DAVID B. BRANDON
The Thompson-Ramo-Wooldridge Products Beverly Hills, Calif.
C
--
I
MATERIAL BALANCE
PROCESS CONDITIONS
D
m
O i v s n m R A B r z EFFORT is currently being expended on the use of process control computers for optimizing or improving plant operations (5). I n the work described here, an effort is made to clarify certain features of a specific computer control application, showing how a detailed study directed a t the computer control of an alkylation plant may be conducted. The study considers a butene-isobutane alkylation plant in a petroleum refinery (7). Alkylate, the product of this process, is a valuable, high-octane gasoline blending stock. The study of alkylation will be approached from a practical point of view, that of the production-oriented person. H e is often most familiar with the numerous details of the plant and the procedures under which it is operated. According to present practice, the plant is run by a shift operator who is given many requirements which must be met. Above all, he is told to maintain each of the important process variables within a range or a t a value specified as reasonable. H e selects particular values of the variables, following regulations and controlling the plant as best he can to fulfill the desired objectives. The process control computer may also be employed to take action aimed a t these same objectives. Let us examine the means by which the computer can direct the plant to approach optimum performance more closely than the shift operator. The basic question at this stage is, “HOW may the control problem be formulated so that the computer can solve it?” When applying computer control techniques to an operating plant, a broad view of the process as a whole is taken. Such a “systems engineering” (4, 6, 70) generally leads to the approach -_ most realistic results. I n this application a minimum computer control system is evolved. Such a course is easily justified, I
814
INDUSTRIAL AND ENGINEERING CHEMISTRY
Co.,
Improved control
will
maintain
plant conditions close to optimum, giving v‘
v‘
more light alkylate
v‘ higher octane numbers since a minimum formulation illustrates many of the useful techniques, avoids proprietary information requirements, eliminates certain aspects of less than general interest, and serves as a foundation for the various embellishments of a particular installation. Type of Control System
The study is directed a t the design of a computer control system based on the “predictive” or mathematical model approach (3, 9). I n a predictive system, the basis for control is a set of equationsthe control equations-upon which plant optimization depends. The computer is given values of the important process variables which it substitutes into the control equations, solves for the corresponding allowable optimum or nearoptimum plant conditions, checks for reasonableness, and transmits the results of the computations to the process. Predictive control with its variety of mathematical manipulations and data storage requirements is usually best carried out with a n on-line, generalpurpose, digital computer. I t will be assumed that this kind of process control computer is to be employed for alkylation. The computer exercises supervisory control-Le., it is connected to conventional flow, temperature, pressure, and other controllers-and accomplishes its control actions by determining the most suitable set-point value for each of these controllers. As process conditions change, the set-point values are altered to compensate for the change(s), after solving for the current optimum conditions. I n the minimum computer control system, plant design and equipment are regarded as essentially fixed. Only modifications to existing instrumentation are to be considered, typically by connecting the computer, associated hardware and including composition analyzers
less steam
v‘ less acid
such as gas chromatographs in the control system. Fundamentals
of Alkylation
Alkylation is based on the reaction of isobutane and butenes in the presence of a strong acid catalyst-sulfuric acid in this case-to produce a complex mixture of hydrocarbons-predominantly isoparaffins such as iso-octane-called alkylate. T h e over-all alkylation reaction may be written : CaHs butenes
+ ( x ) CIHIBisobutane
...
( y ) alkylate
iso-octane
means of favoring alkylation and reducing the extent of reaction occurring via the olefin polymerization route, certain environmental conditions are specified in the design of alkylation plants. Some of these are: 0 A strong acid catalyst consisting of sulfuric acid at about 9Oyo minimum by weight (typically containing 2y0 water and the remainder, polymer)
Efficient agitation, because the catalyst and the reacting hydrocarbons normally exist almost completely in separate phases
+
(2) “polymer”
(1)
(acid soluble) The acid soluble “polymer” appearing as a by-product in Equation 1 is undesirable on two counts: its formation correspondingly reduces the yield of alkylate; and of greater consequence, its presence decreases catalyst activity and selectivity. Over-all stoichiometry is obtained by observing plant performance rather than from theoretical considerations, because of the numerous chemical species (formed by a variety of reaction mechanisms) present in the alkylate. The actual stoichiometry applying a t a particular time is variable. I t may be expressed relative to the olefin consumption rate as denoted in Equation 1 by the letters x , y , and z. Average stoichiometry figures are available and customarily employed in the industry (7, 8) usually being given on a volume basis as 1.1, 1.72, and about 0.01, respectively. The last figure for polymer was calculated for an assumed average acid consumption rate of 0.5 pound of fresh acid per gallon of alkylate. A side reaction, that of self-polymerization of the butenes, can occur under alkylation conditions. This reaction is undesirable since it competes with alkylation, among other effects forming unsaturated products of inferior quality relative to the intended alkylate. As a
I
0 A suitable refrigeration system to remove the heat of reaction and maintain reaction temperature at a low value, generally in the range of 35 to 55’ F.
