MANESH J. SHAH A N D RICHARD E. STILLMAN
Computer Control and Optimization of a Large Methanol Plant Recent design advances combine efficient heat utilization and employ centrifugal compressors to produce large capacity of methanol at lower cost, The authors discuss the problem of control of these interacting plants and present results of simulation and optimization of these large capacity methanol production units -
n recent years several large capacity, single train
I methanol plants have been constructed or planned, following the success in design of large-scale ammonia plants. The major engineering feat in these plants has been the use of centrifucal compressors in the synthesis section where pressures of 150-350 atm are required. I n addition, recent improvements in synthesis catalysts have prompted Imperial Chemical Industries (ICI) to design methanol plants that employ low-pressure synthesis units. However, the IC1 design yields low methanol , conversion in the synthesis converter and therefore requires a modification of the centrifugal compressor to pump large volumes of gas in the recycle stage. As in the arnnionia plant design, the synthesis centrifugal compressor is driven by steani or gas turbines. T h e steam is produced by recovering furnace flue-gas sensible heat as well as the reaction-product sensible heat in the reformer section of the plant. A corniination of efficient heat utilization and eniployment of centrifugal compressor make possible large capacity of methano1 production at lower cost than with the older reciprocating compressor plants. Shah and Weisenfelder (25) have described how the operation of interacting plants, such as the one used in production of ammonia, creates certain inherent control problems. They showed that a corrective action by an operator to offset a disturbance in one section of the plant, for example, can produce diverging disturbances in the entire plant operation owing to the interacting nature of the various plant units. I t was shown that the optimum trouble-free operation of a n ammonia plant depends on control of the sensitive centrifugal compressor and the plant interactions. Because of the similarity in the design of the methanol and ammonia plants, it is reasonable that the methanol plant control poses similar control characteristics. Figure 1 is a simplified flow diagram of a methanol Y
VOL. 6 2
NO. 1 2 D E C E M B E R 1970
59
Figure 1. SimpiiJied flow diagram of methanol plant
plant. Examine the effect of a disturbance in inert composition in the synthesis loop stream 6 in Figure 1. A corrective action may be to adjust either the purge rate or to change makeup gas composition by control actions in the reformer. The purge rate adjustrilelit affects the reformer furnace firing because about 30Yc vol of the reformer fuel is purge gas, the reiiiainder being the natural gas. Thus, the process gas exit temperature from the reformer, and hence methane leakage, is disturbed by purge rate changes. The furnace fuel change also affects the superheated stream production that uses the flue-gas heat; thus the purge change not oiily creates a n immediate perturbation in the synthesis gas conipressor, owing to a flow change in the recycle stream, but in addition, a tiime-delayed disturbance in makeup gas composition as well as steam supply to the compressor drive turbines have been set up owing to upsets in the reformer. If, on the other hand, no action is taken to correct the composition disturbances in the synthesis loop, the performance of the autothermic synthesis reactor may be strongly affected. I t has been shown (23) that a certain combination of inlet temperature, pressure, and composition and reactor quench distribution can cause a quench or blowout condition requiring a reactor restart. It is also known that upsets in flow to the synthesis reactor can cause strong temperature variations in the beds. If the temperatures exceed critical values, side reactions such as methanation and ether formation take 60
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
over, and in turn lead to higher temperatures because of thcir largc heat of reaction. Since the flow-temperature effect has an inherent dead time owing to the largc volume of catalyst, perinanent damage to thc catalyst may occur before the reactor can be brought under control. T o achieve good regulation of the niethanol plant, a scheme of combining feedback, fecdforivard, and deadtime control, which considers the interaction of various plant units, is much more desirable than the schcine of simple feedback control. A digital computer not only can perform the more sophisticated control desired. but also the computer can be used to inonitor and control the centrifugal compressor to achieve its smooth operation. Experience b i t h various ammonia and methanol plants suggests that efficient trouble-free operation of the centrifugal compressor in itself can contribute to large economic incentive of cornputcr control. Methanol Process
Methanol is produced by catalytic reaction bctwccn carbon monoxide and hydrogen
CO
+ 2 Hz
-+
CH30H
(1)
The reaction mixture also contains COZ and CHI. Thus, the following side reactions may also occur :
COz CO 2 CO
+ Hz0 CH4 + E120 CH3OCH3 + HzO
+ Hz+
-t 3 Hz
+ 4 Hz
-+
-+
CO
(2)
(3)
(4)
Small quantities of higher alcohols may be formed when the catalyst reachcs temperatures in excess of the normal reaction temperature of methanol formation. Reactions 1, 3, and 4 are exothermic with heat of reaction of about 25,000-27,000 cal/g-mol CHaOH for reaction 1, and of about 50,000 cal/g-mol ether for reaction 4. Reaction 3 is endothermic. T h e extent of reaction 3 producing CO for reaction 1 depends on the catalyst used. With newer catalysts a larger amount of co, is reacted than with the older catalyst. I n any case, a large amount of heat is generated in the reactor as a net result. The reactor design then follows the steps of amnionia rcactor, wherein the heat of reaction is used to raise the reactor-feed temperature to the point where rates of reactions 1 and 2 become significant. The autotherrnic nature of these rcactors poses some operational problcnis as discussed i n some detail by various authors ( 7 , 27, 22). It is known that stable reactor pcrlormance is obtained for these reactors in a limited operational range. For the methanol reactor, feed temperature, feed rate, feed pressure, inert level, feed H 2 / C 0 ratio, and quench distribution are the Ixariables, the range of which is limited for sustained stablc reactor operation. Outside this range, thc rcactor may drift to blowout condition or quench condltion, resulting in a complete termination of methanol production. As shown in Figure 1, the methanol reactor is part of the synthesis loop in the plant. The mixture of CO and Hz required to produce methanol is obtained by reacting natural gas and steam in a reformer at elevated temperatures
+ n H 2 0 % nCO + (2% + l ) H z
CnHzn+2
(5)
CO
+ HnO % COz + Hz
(6)
At the temperatures employed in the reformer furnace, all hydrocarbons heavier than CH rcact completely to CO and Hz. I n addition, C O z is added to the natural gas feed to favor the equilibrium in formation of CO, as well as to give a Hz/CO ratio greater than 2. The reformer exit stream is quenched and partially or completely stripped of CO:! (depending upon the plant design) before being sent to the synthesis gas compressor as makeup gas. In the synthesis section, the compressed gas passes through the converter, the product is cooled, and liquid methanol is separated. The vapor from the separator is partially purged; the remainder is sent to the recycle stage of the compressor. Thc purge is used as fuel in the reformer. The crude methanol is sent to the distillation section to remove ether, water, and other alcohols. Figurc 2 shows a typical control computer system for thc mcthanol plant, whilc Figure 3 describes a typical prograrnniing system. The programming system incorporates an executive monitor. This monitor allocates various sections of computer core memory and time to various process-control functions on a predetcrrnincd priority basis. I n addition, a supervisory control program performs the monitoring, alarm scan, and supervisory interacting control function on the process, either uza set point adjustment or direct v a k e control (Direct Computer Control). Additional functions such as data logging, optimization, and engineering calculations, which are performed less frequently than control func-
Figure 2. Control computer interface with methanol plant VOL.
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DECEMBER 1970
61
Time Shared I -
Figure 3. PrograrnmirIg system for control computer
tions, use the nonprocess monitor system free time. (See 6, 7, 8, 24 for details.) As indicawd in (4, 24), some basic information regarding the variables in the methanol plant is required to implement supervisory control and optimization. For interacting supervisory control, a generalized adjustment equation that relates the manipulated variable MI, the uncontrolled or disturbancc variable 15-1, and the variables froni other loops VI, V,, . . , wid1 the target variable TI, may bc specified as FIAT1
=
F2AM1 $- F3AL’l
+
+
F;AL71
F3A7’3
... (7)
T h e coefficients F1, F 2 , F,, . , . , erc., in Equation 7, themselves may be nonlinear functions in orher constants A , , B,, C,, D,,E+:G , as expressed by
Ft = A i
+ B,[Ci+ ( D J G J ] ” ~
, ,
.
