Steady-state model boundary conditions

Some models, however, take the form of second-order differential equations, which often give rise to problems of the split boundary type. In order to solve this type of problem, an iterative method of solution is required, in which an unknown condition at the starting point is guessed, the differential equation integrated. After comparison with the second boundary condition a new starting point is estimated, followed by re-integration. This procedure is then repeated until convergence is achieved. MADONNA provides such a method. Examples of the steady-state split-boundary type of solution are shown by the simulation examples ROD and ENZSPLIT. 

Equation 6.42 together with the boundary conditions given by Equation 6.43 form the steady-state model, and its solution is given as... 

A shooting method was used to solve the steady-state model. The wall temperature at 2 = 0 is guessed and the boundary conditions of / = 0 and l2=o are applied. The solution is marched forward using the wall equation (7.8.11) and the two-phase equation (7.8.13) until / = 1.0. The one-phase equation (7.8.12) is applied until z = 2g d. This procedure is repeated until the boundary conditions at 2 = Zgnd are met. 

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. 

Flowever, with CFD, configurations with mostly known or at least steady-state boundary conditions and surface temperatures are calculated. In cases where the dynamic behavior of the building masses and the changing driving forces for the natural ventilation are of importance, thermal modeling and combined thermal and ventilation modeling mu.st be applied (see Section 11..5). 

There is a general trend toward structured packings and monoliths, particularly in demanding applications such as automotive catalytic converters. In principle, the steady-state performance of such reactors can be modeled using Equations (9.1) and (9.3). However, the parameter estimates in Figures 9.1 and 9.2 and Equations (9.6)-(9.7) were developed for random packings, and even the boundary condition of Equation (9.4) may be inappropriate for monoliths or structured packings. Also, at least for automotive catalytic converters. 

The program can solve both steady-state problems as well as time-dependent problems, and has provisions for both linear and nonlinear problems. The boundary conditions and material properties can vary with time, temperature, and position. The property variation with position can be a straight line function or or a series of connected straight line functions. User-written Fortran subroutines can be used to implement more exotic changes of boundary conditions, material properties, or to model control systems. The program has been implemented on MS DOS microcomputers, VAX computers, and CRAY supercomputers. The present work used the MS DOS microcomputer implementation. 

Meiron (12) and Kessler et al. (13) have shown that numerical studies for small surface energy give indications of the loss-of-existence of the steady-state solutions. In these analyses numerical approximations to boundary integral forms of the freeboundary problem that are spliced to the parabolic shape far from the tip don t satisfy the symmetry condition at the cell tip when small values of the surface energy are introduced. The computed shapes near the tip show oscillations reminiscent of the eigensolution seen in the asymptotic analyses. Karma (14) has extended this analysis to a model for directional solidification in the absence of a temperature gradient. 

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. 

In this model there is no stomach compartment but possible dissolution in the stomach can be partly accounted for by defining the boundary conditions (C and r ) in the beginning of the intestine. If fast dissolution is expected in the stomach, the boundary conditions for C can be set to 1, that is, the luminal concentration when entering the intestine equals the saturation solubility in the intestine. If no dissolution is expected in the stomach, the starting value for C will be 0. C can then have all values between 0 and 1 as the boundary condition and this results in the restriction that super-saturation in the intestine due to fast dissolution and high solubility in the stomach can not be accounted for. Assuming that the difference between the mass into and out of the intestine is equal to the mass absorbed at steady state, the Tabs can be calculated from Eq. 17 [40]... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

The steady-state heat equation (Eq. 3.284) is often used as the model equation for an elliptic partial-differential equation. An important property of elliptic equations is that the solution at any point within the domain is influenced by every point on the boundary. Thus boundary conditions must be supplied everywhere on the boundaries of the solution domain. The viscous terms in the Navier-Stokes equations clearly have elliptic characteristics. 

This is the kinetic equation for a simple A/AX interface model and illustrates the general approach. The critical quantity which will be discussed later in more detail is the disorder relaxation time, rR. Generally, the A/AX interface behaves under steady state conditions similar to electrodes which are studied in electrochemistry. However, in contrast to fluid electrolytes, the reaction steps in solids comprise inhomogeneous distributions of point defects, which build up stresses at the boundary on a small scale. Plastic deformation or even cracking may result, which in turn will influence drastically the further course of any interface reaction. 

We have seen that the basic P model has the form of a first-order partial differential Eq. (22) describing each narrow slice as a little batch reactor being transported through the reactor at constant speed. This equation was so elementary that it could be solved at sight in Eq. (30). When we added a longitudinal dispersion term governed by Fick s law and took the steady state, Eq. (40), we had a second-order o.d.e. with controversial boundary conditions. This is the model with ( ) = c(z)lcm and Pe = vLID, Da = kL/v,... 

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