M. J. SHAH
COMPUTER CON The key t o successful process control in t h e manufacture of ethylene is control of t h e thermal cracking furnace.
In this paper t h e author
omputer control of chemical plants has gained wideC spread attention in the chemical industry, especially since the recent advances in computer hardware technology and increase in capabilities of control computers. With the new generation machines, the computers perform the routine tasks of data logging, alarm checks, operator guides, and management information with ease. Furthermore, with time-sharing programs, the computers allow sufficient time for pl'ant optimization and for supervisory control (8) of the plant while performing the routine tasks on a priority basis. These programs allow simultaneous use of the com-
describes t h e control strategy t h a t results in increased production and t h e simulation of t h e tubular heater in t h e cracking furnace. He also
E 5 FURNACE I
discusses numerical results and model simplification f o r on-line
FURNACE II
control
FURNACE 111
70
INDUSTRIAL AND ENGINEERING CHEMISTRY
ETHYLENE PRODUCTION puter by various in-puts which perform different functions for a control computer. In effect, the time sharing prcgram handles the optimum time allocation problem for the computer, based on a set of established priorities for the various functions required in process control. Thus, for example, when the computer is performing a level 6 priority calculation, it can be interrupted by level 5 priority calculation. The level 6 calculations are transferred to an auxiliary storage area, level 5 program is brought in the main working storage of the computer, and calculations can then be performed for this program. When level 5 calculations are completed, level 6 calcula-
tions are retrieved from the auxiliary storage and the computer resumes the calculations from the stage where it was interrupted. Normally, a control computer allows several priority levels (typically 15-20), so that in the above example level 5 calculations can be interrirpted by level 4, level 4 by level 3, and so on. Much of the routine scanning and checking functions required in plant control are performed with a computer clock on an interrupt basis. For details of a typical time-shared pwam (75). To perform optimization, one needs some sort of plant description in the form of an objective function (such as
=
r
s
TOWER
i
DEPROPANIZER
VOL 5 9
NO. 5
M A Y 1967
71
plant production rate) or a profit function, which is to be optimized, in terms of a number of independent plant variables. I n the past, historic plant data or plant data logged by a computer have been used to derive mathematical models representing the objective function by means of regression analysis. In some instances, the plant has been deliberately disturbed to attain the desired range of independent variables in which the model is to be fitted. Apart from the objections of plant operating personnel to the latter experimental method, the reliability of the models so obtained is subject to question in view of plant and instrument noise. T h e problem can become acute if the number of independent variables in the plant is large and one has no mathematical form of an objective function to start with. An alternative approach is to simulate the plant by Ivriting doum the process equations based on the physical and chemical operations involved in the plant. T h e simulation model achieved in this nianner is then programmed on a relatively large off-line digital computer (which as opposed to control computer is operated in a batch mode) and experiments are performed on the simulated plant to observe the effect of c h a q i n g independent variables on the product yield or profit function. Computer simulation is not only faster and cheaper than experimentation on a running plant, but it also provides systems engineers with “operating experience” on the plant via the mathematical model. In addition, the range of independent variables for simulation can be larger than that observed normally in a plant in a day-to-day operation. In this paper a simulation study on an ethylene furnace is presented with the purpose of developing an optimization and control strategy for digital computer control of an ethylene plant.
