Moments from Laplace transforms

Equation (15.39) allows moments of a distribution to be calculated from the Laplace transform of the dilferential distribution function without need for finding f t). It works for any f t). The necessary algebra for the present case is formidable, but finally gives the desired relationship ... 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

From this equation (or from its Laplace transform) the first and second moments are found to be,... 

It can be seen, from the complexity of Eq. (27), that to find the inverse Laplace transform in the general case would be exceedingly difficult, if not impossible. Yagi and Miyauchi (Yl) have presented a solution for the special case where Da = Dt = 0. Fortunately, the moments can be found in general from Eq. (27) without evaluating the inverse transform. They are,... 

A number of kinetic models of various degree of complexity have been used in chromatography. In linear chromatography, all these models have an analytical solution in the Laplace domain. The Laplace-domain solution makes rather simple the calculation of the moments of chromatographic peaks thus, the retention time, the peak width, its number of theoretical plates, the peak asymmetry, and other chromatographic parameters of interest can be calculated using algebraic expressions. The direct, analytical inverse Laplace transform of the solution of these models usually can only be calculated after substantial simplifications. Numerically, however, the peak profile can simply be calculated from the analytical solution in the Laplace domain. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

For temporal moments it is best to consider a semi-infinite tube and let c(0, t) = S(t) be the ideal tracer. Initially, c(z, 0) = 0 and c must remain finite as z — 00. These are not the most sophisticated conditions, as we know from Example 3, but they give the swiftest answer. If we take the Laplace transform of Eq. (261), we have... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

The Laplace transform may be inverted to provide a tracer response in the time domain. In many cases, the overall transfer function cannot be analytically inverted. Even in this case, moments of the RTD may be derived from the overall transfer function. For instance, if Go and GJare the limits of the first and... 

As mentioned earlier, a closed-form, analytical solution of the inverse Laplace transform has not been derived yet in the case of the general rate model for a pulse injection. However, Kucera [30] and Kubin [31] have shown that the first five moments of the solution can easily be calculated from their solution of the general rate model giving the band profile in the Laplace domain (Eqs. 6.58 to 6.64a). 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

These, of course, are experimentally measurable quantities from the output C L, t) measurements. The heart of the analysis is to recognize that such moments are related to the Laplace transform as... 

The second technique is that of using time moments of the data to be compared with those of the model. The advantages are that they can be easily computed by numerical integration of the data, and the integrations tend to somewhat smooth the data. Also, the moments can be readily found from the Laplace transform of the model, without the necessity of a time-domain solution— recall the formula of van der Laan [65] utilized in Sec. 12.5.b. The main disadvantage is that the moments tend to emphasize the data for large values of time, and this data in the tail of the curve is commonly not very accurate. Also it is often difficult to decide just where to truncate the tail of the data curve. In addition, the number of moments must equal the number of parameters. 

Solving the mass balance equations for the case of linear isotherm subject to the boundary conditions (13.2-17) by the method of Laplace transform and from the solution we obtain the following moments when the input is an impulse (Dogu and Ercan, 1983) ... 

In the first situation, a carrier fluid (which is usually an inert fluid but this is not necessary) is passed through the column, and once this is stabilised a tracer is injected into the column with a concentration of Co(t) at the inlet. The concentration is chosen such that the adsorption isotherm of this tracer towards the solid packing is linear. This results in a set of linear equations which permit the use of Laplace transform to obtain solution analytically. Knowing the solution in the Laplace domain, the solution in real time can be in principle obtained by some inversion procedure whether it be analytically or numerically. However, the moment method illustrated in Chapter 13 can be utilised to obtain moments from the Laplace solutions directly without the tedious process of inversion. 

These equations are linear and are susceptible to Laplace transform analysis, from which we can obtain the theoretical moments. The first normalised moment and the second central moment for this model are ... 

The solutions given in Table 8.1 were all obtained by Laplace transformation. To obtain the solution of the model equations in the Laplace domain is straightforward but inversion of the transform to obtain an analytic expression for the breakthrough curve or pulse response is difficult. Simple analytic expressions for the moments of the pulse response may, however, be derived rather easily directly from the solution in Laplace form by the application of van der Laan s theorem... 

The zero, first and second moment of the transient system can be calculated via the Laplace transform [7,8]. With a Thiele modulus greater than five, e.g. strong difhision limitation, the hyperbolic functions in the moments equations tend to their asymptotic values. It can be shown theoretically that the moments become linearly dependent and the number of model parameters reduces to two, where DrHr = constant and kr. If additionally the effect of adsorption on the dynamic response becomes negligible, the number of independent parameters, which are needed to describe the response curve, reduces to one DrH,kr = constant. The latter case can therefore not extract more parameters than could be obtain from a steady state measurement. 

A more advanced method is the inverse (discrete) Laplace transformation (or probability generating function [pgf]) method (Asteasuain et al., 2002a,b, 2004). In this method, the CLD is reconstracted from the integrated moment equations, as illustrated in Fig. 10.6. A number and mass probability are first introduced for the living polymer molecules ... 

When the inverse Laplace transformation is applied to the transmit-tances given in Table 4.4, the pulse responses of the calorimetric system given in Table 4.5 are obtained their plots are shown in Fig. 4.8. Figures 4.8a and 4.8d reveal that, when the heat source and the temperature sensor are situated in the same domain, the shape of the pulse response of the calorimetric system is reminiscent of the response of the calorimetric system of first order. From the shapes of the curves given in Figs 4.8d and 4.8c, it is clear that, when the heat source and temperature sensor are situated in different domains, the pulse response of the calorimetric system at the initial time moment is equal to zero, next increases to a certain maximum value h2 max and h 2, respectively), and then decreases to zero. 

The cumulant expansion, like the Laplace transform, is intrinsically capable of describing any physically-possible A(F). Here A is a large number determined by experimental conditions. The cumulants are related to central moments of A(F), e.g., k = r. The cumulants may be obtained from linear least-square fits to InCg P),... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

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