First-order time lag

Include a first-order time lag in the temperature measurement, using the... 

Process elements which are describable by Eq. (3) are often called first order time lags or first order RC stages, since they cause the output signal to lag behind the input signal and since the time constant in each case is the product of a resistance term and a capacitance term. For... 

In case of a first-order time lag of Eqn. (5.10), the pulse transfer function can be found as follows (Fig. 5.5). 

Process dynamics. Emulation of a first-order process lag (or lead) and time delay. 

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 ... 

The experimental curve in Figure 3 demonstrates overshoot in the tissue oxygen response. It was determined previously (22) that a term representing pure delay along with the steady-state blood flow vs. arterial oxygen tension data would cause overshoot. In this investigation it was found that a first-order time constant delay would also produce overshoot. Therefore, since exact controller mechanisms are not being postulated, the flow controlled dynamics used in this study include pure delay and time constant lag. To consider the problem of sensor location, feedback and feedforward control loops were superimposed on the capillary-tissue model. 

The process flow diagram in Figure 10.1 shows the concept of an open-loop process response that includes a dead time plus two first-order capacitance lags. The process water that is heated with live steam injection moves in plug flow through a... 

Figure 10.1 Process concept for modeling dead time plus two first-order capacitance lags.

A computer simulation of a process with a dead time, DT, of 2 min and a first-order capacitance lag, 7, of 22 min gave an open-loop response shown in Figure 10.2. This was compared with an open-loop response with DT = 2, 7 = 20, and 7 = 2 min. Adding the second capacitance lag, 7, caused the process variable response to be rounded a little after the dead time, which is typical of a plant process response. 

The two most common temporal input profiles for dmg delivery are zero order (constant release), and half order, ie, release that decreases with the square root of time. These two profiles correspond to diffusion through a membrane and desorption from a matrix, respectively (1,2). In practice, membrane systems have a period of constant release, ie, steady-state permeation, preceded by a period of either an increasing (time lag) or decreasing (burst) flux. This initial period may affect the time of appearance of a dmg in plasma on the first dose, but may become insignificant upon multiple dosing. 

First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives ... 

Higher-Order Lags If a process is described by a series of n first-order lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by ... 

In an ideal first-order system, only one capacity causes a time lag between the measured quantity and the measurement result. Typically, an unshielded thermometer sensor behaves as a first-order system. If this sensor is rapidly moved from one place having temperature Tj to another place of temperature T2, the change in the measured quantity is close to an ideal step. In such cases, the sensor temperature indicated by the instrument has a time histoty as shown in Fig. 12.13. 

Figure 4.2 shows the computer simulation results of such a dynamic aeration experiment. The y-axis shows the response fraction with respect to time. The gas phase response is typically first-order, and the liquid phase shows some lag or delay on the signal. The electrode response is much more delayed for a slow-acting electrode.4... 

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. 

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 model developed by Leksawasdi et al. [11,12] for the enzymatic production of PAC (P) from benzaldehyde (B) and pyruvate (A) in an aqueous phase system is based on equations given in Figure 2. The model also includes the production of by-products acetaldehyde (Q) and acetoin (R). The rate of deactivation of PDC (E) was shown to exhibit a first order dependency on benzaldehyde concentration and exposure time as well as an initial time lag [8]. Following detailed kinetic studies, the model including the equation for enzyme deactivation was shown to provide acceptable fitting of the kinetic data for the ranges 50-150 mM benzaldehyde, 60-180 mM pyruvate and 1.1-3.4 U mf PDC carboligase activity [10]. 

It is possible to distinguish between direct and indirect nOes from their kinetic behavior. The direct nOes grow immediately upon irradiation of the neighboring nucleus, with a first-order rate constant, and their kinetics depend initially only on the intemuclear distance r" indirect nOes are observable only after a certain time lag. We can thus suppress or enhance the indirect nOe s (e.g., at He) by short or long irradiations, respectively, of Ha- a long irradiation time of Ha allows the buildup of indirect negative nOe at He, while a short irradiation time of Ha allows only the direct positive nOe effects of Ha on He to be recorded. 

Complex models are often slow in execution owing to the large number of equations involved and the large range of time constants. Under these circumstances it is often useful to approximate the transient behaviour of the full model by a simpler model representation which is faster to compute. Such simplifications are commonly achieved by a combination of first-order lags and time delays and are often represented in Laplace transform form, especially when the sub-model is to be used as part of a control engineering application. 

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. 

Experience has shown, that most chemical processes can often be modelled by a combination of several first-order lags in series and a time delay (Fig. 2.22). 

L(f(c)) = e-tDS Hence for a single first-order lag with time delay... 

Figure 2.23. Fitting of measured response to a first-order lag plus time delay.

For a better fit of the system response, the method of Oldenbourg and Sartorius, as described in Douglas (1972), using a combination of two first-order lags plus a time delay, can be used. The method is illustrated in Fig. 2.24. and applies for the case... 

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... 

K and KH in (2-49a) are referred to as gains, but not the steady state gains. The process time constant is also called a first-order lag or linear lag. 

Example 8.6. What are the Bode and Nyquist plots of a first order lag with dead time ... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

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