(8)
Once the ~ ~ a l uofe A M i is calculated at a given time from Equation 7, a programmed set of sequences of set point or valve adjustnient in a given time inter\Tal needs to be specificd for achie\~ingthe dcsired target change. Clearly, this sequence is determined by the dynamic sensitivity of thc 1.ariablcs in\rol\-cd to changes in manipulated variable M i . T h e readcr is directed to (4, 8, 24, 25) for further discussion on computer supervisory control. In order to iniplenient optimization, relationships between dependent constrained plant variables and independent control variables is required. A general form follows. Profit
=
f(X,Y)
and
(9)
Y,
> Y > Y,
(12)
where x’ and Y represent vectors of indepcndent and dependent variables, and the subscripts L and I; designate lower and upper limits. W e tvill describe, in the following section, how the relationships required for supervisory control and optimization were obtained by mathelmatical simulation of the 1-arious plant units, Lvherein the basic material, energy, and pressure balancc equations were solved. T h e unirs considered most important for siniulation are :
(1) (2) (3) (4)
Reforiiier Synthesis compressor Synthesis reactor Methanol separation section, including purgc and rec)-cle
T h e quciich boiler and thc various heat exchangers are treated as dependent constraints both for control and optimization. Simulation of Individual Plant Units
Synthesis reactor. hlethaiiol is produced in this reactor, and since it is not only the most important, but also the most diffirult plant unit, it is treated first. Major chemical reactions (1-4) take place in the reactor v,ith a large net heat of reaction. There arc various types of autothermic reactors available-e.g., TVA-type reactor with heat exchange tubes in the catalyst bed (22), multiple adiabatic catalyst beds with quench gas mixing between beds (23), and Topsoe radial converter (28). Some other designs (3) use combinations of adiabatic and nonadiabatic beds, or use adiabatic hcds with external cooling of reaction mixture between beds by means of boiler feed water.
and AUTHORS Manesh J . Shah and Richard E. Stzllman are wzth IBAM Cor$., 2670 Hanover Street, Palo Alto, Calif. 94304.
and 62
I N D U S T R I A L AND ENGINEERING C H E M I S T R Y
I' The basic model equations representing material, energy, and pressure-balance equations are similar for all the designs. The differences between the various designs are seen in the procedure for iterative solution of these equations, and in the various boundary conditions at the entrance of each bed. Figure 4 shows a generalized diagram of methanol synthesis reactor, consisting of n number of beds, with or without heat exchange tubes in the catalyst section. All designs have a terminal heat exchanger to cool the reaction products by means of the feed gas. I n addition, a portion of the reactor feed is used as quench between reactor beds. The quench favors the reaction equilibrium to C H 3 0 H by not only lowering the tcmperature of the mixture, but also by increasing the CO/ C H 3 0 H ratio entering the next bed. Some reactors also use a quench to control temperature of feed gas entering the first catalyst bed. Material, energy, and pressure balance equations for the synthesis reactor will now be derived. The following assumptions are made in order to arrive at equations that are manageable from a computational standpoint.
(1) Only the steady-state case is considered since the gas residence time in the reactor is fairly short (two seconds or less in cach bed). (2) The catalyst bed is adiabatic. All reactors are designed to circulate a thin film of cool feed gas mixture surrounding the outer wall of the catalyst basket. This gas acts as an insulator, and also reduces hydrogen embrittlemelit of the reactor metal at high pressure and temperature. ( 3 ) As a result of (2), radial temperature gradients in the catalyst bed are small. Since the radial teniperature gradients are small, the radial concentration gradients can be neglected. (4) Pressure drop in each bed is small relative to the total pressure, and can be expressed in a linear or quadratic form. The material balance for methanol is described by
dnj/dZ =
r l . 7rdZ2/4
(13)
where is the rate of reaction 1, and d, is the diameter of bed i. The subscripts for components n, are, respectively, I-Hn, 2 - C 0 , 3-C02, 4-CH4, 5-CH30H, 6-ether, 7-Hz0 (gas), and 8-Nz. The rate of reaction for methanol formation has been studied by Natta et al. (77), Uchida and Ogino (29), Vlasenko et aE. (37), Werrnann et al. (33),as well as Bakemeir et al. (2). Although there is considerable variation in the reaction rate expressions used in these references, the modified Temkin-Pyzhev expression used by Uchida and Ogino (29) has the least number of empirical and semiempirical parameters. Therefore, their expressions are used for the rate of reaction 1.
d 1 Figure 4. Simpl8ed frow diagram of methanol sjinthcsis conuerter
While Uchida and Ogino (29) have used 0.7 as the value of al, the expericncc of Shah and Weisenfelder (25),Kjaer ( l o ) ,and Nielson (78)with simulation of the ammonia reaction suggests that this may be a variable paraiiie ter. T h e rate constant k l is cxprcssed as
k l = A1 exp ( - Ei/RT)
(15)
while the equilibrium constant K , is expressed in the following form, using the data of Thomas and Portalski (27)
K,
= exp
{-
[u
+ b ( l / T - c)
- 0.003(300
- p ) ]1 (1 6)
where a, b , and c are constants. f describes a catalyst effcctivencss factor, which is a function of catalyst age. A variation of catalyst activity along the bed depth is cxprcssed by making J a linear function of z . The rate of ether generation is described, using a form similar to the methanol equation
Since the amount of ether formed is relatively small, K4 VOL. 6 2
NO. 1 2 D E C E M B E R 1 9 7 0
63
TABLE I . EQUILIBRIUM CONSTANTS FOR REFORM REACTIONS I N SYNTHESIS CONVERTER
+
COP
Reactzon H? e C O
+
Expiemon f o i K K = exp[(9678 T - 2.158 log T 0.0065T - 0.187 X l O - b T * 0.247 X 1 0 - Q T 3 0.924):RI
H20
+
change. T’(O),is given by
+ +
C 0 + 3 H 2 i i C I H ~ + H 2 0 I.;= e x p [ ( - 4 5 3 6 4 ; T + 1 5 . 1 3 2 l o g T - 0.00677’ - 0.233 X 10-6T2 0.194 X 1 0 - Q T 3- 4 5 . 9 ) / R ]
+
R
=
1.987, T in “K
= k2[(PCO2.PHJa2 - (1/Kd
+ T,(O). (1 - 4)
11( P I I P S -
4)
(30)
while the temperature of the reaction product after heat release is given by T,,JL) = T,(O) - [T’(O) - T , ] P , / P 2
is expressed only as a function of temperature in the range 3 of normal operatioiial temperatures and pressures. The rate of CO and CH4 forniation in reactions 2 and 3 is described by r2
T’(0) = T,[(Pl’P2 - 1)
(31)
T , is the temperature of the feed to the synthesis convcrtcr and T,(O) is thc tcrnperature of the product leaving the last catalyst bed aiid entering the heat exchanger. In Equations 30 and 31 P1
=
Cn,C,* 1
P2
=
CW,, z
X
1
( P C O ,PHzOIiPC0,.PH2)1-a2) (19)
where ni and F, represent, respectively-, the number of nloles of component i in the product gas and feed gas r3 = IC3[(PC0.PI{;)aq- ( 1 / K 3 ) x entering the heat exchanger. /i’ is the heat-transfer times the heat exchange area per unit length, ( P C ~ ; . P I ~ I ~ l ( P ~ ~ .1 P (20) ~ i ; ) l -coefficient as) and L’ is the heat exchanger length. Since the heat Expressions for K2 aiid K3 are shown in Table I, capacities C P i are functions of temperature, a trial and whereas kz and k3 are described by expressions of the error procedure is required for the solution of Equations form in Equation 18. 30 and 31. Material balances for the various components can now The material and energy balance equations for the be written down. synthesis reactor are solved subject to certain boundary conditions. For the firsr catalyst bed, the composition d n l / ’ d Z = - ( 2 rl re 3 r3 4 74) . ~ d , ~ /H4 a (21) entering the bed is known. However, the temperature dnz/dZ = -(rl - r2 r3 2 r4). ~ d , ~ / 4 co ( 2 2 ) at that point is unknown. The only temperature specified is that of the feed entering the synthesis converter. d n 3 / d Z = - rr.m?,2/4 C02 ( 2 3 ) This two-point, boundary value problem requires trial d n d / d Z = r3.n-d?