Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
sorption, this VCM level during drying may become much lower as the degree of steam superheat during regeneration increases. Conclusions Based on this work, the following processing scheme is recommended. The carbon should have some residual water on it a t the beginning of adsorption. After the adiabatic adsorption, the steam regeneration step should be done with superheated steam. The drying step may be done with either a vacuum or a drying gas. The amount of residual water on the bed at the beginning of adsorption should be optimized to maximize VCM loadings while maintaining low temperatures in the bed. This amount of water can be attained by the superheat of the regeneration steam and the drying conditions. The outlet gas from the adsorption step may need to be diverted to a small, dry bed before release to the atmosphere. The outlet vapor from the drying step (when a drying gas is used) may need to be diverted to the operating adsorption bed because of high VCM concentrations. However, these extra steps may be avoidable depending on the specific conditions used. I t is not desirable to set up exact optimum conditions because the capabilities of each production facility will differ. For instance, the degree of vacuum or the superheat of the available steam may vary considerably. Also, the bed heat loss and, therefore, proximity to ideal adiabatic behavior will be an important factor. The data presented should predict the general range of optimum operating conditions for a commercial unit. Acknowledgment The following people are acknowledged for aiding the author in this study: Kang Yang, D. V. Porchey, L. M. Henton, J. D. Reedy, R. C. Lindberg, and J. A. Wingrave. Continental Oil Company is thanked for allowing publication of this work.
217
Nomenclature b = constant, g of VCM/(g of carbon (mmHg)'/") CG = water content in the vapor, g of water/ft3 CG" = water content in the inlet vapor, g of water/ft3 C G = water ~ ~ content ~ in the outlet vapor, g of water/ft3 CL = amount of water adsorbed, g of water/g of carbon CL' = CL at zero time, g of water/g of carbon C,, = gas heat capacity, kcal/g K C,c = carbon heat capacity, kcal/g K C p V c M = VCM heat capacity, kcal/g K D = mass diluant/mass VCM, g / g F = vapor flow rate, ft3/min KO = overall mass transfer coefficient, min-' Kw = rate of water desorption, g of water/g of carbon min L = VCM loading on carbon, g of VCM/g of carbon MVCM = molecular weight of VCM, g/g-mol n = constant P = partial pressure of VCM, mmHg q = isosteric heat of adsorption, kcal/g-mol of VCM 41 = integral heat of adsorption, kcal/g-mol of VCM R = gas constant, kcal/g-mol K t = time, min T = absolute temperature, K Tf = feed gas temperature, K To = initial bed temperature, K W , = amount of carbon in bed, g of carbon L i t e r a t u r e Cited Adamson, A. W., "Physical Chemistry of Surfaces", 3rd ed, p 594, WileyInterscience, New York, N.Y., 1976. Fed. Regist,, 39,35890 (1974). Fed. Regist., 41, 46560 (1976). Iammartino, N. R., Chem. Eng., 82, 25 (Nov 24, 1975). Mantell, C. L., "Adsorption", pp 601-603, McGraw-Hill, New York and London, 1951. Matheson, Co., "Matheson Gas Data Bock", p 413, East Rutherford, N.J., 1961. Patel, P. J., Thompson, C. G., Hourihan, E. J., Stutts, C. S., U S . Patent 3984218 (Oct 5, 1976). Raduly, L., U.S. Patent 3 796 023 (March 12, 1974). Young, D. M., Crowell, A. D., "physical Adswption of Gases", p 1IO, ButterswMths, Washington, D.C., 1962.
