Ind. Eng. Chem. Process Des. Dev. 1983, 22, 22-31
22
it may be important in forming protective surface layers. Acknowledgment The authors are indebted to the operators of the SRC-I Pilot Plant at Wilsonville, AL, for their help in providing the tower liquid samples and process information. The work reported herein is a part of the Kentucky Energy Resource Utilization Program conducted by the University of Kentucky Institute for Mining and Minerals Research (IMMR) for the Kentucky Department of Energy (KDoE).
Mining and Minerals Research, Lexington, KY, Apr 1980. Davis, B. H.; Sagüér, A. A., to be presented at Corroslon/83, National Association of Corrosion Engineers Annual Meeting, 1983. Fontana, M. G.; Greene, N. D. "Corrosion Engineering", 2nd ed.; McGraw-Hill: New York, 1978. Jewltt, C., presentation at the workshop on Corroslon/Eroslon In Coal Liquefaction Pilot Plants, Oak Ridge, TN, Nov 29, 1979. Johnson, T.; Sagüés, A. A.; Davis, B. H. DtetiHatton Tower Corrosion: Synergistic Effects of Chlorine and Phenolic Compounds In Coal Liquids, IMMR48-PD23-80, Institute for Mining and Minería Is Research, Lexington, KY, Mar 1980. Kelser, J. R.; Judkins, R. R.; Baylor, V. B.; Canfield, D. R.; Barentt, W. P. Mater. Perform. May 1982, 21, 47. Sagüés, A. A.; Davis, B. H.; RHey, D.; Spencer, D.; Furman, R.; Clagget, J. "Annual Report, A Kentucky Energy Resource Utilization Program, July 1, 1980-June 30, 1981"; IMMR811081 Institute for Mining and Minerals Research, Lexington, KY, Dec 1981, pp 2-15. Sagüés, A. A.; Davis, B. H. Mechanisms of Corrosion and Alloy Response in Coal Liquid Systems Containing Chlorides, In "Corroslon/Eroskm/Wear In Fossil Energy Fuel Systems”; National Association of Corrosion Engineers, Houston, TX, to be published In 1982a. Sagüés, A. A.; Davis, B. H. Hydrocarbon Process. Jan. 1982b, 61, 98. Shalvoy, R. B.; Davis, B. H.; Freeman, G. B.; Sagüés, A. A. J. Vac. Scl. Techno! 1982, 20, 1080. Sorell, G.; Lendval-Untner, E.; Buchheim, G. M. Mater. Perform. Sept 1982, 21, 23. Materials Performance In the EDS Coal Liquefaction Pilot Plant: Illinois No. 6 Coal. Wachter, A.; StHlman, N. Trans. Electrochem. Soc. 1945, 87, 183.
Registry No. NH4CI, 12125-02-9; (NH^COg, 506-87-6; 1050 carbon steel, 12731-95-2; 304L stainless steel, 12611-86-8; cresol, 1319-77-3; 3,5-dimethylphenol, 108-68-9; 3,4-dimethylphenol, 95-65-8; 3,4-lutidine, 583-58-4.
Downloaded via UNIV OF LOUISIANA AT LAFAYETTE on January 11, 2019 at 15:02:54 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
Literature Cited Barnett, W. P.; Sagüés, A. A.; Davis, B. H.; Baumert, K. L; submitted for pubflcatlon In Fuel, 1982. Baylor, V. B.; Kelser, J. R.; Leslie, B. C.; Aden, M. D.; Swlndeman, R. W. Analysis of T-105 Fractionation Column Failure at the WHsonvHle, AL, Solvent Refined Coal (SRC) Pilot Plant. Oak Ridge National Laboratory, Oak Ridge, TN, ORNL/TM-7327, July 1980. Canfield, D. R.; Ibarra, S.; McCoy, J. D. Hydrocarbon Process. July 1979, 58, 203. Davis, B.; Thomas, G.; Sagüés, A.; Jewltt, C.; Baumert, K. Distillation Tower Corrosion: Chemical Analysis of Solvent Refined Coal Process Samples for Chloride, Acidic, and Basic Compounds, IMMR/80-053, Institute for
Received for review February 20, 1981 Accepted June 7, 1982
Control of FlufcKzed Bed Reactors. 1. Modeling, Simulation, and Single-Loop Control Studies Randall C. McFarlane, Terrence W. Hoffman, Paul A. Taylor, and John F. MacGregor* Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
In this paper a dynamic model is developed for a pilot plant fluidized bed reactor carrying out highly exothermic hydrogenolysis reactions. The model is used to tune and to examine the performance of simple single-loop cascade controllers, and some preliminary on-line computer control runs are performed to evaluate further the problems In controlling these reactors over an extended range.