A high isobutane-to-olefin ratio. T o minimize self-polymerization of the butenes, a volume ratio (external ratio) of about 6.0 is often employed. A high ratio may also be achieved by maintaining the isobutane concentration in the reactor effluent in the approximate range of 45 to 70y0by volume Under normal operating conditions, the olefins react to extinction. Developing the Predictive Control System
The effect of the improved, computer control system for alkylation is to maintain plant conditions close to optimum and thus produce: greater yield of light alkylate, increased quality of alkylate (higher octane number), and reduced steam and acid consumption. I n this process, it is noteworthy that these desirable features occur togetheri.e., an improvement which benefits any one area also benefits the other two. Emphasis is naturally placed on the VOL. 52, NO. 10
OCTOBER 1960
815
production of light alkylate because of its significantly greater value relative to that of heavy alkylate. This follows whether or not the two are actually separated in a given installation. The basis for a predictive control system for alkylation is developed in logical order by considering the plantsystem boundaries, a schematic flow diagram, the objective of operation, the mathematical model or process equations, constraints on the variables, the optimization scheme, and adaptive control features. A minimum control system is derived in keeping with the philosophy described earlier. Plant-System Boundaries. The first step in developing the new control system is to specify those parts of the physical plant which are to be controlled by the computer. It is to be understood that the computer will exercise control only within the specified boundaries. These may or may not coincide with geographical or operating unit boundaries and may include more than one alkylation unit or less than an entire plant. One reactor and recovery system will serve the purpose-i.e., a single unit. Simplifying Assumptions. Before proceeding with the development of
additional equations, several assumptions: simplifications, and operating principles will be stated as an aid to general understanding and for convenience in specifying the particular system of interest. 0 Acid and hydrocarbon phases are regarded as immiscible, except for the polymer which is present only in the acid phase
Butene-isobutane feed B contains only butenes, isobutanc, and n-butane, components 1 , 2, 3, respectively 0
0
Butene isomers behave similarly
0 Two products of fixed composition, light alkylate L and heavy alkylate H. are produced by the process; the relative quantities of L and H vary depending on operating conditions 0 Makeup isobutane M , isobutane recycle I, and n-butane A T streams are pure (perfect separations) 0 Following a frequent practice, the ratio of acid to hydrocarbon in the reactor is maintained constant-Le., total hydrocarbon feed rate D and total acid feed rate F a r e proportional, F = kD where k is held constant
Polymer and water are the sole diluents in the acid phase, and water is introduced only in the fresh acid, taken as 98yoby weight sulfuric acid Effluent acidity cy, is given as a weight per cent by the expression 0.98A
where d is the fresh acid feed rate and where S* denotes the polymer flow rate in the spent acid Agitation, volume of reaction mixture, and reactor pressure are maintained cons tan t Makeup isobutane M and minimum reactor temperature T,, obtainable at a particular time serve to limit production (the capacity of the refrigeration system can limit reactor temperature T , especially in the summer with cooling water at a relatively high temperatui e ) 0 Source o l butenes is essentially unlimited 0 All additional alkylate produced by improved control of the process may be marketed
Schematic Flow Diagram
N, n - B U T A N E
T = REACTOR TEMPERATURE
E, EFFLUENT ACID
1
R. ACID RECYCLE
S, SPENT ACID Acid and mixed hydrocarbon feed streams are sent to the reactor where alkylation takes place. The alkylateacid mixture i s separated, with the bulk of the acid recycled to the reactor and the hydrocarbon phase passed on to recovery. The excess isobutane-required to maintain the high isobutane-to-olefin ratio in the reactor-is fractionated from the product mixture and recycled to the reactor. Any other light components are also distilled off, leaving the alkylate as a bottoms product. The total alkylate i s then rerun, which divides it into a major overhead fraction, “light alkylate,” and a minor bottoms cut, “heavy alkylate.” In the figure above, the upper case letters are used to name each of the important streams and symbolize their flow rates. Where appropriate an integer subscript denotes a particular component-e.g., 1 represents butene;
8 16
INDUSTRIAL AND ENGINEERING CHEMISTRY
2, isobutane; 3, n-butane; 4, polymer. When applied to an upper case letter, a particular subscript refers to the flow rate of the specified component in the indicated stream. Corresponding lower case letters are also employed to represent concentrations of specific components. When applied to a lower case letter, the subscript denotes the concentration of the component in the stream symbolized by the letter. For example, Dz refers to the isobutane flow rate at the reactor inlet, while bl denotes the butene concentration in the fresh feed. In general
vi
=
vvi
12)
where V,v refet to the general variable and the subscript i, represents the component.
