Mecklenburgh, J. E., Hartland, S., I. Chem. E. Symp. Ser., No. 26, 130 (1 967). Miyauchi, T., U S . Atomic Energy Commission Rept. UCRL-3911 (1957); see also Miyauchi. T., Vermeulen. T..Ind. Eng. Chem., Fundam., 2 , 113 (1963). Pratt, H.R. C., Ind. Eng. Chem., Proc. Des. Dev., 14. 74 (1975). Wilburn, N. P., Ind. Eng. Chem., Fundam., 3, 189 (1964). Wilburn, N. P., Nicholson, W. L., A.I.Ch.€.-l. Chem. E. Symp. Ser., No. 1, 105 (1965).
t = top section of column x = X phase (feed) y = Y phase (extractant) Literature Cited Gayler, R., Pratt, H. R . C., Trans. hst. Chem. Eng.. 35, 267 (1957). McMullen. A. R., Miyauchi, T., Vermeulen, T., U S . Atomic Energy Commission Rept. UCRL-3911-Suppl. (1958).
Received f o r review August 27,1974 Accepted June 20,1975
Derivative Decoupling Control Roger E. Palmenberg and Thomas J. Ward' Chemical Engineering Department, Clarkson CoNege Df Technology, Potsdam, New York 13676
Derivative decoupling control, a nonlinear noninteracting control approach, is discussed in terms of recent work in order to highlight some limitations and to provide a comparison with the linear structural analysis control approach.
Some time ago Liu (1967) presented an approach for noninteracting process control based on the decoupling of the state derivatives, rather than the state elements themselves. Since this has received new attention (Hutchinson and McAvoy, 1973; Rich et al., 1974), a detailed examination of this control method in terms of the recent work is warranted.
Background Liu assumed that the uncontrolled process could be characterized by a nonlinear state-vector differential equation model of the form Y = F(Y,
x,t )
where Y = n X 1 column vector of state variables; X = n X 1 column vector of inputs; F = column vector of nonlinear functions f j ( Y, X,t ) ; t = time (the overdot represents differentiation with respect to t ) . The control object was the control of Y by the manipulation of X. With control, the manipulative input X became a function of the measurable state Y and the setpoint input vector R . As a result, the controlled process dynamics could be written as a function of the error E , where E = R-Y Y = G ( E ,t )
If G ( E , t ) were specified by design as a column. vector whose ith component was a function of only the corresponding error element ei and t , then the state derivatives would be decoupled. This also meant that the state elements themselves were decoupled. Such derivative decoupiing, which was analogous to the state decoupling used in early noninteraction approaches (Kavanagh, 1957), gave the noninteraction condition as a set of nonlinear algebraic equations
F(Y, X , t ) = G ( E , t )
(3)
where G ( E ,t ) was to be specified as above.
Possible Difficulties There are three general difficulties that can be encoun-
tered in obtaining the xi controller equations from this system of equations. The first difficulty is the specification of G ( E , t ) . There are an infinite number of choices for each of the gi elements and there is no straightforward procedure for finding an effective, realizable choice. Liu arbitrarily assumed that a suitable form for all of the state derivatives was a type of proportional response 3.1 = a 1. e1. (4) where the a; are unspecified proportional coefficients. Since large values of the a; would provide faster response dynamics, Liu then proposed an algorithm that, starting from initial arbitrarily large values, would iteratively reduce the magnitudes until the input constraint limits were satisfied. This algorithm can be utilized as an off-line calculation that requires the complete solution of the nonlinear process equations or as an on-line approach involving integration through each time step. This calculation may not give any acceptable values of the coefficients in some cases without modification of the algorithm. The second difficulty is that it may not be possible to solve the equations for each of the x i as explicit functions of the state variables and setpoints. If eq 1 is linear in the xi, it is possible, as noted by Liu, to obtain an analytical solution. If an on-line numerical approach is required, there is no assurance of convergence to the correct xi values in a reasonable time. In order to overcome this general difficulty, Liu proposed a simplified procedure in which each equation of the set is solved separately for a single x i input as a function of the state variables, setpoints, and the other inputs. While this is an effective procedure for elementary systems, there is no general guide as to which input should be developed from a particular equation. Even more important, the procedure may fail in some cases. The third difficulty occurs in the common process case in which only a small subset of the inputs are manipulative and only a small subset of the state elements are actually controlled. It would often be impossible t o obtain the x i controller equations for this case without modification of the method or a careful reformulation of the process equations. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
41
a.
a. b.