Introduction In recent years considerable research on the control of
of the parameters from reactor data; (ii) to investigate the unsteady-state and nonlinear characteristics of the reactor through simulation; (iii) to tune and to evaluate the performance of some simple cascade controllers for temperature and selectivity using the model, and (iv) to perform some preliminary on-line computer control runs on the pilot plant using these simple controllers. All of the above investigations will illuminate some of the difficulties in controlling these reactors over a range of operating conditions and will provide a base case for the control of the reactor using single-loop conventional controllers against which more advanced controllers can be compared. Subsequent papers will deal with adaptive and suboptimal nonlinear controllers aimed at overcoming some of the deficiencies exhibited by these preliminary some
temperatures, conversion, and selectivities in packed bed and tubular reactors has been published. Much less has been reported on the control of fluidized bed reactors with the exception of the literature on controlling the temperature and overall conversion in fluidized catalytic crackers where the emphasis has been on investigating the interactions between the regenerator and the reactor. Although fluidized bed reactors are in some ways less sensitive to process disturbances, they do exhibit nonlinear and time varying characteristics which give rise to control problems over an extended range of operation. This is the first paper in a series which will deal with the control of a fluidized bed reactor carrying out some complex and highly exothermic hydrogenolysis reactions. The model considered is general enough and the pilot plant reactor used is large enough that the results should represent reasonably well the main features of industrial fluidized bed reactors. The main objectives of this paper are: (i) to develop a dynamic model for the reactor system based on material and energy balances and to estimate
0196-4305/83/1122-0022$01.50/0
schemes.
Process Description The Chemical Reaction Network. The reactions considered are those of the hydrogenolysis of normal butane in a fluidized bed of nickel-impregnated silica gel catalyst. The butane hydrogenolysis reactions were extensively studied by Orlikas et al. (1972). They showed ©
1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
1,
1983
23
oil system. This system consists of 1.27-cm tubing spiralled at 5-cm centers around the reactor wall and around approximately half of the disengaging section. Heat transfer between the coils and the reactor wall is enhanced by a heat-transfer caulking. To heat up the reactor and to maintain its temperature during reaction, a number of electrical immersion heaters are installed in the circulating oil tank. Fine control over the temperature of the oil entering the coils is achieved by control of the cooling air flow in an air-cooled heat exchanger. The entire reactor system can be operated either manually at the site or remotely under computer control using one of the minicomputers in the department’s computer control laboratory (a NOVA 1200 in this case). Under either mode of control all data collection is performed by the computer. A more detailed description of the reactor is given by Shaw (1974).
A Dynamic Model for the Reactor System When developing a model for a system it is important to keep in mind to what use the model will be put. For control purposes a high level of model complexity is not usually required; a model which is able to predict the dominant dynamic effects will usually suffice. With this in mind and knowing that eventually the model is to be used in on-line computer control schemes, we
