Diffusion equation Laplace transforms

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

The limitation of the prescribed diffusion approach was removed, for an isolated ion-pair, by Abell et al. (1972). They noted the equivalence of the Laplace transform of the diffusion equation in the absence of scavenger (Eq. 7.30) and the steady-state equation in the presence of a scavenger with the initial e-ion distribution appearing as the source term (Eq. 7.29 with dP/dt = 0). Here, the Laplace transform of a function/(t) is defined by... 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

Laplace transformation, 1215 Nemst s equation and. 1217 non-steady, 1254 as rate determining step, 1261 Schlieren method, 1235 semi-infinite linear, 1216, 1234, 1255 in solution and electrodeposition, 1335 spherical. 1216. 1239 time dependence of current under, 1224 Diffusion control, 1248... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

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). 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

To solve the diffusion equation (9) or (10) for the density p(r, f) with the random initial condition (3), the outer boundary condition (4) and the partially reflecting boundary condition (22) is straightforward. Again, the solution follows from eqn. (12), but the Laplace transform of eqn. (22) is... 

The term propagator is used to describe the motion of a particle A within the solvent (elastic collisions between A and S). No such term as diffusion is used to describe this motion since that rather determines what is being sought, hardly the beet approach A diffusive propagator is Gn(r, z) — l3 — Dv ] 1, which converts the probability of a particle being at r at the initial time, sa,y p(r), to that at the Laplace transform parameter value z, say p(r, z), i.e. p(r, z) — Go(r, ) p(r), as may be seen by Laplace transforming the diffusion equation. 

Before discussing the solution of these equations, it is prudent to known where we are aiming to go In Chaps. 6 and 7, the quantities of interest were shown to be the survival and recombination probabilities. From the diffusion equation, the former was found to be the volume integral of the probability density, p(r, and was discussed in Chap, 6, Sect. 2. Recombination probabilities are similarly the time and volume integrals of the rate of reaction of the pair. In a similar manner, the survival probability in Laplace transform space is... 

To show how the lifetime distribution of eqn. (346) is directly related to the diffusion equation analysis, first substitute eqn. (343) into eqn. (346) and take the Laplace Transform... 

The mathematics of semi-infinite linear diffusion were given by eqns. (19a—d). The Laplace transforms of these equations are... 

Applying the Laplace transform to both sides of the diffusion equation yields... 

Analytical solutions of Fick s laws are most easily derived using Laplace transforms, a subject described in every undergraduate book on differential equations. The solution of diffusion equations has fascinated academic elec-troanalytical chemists for years, and they naturally have a tendency to expound on them at the slightest provocation. Fortunately, the chemist using electrode reactions can accomplish a great deal without more than a cursory appreciation of the mathematics. Our intention here is to provide this qualitative appreciation on a level sufficient to understand laboratory techniques. 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

Equation (5.64) is equivalent to the following expression deduced by Reinmuth by using Laplace transform and assuming equal diffusion coefficients of species O and R and = 0 [27],... 

The coupled diffusion equations (6) together with the boundary conditions (9) and (10) can be solved in close form in the Laplace transform space, and numerically inverted to the time domain. At early time, the solution behaves according to the solution for a semi-infinite domain. At large time, the solution... 

Equation (9.17) is solved by a Laplace transformation. In chemical kinetics and diffusion, the problems may often be formulated in terms of partial differential equations... 

The solution of Fick s second law gives the variation of flux, and thence diffusion-limited current, with time, it being important to specify the conditions necessary to define the behaviour of the system (boundary conditions). Since the second law is a partial differential equation it has to be transformed into a total differential equation, solved, and the transform inverted1. The Laplace transform permits this (Appendix 1). 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

In electrochemistry, the most frequent use of Laplace transformation is in solving problems involving -> diffusion to an electrode. -> Fick s second law, a partial differential equation, becomes an ordinary differential equation on Laplace transformation and may thereby be solved more easily. 

Using the Laplace transforms described in Appendix B, we are able to solve the two diffusion equations (3.14) and (3.15) under the given initial conditions (Equations 3.16 and 3.17) and boundary conditions (Equations 3.18 and 3.19). Then we have... 

When discussing diffusion, one inevitably needs to solve diffusion equations. The Laplace transform has proven to be the most effective solution for these differential equations, as it converts them to polynomial equations. The Laplace transform is also a powerful technique for both steady-state and transient analysis of linear time-invariant systems such as electric circuits. It dramatically reduces the complexity of the mathematical calculations required to solve integral and differential equations. Furthermore, it has many other important applications in areas such as physics, control engineering, signal processing, and probability theory. 

The main properties of the Laplace transform that can be used in solving diffusion equations are as follows ... 

Besides the above properties, an inverse Laplace transform is often applied by using convolution to solve the diffusion equations, thus,... 

Here is an example solving a non-steady diffusion equation using the Laplace transform and the inverse Laplace transform. According to Fick s second law, the diffusion equation can be expressed as... 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

Since the method is based on the operation of the Laplace transformation, a digression on the nature of this operation is given before using it to solve the partial differential equation involved in nonsteady-state electrochemical diffusion problems, namely. Pick s second law. 

The solution of Pick s second law is facilitated by the use of Laplace transforms, which convert the partial differential equation into an easily integrable total differential equation. By utilizing Laplace transforms, the concentration of diffusing species as a function of time and distance from the diffusion sink when a constant normalized current, or flux, is switched on at f = 0 was shown to be... 

When the transport of reactants is controlled by linear diffusion, the kinetic analysis can be performed using convolution potential sweep voltammetry [182]. Here it is more convenient to choose one of the reactant concentrations to be equal to zero, i.e., the initial conditions are recovered at sufficiently negative or positive potentials as in linear potential sweep voltammetry. By using the Laplace transform and the convolution theorem in solving the second Fick equation for each reactant, the convolution current m. 

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