C.
Figure 2. Algebraic reduction of elementary system.
b. Figure 1. Signal flow diagrams of elementary system.
This derivative decoupling control approach would appear to be an interesting possibility in cases where the above difficulties are not significant or can be effectively resolved. Heat Exchanger Application Hutchinson and McAvoy (1973) applied this derivative decoupling method to the servo control of an experimental heat exchanger system in which the objective was the control of the two state variables by manipulation of two inputs. The proportional response form of eq 4 was assumed to be adequate. Constant ai values were selected for each experiment by trial-and-error so as to avoid input constraint saturation. Analytical forms for the controller equations were readily obtained by the simplified procedure of Liu. The experimental results indicate that the interaction between the two state variables was largely eliminated and that the responses were reasonable. As expected, larger values of an ai constant reduced the response time of the corresponding state variable without introducing oscillation. However, the process responses all exhibited steadystate offset. As pointed out elsewhere (Cheng and Ward, 1975), this offset is probably due to the proportional form specified for G(E, t ) , The offset could be eliminated by selecting other forms for the derivative response G (E, t ) . Following conventional practice, one might logically select a proportional-integral response form yi = ajei
+ bi J” ei dt
(5)
An analysis of the heat exchanger equations indicates that the P I response form would eliminate offset in this case. However, the controller parameters may not correspond to the usual feedback controller constants and their specification could be a problem. If this P I response form is differentiated, the result is yj = aiej
If i.; = 0, this can be written as jji + IYI
+ bjei
(6)
- b.e.
(7)
I
1
or
yj
+ aiyj + bjyj = biri
(8)
This indicates that the P I response form relaxes the response requirements so that the controlled subsystems can 42
Ind. Eng. Chem., Process Des. Dev., Vol. I S , No. 1, 1976
exhibit a second-order response. However, this does not necessarily simplify the design task. The format of eq 8 requires a search for suitable and realizable responses for each of the subsystems. Direct Forcing Generalization Rich et al. (1974) have presented a “direct forcing” generalization of the derivative decoupling concept for higherorder system models and have introduced a control effort algorithm to treat constraints. This generalization will be outlined here. If the uncontrolled process equations are structured so that all derivatives y,, y , , . . . of a state variable y , can be isolated on the left-hand side of a single equation, then the right side is the “direct forcing function” for the derivatives of y l .
h,(y,,y , . . . ) = j , ( ~ Y, . . . ;XU,X u . . . ;X M , X M . . . ;t ) (9) where i = 1 to n; j , is not a function of the derivatives j , , y , . . .,Xu = disturbance input vector; and X M= manipulative input vector. This equation set can be treated by Liu’s method to give the controlled process dynamics in the form
hI(yl, Y, . . .I = g L ( e I )
(10)
The noninteraction condition now is
This would be directly applicable to processes formulated in terms of higher-order derivatives. In some cases, however, it might require that input variables and their derivatives, as well as uncontrolled state elements, be measured or estimated in some way. The input constraint treatment appears to be a direct search algorithm that attempts to allocate the available control effort between the various priority-classified output objectives. While this would be a very desirable goal, there is not yet enough evidence to assess the general effectiveness of the constraint treatment. Implicit Coupling As noted above, the direct forcing generalization would be directly applicable to process models containing higherorder derivatives. However, Rich et al. have proposed this for the first-order state vector model case in which the only signal path between a manipulative input and a controlled variable is through another state variable. This situation, which the cited authors call implicit coupling, is a fairly common occurrence in chemical process models. One example is the control of a composition by manipulation of a
Figure 4. Binary flash separator.
b.
