Boundary Conditions for Concentration

A zero net flux is used at all at symmetric surfaces. This condition is given as 

At the channel and gas diffusion layer interfaces, a continuity condihon is applied  

The mass flux discontinuity condition is used at the electrode-membrane interface 

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A double trial-and-error procedure is needed to determine uq and Tq. If done only once, this is probably best done by hand. This is the approach used in the sample program. Simultaneous satisfaction of the boundary conditions for concentration and temperature was aided by using an output response that combined the two errors. If repeated evaluations are necessary, a two-dimensional Newton s method can be used. Dehne... 

Before we turn to particular instances of system (3.1.1), (3.1.2) let us make a few general observations. First, note that the equilibrium in (3.1.1), (3.1.2) is linearly stable and is approached monotonically in time. Indeed, consider (3.1.1), (3.1.2) on the segment —L < x < L. Assume N = const. The boundary conditions for concentrations compatible with equilibrium are... 

The boundary conditions for concentration reflect the asymptotic approach to the fluid composition at the inlet and the permeability properties of both walls... 

Consider diffusion with a first order isothermal reaction in a rectangular pellet.[ll] [8]. The governing equation and boundary conditions for concentration in dimensionless form are ... 

Similar to the four fundamental thermal boundary conditions for concentric annuli, the four kinds of fundamental conditions for parallel plate ducts are shown in Fig. 5.20. The fully developed Nusselt numbers for the four boundary conditions follow [1] ... 

Closed boundary conditions for concentration and activities are used at other boundaries of the... 

H2. It may be noted that the HI and H2 boundary conditions for the symmetrically heated passages with no sharp corners (e.g., circular, flat, and concentric annular channels) are identical they are simply designated as H. 

The boundary conditions for engineering problems usually include some surfaces on which values of the problem unknowns are specified, for instance points of known temperature or initial species concentration. Some other surfaces may have constraints on the gradients of these variables, as on convective thermal boundaries where the rate of heat transport by convection away from the surface must match the rate of conductive transport to the surface from within the body. Such a temperature constraint might be written ... 

At time t = 0 an electric current of constant strength begins to flow in the system. At this time the uniform initial concentration distribution is still not disturbed, and everywhere in the solution, even close to the electrode surface, the concentration is the same as the bulk concentration Cyj. Hence, the first boundary condition (for any value of jc) is given by... 

The above procedure is applied to all the finite-difference segments in turn. The end segments (n=l and n=N), however, often require special attention according to particular boundary conditions For example, at Z=L the solid is in contact with pure water and Cisf+i=Ceq, where the equilibrium concentration Cgq, would be determined by prior experiment. 

According to the boundary conditions, the concentration profile for A must change with a discontinuity at the reactor entrance, as shown in Fig. 4.16. 

Another example is linear diffusion, with a prescribed concentration gradient at the reference plane, i.e. a prescribed material flux through the reference plane. This type of diffusion transport is important mainly for electrode processes (see Section 5.4). The point of interest in this case is the concentration at the reference plane. In the simplest case, the material flux is constant, so that the boundary condition for x = 0 (Eq. 2.5.5) can be replaced by... 

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 ... 

The boundary conditions require knowledge of the interface concentration of hydrogen ChjL to compute E (see below). For hydrogenations, the equilibrium concentration (ChjL= CfJ L) can be used, albeit with the assumption of no mass transfer resistance on the gas side. Otherwise, it must be determined using Eq. (4). The boundary conditions for the substrate S state that it is not transferred to the gas phase - that is, S is not vaporized. This assumption is most often... 

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. 

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. 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

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... 

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