Received f o r reuieul April 14, 1977 Accepted October 23, 1978
Optimum Behavior of a Third-Order Process under Feedback Control Thomas W. Weber* Department o f Chemical Engineering, State University o f New York a t Buffalo, Buffalo, New York 14214
Mohan Bhalodia Exxon Co. U.S.A., Linden, New Jersey
The behavior of a third-order overdamped process under proportional-integral control was studied using the integral of the square of the error (ISE) as the performance criterion. The process is characterized by its maximum gain and ultimate period, as found by the Continuous Cycling Method (CCM) of controller tuning by Ziegler and Nichols (1942) rather than by its time constants as used by Jackson (1958). The CCM performance was compared with the optimum for four different disturbance locations. For some processes, the CCM leads to unstable behavior, but for many processes, the CCM gives quite satisfactory results when disturbances occur near the end of the process elements. The CCM controller gain recommendation is conservative and an excellent compromise, but the reset time recommendation is about half as large as that dictated by the ISE. The optimum ISE varies with about the inverse square of the critical frequency, but is nearly independent of the maximum gain.
Introduction A number of different models have been used to characterize the behavior of processes for the purpose of estimating suitable controller settings. The simplest model 0019-7882/79/1118-0217$01.00/0
consists of a first-order element plus a pure time delay, and has been used in a number of studies (Cohen and Coon, 1953; Murrill and Smith, 1966; Lopez et al., 1967; Miller et al., 1967; Smith and Murrill, 1966). These studies 0 1979 American Chemical Society
218
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
In Jackson’s poper
[
T, : r i r, T2 :nr :r2 T, C m T . 7 3
Figure 1. Block diagram of control system
basically differed in the criterion of optimization used in determining the best controller settings. A more sophisticated model based on a second-order element plus a pure time delay has been used by other investigators (Latour et al., 1967; McAvoy and Johnson, 1967). Weigand and Kegerreis (1972) compared controller-tuning methods for this second-order dead-time model on the basis of peak height, decay ratio, integral of the absolute error, and the integral of the time times the absolute error. They found that tuning methods based on first-order dead-time models are too conservative when applied to second-order dead-time models. This paper is concerned with a process that can be described by an overdamped third-order model. This model is reasonable since some processes consist of a series of essentially first-order noninteracting elements. Furthermore, it has several computational advantages. In particular, if the criterion for optimization is the integral of the square of the error, abbreviated ISE, this can be expressed in terms of the ratio of two polynomials and is easily calculated. Also, the effect of location of a disturbance in a control loop is easily evaluated. This model was studied by Jackson (1958) using the ISE criterion. As noted above, other investigators have used criteria such as the integral of the absolute error (IAE) and the integral time weighted absolute error (ITAE), and each of these of course results in different optimum controller settings. In spite of these several criteria, from a practical point of view, the determination of satisfactory settings is normally not a problem because an optimization criterion function is usually quite flat in the region of the minimum. The importance of optimum controller studies lies in showing how the minimum value of the function varies with the parameters of the plant-in this case, the relative magnitudes of the three time constants. This knowledge provides a basis for comparing various plant designs. A block diagram of the system is shown in Figure 1. A two-mode controller with proportional and integral actions is assumed since the addition of derivative action could lead to perfect controllability. Jackson confined his study to a disturbance just before the last element a t location 3. He employed a computer and search technique to determine the optimum values of controller gain and reset time to minimize the ISE. This minimum value of the ISE he termed the “controllability, I,” and he suggested that it would be useful in design for evaluating modifications for the improvement of control system