C. I
Figure 3. Direct forcing reduction of elementary system.
thermal input, where the only signal path is through a temperature variable. The lack of a direct signal path can complicate the control analysis (Kleinpeter, 1968). The procedure proposed for this problem by Rich et al. involves the differentiation of the first-order subsystem equation of the controlled variable yi to obtain the second derivative y i . This often has an explicit dependence on the derivatives of other state variables and, as a result, will be explicitly linked to an appropriate manipulative input. In order to ensure stability, a linear combination of y i and y i is utilized in eq 10 to give jii
+ biyi = g i ( e i )
(12)
where bi is an arbitrary response parameter to be selected as part of the control design. If g i ( e i ) is selected as a proportional response form, as was done by Rich et al., then jj.
1
+ blY1 .' . - a 1. eI .
(13)
I t is interesting to note that this procedure appears to give the same dynamic behavior as the PI response form in eq 7, even though the controller algorithms would be different. Another solution for this implicit coupling problem can be found in the structural analysis control method of Greenfield and Ward (1967). This is a linear control approach that includes uncoupling of the linearized derivatives of only the controlled variables, while retaining the model information of the entire state. The structural reduction step of this approach can be used to treat the implicit coupling situation. This has been demonstrated for a process problem by Palmenberg and Ward (1971). Three examples will be examined in order to provide further insight into this implicit coupling complication. Example I. Consider the hypothetical linear process (Rich et al., 1974) defined by the equation pair Y 1 = Y2
3'2
= y2
+x
(14) (15)
where the specified single-variable objective is the control of y1 by manipulation of x. This process can be represented in either the time or Laplace domain by the signal flow diagrams of Figure 1. Note that the only information flow be-
tween x and y1 is through y2. In other words, y 2 is on the primary manipulative-controlled signal path. In order to emphasize the explicit second-order dependence of y1 on x: that is evident in Figure 1, the process structure can be drawn as shown in Figure 2a. An algebraic simplification of this structure leads to the equivalent structures of Figures 2b and 2c. The variable y1 is an integral part of the primary process signal path. It does not have to be measured or computed. The structural reduction step of structural analysis control design includes a matrix genralization of the illustrated graphical evolution. Incidentally, if the y1 variable were measurable, it could be effectively used in a classical cascade control configuration. Rich et al. would treat this case by differentiating eq 14 to obtain yl. This is equivalent to introducing y1 as a new state variable defined by the state equation y 1 = yz
+x
(16)
This augmented process model is shown in Figure 3a. Since is now defined by two parallel paths, the original signal path can be replaced by the alternate path to give the structure of Figure 3b. However, y z is identically equal to y 1 so that the process can be represented by the secondorder model of Figure 3c. This seems like an indirect routine to develop the second-order behavior that is already evident in the original process model of Figure 1. Example 2. The flash separator process shown in Figure 4 has been used by Rich et al. to illustrate how derivative decoupling can be applied to a realistic case involving implicit coupling. The process model (Kleinpeter and Weaver, 1971) consists of six first-order nonlinear differential equations, as well as auxiliary relations between the variables. The state vector model can be written as
y1
Y = F [ F , Zi,Pf. . . ; Ai, A,,
Q; P, TI,T,;xi, ~ 2 HI] , (17)
where state vector Y = [P, TI, T,,XI, y2, Hi] T; known input disturbances = F, Zi,Pf;manipulative inputs = AI, A,, Q; controlled variables = xi, y2, H I ;and F [ . . .] = column vector of nonlinear functions. The specified control problem involves the control of X I ,y2, and Hi by manipulation of Ai, A,, and Q. Derivative Decoupling. In applying derivative decoupling to this model, the liquid level H1 poses no problem since it explicitly depends on the manipulative inputs. However, the state equations for the two controlled composition variables XI and y~ display no explicit dependence on any of the manipulative inputs. Yet there is dependence of the derivatives of the other state variables on the manipulative inputs. Since i 1 and y p involve these other state derivatives, then f l and y 2 can be expressed as functions of the manipulative inputs. The procedure corresponding to eq 12 can be applied to each of these composition variables. This is discussed in more detail by Rich et al. (1974), who assume that the controller terms involving derivatives of the disturbance inputs can be neglected. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
43
-$
Cf
“L
Figure 5. Linearized flash separator model.
