Time-lag method

There are, however, other options for treating data from both first- and second-order kinetics. Collectively, they are known as time lag methods. These methods are primarily of historical interest, although the mathematical rearrangements provide insights into the nature of the functions involved. 

Nonlinear least-squares programs have made time lag methods much less important. They are less accurate, for one thing. For another, the linearity of the appropriate plots, although a necessary consequence of first-order kinetics, does not constitute a proof of first-order kinetics. Certain other kinetic equations also lead to linear plots of either function. For example, Problem 2-11 presents data for a product-catalyzed reaction. The data in this case can be plotted linearly according to the Guggenheim equation, although the reaction does not follow first-order kinetics and the plot of In [A] versus time is decidedly nonlinear. 

A time lag method can also be used. With the usual symbols, this can be written... 

The use of the time lag method for extracting the diffusion coefficient is straight-forward with constant D. It can be shown that the steady-state portion of the q(t) vs. t plot can be expressed by... 

The advantage of using the time lag method is that the partition coefficient K can be determined simultaneously. However, the accuracy of this approach may be limited if the membrane swells. With D determined by Eq. (12) and the steady-state permeation rate measured experimentally, K can be calculated by Eq. (10). In the case of a variable D(c ), equations have been derived for the time lag [6,7], However, this requires that the functional dependence of D on Ci be known. Details of this approach have been discussed by Meares [7], The characteristics of systems in which permeation occurs only by diffusion can be summarized as follows ... 

Steady state methods used to estimate transport parameters [150,151], require the use of the general fate and transport equations, which include three different techniques (1) decreasing source concentration, (2) time-lag method, and (3) root time method. The next sections present these methods. 

This method is commonly used to obtain the diffusion coefficient through porous membranes. The schematic diagram illustrating the best technique for application of the time-lag method for determination of diffusion transport is shown in Fig. 4. As in the test setup shown in Fig. 4 a, the soil is contained between the source and collection reservoirs. Using this technique for diffusion coefficient determination of pollutants requires that the following conditions are satisfied ... 

Fig. 4a - c. Schematic diagram illustrating the time-lag method for determination of diffusion transport of organic pollutants, as follows a column setup b pollutant concentration vs time in source and collection reservoirs during the test c 2 amount of pollutants (i.e., Qt) transported through solid particles with time after the test... 

Tiag = the time needed to achieve e - )le (63%) of the steady-state permeation level (time-lag method),... 

Example 9-5. When measuring the gas permeation through a film one obtains a time-axis intercept of the steady-state permeation asymptote of 0 = 254 min using the time-lag method. The thickness of the film being studied is 75 pm. The pressure difference (Ap) between the two sides of the film remains constant at 0.2 bar and the flux through the film is 2 cwW h. Calculate the value of the solubility coefficient 5. 

Procedures to remove the restrictions of the permeation technique, also inherent in the time lag method, have been described by Grachev et al. [13] and Gibilaro et al. [14], As with the Wicke-Kallenbach method, they are based on the application of a carrier gas. Details of these methods may be found in Ref. 1. 

The described method above is known as the time lag method. It was proposed first by Grachev et al. [19] as a modification of Barrer s method for the determination of the permeability or Knudsen diffusivity [20]. His measurements were conducted with a single gas imposing a pressure drop across the sample. Note that, for effective diffusivity measurements, equal total pressures must be maintained on the two sides of the diffusion cell unless Knudsen diffusion prevails. 

In the characterization of porous membranes by liquid or gaseous permeation methods, the interpretation of data by the hyperbolic model can be of interest even if the parabolic model is accepted to yield excellent results for the estimation of the diffusion coefficients in most experiments. This type of model is currently applied for the time-lag method, which is mostly used to estimate the diffusion coefficients of dense polymer membranes in this case, the porosity definition can be compared to an equivalent free volume of the polymer [4.88, 4.89]. 

Since the product Dk is known from the steady state rate of permeation, kp can also be obtained. This time lag method is the basis of most of the gas and some of the vapor transport studies made today. Little application of the time lag method was made until Barter introduced the use of vacuum on the downstream side of the membrane and measured the gas permeation rate by monitoring the increase in pres-arre in a fixed downstream receiving volume Recently the original isobaric method has been reintroduced in a number of commercial permeability instruments. 