COMPUTER C O N T R O L Some of these assumptions depart from reality; however, they may be justified because they simplify interrelationships without detracting from the techniques employed. Actually additional components-i.e., Ca’s and (25’s-and imperfect separations may be taken into account, with but minor changes from the control equations developed for this four-carbon, perfect separation system. Objective of Operation. Though several operating objectives may apply to the alkylation unit at a particular time, it is best, if not essential, to define a single objective which possesses the desired properties. This objective is usually expressed in monetary terms. For the alkylation unit the objective is to maximize the dollar gain from alkylation subject to various constraints on operating conditions. Both the quantity optimized (maximized in this case) and the various constraints are specified explicitly in terms of the important process variables. This statement will be further amplified as the analysis progresses. As an initial step, the Dollar Gain may be written in terms of several quantities:
Characteristics of an Alkylation Plant The alkylation plant may consist o f a single alkylation unit or a group o f more or less separate units in a series and/or parallel configuration. Considering the entire plant or an individual alkylation. unit, certain characteristics may be observed:
Dollar Gain Value of from Alkylation (Products) Variable Fixed
0
Normal operations are successful. Production requirements are met and the process i s kept under control, even in the face of abnormal situations.
0
The operating level changes as a function of time because of relatively frequent changes in the values of the process variables.
0
The important process variables are known. Experience has demonvariables which strated those to which the process i s most sensitive-the must be involved in any realistic control system. That an unknown variable may exist cannot be denied, but the degree of importance of the unknown variable must be of a lower order of magnitude than that of the known variables. It stands to reason that a truly important unknown variable would not remain so very long and still allow the plant to be controlled properly. The experienced operator would detect its existence and discover suitable means for applying compensating changes to maintain effective control.
0
The interrelationships among the variables are only partly understood. Some quantitative information i s available to the shift operator, but most of his decisions depend upon various experience-based rules which are often of a semiquantitative or qualitative nature.
0
Steady-state conditions are frequently reached. Transitions between successive steady-states are accomplished satisfactorily.
0
The present control system i s based to a great extent on incremental changes from the immediately previous conditions. On detecting a change in the process, the operator notes the previous conditions, predicts the effect of the change, and generally makes an adjustment in the value of one or more o f the variables to compensate for the change.
Various costs are divided into variable and fixed, to distinguish between those elements of cost which are a function of production rate (variable cost) and those which are independent of production rate (fixed cost). The variable costs which are most significant in determining Dollar Gain are included in the expression : Dollar Gain
’ from Alkylation
Value of = (Products)
-
cost of (Butene)
Each term on the right side of Equation 6 may be comprised of more than a single cost item, as necessary to produce a reasonable result for the Dollar Gain. For example, the isobutane term includes the cost of isobutane which reacts and leaves the system as product, the variable handling costs for isobutane, and the variable steam costs necessary to recover isobutane from the product mixture. These costs are all a function of the quantity of isobutane which reacts a t a particular time. Such costs may be lumped together into a variable cost per unit quantity or “incremental” cost of isobutane. This incremental cost when multiplied by the quantity of isobutane which reacts equals the isobutane cost term of Equation 6. Similarly the fixed handling costs and the fixed steam costs
The operating level of the process i s typically suboptimum. The shift operator continually attempts to approach optimum conditions but i s handicapped b y his lack of complete, quantitative information at each time of decision.