performance. He presented contour plots of the optimum controller gain, reset time, and controllability with the dimensionless time constant parameters, m and n, as coordinates. In past studies in which the process models have been formulated in terms of time constants, the optimum controller settings have been related directly to these constants. For a third-order process there is a unique relationship between them and the maximum gain, K,, and the critical frequency, w,. These two parameters are and simply related to the maximum controller gain, K,,, the ultimate period, P,, which are easily determined by a few simple closed-loop tests in the field. The simplicity of these measurements was responsible for the development of the Continuous Cycling Method, denoted CCM,
of controller tuning by Zjegler and Nichols (1942). The maximum open-loop gain and critical frequency can also be obtained from a Bode diagram for the process. The shape of the phase and amplitude curves in the region of crossover a t a phase lag of 180’ can be helpful in guiding the selection of controller modes and parameters (Harriott, 1964; Weber, 1973). Furthermore, the diagram can indicate the effects of changing the various time constants. Harriott (1964) suggested that the product of the maximum gain and the critical frequency can be used as an index of the controllability for comparing various control systems when the criterion of optimization is the integral of the absolute error, IAE. Since this controllability index is also one of the subjects of investigation in this paper, it is pertinent to review some of the reasoning behind it. The critical frequency is expected to be an important factor in the controllability since all of the customarily used optimization criteria lead to underdamped closed-loop behavior. The frequency of a damped oscillation is usually 10 to 30% less than the undamped critical frequency. Furthermore, a t the optimum settings, only a few cycles are required before the output variable damps down to within a few percent of its final value. Therefore, the higher the critical frequency, the sooner the underdamped transient will be essentially completed, and the smaller the IAE. Harriott suggested that the IAE is inversely proportional to w,. The importance of the maximum gain comes about through its effect on the heights of the peaks of the oscillations following a disturbance. The argument goes that the ISE is proportional to the height of the first peak. This height depends on the gain factor for the load variable, KL, and the location of the disturbance. Harriott concludes that the peak error is about 1.5K~/(1+ K ) . The load gain is usually fixed by the design of the process and will not be a variable. Hence, in examining various alternatives for controlling a process, the IAE is inversely proportional to 1 + K. Since the overall gain is usually 10 or higher and the optimum gain is about half the maximum, the IAE is roughly inversely proportional to K,,,. Parenthetically, it should be mentioned that Prinz (1944) considered K,, to be an index of controllability. Accounting for the effects of both K,,, and w,, Harriott concludes the IAE is inversely proportional to their product. In a recent paper, Jeffreson (1976) investigated the relationship between this index and three commonly used indices of closed-loop system performance for several process models. Harriott cautions that the index is only approximate and should be used for quick comparisons. It could lead to serious errors if measurement lag is one of the major lags in either of the systems being compared since the actual and measured errors would be different. Furthermore, the method can be misleading if the major load disturbances occur near the end of the series of processing elements. The reason for this is that the closer to the end a disturbance occurs, the higher the first peak because initially there is very little damping by the processing elements. As noted earlier, the CCM is based on the two parameters, K,,,, and p,. In implementing the method, the controller is set so that only the proportional action is operative; therefore, if the controller has either or both reset and derivative actions, these are set to their minimum values. The controller gain is gradually increased until the system is on the verge of instability. The controller gain a t this point is K,,,,, and the period of oscillation is the ultimate period, P,. These parameters are related to K,, and w , as follows
