Signal Flow Diagrams for Process DONALD P. CAMPBELL Department of Electrical Engineering, Massachureffs lnsfifute of Technology, Cambridge, Moss.
Signal flow diagrams are not analogs. They are cause and effect diagrams constituting as exact a mathematical model as can be set down for a particular problem. Pertinent physical variables, their interrelationships,and dynamical aspects of their cause-effect relations become clear from the signal flow diagrams.
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HE signal flow diagram is the working diagram for the study of dynamical behavior of processes. The mathematical op-
erations assigned to the various blocks in signal flow diagrams derive from the physics of the unit processes. The manner of interconnecting these groups of blocks establishes not only the nature of a unit process but also the precision of control and regulation that may be expected for the particular unit process, as well as the stability of the system in which it is used. To a large extent signal flow diagrams can also be used to determine what constitutes the optimum combination of functional operations needed to render a complex proceas control system workable. To prepare a signal flow diagram one must know the physical behavior of the particular unit operation, portion of an operation, or portion of a system coniprising several unit operations well enough to trace the dynamics of cause and effect throughout in a quantitative manner. This means the ability to express cause and effect relationships as functions of time by means of differential equations. At the very least, approximate equations supported by empirical data are needed to define the various steps of processing action and hence define the operators of the different blocks in the diagram. Thus, the preparation of the signal flow diagram has the doubly important role in process control work; it ensures a quantitative working knowledge of the physical and dynamical effects that are important in the unit processes; it provides a network or diagram with which extensive analysis, study, and synthesis or processes and their controls can be made. Preparation of Signal Flow Diagrams General. Signal flow diagrams are set up from the differential equations that govern the dynamical response of a process, ,The independent variables in the equations which express process behavior become the commands, loads, disturbances, and manipulation signals in the diagrams; thk dependent variables become the response signals. The particular integrodifferential equations which relate dependent variables to the independent variables serve to define the operators to be placed in the blocks, These operators identify the action taken on signals as they pass through blocks in the diagram. Consecutive cause and effect takes on the aspect of cascades of operations in which signals may be magnified, attenuated, filtered, shifted in time phase, freed from noise, and changed in power level or energy content. Signal flow diagrams should first be laid out in their primitive form. The primitive form of the signal flow diagram includes every cause and the effect relationship inherent in the physical process being studied. No simplifications or omissions of portions of the diagram are permitted. In fact, linearization should not be introduced in the primitive form of the diagram until it has been justified. Care taken in preparing the primitive diagram will avoid omissions through error of too early simplification. Furthermore,
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since the primitive forms can be reduced according to definite simplifying procedures, two or more simplified versions of the signal flow diagram to represent the same process may often occur. Because these variations arise from the same primitive, they represent equivalent processes which possess the same dynamical performance. All the physical variables of the primitive signal flow diagrams become recognizable a t once, Some represent points for making direct measurement and manipulation, others represent variables for making only indirect measurement and manipulation. For process control work it becomes important to know the extent to which measurement can be made on an operation, the likelihood of interaction on the process by a measuring instrument, and the possibility of effecting control or regulatory manipulations over particular variables in the process. Often when preparing a signal flow diagram a portion of a unit process or operation may not be understood. This does not mean that one cannot proceed with the diagraming. A general operator that interconnects the cause and effect variables can be chosen. To provide proper definition for the unknown, it may become necessary to study this particular part of the process further to establish the nature of the missing relationships. Consultation may be needed, or as is true in many instances, experimental study may be necessary to obtain more data. The fact that nonlinear equations describe the process behavior does not prevent operators from being used in the cause and effect sense while forming signal flow diagrams. General operators A , B, and C, can be assigned the various relationships between disturbance and response. However, when one approaches the problem of working with the mathematical relationships that govern the system operation in contrast to those that govern the operation of a single block, no general rules are available by means of which the primitive diagrams of the nonlinear type can be reduced in complexity. When a signal flow diagram in primitive form cannot be prepared for a particular operation, the probability is very high that sufficient information is not a t hand to proceed in a sound quantitative manner with process, regulator, or control equipment design. Symbolism. A set of symbols is needed for the development of signal flow diagrams. To unify and standardize the symbols would impose considerable complication, but for the task a t hand six or eight relatively simple symbols will suffice to demonstrate the technique and use of the signal flow diagrams. To designate signal flow, an arrow is used, The arrow head indicates the direction in which the signal flows as it passes from one mathematical operation to another. Whenever two or more signals add or subtract, the signals arc directed to a circle containing a 2. Each arrow is labeled with the appropriate algebraic sign to designate addition or subtraction. In contrast to the practices used in conventional network analysis, a peculiar property of the signal flow is that often signale
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ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT
Plate A
must go t o many different points in a signal flow diagram. .I split-off point is a place from which the same signal goes to tn-o or more points in the diagram. A split-off point is not a subtrwtion of the signals that leave the split point from the signal that enters. Thus
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Aa Plate
B
In many processes multiplication or division is required to represent the resultant signal from two or more incoming signals. The standard multiplication symbol encircled designates multiplication] and the standard division symbol encircled signifit?? division in the time domain. When more than two incoming signals are involved in division, appropriat,e rombina,tions of t,he operations X and t may be joined.
