Control of a Fixed-Bed Chemical Reactor

Control of a Fixed-Bed Chemical Reactor B. E. Stangeland' and A. S. Foss

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Department of Chemical Engineering, University of California,Berkeley, Calif. Q,$?so

Disturbances in a fixed-bed reactor are controlled b y exploiting the interaction between the traveling waves of temperature and concentration within the bed. A secondary reactant stream, injected at an intermediate point along the bed and modulated in accordance with temperature measurements only, provides a localized and traveling corrective action that annihilates a disturbance by the time it reaches the reactor effluent. Calculations for a first-order, homogeneous, exothermic reaction show that a temperature upset i s the most difficult disturbance to control. Further, because of the dominance of the slower thermal wave in these reactors, the task of composition control i s similar to that of temperature control. An approximation technique, developed to match both the short- and long-time behavior of the reactor and its controllers, simplified the study of the dynamics of the reactor under control.

THE

COKTROL of temperature and concentration disturbances in fixed-bed reactors presents perplexing problems, many of which derive from the n-ave-like propagation of these disturbances through the bed. Upsets whose duration is on the order of, and less than, the reactor's characteristic response time are particularly difficult to control; nonetheless their control is often necessary to ensure, for example, the reactor's structural integrity and t o protect catalyst activity. The effective control of such disturbances requires a proper accounting of the dynamic interplay of the temperature and concentration waves within the reactor. Slowly varying disturbances, on the other hand, can usually be controlled by considering the reactor to pass through a sequence of quasisteady states for which dynamic considerations are unnecessary. The method of control discussed here exploits the intrinsic nature of the reactor and its dynamic characteristics. Because considerable time is required for the transit of wave-like disturbances of temperature and concentration through fixedbed reactors, there exists an unparalleled opportunity to shape and correct those disturbances before they emerge in the product stream. By contrast, processes whose state is uniform everywhere, as it is in ideal stirred-tank reactors, do not present such a n opportunity; disturbances there make their way immediately to the process effluent. The localized and traveling nature of the disturbances in fixed beds naturally suggests, therefore, that the corrective action be of a localized and traveling nature as well. Further, to modify the disturbances as they travel through the bed, the dynamic characteristics of the corrective action must be not unlike those of the disturbances themselves. I n this work, the dynamic characteristics and the propagation speeds of the two are made identical simply by employing the reacting fluid itself as the source of the corrective action. This is achieved here by the injection of a niariipulable secondary reactant stream at some intermediate point in the bed, a technique sometimes found in industrial practice. This technique demands, of course, precise detection of the traveling disturbances early in their transit through the bed. I n this work, measurements are made a t the two locations indi1 Present address, Chevron Research Co., Richmond, Calif. 94802.

38

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VOL. 9 NO. 1 FEBRUARY 1970

cated in the schematic representation of this control technique shown in Figure 1. Based on these measurements, the concentration or temperature of the secondary reactant stream is modulated by the two controllers, F and B. These controllers are designed utilizing a knowledge of the dynamic interactions of the traveling waves in the bed, such that a traveling disturbance in the bed is completely annihilated by the time it reaches the end of the reactor. The study of the feasibility, merits, and performance of controllers designed to theqe specifications is one of the principal contributions of this work. I n these studies, the reactor is considered to operate close to its steady-state conditions; circumstances under which transients are large in m a g n i t u d e f o r example, during reactor startup-are not considered here. Further, only upsets in the concentration and temperature of the reactant feed stream are treated, as these are two of the most important encountered. Pressure, coolant temperature, and the momentary occurrence of catalyst poisons may occasionally assume importance, and for these, the method of control discussed here would require extensions. The feasibility and merits of the control method proposed here are studied through a calculational analysis using a simple yet realistic mathematical model for the dynamics of the reactor. The irreversible chemical reaction considered is an exothermic, first-order, homogeneous, liquid-phase reaction. The reactor model, which is similar to that studied by Crider and Foss (1968), is a simple one-dimensional continuum model that accounts for the distributed thermal capacity of the packing and finite rates of heat exchange with the packing, but neglects axial fluid mixing. I n this work, the reactor is considered adiabatic. Because only small excursions about the steady state are considered here, the nonlinear aspects of the reaction rate in the dynamic model may be replaced by linear approximation through local linearization about the steady-state conditions. To achieve faithful modeling, representations of nonlinear effects are retained, however, in determination of the steady state about which the linearization is made. This type of model has been the subject of several experimental and computational investigations and has been found to represent accurately the principal dynamic effects observed in these reactors (Crider and Foss,

=0

II

S, %

-I-’&-

Mixinq valve and va I ve i t i one r f

-pas


32

Controlier

II

Ir“.‘ uInjected stream

--(

e .-c0

:=b . = c

I‘

-:

-0

0 c

-I V

0

0.1

0.2

0.3

0.4

0.5

0.6-0.7

0.8

0.9

1.0

Dimensionless d i s t a n c e , x

Figure 2. Temperature and concentration response of uncontrolled reactor to a unit impulse in feed concentration

‘ P

Figure 1 . Schematic diagram of reactor control system using modulation of secondary reactant stream In general, SI and Sz are reactant streams a t different temperature or concentration.