/4 CHa (24) aiid error solution. The procedure employed for the solution depends on CH3OI-s (25) whether the reactor design employs adiabatic beds or ether (26) catalyst beds With heat exchange tubes. In the latter case, the gas teinperaturc entering the first catalyst bed dn,/dZ = (r2 r3 r.1) ‘7rd,2/4 H20 (27) is assumed and the solution of the differential equations The energy balance in the Zth bed is gi\-en by is carried out for all beds. Subsequently, the energy balance relationships for the heat exchangers are used to S dT .I niC,,>(T)ZZ = r,H,(T) C‘LT’ - T) (28) calculate T,.The calculated value is then compared i=l j=1 with the reactor feed teniperature. If the difference where T’is the gas temperature in the heat exchange between the two iyalues is greater than one degree, the rubcs in the catalyst bed. For an adiabatic reactor lYz feed temperature to the first bed is corrected and the is zero. solution is repeated. The heat of reaction for the various reactions is given \\Then no heat exchange occurs in the catalyst beds, a in (27). The pressure in the bed is expressed linearly better procedure is to (1) assume the exit temperature as from the last catalyst bed, aiid (2) perform the heat exchanger calculations with Equations 30 and 3 1 to yield P = PO(L) - W L Z (29) the temperature T ’ ( 0 ) . Since T’(0) is also the temwhere PO(L) is the pressure entering rlie Lth bed and w L perature of reactor feed gas entering the first catalyst is a pressure-drop coefficient. bed, the solution of the material and energy balance T h e energy balance around the terminal heat exequations can proceed. Shah et ul. (22) illustrated the changer is given by the expression derived for an arnfirst iterative procedure in an ammonia reactor sirnulaiiionia reactor (23). tion for T V A reactor. The second iterative procedure The temperature of the converter feed after heat exfor adiabatic reactor has been illustrated for a quench and
+ + + + +
+ +
c
64
c
+
INDUSTRIAL A N D ENGINEERING CHEMISTRY
bed-type ammonia reactor by Shah (23). In both cases a modified Regula Falsi technique (72) was used for rapid convergence on the assumed temperature boundary condition. For illustrative purposes, this paper will deal with the case of adiabatic beds and use the second iterative procedure of assuming exit temperature from the last catalyst bed. We will also use the case of a reactor with four catalyst beds and four quench shotsone for the reaction gas entering each bed. T h e boundary conditions for the solution of the differential equations of material and energy (Equations 21 through 28) for each bed then become
+
[n,(O)],= [na(L)]l-l Q,F, for each component i
(34)
and
T(0)
=
+
[C~,(L>Z-I.C,~(T(L)Z-~) * T(Lj1-1 C Q , F , c , ( T Q J TQzI z
3
a
Cn%(0)l~C,%(T(O)z) z
(35) C, on the right-hand side of Equation 35 is dependent on T(O), so a trial and error procedure is required to evaluate the equation. I n addition, for the first bed n
n,(L), = Fi(1 -
C Qz) 131
(36)
and T ( L ) ,is equal to T’(0)from Equation 30. Figure 5 is a logic diagram for the solution of the reactor niodel equations. Reformer. The simulation of the primary reformer for an ammonia plant has already been described by Shah and Weisenfelder (25). T h e methanol plant employs a very similar reformer, with the exceptions that the reformer gas exit temperature is higher and COz is added to the reIormer feed to balance the ratio of Hz/CO required for the methanol reaction. T h e two reactions
CnHzn+z
+ nHzO @ n C O + (2n + 1)Hz
(5)
and
CO
+ HzO
COz
+ Hz
(6)
take place in catalyst filled tubes that are heated in a direct-fired furnace. At the operational temperatures, all hydrocarbons higher than methane react completely. Moe (15, 16) demonstrates that the second reaction is at equilibrium, whereas the methane steam reaction is close to equilibrium at exit. As in the case of ammonia reformer simulation, we will derive the material balance equations based on an assumption of an approach to equilibrium temperature for the steam-me thane reaction. With the reformer gas exit temperature as a n independent control variable, and the entrance temperature of the reaction mixture known, the reaction heat load [or the reformer is calculated from the total reaction.
Figure 5. Logic diagram of solution of synthesis converter model equations
Let x be the fraction of methane reacted, and y be the fraction of steam converted to COz. T h e material balance for the various components follows.
+ yF7 + 5 F, + 7 Fiz + 9 = xF4 - yF7 + F Z + 2 Fii + 3 Fiz + 4 = Fz + YFS
H2
nl = 3 xF4
CO
n2
CO:!
723
VOL. 6 2
Fi3
(37)
F13
(38)
NO. 1 2 D E C E M B E R 1 9 7 0
(39)
65
where H p is the polj’tropic head, subscript n indicates 1 conditions of stream entering compressor case n, and n indicates conditions of stream leaving case n. E , is the efficiency that is 1 Mhen the compressor is truly adiabatic. T h e efficieiicy E , may also bc a function of gas flow in each stage. T h e horsepower for each stage is given by (79)
+
Ft specifies moles of component z in the feed to the reformer. If TOis the temperature of process gas exit primary reformer, then n2.nl3 p 2 Kj(‘TO’) = __ -- for the methane-steam reaction nd‘n7 n T
(45) and
Kj’(To)
=
n3,nl/’nz,n7 for the
CO 2 C O S reaction
(46)
equililx-iuni teinpcrature)
(47)
TO’ = To- (approach to Equations 37-47 must be solved simultaneously for x and y . Because these equations are implicit, a trial and error solution is required. T h e reaction heat load in the priniary rcforiiier is given by the energy balance 1s
Heat load
13
FiHI(TJ
n,Hi(Tn)=
= ?=I
(48)
i=l
where Hirepresents d i e enthalpy, and T,is the tcniperatiire of reformer inlet process gas. Synthesis compressor, Since the centrifugal conipressor is an important unit from an opcrational a s well a s from a n optimum production standpoint, its siiiiulation is treated here. Regulation of the compressor is required to avoid surge problems. In addition, total available stcani for conipressor turbine drive, and hence rhc available horseporver from the rcforrii section of the plant, is liiiiited. T h c surge control can be accomplished using a scheme suggested by Magliozzi (73). Since the digital coiiiputer can easily perform nonlinear control, the surge control can be impleiiiented on the coinpressor without thc elaborate hardtva re required for analog controllers alone. For plant optimization a conipressor model is required to predict the required horsepower for a given flow of makeup and recycle streams, whose teniperatures and pressures arc specified. Changes in coinposition of makeup gas and recycle gas affect the compressor performance tiia the change in gas density (average mol wt), because of the fixed volumetric displacement a t a given speed. Some variation in the compressor speed is permitted, although the range is rather IiarroLv. T h e compressor model used in this investigation is based on curves supplied by the manufacturer relating polytropic head to inlet flow a t each stage for various rotational speeds. Similarly, the stage efficiency is plotted a t various inlet flows with speed as a parameter. T h e compressor equation is the modified form of the standard adiabatic compressor relatioilship (19) 66
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Horsepower = (1
+ M,) , U p WnlE,!1.98 . X IO6
(50)
where T17n is the flow in pounds per hour, and M , is the fractional mechanical loss in stage n. Separators. T h e reaction mixture leaving the synthesis reactor is cooled by one or more cold water coolers to condense the riiethanol. The mixture is sent to the high-pressure scparator, ivhich contains baffles to facilitate the separation of crude methanol condcnsatc from the gas stream. T h e vapor from the separator is recycled to the compressor after purging a siriall portion. Crude methanol is sent to a let down tank and then the methanol purification section. T h e material balance equations for the separator were developed from the gas-liquid equilibriuin data of Michels et al. (74), KricheTrskji (17), arid in (5). This data was then fitted into quadratic relationships for temperature and pressure in the separator as follows
% gaseous coinpoiient i in liquid
methanol
yGinethanol in total vapor = d f eP,
-/- f T , $- g T , z
(52)
where ?’,< aiid P, are, respectively, the temperature and pressure in the separator. Total plant simulation. An exariiiriatioii of the flow diagram of the total plant in Figure 1, indicales that the various stream recycles \vi11 pose difficulties if the total plant simulation were attempted by simple coinbination of indilridual plant units. Trial and error solutions are required at several steps of the calculation. I n the synthesis loop, the pressure, flow,aiid coinpositioii of the synthesis reactor feed depend not only on the iiiakeup stream conditions entering the synthesis coiiipressor, but also on the stream conditions entering the recycle stage of the compressor. Since the recycle stream depends on the synthesis reactor outlet, as well as the amount of purge, one ends up with implicit relationships. Furthermore, the purge is dependent on the composition of the makeup stream, Ivhich in Turn is a function of the operation of the entire front end. The amount of natural gas fed to the primary reformer depends on the Hz and CO requirement at the makeup sIage of the conipressor. The Hz and CO requirement is dependent on the extent of the reactions in the methanol converter. I t is clear therefore that several trial and error calculations are required in total plant simulation. Based on our experience with ammonia plant siniula-
