3.0 context and direction3.1 math model of a simple continuous holding tank3.2 response of system to steady input3.3 leaning on the system - response to step disturbance3.4 kicking the system - response to pulse disturbance3.5 shaking the system - response to sine disturbance3.6 frequency response and the Bode plot3.7 stability of a system3.8 concentration control in a blending tank3.9 use of deviation variables in solving equations- makes conceptual sense for process control because they indicate deviations from desired states - makes the mathematical descriptions simpler 3.10 integration from zero initial conditions3.11 response to step changes3.12 developing a control scheme for the blending tank3.13 step 1 - specify a control objective for the process3.14 step 2 - assign variables in the dynamic system3.15 step 3 - introduce proportional control for our process3.16 step 4 - choose set points and limits3.17 type of equipment needed for process control3.18 closing the loop - feedback control of the blending process3.19 integration from zero initial conditions3.20 closed-loop response to pulse disturbance3.21 closed-loop response to step disturbance - the offset phenomenon3.22 response to set point changes3.23 tuning the controller3.24 stability of the closed-loop system3.25 conclusionSpring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank 3.0 context and direction A particularly simple process is a tank used for blending. Just as promised in Section 1.1, we will first represent the process as a dynamic system and explore its response to disturbances. Then we will pose a feedback control scheme. We will briefly consider the equipment required to realize this control. Finally we will explore its behavior under control. DYNAMIC SYSTEM BEHAVIOR 3.1 math model of a simple continuous holding tank Imagine a process stream comprising an important chemical species A in dilute liquid solution. It might be the effluent of some process, and we might wish to use it to feed another process. Suppose that the solution composition varies unacceptably with time. We might moderate these swings by holding up a volume in a stirred tank: intuitively we expect the changes in the outlet composition to be more moderate than those of the feed stream. F, CAiF, CAovolume VF, CAiF, CAovolume V Our concern is the time-varying behavior of the process, so we should treat our process as a dynamic system. To describe the system, we begin by writing a component material balance over the solute. AoAiAoFCFCVCdtd−= (3.1-1) In writing (3.1-1) we have recognized that the tank operates in overflow: the volume is constant, so that changes in the inlet flow are quickly duplicated in the outlet flow. Hence both streams are written in terms of a single volumetric flow F. Furthermore, for now we will regard the flow as constant in time. Balance (3.1-1) also represents the concentration of the outlet stream, CAo, as the same as the average concentration in the tank. That is, the tank is a perfect mixer: the inlet stream is quickly dispersed throughout the tank volume. Putting (3.1-1) into standard form, revised 2005 Jan 13 1Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank AiAoAoCCdtdCFV=+ (3.1-2) we identify a first-order dynamic system describing the response of the outlet concentration CAo to disturbances in the inlet concentration CAi. The speed of response depends on the time constant, which is equal to the ratio of tank volume and volumetric flow. Although both of these quantities influence the dynamic behavior of the system, they do so as a ratio. Hence a small tank and large tank may respond at the same rate, if their flow rates are suitably scaled. System (3.1-2) has a gain equal to 1. This means that a sustained disturbance in the inlet concentration is ultimately communicated fully to the outlet. Before solving (3.1-2) we specify a reference condition: we prefer that CAo be at a particular value CAo,r. For steady operation in the desired state, there is no accumulation of solute in the tank. r,Aor,AirAoCC0dtdCFV−== (3.1-3) Thus, as expected, steady outlet conditions require a steady inlet at the same concentration; call it CA,r. Let us take this reference condition as an initial condition in solving (3.1-2). The solution is dt)t(CeeeC)t(CAit0tttr,AAo∫ττ−τ−τ+= (3.1-4) where the time constant is FV=τ (3.1-5) Equation (3.1-4) describes how outlet concentration CAo varies as CAi changes in time. In the next few sections we explore the transient behavior predicted by (3.1-4). 3.2 response of system to steady input Suppose inlet concentration remains steady at CA,r. Then from (3.1-4) revised 2005 Jan 13 2Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank r,Attr,Atr,At0tr,Attr,AAoC1eeCeCeCeeCC=⎟⎠⎞⎜⎝⎛−+=ττ+=ττ−τ−ττ−τ− (3.2-1) Equation (3.2-1) merely confirms that the system remains steady if not disturbed. 3.3 leaning on the system - response to step disturbance Step functions typify disturbances in which an input variable moves relatively rapidly to some new value and remains there. Suppose that input CAi is initially at the reference value CA,r and changes at time t1 to value CA1. Until t1 the outlet concentration is given by (3.2-1). From the step at t1, the outlet concentration begins to respond. ⎟⎠⎞⎜⎝⎛−+=⎟⎠⎞⎜⎝⎛−+=>ττ+=τ−−τ−−τττ−τ−−ττ−τ−−)tt(1A)tt(r,Attt1A)tt(r,A1ttt1At)tt(r,AAo111111e1CeCeeeCeCtteCeeCC (3.3-1) In Figure 3.3-1, CA,r = 1 and CA1 = 0.8 in arbitrary units; t1 has been set equal to τ. At sufficiently long time, the initial condition has no influence and the outlet concentration becomes equal to the new inlet concentration. After time equal to three time constants has elapsed, the response is about 95% complete – this is typical of first-order systems. In Section 3.1, we suggested that the tank would mitigate the effect of changes in the inlet composition. Here we see that the tank will not eliminate a step disturbance, but it does soften its arrival. revised 2005 Jan 13 3Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank 0.70.80.911.1012345t/τresponse600.51012345disturbance6 Figure 3.3-1 first-order response to step disturbance 3.4 kicking the system - response