1966, 1968; Hoiberg, 1969; Simpkins, 1966; Sinai and Foss, 1970; Tinkler and Lamb, 1965). Even though this analysis is restricted to the type of reactor model described above, the concepts advanced should find general applicability in the control of reactors of similar type. Certain catalytic reactors operating, for example, with gaseous reactants under conditions that are not extreme exhibit dynamic characteristics similar to those of the liquid phase reactor treated here (Hoiberg, 1969) and thus would be amenable to control by this technique. This control method, however, is not to be considered a complete system, for, because of its feed-forward nature, it lacks the ability to correct for model inaccuracies and upsets other than those feed disturbances considered. Rather, it is viewed as a first line of defense capable of removing a major burden from the responsibilities of a feedback control system. The synthesis of a feedback control system effectively coordinated with the system under investigation here is a topic for another study, however. Control Problem

Because this fixed-bed chemical reactor is a distributedparameter process, the major disturbances of temperature and concentration propagate through the bed as traveling waves. These waves of heat and reactant interact through the chemical reaction and, in contrast to an empty-tube reactor, move a t different velocities because the exchange of heat between the fluid and packing slows the thermal wave. These phenomena make difficult both the characterization and control of such a reactor. The existence and behavior of such waves are demonstrated in Figures 2 and 3. The response of the reactor model (Crider and FOSS,1968) to unit impulses in feed temperature and concentration is plotted as a function of reactor length, with

2 IL

(I; - 8 . 0

0

I

I

I

I

I

I

I

I

I

01

03

04

05

06

07

08

I

02

09

IO

Dimensionless distance, x Figure 3. Temperature and concentration response of uncontrolled reactor to a unit impulse in feed temperature

time in fluid residence times as a parameter. Not shown here are the remnants of the input impulses and the induced impulses (Stangeland, 1967). Here the dependent variables represent deviations from local steady-state values. Thus, as shown in Figure 2, positive concentration upsets always generate waves of higher temperature and reduce concentration throughout most of the bed. However, as shown in Figure 3, both the temperature and concentration responses to a positive impulse in feed temperature are positive in some parts of the bed and negative in others. This behavior, which is a consequence of the velocity difference between the thermal wave and the fluid, has been observed by Boreskov and Slin’ko (1965) and Hoiberg (1969) and discussed by Crider and Foss (1966, 1968). As discussed by those authors, a temperature decrease occurs momentarily near the end of the bed because the reactant concentration there, and hence the rate of heat generation, fall below their values a t steady state. The decreased reactant concentration results from the passage of fluid through the slowly moving zones of high temperature early in the bed in which the reaction rate is higher than normal. With this behavior known, a simple “thought experiment” can show what would happen if a feedback controller were used to maintain constant effluent temperature by adjusting feed temperature. For a positive temperature upset in the feed, a VOL. 9 NO. 1 FEBRUARY 1970

I&EC FUNDAMENTALS

39


temperature detector a t the effluent would sense nothing for one fluid residence time. Then, as shown in Figure 3, it would sense a decrease in temperature. This would require the feedback controller to raise the feed temperature for about another residence time until the larger positive temperature wave appears in the effluent. But the original disturbance was an increase in feed temperature; the controller therefore has added to the disturbance. With a gain of unity, the controller would actually add more than twice the heat entering in the original impulse. Thus, because of both the time delay due to fluid transport lag and the momentary “wrong-way” behavior of temperature in the reactor, this type of feedback control is not suitable. Although the above discussion has considered disturbances only in feed temperature and concentration, a wide spectrum of other possibilities must be recognized before designing an industrial control system-for example, the reactor may be subjected to flow and pressure upsets as well as variation in the temperature of the cooling jacket. Even the characteristics of the reactor may change with time because of catalyst poisoning and changes in free surface. Rut for this initial study of this control problem, consideration is given only to disturbances in the temperature and concentration of the feed stream. The choice of the control objective is equally involved, and is influenced on the one hand by the ideal of perfect control of all process variables and on the other hand by what variables can be measured and what is economically and mechanically feasible. Usually a workable compromise can be reached by controlling only a few variables within prescribed ranges or by minimizing an objective function which includes a trade-off between quality and cost. Here, either of two objectives is sought: the maintenance of perfect invariant control of either effluent temperature or concentration while allowing the other t o vary freely. Control Method

I n the control method used here, disturbances are detected early in the bed. The information is then transmitted to a control device which, through modulation of the secondary stream, generates a traveling control wave of reactant that annihilates a disturbance by the time it reaches the end of the bed. I n this sequence of actions, the rapid measurement of both temperature and concentration is critical to the success of the method of control. Because the measurement of concentration, when feasible at all, is usually a very time-consuniing process, it is not used here; rather concentration disturbances are inferred from measurements of temperature alone. As shown in Figure 1, temperature is measured a t two locations: the reactant feed and an interior point, z = b. )Then temperature disturbances arrive a t the interior measurement point, they are compared with the temperature that would appear M-ere only a temperature disturbance present in the feed. Any discrepancy can be attributed to the teniperature effect resulting from changes in feed concentration. Thus, by the time any disturbance reaches point b, knowledge of feed disturbances becomes available to the controllers and knowledge also of those disturbances in concentration as well as temperature that have arrived at point b. Such a calculation requires a n accurate knowledge of the dynamic behavior of the reactor; this information is considered available in this work. A short section of the reactor may be needed between the point of measurement and the point of reactant injection to account for a finite calculation 40

