Time-lag analysis

This chapter has addressed the method of time lag, and we have shown its application to a large number of diffusion and adsorption problems to show its utility in the determination of the diffusion coefficient as well as adsorption parameters. The central tool in the time lag analysis is the Frisch s method, and such a method has allowed us to obtain the expression of the time lag without any recourse to the solution of the concentration distribution within the medium. We shall present in the next few chapters other methods and they all complement each other in the determination of parameter. 

Time lag analysis of restoration processes (autocorrelations and crosscorrelations of restoration curves)... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

Its main features are given by the use of a stream of inert carrier gas which percolates through a bed of an adsorbent covered with adsorbate and heated in a defined way. The desorbed gas is carried off to a detector under conditions of no appreciable back-diffusion. This means that the actual concentration of the desorbed species in the bed is reproduced in the detector after a time lag which depends on the flow velocity and the distance. The theory of this method has been developed for a linear heating schedule, first-order desorption kinetics, no adsorbable component in the entering carrier gas (Pa = 0), and the Langmuir concept, and has already been reviewed (48, 49) so that it will not be dealt with here. An analysis of how closely the actual experimental conditions meet the idealized model is not available. 

Cohen and Coon observed that the response of most uncontrolled (controller disconnected) processes to a step change in the manipulated variable is a sigmoidally shaped curve. This can be modelled approximately by a first-order system with time lag Tl, as given by the intersection of the tangent through the inflection point with the time axis (Fig. 2.34). The theoretical values of the controller settings obtained by the analysis of this system are summarised in Table 2.2. The model parameters for a step change A to be used with this table are calculated as follows... 

We know from frequency response analysis that time lag introduces extra phase lag, reduces the gain margin and is a significant source of instability. This is mainly because the feedback information is outdated. 

Hence, film rupture is much faster than either film transport or film drainage and filling. In this initial analysis we therefore neglect any time lag for the rupture event and assess breakage as... 

Barrer (19) has developed another widely used nonsteady-state technique for measuring effective diffusivities in porous catalysts. In this approach, an apparatus configuration similar to the steady-state apparatus is used. One side of the pellet is first evacuated and then the increase in the downstream pressure is recorded as a function of time, the upstream pressure being held constant. The pressure drop across the pellet during the experiment is also held relatively constant. There is a time lag before a steady-state flux develops, and effective diffusion coefficients can be determined from either the transient or steady-state data. For the transient analysis, one must allow for accumulation or depletion of material by adsorption if this occurs. 

A seminal contribution to understanding the role of money in expanded reproduction has been provided by Foley (1986) in chapter 5 of his book, Understanding Capital. This is an extremely detailed model of all the facets of Marx s system of reproduction, including the complex role of time lags between various activities and the way in which capital is transformed into its different forms. In the analysis that follows a stripped down version of this model is presented. 

Figure 8.8 In impedance analysis, a sinusoidally varying potential V is applied across a sample, and the time-dependent current / is measured as a function of the frequency co. The current induced in response to the varying potential will be out of phase, by a time lag 9, and of different magnitude.

IV.b. Drug Distribution and Analysis of Time Lag Between Concentration and Effect... 

When analysing the discrepancies (L, etc.) between the observed time lag parameters (La, etc.) and the calculated ideal values (La, etc.), one must take into account possible contributions from causes other than dependence of S, DT on X. Analysis of the case of simultaneous dependence of S, Dx on X and t shows that 4 135>... 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

The resulting trajectories are usually analyzed for the mode of motion executed by the particle as the motion provides information on the location and status of the particle as described in this review. The most common analysis starts with the calculation of the so-called mean-square displacement (MSD). Then, the time dependence of the MSD is plotted. This plot allows a mode of motion analysis [35]. A simplified way to calculate the MSD is depicted in Fig. 2b. The mean square displacement describes the average of the squared distances between a particle s start and end positions for all time-lags of certain length At within one trajectory. 

The seriousness of this oversight is apparent in Sefcik and Schaefer s analysis of Toi s transport data (24) in terms of their NMR results (28) The value of the so-called "apparent" diffusion coefficient calculated from Toi s time lag data increases by 25% for an upstream pressure range between 100 mm Hg and 500 mm Hg On the other hand, the value of Deff(c) calculated from Toi s data changes by 86% over the concentration range from 100 to 500 mm Hg The difference in the two above coefficients arises from the fact that Da is an average of values corresponding to a range of concentrations from the upstream value to the essentially zero concentration downstream value in a time lag measurement Deff > on t le other hand, has a well-defined point value at each specified concentration and is typically evaluated (independent of any specific model other than Fick s law) by differentation of solubility and permeability data (22) ... 

The processing of the data consisted of extracting the steady state slopes and the time lags from the pressure versus time traces, converting the slopes to permeances using the equations given above, and tabulating the results for subsequent statistical analysis. 

Note that Q(23)=e1JQ + a. A measurement on an individual specimen can be expected to deviate from this estimate because of measurement error,and because of the specimen effect embodied in the S. Values of Q(23) and 3 obtained from the analysis of the permeance and the time-lag data are given in Tables I and II respectively. Since the time-lag for carbon dioxide depends upon the upstream pressure it is necessary to multiply the estimates obtained from equation 8 by a term of the form ... 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

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