Figure 6. Reduced separator model. S t r u c t u r a l Analysis. Kleinpeter and Weaver (1971) attempted to apply the structural analysis control method to this same flash separator problem. However, the procedure failed because the uncontrolled variables were arbitrarily classified without regard for the implicit coupling that existed. The correct procedure will be outlined here. The linearized state-vector process model structure is illustrated in Figure 5, where all branches represent constants except for the six integrator branches connecting derivatives to their respective state variable. From this diagram it is evident that and Q affect 9 2 and f l only through the other state variables Tb7,PI, and P. If these other state variables are classified as “not-known state variables” ( YN), the structural reduction step gives the process model structure shown in Figure 6. The branches are now functions of the Laplace variable s. This structure can be used to design the feedforward, uncoupling, and primary feedback controllers of structural analysis control. The controller functions become functions of the Laplace variable s. These often can be used directly, particularly in digital systems. If the functions are unrealizable or too complex in form, conventional frequency domain techniques can used to obtain simpler, realizable approximations (Greenfield and Ward, 1967) without entirely neglecting derivatives. Note that this approach utilizes the available structural information to avoid the necessity of directly measuring or estimating the uncontrolled state variables and their derivatives. If measurable, these variables could be used in a cascaded feedback scheme. Example 3. The double-effect evaporator process shown in Figure 7 was first analyzed by Andre and Ritter (1968) 44
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
’h‘
b-7Q
FIRST EFFECT
SECOND EFFECT
Figure 7. Double-effect evaporator process. and later by Newell and Fisher (1972). It has been used in a number of control studies (Ritter and Andre, 1970; Palmenberg and Ward, 1971; Newell and Fisher, 1971, Wilson et al., 1974). A suitable nonlinear process model is given in Table I. The objective is the control of Cp, W1, and Wp by manipulation of Si, B1,and B p . This involves the implicit coupling complication because CZ, W,, and W p do not exhibit an explicit dependence on Si. Newell and Fisher (1972) have noted that this process cannot be decoupled by state variable feedback. Derivative Decoupling. The simplified derivative decoupling procedure of Liu can be directly applied to the W l and W 2 equations to obtain the B1 and B2 controller equations given in Table 11. However, when the Cp equation is used to obtain the third controller function, it is found that the direct-forcing generalization must be used in the form of eq 1 2 to give
Cp
+ blCp = g l ( C p s - C p )
(18)
The resulting Si controller, as a function of the measured variables, is an extremely complex analytical form. However, it can be readily evaluated in an on-line digital system. The functional forms involved in the algorithm are given in Table I1 and can be evaluated from the model equations in Table I. While C p does exhibit an explicit dependence on h l and, in turn, Si, it introduces an additional difficulty because it will also be a function of &. Rich et al. note that measurement problems may be serious if the manipulative input derivatives are included in the controllers. Accordingly, the derivative decoupling approach now must neglect derivatives of both the disturbance and manipulative inputs. S t r u c t u r a l Analysis. The structural analysis control design for this example starts with the linearized state vector process model given in Table 111. This is illustrated by the signal flow graph of Figure 8. The implicit coupling is clearly evident since there is no direct signal path from Si to C 2 . If hl and C1 are classified as “not-known state variables”, the process structure can be reduced as in the previous example to that of Figure 9. The structural analysis control configuration for this process model is shown in Figure 10 and the controllers are given in Table IV. The optional given in the table are feedrate disturbance controllers FMK omitted from Figure 10 for clarity. It is evident that some of the controllers of Table IV involve derivatives of input variables. However, the operational forms of Table IV permit various levels of functional approximation without completely neglecting the derivatives. Palmenberg (1973) showed that even the zero-frequency approximations,
Table I. Nonlinear Evaoorator Process Model
). Q ; = UiAi(T,'- T,') hi = Tj(l.0- 0.454Ci) + 6.0Ci= -32.0 + 1 . 0 ~ .(i ~= 1, 2 j Hoi = 1066 + 0.4Ti ( i = 1,2) Si(Hsi + 32) + U,A,T,
ki
1
32.1 ( i = 1, 2, F)
h
Figure 8. Linearized evaporator model.