1. Schematic of the pilot plant equipment and control configurations.
Figure
that these highly exothermic hydrogenolysis reactions could be represented by the following set of four seriesparallel equimolar reactions (three of which are stoichiometrically independent) C4H10 +
h2-> C3H8 +
CH4 —* + C4H10 H2 2C2H6 — + c3h8 h2 c2h6 + ch4 C2H6 + H2
—
2CH4
(1)
The fluidized Bed Reactor System. A schematic diagram of the fluidized bed reactor system is shown in Figure 1. The reaction section containing the catalyst is 0.20 m in diameter and 1.83 m high. Above this is a 0.46 m diameter disengaging section. An external cyclone removes any solid that leaves this section and returns it via a dip leg. The butane and hydrogen feed rates are controlled separately from the minicomputer and then combined and preheated to reaction temperature (=* 255 °C) in a hot oil preheater. The hot feed gas then enters the bottom cone of the reactor which is packed with stainless steel rings and passes into the reaction section through a distributor plate drilled with 230 0.14-cm holes. The nickel catalyst is supported on silica gel particles (10% Ni by weight) with diameters ranging from about 70 to 300 µ . The depth of the catalyst bed under static conditions is approximately 0.47 m. The reactor bed temperatures are measured using thermocouples inserted at several heights. Radial traverses with these thermocouples indicated that the bed was nearly isothermal, except in the immediate vicinity of the wall, as long as the gas velocity was in excess of ten times that required for minimum fluidization. The reactor is maintained under constant pressure, and the effluent gas is sampled every 360 s and analyzed with an on-line Beckman process gas chromatograph. The complete chromatographic analysis cycle requires 360 s for completion, thereby introducing a considerable dead-time in obtaining the concentration measurements. This is an important consideration later in the control studies. Cooling of the reactor is accomplished with a circulating
try to strike balance between the requirement of an accurate mechanistic model of the fundamental transport phenomena in the system and a simple and easily solved model suitable for use in a computer control scheme. Reaction Kinetics. Orlikas et al. (1972) developed a mechanistic kinetic model for the reaction system described in eq 1 and showed that it gave very good predictions of conversion and selectivities over a wide range of conditions. These expressions and their parameters were used directly in this study. Only the catalyst activity term had to be reestimated for our system. The reader is referred to this reference for the detailed kinetic expressions. The main characteristics of the reactors are that -40 kcal/mol) and they are all highly exothermic (AHR =a 40 to 60 have large activation energies ( kcal/mol), leading to a doubling of the reaction rates for about every 3 °C increase in reaction temperature. In very simplified terms the kinetic expressions behave roughly to a firstorder dependence on the butane concentration and to a negative second-order dependence on the hydrogen cona
centration.
Fluidized Bed Reactor Model. Prior experimental
and steady-state modeling studies were carried out on this reactor by Shaw et al. (1972,1973). They found that the simple, two-phase model of Orcutt et al. (1962) described the conversion and product distribution (selectivity) with acceptable accuracy even though other models provided a better representation of the reactor under certain conditions. We have based our dynamic model upon the Orcutt model because of its simplicity. The model contains the following assumptions: (1) There are two phases in the reactor: a bubble phase and an emulsion phase. (2) Reaction occurs only in the emulsion phase. (3) The emulsion phase is perfectly mixed. (4) The bubbles are of constant diameter and pass through the bed in plug flow at velocity ub, and there is no breakup or coalescence of bubbles. (5) All gas in excess of that required for minimum fluidization flows as bubbles. (6) The bed is isothermal. (7) Mass transfer between the bubble and emulsion phases occurs at a uniform rate over the bed height. This two-phase representation is illus-
trated in Figure 2. Material Balances: Bubble Phase. For any compo-
24
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
product
1,
1983
Simultaneous solution of this set of differential equations (5) gives the emulsion phase concentrations through the reactor. If one knows the division of flow into each phase, the bed exit concentration can then be calculated from the solution of the set of eq 3 and 5. Disengaging Section. Since the concentrations are measured by a process chromatograph at the exit of the reactor, the dynamics of the disengaging section are also important, and they were modeled as a perfectly mixed tank. Empirical Correlations. Several empirical correlations were available from the literature for predicting certain physical phenomena in the bed. (i) Bubble diameter as a function of bed height was given by Hato and Wen (1969) as
gas
BUBBLE PHASE
EMULSION PHASE |
V
V
^
"b ,h 1
volume
volume
=
=
|
s v
Víl-N
i 1
Q
V.
D
Db-1.W(^)+[6(^)]VM6.
)
_
Since the Orcutt model assumes a constant bubble diameter with respect to height, an integrated average value of Db was calculated from this expression as described by Shaw et al. (1973). (ii) The bubble rise velocity was predicted from (Shaw et al., 1973)
i
%
ub
0.7ll(gDb)1/2 + (u,
=
(7)
umf)
-
(iii) The interchange rate between the bubble and the emulsion phases was calculated from the correlation (Kobayashi et al., 1965)
9 feed
gas,
C
2.