+a
+a
Plate C
Signals are magnified or diminished by passing them through a block that contains a constant. T o distinguish the concept of signal magnification from the concept of the signals’ being operated upon in the dynamical sense, a triangle is often a more convenient symbol t o use for magnification or attenuation. When a general undefined magnification takes place, the constant K is placed in the triangle and the action is spoken of as magnification of the signal by K or:the block has a gain K .
“b Plate
D
Dynamical Actions. Dynamical behavior is translated in the signal flow diagram by having the cause signal pass through a block containing a mathematical operator and then emerge as the effect signal or response. The input-output idea is well known. The real issue is the establishment of the operators that go in the block. As a start two basic mathematical operations 410
are needed to express various dynamieal actions in processing: the derivative of a variable with respect t o time d l d t and the integral of a variahle with respect to time f dt. The derivative operator designates rate of change. The integral operator designates accumulation or growth. Although more complex forms of higher order differentiations and integrations will be encountered in the preparation of signal flow diagrams, these develop naturally from the primitive diagrams in whirh the d / d t and f dt forms suffice. Examples will demonstrate the primitive forms using the d / d l and f d t . Operators can give complex combinations: of hloclcs surh as shown in plate E. Rules of simplification or reduction may be applied and a more compact diagram results. (See plate F.) However, the primitive aspect of the diagram is gone. 411 the variables do not appear, or a t least all operations taken on signals cannot be identified.
Signal Flow Diugrams for a Simple Thermal Brocess A thermal process comprises a t,ank, an inflowing $cream of liquid that carries heat, and an outflo.iving stream of liquid that also carries heat. The density of the liquid is p , its specific heat C,. The volume of the fluid held in the tank is V,. T l ~ e t,ank is made of a material whose density is p t and whose specitic heat is C,t. The volume of the material comprising the tank is Vi.The area of contact between the liquid and the inner SUYface of the tank is Ati. The area of contact between the t a ~ r k exterior and the medium on the outside is Ate. Basically when fluid enters it brings heat. This heat is addoti to the heat already held in the material in the tank. M’hrn fluid leaves, it carries away heat. Heat is lost through the wall of the tank; first, by the fluid to the vall itself; then, from the wall to the outside medium. Generally, partial differential equations must be used to define es because of the distrihutd the behavior of thermal pro nature of thermal capacity and thermal conductivity. Temperatures have gradients through regions of heat flow. That iq, temperature T is a function of 2 , y, 2, and t. However, for this study, assumc that the thermal capacity of the liquid and thr material of the tank can lie r.epreeent,edby single lumped rapacities. Thus, the capacity of the liquid in the vessel ip C = pC, C’ and the thermal capacity of the tank is Ct = ptCptVt. Between the tank and t,he liquid heat transfers through a lumped conductance G = g.4ti and between the vessel and t,he outside medium there is another lumped conductance Gt, = gAi,. Also for the sake of simplicity, it may be assumed that the mixing of nrw fluid with that already in the tank is perfect and instantaneous: that is, the temperature of the liquid T i s uniform throughout the tank. Further, 75-ithin the vessel wall no thermal gradients exist, so that T t represents the temperature of the material comprising the tank at any point. The primitive signal flow diagram i.; shown in Figure 1. Two important points arise in the interpretation of the signal flow diagram in Figure 1. First the temperature of the outside medium T o and the inflow of heat &; become signals of a disturbance or manipulation type. Therefore, two ways exist by means of which the liquid temperature may be altered. The heat inflow in the liquid stream may be changed. The heat losses to the outside may be changed. Hence to control the tank liquid temperature one might resort to a feed stream heater or a jacketed vesrd in which the temperature of the liquid medium within the jack(9t is changed. Secondly, as some mathematical manipulations show, the transient variation of the temperature of the liquid i n the vessel bear exponential relationships of a decaying type with respect t o sustained incremental values of the net heat flow ZQ or outside temperature To. Runaway operation in the sense of uncontrolled rise of temperature toward very high values caused by a sustained heat flow unbalance cannot exist. As the tem-
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ANALOGS
Plate E
Plate F
Figure 1.