-
cost of
cost of
Fixed
(Isobutane) - ( Acid ) - (Cost )
associated with isobutane are included in the fixed cost term of Equation 6. All fixed cost factors are combined into the single fixed cost term of Equation 6. Defining some new symbols, substituting certain flow rates given in the figure, and using the notation described earlier, Equation 6 becomes G =
+ HUH- B I U I-
LUL
(BI
+M)
uz
- A U A - uf (7)
where G = dollar gain from alkylation per unit time L = light alkylate production rate H = heavy alkylate production rate B1 = butene flow rate Bz = isobutane flow rate in fresh butene feed
(6)
M = mabeuu isobutane flow rate A = acid addition rate u L = incremental value of light alkylate
uR = incremental alkyl at e u1 u2
value
of
heavy
= incremental cost of butene = incremental cost of isobutane
net cost of acid (net refers to the difference between cost of fresh acid and salvage value of spent acid) u, = total fixed cost per unit time ud = incremental
Each ui represents composited variable cost (or value) per unit quantity of material i, which enters into Equation 7. These factors are sometimes difficult to determine with accuracy, but they usually can be estimated. The various VOL. 52, NO. 10
0
OCTOBER 1960
817
u; are not generally constant, because
they change as a function of source of supply, level of usage, seasonal conditions, production requirements, quality of alkylate produced, and other factors. Each u; is preferably expressed explicitly in terms of its determining factors. Thus, U L and U H may be given as functions of octane number of alkylate, while may be determined from two quantities, the incremental cost of steam for
separating isobutane from the product mixture and the incremental cost of fresh isobutane. As quality, economic, and other factors change so must the ud. However, the ui are expected to undergo significant change less often than other process variables, such as flow rates and concentrations of important streams. Mathematical Model of Alkylation. The mathematical model of alkylation serves as a means to predict values
How to Set Up for Computer Control
.
Major Variables
Additional Equations. .interrelate
major variables and material balance variables
Eight variables, I, M , 6 1 , b,, B , A , a , and T. are usuallv involved in the control of the unit ’
a = - 0.98 A
Bi
Eight variables, I , M , B I , , B 2 , B,, A , S d , and L , define the material balance
B = BI
D = B
Bz = Bbz
+ Bz + B S
S=A+S,
E = R + S B3
-I
Yr, = 4 0 ( d d d i , d,, D,a , T )
L
(8)
+M +I
H = D $- F --E
-
3
N
- 61 - 6 2 )
Y L = L/Bi (3)
R = F - A
A‘
Bbi
B I = B (1
Material Balance Equations
F = kD
+ Sa
A
Material Balance Variables
i d S / d i , d,, D , a, T )
B,
=
(9) (lo)
D1 = B1
(I1)
dl = D i / D
(12) (I3)
dz = D z / D
D z = B2
+I +M
Y L = 4 ( B , b,,
62,
M , I , a, T )
-*-L ( 1 4 )
.
The Objective. .maximize Dollar Gain
+ HUH- Blui - (Bz + M)uz - A U A - u f
G = LUL
The Control Equations. , .result from substituting the Major Variables into Equation 7 G = B [ ~ I ~ (-u ULH )
4- ( b i 4-
b 2 ) ~ ~blul -
M ( U H-
.
Optimization. .occurs when.
uz)
-
4-
6 2 ~ 2 1
a)uH]
A
-
uA
-
u,
(31)
.. (35) (36)
(37) Solving. a is
..
determined directly from Equation 35.
B is determined from Equation 36, for the now fixed values of CY and the remaining specified variables. An iterative technique may be employed, in which the left and right sides of Equation 36 are evaluated separately for an assumed value of B. When both sides differ by less than the allowable errar, the value for B is established. A is determined from Equation 37, since both Iteration may again be employed, as above.
818
INDUSTRIAL AND ENGINEERING CHEMISTRY
cy
and B have been specified.
