First-Order Lag Model

Well-mixed tank systems (Fig. 2.18) are characterised by a first-order lag response. 

Taking transforms where the variables Ci and Cq now represent deviations from initial steady state, gives 

Inversion of the Laplace transform for a step change in Cq gives the analytical solution derived previously. 

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. 

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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. 

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. 

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). 

Model 3 (three equal first-order lags) ... 

A = first-order lag and two equal first-order lags) ... 

A Show how the ATV method can be used to determine the time constants and damping coefTicient of a third-order model one first-order lag and a second-order underdamped lag, Note that there arc three unknown parameters and only two equations, but the third relationship is that a model with the largest possible damping coefTicient is desired. 

First-order absorption model with or without lag time... 

The response of electrodes used to measure pH is commonly modeled as a first-order lag ranging from one second to several minutes. The actual behavior of electrodes is known to be very complex and a number of researchers have... 

The most commonly used model of a mixed vessel is the fractional tubularity (delay-lag) model in which some part of the reactor is taken as exhibiting plug-flow conditions and contributing a delay and the rest of the reactor is taken as perfectly mixed (uniform concentrations) contributing a first-order lag (/( ,). The delay and lag in series are taken as describing the reactor residence time distribution (RTD). The delay-lag representation was validated using both CFD analysis and experimental residence time distributions (Walsh, 1993). 

To avoid the need for special procedures and modification of the integration algorithm, delays may be modeled by using rational approximations, e.g., Fade functions or multiple first-order lags in series. Experimentation suggests that 10 series lags is adequate for most applications, so this is used as a default. The approximation should be checked by comparison with a more detailed model where it is believed to be particularly significant. 

All delays were represented by 20 identical first-order lags in series. This representation is more precise than the default 10-lag model as the time delays... 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their linearized approximations and is achieved by a combination of first-order lag function and time delays. This limitation together with additional complications of modelling procedures are the main reasons for not using this method here. Specialized books in control theory as mentioned above use this approach and are available to the interested reader. 

Many chemical engineering systems can be modeled by a transfer function involving a first-order lag with deadtime. Let us consider a typical transfer function ... 

Perform a step test on the three-heated-lank process and fit a first-order lag plus deadtime model to the response curve. Calculate the ultimate gain and the ultimate frequency from the transfer function and compare with the results from Problem 16.1. 

The transfer function between the actual feed and the peak resultant cutting force is usually modeled as a first-order lag in a stable cutting process. The peak force is detected usually at one spindle period in order to avoid run-out effects in multipoint cutting operations such as drilling and milling. The overall discrete transfer function between the feed command (fc) and peak force (Fp) is considered to represent the combined dynamics of CNC, feed drives, and cutting process, and may be represented as follows ... 

Step 3 Decomposition into component parts. It has been demonstrated that a first-order lag is a reasonable approximation for the dynamics of a distillation column (Skogestad, 1987). Thus, the LSF configuration is decomposed into two component parts, one for each column. Four intermediate variables are identified to model the information transfer between the component parts Xbh B, Tg, and Qch (= Qrl)- Note that both Tg and Qch are needed for the energy balance in the reboiler because partial vaporization occurs. 

Models Considering Membrane Diffusion. The following model has been used when assuming that the electrode response is a first-order lag function, the liquid and gas phases are perfectly mixed, there is negligible nitrogen diffusion, and the interfacial area and oxygen concentration in the gas phase are constant (Blazej et al., 2004a Chisti, 1989 Freitas and Teixeira, 2001 Fuchs et al., 1971) ... 

We have also examined the IMC-PID design from Rivera et al. (1986) for this example. These authors suggest that for processes with a left half plane zero, the PID controller should be augmented with a first order lag to cancel this zero. We have designed an IMC-PID controller using their Case R which is for a model of the form... 

Two of the most common personalities are those for first and second-order systems. First-order systems may also be called first-order processes or first-order lags and can be mathematically modeled through the use of a first-order differential equation. Shown in Figure W2.1 is the typical step response of a first-order process. The time constant r was discussed in Chapter 3 and is related to the speed of the process response the slower the process, the larger the value of r. 

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. 

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. 

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]. 

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... 

It is commonly asserted that the Durbin-Watson statistic is only appropriate for testing for first order autoregressive disturbances. What combination of the coefficients of the model is estimated by the Durbin-Watson statistic in each of the following cases AR(1), AR(2), MA(1) In each case, assume that the regression model does not contain a lagged dependent variable. Comment on the impact on your results of relaxing this assumption. 

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