Signal flow diagram for simple thermal process
Three manipulation variables exist, Qz, Qo, and To. However, altering stirring rate changes G; making the outside medium move alters Gto; hence, two more possibilities for manipulation occur
= heat inflow rate, B.t.u./min. = heat outflow rate, B.t.u./min. T = temperature of liquid in vessel Tt = temperature of vessel 1, = temperature of outside medium C = thermal capacity of liquid Ct = thermal capacity of vessel
Qi
Q,
Qo
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= thermal conductance liquid to vessel Gt, = thermal conductance vessel to outside
G
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ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT +2A.
U
t
Signal flow diagram showing heat released due to reaction
= catalyst flow rate = catalyst concentration = rate of reaction
Qc C
R
r
= temperature
Qr A, B,
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+2B.
= heat generation rate
D =
undefined operators to account for mixing, reaction kinetics, and chemicol to thermal conversion
Composite signal flow diagram incorporating signal flow diagram of Figure 1 with that of Figure 2A. Signal flaw loop designated b y ZQ-T-CT-R-Q,-ZQ constitutes a positive feedback path or a regenerative condition which could make the process in question an oscillator
Figure 2.
Signal flow diagram for thermal process with chemical reaction
perature of the liquid rises, the losses through the tank wall increase and in doing so, self regulate the thermal process.
brief discussion of the meaning of the signal flow diagrams in Figures 1 and 2 should serve to demonstrate the value of the signal flow diagram technique for process evaluation.
Signal Flow Diagram for Thermal Process with Chemical Reaction I n this example a runaway situation can be encountered because of an exothermic reaction within the vessel: Let the rate of reaction be proportional to a function of the product of the temperature of the vessel contents and the concentration of catalyst in the vessel. Development of the new primitive signal flow diagram requires that the rate a t which heat is generated bv the reaction be added to the thermal signal flow diagram of Figure 1 a t the heat flow summation point. As the catalyst flon-s to the vessel, the manner of its mixing with the contents of the vessel (as the concentration of catalyst) influences reaction rate; hence, rate of heat generation must also be taken into account. The general mathematical relationship among reaction rate, temperature, and catalyst concentration tend to be rather complicated and nonlinear. A functional form of primitive is adopted to relate heat generation rate due to the reaction rate which in turn is a function of temperature and catalyst concentration. However, the concentration of catalyst in the vessel cannot be changed instantly by changing the flow rate of the catalyst stream. A mixing lag is present even in the presence of perfect stirring action. Thus the signal flow diagram of Figure 2A is needed to express the catalyst concentration within the vessel aa a function of the flow rate of catalyst to the vessel. The composite signal flow diagram shown in Figure 2B revrals many facts pertinent to process design and process control. A 412
Interpretation
of Diagrams
Figure 1 shows that the thermal process can be manipulated or disturbed by alteration of the heat inflow rate &$or by alteration of the outside medium temperature To. A change in rither Qa or T oor both will produce a change in the liquid temperature T . TWOintegrations f d t appear in the signal flow diagram associated with thermal capacities C and Ct. Hence a second order (linear) differential equation will approximate the relation between T and the variables &.. and To. A specific form of exponential overdamped transient response occur8 for changes in the manipulation or disturbance variables. By changing the liquid throughput or by changing the flow rate of the external medium over the surface of the vessel, t u o more manipulations or disturbances occur because the condurtances G and Gt, vary with flow. However, this form of parameter variation produces a nonlinear process response. T o investigate the process dynamic behavior calls for a specific knon ledge about the manner of variation of the conductance with flow and a solution for the nonlinear equation which relate T and G or Gt, must be effected. A flow increase also varies Q 0 since Qo =
KT. Two characterizing parameters appear in the signal flow diagram. They are the time constants, C/G and CtlGt,. These quantities are fundamentally related to the material and geometry of the vessel and to the nature of the liquid being processed.
