Dispersion Laplace transform

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

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. 

As was done with the dispersion models, the mean and the variance can be found either directly from Eq. (114) or from the Laplace transform, Eq. (112). The results are,... 

From these relations we see that the width and shift of the power spectrum and consequently the spectroscopic lines are related through the Kronig-Kramers dispersion relations. Exactly the same arguments apply to the Laplace transform of the time-correlation function, H(/co). The real and imaginary parts, C H(co) and C"//(/(0), are related by Kramers-Kronig dispersion relation. 

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 . 

The relationship for the interpretation of data measured for RTD can be easily derived with Laplace transformation [54]. However, the equations so obtained are not convenient for time-dispersed measurement because they still need Laplace transformation to deal with the data. In the following, another relationship will be derived. 

The exit concentration Cm(z — 1, t) for the case of a unit impulse (Delta function) input (f(z) — 0, git) — S(i)) is known as the dispersion (or RTD) curve. For the hyperbolic model, this can be found either by Laplace transformation or from the general solution of the model [see Balakotaiah and Chang (2003), for a general analytical solution of Eqs. (57)—(59)]. It is easily seen that the Laplace transform of the dispersion curve is given by... 

Another stochastic model (4.27)-(4.32) treatment can be made tvhen the aim is to calculate the average time of residence and the axial dispersion coefficient. In this problem, we use the properties of the characteristic function, ivhich is associated ivith the distribution function of the average time of residence [4.28, 4.29]. For this analysis we start ivith the Laplace transformation of the stochastic model ivhen the system (4.31)-(4.32) is considered ... 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

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. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

More complicated reactions can be easily treated by the methods outlined in the preceeding sections, that is (a) determine the coupled diffusion-chemical reaction equations, (b) linearize the equations in the concentration fluctuations, (c) solve the linearized rate equations by Fourier-Laplace transforms, (d) solve the dispersion equation... 

For a general rate model including axial dispersion mass transfer (Equation 6.80), (apparent) pore diffusion (Equations 6.82, 6.84 and 6.85), and linear adsorption kinetics (Equations 6.32 and 6.33), Kucera (1965) derived the moments by Laplace transformation, assuming the injection of an ideal Dirac pulse. If axial dispersion is not too strong Pe S>4), the equations for the first and second moments can be simplified to (Ma, Whitley, and Wang, 1996)... 

Laplace transform of C axial dispersion coefficient, m /s Henry s law solubility constant, Cl/Cq. adsorption equilibrium constant radial distance in the catalyst particle, radius of the catalyst particle Laplace transform variable t ime, s. 

In the case of a dispersion model, for instance, transfer function G(p) is given as Eq. (6-59). Transfer function is defined as the ratio of the Laplace transform of the elution concentration curve and that of the input concentration curve, the latter of which is a constant in the case of impulse input. Laplace parameter, p, is a complex variable but if a response curve, C(r), is transformed by using Eq. (6-8) by assuming p as a real parameter, then the resultant C(p) gives a transfer function G(p) by dividing by the size of pulse, M, in a real plane. C(p) is then compared with the solution of basic equations obtained in a Laplace domain. 

After assuming a flow model based both on the physical structure of the reactor and the characteristic features of the experimental RTD curve, mass balance equations are written for the dispersion of an inert tracer among the various zones involved in the model. These equations are linear differential equations (ODE or PDE) owing to the linear character of mixing processes. By solving the equations in the Laplace domain, the theoretical transfer function G (s,p ) is obtained, which is nothing but the Laplace transform of the - theoretical RTD E (t,pj ) where pj are the parameters of the model... 

This finding offers the opportunity to tailor the thermal and mechanical properties of the photoconductor without seriously affecting the transport properties. This can be achieved either by variation of the spacer length between the chromophore and the backbone or by variation of the backbone itself A typical example for this group is PVK [12, 32, 34, 35]. Here, the transition from dispersive to non-dispersive transport could be observed [34]. Due to the fact, that the TOP curves have been measured over 10 decades in time, it was possible by means of a numerical inverse Laplace transform [34] to calculate from the measured photocurrent the trapping rate distribution, which is a measure for the density of localized states. 

Proof. The existence can be established by applying Volterra integral equation techniques to the integral version of problem (3.21-25) or by Laplace transform methods [ 6,thm.IV.B1]. Here we shall concentrate only on the asymptotic behavior which can- be observed directly from the dispersion relations obtained in the... 

Moench, A.F. 1989. Convergent radial dispersion A Laplace transform solution for aquifer tracer testing. Water Resources Research 25(3) 439 47. 

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