Transient diffusion boundary conditions

Consider the case of transient diffusion at constant potential (constant surface concentration). The first boundary condition, (11.2), is preserved and the second boundary condition can be written (for any time t) as... 

With these boundary conditions, the differential transient-diffusion equation (11.1) has the solution... 

A related technique is the current-step method The current is zero for t < 0, and then a constant current density j is applied for a certain time, and the transient of the overpotential 77(f) is recorded. The correction for the IRq drop is trivial, since I is constant, but the charging of the double layer takes longer than in the potential step method, and is never complete because 77 increases continuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusion equation only for the case of a simple redox reaction and the range of small overpotentials 77 [Pg.177]

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

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... 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

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. 

Consider now the effect of uncompensated iR on the shape of the potentiostatic transients. This was shown in Fig. 6D. The point to remember is that although the potentiostat may put out an excellent step function - one with a rise time that is very short compared to the time of the transient measured - the actual potential applied to the interphase changes during the whole transient, as the current changes with time (cf. Section 10.2). This effect is not taken into account in the boundary conditions used to solve the diffusion equation, and the solution obtained is, therefore, not valid. The resulting error depends on the value of R, and it is very important to minimize this resis-tance, by proper cell design and by electronic iR compensation. 

The set of diffusion equations (8) along with the appropriate initial and boundary conditions (6), (7), solved using a computer, gives transient concentration proffles, C as functions of r and t values. The simplifying assumption that the concentration, c , of free B ions in solution and at the boundary, r, in the surface layer of the associated exchanger bead is the same is justified by the absence of the Donnan exclusion of electrolyte in the associated layer RB. The same assumption has been adopted for calculations elsewhere [32,34,42,43,52-54,56]. 

A major breakthrough in the study of gas and v or transport in polymer membranes was achieved by Daynes in 1920 He pointed out that steady-state permeability measurements could only lead to the determination of the product EMcd and not their separate values. He showed that, under boundary conditions which were easy to achieve experimentally, D is related to the time retired to achieve steady state permeation throu an initially degassed membrane. The so-called diffusion time lag , 6, is obtained by back-extrapolation to the time axis of the pseudo-steady-state portion of the pressure buildup in a low pressure downstream receiving vdume for a transient permeation experiment. As shown in Eq. (6), the time lag is quantitatively related to the diffusion coefficient and the membrane thickness, , for the simple case where both ko and D are constants. 

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

Fig. 9.30. Spatial propagation of a sharp Cef front of the type seen in eardiae cells (type 1 wave). Shown are six successive stages of the transient pattern obtained by numerical integration of eqns,(9.11) of the model based on CICR, from which the term Vj/S related to stimulation has been removed and to which the diffusion of cytosolic Ca has been added. In these simulations, the Ca -sensitive Ca pool is assumed to be distributed homogeneously within the cell. The latter is represented as a two-dimensional mesh of 20 x 60 points and diffusion is approximated by finite differences boundary conditions are of the zero-flux type. The terms related to influx from (vq) and into kZ) the extracellular medium only appear in the points located on the borders of the mesh. The diffusion coefficient of is equal to 400 pmVs other parameter...

If one constructs the appropriate dimensionless equation that governs the molar density profile fi for component i, then xj/i depends on all the dimensionless independent variables and parameters in the governing equation and its supporting boundary conditions. Geometry also plays a role in the final expression for in each case via the coordinate system that best exploits the summetry of the macroscopic boundaries, but this effect is not as important as the dependence of on the dimensionless numbers in the mass transfer equation and its boundary conditions. For example, if convection, diffusion, and chemical reaction are important rate processes that must be considered, then the governing equation for transient analysis... 

TRAFIC (Ref. 6) A core-survey code for calculating the full-core release of metallic fission products. TEIAFIC is a finite-difference solution to the transient diffusion equation for the multihole fuel element geometry with a convective boundary condition at the coolant surface. The temperature and failure distributions required as input are supplied by an automatic interface with the SURVEY/PERFOR code. 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

Integration of Eq. (61.1) for the desired geometry and boundary conditions yields the total rate of permeation of the penetrant gas through the polymer membrane. Integration of Eq. (61.2) yields information on the temporal evolution of the penetrant concentration profile in the polymer. Equation (61.2) requires the specification of the initial and boundary conditions of interest. The above relations apply to homogeneous and isotropic polymers. Crank [3] has described various techniques of solving Pick s equations for different membrane geometries and botmdary conditions, for constant and variable diffusion coefficients, and for both transient and steady-state transport. 

Equation 4.15 is a second-order partial differential equation. When treating diffusion phenomena with Pick s second law, the typical aim is to solve this equation to yield solutions for the concentration profile of species i as a function of time and space [Ci(x,f)]- By plotting these solutions at a series of times, one can then watch how a diffusion process progresses with time. Solution of Pick s second law requires the specification of a number of boundary and initial conditions. The complexity of the solutions depends on these boundary and initial conditions. Por very complex transient diffusion problems, numerical solution methods based on finite difference/finite element methods and/or Fourier transform methods are commonly implemented. The subsections that follow provide a number of examples of solutions to Pick s second law starting with an extremely simple example and progressing to increasingly more complex situations. The homework exercises provide further opportunities to apply Pick s Second Law to several interesting real world examples. 

Transient Semi-Infinite Diffusion The simplest transient diffusions problems are generally those that involve semi-infinite or infinite boundary conditions. Consider, for example, the situation illustrated in Figure 4.6, which represents diffusion of a substance from a surface into a semi-infinite medium. 

When solving Pick s second law for any specific problem, the first step is always to specify the boundary and initial conditions. For the semi-infinite diffusion process illustrated in Figure 4.6 as an example, the concentration of species i is initially constant everywhere inside the medium at a uniform value of c°. At time f = 0, the surface is then exposed to a higher concentration of species i (c ), which causes i to begin to diffuse into the medium (since c > c"). It is assumed that the surface concentration of species i is held constant at this new higher value c during the entire transient diffusion process. Based on this discussion, we can mathematically specify the boundary and initial conditions as follows ... 

The mathematics to obtain the analytical solution to Pick s second law under these conditions [5] are actually fairly involved. However, generalized analytical solutions for this and many other diffusion problems have been obtained and compiled in extensive reference books. In particular. Crank s handy reference text. The Mathematics of Diffusion [5], provides solutions to a wide range of transient diffusion problems. For many diffusion problems, it is often sufficient to consult such a reference text in order to obtain a generalized solution and then apply a particular problem s specific boundary and initial conditions to obtain a full solution. 

Exact solution to the transient diffusion problem illustrated in Figure 4.6 may be obtained by applying the boundary and initial conditions to the general solution provided in Equation 4.20. First, applying the initial condition [c,(jc, t = 0) = cp] yields... 

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