SCOPE OF INVESTIGATION To describe the plant conditions as accurately as possible, a simulation model should take into consideration all the physical and chemical processes occurring in the plant. However, this leads to an extremely complex mathematical model whose solution presents formidable difficulties even on a large scientific computer such as IBM 7090. T h e purpose of a simulation model is to predict the correct trends of plant behavior \then upsets in the plant independent variables are observed. Hence, a mathematical model which approximates a plant to the point that it predicts these trendswith reasonable accuracy is adequate for our purpose. T h e approximations are, in most cases, necessary to obtain numerical solution to the model equations within reasonable computer time. In almost all instances, even this model ill be unsuitable for digital computer control of the plant, and further simplification of the model must be carried out as a second step to provide the so-called “on-line control” relationships. T h e simplification may be achieved by examining the results predicted by the first model and then by attempting to fit these results to a simplified 72
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
relationship, either derived from a model simplification scheme (28) or via regression analysis. Ethylene Process
Ethylene is an important basic raw material in the synthetic organic chemical industry. An average sized plant produces about 150 million pounds per year, and, on this basis, together with the prevailing ethylene price, it is estimated that an additional revenue of about $1000 a day can be achieved by a mere 5% increase in production capacity. Almost all large capacity ethylene plants have several furnaces (10 to 25) Ichich operate \cith one or more different types of feedstock and are, in general, at different coking conditions. T h e products from the various furnaces are fed to a distillation unit via a single compressor. T h e operation of the various furnaces and the distillation unit is tricky owing to the various interactions and constraints leithin the plant. An optimization scheme lehich can decide the best operating conditions of each furnace, in view of the interactions and constraints of the plant, can be easily shoicn to perform better than human operators whose intuition is generally limited to no more than three variable interactions. Depending upon the stability of the plant and frequency of disturbances in the plant, a computer can not only obtain the increase in production of 5% or more with an optimization scheme, but the computer can also save money b>- reducing the maintenance cost of the plant by a superior monitoring system as well as via a maintenance cost penalty factor in the objective function used in the optimization scheme. T h e increase in revenue is often much larger than the figure quoted above, as experienced in a coniputer control of an ethylene plant using naphtha feed (4, 5). T h e ethylene process can be divided into two sections: (1) furnaces, where the steam cracking reactions take place and (2) the distillation train where separations of various products take place. In this paper we will discuss the furnace section and, in particular, the simulation of a fixed tubular heater-type furnace most common in ethylene plants. Figure 1 shows two furnace designs. T h e furnaces are divided into two zones: a preheating or convection zone and a radiation zone, \\There radiant heat is transmitted to the gases in the cracking tubes, either directly from the flame or indirectly via reradiating walls. In the convection section the hot flue gases transfer the heat mainly by convection, and within the pyrolysis tubes virtually no reaction takes place. In the radiation section, the gases in the tubes are rapidly raised to reaction temperature, and the endothermic reactions produce ethylene and other by-products. T h e exit gases are quenched to prevent further degradation of products. An average residence time in the pyrolysis tubes i s of the order of 1 to 1.5 sec. so that, for a 400-ft. long pyrolysis tube, the average gas velocity is 300 to 400 ft./sec. T h e product gases from the furnace contain ethylene, methane, acetylene, propylene, hydrogen, and aromatics,
GAS
FUEL I
I 1
1
FEED GAS
STEAM
... ..,
.. ...e
Figure 7.
STEAM FEED
Furnace heaters used in ethylene manufacture
1
1
I
0 DISTILLATION TOWERS AND PARATION UNITS
CONDENSATE
Figure 2. Simplified diagram of an ethylene process
together with tinreacted feed. T h e by-products are removed in the distillation train, and unconverted ethane and propane are recycled to the furnace. T o arrive at an optimization scheme for an ethylene plant, one should simulate the entire plant, including the furnaces, distillation train, compressor, and the refrigeration section. Sormally, the latter three sections are capacity liriiitinq and function as constraints to the plant and, hence, to the optimizer. Although use of dynamic programming (25) and the maximurn principle (19) have been illustrated in handling optiiiiization of interacting systems, these methods are unable to handle large number of variables encountered in ethylene plants. Furthermore, when constraints are placcd on dependent as well as independent variables in the plant, serious difficulties are encountered in solving the optimization problem. I n the present study, a nonlinear Optimization technique (.38)Jvhich handles constraints on both dependent and independent variables has been used. This method allo\rs the convenience of separation of the furnace from the rest of the plant, in that the rest of the plant may be treated as a set of movable constraints on the dependent variables. For example, if the optimum solution for the furnace calls for exceeding the capacity of some unit in recovery section, the optirnization can simply be repeated Ivith modified constraints on furnace dependent variables. It is not implied here that this scheme eliininates the siniulation of the distillation section entirely, for the amount of recycle ethane and propane to the furnace depends on the performance of the separation section. The optimization scheme used here, ho\ve\-er, makes it feasible not only to separate simulation of the furnaces from the rest of the plant but also alloxvs a much less elaborate simulation of the distillation section as exemplified in (,?). T h e present paper, then, will be restricted to simulation of the furnace section, with the aim of presentinq the results of optimization of the entire ethylene plant at a subsequent date.
