Crank’s equations

Our experimental data, when fitted to this equation, yield a value of 1.0 x 10-11 cm2/sec for Di. Least square fit of the same data to Crank s rate equation, which takes into account only the diffusion in the matrix, evaluates Dj to be 2.2 x lO-1 1 cm2/sec. The integral form of Crank s equation (13), giving the cumulative amount released per unit surface area,... 

Miller has shown that TBTO will prevent fouling attachment at leaching rates as low as 1.25 yg/cm2/day (18). It is thus reasonable to assume that fouling commences when the rate of release falls below 0.5 yg Sn/cm2/day. Based on this, the effective dif-fusivities are calculated, using Crank s rate equation. The calculated effective diffusivities are then substituted in the integral form of Crank s equation to estimate the amount of Sn lost. 

The diffusion coefficient in the solid layer of thickness 1 is connected with time (ti/2) required to observe a half of bulk concentration of the surface product pt = O.Spoo by Crank s equation... 

The lEDP method can be combined with gas-phase analysis (GPA) as alternative methods for measuring the self-diffusion and surface self-exchange coefficients. In this analysis, the sample placed in a flxed bed is first equilibrated with a controlled atmosphere kept at a well-defined 02 partial pressure. After equilibration, the feed stream is switched to 02 at the same partial pressure and the outlet gas composed by 02 and 02 leaving the reactor is monitored (i.e., breakthrough curve analysis). The self-diffusion and surface self-exchange coefficients can be measured using the Pick s second law (i.e.. Crank s equation). 

Many texts, such as Crank s treatise on diffusion [2], contain solutions in terms of simple functions for a variety of conditions—indeed, the number of worked problems is enormous. As demonstrated in Section 4.1, the differential equation for the diffusion of heat by thermal conduction has the same form as the mass diffusion equation, with the concentration replaced by the temperature and the mass diffusivity replaced by the thermal diffusivity, k. Solutions to many heat-flow... 

As a result of this many solutions to the heat conduction equation can be transferred to the analogous mass diffusion problems, provided that not only the differential equations but also the initial and boundary conditions agree. Numerous solutions of the differential equation (2.342) can be found in Crank s book [2.78]. Analogous to heat conduction, the initial condition prescribes a concentration at every position in the body at a certain time. Timekeeping begins with this time, such that... 

Solutions to Pick s equations for a variety of boundary conditions and grain geometries, i.e. infinite plate, cylinder, sphere, etc., have been derived by numerous workers (e g. lost 1960, Crank 1975). Many of these solutions demonstrate how the degree of equilibration, F (see e g. Eqn. 97), is related to the diffusion coefficient, as well as grain size, grain geometry and solution to solid volume ratio. Diffusion rates are sensitive to a number of factors which can be broadly divided into (a) environmental and... 

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. 

Flanagan and Marcoux [29] were the first to attempt a UMDE time-marching simulation, in order to find the constant in the approximation of Lingane s equation (12.1) they used the explicit method. Crank and Furzeland [40] addressed the steady state for the UMDE and described some of the problems they also briefiy mention time-marching simulations. Their work appears to have come just after that of Evans and Gourlay [101], who used hopscotch. They also found some oscillatory behaviour of the solution, which is not always mentioned. As Gourlay realised [99], hopscotch is mathematically related to ADI, which in turn approximates Crank-Nicolson, known to be oscillatory in response to initial discontinuities such as a potential jump (more on this problem below). 

Pick s second law of diffusion is for a non-steady state or transient conditions in which dCj/t 0. Using Crank s model [23] for the rectangular element shown in Figure 4.1 yields the fundamental differential equations for the rate of concentration. Consider the central plane as the reference point in the rectangular volume element and assume that the diffusing plane at position 2 moves along the x-direction at a distance x-da from position 1 and x-Hdi to position 3. Thus, the rate of diffusion that enters the volume element at position 1 and leaves at position 3 is... 

Solutions to this differential equation for a number of boundary conditions are available in Crank s text The Mathematics of Diffusion and in Carslaw and Jaeger s Conduction of Heat in Solids " ... 

All calculations used here were based upon use of the exact calculation based upon Equation 14.11 which corresponds to the solution given in Crank s Mathematics of Diffusion for the case of a finite polymer in contact with a finite liquid and is the same as the Piringer model used by O Brien and Cooper in their earlier work [13] (see Equation 14.9). 

Several important assumptions have been implicitly incorporated in Eqs. (15) and (16). First, these equations describe the release of a drug from a carrier of a thin planar geometry, equivalent equations for release from thick slabs, cylinders, and spheres have been derived (Crank and Park, 1968). It should also be emphasized that in the above written form of Fick s law, the diffusion coefficient is assumed to be independent of concentration. This assumption, while not conceptually correct, has been... 

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 subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

As mentioned in Chapter 1, Section 1.3.6.1, Fick s second law (including Equation 1.130.) cannot be solved generally, only partial solutions exist under a well-determined boundary and initial conditions (Crank 1956). In Figure 3.5, the so-called migration cell type, a frequently used experimental setup, is shown. 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

Referring to the nondimensional equation of convective diffusion (3.3), it is of interest to examine the conditions under which the diffusion term, on the one hand, or convection, on the other, is the controlling mode of transport. The Peclet number, dUfD, for flow around a cylinder of diameter r/ is a measure of the relative importance of (he two term.s. For Pe 1, transport by llte flow can be neglected, and the deposition rate can be determined approximately by solving the equation of diffusion in a non flowing fluid with appropriate boundary conditions (Carslaw and Jaeger, 1959 Crank, 1975). 

Most kinetic studies of diffusion into keratin fibers employ equations derived from this form of Pick s law and provide approximate diffusion coefficients, assumed to be constant throughout the diffusion reaction. Pfowever, Crank [82] has provided equations for evaluating diffusion data under a wide variety of circumstances, including a variable diffusion coefficient described later in this chapter in the section entitled The Case of a Variable Diffusion Coefficient. ... 

The diffusion equations described in the previous section have been derived from Pick s second law for unidirectional diffusion with the assumption that the diffusion coefficient is constant throughout the reaction. Crank [82] has also derived equations for evaluating diffusion data for systems with a variable diffusion coefficient that can be used to test one s data. 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

The movement of the rubbery-solvent interface, S, was governed by the difference between the solvent penetration flux and the dissolution rate, derived earlier. An implicit Crank-Nicholson technique with a fixed grid was used to solve the model equations. A typical concentration profile of the polymer is shown in Fig, 24. Typical Case II behavior was observed. The respective positions of the interfaces R and S are shown in Fig. 25. Typical disentanglement-controlled dissolution was observed. Limited comparisons of the model predictions were made with experimental data for a PMMA-MIBK system. 

These equations are named Pick s law and Fourier s law, respectively, and can be solved with suitable boundary and initial conditions. Literature on solving these equations is abundant, and for diffusion a classic work is that of Crank (1975). It is worth mentioning that, in view of irreversible thermodynamics, mass flux is also due to thermodiffusion and barodiffusion. Formulation of Equations 3.22 and 3.23 containing terms of thermodiffusion was favored by Luikov (1966). 

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