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 219
where K p = process gains (= K1K2K3 in Figure 1) and
P, = 2ir/w,
= 0.45Kc,m,
&,CCM
TR,CCM = PJ1.2
(4)
(5)
K c , c c= ~ 0.6KC,,, (6)
proportional-integral-derivative
TR,CCM = PUP
(7)
TD,CCM = Pu/8
(8)
With this background in mind, this study is concerned with the following: (a) recasting the optimization study of Jackson in terms of the maximum gain and critical frequency, and extending it to include cases involving disturbances L1, L2,and L4 in Figure 1; (b) investigating the suitability of the CCM over a large range of maximum gains and critical frequencies; (c) determining the optimum parameters and corresponding optimum ISE over the range of maximum gains and critical frequencies and investigating the suitability of Harriott's controllability index in terms of the ISE; and (d) comparing the CCM ISE with the optimum ISE over the range of maximum gains and critical frequencies. Theory For the purposes of this study, the process gains, K1,K2, and K,, in Figure 1 can be combined with controller gain, K,, into the open-loop gain, K. This gain can be considered to be located a t the controller. The parameters, K,,, and w,, are related to the ratios of the time constants (Harriott, 1964) 1
+ K,,
= (1
+ R1 + R,)
C2Kma:
(2)
The following are the recommended settings for the three main types of controllers proportional K,CCM = 0*5Kc,m, (3) proportional-integral
R1 and R2. It is apparent from eq 13 that feasible solutions exist only if
( i1 i2) 1+ - + -
-
4(C2 + C)(C + 1 + Kmax)2 0
(16)
It can be shown that when R2 is real and positive, R1 will also be real and positive. Furthermore, from eq 16, it follows that K,, cannot be less than 8 since C is real and positive. Summarizing to this point, if a value of K,,, equal to or greater than 8 is specified, together with w, and T3,unique values of R1 and R2 will result, from which T l and T 2 can be calculated. The next step concerns the calculation of the criterion function y
ISE = l e0 e 2 d t
(17)
where e = error shown in Figure 1. A unit step input is assumed a t each of the four disturbance locations of Figure 1. The transform of the resulting error has the following general form c3s3
e =
+ c g 2 + C I S + co
d4s4 +d3s3
+ d2s2 + dls + do
(18)
The denominator is the same in all cases and the coefficients are d4 = T ~ T ~ T ~ T R d3
+
=TR(T~T~ TIT3
d2 = TR(T1
+ T2T3)
+ Tz + T3)
d l = TR(1 + K )
do = K
(19)
The numerators for the four disturbances are L1: CO = T R ; CI = C Z = ~3 = 0
L2:
(9) L3:
CO
= TR; CI = TRT1; ~2 =
~3
=0
+ T2);CZ = TRTiT2;~3 = 0 = TR; ~1 = TR(T1 + T2 + T3);~2 = TR(T1T2 + T2T3 + TSTJ; ~3 = T R T ~ T ~(20) T~
CO
= TR; ~1 = TR(T1
and
L4:
where
Using Parseval's theorem and the tables given by Newton et al. (19571, the criterion function can be calculated from the equation
R1 = and
(12)
From eq 9 and 10
Rz =
CK,,
f [C2Kma; - 4(C2
+ dodld,) + (cZ2 - 2~1~JdOdld4 + (cl2 2cocz)dod3d4 + ~ o ~ ( - d 1 d+4 ~d2d3d4))/ (2dod4(-dod3' - di2d4 + dld2d3)l (21)
y = (~,2(-d:d3
T1/T3
RZ = T2/T3
CO
+ C)(C + 1 + Km,,)]1'2
2(C + 1 + K m a J
The maximum gain, K,,,, is the reciprocal of the normalized amplitude ratio of a Bode diagram at the frequency where the phase lag is 180'. The value of the ratio a t K,,, is denoted fmax;thus fmax
and
where C=
1 + Kmax w,2T32
The choice between the + or - sign in eq 13 is of no particular importance since it just reverses the values of
= I/Kmax
(22)
Also, the frequency is made dimensionless by multiplying it by T 3 so that the system is uniquely defined. A computer program was written in which ranges off,, and w,T3 were specified and calculations were carried out a t a large number of points within the two ranges. At each point the following calculations were made. (1) The recommended controller settings of the CCM were deA Routh test for termined, namely Kc,CCM and r,,,,,. stability was then made for the denominator polynomial of eq 18. If the system was stable, then the corresponding criterion, yCCM, was calculated from eq 21. (2)The search
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2. 1979
r 6
Continuous
f 5 3 4
3
2 1 Infeasible Repion
d4
b5
d6
d7
b8
&3
;0
1;
1;
/3
‘0
01
02
03
04
fmax
Figure 2. Various regions of the fmax-u,T3plane.
05
06
07
08
09
10
11
I2
I3
fmox
Figure 3. Contours of R , and R2.
routine of Hooke and Jeeves (1961) was used to find the optimum open-loop gain, K*, optimum reset time, TR*, and the optimum value of the criterion function, y*,for each of the four disturbance locations.
