The Reference Time

If the reference time f and the current time t coincide then the reference and current positions will also coincide and the right-hand side of Equation (3.77) can be replaced by the reference position defined as x in Equation (3.76). In a velocity field given as = u x,t ) the motion of a material point can be described... 

In this example we will repeat the previous study but now slow down the action by reducing the free-moving probability of the ingredient to Pm = 0.5. Repeat the same calculations as in Study 2.1a, but now using 100 iterations as the reference time. In this case plot d) against for = 0, 10, 20, 30,..., 100. Again, comment on your results. How does d) after 60 iterations compare with the corresponding result from Study 2.1a above Explain any difference. 

Some BOMs have validities start and end of validity interval and the reference time used to determine if the product flow is valid or not earliest start, due date, start or end of production. 

To elucidate the possible causes of the decrease in suppression potential, the effects of flow residence time and relative pulsating fuel amount were examined. One possible explanation for the above trend is the reduction in flow residence time as the flow rate was increased. At these conditions some of the larger fuel droplets that persisted in the downstream may not have had enough time to react completely if the residence time became very short. When the residence time was estimated by the reference time scale which is the combustor length divided by inlet velocity (Fig. 21.12), the general trend appears to be consistent with the expectation. The scatter in the plot reflects the crudeness of the estimation larger droplets do not follow the carrier flow very well. 

The reference time x0 in Equation 13.14 represents the resource depletion time at which the abundance factor a, is at the value of exactly one half (i.e., 50%). This reference time x0 must be given an appropriate value at which the factors a, adequately reflect the differences in the abundance of real resources. For example, a reference time x0 of 1000 years will give fossil fuels abundance factors roughly ranging from 0.1 to 0.5, while sunlight, probably available for billions of years, will have an abundance factor approaching unity. 

Note the close resemblance between the reference time of the internal rotary diffusion of the magnetic moment and the Debye time... 

In the opposite limit, a —> 0, all the decrements, including A], tend to the sequence A/ 1(1 I lj and thus become of the same order of magnitude. This regime corresponds to a vanishing anisotropy so that the difference between the inter- and intrawell motions disappear, and the magnetic moment diffuses almost freely over all the 4n radians with the reference time xD given by Eq. (4.91). 

The reference time scale r depends on the system to be simulated, as will be seen in the next section, where some model systems are described. There, the characteristic distance 6 will also be defined as used in this book (Sect. 2.4.1). Other variables that are normalised are the current and electrode potential. Current i is proportional to the concentration gradient, by Fick s first equation (2.2), as expressed in (2.9). We introduce the dimensionless gradient or flux, defined as... 

The reference time may be an arbitrarily chosen time, but normally it is connected with the nature of the phenomenon observed. In these cases the reference time has a very specific meaning ... 

Compounding factors to give future worths for cash flows which occur uniformly before the reference point. The basis for these factors is a uniform and continuous flow of cash amounting to a total of one dollar during the given time period of T years, such as for construction of a plant. The factor converts this one dollar to the future worth at the reference time and is based on Eq. (23). 

As an example, for the case of continuous compounding at r equivalent to 20 percent for a period from 3 years before the reference time, the appropriate factor, as shown in Table 3, is... 

OCP, or the rest potential of an electrode, is the potential of a freestanding electrode without electrical connection to any other conducting materials. Thus, at OCP there is no net current flow in or out of the electrode. OCP of an electrode is determined by the kinetic state of the electrode. It is the most easily measurable electrochemical parameter and at the same time is the most convoluted quantity as it is determined by aU the kinetic factors in the system. The electrode at OCP can be at an equilibrium state or a nonequiUbrium state depending on the nature of the particular electrode/electrolyte system and the reference time scale. 

To make this result more general we can nondimensionalize both axes. The concentration of dye can be referenced to the maximum concentration at time zero Cdya[0]. But what of the time axis How shall we nondimensionalize this We will use the "holding time" as the reference time. The holding time is the time required for a volume of liquid equal to the volume of the unit to pass entirely through the unit. This is the ratio of the volume to the flow rate, that is, T = We can remake the graph in nondimensional form ... 

This is a discontinuous method necessitating the preparation of a great number of samples, cured at various values of temperature and time so as to be able to obtain a law expressed in terms of temperature and time. The principle of the reference temperature or of the reference time was used many times by various authors, and considered as the basis of their method. A modem method of calculation was used at that time, in spite of the fact that an analytical solution exists for the profile of temperatme developed through the thickness of the sample when heated in the mold. 