of specified dependent quantities, given values for the independent variables of the process. The independent variables used for control are determined by the process itself, available plant facilities, and convenience (where a particular independent factor may be expressed in one of several ways). The dependent quantities associated with the control system are generally selected for convenience; they aid in understanding the process and are involved in the evaluation of plant performance. Product yield and quality are dependent quantities frequently employed as measures of plant performance. As considered here, optimum values of specific variables do not result directly from the use of the mathematical model. The model provides a basis for making calculations of process performance corresponding to particular operating conditions. These conditions will not be optimum conditions, except by coincidence. The group of equations comprising the model, as so regarded, will be called the “process equations.” Process equations are later combined with additional equations which taken collectively make possible the determination of optimum or near-optimum conditions. The computer control system will operate upon the complete set of equations, the control equations. Process equations may exist in several forms, notably linear or nonlinear, steady-state or dynamic, empirical or theoretical, or a suitable combination of these factors. The process equations for the alkylation unit developed in this paper are based on a nonlinear, steadystate, partly empirical, partly theoretical, approach. The steady-state aspects of this treatment are virtually a necessity if normal operating data are used to synthesize the process equations A satisfactory model of the alkylation unit may be obtained if all of the independent variables appear in and are adequately interrelated by the process equations. These equations must reflect a material and/or heat balance of the major components, and must take into account the chemical transformations which occur. The relationships expressing the chemical transformations serve to bring into the set of process equations those variables employed in controlling the process which are not directly involved in material balance. Material Balance. There are eight independent variables Ivhich are necessary to define the operating level or state of the alkylation uni:. This fact is most readily observed by noting the minimum number of variablrs whose values must be specified in order to establish a material balance around the unit, subject to the various operating practices employed For example, given values Bs,A , 5‘4, for the variables, Z: M , B I , Bz., and L, the other variables involved in
e
COMPUTER C O N T R O L material balance are , determined and may be calculated for the steady-state condition using Equations 3, 8-14 (see the figure caption). Thus, specifying values for the given set of eight variables determines all of the remaining variables directly associated with material balance. I n addition, it will be noted that effluent acidity a, is also determined by the set of eight variables, since prescribed values for A and S d may be substituted into Equation 4. Further, it is apparent intuitively that the values of the set of eight variables must also determine the value of the single remaining variable of this system, reactor temperature T. Were this not so, T could vary without affecting the other variables of the process. Given values of seven variables or less, it would not have been possible to specify all of the remaining variables. The above discussion could have been conducted by means of a degreesof-freedom analysis. This was avoided to maintain the practical vein. Naturally, these same results would have been obtained. The reactor-separator-recovery system exclusive of the multiple feed points in both hydrocarbon and acid phases has five degrees of freedom. This is apparent later in Equation 19 where Y , is expressed as a function of five and only five independent variables. Since there are two extra streams contributing to D and one extra stream contributing to F, three additional degrees of freedom are provided, for a total of eight in the complete unit. Major Variables. The variables above form one of several possible sets of eight variables which can specify the state of the process. I n particular, a second set comprised of I , M , 6 1 , bz, B, A , a, T may also be employed. This set of variables, called the “major” variables, includes those quantities which are usually involved in the control of the unit. The present control system deals with these variables, and the computer also is expected to employ them. Process Equations. A useful set of process equations results when the equations interrelating both of these sets of variables are written. I n this way when any eight independent variables are specified, all of the remaining variables pertinent to the control system may be calculated. I t is not yet possible to attain this goal. O n examining both sets of variables, it is apparent that the variables I, M , and A appear in both sets. The combination B1, Bz, B3 of the first set and bl, bz, and B of the second set are equivalent, employing Equations 2 and 8 to derive the sequence of Equations 15-17. I n a similar fashion A and S, of the first set and A and CY of the second set are interrelated by Equation 4. But no relation-
