separating different phases, for instance, water and air Obviously, the equations have to be modified since we cannot just subtract two concentrations, CA and CB, which refer to different phases, for instance, mole per m3 of water and mole per m3 of air. In such a situation the equilibrium (no flux) condition between the two phases is not given by CA = CB. 

Fig. 1.18. Distribution of A and B particles on the surface in the annihilation reaction A + B —> 0. For clarity, the distributions of A s and B s have been separated and are shown in the left-hand column and in the right-hand column of the figure, respectively. The results shown correspond to constant and equal fluxes of A and B. The simulation were carried out on a 100 x 100 square lattice, (a) The A and B distribution are complementary. A narrow lane of empty sites separates between them, (b) The long-time (near steady-state) structure of the overlayer developing from the initial condition in (a), (c) The long-time overlayer pattern developing from an initially empty lattice.

Bader has shown that the topological partitioning of the molecules into atomic basins coincides with the requirements of formulating quantum mechanics for open systems [93], and in this way all the so-called theorems of quantum mechanics can be derived for an open system [94], Furthermore, the zero-flux condition, Eq. 1, turns out to be the necessary constraint for the application of Schwinger s principle of stationary action [95] to a part of a quantum system [93], The successful application of QTAIM to numerous chemical problems has thus deep physical roots since it is a theory which expands and generalises quantum mechanics themselves to include open and total systems, both treated on equal formal footing. 

Case (j8). Two distinguishable species A and B, characterized by equal V and occupying the same sites (equal 8) are present. No gradient in the sum of the concentrations exists, but we have opposite and numerically equal gradients in concentration for either species. There will be no net flow of particles, since (c + c ) is constant in space and time, but there can be opposite equal fluxes of both species. Under these conditions of equal mobility for both components, the rate of binary diffusion must be equal to the rate of self-diffusion for either pure component at the same total concentration. The average jumping frequency v of any particle would be... 

At the liquid-solid boundary no-penetration and no-slip boundary conditions are used for Navier-Stokes equations and continuity conditions for the temperature and heat fluxes. At the air-solid boundary the no-penetration condition is used for the diffusion equation and continuity for the temperature and the heat fluxes. Conditions of axial symmetry are applied at r=0. At the outer boundaries of the system vapour concentration and temperature are equal to their far-field values. The velocity component normal to the liquid-air boundary is negligible in comparison with its tangent component, caused by thermal Marangoni stress. Hence ... 

Equation (12.94a) is the initial condition—at the beginning of the computation (or experiment) the concentration in the whole domain is equal to the bulk concentration. Equation (12.94b) is the boundary condition at the electrode, where the electroactive species reacts instantly, so that the concentration is zero. Equations (12.94c) and (12.94d) are the no-flux conditions at the insulating plane surrounding the electrode. Equations (12.94e) and (12.94f) are symmetry conditions along the XZ and YZ plane, respectively. Finally, the last three equations are the far field conditions where C remains at its normalised bulk value. 

We now have two situations, with two different boundary conditions. If 2H < 6/V, then Ymax must be set equal to 2H, and we apply a no-flux condition to the channel roof. If however 2H > 6/ V, then we can apply the constant concentration (bulk value) to the level Y ax-... 

These boundary conditions wiU usually include an initial condition, a symmetry condition and a surface flux condition that will depend upon the surface exchange coefficient and a surface, a surface rate constant controlling the rate of exchange between the tracer in the solid and the tracer in the surrounding atmosphere. The tracer diffusion coefficient is related to the diffusion coefficient through a correlation factor or the Haven ratio (the tracer diffusion coefficient is equal to the diffusion coefficient divided by the Haven ratio). 

Equation (9.23) is to be compared with the Feng and Stewart relations (9.4) which describe the fluxes in the same system under non-reactive conditions, The factor (BA coth BA - 1) has the form sketched in Figure 9.2. From the definition of B given by equation (9.19) it is seen chat 9- 0 as and each tend to zero, their ratio remaining equal to Che... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

If a sedimentation experiment is carried out long enough, a state of equilibrium is eventually reached between sedimentation and diffusion. Under these conditions material will pass through a cross section perpendicular to the radius in both directions at equal rates downward owing to the centrifugal field, and upward owing to the concentration gradient. It is easy to write expressions for the two fluxes which describe this situation ... 

Under conditions of sedimentation equilibrium, the sum of these two fluxes equals zero, or... 

At heat fluxes greater than or equal to 300 kW/m2 it is recommended that caustic alkalinity be maintained at 10 to 15% of the TDS concentration, if phosphate conditioning is practiced. If carbonate conditioning is practiced, see note 11. 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

It is shown that the stability of the flow, with evaporating meniscus, depends (other conditions being equal) on the wall heat flux. The latter determines the rate of liquid evaporation, equilibrium acting forces, meniscus position, as well as the heat losses to the cooling inlet. The stable stationary flow with fixed meniscus position corresponds to low wall heat fluxes (Pe heat fluxes (Pe > 1) an exponential increase of small disturbances takes place. That leads to the transition from stable stationary to unstable flow with oscillating meniscus. 

Consider first the cases where the mixed-convection parameter, Gr/VR , equals 6.4. When the reactor walls are insulated, the susceptor heat flux is clearly not uniform, especially over the outer half of the disk. However, when the walls are cooled (all other conditions being the same) the heat flux is highly uniform and agrees with the one-dimensional result. Furthermore, with the cool walls, the mixed-convection parameter can even be increased (shown here... 

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. 

In an irreversible reaction that occurs under kinetic or mixed control, the boundary condition can be found from the requirement that the reactant diffusion flux to the electrode be equal to the rate at which the reactants are consumed in the electrochemical reaction ... 

As the electric field always points in the direction of the electrode, the densities of the electrons and negative ions are set equal to zero at the electrode. It is assumed that the ion flux at the electrodes has only a drift component, i.e., the density gradient is set equal to zero. The conditions in the sheath, which depend on pressure, voltage drop, and sheath thickness, are generally such that secondary electrons (created at the electrodes as a result of ion impact) will ionize at most a few molecules, so no ionization avalanches will occur. Therefore, secondary electrons can be neglected. 

Mass conservation at the rhizoplane means that the diffusive flux towards the root, Eq. (4), must equal the rate of extraction by the root, Eq. (9), leading to the boundary condition... 

Local quality method Dryout occurs when the local nonuniform heat flux equals the uniform heat flux dryout value at the same local conditions (quality, etc.). 

Since it is assumed that the only limiting resistance to moisture uptake is mass transport resistance, the basis for the model is contained with Eq. (39). It is assumed that the system is at steady state and that rectangular coordinates (uptake in one dimension) are appropriate. Since the system is at steady state and we are dealing with transport in one direction, the flux into a volume element must be equal to the flux out of that element. This condition is expressed as... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

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