Results (1) Regions of the f, - wcT3Plane. Calculations were made in the region 0.2 5 u,T3 5 10.0 and 0.0025 5 f,,, 5 0.125. This region covers most of that of Jackson’s study and includes some regions that his study excludes. Referring to Figure 2 , the feasible region is that in which positive real values of R, and R2 exist for given values of wC, T3, and fmax. Although a particular combination off,,, and ucT3is feasible, it may result in CCM settings that lead to unstable behavior. There are two regions within the feasible region that exhibit this, one very small and the other quite large. This property of the CCM has received little emphasis in textbooks or papers in which it is described, although some have noted that the settings are not always satisfactory. Since the CCM method was developed empirically, it is not surprising to find that it fails in some cases. Of course, this instability problem would not be a serious limitation in practice since the system could be stabilized by increasing the reset time and/or reducing the controller gain. On the other hand, it is worth noting that the CCM settings are not always conservative. T o understand why the CCM fails in some regions, contours of R1 and R2 are superimposed on the f,, -- ucT3 plane in Figure 3. Each point on the plane corresponds to a particular set of values of R1 and RZ. For example, f, = 0.0505 and wcT3= 5.66 corresponds to R1 = 0.1 and R2 = 0.5 (or to R1 = 0.5 and R2 = 0.1). Hence, if T3 = 1, then TI = 0.1 and T 2 = 0.5 (or T , = 0.5 and T 2 = 0.1). Unstable cases in the large unstable region occur because two of the three time constants are of the same order of magnitude while the third is much smaller. This means that on a Bode diagram, the slope of the phase curve is rather low in the region of crossover a t 180’ of phase lag. The CCM calls for a reset time of PJ1.2 = .lr/(0.6 a,) which introduces about 10.8’ of lag at the former crossover frequency for the process itself. With this low slope, the critical frequency is substantially reduced. At the same time, the amplitude factor, f , on the Bode plot is falling with a slope of about -2. Thus, the proportional gain would have to be appreciably reduced if instability is to be avoided. The gain and reset time recommendations for the CCM are not sufficiently conservative for this situation,
0 45 08 I 5
‘&.-.-.-32 (-6J
and instability results. In general, CCM settings are stable for f,, greater than about 0.03; in other words, if K,, is found to be between 8 and about 33, stability will be achieved. (2) Optimum ISE and Corresponding Controller Settings. (a) Effect of Disturbance Location on ISE. The optimum ISE is a function of the location of the disturbance. In most process control installations, the behavior of the system as a regulator is more important than the behavior as a servo, so location 4 may be of only peripheral interest. Furthermore, disturbances near the end of the process-L, in this case-are usually more important than those occurring a t the beginning, such as L, and L2. Referring to Figure 2 , the “center of mass” of the feasible region for stable CCM settings is roughly at f, = 0.05 and ucT3= 2.0. The effect of disturbance location on the optimum ISE was examined, first with u,T3 constant a t 2.0, and then with f,, constant at 0.05. At constant ucT3, the optimum ISE for each location is nearly constant as f,, varies. It increases as the location of the disturbance moves from the beginning of the process to the end, as would be expected from physical considerations. Thus, the ISE for location 4 is very large because a disturbance there suffers no damping by processing elements at the instant that it occurs, whereas disturbances a t all other locations encounter some damping. At constant f,,,, the optimum ISE decreases rather rapidly with increasing frequency, and again, there is an increase in the ISE as the location moves from the be-
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
O I b - - -
i \ 1 wcT, = t
' . \-
---___ , -----,
\infeor#ble ReQlon Bounk'ry
I
I 10
I
(0
wc TI
Figure 5. Optimum criterion function related to frequency for location 3; f,, = 0.05.
