Boundary and initial conditions

The initial conditions represent mathematically the nature of the profiles of concentration initially throughout the sheet. Either the profile of concentration is uniform or not. In the first case, an analytical solution can be found, while in the second case, generally a numerical method should be used. 

With constant initial concentration, it is written, with a sheet of thickness 2 L  

The boundary conditions represent mathematically the conditions on the surfaces of the sheet. Generally, three conditions may appear  

When the rate at which the diffusing substance enters an external fluid by convection is equal to the rate at which the substance is brought by internal diffusion to the surface  

Note that the concentration in a fluid, because of the couvecliou which acts much fester than diffusion through a soUd, may be cousidered as uniform. 

The initial and boundary conditions associated with the partial differential equations must be specified in order to obtain unique numerical solutions to these equations. In general, boundary conditions for partial differential equations are divided into three categories. These are demonstrated below, using the one-dimensional unsteady-state heat conduction equation 

This is identical to Eq. (6.18). It is derived from Eq. (6.8) by assuming that the temperature gradients in the y and z dimensions are zero. Eq. (6.20) essentially describes the change in temperature within a solid slab (e.g., the wall of a furnace), where heat transfer takes place in the x-direction (see Fig. 6.2). 

Dirichlet conditions (first kind) The values of the dependent variable are given at fixed values of the independent variable. Examples of Dirichlet conditions for the heat conduction equation are 

These are alternative initial conditions that specify that the initial temperature inside the slab (wall) is a function of position/(x) or a constant Tq (Fig. 6.2a). 

Numerical Solution of Partial Differential Equations Chapter 6 

The next several sections discuss initial and boundary conditions, and methods to solve Equation 3-10 given such conditions. In the process of learning the methods, typical solutions to the diffusion equation will be presented. These solutions will be encountered in later discussions. More solutions are presented in Appendix 3. 

Many solutions exist for a partial differential equation such as Equation 3-10. For example. 

The initial condition for Equation 3-10 generally takes the form C t o = f(x). That is, the concentration distribution is given at t=0. The simplest initial condition is that C t=o = constant. 

The boundary conditions are more complicated and three cases may be distinguished  

Because an infinite or a semi-infinite reservoir merely means that the medium at the two ends or at one end is not affected by diffusion, whether a medium may be treated as infinite or semi-infinite depends on the timescale of our consideration. For example, at room temperature, if water diffuses into an obsidian glass from one surface and the diffusion distance is about 5 /im in 1000 years, an obsidian glass of 50 / m thick can be viewed as a semi-infinite medium on a thousand-year timescale because 5 fim is much smaller than 50 /im. However, if we want to treat diffusion into obsidian on a million-year time-scale, then an obsidian glass of 50 fim thick cannot be viewed as a semi-infinite medium. 

As for all partial differential equations, it is also necessary to complement the mass balance equation with initial and boundary conditions, as explained in detail by Guiochon et al. [13]. The initial condition describes the state of the column when the experiment begins, i.e., at t = 0. In this case, the initial condition corresponds to a column empty of sample and containing only mobile and stationary phases in equilibrium  

Where C, (t, z) is the concentration of the the / th component at position z and L is the column length. Boundary conditions characterize the injection and if the dispersion effects are neglected they can be described by rectangular pulses with duration tp at the column inlet. Assuming that the sample concentration is C0 i 

The systems of mass balance equations with the proper isotherm equations are integrated numerically to obtain the concentration profiles at the column outlet. 

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R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Pick s second law of difflision enables predictions of concentration changes of electroactive material close to the electrode surface and solutions, with initial and boundary conditions appropriate to a particular experiment, provide the basis of the theory of instrumental methods such as, for example, potential-step and cyclic voltanunetry. 

Potential-step teclmiques can be used to study a variety of types of coupled chemical reactions. In these cases the experiment is perfomied under diffrision control, and each system is solved with the appropriate initial and boundary conditions. 

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

The crack is said to have a zero opening in this case. As it turned out there is no singularity of the solution provided the crack has a zero opening. What this means is the solution of (3.144), (3.147), (3.148) coincides with the solution of (3.140)-(3.142) found in the domain Q with the initial and boundary conditions (3.144), (3.145) (and without (3.143)). In the last case the equations (3.141), (3.142) hold in Q. This removable singularity property is of local character. Namely, if O(x ) is a neighbourhood of the point and... 

The emission inventory and the initial and boundary conditions of pollutant concentrations have a large impact on the ozone concentrations calculated by photochemical models. 

Preprint 2000-041. .. which together with initial and boundary conditions determines a dynamic cake filtration... 

To start the numerical solution, initial and boundary conditions must be specified. 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

The initial and boundary conditions are the same presented before and, for x = 0, Equation (20) becomes ... 

Solutions for this second-order differential equation are known for a number of initial and boundary conditions [4]. 

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. 

The absorption is assumed to occur into elements of liquid moving around the bubble from front to rear in accordance with the penetration theory (H13). These elements maintain their identity for a distance into the fluid greater than the effective penetration of dissolving gas during the time required for this journey. The differential equation and initial and boundary conditions for the rate of absorption are then... 

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)... 

The resulting equations, arrived at by setting appropriate initial and boundary conditions (depending on the particular electrode), are given in Table 3-4. 

These equations are solved under the initial and boundary conditions as follows Since all the fluctuations at 1 0 are produced by electrode reactions, the initial components induced by diffusion are equal to zero. Therefore,... 

Before going further, we must append to (7) the initial and boundary conditions. These depend on the range of variables thus, the boundary conditions and the first initial condition are specified exactly ... 

Computing region, notation of initial and boundary conditions. 

This book is a supplementary source of knowledge on combustion, to facilitate the understanding of fundamental processes occurring in flames during their formation, propagation, and extinction. The characteristic feature of the book lies in the presentation of selected types of flame behavior under different initial and boundary conditions. The most important processes controlling combustion are highlighted, elucidated, and clearly illustrated. 

For the HMDE and for a solution that contains only ox of a reversible redox couple, Reinmuth102, on the basis of Fick s second law for spherical diffusion and its initial and boundary conditions, derived the quantitative relationship (at 25° C)... 

Application of the Balzhinimaev model requires assumptions about the reactor and its operation so that the necessary heat and material balances can be constructed and the initial and boundary conditions formulated. Intraparticle dynamics are usually neglected by introducing a mean effectiveness factor however, transport between the particle and the gas phase is considered. This means that two heat balances are required. A material balance is needed for each reactive species (S02, 02) and the product (SO3), but only in the gas phase. Kinetic expressions for the Balzhinimaev model are given in Table IV. 

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