Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems 1.0 context and direction Process control is an application area of chemical engineering - an identifiable specialty for the ChE. It combines chemical process knowledge (how physics, chemistry, and biology work in operating equipment) and an understanding of dynamic systems, a topic important to many fields of engineering. Thus study of process control allows chemical engineers to span their own field, as well as form a useful acquaintance with allied fields. Practitioners of process control find their skills useful in design, operation, and troubleshooting - major categories of chemical engineering practice. Process control, like any coherent topic, is an integrated body of knowledge - it hangs together on a multidimensional framework, and practitioners draw from many parts of the framework in doing their work. Yet in learning, we must receive information in sequence - following a path through multidimensional space. It is like entering a large building with unlighted rooms, holding a dim flashlight and clutching a vague map that omits some of the stairways and passages. How best to learn one’s way around? In these lessons we will attempt to move through a significant portion of the structure - say, half a textbook - in about two weeks. Then we will repeat the journey several times, each time inspecting the rooms more thoroughly. By this means we hope to gain, from the start, a sense of doing an entire process control job, as well as approach each new topic in the context of a familiar path. 1.1 the job we will do, over and over We encounter a process, learn how it behaves, specify how we wish to control it, choose appropriate equipment, and then explore the behavior under control to see if we have improved things. 1.2 introducing a simple process A large tank must be filled with liquid from a supply line. One operator stands at ground level to operate the feed valve. Another stands on the tank, gauging its level with a dipstick. When the tank is near full, the stick operator will instruct the other to start closing the valve. Overfilling can cause spills, but underfilling will cause later process problems. revised 2006 Jan 30 1 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems To learn how the process works, we write an overall material balance on the tank. i FV dt d ρ=ρ (1.2-1) The tank volume V can be expressed in terms of the liquid level h. The inlet volumetric flow rate F i may vary with time due to supply pressure fluctuations and valve manipulations by the operator. The liquid density depends on the temperature, but will usually not vary significantly with time during the course of filling. Thus (1.2-1) becomes )t(F d t dh A i = (1.2-2) We integrate (1.2-2) to find the liquid level as a function of time. ∫ += t 0 i dt)t(F A 1 )0(hh (1.2-3) 1.3 planning a control scheme Clearly the liquid level h is important, and we will call it the controlled variable. Our control objective is to bring h quickly to its target value h r and not exceed it. (To be realistic, we would specify allowable limits ± δh on h r .) We will call the volumetric flow F i the manipulated variable, because we adjust it to achieve our objective for the controlled variable. The existing control scheme is to measure the controlled variable via dipstick, decide when the controlled variable is near target, and instruct revised 2006 Jan 30 2 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems the valve operator to change the manipulated variable. The scheme suffers from • delay in measurement. Overfilling can occur if the stick operator cannot complete the measurement in time. • performance variations. Both stick and valve operators may vary in attentiveness and speed of execution. • resources required. There are better uses for operating personnel. • unsafe conditions. There is too much potential for chemical exposure. A new scheme is proposed: put a timer on the valve. Calculate the time required for filling from (1.2-3). Close the valve when time has expired. The timing scheme would no longer require an operator to be on the tank top, and with a motor-driven valve actuator the entire operation could be directed from a control room. These are indeed improvements. However, the timing scheme abandons a crucial virtue of the existing scheme: by measuring the controlled variable, the operators can react to unexpected disturbances, such as changes in the filling rate. Using knowledge of the controlled variable to motivate changes to the manipulated variable is a fundamental control structure, known as feedback control . The proposed timing scheme has no feedback mechanism, and thus cannot accommodate changes to h(0) and F i (t) in (1.2-3). An alternative is to build on the feedback already inherent in the two- operator scheme, but to improve its operation. We propose an automatic controller that behaves according to the following controller algorithm: near i max r near i max r near hh F F hh hh FF hh <= − >= − (1.3-1) Algorithm (1.3-1) is an idealization of what the operators are already doing: filling occurs at maximum flow until the level reaches a value h near . Beyond this point, the flow decreases linearly, reaching zero when h reaches the target h r . The setting of h near may be adjusted to tune the control performance. 