l&EC FUNDAMENTALS


rate, so that b y the time the disturbances have traveled from point b to point c in Figure 1 the controllers are ready to act on the waves as they pass the injection point. The use here of a traveling control wave, achieved b y injection of secondary reactant, is the most direct and natural corrective action that may be taken. Yet, with the exception of the work of Powell (1963), i t does not seem to have been exploited in earlier studies of this problem. Proposals to manipulate the total feed rate or the external coolant temperature (Tinkler and Lamb, 1965) are less effective as control measures for the short-term solitary disturbance because such control action is not of a local nature. I n this work, the concentration and temperature of the secondary stream are considered as the manipulated variables. I n industrial practice, however, it is more common to find the flow rate of the secondary stream as the manipulated variable. With such a technique, the control action is no longer a simple traveling correction but includes also the often undesirable effects of flow changes throughout the reactor. I n general, the design of the controllers depends upon and is complicated by the nonlinear behavior of the reactor, I n this work, however, nonlinear dynamic effects in the reactor have not been considered because the principal dynamic characteristics of these reactors operating near the steadystate conditions are well represented by locally linearized dynamic models (Simpkins, 1966; Sinai and Foss, 1970; Tinkler and Lamb, 1965). Thus, with a linear dynamic model of the reactor, the control functions may be expressed in terms of the transfer functions of the various sections of the reactor. I n this work, therefore, controllers F and B of Figure 1 are restricted to linear controllers. Determination of Control Functions

To derive representations for these controllers, transfer functions for the reactor are needed that describe the effects upon temperature and concentration at any point in the reactor of disturbances in these same variables introduced at any point upstream from the point of interest. The responses at z are related to disturbances a t point u < z by the following two equations: T(s,z;u) = Grc(s,z;u).C(s,u)

+ Grr(s,z;u).T(s,u)

(1)

Here T and C represent, respectively, deviations of temperature and concentration from local steady-state values. The transfer functions, G, are functions of the complex variable, s, the location, u , of the disturbance, and point z a t which the response is desired. Each of the three sections of the reactor shown in Figure 1 can then be represented by its appropriate set of four transfer functions and the two controllers can be represented in terms of these functions. There are four control cases of interest because either T, or C, may be controlled by modulation of either the temperature or concentration of the injected secondary stream. Using the notation 6T or SC, respectively, for the temperature and concentration excursions in the injected stream, the four cases may be enumerated as follows: Case 1. Case 2. Case 3. Case 4.

Control Control Control Control

C, with SC, define F C, with ST, define F T , with SC, define F T , with 6T, define F

= FCCand

= BCC

3 FTC

E

B and B 3 FCT and B = F T T and B

BTc BCT 3 BTT 3

Evaluation of Equations 1 and 2 a t points 2 = b, c, and p provides the relationships needed to define the controllers.

The disturbances entering the third section of the reactor (the sum of the original and injected disturbances) are defined as AC(s) and A T ( s ) . The responses a t point p are given by

Cp(s)

=

AC(s) .GTC(S,~;C) f AT(s) .GTT(s,P;c)


=

f x 1-f

+ (1 - f)*GC(s) + (1 - f ) . b T ( s )

1

(4)

Since the control objective is to hold either of the two variables C , or T , identically equal to zero, the task is reduced to finding the appropriate expression for AC(s) and A T ( s ) in terms of the feed disturbances Co(s) and To(s).The fraction of total mass t h a t enters the reactor in the feed stream is defined to be f, and 1 - f is the remainder that enters in the secondary feed stream. If these two streams have equal heat capacities, heat and mass balances around the injection point take the form

AC(S) = f*CC(s)

(5)

AT(s)

(6)

= f.Tc(S)

Fcds)

+ AT(s)* G c T ( ~ , P ; ~ ) (3)

AC(s) .Gcc(s,p;c)

Tp(8) =

FORCASE1

Similar treatment of the other three cases gives the controllers:

FORCASE2 FTC(S) =

f

__

1-f

x

1

Additional constants would be included here if these two streams had different heat capacities. These equations thus relate the disturbances entering the final section of the reactor, appearing in Equations 3 and 4, to those leaving the second section of the reactor and those added at the injection point, 2 = c.

Similarlj-, the disturbances leaving the second section of the reactor can be described in terms of those entering in the feed to the first section of the bed.

+ To(s).Gc~(s,c;O) CO(S)+G~c(s,c;O) + To(s).Grr(s,c;O)

C,(S) = CO(S). Gcc(s,c;O)

(7 1

Tc(s) =

(8)

(15)

FORCASE3 Fcds)

f

= -

1- f

x

The injected disturbances are determined by the two controllers and the two temperatures they measure. 6C(s) or 6T(s) = F ( s ) .To(s)

+ B(s).Tb(s)

(9)

The temperature at 2 = b is defined in terms of the feed disturbances.

+

T ~ ( s=) CO(S) .G~c(s,b;O) To($).G~r(s,b;O)

(10)

For each of the four possible cases, Equations 3 to 10 define the control functions F ( s ) and B(s) that w ~ l maintain l either C,(s) or T p ( s )invariant for all temperature and concentration disturbances in the primary reactant stream. The procedure for the determination of F and B involves the solution of these equations for C, and T, in terms of the two measured temperatures. For Case 1 control, in which effluent concentration is controlled by modulating concentration, this expression is C,(S)

=

To(s).[Gcc(s,i’;”:. ((1 - f ) F c c

fGrc(s,C;o)

Explicit representations of these control functions in terms of the basic parameters of the reactor are given following the determination of the reactor transfer functions in the next section. Determination of Reactor Transfer Functions

Defining the controllers to make the two brackets in this equation identically zero will make C p 3 0 for any values of Toand Tb (and thus also of CO).The expressions for these two controllers are

T o demonstrate the validity of this control method and its effect upon the dynamic behavior of the reactor, an appropriate model for the reactor must be available. The model must be comprehensive enough to describe the process adequately and realistically and yet be simple enough mathematically VOL. 9 NO. 1 FEBRUARY 1970

