Diffusion equation time-dependent boundary conditions

Solving the diffusion equation in environmental transport can be challenging because only specihc boundary conditions result in an analytical solution. We may want to consider our system of interest as a reactor, with clearly defined mixing, which is more amenable to time dependent boundary conditions. The ability to do this depends on how well the conditions of the system match the assumptions of reactor mixing. In addition, the system is typically assumed as one dimensional. The common reactor mixing assumptions are as follows ... 

We use the time dependent "rate constant", k(t), obtained by solving diffusion equation under appropriate boundary conditions 2-4. 

In the case of a diffusion-controlled reaction a current-potential curve can be evaluated quantitatively. The diffusion equation has to be solved again by using time-dependent boundary conditions. The mathematics, however, are very complicated and cannot be shown here. They end up with an integral equation which has to be solved numerically [11]. The peak current, /p, for a diffusion-controlled process (reversible reaction) is found to be... 

Equation 3-10 is the most basic diffusion equation to be solved, and has been solved analytically for many different initial and boundary conditions. Many other more complicated diffusion problems (such as three-dimensional diffusion with spherical symmetry, diffusion for time-dependent diffusivity, and... 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

The relationship between the concentration and the current can be obtained by solving Pick s equation under appropriate boundary conditions. Time-dependent diffusion equations can be easily solved numerically, for instance by finite-element methods. However, such techniques normally require medium or large pla tforms to obtain accurate results with reasonable run times. This is particularly true when dealing with structured devices. A different approach was used here. Analytical methods2> show that the time dependence of concentration profiles is mostly exponential in character. Solutions of Pick s equation of the form... 

Previous theoretical works have addressed these questions by adding appropriate assumptions to the theory. Sueh models can be roughly summarized by the following scheme (i) consider a diffusive transport of surfactant molecules from a semi-infinite bulk solution (following Ward and Tordai) (ii) introduce a certain adsorption equation as a boundary condition at the interface (iii) solve for the time-dependent surface coverage (iv) assume that the equilibrium equation of state is valid also out of equilibrium and calculate the dynamic surface tension [10]. 

The time evolution of the density field pi r) can be described by a time dependent Landau-Ginzburg type equation (11). The boundary conditions that are used on the simulation box are periodic boundary conditions. For the diffusion flux in the vicinity of the filler particles, rigid-wall boundary conditions are used. A simple way to implement these boundary conditions in accordance with the conservation law is to allow no flux through the filler particle surfaces, i.e.,... 

Semi-infinite linear diffusion conditions The rate of an electrochemical process depends not only on electrode kinetics but also on the transport of species to/from the bulk solution. Mass transport can occur by diffusion, convection or migration. Generally, in a spectroeiectrochemicai experiment, conditions are chosen in which migration and convection effects are negligible. The solution of diffusion equations, that is the discovery of an equation for the calculation of oxidized form [O] and reduced form [R] concentrations as functions of distance from electrode and time, requires boundary conditions to be assumed. Usually the electrochemical cell is so large relative to the length of the diffusion path that effects at walls of the cell are not felt at the electrode. For semiinfinite linear diffusion boundary conditions, one can assume that at large distances from the electrode the concentration reaches a constant value. 

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. 

Lamm I-L, C. Samuelsson, and H. Pettersson, Solution of the One-Dimensional Time-Dependent Diffusion Equation with Boundary Conditions Applicable to Radon Exhalation from Porous Materials, Coden Report LUNFD6(NFRA3042), Lund University, Lund (1983). 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

The first component h2/Z) is the period of time required to traverse a distance b in any direction, whereas the second term/ (alb) strongly depends on the dimension-ahty. Adam and Delbrtlck define appropriate boundary conditions and equations describing the concentration of molecules in the diffusion space in terms of space coordinates and time. They treated four cases (1) onedimensional diffusion in the linear interval a < jc < h (2) two-dimensional diffusion on the circular ring a < r < b (3) three-dimensional diffusion in a spherical shell a < r < b, and (4) combined three-dimensional and surface diffusion. They provide a useful account of how reduced dimensionahty of diffusion can (a) lower the time required for a metabolite or particle originating at point P to reach point Q, and (b) improve the likelihood for capture (or catch) of regulatory molecules by other molecules localized in the immediate vicinity of some target point Q. 

Because D increases with increasing temperature (the Arrhenius equation 1-73), time-dependent D is often encountered in geology because an igneous rock may have cooled down from a high temperature, or metamorphic rock may have experienced a complicated thermal history. If the initial and boundary conditions are simple and if D depends only on time, the diffusion problem is easy to deal with. Because D is independent of x. Equation 3-9 can be written as... 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

The exact solution for the time-dependence of the current at a planar electrode embedded in an infinitely large planar insulator, the so-called semi-infinite linear diffusion condition, is obtained. Solving the diffusion equation under the proper set of boundary and initial conditions yields the time-dependent concentration profile. 

Example calculation for diffusion of multiple chemicals Consider a mix of chemicals that differ greatly in molecular mass (and hence in their diffusion coefficients), such as a 3 1 ratio of ethanolihexadecanol in the air surrounding a sensory hair or filiform antenna. The 16-carbon alcohol will be approximately eight times as massive as the 2-carbon alcohol. The diffusion coefficients (D) are 1.32 x 10 5 m2/s for ethanol (Welty el al., 1984) and 2.5 x 10 6 m2/s for hexadecanol (using the value for bombykol), both in air at 298 K. What will the rate of interception be at the level of a sensory hair for these two chemicals The answer (and choice of equation) depends on the boundary conditions. 

In Example 10.1 the case where the aerosol concentration does not change with time was considered. In many practical situations, however, the aerosol concentration does change with time, possibly as a result of diffusion and subsequent loss of particles to a wall or other surface. In this event, Fick s second law, Eq. 9.2, must be used. Solution of this equation is possible in many cases, depending on the initial and boundary conditions chosen, although the solutions generally take on very complex forms and the actual mechanics involved to find these solutions can be quite tedious. Fortunately, there are several excellent books available which contain large numbers of solutions to the transient diffusion equation (Barrer, 1941 Jost, 1952). Thus, in most cases it is possible to fit initial and boundary conditions of an aerosol problem to one of the published solutions. Several commonly occurring examples follow. 

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