1.4 choosing equipment We need a sensor to replace the dipstick, a valve actuator to replace the valve operator, and a controller mechanism to replace the stick operator. We imagine a buoyant object floating on the liquid surface. The float is linked to a lever that drives the valve stem. When the liquid level is low, the float rests above it on a structure so that the valve is fully open. revised 2006 Jan 30 3 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems 1.5 process behavior under automatic control Typically these things work quite well. We predict its performance by combining our process model (1.2-2) with the controller algorithm (1.3-1), which eliminates the manipulated variable between the equations. We take the simple case in which F max does not vary during filling due to pressure fluctuations, etc. For h less than h near , t A F )0(hh known)0(hF dt dh A max max += == (1.5-1) Equation (1.5-1) can be used to calculate t near , the time at which h reaches h near . For h greater than h near , () () r max near near rnear r near r r near fill r near hhdh AF h(t)h dt h h h(t t ) hh h h exp thh − == − ⎡ −− =− − ⎢⎥ − ⎢⎥ ⎣⎦ ⎤ (1.5-2) revised 2006 Jan 30 4 Content removed due to copyright restrictions. (To see a cut-away diagram of a toilet, go to http://www.toiletology.com/lg-views.shtml#cutaway2x) Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems where the parameter t fill is the time required for the level to reach h r at flow F max , starting from an empty tank. r fill max Ah t F = (1.5-3) The plot shows the filling profile from h(0) = 0.10h r with several values of h near /h r . Certainly the filling goes faster if the flow can go instantaneously from F max to zero at h r ; however this will not be practical, so that h near will be less than h r . 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t/t fill h/hr h near /h r = 0.95 0.75 0.50 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t/t fill h/hr h near /h r = 0.95 0.75 0.50 1.6 defining ‘system’ In Section 1.2, we introduced a process - a tank with feed piping - whose inventory varied in time. We thought of the process as a collection of equipment and other material, marked off by a boundary in space, communicating with its environment by energy and material streams. 'Process' is a good notion, important to chemical engineers. Another useful notion is that of 'system'. A system is some collection of equipment and operations, usually with a boundary, communicating with its environment by a set of input and output signals. By these definitions, a process is a type of system, but system is more abstract and general. For example, the system boundary is often tenuous: suppose that our system comprises the equipment in the plant and the controller in the central control room, with radio communication between the two. A physical boundary would be in two pieces, at least; perhaps we should regard this revised 2006 Jan 30 5 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems system boundary as partly physical (around the chemical process) and partly conceptual (around the controller). Furthermore, the inputs and outputs of a system need not be material and energy streams, as they are for a process. System inputs are "things that cause" or “stimuli”; outputs are "things that are affected" or “responses”. system inputs (causes) outputs (responses) To approach the problem of controlling our filling process in Section 1.3, we thought of it in system terms: the primary output was the liquid level h not a stream, certainly, but an important response variable of the system and inlet stream F i was an input. And peculiar as it first seems, if the tank had an outlet flow F o , it would also be an input signal, because it influences the liquid level, just as does F i . The point of all this is to look at a single schematic and know how to view it as a process, and as a system. View it as a process (F o as an outlet stream) to write the material balance and make fluid mechanics calculations. View it as a system (F o as an input) to analyze the dynamic behavior implied by that material balance and make control calculations. System dynamics is an engineering science useful to mechanical, electrical, and chemical engineers, as well as others. This is because transient behavior, for all the variety of systems in nature and technology, can be described by a very few elements. To do our job well, we must understand more about system dynamics how systems behave in time. That is, we must be able to describe how important output variables react to arbitrary disturbances. 1.7 systems within systems We call something a system and identify its inputs and outputs as a first step toward understanding, predicting, and influencing its behavior. In some cases it may help to determine some of the structure within the system boundaries; that is, if we identify some component systems. Each of these, of course, would have inputs and outputs, too. system inputs outputs 1 2 revised 2006 Jan 30 6 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems Considering the relationship of these component systems, we recognize the existence of intermediate variables within a system. Neither inputs nor outputs of the main system, they connect the component systems. Intermediate variables may be useful in understanding and influencing overall system behavior. 