CHEMISTRY OF PYROLYSIS
AUTHOR iM. J . Shah is a Senior Industry Analyst in the Control Systems Development Center, International Business Machines Corp., San Jose, C a y . He acknowledges the assistance of K . Johnson and N . Turner in making the computer runs, of D r . B. Dauidson in making the furnace calculations, and of W. M . Syn with the DSL/SOprogram.
T h e products emerging from a thermal cracking furnace vary from feed to feed. For ethane feed, ethylene, acetylene, methane, h) drogen, coke, some aromatics, tar, CO, and COS are ohsenTed. For propane and butane feed, one finds, in addition to the products described above, propylene and butylene. There IS a good deal of arqument in the literature (32, 34) about the exact mechanism ivhich yields these products from a given feed. For design purposes, most ivorkers (21, 23, 26) have used some serniempirical stoichiometric equations to describe various chemical reactions. These equations \Yere devised essentiall) to fit a particular set of plant data. For the purpose of simulation, the stoichiometric equations should be qeneral in nature, since the siniulation model does not exactly duplicate a particular plant. Hence, the chemical reaction equations that are therlnoVOL. 5 9
NO. 5
MAY 1967
73
dynamically feasible and give rise to the products observed at the pyrolysis tube exit for a particular feed are used here, rather than any semiempirical stoichiometric equations which may fit a particular plant. Sl'e postulate, then, the following reactions for ethane feed : (1) (2) (3) (4)
+ + + +
CzH6 * C2H4 Hz C2H6 ++ ' / 2 C Z H ~ CH4 CzH4 tf C2H2 Hz CzH2 * 2 C HS
(5) C3H8 * C3H6 HZ (6) C3H8 * C2H4 CH4 ( 7 ) C3Hs * '/2 C2H6 '/z C4Hia (8) C3Hs * '/z C3H6 '/z CzHs plus Reactions 1, 2, 3, and 4
+ +
+
'/z
CH,
For butane feed :
+ + +
(9) C4Hio t+ C2H4 C2H6 (10) C4Hio tf C3H6 CH4 (11) C4H10 tf C ~ H B HZ
plus Reactions 1, 2, 3, and 4 T o account for the small amount of C4 and CSpolymers in the products, we postulate:
(12) 2 C2Hz + C4'S (13) 2 C2H4 + C4'S (14) C3H6 '/? c6's without correctly assessing the stoichionietr) of equations, since the products from these reactions represent only a small fraction of the total yield. Values of free energy of formation, enthalpy, and heat capacity for various temperatures comprise the thermodynamic data for the reactions. Flirthermore, one needs to know the order of the reactions, the activation energy, and the Arrhenius constant. Lt'hether the reverse reaction in each case is significant, at the temperature under consideration, can be determined from the free energy data. Thermodynamic Data
T h e free energy, heat capacity, and heat of reaction data were obtained from Smith (37) and Lewis and Randall (20). T h e heat capacity values were checked
TABLE I. T H E R M O C H E M I C A L DATA FOR PYROLYSIS REACT IONS Reaction
+
CZH6 +. CZH4 HZ C2H6 +. CH4 '/2 CzH4 CZH4 C Z H Zf H Z C3Hs + CBH6 HZ C3H6 +. CzH4 CH4 C HZO+CO Ha CzHz -C 2 C Hz
+
+
+
74
+ +
+
+
Reaction
AHzQso
Ao,
34,764 8,600 43,714 31,724 19,430 33,145 51,676
7.530 2.549 11.448 7.790 3.801 2.011 8.61
AP X 103
-
9.8 - 5.857 -16.179 -12.279 -10.55 - 1.853 - 6.04
Ay X 108
2.804 2.386 5.318 4.274 4.507 0.212 0.432
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Temp., T a K.