I&EC FUNDAMENTALS

41

Numbered positions in the reactor 2 3 4 5 6 7 8 9

c IO

The steady-state conditions are obtained by solution of Equations 20 to 22 with the time derivatives set identically to zero. A further approximation that serves well here and simplifies both the steady-state representation and the transfer functions is the use of the term exp(aT’) in place of exp{AT’/(T* 2”’)) for the reaction rate temperature dependence in Equations 20 and 21. This simplification could be unsatisfactory for a study of reactor stability, as noted by Aris (1966), but it is sufficiently accurate for studies of reactor dynamical behavior when questions of stability and parametric sensitivity are not of concern (Crider and FOSS, 1968). With this approximation, the steady-state relations are (Douglas and Eagleton, 1962) C,’ T,’ = 1 (23)

+

+


Ei(otC,’)

=

E i ( a ) - e”kz

(24)

0 01 0 2 0 3 04 05 06 07 08 0 9 I O Dimensionless distance, X Figure 4. Steady-state profiles of concentration, reaction rate, and sensitivity functions ec and r T

to be useful. There are several methods by which such a model could be obtained. Complex systems are often characterized by an experimental determination of their frequency response. A set of phenomenological differential equations can be used as a model, as can the frequency response of the equations, obtained perhaps by numerical integration. hloments of the impulse response, determined experimentally or from these equations, can also serve as a system model. Used here is a set of transfer functions (Crider and FOSS, 1968) that applies to a packed-bed reactor in which an irreversible exothermic first-order reaction occurs in the fluid phase. The dynamics of this reactor are known t o be well represented by the following set of three equations that describe in dimensionless form a component balance and heat balance for the fluid and a heat balance for the packing.

These equations include the important effects of concentration-temperature coupling through the reaction rate term, r, and finite rates of heat transfer between the fluid and solid phases. The characteristic response time for this reactor is determined by the ratio of fluid to solid heat capacitances, p. I n the light of present experimental and computational evidence, the influence of radial gradients, longitudinal mixing, and nonconstant fluid properties appears secondary and therefore has not been included in this model. Upon local linearization about the steady-state conditions (Bilous and Amundson, 1956), these equations may be solved for the reactor transfer functions (Crider and FOSS,1968). 42

I&EC FUNDAMENTALS


where

The function 6 is associated with the dynamic thermal coupling between the fluid and packing, rs is the steady-state reaction rate, and r c and T T , are respectively, the reaction rate sensitivity to concentration and to temperature. The impulse responses of these four transfer functions were determined by Stangeland (1967) for a particular set of parameters (Figures 2 and 3). The values of the parameters used in this study are typical of those found in liquid-phase reactors (Sinai and FOSS, 1970): the inlet reaction rate k = 0.454, the heat transfer parameter H = 19.976, the ratio of heat capacitances p = 0.92, and the reaction coupling parameter CY = 3.87. The steady-state profiles of the functions Cs‘, r,, re,and T T under these conditions are shown in Figure 4. Explicit transfer functions for the four pairs of controllers may now be obtained by substitution of Equations 25 to 28 into the definitions, Equations 12 to 19:

FORC A ~ 1E Fcc(s) =

(&)

e -see - 8 c

kl(s,b;0) X


Time, t , i n f l u i d residence times Figure 5. Exact and approximate response of reactor temperature to a unit impulse in feed temperature location of numbered positions shown in Figure

FORCASE2

4

ing accurate and yet simple approximations for complex processes is iiot a new one. The usual approach enbails finding the simplest form of approximation that will adequately represent the slower response characteristics of the process; the parameters of the approximation are then adjusted t o obtain the best fit of the low frequency (or large time) response. This method gives good results for systems whose frequency response is highly damped a t the higher frequencies (Law et al., 1965), as is the case for diffusion-limited processes. For t'he reactor studied here, however, t,he behavior a t both large times (during the approach to steady state) and short times (near the wave front of the initial disturbance) is important. For this reason, the approximation for reactors with liquid reactants is perhaps the most difficult; with gaseous reactants, the short-time response is usually considerably less significant. The approximations used here consist of the sum of two parts; the parameters in one are adjusted to produce matching a t small values of t,ime (or a t high frequency) and the other parameters are then chosen using the method of moments (Paynter, 1987) to give good agreement a t large values of time (or a t low frequency) (Stangeland, 1967). This approximation has the form

FORC.4SE 4 ; ~ ( s )=

e-..[

5

i=O

BTT(S)

=

-

(&)

rs(C)e-s(c--b)

e --6c X r,(b)Z(s,b;O)

Each controller transfer function contains a fluid transport term, e-'' or e - s ( c - b ) ' which is here a because

czb2O.

Approximations to Transfer Functions

Because the transfer functions of the reactor and the controllers are so complex, there is considerable incentive to seek effective approximations of simple form. The problem of find-

ai/(s

+

+

The impulse responses of the approximate and exact reactor transfer function GTT(s,z;O)are compared in Figure 8. For positions 1 to 7, the approsimate response is so closely coincideiit with the exact response that the two may iiot be distinguished in this figure; a t positions 8 to 10, the approsimation is iiot so accurate, but nevertheless adequate. The approsimation used here for positions to has the of Equation 37; for positions 5 to 10, the approximation consists of the product of two terms of the form of Equation 37 (Stangeland, 1967). This approximation method also matches the frequency response accurately over the entire frequency range. A similar comparison of the controller's approximate transient responses with the exact response could not be made, however, because the latter is not known. VOL. 9 NO. 1 FEBRUARY 1970

l&EC FUNDAMENTALS

43

r---.