tion, the number of iterations in the overall calculations are best reduced by starting the calculations with given reactor feed conditions. Although the reactor feed pressure is obtained from the compressor calculations, a repeat calculation for the synthesis reactor is made only if the difference between assumed pressure and pressure obtained from the compressor calculations is greater than 3 atni. I n the optimization calculations, this pressure difference is maintained within h i t s by treating it as a tightly constrained dependent variable. Thc riiethanol converter calculations take up the greatest computer time. Therefore, using the reactor feed as the starting point for reactor calculation makes possible only one pass through the reactor simulation, resulting i n considerable saving of coniputer execution time. Referring to Figure 1, then, the procedure for the plant calculation is as follows. (Items marked with an asterisk require trial and error calculations.) (1) Assume reactor feed pressure", specify fced flow and composition, and calculate reactor exit flow, ternperature, and composition. This step involves the methanol con\wter simulation model, which involves multiple iterative calculations. (2) Calculate exit temperature of the gas mixture and the boiler water in the boi*jerfeed water heater. (3) Proceed with the water-cooled heat exchanger calculations to evaluate temperature of thc gas mixture entering the separator. (4) Calculate components in liquid and vapor stream leaving the separator. (5) Assume inerts* (CH,, N?, A) in makeup stream, assume purge split* and calculate inerts leaving in purge stream. Compare the inerts leaving the synthesis loop with the iiierts entering the makeup stream, and correct the purge split until their difference is small. (6) Based on Hz, CO, and C O Z consumed in the synthesis converter and losses in purge, calculate H1, CO, and GO2 required in makeup stream*. ( 7 ) Now calculate a first guess of required natural gas* to provide the required makeup gas. (8) Proceed with the primary reformer calculations, with specified inlet/outlet temperatures, COz/natural gas ratio, and steam/gas ratio. (9) Calculate makeup stream flow and coinposition based on primary reformer outlet conditions and CO1 absorber efficiency. (10) Compare the calculated Hn with that in step 6, correct the required natural gas and the new value of inerts in makeup. Repeat steps 5 through 10 until the required Hz matches that calculated in this step. (1 1) Perform compressor calculations and evaluate feed pressure to the reactor. If the difference between this value and the value assumed in step 1 is larger than 3 atm, repeat steps 1 through 11. A typical computer printout simulating the plant is shown in Appendix A. This program was run on an IBM 1130 with two microsecond memory cycle time. I t took approximately two minutes per pass through the methanol converter calculations, and about three seconds
per pass through the reformer calculations. An average time of less than seven minutes was requirrd for the entire plant simulation. For supervisory control of the various plant units the total plant simulation cannot be directly used because of the large coniputation time requirements. However, in the operating range of the variables, simple control relationships of the type in Equation 7 follow directly from the simulation results with the use of multiple computer runs. The off-line optimization of the plant is not bound by the coniputer tinie requirement. Thus the entire siniulation niodel can be used for performing optiniization on a large computer. For this investigation, the process optimization program POP I1 (26) was used because of its capability of handling large size nonlinear problems with constraints on dependent variables. Any worker in the field of optimization is aware of the numerous methods of optimization available, each one requiring a great deal of trial and error before obtaining successful results. Although the method of nonlinear optimization used in P O P I1 does not guarantee a global optimuiii, a comparison with several other methods in this class of problcrns, by us, showed that the POP I1 program gave consistent results with higher optimum than other methods. The program POP requires that a gradient matrix D Y / D X be evaluated at X 3 , where the superscriptj indicates a condition of the variables x , . The process is then assumed to be linear in a range of X that is larger than the range of AXused to evaluate the gradient matrix. By using a linear programming algorithm a new more profitable value of X3+' is calculated, and a corrected profit ylJ+' at the new X-' is compared with the last value of profit, y?. If y ~ J + > l y?, a new gradient D Y / D X at XI+' is evaluated and the above procedure repeated. The details of the optimization method and the program are discussed in (26). I t is obvious that the evaluation of the gradients D Y / D X at various points requires many computation passes through the simulation model, necessitating the use of a computer considerably faster than an IBM 1130. The off-line optimization computation was perfornied on an IBM 360 Model 91. In almost every case, an optimization casc study required between two and three minutes of computer time. Discussion of Results
We will discuss the results of thc reactor and total plant simulation and then show the computer results of off-line optimization studies on two different plants-one with synthesis reaction at 350 atni and 600"-700"F, and, in the second case, the niethanol reactor operated at 200 atm and 500"-600"F. I n a n earlier paper (27), we presented results of the methanol reactor simulation where only reactions 1 and 4 were considered. I n this investigation, we ha\re conducted the reactor simulation considering all four reactions. The siniulation equations were solved using a variable step, fourth-order, Runge-Kutta integration procedure. VOL. 6 2
NO.
12
D E C E M B E R 1970
67
LT
.-0 2
a,
8
s
0 0 Y
C
a,
2 a, e
44,000
52,000
48,000
110
90
Feed Rate, Lb Mol/Hr
130
Feed Temperature, " F
Figure 6. Effect of changes in feed late on methanol pmduction
Figure 7. Methanol production onriation zeith .rpthesis reactor feed temperature chatges
690 x m
660
630
-
c 5
-
+I
8
a,
- 33 6(30 ---
I
I
I
0.1
0.2
0.3
0 0
a
I 0.4
5.0
CO,/CO Ratio in Feed
Figure 8. EJect of uniution in CO,/CO ratio o n CO conver.rion in sjntkesis rcactor
Results of the high pressure reactor siiiiulation are s h o ~ v i iin Figures 6, 7, 8, 3, and 10. as well as Table 11, to illustrate the effect of changes in the reactor variables. Not all of the variables sludied are actuall) control variables in the methanol plant. Often they are disturbance variables or variables that can be indirectly coiitrolled. Per ccnt ineris and H2/CO ratio in the rcactor feed are two such variables. O u r simulation model was fitted to soiiie of the literature data (2, 20) available on niethanol plants. Extcnsive fit of the results could not be niade because of the lack of sufficient published data Howecer, the purpose of this investigalioii is not to rcproducc absolute T alues of variables 111 a particular methanol plant, but to exaiiiine the effect of upsets 111 plant \rariables o n the iiiethaiiol plant production and efficiency. For control as M ell as optiiiiization, this translates into prediction of incremental changes in the dependent or constrained plant variables as either a result of variation of controlled 7 ariables or of upsets in disturbance T ariable values. Thcreforc, the present analysis, M ith a limited fit of our model to plant data, \vi11 be niore than adequate if the computer results agree ~ i t h observed plant beha\ ior . The profiles of temperature and methanol conccntra68
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
6.0
6.6
7.0
H,/CO Ratio in Feed Figure 9. Gflect of variation in I&/CO ratio
28
OR
32 38 Total Quench Fraction of Feed
CO conuersion
40
Fijui-e 70. S:,rrflresis rcrictcr pet~orormancemriation with changes in total quench 0 First bed; A S P C Obed; II~
Th+d bed;
X FQUY~J~ bed
tion in the first two beds are shown in Figures 11 and 12. The reaction procceds rapidly and rcaclies equilibrium, especially in t h e second bed. Figures 6-10 arid Table IT sho\v thc effect of Lariation of feed pressure, H2/CO ratio, C H L / C O ratio, inethanol conccntratiun in feed, and \rariahles of feed tempcrature, feed pressure, quench distriburion; and feed rate. I n Table 11, the first run
TABLE II. RuIl no.