Two-phase representation of
a
fluidized bed reactor.
nent i in the bubble phase its concentration as of the height (h) through the bed is given by dC
7,·
(8)
(Vb/Db)
a parameter that was estimated from reactor conversion/selectivity data. Enthalpy Balances. Reactor Contents. On the assumption that the gas and the catalyst particle temperatures are equal (TR), a combined gas/solid phase enthalpy balance can be written as
where t, is
0
Figure
=
a
function
1
= {i = h 2’ 3) (2) Q(Ce‘ Cb¿) UbVh~dt with initial conditions CJ = CJ at h = 0. Q is the transfer or interchange rate for any component between a bubble of volume Vb and the emulsion. Since Vb, ub, CJ, and Q are assumed to be independent of height in the bed, the bubble material balance equations in (2) can be integrated from h = 0toh=Hto give the exit bubble concentrations ~
as
Cbjj
CJ + (CJ
=
QH
CJ) exp
-
(i
ubVb
=
1, 2, 3)
(3)
Emulsion Phase. For any component i balance
over
a
material
the entire perfectly mixed emulsion phase
yields dCe‘
b ^
~
^mf Aa(C0‘
-
NbubVb AS(C0‘
CJ) + -
Cb,H‘)
-
ÁVe
(i
=
1, 2, 3)
(4)
Substituting (3) into (4) then gives the emulsion phase
concentrations dCe‘
Ve—
=
as
Aa(C0-
-
Ce‘)|umf
+ NbubVb\ 1
-fV.
-
(i
=
1, 2, 3) (5)
(WCpb +
=
eVpgCpe)-^-
voCpgPg(T0
-
7r)
+
Mw(Tw
-
+
7r)
ZRjMijW
;=i
(9)
Reactor Wall. The reactor wall, cooling coils, and supporting structure in contact with the wall were judged to have a mass sufficient to make their dynamic effect an important consideration in the temperature dynamics of the overall system. Lumping these elements into a single mass (Mw) at a uniform temperature (Tw), an enthalpy balance yields MwCpw(dTw/dt)
=
h„AJTn
-
Tw) +
Mc(Toil
'
Tv)
-
9ioss
(10)
Cooling Coils and Air-Cooled Heat Exchanger. The temperature response of these heat-exchange units was shown to be rapid in relation to the rest of the system, and therefore the instantaneous outlet oil temperatures were calculated from the steady-state relationships for a heat exchanger with a constant heating/cooling source temperature.
Parameter Estimation Although many of the parameters in the foregoing model were readily available in the literature, or had been previously well estimated from statistically designed kinetic
experiments (Shaw et al., 1972; Orlikas et al., 1972), there
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
Figure
3. Comparison between the model
coefficient coil heat transfer coefficient apparent mass of the reactor wall heat loss to environment catalyst activity
^wAw Ac
1983
25
predictions for various reactor temperatures and experimental data.
Table I. Parameters of the Model and Their Least-Squares Estimates Obtained from Unsteady-State Reactor Data estimate parameters symbol
wall heat transfer
1,
21.4 cal/(s K)
17.36 cal/(s K)
MwCpw
3000 cal/K
Qloss
0.0165 cal/s
k/k0
1.87
still remained a number of unknown parameters that had
to be estimated from data collected on the fluidized bed reactor itself. These are listed in Table I. As is usual with exothermic (or endothermic) reactions, some of these reactor model parameters will be very highly correlated if only temperature data under reaction conditions are used in the estimation (Jutan et al., 1977). This can be seen from the structure of eq 9, where the parameters (hwAw) and k/k0 appear in the last two terms, respectively, which represent the rate of heat removal and the rate of heat generation. Any parameter values for (hwAw) and k/k0 which make these terms almost equal in magnitude and opposite in sign will provide an almost equally good fit to the data. To help to reduce this correlation, two sets of data were used, one taken under nonreacting conditions (i.e., the butane feedrate was suddenly set to zero and maximum cooling applied to the reactor) and another taken under reacting conditions where both the hydrogen/butane feed ratio and the cooling oil temperature were varied. At this preliminary stage of the investigation the exit concentration data were not available from the chromatograph, and so the parameter estimates were obtained by minimizing the sum of squares (Z- Z)T(Z- Z) where ZT = (THT, TWT, ToilT) and Z is the estimate of Z from the model. The parameter estimates obtained are listed in
Table I, and in Figure 3 it is shown that the resulting model predictions are quite reasonable.