ship has been expressed as yet between L of the first set and T of the second set. This relationship is presently unknown and must be developed. A practical way of expressing the unknown relationship is to employ a parameter, such as yield of light alkylate, Y L . This parameter may be written in terms of both variables to be interrelated as in Equations 18 and 19. Equation 18 serves to define YL as the ratio between light alkylate production rate and olefin addition rate. Equation 19 expresses in functional form the fact that YLis determined in this system by the isobutane-butene ratio at the reactor inlet, butene concentration in the hydrocarbon stream entering the reactor, total flow rate into the reactor (a measure of residence time in a constant volume reactor), effluent acidity, and reactor temperature, respectively. The variables on the right side of Equation 19 appear as the result of a thorough analysis of the alkylation process, taking full account of the wealth of knowledge and experience which is available. The function may be established empirically by means of standard regression techniques or by other methods appropriate for use in conjunction with the process control computer (2, 73). The explicit evaluation of the function $10 may be tedious and will be regarded as being obtainable but by methods beyond the scope of this paper. Eliminating Y L from Equations 18 and 19 gives Equation 20. With $IO known, the two sets of eight variables may be interrelated. For example, given values for the second set of variables, the first set is determined by the sequence of Equations 15-17, 4, 9, several new Equations: 21, 22, 23, 24, and finally, Equation 20. The above sequence of 10 equations together with Equations 3, 8, 10, 11, 12, 13, and 14 for a total of 17 form a set of process equations for the alkylation unit. It becomes clear that with the values for any eight independent variables known, the values of all remaining variables may be calculated. Constraints on Process Variables. In order to ensure that only reasonable values for the manipulated variables will be transmitted to the process in the form of set-point signals to conventional controllers, a number of constraints must be specified. Constraints may take two forms : Upper and/or lower limits on the manipulated variables 0 Restrictions on functions of the variables Some examples of both types are quite apparent. The capacity of the deisobutanizer column determines an upper limit on the flow rate of isobutane recycle I . Similarly, theoretical considera-
tions require that the effluent acidity, be at least 85% to avoid unwanted side reactions; this figure may serve as a lower limit for a. The minimum reactor temperature attainable T , is representative of the second kind of constraint since it is a function of the capacity of the refrigeration system (itself a function of several variables) and the throughput and composition of the feed tu the reactor. If CY,
where
To
=
T,M=
Practical lower temperature limit (to prevent the reaction mixture from freezing) Practical upper temperature limit (to minimize unwanted side reaction products)
reactor temperature T can be set equal If T , < TO,T to or greater than T,. may be set equal to or greater than TO, in the latter case by using less than the full capacity of the refrigeration system. If T , > T M feed , rates are decreased to a point where T can be set equal to or less All practical constraints are than T,. included in the computer control system. Setting u p the Mathematical Model. One of the major items leading to the economic justification of on-line computer control is the fact that improved control provides for plant operation more closely approaching optimum than is possible by conventional means. Optimum operating conditions for the alkylation unit are determined by the best set of values of eight independent variables. The second set of variables given above was originally selected for discussion because it contains readily measured variables of the. process and anticipates practical control actions This set, comprised of I , M , b l , b2, B, A , C Y , and T , will serve as the basis for the computer control system and the above are thus the “major” variables of the unit.
I
Maior variables are divided into: Manipulated Variables-controllable Quantities whose values are intentionally changed (within limits) by the control system. Disturbonce Varia bles-uncontrollable Quantities whose values may b e m e a s u r e d b u t not changed by the control system.
I
In this system I, B, A , a, T are the manipulated variables, capable of being controlled directly by a human operator VOL. 52, NO. 10
OCTOBER 1960
819
or a computer. The remaining variables, M , b l , and bn, are the disturbance variables, controlled usually by the operation of up-stream process units and thus external to the alkylation control system. The control problem consists of measuring the values of the disturbance variables at each time of interest and adjusting the values of the manipulated variables in such a way as to maximize G of Equation 7, while ensuring that all constraints on operating conditions are met. One of the possible optimizing schemes for this system can be illustrated once Equation 7 is expressed in terms of the major variables. This may be accomplished after suitable algebraic manipulations have been performed. Making substitutions for the first three variables of Equation 19 results in the equation B62
+M +I
Bb1
B
+M +