AHmp = 1098 - 0.6T, = Hol - h o l U,A,(T, - T,) + B , ( h , - h , ) + X B , ( C , - C,)-HL, 0, = Ho, - h , + X C , ah, X = -= constant aC2
Table 11. Direct-Forcing Controllers for Evaporator Decoupling condition
W, Controller W, = a,(W,, - W , )
W, Controller
Decoupling condition
W, = a,(W,s
Figure 9. Reduced evaporator model.
- W,)
1 B , = B, - __ (20.912T1- 3120 1009 + BIT, - 0.454B1T,C, 149B, - 55.7B,C1 - 55.51 - a,(W,s - W,) C, Controller Decoupling condition C, + b,C, = a,(C,, 7C,) . Controller functions Si = f , ( ~ , ,c,., h i , c,; h , , w,, c2, 0 2 , c2, 0 2 , w2, F , hf) a Where the derivatives and functions can be evaluated from the model equations in Table I.
Controller function
a,,
Table 111. Linearized State Vector Model of Evaporator
cl
0.087%,- 0.043?, - 0.020F + 0.045& - 0.0142, - 0.04?+ 0.1696 + 0.025hf Ch, = 0.125h: + 0.039c1 - 0.036c2 - 0.02&, = -0.087$, - 0.002?1 + 0.0652- 0.045B^,A W, =-0.125$, - 0.003< - 0 . 0 0 0 1 + ~ 0.061B1- 0.036h2
g,
=
= -0.571$,
t,
which neglect all derivatives, provided effective regulation of the nonlinear process model. This example illustrates how individual controller elements can be deleted. The f31MC controller is omitted in the design of Table IV because it is a relatively insignificant element. Note that this design again utilizes the process structure to avoid calculating h l and C1. If C1 is measurable, it could be used effectively as an internal variable (Greenfield and Ward, 1967). Since h l FS: f(TI), it would also be possible to
B2
Figure 10. Structural analysis control of evaporator. utilize 2'1 measurements in a cascade arrangement. However, the above design requires neither measurement nor calculation of these variables.
Discussion Both the derivative decoupling control approach and the direct forcing generalization to higher derivatives offer the advantage of being based on a nonlinear process model. This may be advantageous if an accurate nonlinear process model is available. However, in addition to the general difficulties noted above, there are several features that may Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. I,1976
45
Table IV. Structural Analysis Controllers for Evaporator Structural Decoupling Controllers Feedforward decoupling controller Controller function 1.149 ( s + 0.66OMs + 0.045) (s + 0.0701) -0.325 3 (s + 0.660) -0.5832 ( s + 0.660) 1.6731 The one feedback decouolina controller to compensate for the C, + W , interaction is neglected for simplification since the gain is negligible.