ginning to the end of the process. (b) Effects off,, and wcT3on the ISE for Location 3. The effects of fmax and w,T3 are illustrated in Figure 4 in which contours of the normalized ISE, y*/TB,are plotted on the w,T~-~,,, plane for location 3, the disturbance location that is usually of most practical interest. At frequencies less than about 4, the optimum ISE is essentially independent off,,,. On the other hand, it is a rather strong function of frequency. This dependency on frequency correlates rather well on the logarithmic plot of Figure 5 where y*/T3is plotted against wcT3for f,,, = 0.05. The slope a t low frequencies is about -1.5 and decreases to about -2.8 a t high frequencies. Figure 5 is of course a rather specialized situation because it was based on a single value off,,,. Physically, this means that although T3was held constant, the values of both T 2and T3had to be varied, one increasing while the other was decreasing, as can be seen from Figure 3. Since Figure 4 indicates that y*/T3 is essentially independent off,,,, this would lead one to suspect that Figure 4 would apply in the more general case where two of the time constants were fixed, while the third varied, resulting in changes of both f,,, and wCT3. Two series of cases were studied, the first in which T3= 1and T2= 0.3, and the second in which T 3 = 1 and T 2= 1.2. For both of these series, the values of y*/ T3fell on the line of Figure 5. Since Figures 4 and 5 show that the optimum ISE is roughly inversely proportional to the square of the critical frequency alone, they do not agree with Harriott's index. On the other hand, Jeffreson (1976) noted that the ISE criterion tends to result in responses that are persistent and slowly decaying. Therefore, one would not necessarily expect the arguments used by Harriott to hold for the ISE criterion. ( c ) Relation of CCM Controller Settings to Optimum Controller Settings for Location 3. Referring to the CCM settings given by eq 4 and 5, it is logical to examine the optimum gain, K*, in terms of its ratio to the maximum gain, Kmm. Similarly, the optimum reset time, TR*, is examined in terms of the ratio of the ultimate period to the reset time, Pu/TR*.The gain ratio was found t o be relatively insensitive to f,, and frequency, falling generally between about 0.42 and 0.6; however, a t fre-
221
222
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 - --CCM Stability Boundary
T 2are much smaller than T3. The time required to reach 63.2% of the ultimate change in output could be used as
L/1I r2 I 8
O'
71
.
For Location 3
4t
I J
wcT3
Figure 7. Ratio of the CCM ISE to the optimum ISE vs. wcT3at location 3 for various values of f m m .
controller gain becomes very large and so does the critical frequency. One reviewer noted that in this case, a purely proportional controller would be the logical choice rather than a proportional-integral controller. This reviewer went on to say that he had used the CCM for the past 15 years and had not run into a case where it gave rise to instability. A second reviewer remarked that he thought that many control engineers were already aware of the potential for instability in using the CCM, but that this problem perhaps needed to be stated more clearly in the literature. For disturbances a t location 3, the response under CCM settings compares favorably with the response under optimum settings except near the region of CCM instability. The CCM settings are more appropriate for regulator operation than for servo operation. As anticipated from physical considerations, the optimum ISE increases as the disturbance moves closer to the end of a process. It is nearly independent of the maximum gain and strongly dependent on the critical frequency. The controllability index in the ISB-sense is that the optimum ISE is about inversely proportional to the square of the critical frequency and independent of the maximum gain. The recommended gain parameter of 0.45 in the CCM is an excellent compromise for an ISE criterion. On the other hand, the reset time parameter of 1.2 is too high, not only because it leads to instability in some cases but also because it results in relatively large values of the ISE. Because of the