Uncontro I led Control led

-Controlled

-E

/'

\ \

0

L

W

n E


c"

Figure 7.

-5

-- I Figure 6.

I

I

I

I

I N I I

I

I

I

1.

Time, t , i n f l u i d residence times Behavior of controlled section of reactor

Feed disturbance. Positive temperature impulse Control objective. Control of temperature at position 8 with concentration injection at position 4 (Case 3)

Reactor under Control

The approximate models for the controllers and reactor transfer functions are used here to study the response of the controlled reactor, to ascertain the quality of control that can be achieved using this method of control, and to investigate the behavior of the uncontrolled variable and the dependence of the reactor's response upon the injection position. These questions have been examined (Stangeland, 1967) for all four types of control configuration and for disturbances in feed temperature and concentration. A sampling of these results is presented here, as well as a more detailed description of one of the four configurations, which demonstrates the use of the coupling between temperature and concentration waves to achieve control. The particular configuration described here employs concentration modulation to achieve invariant effluent temperature (Case 3). Rioreover, in this example, the point of measurement in the bed is made coincident with the point of injection-that is, b = c. This condition merely simplifies the presentation of the results; the performance of the controlled reactor under this condition differs in no fundamental respect from that of the more general case, b f c. I n the calculations reported here the controller transfer functions FCT and B c T were approximated by expressions of the form given in Equation 37. While the calculations reported here apply to the case of liquid reactants for which parameter p is of order unity, similar behavior under control may be expected for smaller values of p that obtain when the reactants are gases. This may be seen from the equations that represent the reactor (Equations 20 to 22) by defining a new time variable 7 = @(t - 2). Thus, parameter 0 is simply a scale factor for time exclusive of the fluid transit time. Controlled Reactor Response. When a n upset in feed 44

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Time, t, i n f l u i d residence times Behavior of controlled section of reactor

Feed disturbance. Positive concentration impulse. Control objective. Control of temperature at position 8 with Concentration injection at position 4 (Case 3)

temperature enters the reactor, disturbances in both temperature and concentration propagate uncontrolled through the bed as shown in Figure 3. Upon the injection of the concentration control wave a t position 4, however, the temperature disturbance is essentially annihilated upon reaching position 8, the point under control in this example. The history of the temperature disturbances in the reactor under control as well as that for the uncontrolled case is shown in the upper portion of Figure 6. The objective in this case is the maintenance of constant temperature a t position 8 rather than a t the end of the bed. This corresponds to minimizing disturbances in the zone of maximum reaction rate, where hot spots are likely to occur under dynamic conditions. The history of the concentration control wave that annihilates the temperature disturbance is shomm in the lower portion of Figure 6. The concentration control wave a t the injection point, position 4, consists of a sizable positive portion followed by a large negative wave. The positive portion of this control action increases the reaction rate and counteracts the initially negative uncontrolled temperature wave a t position 8 shown in the upper plate. Similarly the later large negative concentration control wave decreases the reaction rate and removes the larger positive temperature wave that passes through position 8 at t = 1.3. The net effect is essentially to annihilate the temperature wave a t the objective point. The small temperature disturbances remaining at position 8 and the small initial dip in concentration a t position 4 can be attributed to inaccuracies in the approximations used. The concentration response at position 8, however, is not invariant, since constancy of concentration is not the control objective. The negative waves in concentration shown at positions 4 and 6 were tailored with the sole objective of removing temperature upsets. Thus a residual concentration disturbance remains a t position 8. Even so, this concentration disturbance is less than it would be were the reactor uncontrolled. Similar results are shown in Figure 7 for an impulse upset of feed concentration. Again the concentration control wave consists of a positive portion followed by a larger negative part.


T h e major negative disturbance decreases the reaction rate and rate of heat generation as it moves through the bed, steadily eroding the positive temperature wave shown in the upper part of Figure 7 . As before, essentially perfect temperature control is achieved a t position 8. The residual negative concentration wave a t this point is smaller than would have been the case with no contvol. 1-1 significant feature of this control method is the improvement of response time for both the controlled and uncontrolled variables throughout the controlled section of the bed. As shown in Figures 6 and 7 , all disturbances in the controlled reactor disappear sooner than they would a t position 8 in the uncontrolled reactor. This is a direct consequence of using injected waves as the corrective mechanism of control. Corrective action occurs simultaneously on all portions of the disturbance waves as they move through the bed. Furthermore, a t any instant in time this corrective action may be a n increasing temperature in one section of the bed while decreasing in another. By contrast, modulation of either coolant temperature or flow rate, as suggested by Tinkler and Lamb (1965), vould upset the entire bed simultaneously and Lvould not utilize the unique features of the traveling corrections studied here. Influence of Position of Injection Point. T h e control waves, formed by the confluence of the secondary stream with the main stream, consist, in general, of a series of waves of alternating sign (Figures 6 and ’7). The detailed structure of these waves depends, however, upon the point a t which they are injected into the bed, since the character of the original upsets is different a t each point in the reactor. These differences are reflected in the history of coiicentration in the secondary stream needed to achieve control; this history exclusive of the impulse a t the fluid front for five different points of injection is shown in Figure 8. In general, less control effort is required if injection occurs near the inlet of the reactor. With early injection more of the bed is available in which interaction can occur betiveen the disturbance and injected wave before they reach the effluent. However, this longer residence time of the controlling wave also means that it has more time to generate additional waves in both temperature and concentration. This, together with the differing propagation velocities of these disturbances, requires the injection of a series of control waves to counteract these induced waves, as shown in Figure 8 for position 1. An additional problem with early injection is the large controller gains required. A s the injection point (and necessarily, the measurement point) is placed closer to the main feed, the length of the “analyzing section’’ decreases. This means that less time is available for the interaction of disturbances needed t o generate measurable temperatures at point b. Thus the smaller temperature signals a t point b require higher gains in the controllers to produce the necessary control stream modulation. On the other hand, injection near the end of the bed requires larger control efforts. When the controlling and controlled variables are of the same type, late injection must depend mainly upon direct cancellation of disturbances, because little of the bed (and little time) remains in which interaction can be used. If these two variables are not of the same t y p e as for example, in Case 3-control cannot be achieved by cancellation and must depend entirely upon interaction. As less of the bed becomes available, the control effort increases to infinity when zero interaction time is left. The ideal situation then would have t h e feed disturbance, the controlling variable, and the objective variable all the same type. I n this

