First-order systems, dynamic response

The time constant is one way of determining the dynamic features of a measurement system. Not all instrument manufacturers use the time constant some use the response time instead. The response time is the time between a step change of the measured quantity and the instant when the instrument s response does not differ from its final value by more than a specified amount.The response time is defined according to a deviation from the final value. Often response times for the relative deviation of 1%, 5%, 10%, or 37% are used. The corresponding response times are denoted by 99%, 95%, 90%, or 63% response time, respectively. The response time for a first-order system can be solved from Eq. (12.15). Note that the 63% response time of a first-order system is the same as the time constant r of the system. 

The question then arises as to how one handles first-order systems with variable time constants and static gains in order to find the dynamic response of such systems. There are two possible solutions ... 

In an ideal binary distillation column the dynamics of each tray can be described by first-order systems. Are these capacities interacting or not What general types of responses would you expect for the overhead and bottoms compositions to a step change in the feed composition ... 

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). 

First-order open-loop response Consider a control system with the following dynamic components ... 

The preceding examples have illustrated a very important point The higher the order of the system, the worse the dynamic response of the closedloop system. The first-order system is never underdamped and cannot be made closedloop unstable for any value of gain. The second-order system becomes underdamped as gain is increased but never goes unstable. Third-order (and higher) systems can be made closedloop unstable. If you remember, these arc exactly the results we found in our simulation experiments in Chapter I. Now we have shown mathematically why these various processes behave the way they do. 

This is the first-order system discussed previously, which is the analog of a resistance-capacitance circuit in electronics. It is mainly apphcable to responses of instruments (sensors and valves) but usually is inadequate to describe dynamic behavior of actual chemical processes. 

Observe that for first-order systems, /xq = 0 with the value increasing as the actual process dynamics (as represented by the step response) exhibits more complex characteristics. Due to scaling, the maximum meaningful value that can be obtained is = 1 as with t —> 0 (essentially becoming can always be chosen as the comparison operator. 

The dynamic behavior of a dynamic system can be well represented through the so-called step response . The dynamic evolution of the output variable can be monitored in response to a step-change of the input. We may ask how this system responds to a step-change of magnitude M in the input flow rate. The first-order system becomes that described by Eqs. (49), where H(t) is the Heaviside step function defined by Eq. (50). 

The ramp response y(t) of a first-order system was obtained in Eqs. 5-19 through 5-21. The maximum deviation between input and output is at (obtained when t j), as shown in Fig. 5.5. Hence, as a general result, we can say that the maximum dynamic error that can occur for any instrument mth first-order dynamics is... 

The time constant r characterizes the response of the first-order system and is discussed in greater detail in the next section. All higher-order systems can be broken down into sets of first-order systems, and the time constants of these LDEs can be used to ascertain the relative importance of each from a dynamic response perspective. That is, the dominant, or largest, time constant will determine the speed of the response. The commonly used rule of thumb is that any subsystem with a time constant an order of magnitude (10 times) less than the dominant time constant can be described by steady-state or algebraic equations. 

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

Chapter 14 and Section 15.2 used a unsteady-state model of a system to calculate the output response to an inlet disturbance. Equations (15.45) and (15.46) show that a dynamic model is unnecessary if the entering compound is inert or disappears according to first-order kinetics. The only needed information is the residence time distribution, and it can be determined experimentally. 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

The phenomenon of hysteresis is encountered in many areas of physics. It is associated with the delay of the dynamic response of cooperative systems to external perturbation. During a heating-cooling process in a system, thermal hysteresis (TH) commonly appears accompanying phase transitions. In particular, it is regarded as a signature of the first-order phase transition. But, the TH is less known than the magnetic hysteresis (MH), which is another type... 

Example 11.4 demonstrates very clearly how the simple first-order dynamic behavior of a tank can change to that of a second-order when a proportional-integral controller is added to the process. Also, it indicates that the control parameters Kc and r can have a very profound effect on the dynamic behavior of the system, which can range from an underdamped to an overdamped response. 

Find the dynamic response of a first-order lag system with time constant tp = 0.5 and static gain Kp = 1 to (a) a unit impulse input change, (b) a unit pulse input change of duration S, and (c) a sinusoidal input change, sin 0.51. Determine the behavior of the output after long time (as / - oo) for each of the input changes above. 

Show that the concentration cA of reactant A in an isothermal continuous stirred tank reactor exhibits first-order dynamics to changes in the inlet composition, cA/. The reaction is irreversible, A - B, and has first-order kinetics (i.e., r = kcA). Furthermore (a) identify the time constant and static gain for the system, (b) derive the transfer function between cA and cA (c) draw the corresponding block diagram, and (d) sketch the qualitative response of cA to a unit pulse change in cAj. The reactor has a volume V, and the inlet and outlet flow rates are equal to F. 

Thus for a first-order uncontrolled process, the response of the closed-loop becomes second-order and consequently it may have drastically different dynamic characteristics. Furthermore, as we have seen in Sections 11.3 and 12.1, by increasing the order of a system, its response becomes more sluggish. Thus ... 

Dividers, 428 (see also Ratio control) Drum boiler, 105, 216-17, 412-13 Drying control, 456 Duhem s rule, 97 Dynamic analysis, 51 qualitative characteristics, 168-72 Dynamic behavior of various systems dead time, 214-16 definition, 51 first-order lag, 179-83 higher-order, 212-14 inverse response systems, 216-20 pure capacitive, 178-79 second-order, 187-93... 

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,. 

There are several approaches that can be used to tune PID controllers, including model-based correlations, response specifications, and frequency response (Smith and Corripio 1985 Stephanopoulos 1984). An approach that has received much attention recently is model-based controller design. Model-based control requires a dynamic model of the process the dynamic model can be empirical, such as the popular first-order plus time delay model, or it can be a physical model. The selection of the controller parameters Kc, ti, to) is based on optimizing the dynamic performance of the system while maintaining closed-loop stability. 

The feedforward controller contains a steady-state gain and dynamic terms. For this system the dynamic element is a first-order lead-lag. The unit step response of this lead-lag is an initial change to a value that is (- KJKm) tmItl), followed by an exponential rise or decay to the final steady-state value — KJKm- ... 

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