Reactor 16 mol/ht
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 71
32 33
46000. 46000. 46000, 46000, 46000 46000. 46000. 46000, 46000. 46000. 46000. 46000, 46000. 46000. 46000. 46000. 46000. 46000. 46000 46000. 46000. 46000, 46000, 46000. 46000. 46000, 46000. 46000, 46000, 48000. 50000 52000. 44000,
SIMULATION RESULTS F O R H I G H PRESSURE METHANOL SYNTHESIS CONVERTER CHION
co
Methanol
Feed
feed,
conuers:
CHdCO
70
5%
Prod, T/day
1 .00 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .00 1 .OO 1 .OO 1 .OO 1 .OO 1 .OO 1 .00 1 .OO 1 .OO 1 .DO 1 .OO 1 .00 1 .OO 1 .OO 1 .OO 1 .OO 2.00 2.70 1.00 1 .OO 1 .OO 1 .OO
39.26 38,45 40.31 40.73 38.66 37.53 40.03 35.37 42.00 46,83 37,74 41 .OO 41.64 34.42 43.60 46.53 40.36 40,67 41 . 3 6 43.59 37,74 37.72 41.79 43.33 38.08 41.24 42,94 39.01 38.83 39.80 40.20 40.42 38.97
640,10 624,60 661.20 669,60 635.30 627,lO 649,40 654.80 626.90 622.20 643.90 640,30 640.50 612.90 667,60 684.20 662.80 677,80 681.90 726.40 609.60 609.30 690.80 725.20 616.00 681 .50 723.80 634.90 630.80 680.10 716.80 752 00 606.40
Quench lied 1,
Quench bed 2,
Quench bed 3.
Quench lied 4,
Total quench
Feed
Ferd
aim
o/o
7%
%
%
SZ
Hz/CO
co2/co
364 364 364 364 340 320 385 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364 364
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 0 0 0 0 0 0 0 0 0 0 0
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 17 22 7 12 12 12 12 12 12 12 12 12 12 12 12
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 7 18 22 12 12 12 12 12 12 12 12 12
7 7 7 7 7
31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 34 36 36 41 26 26 37 41 27 36 41 31 31 31 31 31 31
5.35 5.35 5.35 5.35 5.35 5.35 5.35 4.50 6.00 7.00 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5,35
0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.15 0.35 0.42 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
1.so 1 .50 1 .50 1.50 1.50 1.50 1.50 1 .50 1 .50 1.50 1 .OO 2.00 2.20 1.50 1 .so 1.50 1 .50 1 .50 1. s o 1 .50 1 .so 1.50 1.50 1 .so 1.50 1.50 1.50 1.50 1 .50 1 .50 1.50 1 .50 1.50
Feed temp.
Feed pressure,
01;
120 140 100 90 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
0
0 0 0
7 7 7 7
7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 3 12 17 7 7 7 7 7 7
is a standard design condition of about 630 tons/day production, subsequent runs being variations froin the design case. The majority of runs was made with a fixed total feed, since the interpretation of results would be too complex if the feed rate of carbon monoxide was fixed. When the total feed is maintained constant for a given synthesis pressure, there is a more or less constant recycle load on the synthesis compressor. Although the total methanol production in tons/day is listed in one column of Table I1 depicting the simulation results, CO coilversion is also shown in the table to clarify the effect of synthesis gas composition changes on the reactor performance. Feed temperature. A lowering of the feed temperature to the reactor favors an increase in methanol production and conversion. However, it decreases the sensible heat available from the reactor product. An increase in methanol production of about one ton per day/one degree reduction in feed temperature is observed from the table. I t is clear that the feed temperature to the reactor is an important variable. The results also indicated that the reactor was unstable for a feed temperature less than 90°F. Feed pressure. I t is apparent from runs 5 , 6, and 7 that pressure has a relatively small effect on methanol conversion. I t was found, however, that if the pressure
Temp XT reactor,
OF 324.10 333,60 308,lO 301,60 318.70 316.70 323.00 323.60 330.10 316.70 320.10 321.50 321.30 326,50 316.50 313.30 329.90 336.00 336.10 352.70 307.50 307.50 339.20 353.10 310.30 336.10 353.10 319.20 317.90 323,20 326,lO 330.90 318.60
was substantially lower than 320 atm, the result was reactor instability. For a well operating compressor, large pressure fluctuations as in runs 5, 6, and 7 are UIIconinion. I t is somewhat reassuring that the reactor will continue to perforni well under reasonably small pressure variations.
600 L
I 1 0
H a
1
50C
5-
LL
0 5
5 W
400
I 300
4
2
6
Bed Length, Ft Figure 7 1. Temperature and methanol concentration proJles in jirst catalyst bed of synthesis converter VOL. 6 2
NO. 1 2
DECEMBER
1970
69
600
590 $
@
3 +
570 Q
$
iv)
550
8
530 Bed Length, Ft
Figure 12. Temperature and niethanol concentration pi ojiiles in second catahst bed, sjnthesis reactor
H,/'CO ratio. h i increase in H2'CO ratio causes substantial increase in conversion of CO to CH3OH. However, the increase in Ha/CO ratio decreases the actual amount of CO circulated in the synthesis loop and, therefore, decreases the production of methanol in the plant. The coinpressor horsepower requirement per ton of methanol production increases with the decrease in H2 circulation. CH,,'CO ratio. An increase or decrease in CH,,'CO ratio has relatively small effect on the reactor performance, a t least in the range of variation studied here. CO,,'CO ratio. 'l'he most dramatic effect on both CO conversion and inethanol production is shown by variation in CO,/CO ratio. A reduction of CO2/CO ratio to 0.15 decreases yc C O conversion by 17y0. However, an increase of CO,/CO ratio to 0.42 causes a substantial increase in methanol prod.uction in spite of the decrease in CO circulation. Two factors contribute to this increase in methanol production; (1) the high heat capacity of C o n , and (2) the increase in CO formation caused by endothermic reaction 2. The recycle load on the conipressor is increased with the increase in GO2 because of the higher molecular weighL of CO2. Quench distribution. The quench distribution in an adiabatic reactor is one set of independent variables that can be used to control reactor performance. As seen in runs 17-27, the methanol production varies from 61 0 tons to 724 tons, depending on the reactor quench disrribution. I t also appears that the total aniount of quench is the key factor iii determining methanol production. T h e distribution of the quench affects basically the high temperature limits reached in individual beds. However, the quench in the first bed is a much larger contributor to the instability of the reactor in comparison to the remaining quenches. Thus, keeping the remaining variables at design condition, up to 417, of the feed could be used as quench to attain a significant iinprovement in methanol production. 70
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Feed rate. Runs 30--33 indicatc that incrcascd feed rate brings about a slight increase in C O conversion. The rate of increase in methanol production is almost linearly proportional to the rate of increase in the fccd rate. Above 52,000 Ib mol/hr, the reactor reaches iristability. Increased feed rate also requircs incrcaseti recirculation compressor load. However, an incrcasc in recirculation horsepower per ton of production is loLver than the horsepower required for the increased makcup gas. For many plants the synthesis compressor capacity is a major bottleneck, and if the steam availability to thc drive turbine frorn the front end is restricted, thc nlcrhanol production of the plant is limitcd. Methanol concentration. Runs 28 and 29 iiidicatr the effect of an increase in methanol concentration oil the reactor performance, showirig a substantial reduction of production as a result of increase of CHROFIi l l the feed. Close control of the separation section i u a methanol plant is therefore necessary. This Incans that the temperature in the separation tank, as well as its efficiency, must be closely monitored. We have discussed the effect of variation of individual parameters on the reactor performance. 111 most cases the direction of the improved performarice was also thc direction of decreased reactor stability. Our intercst is in determining stable as well as optiinuin rcactor pcrformance of the reactor by means of a combination of variations in the control parameters. 'Thc optiiriizatioii of reactor performance can be best exaniined by running the reactor program as a subset of an optimization program. Since it is of primary intercst for us to invcstigate optimum performance of the eiirire inethanol plant. with its interaction of the various srctions of thc I)la?nt, including the synthesis reactor, we will not discuss the reactor optimization alone. m I he simulation of the reactor for lower prcssurc synthesis reaction (200 atni and 500"-600"7:) also gave results similar to those discussed abovc. Thercforc, instead of showing the results for thc low-pressure plant, we will discuss the results of optimization of the low-pressure plant together with the optimization of the plant where synthesis reaction is conducted at a prcssure of 350 atm. I n simulation of the total plant the reformer, the synthesis compressor, and the separator presented no specific computational difficulties. In almost all cases the iterative procedure for the total plant siinulation converged satisfactorily. A typical total plant simulation run is shown in Appendix A, where the various feed corriponents to the synthesis reactor, reactor gas composition and temperature at the end of each bed, and, finally, the various stream composition in each stream of the plant are shown. T h e total plant simulation is set up to run as a subset of the optimizer program POP 11. The results of off-line optimization are shown in Tables 111, IV, and V- for the high-pressure plant. Each table represents a computer run under a different plant constraint: (1) Market limited optimization with production held to 650 tons/day, Table V ; (2) Avail-