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They determine the dynamical response of the process to both manipulation and disturbance. Their relative size also determines the precision with which the liquid temperature T can be regulated and the speed with which the controlled process can recover from an upset. Figure 2B shows that the combined chemical reaction and thermal process can be manipulated or disturbed by several variables, As before, the heat inflow Qiand the outside temperature T o are process manipulators. They can be used to alter the temperature T which in turn alters the rate of chemical reaction. Another manipulation variable is the catalyst inflow rate Qc. Changes in Qc can produce changes in the catalyst concentration C which in turn alters the reaction rate R. The heat generation rate Q,. may be regarded as a uniformly distributed heat source in the vessel. The heat flow Qp must therefore be added to the heat flow summation point so that Z& which raises the liquid temperature T takes into account heat released due to the reaction. An important consideration in process design and in process control occurs in connection with the signal flow path ZQ T CT R &, - ZQ. This constitutes a positive feedback or regenerative signal loop. It gives rise to the possibility of sustained oscillation or even a runaway condition of the liquid temperature T. A careful study of the stability of the process is needed to determine how to counteract by means of control or by means of process redesign an inherent runaway tendency. When considering the process design and the effectiveness of control over the chemical-thermal process, the mixing lag associated with operator A , the time constant C/G, and the time con-
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stant Ct/Gt, figure prominently in all decisions. A large mixing lag tends to interfere with process control by means of manipulating the catalyst inflow rate Qc. A large thermal lag in the vessel tends to render jacket-type heat exchange to control liquid temperature ineffective. A large mixing and thermal lag in the liquid makes temperature control difficult when heat inflow rate becomes the manipulator.
Conclusion This paper has presented in brief the concept of the signal flow diagram. The examples of the thermal and thermal-chemical processes were chosen particularly to demonstrate: Dynamical behavior of a process, as it is pertinent to regulation and control, begins with the process design. The signal flow diagram is not an analog; rather it is as exact a mathematical model as one can set down for a particular problem. The resulting primitive diagram, however, can be converted to any convenient analog form. The signal flow diagram technique is a powerful analytical tool for studying almost any kind of physical process. I t s greatest value seems to be the striking focus it puts on lack of information or hazy concepts about process dynamics. References ( I ) Brown, G . S., and Campbell D. P., “Principles of Sorvomecha-
nisms,” Wiley, New York, 1948. (2) Mason, S. J., Proc. I.R.E., No. R363.23 (September 1952). (3) Stout, T. M., Am. Inst. Elec. Ewrs., Paper No. 52-254 (No-
vember 1952).
RECEIVED for review September 28, 1964.
ACCEPTEDJanuary 31, 1955.
Analysis of Unsteady Fluid Flow Using Direct Electrical Analogs S. E. ISAKOFF,
Engineering Reseorch Department,
E. 1. du font de Nemourr & Co., Inc., Wilmington, Del.
A direct analogy between pressure and flow fluctuations in fluid systems and voltage and current oscillations in electrical networks is discussed. The validity o f this analogy i s illustrated b y comparing the results o f experimental studies of pulsation dampers located at the discharge of a reciprocating compressor with tests of equivalent electrical filter networks, Advantages, limitations, and practical considerations involved in using electrical networks for solving analogous unsteady fluid flow problems are given. Examples are cited of use of electrical analogs in the analysis of factors affecting the tendency of a system to oscillate, in development of frequency response data to aid in instrumentation and control, in study of pressure transmission in process and measurement systems, and as an aid in determining fluid meter requirements.
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HE visualiaation, formulation, and solution of technical problems through the use of familiar analogous systems is a technique long known in the physical sciences. Because the mathematical equations which define many different physical systems in pature are similar, it is possible to simulate the behavior of a system in a field in which computations are tedious, models expensive, and experimentation difficult, by transforming to an analogous system which can be readily assembled and tested. Results obtained with the analog can then be translated into the terms of the original physical system.
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Although many different physical Bystems have been used as analogs, electrical circuits have proved the most generally applicable. Widespread knowledge of electrical circuit theory and measurement techniques, coupled with the availability of high quality, easily assembled, moderate-cost components, makes the electrical analog computer a convenient working tool. Chemical engineers, as a group, have made little use of electrical analogs, however, and articles in the technical journals describing their application t o chemical engineering problems are few. This is due, in part, to a lack of familiarity with analog techniques and
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