f3zMM
Feedback decoupling controller
r
-
Feedforward Control of Known Feed Rate Disturbance Feedforward controller fllMK
f 2
IMK
Controller function 0.2718(s + 0.1433) s + 0.0701 1.444(s + 0.7132) s + 0.6597 0.13875 s + 0.6597
limit the applicability of the method in treating the implicit coupling complication. 1. The derivatives of both manipulative and disturbance inputs may have to be measured, estimated numerically, or neglected. If these derivatives are neglected, as suggested by Rich et al., it may be difficult to assess the degree of approximation involved or the significance of the terms involved. If these derivatives are computed, then measurements of the input variables are required and the computation tends to amplify signal noise. 2. The method requires the measurement of all state variables. While numerical integration of the state vector model would provide an estimate of the unmeasured state elements, this might still require measurements of “unknown” inputs. 3. The nonlinear process model provides no clear guide as to how the control problem should be structured, particularly when implicit coupling and other complications exist. In even moderately complex situations, the development of effective controllers may become a formidable task. 4. The significance of the terms in the nonlinear controllers is not always evident. As a result, there is no straightforward way to simplify the controllers to more tractable forms or to relate to conventional multiloop control. 5 . The method may exhibit considerable sensitivity to model and measurement error since all of the controllers are based on the nonlinear model. There is no independent feedback to reduce the sensitivity. In contrast, the structure analysis control method is based on a linearized process model. While linearization about some steady state would seem to be a limitation in servo control (setpoint changes), it has not been a serious problem in several laboratory and industrial studies. This is because the essential control is multiloop feedback. All of the additional controllers of this method merely serve to reduce the demands on the loop controllers. Other features of structural analysis control are the following. 1. The method was designed expressly for chemical process systems in which (a) a linearized model is often avail46
Ind. Eng. Chern., Process Des. Dev., Vol. 15.No. 1, 1976
able and identifiable, (b) only a small subset of variables are manipulative and controlled, and (c) numerous additional measurements are often available. 2. The uncoupling, feedforward, and internal-variable controllers all serve distinct and independent supporting roles. The significance of each controller can be assessed separately. This offers flexibility in deciding which variables should be measured in addition to the controlled variables. 3. The matrix generalization of this approach permits a direct extension to large-scale systems. 4. The companion signal flow graphics provide a powerful visual aid that is useful in every step from control problem specification to process operation. 5. Systematic modification of complex or unrealizable control functions can be accomplished by classical frequency response or numerical fitting procedures. Rather than neglecting derivatives entirely in the controller functions, the method will give realizable approximations to the functions. 6. While model and measurement error are neglected in the analysis, these do not seem to be severe limitations since the primary loop feedback tends to account for nonidealities. 7. Time delays and constraints are also neglected. Since all of the auxiliary control is feedforward with respect to internal disturbance propagation, the neglect of time delays should not be a significant limitation. If the delays are capable of being isolated, as in the case of measurements, then the analysis can be extended to consider the effects of delays. In a well-designed process, constraint saturation or some override logic should provide an adequate treatment of constraints. 8. The primary loop feedback is independently specified and tuned by classical methods. This permits the approach to be viewed as an extension of conventional multiloop feedback control in which the uncoupling, internal-variable, and feedforward disturbance control are all flexible options to be selected as needed.
Conclusions The derivative decoupling of Liu and Rich et al. offers the significant advantage of directly using the nonlinear process model for controller design. In comparison, the structural analysis approach suffers from being based on a linearized model. However, if such a model is adequate, the structural analysis approach utilizes all available measurements and model information in a scheme that maximizes the feedforward compensation of disturbance inputs, as well as providing independent feedback loop control to account for nonidealities. Acknowledgment The comments of T. J. McAvoy, University of Massachusetts, and V. Van Brunt, University of South Carolina, were both helpful and stimulating. Nomenclature The generalized nomenclature is defined in the text. The nomenclature for the two process examples is defined separately below. Flash Separator Example A = valve flow area (or flow rate through valve) F = feed rate H = enthalpy P = pressure Q = heat addition T = temperature