Control Objectives and Univariate Controllers A desirable overall objective would be to control the production or consumption rates of all five species at de-
sired values specified at any given time by optimizing some profitability function. Since only three of the reactions in (1) are stoichiometrically independent, this can be accomplished effectively by controlling the production rates of only three of these species. There are potentially four manipulated variables: the oil temperature, the inlet feed temperature, and the hydrogen and butane feed rates. However, there are operational constraints which limit the freedom of control action: the reactor temperature must be maintained within certain bounds to avoid quenching or runaway of the reactions; the effective speed of response of the system to changes in the various manipulatable inputs is very different (that due to the reactant feed rates being very much faster than that due to either of the temperature variables); the temperature difference between the cooling oil and reactor should be large enough to provide sufficient heat removal but not so large as to produce excessive heat losses; and the total flow rate of the gases should be high enough to ensure adequate fluidization of the catalyst without being so high that slugging or excessive bypassing occurs. In this paper a first attempt is made at controlling this nonlinear, coupled, multivariable process by using a set of univariate controllers as shown in Figure 1. The highly exothermic nature of the reactor coupled with the large activation energies meant that direct control of the reactor temperature, TR, was a necessity for safe and stable operation. This was accomplished by manipulating the ratio of hydrogen to n-butane feed rates. The total flow rate was kept constant so as to keep the state of fluidization constant. The other manipulatable variables, oil temperature and inlet feed temperature, could not be used for this purpose since their effect on TR was slowed by heat
26
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
1,
1983
270
260
250
240
.3
•
r-r
K.
.1
0 15
11
7
3
No. of control
Intervals,
temperature loop (30s)
Figure 0 and w
No. of control =
intervals,
temperature loop (30s)
^
t·’
=
1)
of the reactor temperature control system to temperature step set point changes with the PI controller tuned for (simulation).
4. Response =
1
w
.4
63
3C
Number of
Figure 1
5. Response
(experimental).
of the reactor temperature control system to
Control
a
Intervals,
Temperature Loop (30s)
temperature step set point change with the PI tuning equivalent to
w
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
27
1,
1983
a
selectivity
transfer limitations. The reaction temperature controller part of a cascade control scheme designed to control the propane selectivity. At the lowest level, the individual flow rates of hydrogen and n-butane were controlled by proportional plus integral (PI) controllers on a 1-s basis; at the next level the ratio of hydrogen to n-butane feed rates was controlled by the reactor temperature controller on a 30-s basis, and at the highest level the reactor temperature setpoint was controlled by the selectivity controller when the product selectivities became available from the Beckman chromatograph, i.e. every 360 s. An independent PI controller, manipulating the air flow rate through an air-cooled heat exchanger, was used to maintain the oil temperature at a constant value. In this study the outer loop of the cascade arrangement was univariate and so only one output variable could be independently controlled. If this was chosen to be the production rate of one of the products, then the selectivity of that product relative to the other products and reactants would not be uniquely defined. Conversely, if the selectivity of one of the products is controlled, then its production rate is not uniquely defined. This is a characteristic of applying univariate control to a truly multivariate process. Faced with this dilemma, we chose to control selectivity of one of the intermediate products. Propane selectivity (S3) was chosen because it was most sensitive to changes in the operating point. Typical variations of propane selectivity with the reactor conversion and temperature are illustrated by Shaw et al. (1972). was
Temperature Control The tuning of the temperature loop was complicated by the nonlinearity of the reaction system and the different dynamics for heating and cooling. Heating is caused by the heat of reaction and heat transfer from the feed preheater while cooling is dominated by heat transfer to the wall and hence to the oil. The expected difference between heating and cooling dynamics was simulated by forcing the dynamic model with step changes in feed rate ratio. The heating time constant was nominally 375 s and the cooling time constant nominally 450 s. However, these time constants were found to be strong functions of the temperature difference between the reactor and the oil. This behavior, together with the other nonlinear characteristics and the possibility of thermal runaway, led us to investigate the temperature loop tuning via simulation before application to the physical reactor. A further consideration in tuning the temperature loop was the relatively fast response of selectivity to changes in both TR and feed-rate ratio. In comparison to the response of TR to changes in the feed-rate ratio the selectivity responses were almost immediate; that is, the material balance (5) could be assumed to be in quasi-steady state with respect to the reactor temperature equation (9). Thus, changes in the feed-rate ratio had an immediate effect on selectivity and a much slower, first-order type effect upon TR. Therefore, in tuning the temperature loop we had to take into account both the temperature and selectivity response. The temperature controller was a discrete proportional integral controller in its velocity form (7,h2
Uh, Uc,
t
+ ^p[eTR,,
U.c4
"
erRiM
+ A"i6Tr J
(11)
where eTR¡ = TR¡txt p®”1 TR tmea8d = the temperature error at time t, UnJUcJt = ratio of hydrogen to n-butane feedrates at time t, Kp = proportional gain, and KT = integral gain. -
No. of control
Intervals,
temperature loop (30s) 6. Response of the cascade control system to step set point change from 0.22 to 0.26 (simulation).