K
800 875 900 900
5.51 0.036 33.8 26.0
+
CZH6 -t C Z H ~ H Z CZH4 -+ C Z H Zf H Z C3H8 C Z H ~ CH4 C Hz0 + CO Ha
+
For propane feed:
+ +
TABLE I I . VALUES OF E Q U I L I B R I U M CONSTANT A T T Y P I C A L PYROLYSIS TEMPERATURE
-
+
+
K
nith some more recent literature data (6) and were found to be sufficiently accurate for our purpose. Table I lists heats of reaction data for reactions with ethane and propane feed. T h e equilibrium constants for the major reactions were calculated using free energy tables of Lewis and Randall (20) Lvith a simple temperature dependence of A ( F - H o ) / T . Table I1 lists the equilibrium constant K-values for typical temperatures in the pyrolysis tube. T h e equilibrium > ields of ethylene become significant for Reaction 1 a t about 730" C. For propane and butane feed the temperature required for decomposition is lou er than for ethane feed to yield significant amounts of ethylene. The free energy data Lvere used to obtain the values of K at various temperatures with the standard relationship :
For reactions in \vhich the reverse reaction \vas significant, AFO \vas fitted to a polynomial in T. Heat capacity data for steam lvere obtained from the steam tables (76).
Kinetic Data
Most industrial plant kinetic data (7, 2, 26) are integrated over a temperature range within the pyrolysis tubes so that the values of A and E determined in this manner are not accurate. T h e data of Towel1 and Martin (36, .37), on the other hand, are nonisothermal reaction rates for Reactions 1 through 8 and 12 and 13. Their values of activation energy were selected for use here, although their Arrhenius factor values were considered to be less reliable. For the propane and butane decomposition, the values of A and E given by Steacie and Puddington ( 3 3 ) , Lichtenstein (27), and Myers and ll'atson (23), as well as the values from the tables of homogeneous reaction kinetic data of National Bureau of Standards ( 7 7 ) were used, especially for side reactions. Of the two steam-carbon reactions (15) C (16) C
+ Hg0 + 2 H2O
- ++
CO H2 COz 2 H?
only Reaction 15 is important at the pyrolysis temperatures. T h e kinetic data of Scott (27) were used for this reaction. Table I11 gives the summary of E and A values as well
Material Balance TABLE 1 1 1 .
VALUES OF Ai AND VARIOUS REACT1O N 9
E,
FOR
Ei, Reaction
Order
+ HZ CHI + CzH4 C?Hz + Hz 2 C + Hz
C2H6 + C2H4
C?Hs + '/z CzH4 CzHz + 2 CzHz + (24)s 2 C2H4 + (24)s C 4- H20 + CO H2 C3HB C3H6 H? C3Hs CHI -t CzH4 -+
-
+
a
+
+
1 1 1 1 2 2 1 1 1
Ai, Set.-' 6.04
1.6
X X X X X
10'6 1OI2
1.8 lOI3 9.7 10'0 3.2 10" 2.6 x 1013 9.255 X lo3 2.9 x 1013 3.2 x 1013
(Cal./Mole) 82.0 67.0 76.0 62.0 45.0 60.0 21.3 63.3 63.0
See (27, 22, 33,34, 36, 37).
as the order of the reactions and the form of kinetic equations for the various reactions. The Simulation Equations
T h e simulation equations consist of material, energy, and rnonientuni (pressure) balance in the cracking furnaces. I n the convection section of the fnrnace, no chemical reactions take place and only the latter t\vo equations are important. In the radiation section of the furnace, the energy balance ma)- be lvritten in t\vo different ways. If information on the rate of heat input Q is available as a function of the length of pyrolysis tubes, the energy balance is simply a function, Q (2). On the other hand, the tube skin temperature profile measurement or the measurement of furnace fuel rate, air rate, stack gas temperature, and radiation characteristics of the furnace may be specified. .4 tube wall temperature profile polynomial can be calculated from these furnace parameters. In any case, the energy balance for the gases in the p)-rolysis tubes is dependent on tube skin temperature. T h e tube skin temperature profile can be determined separately by using the furnace variable values and kno\ving the furnace design characteristics. T h e solution of the equations for the various balances in the pyrolysis tubes may be carried out subsequently. T h e tube skin temperature profile calculations are bypassed if tube skin temperature measurements are specified. T h e changes in furnace parameters (such as fuel flohv, secondary air flow, flue gas recirculation, etc.) are reflected as changes in the values of the coefficients in tube skin temperature profile polynomial. T h e effect of changes in furnace variables on the tube skin temperature is much slower than the effect of feed changes in the pyrolysis tube. Thus, \.\.hen dynamic control of the cracking unit is considered, this division of calculation is convenient. For example, when the furnace variables are time variant, the pyrolysis reaction mixture may be assumed to respond to the variation in tube wall temperatures without any additional time delay.