Experience indicates that under all practical conditions maximum isobutane recycle and minimum reactor temperature are optimal. Thus, I is set equal to its maximum value permitted by the capacity of thc unit (the capacity of the deisobutanizer tower) T is set equal to its minimum value permitted by the capacity of the refrigeration system T , (based on the previous steady-state value) and the practical upper and lower limits T,, and To The ui are taken as the specified current values or the previous steady-state values. Under these conditions with the u i and five of the eight major variables specified, Equation 31 relates G as a function of only the variables A: B , and CY. Optimization Scheme. The optimization of G will be described for the simple case where operating conditions fall within all constraints. The quantity G is maximized when its partial derivatives with respect to the variables A, B, and CY vanish. Thus
which establishes a different function $ of the variables in Equation 25, YL
=
+ ( B , 61,
62,
h1, I ,
a,
T ) (26)
From Equations 18, 15: and 26 L
B61+
=
(27:
Eliminating R from Equations 10 and 12 and substituting Equations 11 and 4 into the result produces
Literature Cited (34)
From Equations 32>33, 34 0.98
Using Equations 9, 28, 13, 27, 8, 15, and 16, Equation 14 becomes,
+
H = Bbi f B62 M (0.98 - 0 1 )_A _ a
a=---
UH
m Bbi+
(29)
-
~I+(ZLL
Bbi+UL
(0.98
-
+ (bi +
UH
(35)
UA
~?)UZI
-
6 1 ~ 1- b m n
(36)
a+
=-CY’
+ LB6 + Bbn + dd -
0.98 AUH Bbi(2L~- U L )
(37)
1
cr)A
01
(6’62
1
- B614 uti - Bbiui
+ .M)uZ - Au.4
-
uj
-
(30)
Collecting terms
+ +
G = B [ ~ s + ( u-L U H j (61 b?)Ux b i ~ i- 6 2 ~ ~ 1A V f ( ~ L ~- U P ) -
+
Thus, Equation 31 expresses G in terms of the major variables and the u,. I n the computer control system a t a particular time when an optimization calculation is to be made, Equation 31 may be simplified considerably oiving to the following : The values for M , h l , and bz are fixed since these are disturbance variables with specified values at t
820
UH)
-
B b i ( ~ a- UL)
Substituting Equations 27, 29, 15, and 16 into Equation 7 produces
G =
valut-s of presently unknown quantities or known variables which have deliberately been omitted from the computer control system. The mechanism for adapting the control equations to follow the changing process rests with a procedure which modifies the values of the coefficients in appropriate equations when such action is warranted, typicall>- when a new steady-state has been reached. The empirically derived function in Equations 19 or 26 is the primary target of the updating procedure. Generally simple but effective modifications are made on-line, while the more complicated, complete reevaluation of coefficients is done infrequently using off-line methods. Online up-dating may be accomplished by comparing earlier predictions of steadystate performance-for example, light alkylate production rate-Ivirh later measured values and then changing preferably a single constant in the function to compensate for an observed discrepancy. The adaptive feature of predictive control is vital to the success of the system. I t builds in a tolerance for error associated with the original formulation and provides the means for the set of control equations to adjust to the process-not the reverse.
Therefore, Equations 35, 36, and 37 establish the best values for CY, B , and A. In actual practice an optimization procedure with constraints included would be employed. Both equality and inequality constraints can be handled satisfactorily using Lagrangian multipliers. Optimization methods based on systematic search ( 73), gradient methods (72. 73), and dynamic programming ( I 7) are being applied to solve the control equations for chemical-type systems. Adaptive Control Peatures. U p to this point the predictive, feed forward aspects of computer control have been emphasized. However, to be realistic the system must also include certain corrective: feed back elements. These are required by the inevitable variability of the process itself and the need to compensate for fluctuations in the
INDUSTRIAL AND ENGINEERING CHEMISTRY
(1) Borthick, G. D.: Durland, L. V., Pope, B. J., Oil and Gas J . 54, No. 23; 88-9 (1956). (2) Brandon, D. B., ZSA J o u r n a l 6 , h-0. 7 , 70-3 (1959). 13) Brandon. D. B.. IRE T r a n s . on Ind. Electronics IE-7, NO. 1, 15-20 ( 1 960). (4) Freilich, A , , ISA Journal 6, No. 7, 47-53 (1959). (5) Zbid,. 6 , hTo. 7, 66-9 (1959). (6) James, E. W . , Boksenbom, A. S., Control Eng. 4,No. 9, 148-59 (1957). (7) Payne, R. E., Petrol. Rejner 37, No. 9, 316-29 (1958). (8) Ibib., 37, No. 9, 251-5 (1958). ( 9 ) Phister. M.. Jr.. “Digital Control \
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Systems, ‘ Present and FGture,” Paper presented at the Conference on Industrial Instrumentation and Control, Armour Research Foundation, Chicago, Ill., April 14-15, 1959. (10) Phister, M., Jr.: Grabbe, E. M., Control Eng. 4, No. 6 , 129-36 (1957). (11) Roberts, S. M., “Dynamic Programming Formulation of the Catalyst Replacement Problem.” Paper to appear
in A.1.Ch.E. Computer Symposium Volume, 1960; presented at the International Congress of Chemical Engineering, Mexico City, Mexico, June 21, 1960. (12) Schrage. R. W., Operations Research 6 . 498-515 11958). (13j Stout, T: M.,’ZSA J o u r n a l 6 , No. 9 > 98-103 (1959).
RECEIVED for review January 25, 1960 ACCEPTEDJuly 9, 1960 Adapted from a talk delivered at a Princeton University Conference on “Computer Control of Industrial Processes” in Princeton,N. J.,Feb. 11,1959.