The reference time, i f and concentration, Cp. f, are chosen for a specific application (e.g., in a flow reactor, the mean residence time and feed concentration, respectively). Equation 5.2.C-6 now permits a solution for the amount of poison, /Cpia, to be obtained as a function of the bulk concentration, Cp, and the physicochemical parameters. In a packed bed tubular reactor, Cp varies along the longitudinal direction, and so Eq. 5.2.C-6 would then be a partial differential equation coupled to the flowing fluid phase mass balance equation—these applications will be considered in Part Two—Chapter 11. 

Fig. 4.11(b)) is produced, which is in general measured with respect to t = 0 as the reference time — i.e. A(0 ) = 0. After a sufficiently long time, the deformation of the fluid reaches a steady state namely, the deformation proceeds with a constant rate-of-strain, Ao . The strain value Aq, obtained by extrapolating the strain in the steady-state region to the time t = 0, is a viscoelastic quantity of special meaning. We define the steady-state compliance as the ratio of Aq to the apphed constant stress [Pg.66]

Following the treatment [101], it is believed that the reference times of random influences on fractal aggregation process of macromolecules are much less than the reference times of the aggregation itself and, consequently, it is possible... 

Normally, the equations as presented above, are rendered dimensionless (normalized) by expressing the variables as multiples of reference values. For the Cottrell system, the reference time value is the observation time, or duration of the experiment, t concentrations are referred to c and distance x to a suitable length scale 5. In view of the solution of Eqs. (1 and 2) (see Chapter 2.2, this volume and Ref. [2]), this scale is conveniently defined as 5 = /Dr, which means that the concentration profile will extend into the solution bulk by only a few 5-units. We then have the following... 

Figure 8.13 Influence of wavelength in the measurement of the reference time obtained by injection of (a) micellar solution, and (b) water. A 0.05 M SDS mobile phase without modifier was used. Wavelengths are given in nm. Reprinted fi om Ref. 25.

For purely batch reactors, the reference time is naturally the residence time, as a function of which the conversion in the reactor is usually described. This batch time can also be used for analysis of semi-batch reactors. Nevertheless, as the reactant introduction in semi-batch reactors can have a drastic influence on the reactor performance, the feed time is more preferably used as the reference time 4]. 

I n contrast to the process fundamental times described below, the reference time is an operational parameter of the system. It can be experimentally modified, which generally leads to a change in the reactor performance. Subsequent steps of this analysis will consist in relating these performance variations to the reference time and the characteristic times of the phenomena involved. 

Regardless of the reference time, all usual physical and chemical phenomena can be described by a fundamental characteristic time, which is peculiar to it From a general point of view, a characteristic time can be defined as the time required for a physical/ chemical system governed by this phenomenon to evolve from a non-equilibrium state to its equilibrium. In practice, they can often be considered as the ratio of a quantity of extensity to the exchanged flux or the transformation rate of this extensity. For example, a reaction time relates a mole quantity to the molar transformation rate and can be simplified as the ratio of the concentration [molm ] to the reaction rate [molm s ]. Similar relations can be developed for the heat-transfer time and mass-transfer time, where the extensities considered are the heat and the mass, respectively. 

For the phenomena presented in Table 2.1, efficiency can be related to characteristic times by writing a balance of the extensity concerned. For a chemical plug-flow reactor (with an apparent first-order reaction or with heat/mass transfer at constant exchange coefficient), the quantity of this extensity is linearly related to its variation with respect to the reference time, yielding ordinary differential equations such as... 

For these first-order cases of plug-flow reactors, the efficiency can be more precisely expressed as a function of the ratio of the reference time x to the concerned characteristic time of the phenomenon top as... 

First, the value of the global operation time can be identified from experimental or simulation data this is made possible by fitting the efficiency as a function of the reference time using expressions such as Equations (2.2) or (2.3) or appropriate expressions corresponding to the apparent system order. For example, for first-order systems, the operation time is the time required to reach 63% efficiency. Then, using appropriate literature results or correlations enables to estimate the fundamental times involved in the studied system, using expressions presented in Table 2.1. 

Whereas process intensification may aim at different objectives, these goals can generally be reformulated as productivity increase or equipment miniaturization. These two aspects are shown to converge towards one goal using the idea of reference time described above. Indeed, for batch and semi-batch processes, productivity increase is reached by reduction of the batch time or feed time. For continuous processes, miniaturization requires the volume reduction, but is also constrained by the fact that the production flow rate must be maintained. This constraint implies that the space time must be reduced. As a result, in all continuous and discontinuous cases, miniaturization and intensification both require to reduce the reference time t. 

As demonstrated above, the efficiency of a system increases with the NOU, i.e. with the ratio of the reference time to the characteristic operation time. So as to satisfy the constant productivity constraint, this time ratio must be maintained. Therefore, the characteristic operation time must be reduced by the same factor as the reference time. 

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