1

8

,

1

1

1

1

1

1

1

1

(

1

1

,

1

Injection Point

4 0

SC -4 -8 -12

0.8

0 6C -0.8 -1.6 -2.4 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Time, t , i n f l u i d residence times Figure 8. Concentration history of injected stream required for control of effluent temperature Case 3.

Reactor subiected to impulse feed disturbances

case, early injection would effectively cancel out most of the feed disturbance as it enters the main section of the reactor. The task remaining for the control system would be to compensate for the still small disturbance generated in the other variable during the passage of the major disturbance through the “aiialyzer section.” Care must be exercised in locating the injection point when concentration is to be controlled at point p by modulation of concentration a t point c (Case 1). It has been shown (Stangeland, 1967) that there is a point in the bed, e*, at which changes in t h e steady-state concentration have no influence on the steady-state concentration a t point p . T h a t is, point p behaves as a concentration node point with respect to concentration changes a t point c*. Control with concentration injection a t point e* is therefore impossible. The location of this singular point in terms of the reactor parameters is (Stangeland, 1967) e* = p - l/ar,(c*) (38) When e* = 0, this relation gives the location of the concentration node point discussed by Crider and Foss (1968). Influence of Type of Disturbance. T h e dependence of control effort upon the type of feed upset is shown by the differences in scales between the sets of responses given in Figures 6 and 7 . The response to temperature impulses and the control effort required is on the order of four times t h a t required for concentration upsets. This is related to the magnitude of the coupling parameter, cy, Iyhich for small disturbances is the ratio of the effects of temperature and concentration disturbances upon the reaction rate. Thus since a = 3.87 here, a change of 4% in the feed concentration has roughly the same effect as a 1% change (in units of adiabatic temperature rise) in feed temperature. This observation emphasizes the importance of maintaining good control over feed temperature. VOL. 9 NO. 1 FEBRUARY 1970

I&EC FUNDAMENTALS

45

I n j e c t i o n Point

-

c 0

-20 -

-

m

L

0

Approximation -

al

-40-

U 2

.-

c

- Exact

O'I c

Approximation

n

-6Ob

\ 5

-


al 0

al

I

1

0.4 0.7 1.0

I l l l 1 l I

I

4

Dimensionless frequency,

7 10

a

Figure 9. Frequency response of controller FCT

0.I

I

0.4 0.7 1.0 Dimensionless frequency,

I 1 1 1 1 1 1

4

7 IO

a

Figure 10. Frequency response of controller BCT Control of temperature at position 8

Control of temperature at position 8

Choice of Controlled Variable. T h e task of controlling concentration a t the reactor effluent is remarkably similar to that of controlling temperature-that is, the control waves in the two cases are similar. Apparently this similarity results from the dominant influence of temperature waves on the reactor's dynamic behavior. Consequently, 11hether the objective be the control of concentration or temperature, the major responsibility of the control wave is the containment of the temperature excursions in the latter portion of the bed. Influence of Type of Control Action. Both advantages and disadvantages may be found for the use of either type of control action. The higher sensitivity of the reaction rate to temperature recommends the use of temperature control waves because excursions of temperature and concentration within the reactor are not extreme in this case. Further, with sustained feed disturbances (step changes), only small temperature changes need be made in the injected stream. The slow propagation velocity of the temperature wave detracts, hon ever, from its general utility as a controlling wave. Often secondary disturbances are induced by the passage of reactant through the slowly nioving temperature waves with the result that the temperature of the injected stream need be highly oscillatory. Moreover, when temperature is used to control effluent concentration, temperature control waves of very large magnitude are needed. Such deniands result because the faster moving concentration disturbance niust be counteracted with the "leading edge" (alternatively, the high frequency portion) of the temperature control wave, and the attenuation of high frequency temperature waves in high. The use of concentration as the controlling wave has the merit that it is the fastest traveling corrective action available but has the shortconiing that its influence on the reaction rate is not nearly so strong as that of temperature. Further, when effluent concentration is to be controlled, the concentration control waves need be impractically large 46