x = liquid product mole fraction y = vapor mole fraction z = feed mole fraction
Subscripts v = vapor 1 = liquid f = feed 1 = component1 2 = component2 = perturbation of indicated variable (from steady state) A
Evaporator Example A = heat transfer area B = bottomsflowrate C = mole fraction F = feed rate f = controller function h = liquid enthalpy H = vapor enthalpy 0 = overheadvapor Q = heat transfer rate si = steam rate T = temperature U = overall heat transfer coefficient W = accumulation (level) in an effect Subscripts 1 = first effect 2 = second effect f = feed stream 01 = overhead vapor from first effect
Si = first effect steam feed A = perturbation of indicated variable (from steady state) Superscripts MC = primary feedback controller MM = feed forward uncoupling controller MK = feed forward disturbance controller Literature Cited Andre, H., Ritter, R . A,, Can. J. Chem. Eng., 46, 259 (1968). Cheng, Y. C., Ward, T. J., lnd. Eng. Chem., Process Des. Dev., 14, 193 (1975). Greenfield, G. G., Ward, T. J., h d . Eng. Chem., Fundam., 6, 564, 571 (1967). Hutchinson. J. F., McAvoy, T. J., lnd. Eng. Chem., Process Des. Dev., 12, 226 (1973). Kavanagh, R. J., Trans. AI€€, 76 [Part 111, 95 (1957). Kleinpeter, J. A,, P h D Dissertation, Tulane University, New Orleans, La., 1968. Kieinpeter, J. A,. Weaver, R. E. C., AlChEJ., 17 (3), 513 (1971). Liu, S. L., lnd. Eng. Chem., Process Des. Dev., 6, 460 (1967). Neweii, R. B., Fisher, D. G., "Optimal, Multivariable Computer Control of a Pilot Plant Evaporator", IFACiiFlP 3rd International Conference on Digital Computer Applied to Process Control, Helsinki, Finland, 1971. Newell, R. B., Fisher, D. G., "Implementation of Optimal, Multivariable Setpoint Changes on a Pilot Plant Evaporator", 2nd iFAC Symposium on Multivariable Control Systems, Duesseldorf, Germany, 197 1. Palmenberg, R. E., M.S. Thesis, Clarkson College, Potsdam, N.Y., 1973. Palmenberg. R. E., Ward, T. J., "Regulator Control of a Double-Effect Evaporator", 21st Canadian Chem. Eng. Conf.. Montreal, Canada, 1971. Rich, S. E., Law, V. J., Weaver, R. E. C., "Model Characteristics and the Control of Nonlinear Multivariable Processes", Proceedings, 1974 Joint Automatic Control Conference, 319 AIChE. N.Y., 1974. Ritter, R. A,, Andre, H., Can. J. Chem. Eng., 48, 696 (1970). Wilson, R. G., Fisher, D. G., Seborg, D. E., AlChEJ., 20 (6),1131 (1974).
Receiued for reuieu September 9, 1974 Accepted September 19, 1975
The Fast Fluidized Bed Joseph Yerushalmi,' Davld H. Turner, and Arthur
M. Squires
Deparrment of Chemical Engineering, City College, City University of New York, New York, New York 1003 1
Fast fluidization is a technique for bringing gas at high velocity into intimate contact with a fine solid in an entrained dense suspension characterized by extreme turbulence and extensive refluxing of dense packets and strands of particles. The technique is primarily oriented toward gas-solid reactor applications where it offers several important advantages over the conventional, low-velocity bubbling fluidized bed. This paper delineates the regime of fast fluidization; it records observation of the fast fluidized bed in equipment of transparent walls; and it presents data on the fluidization characteristics of a cracking catalyst in a 3-in. round fast bed. Solid concentrations approaching 25% of the bed volume were typically achieved at gas velocities around 10-15 ft/sec. Corresponding slip velocities were an order of magnitude greater than the free-fall velocity of the largest particle in the test solid.
Introduction Fast fluidization is a technique for bringing a high velocity gas into intimate contact with a fine solid in what is essentially an entrained, dense suspension. The technique is primarily oriented toward gas-solid reactor applications, catalytic and noncatalytic. The solid in the fast fluidized bed may typically occupy up to 25% of the bed volume and is in a state of extreme turbulence marked by extensive refluxing of dense strands and packets of particles. Notwithstanding some commercial experience, the phenomenon of fast fluidization remains virtually unexplored. Fast fluidization offers several important advantages, to be described presently, over contacting of gas and solid in a
conventional, low-velocity bubbling fluidized bed; we have accordingly undertaken its investigation. In this paper we describe the phenomenon of fast fluidization, noting its historical roots and its commercial use to date. We record observations of the fast fluidized bed in a transparent two-dimensional rig, including observations derived from movies filmed at high speed, and we present data on the fluidization characteristics of a cracking catalyst in a round, 3-in. i.d. fast fluidized bed. The closing discussion draws distinctions between the regime of fast fluidization and other gas-solid transport regimes and sets forth the qualities and advantages of the fast fluidized bed for reactor applications. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
47