The concept of two or mote modes of sorption of penetrants in polymers is very familiar to cellulose and protein chemists for the case of water vapor. In fact combined Lan uir and Henry s law sorption was proposed and correctly formulated by Matthes in 1944 for water in cellulose The discovery of dual mode sorption of gases in assy polymers and the aibsequent realization that diffusion constants determined by the time lag method did not have the same ple fundamental significance a ociated whh these parameters for mbbery polymers was of profound importance. Not only were the many carefully determined diffusion coefficients in the literature of questionable value for polymers below their ass transition but a good deal of the careful peculation about solution and diffusion and the effect of... 

To check the validity of the time-lag method for estimating solubility coefficients in these systems, static sorption measurements were conducted with CO2. Such measurements are difficult owing to the very low level of sorption. The solubility coefficient obtained is given in parenthesis in Table 2. Clearly, the agreement with those obtained from transient permeation experiments is not perfect however, the values obtained by the two approaches are close enough to confirm the general conclusions discussed above. 

Some of the parameters used to evaluate membranes are the diffusion coefficient, D the permeability coefficient, P the solubility constant, S the filtration coefficient, L, the solute permeability coefficient, to and the reflection coefficient, o. The diffusion coefficient can be obtained by the time lag method illustrated in Figure 35-9. 

Figure 35-9. Illustration of the time lag method for diffusion through a polymer membrane. (Courtesy - Garbarini, G.R., Eaton, R.F., Kwei, T.K., and Tobolsky, A.W.,J. Chem. Ed., 48 (4), 226, 1971)...

Radiation heat transfer, as used in the simplified time lag method for creating furnace heating curves (temperature vs. time) is really an average condition of the gas blanket temperature, gas blanket thickness, and vapor pressure of triatomic gases. With high excess air, the heat transfer will be less due to lower percentages of the... 

To increase the rate of heat transfer above that determined by the simple time-lag methods ... 

Macroscopic, such as the analysis of uptake curves, Wicke-Callanbach methods based on steady-state or transient diffusion cell, time lag method, chromatographic methods, zero length column (ZLC) method, and FR method... 

The remainder of the book deals with various methods commonly used in the literature for the measurement of diffusivity. We start with Chapter 12 with a time lag method, which belongs to the class of permeation method, of which another method employing a diffusion cell is presented in Chapter 13. The time lag method was pioneered by Barrer in the early 50 s, and is a very useful tool to study diffusion through porous media as well as polymeric membranes. Chromatography method is presented in Chapter 14, and finally we conclude with a chapter (Chapter 15) on the analysis of batch adsorber. 

We will first illustrate the time lag method with a simple case of non-adsorbing gas and conditions are chosen such that the transport mechanism is due to the Knudsen mechanism. Diffusion of oxygen, nitrogen, argon, krypton, methane and ethane through inert analcite spherical crystals (Barrer, 1953) at low pressure is an example of non-adsorbing gas with Knudsen flow. Conditions of the experiments are chosen such that the diffusion into the crystals does not occur and flow is restricted to the Knudsen mechanism around the individual crystallites in the bed. The time lag method can be used to complement with the steady state method by Kozeny (1927), Carman (1948) and Adzumi (1937). 

This time lag (12.2-32) using the pressure response of the supply reservoir or the time lag (eq. 12.2-24) using the pressure response of the receiving reservoir can be used to determine the diffusivity. Thus, the time lag method provides a very convenient if not a straightforward method to determine the diffusion coefficient. The method is not restrictive to the simple Knudsen diffusion mechanism, it is also applicable to other situations, for example... 

The two time lags in eqs. (12.2-38) and the two in eqs. (12.2-39) make up four time lags this method can provide. This is commonly referred to in the literature as the four time lag method, and either one of them can be used to determine the diffusivity or for the certainty of parameter determination, more than one time lag can be used. 

The analysis so far dealt with the time lag method for diffusing (non-adsorbing) gases. The method can be applied to adsorbing gases or vapours as well. This section and the subsequent sections will show the applicability of the time lag method to adsorption systems and how adsorption and diffusion parameters can be extracted from the analysis. 

---