l&EC FUNDAMENTALS


when the injection point is close to the singular point of Equation 38. Characteristics of Controllers and Their Approximations. The validity of the results of this study depends heavily upon the accuracy of the approximations of the transfer functions of the reactor and the controllers. As shorvn in Figure 5, the approximations of the reactor transfer functions provide an accurate representation of the impulse response of the system equations. The same comparison could not be made for the controllers, however, since their impulse responses are not known. A comparison mas made therefore of the frequency response of the exact controllers and their approximations (shown in Figures 9 and 10 for injection a t positions 2 and 4). The match in gain and phase lag is good a t all values of the dimensionless frequency Q = w / H P . The major discrepancy occurs a t position 4 for controller FCT, nhere a t high frequency the phase lag of the approximation approaches -360' instead of 0'. This is not serious, because the gain is small in this region. The accuracy of the approximation procedure used here was further confirmed through comparison of transients in a reactor for which a = 0 (Stangeland, 1967). The impulse response of the controlled reactor approximated with functions of the form of Equation 37 was in excellent agreement with the analytic solutions found for this special case. The frequency response of controller F C T closely resembles that of the simple heat regenerator; this similarity is a further manifestation of the dominance of thermal effects in these reactors. The sharp change in magnitude and phase of controller Bcr a t i2 = 0.4 results from the similar behavior of the integral I(jw,p;O) a t point 8. Even with these rapid changes, the approximation to controller BCTis remarkably accurate. These control functions contrast with those found by Tinkler and Lamb (1965) for the same type of reactor. The frequency response of the controllers determined by those


investigators exhibited a n infinite number of sharp resonance peaks. The simpler control functions determined here result from the use of a traveling corrective action as distinguished from the uniformly applied flow rate modulation employed b y Tinkler and Lamb. The implementation of these control functions requires the availability of a holding device to represent the transport delay term in the F controller. I n gas phase reactors, however, the delay is small compared with the thermal lag of the bed and can be neglected. The other part of the control functions consists of sums of simple lag terms, which can be easily simulated with either analog or digital equipment. Since gas-phase reactors respond much more slowly than liquid-phase reactors, the time scale of events may be too large for accurate analog representation, in which case digital implementation seems appropriate.

composition measurements are necessary and the control functions are simple enough to be implemented easily. This assessment may be expected to hold for any type and combination of feed disturbances. I n fact, the test of the control system’s performance in the face of impulse upsets is the most severe test that can be made. Moreover, this assessment should hold for many other kinds of chemical reactors having appreciable coupling between temperature and reactant concentration. Comparable control performance may be expected for other reactions because the thermal effects, which are common to all reactors of this type, usually dominate the dynamic behavior. Nomenclature

Unless otherwise noted, all symbols are dimensionless.

A

2

E/RT,* constant in i t h short-time term of approximate transfer function, Equation 37 = reactor controller = reactant concentration, fraction of feed concentration = Laplace transform of concentration deviation = fluid heat capacity, cal/(g) (“C) = solid heat capacity, cal/(g) (“C) = activation energy of reaction, cal/g mole

Ei

= exponential integral, Ei(z) =

Conclusions

a,

This nork has examined a control method which perhaps is the most natural and direct method for the control of fixedbed reactors subject to feed upsets. The essential feature of the method is the exploitation of concentration-temperature interactions inherent in the reaction system to annihilate a disturbance by the time it reaches the reactor effluent. The traveling nature of the corrective action, achieved through the injection of a secondary reactant stream, is a n equally essential feature, without which the control objective could not be achieved easily. I n any specific application of this control method, the benefits of control must be balanced with the disadvantages and difficulties of secondary reactant injection-for example, it may be impossible to tolerate the concentration variations that accrue in the effluent upon control of temperature. Moreover, longer beds may be needed to achieve a desired conversion, and the features of the reactor’s structural design may all but preclude the injection of a secondary stream. l l a i i y industrial reactors nolv in use employ secondary stream injection a t one or more locations along the bed, but it is seldom that these streams are manipulated to achieve control in the sense discussed here. The characteristics of this control system have been examined here through coiisideration of a locally linearized representation of the reactor’s dynamic behavior. Such an approach is justifiable under conditions of reactor operation remote from regions of parametric sensitivity and from conditions that engender instability. Even when these conditions are satisfied, the reaction rate nonlinearity will preclude the perfect annihilation of a disturbance by the linear control system devised here. The magnitude of the imperfections in the control system performance caused by the nonlinearity of the process could be determined, of course, by numerical solution of the nonlinear differential equations for the reactor under control. This has not been attempted here, because the experiments and calculations of prior investigators have shown that for the small excursions normally tolerated in a controlled process, nonlinear dynamic effects are of secondary importance. Konetheless, this imperfection of the reactor model and those arising from inaccurate knon-ledge of process parameters suggest the use of a feedback control system coupled with the feedforward system proposed here. A feedback system is also needed to correct for disturbances other than those in the reactant feed and for the influence of catalyst poisons. I n spite of these shortcomings, calculations show that this control system has much to recommend it as a n important part of a complete system. I t s ability to control high frequency disturbances is perhaps its principal contribution. This task is accomplished with relative ease, because no

B C’

C CPf

= =

f-2

J

eUdy/y -m

= fraction of effluent mass that enters in feed = reactor controller F GTC,G T T , Gee, GCT = reactor transfer functions given in Equations 25 through 28 = heat transfer parameter, h L/e ( p c , ) / v H = fluid-solid heat transfer coefficient, cal/(sec)h (cm”)”C,

f

I

j k* k

=

e-@u J m ” y

=& = pre-exponential factor in reaction rate term, sec-1 = dimensionless reaction rate constant, k * ( L / v ) e - A ,

K

=

L 6 R r

=

Tc

= = = =

TT

=

S

Si

= =

T,*

=

T’

T*

= =

TP’

