Exponential lag

These lags are cumulative as the liquid passes each tray on its way down the column. Thus, a 30-tray column could be approximated by 30 first-order exponential lags in series having approximately the same time constant. The effect of increasing the number of lags in series is to increase the apparent dead time and increase the response curve slope. Thus, the liquid traffic within the distillation process is often approximated by a second-order lag plus dead time (right side of Figure 2.82). 

Each valve will be driven by a valve-positioner, which is a servomechanism designed to drive the valve travel, jc, to its demanded travel, xj. This valve positioner will take a certain time to move the valve, and we will use the simplest possible model of the dynamics of the valve plus positioner, namely a first-order exponential lag ... 

Referring to the box marked Measurement in Figure 22.1, the measurement signal, 9 , will follow the true plant signal. Bp, with a small lag, depending on the physics of the measurement system. It will normally be sufficient to model this as a simple exponential lag ... 

As before, it is possible to model the behaviour of R by a first-order exponential lag if necessary ... 

The dynamics of the valve-positioning system may be described in terms of its small-signal and large-signal characteristics. The small-signal characteristics may be represented to reasonable accuracy by a first-order exponential lag, which will obey the canonical equation ... 

The measurement of differential pressure will be an inherently linear operation, subject to an exponential lag of typically about 0.5 seconds. The mathematical description is thus ... 

A valve may usually be modelled as an exponential lag, subject to rate limits. For small-signal, linear analysis, the rate limits will not be breached, so the model is simply ... 

Ri exponentially lagged version of X R-dimensional vector of system ... 

An exponential approximation of slewing is now considered. Assume that the valve can move from its zero position to its 100% position in Tioo seconds, at a constant rate, when a step input signal is applied at t = 0 seconds. Assume that an equivalent exponential lag term responds to the same step input over the same period of Tioo seconds. Figure 2.18 shows the two responses referred to a common base of time. A good measure of fit can be made by choosing the time constant Tfa such that the area represented by the lower part (A) equals that represented by the upper area (B). This is determined by equating these two areas. The areas are found by integration. Area (A) is found by... 

The death rate coefficient is usually relatively small unless inhibitoiy substances accumulate, so Eq. (24-10) shows an exponential rise until S becomes depleted to reduce [L. This explains the usual growth curve (Fig. 24-21) with its lag phase, logarithmic phase, resting phase, and declining phase as the effect of takes over. 

Batch fermentation is the most widely used method of amino add production. Here the fermentation is a dosed culture system which contains an initial, limited amount of nutrient. After the seed inoculum has been introduced the cells start to grow at the expense of the nutrients that are available. A short adaptation time is usually necessary (lag phase) before cells enter the logarithmic growth phase (exponential phase). Nutrients soon become limited and they enter the stationary phase in which growth has (almost) ceased. In amino add fermentations, production of the amino add normally starts in the early logarithmic phase and continues through the stationary phase. 

The logistic equation leads to a lag phase, an exponential initial growth rate and a stationary population of concentration (xm). In a population, it is often the case that the birth rate decreases as the population itself increases. The reasons may vary from increased scientific or cultural sophistication to a limited food supply. 

Figure 3.7 shows the growth of R. rubrum in a batch fermentation process using a gaseous carbon source (CO). The data shown follow the logistic model as fitted by (3.14.2.11) with the solid lines, which also represent an unstructured rate model without any lag phase. The software Sigma Plot was used to fit model (3.14.2.11) to the experimental data. An increase in concentration of acetate in the prepared culture media did not improve the cell dry weight at values of 2.5 and 3 gT-1 acetate, as shown in Figure 3.7. However, the exponential growth rates were clearly observed with acetate concentrations of 0.5-2 g-F1 hi the culture media.

The objective of a good process design is to minimise the lag phase period and maximise the length of exponential growth phase. 

For each run, calculate and plot the cell biomass concentration, glucose concentration, ethanol concentration, and pH as a function of time. Identify the major phases in batch fermentation lag, exponential, stationary and death. 

A simple way to model the lag phase is to suppose that the maximum growth rate fimax evolves to its final value by a first-order rate process jUmax = Moo[l — exp(—af)]. Repeat Example 12.7 using a=lh. Compare your results for X, S, and p with those of Example 12.7. Make the comparison at the end of the exponential phase. 

Often an instrument response measurement can be fitted empirically to a first-order lag model, especially if the pure instrument response to a step change disturbance has the general shape of a first-order exponential. 

Thus as shown previously in Sec. 2.1.1.1, if the step response curve has the general shape of an exponential, the response can be fitted to the above first-order lag model by determining x at the 63% point. The response can now be used as part of a dynamical model, either in the time domain or in Laplace transfer form. 

In the controlled (constant) potential method the procedure starts and continues to work with the limiting current iu but as the ion concentration and hence its i, decreases exponentially with time, the course of the electrolysis slows down quickly and its completion lags behind therefore, one often prefers the application of a constant current. Suppose that we want to oxidize Fe(II) we consider Fig. 3.78 and apply across a Pt electrode (WE) and an auxiliary electrode (AE) an anodic current, -1, of nearly the half-wave current this means that the anodic potential (vs. an RE) starts at nearly the half-wave potential, Ei, of Fe(II) - Fe(III) (= 0.770 V), but increases with time, while the anodic wave height diminishes linearly and halfway to completion the electrolysis falls below - / after that moment the potential will suddenly increase until it attains the decomposition potential (nearly 2.4 V) of H20 -> 02. The way to prevent this from happening is to add previously a small amount of a so-called redox buffer, i.e., a reversible oxidant such as Ce(IV) with a standard... 

Figure 8.1. Schematic response (solid curve) of a first order function to a sinusoidal input (dashed). The response has a smaller amplitude, a phase lag, and its exponential term decays away quickly to become a pure sinusoidal response. 

An exponential (or logarithmic) growth phase follows the lag phase, and during this period the cell mass increases exponentially. The growth rate is at a maximum during this phase, and the population of cells are fairly uniform with respect to chemical composition and metabolic composition. 

Suitable rate expressions for rs and rx and the specification of the initial conditions would complete the batch fermenter model, which describes the exponential and limiting growth phases but not the lag phase. 

However, precise fits require the use of at least second order models, which include three characteristic phases in the biodegradation process [4] (i) an acclimatisation or lag phase (ii) an exponential phase with most of the effective degradation and (iii) a final phase in which the microorganism population is stable and some portion of the substrate could persist (see Fig. 5.3.1(a) [21]). 

Initially, the cells require a period of time, the lag phase, to adjust to the altered environment. No cell multiplication occurs until the individual cells have almost doubled in size. After the adaptation is complete, the cells begin to reproduce exponentially. With each period of cell division, the population doubles. Some cells... 

If the per os meal consists of liquid and solid components, gastric emptying exhibits a biphasic mechanism. With the exception of emptying of solid particles in MMC phase III, gastric emptying of solids into the duodenum takes place only if these particles are smaller than 1—3 mm in diameter (43,52). These particles are emptied, after a short lag phase, according to linear kinetics, whereas the liquid fraction often exhibits exponential or biphasic-(exponential) release kinetics (53-55). 

The feedforward controller contains a stcadyslate gain and dynamic terms. For this system the dynamic element is a first-order lead-lag. The unit step reaponae of this lead-lag is an initial change to a value that is (—followed by an exponential rise or decay to the final steadystate value... 

Note that this is Jiot a hrsl-order lag because of the negative sign in the denominator. The system has an openloop pole in the RHP at s = -I- 1/tj,. The unit step response of this system is an exponential that goes olT to inhnity as time increases. 

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