Transfer functions process characteristics

Process Characteristics in Transfer Functions In many cases, process charac teristics 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. 

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. 

If the transfer functions GL and Gp are based on a simple process model, we know quite well that they should have the same characteristic polynomial. Thus the term -GL/Gp is nothing but a ratio of the steady state gains, -KL/Kp. 

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]. 

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). 

The transfer function relates two variables in a process one of these is the forcing function or input variable, and the other is the response or output variable. The transfer function completely describes the dynamic characteristics of a system. The input and output variables are usually expressed in the Laplace domain and are written as deviations from the set-point values. 

The frequency response characteristics of a process element or a group of elements can be computed readily from the corresponding transfer function merely by substituting ju for s, where j is the imaginary number, /— 1, and is the angular velocity. Thus the frequency response characteristics of a simple first order lag are given by... 

In Chapter 14 we examined the dynamic characteristics of the response of closed-loop systems, and developed the closed-loop transfer functions that determine the dynamics of such systems. It is important to emphasize again that the presence of measuring devices, controllers, and final control elements changes the dynamic characteristics of an uncontrolled process. Thus nonoscillatory first-order processes may acquire oscillatory behavior with PI control. Oscillatory second-order processes may become unstable with a PI controller and an unfortunate selection of Kc and t,. 

---