Boundary conditions simplification

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

The preceding analysis is made simpler if, as often the case, the first as well as the electron transfer step may be regarded as totally irreversible, and dimerization is so fast that pure kinetic conditions are fulfilled. The last simplification implies that Qb/Qr = 0 in equation (6.58). Integration of this equation, taking into account initial and boundary conditions (6.59) and equations (6.63), leads to... 

In order to describe the fluorescence radiation profile of scattering samples in total, Eqs. (8.3) and (8.4) have to be coupled. This system of differential equations is not soluble exactly, and even if simple boundary conditions are introduced the solution is possible only by numerical approximation. The most flexible procedure to overcome all analytical difficulties is to use a Monte Carlo simulation. However, this method is little elegant, gives noisy results, and allows resimulation only according to the method of trial and error which can be very time consuming, even in the age of fast computers. Therefore different steps of simplifications have been introduced that allow closed analytical approximations of sufficient accuracy for most practical purposes. In a first... 

Figure 3. Two-dimensional simplification of the Feynman ratchet, consisting of one vane (flat sheet in lower reservoir) and one ratchet (triangular shape in upper reservoir), that is free to move as a rigid whole along the horizontal direction x. The boundary conditions are periodic both left and right and up and down in each container.

In terms of modeling, the equations are the same as those in section 4, with perhaps some simplifications. Additional boundary conditions are required due to the higher dimensionality of the equations, but these are relatively straightforward, such as no fluxes of gas species across the external boundary of the gas channels. 

Fortunately, for small extents of dispersion numerous simplifications and approximations in the analysis of tracer curves are possible. First, the shape of the tracer curve is insensitive to the boundary condition imposed on the vessel, whether closed or open (see above Eq. 11.1). So for both closed and open vessels... 

With this simplification of the two dimensional step flow problem, we can study the long time behavior of the step train well beyond the initial onset of instability. We start with an array of 40 steps with small perturbations from an initial uniform configuration. We discretize the y coordinate so that each step has 2000 segments. Periodic boundary conditions are used in x and y direction. The time evolution problem of Eqs. (15) using (16) is converted into a set of difference equations. We control the time step so... 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

Zwanzig s diffusion equation [444], eqn. (211), can be reduced to the stochastic equation used by Clifford et al. [442, 443] [eqn. (183)] to describe the probability that N identical reactant particles exist at time t (see also McQuarrie [502]), Let us consider the case where U — 0, with a static solvent, for a constant homogeneous diffusion coefficient. This is a major simplification of eqn. (211). Now, rather than represent the reaction between two reactants k and j by a boundary condition which requires the... 

When Amundson taught the graduate course in mathematics for chemical engineering, he always insisted that all boundary conditions arise from nature. He meant, I think, that a lot of simplification and imagination goes into the model itself, but the boundary conditions have to mirror the links between the system and its environment very faithfully. Thus if we have no doubt that the feed does get into the reactor, then we must have a condition that ensures this in the model. We probably do not wish to model the hydrodynamics of the entrance region, but the inlet must be an inlet. One merit of the wave model we have looked at briefly is that both boundary conditions apply to the inlet. 

Simplification of the Model Triple-Phase-Boundary Reactions as Boundary Conditions... 

In the present section, boundary and interfacial conditions are presented for the three modeling approaches given in Sections 3.3, 3.4, and 3.5. Since the modelling approaches described in Sections 3.4 and 3.5 can be considered simplifications of the model presented in Section 3.3, boundary conditions are presented first for Section 3.3, and, consequently, for Sections 3.4 and 3.5. 

The appropriate boundary conditions need to be included before a solution can be achieved. The solution to this problem remains complex, therefore, simplifications are necessary. 

A number of solutions exist by integration of the diffusion equation (7-12) that are dependent on the so-called initial and boundary conditions of special applications. It is not the goal of this section to describe the complete mathematical solution of these applications or to make a list of the most well-known solutions. It is much more useful for the user to gain insight into how the solutions are arrived at, their simplifications and the errors stemming from them. The complicated solutions are usually in the form of infinite series from which only the first or first few members are used. In order to understand the literature on the subject it is necessary to know how the most important solutions are arrived at, so that the different assumptions affecting the derivation of the solutions can be critically evaluated. 

These three equations in the three variables o>, and T are the set of equations that must be simultaneously solved subject to the correct boundary conditions to obtain the heat transfer rate. The original set of Equations from which they were deduced, i.e., Eqs. (2.70), (2.71), (2.72). and (2.77), contained four unknowns u, v, p, and T and this reduction in the number of variables and, therefore, the number of governing equation in itself constitutes a considerable simplification. The governing equations in terms of u, v, p, and T are often said to be expressed in primitive variable form. 

Here, Kt are the mass transfer coefficients (permeabilities) for each wall, and CL, and CUi are the ambient concentrations of each component i outside the lower and upper walls, respectively. Sometimes, selective membranes may be used as the walls. These membranes may be permeable to selected components only. For example, in a purification process, the membrane would be permeable to one of the solutes only. In a concentration process, both walls can be impermeable to the selected solute. Equations (7.151) and (7.152) describe the thermodynamically and mathematically coupled heat and mass flows at stationary conditions and may be solved with boundary conditions and with some simplifications (Coelho and Telles, 2002). 

Integral Equation Solutions. As a consequence of the quasi-steady approximation for gas-phase transport processes, a rigorous simultaneous solution of the governing differential equations is not necessary. This mathematical simplification permits independent analytical solution of each of the ordinary and partial differential equations for selected boundary conditions. Matching of the remaining boundary condition can be accomplished by an iterative numerical analysis of the solutions to the governing differential equations. 

With the equations of secs. 4.6a-c the problem of computing f from mobilities at given Ka is in principle soluble. But not in practice The mathematics are very complicated and require a number of finesses, whereas simplifications is dangerous because the various fluxes and forces are coupled, so that approximating only one of these may offset the balance and misrepresent characteristic features. Moreover, the boundary conditions are in part determined by the composition of, and the mobilities of ions in the various parts of the double layer, for which model assumptions must be made. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

More detailed information on cell operation is given by 2D models. These models can be subdivided into two groups. The first group constitute the so-called along-the-channel models, in which equations are written in a plane, directed along the z axis (Fig. 19) [10,159-166]. Further simplification comes from an assumption that the z-direction components of the fluxes and currents in the MEA are small compared to x-direction components. The 2D problem then is reduced to a set of 1D problems along x direction and to the problem of gas flow in the channel, which supplies the boundary conditions for the ID problems [10,159,161,166],... 

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