=

T

=

t

=

0

=

2

=

equal to steady-state reaction rate at reactor inlet constant in approximate transfer function given by Equation 37 reactor length, cm heat transfer function, H s / ( s H b ) gas constant, cal/g mole OK reaction rate, kC’w T’ steady-state concentration sensitivity of reaction rate, k exp(cuT,’) steady-state temperature sensitivity of reaction rate, k cy C8’,exp(aT,’) a complex variable i t h pole in approximate transfer function, Equation 37 a constant absolute reference temperature taken as steady-state feed temperature, OK T,* divided by adiabatic temperature rise fluid temperature in excess of feed temperature, expressed as fraction of adiabatic temperature rise solid temperature in excess of feed temperature, expressed as fraction of adiabatic temperature rise Laplace transform of fluid temperature deviation time, in unit of fluid residence time fluid velocity, cm/sec distance measured from reactor inlet, fraction of reactor length

+

GREEKLETTERS cy = coupling parameter = heat capacity parameter, ( p c s ) f ! / ( p c p ) P ( l - e) P = concentration modulation of injected stream SC = temperature modulation of injected stream ST VOL 9 NO. 1 FEBRUARY 1970

l&EC FUNDAMENTALS

47

AC

=

AT

=

E

=

PI

=

PP

= = =

0

Q

concentration deviation entering controlled section of the reactor temperature deviation entering controlled section of the reactor void fraction of packed bed fluid density, g/cnia solid density, g/cnia frequency, radians/fluid residenre time frequency, w / H P

SUBSCRIPTS

'1

=

evaluation a t positions of z in bed when used as subsciipts. Otherwise values of z a t these positions

S

=

denotes steady-state quantity

P

Crider, J. E., FOSS, A. S., A.I.Ch.E. J. 14,77 (1968). Douglas, J. M., Eagleton, L. C., IXD.ENG.CHEM.FUNDAMENTALS 1, 116 (1962). Hoiberg, J. A., Ph.D. thesis, University of California, Berkeley, 1969. Law, V. J., Rodehorst, C. W., Appleby, A. E., von Rosenberg, D. U., paper 15c, 56th National Meeting of A.I.Ch.E., May 1965, San Francisco, Calif. Pavnter, H. M., "flegelungs-technik: Moderne Theorien und ihre verwendbarkeit, Verlag R. Oldenbourg, Munchen, 1957; Report of Heidelberg Meeting, September 25-29, 1956. Poxell, B. E., ISA J . 10,45 (1963). Simpkins, C. R., Ph.D. thesis, University of Delaware, 1966. Sinai, J., Foss, A. S., A.I.Ch.E.J., tobepublished, 1970. Stangeland, B. E., Ph.D. thesis, University of California, Berkeley, 1967. Tinkler, J. D., Lamb, D. E., Chem. Eng. Progr. Symp. Ser. 61, No. 55, 155 (1965). RECEIVED for review February 27, 1969 ACCEPTEDAugust 7, 1969


literature Cited

Ark, It., Ind. Eng. Chem. 58, 32 (1966). Biloiis, O., AInundson, N. R., A.Z.Ch.E. J . 2 , 117 (1956). Boreskov, G. K., Slin'ko, h l . G., Pure A p p l . Chem. 10, 611 (1965). Crider, J. E., FOSS,A . S.,il.I.Ch.E. J . 12,514 (1966).

Work supported by fellowships and research grants from the National Science Foundation and by a fellowship from the Standard Oil Co. of California. Computational facilities were provided by the Computer Center, University of California at Berkeley.

Branch and Bound Synthesis of Integrated Process Designs K. F.

Lee,' A. H. Masso,Z and

D. F. Rudd

Chemical Engineering Department, University of Wisconsin, Madison, TVis. 65706

The branch and bound techniques of problem solving can b e used to guide the design engineer during the invention of integrated process designs. When confronted with a design problem for which no method of design i s known, the engineer gainfully can branch to simpler design problems which bound the original problem. Often, the optimal solution to the original unsolvable design problem can b e inferred from the solutions to bounding problems. These ideas are illustrated in the design of an energy exchange system.

Rm.m~.ux,T little theoretical guidance is currently available during the synthesis of integrated process designs (Hwa, 1965; Kesler and Parker, 1969; hlasso and Rudd, 1969; Rudd, 1968; Siirola et al., 1970). For the most part', the selection of process equipment and its integration into a process sheet are left to experience. The engineer invents processes with the full knowledge that more efficient processing schemes may have escaped him. I n this report we examine the branch and bound method of problem solving (Lawler and Wood, 1966) and demonstrate how these ideas can be used by the engineer as guidelines during the invention of process designs. Using branch and bound theory, methods have been devised for the systematic synthesis of energy exchange systems. The systems generated are known to be the best attainable acyclic designs. Branch and Bound Strategy

An engineering design problem consists of a statement of a task to be performed, such as the maiiufact'ure of certain Present, address, Amoco Chemical Corp., Whiting, Ind. 46394 Present address, Shell Development Co., Emeryville, Calif. 94608 1 2

48

l&EC FUNDAMENTALS


products from crude feedstock; pertinent technical information, such as the performance characteristics of process equipment, limitations on sources of energy and raw materials, etc. ; and pertinent economic information including a n economic objective to be reached, such as maximum venture worth. For most industrial design problems there are no formal procedures which lead directly to the process design that accomplishes the design objective. Should we be confronted with such an unsolvable or excessively difficult design problem (problem A) we look for simpler design problems for which methods of design esist (problems B). Suppose that one can invent a design problem B which is more easily solved. This simpler problem might arise from a clever manipulation of the original design task, the technical constraints on the original problem, or the economics. What information can be gained about the solution to the original unsolvable problem A by examination of the solution to alternate problems B? If problem B has been constructed to exhibit certain bounding properties, and if its solution sat'isfies the original problem in certain critical ways, considerable information

Control of a Fixed-Bed Chemical Reactor

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