able compressor horsepower restricted to 20,000, (production limited), Table I11 ; (3) Compressor horsepower restricted to 18,700 which is slightly higher than the base case, Table IV. The tables show the various independent and dependent variables for the off-line optimization. The independent variables in the off-line optimization do not all necessarily represent directly controllable variables. For example, the CO/H2 and C02/H2 ratios are controlled by manipulating CO/H2 and C O Z / H Zratios in the makeup stream over a period of time. (The symbols U and L represent upper and lower limits, respectively, on the values of these variables. The asterisk after the value indicates that although the linear solution was within these limits, the optimum move after linearity correction violated one of these limits slightly.) However, it is of interest to examine the extent to which the optimum conditions differ from the recommended design condition. We see that in all three cases of optimization the profit is substantially increased over the design conditions, even though the optimization was stopped by
limiting the number of optimizer loops to 6. I n the market limited case, the optimum is at loop 5 , a profit increase of 2470, while the production limited situation indicates a profit increase of more than 50%. I n the offline optimization in the latter case, some plant units such as the reformer quench boiler or the reformer furnace firing box heat load will probably constrain production before such a large increase in production and profit can be attained. T h e result for compressor horsepower limited optimization is similar to the market limited situation, with slightly increased profit. We note that in all three cases the optimum lies at a corner of two or more constraints on the independent variables x t . Also, the constraints on all dependent variables are also respected by the optimizer within a small fraction % deviation. Interestingly, Tables I11 and IV, which show the optimization for the compressor limited plant, indicate that the plant should generally be run with the following conditions :
TABLE 1 1 1 . OPTIMIZATION OF H I G H PRESSURE CHIOH PLANT-PRODUCTION L I M I T E D CASE
TABLE IV. H I G H PRESSURE OPTIMIZATION W I T H COMPRESSOR HORSEPOWER L I M I T E D T O DESIGN
(1) Maximum quench for synthesis converter (2) Lowest synthesis converter feed temperature
Injut Fznal Independent Variables 1 Reactor feed lb mol/hr 46000,000 43750.000 2 HZ/CO ratio reactor feed 5.350 4.500L 3 C O Z / C O ratio reactor feed 0 260 0.450U 4 CHd/CO ratio reactor feed 1.500 1.455 5 70CH30H reactor feed 1,000 0.100 6 Reactor feed temp, O F 120.000 90.000L 7 Reactor quench fraction bed 1 0.006 0.0 8 Reactor quench fraction bed 2 0.120 0.200u 9 Reactor quench fraction bed 3 0.120 0.150U 10 Reactor quench fraction bed 4 0.070 0.050 11 Quench gas temp, O F 120.000 95.0001, 12 Synthesis compr speed, rpin 10500.000 10837.500 13 T e m p exit reformer, "F 1450,000 1500. OOOU 14 Steamigas reformer 3,200 3.149 15 Inlet temp reformer, O F 730.000 834.740 16 COz fraction reformer feed 1.020 0.832
Input Final Independent Variables 1 Reactor feed Ib inol/hr 46000.000 43500,000 2 H * / C O ratio reactor feed 5.350 4.500~ 3 C O , / C O ratio reactor feed 0.260 0.450U 4 CHd/CO ratio reactor feed 1.500 1.860 5 70CHIOH reactor feed 1 .ooo 0.400 6 Reactor feed temp, O F 120.000 02.920 7 Reactor quench fraction bed 1 0.006 0.0 8 Reactor quench fraction bed 2 0.120 0.120 9 Reactor quench fraction bed 3 0,120 0,150U 10 Reactor quench fraction bed 4 0,070 0.130U 11 Quench gas temp, "F 120.000 113.166 12 Synthesis compr speed, rpin 10500.000 10874.977 13 T e m p exit reformer, O F 1450.000 1500.000U 14 Steam/gas reformer 3.200 3.344 15 Inlet temp reformer, "I; 730.000 668.290 16 C O Pfraction reformer feed 1.020 0.804
Dependent Variables 1 Profit $/day 4250 727 7588.527 2 T e m p reactor bed 1, top 607.200 626.410 3 T e m p reactor bed 1, exit 677,300 715.100 4 T e m p reactor bed 2, exit 684.500 682.800 5 T e m p reactor bed 3, exit 674.200 669.900 6 T e m p reactor bed 4, exit 669.300 666 500 7 T e m p converter, exit 330.800 337.900 8 T e m p to cold water exchanger 80.840 87.900 25 Compressor horsepower 18860.547 20020 180U Hz makeup stream 4164.156 4537.949 33 34 CO makeup stream 1865.910 1912.590 COa makeup stream 405.140 334 120 35 36 CH4 makeup stream 146.500 117.500 39 Pressure reactor feed 373.250 380.870 40 Pressure diff compr exitreactor 3.250 1.977* 41 Natural gas reformer feed 1612.890 1687,120 42 Heat recovery syn reactor, mil Btu 82.350 96,100 44 C H I O H production tons/day 649.400 778.890 47 Fuel required reformer 2586.300 2607.830 48 COtexitrefornier Ibmol/hr 1306.910 1077.890 50 Total quench converter 0.320 0.400U
Dependent Variables 1 Profit $/day 4250.727 6795.309 2 T e m p reactor bed 1 , top 607.200 616.020 3 Temp reactor bed 1, exit 677.300 690.740 4 Temp reactor bed 2, exit 684.500 678.430 674.200 663.200 5 T e m p reactor bed 3, exit 6 Temp reactor bed 4, exit 669.300 654.150 7 Temp converter, exit 330.800 337.000 8 T e m p to cold water exchanger 80.840 87.000 25 Compressor horsepower 18860.547 18718.430U 33 H P makeup stream 4164.156 4317.227 34 C O makeup stream 1865.910 1732.610 35 CO1 makeup stream 405.140 328.230 36 CHd makeup stream 146.500 94.400 39 Pressure reactor feed 373.250 365.580 40 Pressure diff compr exitreactor 3.250 2.080* 41 Natural gas reformer feed 1612.890 1566.910 42 Heat recovery syn reactor, mil Btu 82.350 95.640 44 C H I O H production tons/day 649.400 740.740 47 Fuel required reformer 2586.300 2634.820 48 COzexitreformer lb mol/hr 1306.910 1058.800 50 Total quench converter 0.320 0.400U
VOL. 6 2
NO. 1 2
D E C E M B E R 1970
71
TABLE V. H I G H PRESSURE METHANOL PLANT OPTIMIZATION FOR MARKET L I M I T E D CASE Irzjut Final Independent Variables 46000.000 43500.000 1 Reactor feed lb mol/hr 4.500L 2 HzjCO ratio reactor feed 5.350 0.320 0.260 3 C 0 2 / C 0 ratio reactor feed 1.950 4 CH4/CO ratio reactor feed 1.500 0.400 5 7cC H a O H reactor feed 1.000 100.000 6 Reactor feed temp, O F 120.000 0.0 7 Reactor quench fraction bed 1 0.006 0.086 8 Reactor quench fraction bed 2 0.120 0.139 9 Reactor quench fraction bed 3 0.120 0.131 10 Reactor quench fraction bed 4 0.070 150.000u 11 Quench gas temp, OF 120.000 12 Synthesis compr speed, rpiii 10500.000 10874.000 1500.000U 13 T e m p exit reformer, OF 1450.000 2.700 14 Steam/gas reformer 3.200 981.130 15 Inlet temp reformer, O F 730.000 0.776 16 C O ?fraction reformer feed 1.020 Dependent Variables 5335.297 1 Profit $/day 4250.727 619.660 607.200 2 T e m p reactor bed 1, top 704.300 3 T e m p reactor bed 1, exit 677.300 691.970 4 T e m p reactor bed 2, exit 684,500 677.230 5 T e m p reactor bed 3, exit 674,200 667.780 6 T e m p reactor bed 4, exit 669.300 331,800 7 T e m p converter, exit 330.800 81.800 8 T e m p to cold water exchaiigcr 80.840 25 Compressor horsepower 18860.547 17857.297 3807.170 33 HP makeup stream 4164.156 1677.750 CO makeup stream 1865.910 34 261.340 COa makeup stream 405.140 35 123.570 36 CH: makeup stream 146,500 385.430U 39 Pressure reactor feed 373.250 40 Pressure diff conipr exit2,384* reactor 3,250 1457,630 41 Natural gas reformer feed 1612.890 42 Heat recovery syn reactor, 89.650 i d Btu 82.350 650.730C 44 C H 3 0 Hproduction tonsjday 649.400 2105.710 47 Fuel required reforiiier 2586.300 843.030 48 CO2 exit reformer lb mol/hr 1306.910 0.357 50 Total quench converter 0.320
Highest CO,/CO ratio converter feed Lowest Hz/CO ratio in converter feed Increased temperature primary reformer feed Highest temperature for primary reformer exit Reduced COn addition to primary rcforiner also to be noted that the increased production is brought about in spite of reduction in total feed to the synthesis converter. T h e market limit optimization shown in Table I r is accomplished by an increase in C H d C O ratio instead of CO2,'CO rario, reduction in steani/gas ratio and C O Z addition to primary reformer, increased feed temperature to primary reformer, and quench temperature to the synthesis converter. Thus, the three sets of values of independent variables appear to be sufficiently different to suggest that under the three values of constraints considered here, optiinum values of X-the independent variable \,ector-are significantly different. I n Tables \'I, V I I , and VI11 optimization resulrs from the low-pressure plant (3000 psia and 500-600'F 72
INDUSTRIAL AND ENGINEERING CHEMISTRY
synthesis reaction) are shown for the three different constraints as in the higher pressure case, except that the horsepower requirements and limits are considerably lower. Here again, not only the profit increase for all three cases is substantial above the design case, but also the optimum lies at the corner of constraints of independent variables. Furthermore, the compressor horsepower limited optimization shows the sarnc direction of plant optimization as in the high-pressure case with further reduction in COZ addition and steainigas ratio to primary reformer feed. It is also of interest to note that when rhe compressor horsepower limit is lowered, the direction of sonic of the independent variables alters significantl}.. For example, feed Hz/CO and feed teniperature to converter are no longer shifted to their lower limits. 'The niarket liinited optimization again shows a different trend, lvith upper limits on CHd/CO to converter feed, coni-erter feed teiiiperature and teniperature entry, and exit primary reformer. I n all three cases, the values of the vector of independent variablcs are substantially different.
