First-order systems in series

It is frequently required to examine the combined performance of two or more processes in series, e.g. two systems or capacities, each described by a transfer function in the form of equations 7.19 or 7.26. Such multicapacity processes do not necessarily have to consist of more than one physical unit. Examples of the latter are a protected thermocouple junction where the time constant for heat transfer across the sheath material surrounding the junction is significant, or a distillation column in which each tray can be assumed to act as a separate capacity with respect to liquid flow and thermal energy. 

Systems or processes operating in series may be considered as non-interacting, as interacting, or as a mixture of both. 

When two or more systems are operating in series and the behaviour of any system is not affected by any system subsequent to it, then those systems are said to be non-interacting. 

Two tanks in series. Consider the two tanks shown in Fig. 7.16. In this system neither the rate of flow through tank 1 nor the level in tank 1 is affected by what occurs in tank 2. Thus the two processes (or capacity systems) are non-interacting and we can model their dynamic behaviours individually. To establish the relationship between the level in tank 2 and the volumetric flowrate entering tank 1 at any instant of time, we need only to determine the individual transfer functions between Q0 and Q, and between 2, and z2. 

By eliminating a, between equations 7.28 and 7.29, the transfer function relating Q0 and z2 is obtained, viz.  

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G2 On the basis of two identical first order systems in series, the following is obtained ... 

A distillation process. The behaviour of liquid and vapour streams in any stagewise process can usually be approximated by a number of non-interacting first order systems in series. For example, Rose and Williams021 employed a first order transfer function to represent the dynamics of liquid and vapour flow in a 5-stage continuous distillation column. Thus for stage n in Fig. 7.17 ... 

Transfer functions analogous to equations 7.33 and 7.34 can be obtained for any number of non-interacting first order systems in series, e.g. for N tanks in series (Fig. 7.18c) having the same time constant r, from equation 7.27 ... 

There are distinct similarities between second order systems and two first-order systems in series. However, in the latter case, it is possible physically to separate the two lags involved. This is not so with a true second order system and the mathematical representation of the latter always contains an acceleration term (i.e. a second-order differential of displacement with respect to time). A second-order transfer function can be separated theoretically into two first-order lags having the same time constant by factorising the denominator of the transfer function e.g. from equation 7.52, for a system with unit steady-state gain ... 

A similar expression is obtained for the response of two first-order systems in series (see Section 7.5.3). 

The downstream region of the UF test can be modeled in terms of a first order system in series to a pure time lag. Dimensionless product concentration at the UF cell outlet is then related to that immediately downstream of the membrane by the system transfer function. 

In Section 11.3 we found that two capacities in series, interacting or noninteracting, give rise to a second-order system. If we extend the same procedure to N capacities (first-order systems) in series, we find that the overall response is of Nth order that is, the denominator of the overall transfer function is an iVth-order polynomial,... 

Which of the following second-order systems are equivalent to two first-order systems in series and which are not ... 

Overdamped are the responses of multicapacity processes, which result from the combination of first-order systems in series, as we will see in Section 11.3. 

Such a process can exhibit underdamped behavior, and consequently it cannot be decomposed into two first-order systems in series (interacting or noninteracting) with physical significance, like the systems we examined in previous sections. They occur rather rarely in a chemical process, and they are associated with the motion of liquid masses or the mechanical translation of solid parts, possessing (1) inertia to motion, (2) resistance to motion, and (3) capacitance to store mechanical energy. Since resistance and capacitance are characteristic of the first-order systems, we conclude that the inherently second-order systems are characterized by their inertia to motion. The three examples in Appendix 11A clearly demonstrate this feature. 

Show that as the number of noninteracting first-order systems in series increases, the response of the system becomes more sluggish. 

Consider a first-order system with a dead time t between the input f(t) and the output y(t). We can represent such system by a series of two systems as shown in Figure 12.2a (i.e., a first-order system in series with a dead time). For the first-order system we have the following transfer function ... 

Multicapacity processes These constitute the large majority of real processes. Consider two first-order systems in series with... 

Equation (16.12) indicates that the process reaction curve has the same dynamic characteristics as the response of a system composed of four first-order systems in series (i.e., it is a sigmoidal curve). 

As can be seen, the response of two first-order systems in series without feedback is a critically damped response. This result could be expected, a disturbance at the entrance propagates through the system without damping and without amplification. 

Figure 5.7 Two first-order systems in series yield an overall second-order system.

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