relatively oscillatory response based on the ISE, this criterion is often not chosen in deference to the ITAE. A reviewer noted that the CCM produces a response which should be near optimum in terms of the ITAE. This being the case, it should not be concluded from the present study that the CCM reset time parameters be changed. However, it is suggested that in applying the CCM, the reset time could be set a t about twice the tecommended value initially, and then reduced in stages until satisfactory behavior is achieved. There would be some problems involved in the direct application of these studies to an actual plant because the maximum gain, K,,,, and the dimensionless frequency, ucT3,must be known. The critical frequency is easily obtained from the CCM procedure since it is equal to 2w divided by the ultimate period. The last time constant, T3,might be estimated from an open-loop response to a step change in controller output if it is known that T 1and
a rough estimate of TB. Regarding K,,,, the CCM procedure leads to a direct determination of K,,, and from an open-loop test, K , could be calculated from the ultimate change in process output divided by the change in controller output. Thus, it would be possible to get a rough idea of the location of the process with respect to Figure 2. Since this study revealed that the CCM can lead to unstable controller settings in certain regions of the O , T ~ - ~ ,plane, ~, this suggests that any empirical tuning procedure should be used with some caution. Each procedure has its practical limitations which are frequently not spelled out. Nomenclature C = constant defined in eq 15 CCM = continuous cycling method of controller tuning c = output variable co, cl, c2, c3 = constants defined in eq 20 do, dl, d,, d S , d, = constants defined by eq 19 e = error f = normalized amplitude factor on a Bode diagram f,,, = normalized amplitude factor at the 180O-phase lag crossover point IAE = integral of the-absolute value of the error ISE = integral of the square of the error K = open-loop gain of controller and process K1, K2, K 3 = gains of individual process elements K* = optimum open-loop gain K,,, = maximum open-loop gain K , = process gain = K1K2K3 K , = controller gain K,,,,, = maximum controller gain K,* = optimum controller gain Kc,CCM = CCM controller gain KL = load disturbance gain factor L1,L,, L3, L4 = load disturbances m = rS/ilused by Jackson n = i 2 / r 1used by Jackson P,, = ultimate period r = setpoint Laplace transform operator Tl,T2,T3= time constants of the three elements of third-order process shown in Figure 1 TD,CCM = CCM derivative time TR = reset time TR* = optimum reset time TR,CCM = CCM reset time t = time y = integral of the square of the error y* = optimum integral of the square of the error Y C C M = integral of the square of the error for CCM recommended controller settings Greek Letters r1 = Jackson's last time constant T3 in this paper i2 = Jackson's middle time constant T2 in this paper i3 = Jackson's first time constant = T1 in this paper o, = critical frequency s =
Literature Cited Cohen, G. H., Coon, G. A,. Trans. ASME, 75, 827 (1953). Harriott, P., "Process Control", McGraw-Hill, New York, N.Y., 1964. Hooke, R., Jeeves, T., J . A . C . M . ,12, 212 (1961). Jackson, R., Trans. Soc. Instrum. Techno/., 10, 68 (1958). Jeffreson, C. P., Ind. Eng. Chem. Fundam., 15, 171 (1976). Latour, P. R.. Koppel, L. 6..Coughnowr, D.R., Ind. fng.Chem. Process Des. Dev., 6, 452 (1967). Lopez, A. M., Miller, J. A., Smith, C. L., Murrill, P. W., Instrum. Techno/., 14, 57 (1967). McAvoy, T. J., Johnson, E. F., Ind. Eng. Chem. Process Des. Dev., 6. 440 (1967). Miller, J. A., Lopez, A. M., Smith, C. L., Murrill, P. W., Conrr. Eng., 14, 72 (1967).
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 223 Murrill, P. W., Smith, C. L., Hydrocarbon Process., 45, 105 (1966).
Newton, G. C., Jr., Goukl, L. A.. Kaiser, J. F., "AnaHcal Design of Linear Feedback Controls", DD 366-381. Wilev, New York, N.Y.. 1957. Prinz, D. G., i . ' S c i . Instrum., 21(4), 53 (1944). Smith, C. L., Murrill, P. W., ISA J . , 13, 50 (1966). Weber, T. W., "An Introduction to Process Dynamics and Control", WileyInterscience, New York, N.Y., 1973.