TABLE VI. LOW PRESSURE METHANOL PLANT OPTIMIZATION-PRODUCTION L I M I T E D CASE Independent Variables 46000.000 49500.000 1 Reactor feed Ib ino1:hr 4.500L 5.350 2 Ha/CO ratio reactor feed 0.450U 0.260 3 C O S / C Oratio reactor feed I . 461 1.500 4 CHd/CO ratio reactor feed 0.200 5 96 C H 3 0 Hreactor feed 1.000 117.150 120.000 6 Reactor feed temp, O F 0.0 7 Reactor quench fraction bed 1 0.006 0.154 8 Reactor quench fraction bed 2 0.120 0.150U 9 Reactor quench fraction bed 3 0.120 0,096 10 Reactor quench fraction bed 4 0.070 106.290 11 Quench gas temp, O F 120,000 12 Synthesis conipr speed, r p m 10500.000 11025.000 1500. 000U 13 T e m p exit reformer, "F 1450.000 2,500L 14 Steamt'gas reformer 3.200 984.760 15 Inlet temp reformer, O F 730.000 0.693 16 COz fraction reformer feed 1.020 Dependent Variables 7858.977 1 Profit $/day 3132.190 552,460 2 T e m p reactor bed 1, top 543,190 619.830 3 T e m p reactor bed 1; exit 590.450 605.450 602.610 T e m p reactor bed 2, exit 4 594.010 5 T e m p reactor bed 3, exit 594.400 589.390 6 T e m p reactor bed 4, exit 590.470 321,570 7 T e m p converter, exit 306.760 71.570 8 T e m p to cold water exchanger 56.760 25 Compressor horsepower 12129.488 14474.918 U 4419.367 33 H Smakeup stream 3679.450 1917.770 1648.720 34 CO makeup stream 270.600 35 COS makeup stream 357.980 168.910 CH4 makeup stream 129.440 36 206.290 39 Pressure reactor feed 194.622 40 Pressure diff cornpr exit1.879* reactor 2.150 1709.600 41 Natural gas reformer feed 1425.140 42 Heat recovery syii reactor, 84.350 mil Btu 69.300 746.210 44 C H B O Hproduction tons/day 551.400 2400.570 47 Fuel required reformer 2285.260 872.910 48 C O ~ e x i t r e f o r m e r l bmol/hr 1154.780 0.400U 50 Total quench converter 0.316
T h e discussion of optimization results presented here clearly indicates that the plant optimum is strongly dependent on the constraints at a given time imposed on the plant variables. These results can then be extended easily to optimization with constraints other than the three cases discussed here-whether the optimization is implemented off-line or on-line. Supervisory Control and On-line Optimization
I n an earlier section we pointed out that the interacting control of the plant can be performed using relationships such as in Equation 7. We also said that constants Fa, are obtained from the results of simulation of individual plant units or the total plant unit. T o illustrate implementation of interacting, supervisory control, consider the control of CO/Hz ratio as well as CH4 in the makeup gas stream 5 in Figure 1. This ratio is affected by:
(1) Reformer exit temperature (2) Natural gas feed (3) Steam/gas ratio reformer feed
(4) CO2/natural gas ratio reformer feed Variable 1 is affected by fuel feed to reformer as well as the purge gas from the synthesis loop. Variable 4 strongly affects the C 0 2 / H 2ratio in the makeup, and hence inerts in the synthesis loop, and therefore, the purge flow. T h e interacting control of these variables is then achieved by simultaneous solution of the interacting control equations such as in the form
FdTt
4-F d M t
+ Fi3Au.1+ F&Vi + F 4 V i '
0 (53)
=
where A T , equals target - (current value of i), AM% equals change in the manipulated variable i (to be calculated), AUi equals last value - (current value of disturbance variable z), and AV,, AV,' equal change in interacting variable for target i (to be calculated). i in this equation is the target variable for which interacting control is necessary. A M c is obtained from simultaneous solution of the equation for i = 1, 2, 3, etc. Table IX shows typical interacting variables in a
Input Final Independent Variables 1 Reactor feed lb mol/hr 46000.000 47350.000 2 Hz/CO ratio reactor feed 5.350 5.569 3 COz/CO ratio reactor feed 0.260 0.450U 4 CHd/CO ratio reactor feed 1.500 1.849 5 % C H 3 0 H reactor feed 1.000 0.200 6 Reactor feed temp, O F 120 000 128.750 7 Reactor quench fraction bed 1 0.006 0.0 8 Reactor quench fraction bed 2 0,120 0.098 0.120 9 Reactor quench fraction bed 3 0.142 10 Reactor quench fraction bed 4 0,070 0.132 11 Quench gas temp, O F 120.000 138.750 12 Synthesis compr speed, r p m 10500.000 10790.598 13 T e m p exit reformer, O F 1450.000 1500. OOOU 14 Steam/gas reformer 3.200 2.577 15 Inlet temp reformer, O F 730.000 650. OOOL 16 COn fraction reformer feed 1.020 0.780
TAKE VIK LOW PRESSURE METHANOL O P T l M IZATION FOR MARKET L I M I T E D CONDITION Final In$ut Independent Variables 1 Reactor feed lb mol/hr 46000.000 43623.566 4.725 2 Hz/CO ratio reactor feed 5.350 3 COz/CO ratio reactor feed 0.260 0.450U 4 CH4/CO ratio reactor feed 1.500 2 . ooou 0,101 5 % CHaOH reactor feed 1 .ooo 6 Reactor feed temp, O F 120,000 145.000 7 Reactor quench fraction bed 1 0.006 0.0 8 Reactor quench fraction bed 2 0.120 0.052 9 Reactor quench fraction bed 3 0.120 0.149 10 Reactor quench fraction bed 4 0,070 0.140U 150.000U 11 Quench gas temp, O F 120.000 12 Synthesis compr speed, r p m 10500.000 10987.477 13 T e m p exit reformer, O F 1450.000 1500.000U 14 Steam/gas reformer 3.200 2.800 15 Inlet temp reformer, O F 730.000 1000. ooou 16 COZfraction reformer feed 1.020 0.772
Dependent Variables 1 Profit $/day 3132.190 5049.438 2 T e m p reactor bed 1, top 543.190 549.740 3 T e m p reactor bed 1, exit 590.450 603,920 4 T e m p reactor bed 2, exit 602.610 606.410 5 T e m p reactor bed 3, exit 594.400 594.110 6 T e m p reactor bed 4, exit 590.470 586.130 7 T e m p converter, exit 306.760 320.600 8 T e m p to cold water exchanger 56.760 70,590 25 Compressor horsepower 12129.488 12516.289U 3751.760 33 Hz makeup stream 3679.450 34 C O makeup stream 1648.720 1696.000 COz makeup stream 357.980 254.350 35 36 CH4 makeup stream 129.440 131.660 39 Pressure reactor feed 194.622 198.380 40 Pressure diff compr exitreactor 2.150 2.015* 41 Natural gas reformer feed 1425,140 1456.350 42 Heat recovery syn reactor, mil Btu 69.300 74.830 44 CHsOH production tons/day 551 .400 627.500 47 Fuel required reformer 2285.260 231 1.340 48 COzexitreformer lb mol/hr 1154.780 820.470 50 Total quench converter 0.316 0,372
Dependent Variables 1 Profit $/day 3132.190 4150.180 551,880 2 T e m p reactor bed 1, top 543.190 3 T e m p reactor bed 1, exit 590.450 611.500 4 T e m p reactor bed 2, exit 602.610 613.840 5 T e m p reactor bed 3, exit 594.400 599.370 6 T e m p reactor bed 4, exit 590.470 590.640 7 T e m p converter, exit 306.760 320.470 8 T e m p to cold water exchanger 56.760 70.470 25 Compressor horsepower 12129.488 11451.500 3231 ,460 Hz makeup stream 3679.450 33 34 C O makeup stream 1648.720 1394.810 223.800 35 C O Zmakeup stream 357.980 98.610 36 CH4 makeup stream 129.440 39 Pressure reactor feed 194,622 205.890 40 Pressure diff compr exitreactor 2.150 2.011* 1223.970 41 Natural gas reformer feed 1425.140 42 Heat recovery syn reactor, mil Btu 69.300 64.640 44 CHaOH production tons/day 551 ,400 550.570U 47 Fuel required reformer 2285.260 1768.220 48 COzexitreformer lb mol/hr 1154.780 721.940 0.340 50 Total quench converter 0.316