1.8 the system of single-loop feedback control When we add a controller to a process, we create a single time-varying system; however, it is useful to keep process and controller conceptually distinct as component systems. This is because a repertoire of relatively few control schemes (relationships between process and controller) suffices for myriad process applications. Using the terms we defined in Section 1.3, we represent a control scheme called single-loop feedback control in this fashion: process controller final control element sensor set point manipulated variable other inputs other outputs controlled variable system process controller final control element sensor set point manipulated variable other inputs other outputs controlled variable system Figure 1.8-1 The single-loop feedback control system and its subsystems We will see this structure repeatedly. Inside the block called "process" is the physical process, whatever it might be, and the block is the boundary we would draw if we were doing an overall material or energy balance. HOWEVER, we remember that the inputs and outputs are NOT necessarily the same as the material and energy streams that cross the process boundary. From among the outputs, we may select a controlled variable (often a pressure, temperature, flow rate, liquid level, or composition) and provide a suitable sensor to measure it. From the inputs, we choose a manipulated variable (often a flow rate) and install an appropriate final control element (often a valve). The measurement is fed to the controller, which decides how to adjust the manipulated variable to keep the controlled variable at the desired condition: the set point. The revised 2006 Jan 30 7 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems other inputs are potential disturbances that affect the controlled variable, and so require action by the controller. 1.9 conclusion Think of a chemical process as a dynamic system that responds in particular ways to its inputs. We attach other dynamic systems (sensor, controller, etc.) to that process in a single-loop feedback structure and arrive at a new dynamic system that responds in different ways to the inputs. If we do our job well, it responds in better ways, so to justify all the trouble. revised 2006 Jan 30 8 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review 2.0 context and direction Imagine a system that varies in time; we might plot its output vs. time. A plot might imply an equation, and the equation is usually an ODE (ordinary differential equation). Therefore, we will review the math of the first-order ODE while emphasizing how it can represent a dynamic system. We examine how the system is affected by its initial condition and by disturbances, where the disturbances may be non-smooth, multiple, or delayed. 2.1 first-order, linear, variable-coefficient ODE The dependent variable y(t) depends on its first derivative and forcing function x(t). When the independent variable t is t 0 , y is y 0 . 00 y)t(y)t(Kx)t(y d t dy )t(a ==+ (2.1-1) In writing (2.1-1) we have arranged a coefficient of +1 for y. Therefore a(t) must have dimensions of independent variable t, and K has dimensions of y/x. We solve (2.1-1) by defining the integrating factor p(t) ∫ = )( exp)( ta dt tp (2.1-2) Notice that p(t) is dimensionless, as is the quotient under the integral. The solution ∫ += t t 00 0 dt )t(a )t(x)t(p )t(p K )t(p )t(y)t(p )t(y (2.1-3) comprises contributions from the initial condition y(t 0 ) and the forcing function Kx(t). These are known as the homogeneous (as if the right-hand side were zero) and particular (depends on the right-hand side) solutions. In the language of dynamic systems, we can think of y(t) as the response of the system to input disturbances Kx(t) and y(t 0 ). 2.2 first-order ODE, special case for process control applications The independent variable t will represent time. For many process control applications, a(t) in (2.1-1) will be a positive constant; we call it the time constant τ. 00 y)t(y)t(Kx)t(y dt dy ==+τ (2.2-1) The integrating factor (2.1-2) is revised 2005 Jan 11 1 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review τ = τ = ∫ t e dt exp)t(p (2.2-2) and the solution (2.1-3) becomes () dt)t(xee K ey)t(y t t tt tt 0 0 0 ∫ ττ − τ −− τ += (2.2-3) The initial condition affects the system response from the beginning, but its effect decays to zero according to the magnitude of the time constant - larger time constants represent slower decay. If not further disturbed by some x(t), the first order system reaches equilibrium at zero. However, most practical systems are disturbed. K is a property of the system, called the gain. By its magnitude and sign, the gain influences how strongly y responds to x. The form of the response depends