Figure
The tuning was accomplished by minimizing a weighted of squares criterion with respect to the controller parameters, Kp and KR Four consecutive set point changes were made while the following criterion was evaluated sum
/ ~
min ,
+
$3,1
S3/et
Sg/*
(12)
where the summation is performed over 200 sampling intervals and = relative weighting of S3 deviations with respect to TR deviations. Figure 4 shows the simulated response when the reaction temperature controller was tuned with = 0 (S3 deviations ignored) and with = 1 (S3 deviations dominating). The asymmetry of the reaction system is clearly seen in the = 0 result. The controller parameters corresponding to = 1 were then implemented on the physical system so that S3 deviations would be minimized. Figure 5 shows the actual reactor response for a step change of 1.5 °C in the TR set point. In this and subsequent figures, the reactor temperature, conversion of butane, and selectivity of propane are plotted with respect to time. The time axis has units of integer multiples of the 30-s control interval. The measured conversion and selectivity are only available every 360 s from the chromatograph and so they appear as discrete data points every 12 control intervals. The conversion and selectivity measurements are plotted at the point corresponding the chromatograph injection time
28
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No.
0
1,
1983
25
50
Number of Control
Figure
7. Response
of the cascade control system to
a
75
Intervals,
100
Temperature Loop (30s)
selectivity step set point change from 0.23 to 0.26 (experimental).
rather than the analysis time, which is 6 min later. Dashed lines represent the appropriate set points. In Figure 5 the ratio of hydrogen to butane feed rates is also shown.
HG(z) and HKc(z), respectively, then the discrete Dahlin controller, D(z), can be expressed as (see Smith, 1972) D(z)
Selectivity Control A dominant feature of the selectivity control loop is the 360 s dead time introduced by the chromatograph analysis time. This is known to cause problems for PI controllers (Shinskey, 1979), and so a controller with inherent dead time compensation was required. Dahlin Algorithm. Dahlin (1968) incorporated dead time compensation by specifying a closed-loop response, K0(s), which was first order plus dead time and required that the closed-loop dead time be at least as long as the open-loop dead time. In terms of propane selectivity control this means S3(s)
Kc(s)
\s +
S3set(s)
1
(13)
where is the time constant and is the dead time of the closed-loop response. The process to be controlled has the reaction temperature set point as its input and propane selectivity as its output. An approximate transfer function, G(s), can be estimated from Figure 5. S3(s)
—--=
TRaet(s)
G(s)
Ke~>s =
——
ts
+
1
(14)
The simulations shown in Figure 4 indicate that this is a gross approximation not only to the shape of the response but also to the asymmetry and nonlinearity of the reaction system.
If the pulse transfer functions of G(s) and Kc(s)
are
i HKc(z) HG(z)
HKc(z)
=
1
-
(15)
where
HG(z)
(1
=
e
-
T!r)z
N
1
(1 -e~T/'Tz~1) (1
HKc(z)
e-7’/x)z-JV-1
-
=
(1
e~T/xz~l)
-
and N
T .
=
= nearest integer number of control intervals in , control interval. Note that has been set equal to
With a measurement delay of 360 s the Dahlin algorithm 360 s, IV 1. were chosen as 360 s, T After substitution and rearrangement, the controller was expressed as a difference equation parameters
=
125
?Vet
=
=
=
02eSlM) + (1
-
~~(