Weigand, W. A.. Kegerreis, J. E., Ind. Eng. Chem. Process Des. Dev., 11, 86 (1972). Ziealer. J.. Nichols, N.. Trans. ASME, 64. 759 (1942).
Receiued f o r review May 27, 1977 Accepted November 22, 1978
Mechanism of Segregation of Differently Shaped Particles in Filling Containers Kunio Shinohara Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
The mechanism by which particles segregate according to shape during the filling of storage vessels is analyzed. The theoretical approach is based on a screening model proposed previously by the author for size segregation effects. The entrainment of angular particles, which have percolated through the flowing layer to collect at the interface between the flowing and stationary layers of the heap, characterizes the shape segregation process. Angular particles are deposited near the central feed point and form a V-shaped zone. While this pattern is similar to that resulting from size segregation, the mechanism of the two processes is different. I n the present analysis, the segregation patterns observed are explained for various initial mixing ratios of feed particles, rates of feed, and apex angles of the vessel.
The segregation of particle mixtures due to size differences is well known, but little information has been reported on the effects of particle shape on segregation within a storage vessel (Denburg and Bauer, 1964). One of the main reasons lies in the fact that, though it may be possible to express particle shape mathematically (Cheng and Sutton, 19711, it is difficult to relate the physical behavior of particles to their mathematical shape definition. As a matter of fact, particle shape is not separable from size geometrically nor is it separable from surface frictional effects kinematically. In the case of segregation within a particle assembly, shape affects particle mobility within and over a solids layer, as does particle size. As is shown below, however, particle shape and size differences cause segregation in different ways. This paper presents an analysis and experiments on the mechanism of shape segregation of particles during filling a storage vessel based on a previous screening model (Shinohara et al., 1970, 1972). Segregation Process T h e following is an idealized description of the segregation process for binary mixtures of differently shaped particles based on video film of an operating hopper. When the particle mixture is poured and flows down a solids heap, nonspherical or angular particles percolate through the flowing mixture until they attain a lower stationary heap surface (see Figure 1). Then the angular particles separated from the mixture slide down slowly on the inclined heap surface under the influence of the upper flowing layer of particles remaining from the original poured mixture. Although in this respect nonspherical particles behave just like small particles in size segregation (Shinohara et al., 1972), in the case of a binary mixture of near sized particles, they cannot fall directly into the interspaces of the spherical particles. Angular particles penetrate into the layer of spherical particles only a t the moment of the latter layer's expansion under shear stress 0019-7882/79/1118-0223$01.00/0
that occurs continuously during the flow process. Let us divide the flowing layer of solids feed along the heap line into several blocks of differential thickness, as shown in Figure 1. Each block is assumed to flow down at a nearly constant velocity, u, during which nonspherical particles already separated also slide down under the block a t a lower speed, u,. To simplify the analysis, stepwise variation is considered during a small time interval, At. Through a material balance on the nonspherical particles in the sliding layer, the net volume, Vm,n,of angular particles left in the nth zone during At is given for the mth block of the flowing layer by
where V is the net volume of nonspherical particles supplied to the sliding layer of the nth zone from the flowing layer above by percolation. These particles are assumed not to move to the ( n + 11th zone but to fill vacancies in the sliding layer of the nth zone during the time interval. The second term corresponds to the volume fraction of nonspherical particles sliding down from the ( n- 1)th zone by entrainment, and the third term indicates the volume of angular particles remaining in the sliding layer of the nth zone after the underlying particles formed by the ( m - 1)th block in the zone have slid down to the ( n 1)th zone. Thus, it is straightforward to calculate V,, one by one from Vm,n-lof the former zone and Vm-l,nof the former block, as indicated in Table I. In the first zone and the first flowing block, Vm,o= 0 and V0,, = 0, respectively. Vm,nis, therefore, induced in general by tabulation as
+
where C denotes the combination and r = u,/u. If ncr 0 1979 American Chemical Society
