Transient diffusion initial conditions

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 Smoluchowski theory of diffusion-limited (or controlled) reactions relies heavily on the appropriateness of the inital condition [eqn. (3)]. Though the initial condition does not determine the steady-state rate coefficient [eqn. (20)] because diffusion of B in towards the reactant A is from large separation distances (>10nm) in the steady-state, it does directly determine the magnitude of the transient component of the rate coefficient because this is due to an excess concentration of B present initially compared with that present in the steady-state. As a first approximation to the initial distribution, the random distribution is intuitively pleasing and there is little experimental evidence available to cast doubt upon its appropriateness. Section 6.6 and Chap. 8 Sect. 2.2 present further comments on this point. 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

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. 

The general solution to the differential equation includes many possibilities the engineer needs to provide initial conditions to specify which solution is desired. If aU conditions are available at one point [as in Eq. (8.1)], then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equation becomes a two-point boundary value problem (see Chapter 9). Initial value problems as ordinary differential equations arise in the control of lumped parameter models, transient models of stirred tank reactors, and generally in models where there is no diffusion of the unknowns. 

In the previous transient unidirectional flow problems that we have considered, the dimensionless governing equations and boundary/initial conditions were completely free of all dimensional parameters. In these problems, however, the only relevant time scale was the diffusion time, i2c/v. Here, in contrast, the flow is characterized by a second imposed time scale that is due to the oscillatory pressure gradient, and the form of the resulting flow is predicted by (3-354) to depend upon the ratio of the diffusion time R2/v to the imposed time scale, 1 / >. The dimensionless ratio of time scales, R can also be considered to be a dimensionless frequency for the flow, and in that context is sometimes called the Strouhal number. 

We introduced the idea of change of variables in Section 10.1. The coupling of variables transformation and suitable initial conditions often lead to useful particular solutions. Consider the case of an unbounded solid material with initially constant temperature Tg in the whole domain 0 < x < oo. At the face of the solid, the temperature is suddenly raised to 7 (a constant). This so-called step change at the position jc = 0 causes heat to diffuse into the solid in a wavelike fashion. For an element of this solid of cross-sectional area A, density p, heat capacity C, and conductivity k, the transient heat balance for an element Ax thick is... 

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. 

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... 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

Initial conditions may also be complex. Suppose, for example, that we wish to solve Pick s equation for transient diffusion into a sphere. The simplest case here is to assume that the sphere is initially "clean," i.e., contains no solute. But what if it is not We would then have to specify an initial concentration distribution Cf=o =/(r,0,(p) and this distribution would have to be entered into the solution process as an initial condition. Evidently there are an infinite number of such distributions hence Pick s equation for this case will have an unlimited niunber of solutions. 

Although the diffusivity can be obtained from an individual transient for the specified charging conditions, multiple values of D can be determined for different charging currents from one experiment. After a steady state is reached, the current is increased so that successive transients are obtained. However, the initial hydrogen concentration for the second and subsequent transients is no longer zero. For the case of two consecutive transients, the initial and boundary conditions are given by" ... 

If the partial pressure on both sides is not maintained constant, the differences in P02 level out (we switch off the gas flows in Fig. 7.2). We designate this as chemical depolarization. Its transient behaviom permits calculation of chemical diflhision coefficients or effective rate constemts of the surface reaction. Similarly and k can be obtained from the transient of the chemical polarization (i.e. one-sided steplike change in the partial pressure of o g gen starting from the homogeneous initial situation). Figure 7.11 shows the stoichiometry profiles for a diffusion-controlled chemical polarization. These profiles are obtainable via Yi dc/dx. and c(x,t) by solution of the second Fick s law with the initial condition c(x,0) = Ci and the boundary conditions c(0,t) = C2 and c(L,t) = ci = c(x,0) (see e.g. [431]). 

It is evident fi-om Eq. (13.3.12) that a measurement of the steady-state flux alone does not yield D, but instead gives the product DS, called the permeability. To obtain the diffiisivity, we have to know the value of the solubility or make one additional measurement this is usually the time lag before a steady state is reached in the permeation experiment [26]. The time lag can be related to D by solving Eq. (13.2.11) for one-dimensional transient diffusion through the initially solute-fi ee membrane, subject to the boundary conditions (see Fig. 13.4) ... 

Transient cavitation is generally due to gaseous or vapor filled cavities, which are believed to be produced at ultrasonic intensity greater than 10 W/cm2. Transient cavitation involves larger variation in the bubble sizes (maximum size reached by the cavity is few hundred times the initial size) over a time scale of few acoustic cycles. The life time of transient bubble is too small for any mass to flow by diffusion of the gas into or out of the bubble however evaporation and condensation of liquid within the cavity can take place freely. Hence, as there is no gas to act as cushion, the collapse is violent. Bubble dynamics analysis can be easily used to understand whether transient cavitation can occur for a particular set of operating conditions. A typical bubble dynamics profile for the case of transient cavitation has been given in Fig. 2.2. By assuming adiabatic collapse of bubble, the maximum temperature and pressure reached after the collapse can be estimated as follows [2]. 

Transient and Steady-State Conditions From the landmine studies we readily conclude that the source term for these molecules has an initial spike, or increased rate, in the days or weeks after the mine is placed. This rate then decreases to some more or less constant level and may remain at that level for years. The initial spike comes from surface contamination, while the long-term rate is primarily from diffusion through the case and seals or leakage through imperfections or damage. The rates are clearly subject to environmental factors, principally temperature and soil wetness. Nevertheless, it seems clear that, at least in the case of landmines, there is a continuing flux of molecules that provide a potential for detection. 

Steady state heat transfer refers to the condition where the rate of heat flowing into one face of an object is equal to that flowing out of the other. If, for example, a slab of metal were placed on a hot-plate, the heat flowing into the metal would initially contribute to a temperature rise in the material, until ultimately a linear temperature gradient formed between the hot and cold faces, wherein heat flowing in would equal heat flowing out and steady state heat transfer would be established. The time involved before steady state conditions axe encountered is dependent on the thermal requirements, that is, the total heat capacity of the material. A useful constant, therefore, in depicting transient, or non-steady state heat transfer is the thermal diffusivity ... 

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... 

This fact has lead to investigations of transient laser irradiation. This is obtained by chopping the laser beam with a rotating wheel at a frequency of 1 kHz. It is observed that the rise time of Ir is of the order of a few hundred microseconds while 13 follows closely the laser irradiation. It is possible to show that the initial dip/dt is related to the time constant of CsH formation, because CsH molecules have a very short lifetime and because diffusion does not play an important role at the very beginning of the laser irradiation. Thus the risetime constant x = Ip (dIp/dt)" is assigned to [CsH] (d[CsH]/dt)" and has been measured under various experimental conditions ([Cs], [Hz]). We have derived from these measurements the quantity k = ([Cs(7P)] [Hz])" d[CsH]/dt = (x[H2]) [CsH]/[Cs(7P)] which represents the rate of CsH formation according to the global reaction (2). This rate coefficient has been found to be independent of both [Cs(7P)] and... 

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. 

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