Boundary conditions flux condition

There is now the additional boundary condition (flux condition),... 

It is through a generalization of Schwinger s principle that one obtains a prediction of the properties of an atom in a molecule. The generalization is possible only if the atom is defined to be a region of space bounded by a surface which satisfies the zero flux boundary condition, a condition repeated here as equation 15,... 

Fig. 2. Diagrammatic representation of steady-state concentration profiles in a membrane (from S61egny et al. [40]). Full lines as well as fluxes Ji and J2 correspond to diffusion reactions dashed lines illustrate the non-reactive Fickian diffusion profiles with the same boundary concentrations. Fluxes, conditions creating profiles of type A, B, C or D, position or value of the minimum can be...

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

It remains to determine the total flux K by introducing into equation (4.19) the value of given by the boundary condition (4.14). Thus we... 

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. 

Given the boundary condition (A.1.6) it is a straightforward matter to integrate the Navier Scokes equations in a cylindrical tube, and hence to find the molar flux N per unit cross-sectional area. The result, which was also obtained by Maxwell, is... 

NUh2 is the Nusselt number for uniform heat flux boundary condition along the flow direction and periphery. 

Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for nnidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions and I Aft at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restric tion of the following equations is that the concentration and diffnsivity are assumed to be constant. As written, the flux is positive from left to right, as depic ted in Fig. 5-25. 

For axial dispersion in a semi-infinite bed with a linear isotherm, the complete solution has been obtained for a constant flux inlet boundary condition [Lapidiis and Amundson,y. Phy.s. Chem., 56, 984 (1952) Brenner, Chem. Eng. Set., 17, 229 (1962) Coates and Smith, Soc. Petrol. Engrs. J., 4, 73 (1964)]. For large N, the leading term is... 

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 presence of a magnetic flux, the boundary condition is changed hy the Aharonov-Bohm effect and the band gap exhibits an oscillation between 0 and 2nylL with period 0q is shown in Fig. 3. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

In most cases, however, heat transfer and mass transfer occur simultaneously, and the coupled equation (230) thus takes into account the most general case of the coupling effects between the various fluxes involved. To solve Eq (230) with the appropriate initial and boundary conditions one can decouple the equation by making the transformation (G3)... 

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... 

Finally, the boundary condition (276) refers to the continuity of the flux J = — ClXd /dr) at the interface. 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

Thermal boundary condition at constant wall heat flux... 

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. 

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,... 

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. 

In the case that the effective diffusion coefficient approach is used for the molar flux, it is given by N = —Da dci/dr), where Dei = (Sp/Tp)Dmi according to the random pore model. Standard boundary conditions are applied to solve the particle model Eq. (8.1). 

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. 

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