Process Characteristics in Transfer Functions

Process Characteristics in Transfer Functions In many cases, process characteristics are expressed in the form of transfer functions. In the previous discussion, a reactor example was used to illustrate how a transfer function could be derived. Here, another system involving flow out of a tank, shown in Fig. 8-6, is considered. 

Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulli s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as 

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

Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except I lial /j no longer depends on hi. For changes in /. the transfer function relating changes in hi to changes in f, is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross-sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held constant, then the level builds up linearly, as shown in Fig. 8-11. Eventually the liquid will overflow the tank. 

Second-Order Element Because of their linear nature, transfer functions can be combined in a straightforward manner. Consider the two-tank system shown in Fig. 8-12. For tank 1, the transfer function relating changes in/, to changes in /, is 

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Section VI). If the investigated sample is known to contain a set of well characterized spin systems, numerical simulations of coherence-transfer functions can help in the assignment process. For example, the aliphatic C spin systems of labeled amino acids form a small number of distinct coupling topologies with characteristic coherence-transfer functions that can be discriminated based on a set of only four experiments with carefully chosen mbdng times (Eaton et ai, 1990). 

Finnigan, J.J., Raupach, M.R. (1987) Transfer processes in plant canopies in relation to stomatal characteristics, in, Stomatal Function, (eds. E. Zeiger, Farquar,G.D and Cowan I.R.), Stanford University Press, Stanford, California, 385-429. 

Another control strategy for treating both disturbances and set-point changes is the analytical predictor, which utilizes a prediction of the process behavior in the future based on the process and disturbance transfer functions, G and G. In the context of Eq. 16-23, if Gc included a term (a perfect prediction 0 units of time ahead), then the time delay would effectively be eliminated from the characteristic equation. However, this is an idealized view, and further details are given in Chapter 17. 

In this rearrangement, xp is the process time constant, and Kd and Kp are the steady state gains.2 The denominators of the transfer functions are identical, they both are from the LHS of the differential equation—the characteristic polynomial that governs the inherent dynamic characteristic of the process. 

Take note (again ) that the characteristic polynomials in the denominators of both transfer functions are identical. The roots of the characteristic polynomial (the poles) are independent of the inputs. It is obvious since they come from the same differential equation (same process or system). The poles tell us what the time-domain solution, y(t), generally would "look" like. A final reminder no matter how high the order of n may be in Eq. (3-4), we can always use partial fractions to break up the transfer functions into first and second order terms. 

As the thermal capacitance and resistance of the hotplate provide a thermal low-pass transfer function (with the dominant pole corresponding to a characteristic time of 10-20 ms, depending on the fabrication process), the ZA modulator driving the hotplate constitutes a linear noise-shaping DAC with an output in the thermal domain. 

This equation shows that closedloop dynamics depend on the process openloop transfer functions (G, Gv, and Gj) and on the feedback controller transfer function (fl). Equation (10.10) applies for simple single-input-single-output systems. We will derive closedloop characteristic equations for other systems in later chapters. 

These are the equations that we will use in most cases because they are more convenient. Keep in mind that the transfer function in Eq. (10.11) is a combination of the process, transmitter, and the valve transfer functions. The closed-loop characteristic eq nation is... 

In Chap. 12 we will show that we can convert from the Laplace domain (Russian) into the frequency domain (Chinese) by merely substituting ia for s in the transfer function of the process. This is similar to the direct substitution method, but keep in mind that these two operations are different. In one we use the transfer function. In the other we use the characteristic equation. 

Example 19.6. The chromatographic system studied in Example 18.9 had a first-order lag openloop process transfer function and a deadtime of one sampling period. The closedloop characteristic equation was [see Eq. (18.100)]... 

The type of dimensionless representation of the material function affects the (extended) pi set within which the process relationship is formulated (for more information see Ref. 5). When the standard representation is used, the relevance list must include the reference density po instead of p and incorporate two additional parameters po. Tq. This leads to two additional dimensionless numbers in the process characteristics. With regard to the heat transfer characteristics of a mixing vessel or a smooth straight pipe, Eq. (27), it now follows that... 

The above values are most often determined via correlations, which allow a scale-up (or down) to different operating states. Along these fines, the liquid and gas phase mass transfer coefficients are usually related to Sherwood number (Sh) as a function of Reynolds number (Re), Schmidt number (Sc) and other dimensionless process characteristics [3, 59-61]. It is important that the correlations are applied within the same parameter range in which they are determined as only there can their reliability be assured. 

In these studies, a modulation of the transport rate is imposed upon a steady-state rate. As noted in Section 10.3, the ideas can be generalised through the concept of a transfer function linking fluctuations in current to fluctuations in the velocity gradient normal to the electrode. There are two distinct themes in the literature one is to impose a flow with known fluctuation characteristics in order to deduce information about electrochemical processes occurring at or near the interface, this being the focus of the present review the other is to use the variations in limiting current to deduce the characteristics of the flow, with an emphasis on analysis of the fluctuations in current to deduce characteristics of turbulent flow [81-85]. 

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