TABLE V I I . LOW PRESSURE OPTIMIZATION W I T H COMPRESSOR HORSEPOWER L I M I T E D TO DESIGN
I
~
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methanol plant. Certain variables in the table cannot be measured reliably and accurately for control; however, they can also be indirectly calculated from other known process conditions to provide counterchecks. Wet test meters for steam and process chromatographs for components such as methane in makeup gas are two instruments where substantial measurement errors may occur. The implementation of changes A M , must consider the dynamic characteristics of the loop associated with each M , . A simple scheme called a “poor man’s dynamic control’’ (8, 25) is to implement sniall step changes in set point over a finite period for a sensitive loop, and large step changes with overshoot in a short time interval for a sluggish loop. Figure 13 illustrates this scheme. The implementation of on-line optimization is somewhat more difficult to achieve than the off-line optimization discussed in the previous section. First, the model equations must be simplified to perform optimization calculations in a reasonable time on the control computer, such as an IBM 1800. Second, the dynamics of the various sections of the plant differ, so that for on-line optimization the problem must be decomposed into two or more parts. I n the methanol plant the synthesis loop takes a shorter time to reach stability following optimizer moves than the reformer section, since the reformer furnace has large thermal lag to respond to changes in fuel flow. The strategy of on-line optimization thus is different from off-line Optimization. In addition, this strategy must also be changed from time to time as the objective function changes. For example, when production is fixed (market limited situation) the synthesis loop optimization is carried out with makeup Hz and GO fixed to minimize compressor load, water cooler load, and to
TABLE I X .
74
tfAt
t+3at Time
tf5At
Figure 73. Method oJ set-point adjustment and response I , S e n s i h e loop; 7R, sensitioe loop resfionse; 2, slugxish loop; 2R, sluggish loop response
maximize heat recovery in boiler feed water heater. O n the other hand, for a production limited situation, the synthesis reactor feed is variable and the synthesis loop optimization calculations are performed as in off-line optimization, but with a constraint on the amount of Hz and CO available in the makeup stream from the reformer section. Finally, the on-line optimization must also consider constraints on the independent and dependent variables that are not absolute-z.e., one may violate the specified constraint at a penalty. As discussed in (9, 32) an optimization program such as C O P (control optimization
SHOWING T H E INTERACTING VARIABLES I N A METHANOL PLANT
Target Ha makeup rate
Mantpulated variables Process gas rate
CO/H2 ratio in makeup Temperature exit reformer
‘Temperature exit reformer. COz/natural gas reformer feed Fuel flow to reformer
Steam/gas reformer
Steam flow rate
C O Z / C Omakeup
CO2 flow rate to reformer as feed
CH4 in makeup
Temperature exit reformer
Methanol conversion in synthesis reactor
Process flow to synthesis reactor
Pressure to synthesis reactor
Compressor speed. Purge flOW Temperature methanol separator
hfethanol escape from methanol separator
t
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Interacting variables COs/natural gas reformer feed
Process natural gas flow rate. Process steam rate. COn flow rate to reformer Process natural gas flow rate. COa flow rate to reformer Temperature exit reformer. ( C O Z absorption efficiency if absorber in design.) Process steam Process natural gas flow rate. Process steam ratc Temperature feed to synthesis reactor. Total quench to reactor CHa/CO, C O Z / C Oratios and argon in makeup gas
Disturbance cartables
Purge gas flow to reformer fuel
CO,/CO, H 2 / C 0 ratios to synthesis rcactor feed Synthesis compressor efficiency Temperature exit synthesis reactor
APPENDIX A.
METHANOL PLANT S I M U L A T I O N RESULTS ON AN I B M 1130
Reactor Feed Components 27765.9
5189.9
1349.4
7784.8
460.0
0.0
0.0
3450.0
0.0
0.0
881.5 991 .O 1115.5 1180.0
5371.5 6305.7 7239.9 7784.8
603.3 1188.4 1634.1 1894.6
0.0 0.0 0.0 0.0
49.6 102.0 139.4 169.4
2380.5 2794.5 3208.5 3450 .O
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
Reactor Components in Various Bed Exits 18537.0 20756,8 23270.2 24727.4
3344.7 3490.0 3759.6 3924.7
Front End Streams Stream no. 1 2 3 4 5 6 7
8 9 10 11
0. 0. 3681, 3681. 3681. 27766, 24727, 0. 24727, 24076. 651.
0. 0. 1649. 1649. 1649. 5190. 3925, 0. 3925. 3821. 103.
0. 1454. 1155. 358. 358. 1349. 1180. 0. 1160. 1149. 31.
.
1337. 1337. 129. 129. 129. 7705. 7785. 0. 7785. 7580. 205.
0. 0. 0. 0. 0. 460. 1895. 1895. 0. 0. 0.
program) provides these facilities of hard and soft constraints.
Optimizer Control Communication T h e results of on-line optimization must be relayed to the supervisory controller for implementation, and the decision about when to call for on-line optimization must still be made. For the methanol plant, the synthesis loop optimization may be done at an interval of one to two hours. T h e reformer section may be optimized a t every four to eight hours, depending on how well the plant is regulated by the supervisory controller. I n either case, the results of the optimizer must be edited before providing them to the control supervisor as new target values. Editing ensures that the new value of the target is significantly larger than the existing value and also that the calculated change is not unreasonably large. Generally, the editing optimizersupervisor communication program is tailored for each individual plant. T h e sequence of target changes recommended by the optimizer is determined by the communication program, according to the control strategy for the plant, and the dynamic characteristics of each loop associated with the set point to be altered. Summary a n d Conclusions I n this paper, we have presented simulation and optimization of a large-capacity, single-train methanol plant with a centrifugal compressor employed in the synthesis loop. The simulation of the most important plant units was carried out by solving the material and energy balance equations. T h e simulation results show sensitivity of the plant units to certain critical variables. T h e optimization results for three different plant constraint conditions show that significant profit improvement can be attained in each case. Finally a method of implementing interacting supervisory control of the plant is suggested.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 169. 169. 0. 0. 0.
17. 17. 17. 17. 17. 3450. 3450, 0. 3450, 3359. 91.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
71. 71. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0.
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d.,
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