on the nature of the disturbance. Example: suppose x is a unit step function at time t 1 . Before we proceed formally, let us think intuitively. From (2.2-3) we expect the response y to decay toward zero from IC y 0 . At time t 1 , the system will respond to being hit with a step disturbance. After a long time, there will be no memory of the initial condition, and the system will respond only to the disturbance input. Because this is constant after the step, we guess that the response will also become constant. Now the math: from (2.2-3) () () () ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+= − τ += τ −− τ −− ττ − τ −− ∫ 1 0 0 0 tt 1 tt 0 t t 1 tt tt 0 e1)tt(KUey dt)tt(Uee K ey)t(y (2.2-4) Figure 2.2-1 shows the solution. Notice that the particular solution makes no contribution before time t 1 . The initial condition decays, and with no disturbance would continue to zero. At t 1 , however, the system responds to the step disturbance, approaching constant value K as time becomes large. This immediate response, followed by asymptotic approach to the new steady state, is characteristic of first-order systems. Because the response does not track the step input faithfully, the response is said to lag behind the input; the first-order system is sometimes called a first-order lag. revised 2005 Jan 11 2 [...]... of y and its derivative on the left-hand side of the equation Dead time revised 2005 Jan 11 10 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review occurs because of a time delay in processing a disturbance on the righthand side 2.6 conclusion Please become comfortable with handling ODEs View them as systems; identify their inputs and outputs, their gains and time... introduce proportional control for our process The controller algorithm dictates how the manipulated variable is to be adjusted in response to deviations between the controlled variable and the set point We will introduce a simple and plausible algorithm, called revised 2005 Jan 13 13 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank proportional control This algorithm... are ready to consider control CONTROL SCHEME 3.12 developing a control scheme for the blending tank A control scheme is the plan by which we intend to control a process A control scheme requires: 1) specifying control objectives, consistent with the overall objectives of safety for people and equipment, environmental protection, product quality, and economy 2) specifying the control architecture, in... 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. .. needed for process control Figure 3.17-1 shows our process and control scheme as two communicating systems The system representing the process has two inputs and one output Of these only one is a material stream; however, we recall that systems communicate with their environment (and other systems) through signals, and in the blending process the outlet composition responds to the inlet composition and make-up... composition? revised 2005 Jan 13 15 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank system representing process outlet composition inlet composition -tank to hold liquid make-up flow -agitator to mix contents -inlet and outlet piping system representing controller and other equipment adjust make-up flow multiply gain and add bias subtract from set point measure... the system variables are assigned to roles of controlled, disturbance, and manipulated variables, and their relationships specified 3) choosing a controller algorithm 4) specifying set points and limits 3.13 step 1 - specify a control objective for the process Our control objective is to maintain the outlet composition at a constant value Insofar as the process has been described, this seems consistent... the inlet composition These limits are determined by the process and its environment No amount of controller design can compensate for a manipulated variable that is unequal to the disturbance task revised 2005 Jan 13 14 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank tolerable variation: ideally the controlled variable would never deviate from the set point... (2.4-3) 5 Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review and write (2.4-1) in three equations We put the initial condition with no disturbances, and each disturbance with a zero initial condition dy H + yH (t) = 0 dt dy τ 1 + y1 ( t ) = K1x1 ( t ) dt dy τ 2 + y 2 (t) = K 2 x 2 (t) dt τ yH (t 0 ) = y0 y1 ( t 0 ) = 0 (2.4-4) y2 (t 0 ) = 0 Equations and initial conditions... description of both equipment and controller algorithms When we do, however, we will find that the overall concept of feedback control is the same as presented in Figure 3.17-1: the controlled variable is measured, decisions are made, and the manipulated variable is adjusted to improve the controlled variable CLOSED LOOP BEHAVIOR 3.18 closing the loop - feedback control of the blending process Our next task . Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems 1.0 context and direction Process control is an application. Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 1: Processes and Systems 1.5 process behavior under automatic control Typically