Laplace transforms method

Then Eqs. (3-130) are substituted into Eqs. (3-128), giving Ca. and cc as functions of time. The final expressions are not written here because we have already derived them by the Laplace transform method they are Eqs. (3-99), (3-101), and (3-103), with X2 and X3 replacing a and p. 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

The solution of this equation has been discussed by DANCKWERTSt28), and here a solution will be obtained using the Laplace transform method for a semi-infinite liquid initially free of solute. On the assumption that the liquid is in contact with pure solute gas, the concentration Cm at the liquid interface will be constant and equal to the saturation value. The boundary conditions will be those applicable to the penetration theory, that is ... 

The discussion so far has been empirical in the sense that Laplace transform method has been utilized in conjunction with an experimentally determined scavenging function without a theoretical model for the recombination kinetics. A theoretical model will be attempted in the following subsections. 

There are several methods for finding particular solutions. Laplace transform methods are probably the most convenient, and we will use them in Part IV. Here we will present the method of undetermined co cients. It consists of assuming a particular solution that has the same form as the forcing function. It is illustrated in the examples below. 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

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. 

Progress has been recently made in constructing an iterative inverse Laplace transform method which is not exponentially sensitive to noise. This Short Time Inverse Laplace Transform (STILT) method is based on rewriting the Bromwich inversion formula as ... 

Using a Laplace Transform method, Rzad et al. [365] and Abell et al. [391] showed that eqn. (174) implies... 

In order to facilitate the way of notation, it is useful to represent the virtual voltage sources by SF = [SF] cos (2coLf) and Sc = [Sc] cos (2col ). Again, the Laplace transform method is most useful for solving... 

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

The derivation is most elegantly performed by the Laplace transform method, starting with the general relationship, valid for semi-infinite linear diffusion. 

Here also, the Laplace transform method is particularly useful [130]. First, the two rate equations for v1 and vu are linearized and subsequently written in the Laplace domain. 

The Laplace transform method is a powerful technique for solving a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain. After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved. The transformed solution is then back-transformed to obtain the desired solution. 

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

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). 

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

To obtain Eqs. (2.137) and (2.140), the Dimensionless Parameter Method (DPM) has been used as described in Appendix A and expressions of the concentration profiles have been obtained [52], In the 1960s, a compact analytical solution for the I-E response was obtained by using the Laplace transform method when the oxidized species was the only present in the electrolytic solution, i.e., for a cathodic wave [53, 54], and non-explicit expressions for the concentration profiles and surface concentrations were obtained. 

The solution to Eq. (2.2) can be deduced by using Laplace transform method (see Appendix B), by obtaining the following expression ... 

The above problem is solved using the dimensionless parameter method developed by Koutecky [1-3] (the Laplace transform method has been addressed in reference [4]). First, we insert the dimensionless diffusion parameter... 

This problem was solved by Delmastro and Smith by using the Laplace transform method, assuming that only oxidized species O was initially present in the solution [13], and they reported the following analytical solution ... 

When c = 0 Eq. (5.51) coincides with the following well-known expression (deduced by Nicholson and Shain at planar electrodes using Laplace transform method see Appendix H and [9]) ... 

To date, there has been no explicit solution for this problem for p > 3, since the surface concentrations of electroactive species O and R are time dependent and therefore the Superposition Principle cannot be applied (see also Sect. 4.3) [1,5]. In these conditions, a non-explicit integral solution has been deduced using the Laplace transform method (see Appendix H). 

Using the Laplace Transform method, this expression can be compared with that deduced by Polcyn and Sham for CV [7] ... 

Appendix B. Laplace Transform Method Solution for the Application of a Constant Potential to a Simple Charge Transfer Process at Spherical Electrodes When the Diffusion Coefficients of Both Species are Equal... 

To obtain p, the Laplace transformation method (see Section 2) can be used, i.e.,... 

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