For any reaction, i, the rate of reaction may be described by rt
=
'4, exp(-E,/RT)
[Cjm- CKPCLQ/K]
(2)
C, is the concentration of the j t h reactant, CK and CL are products, and m,p , and q are the powers for stoichiometer balance. A , and E , are, respectively, the Arrhenius parameter and activation energy for the reaction. K is the equilibrium constant. It is important to differentiate between the Lth reaction and j t h reactant, for as the Reactions 1 throiigh 1 5 indicate, the reactions are coupled, and each reactant is involved in several reactions. T h e material balance for a single component may then be described by
where the summation is taken over the reactions in which the j t h component takes part and ail gives the stoichiometric ratio for the reaction. T h e material balance for ethylene, for example, when mixed ethane-propane feed is used in the furnace, is dn - ethylene = dr
Tr2
(rl
+
'/z r 2
- r3
+
r6
-2
113)
(4)
where rl is expressed as rate of depletion of ethane which is r l = - _dnethane _ _ _ = Alexp - (E1/RT)
dv
T h e concentration is expressed using the ideal gas law for the low pressures of cracking operation. T h e pattern of rate expressions for the rest of the reactions is clear from the above equations. Table I V gives the material balance for each species and rate of reaction expression for each reaction. T h e rates r j , r7, rs, and r14 'are expressed in terms of partial pressures rather than concentration, as in (23), from which we obtained the Ai,and E , values for the Reactions 5, 7, 8, and 14. Reactions 5 and 7 are considered reversible. T h e expressions for equilibrium constant K for these two reactions are also described in Table IV. T h e expression for equilibrium constant for the first reaction was obtained from free energy data of Lewis and Randall (20). I t is noted in Table I V that we have lumped the C 4 and CS components simply as C4's and C6 rather than considering individual reactions giving aromatic components. If one desires, the secondary and tertiary reactions postulated in (1), ( 2 ) , and (23) may be taken into account. For the present stud) these reactions only complicate the material balance without contributing significantly to our final objective. T h e complexity of the material balance expressions VOL. 5 9
NO. 5
MAY 1967
75
T A B L E IV. dnrnethane r2
=
=
[-
7
EXPRESSIONS
.
F O R K I N E T I C RATES A N D M A T E R I A L BALANCES
A]
Aze-EzlRT
and
is reduced to some extent if the feed is either pure ethane or pure propane. T h e equations then reduce to those given in our previous publication (29) and will not be repeated here. I n the rate expressions 4 through 15 as well as in the equilibrium relations 17 and 18, both temperature T and pressure P are functions of 2, the tube length. T h e function T ( z ) must be obtained from the energy balance equation, whereas P ( z ) will be expressed simply as a polynomial in z. Energy Balance
T h e energy balance of a section of the pyrolysis tube is:
measurement the heat input function to the pyrolysis tubes is given. Q ( z ) is generally known as multiplestep function, and hence a certain amount of smoothing of the function should be carried out. T h e first summation in the energy balance Equation 31 is over all the reactants and products present in the reaction mixture and the second summation is over all reactions. Preliminary calculations showed, however, that when the feed contains primarily ethane, propane, and steam, with only a small per cent of products from recycling, the convection term can be closely approximated by taking the first summation over ethane, propane, and steam with the stipulation that in the summation n e t h a n e = n e t h a n e (2 =
-
i
AH,
dn
+2
T T
h ( T ) (T, - T )
(31)
where the second term on the right-hand side of the equation is for the case when T,, the tube skin temperature is specified a priori as T,(z). This term will be replaced by Q ( z ) , if instead of tube skin temperature 76
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
o), n p r o p a n e
-
npronane (2
=
O ) , and so forth'
T h e second summation is taken over Reactions 1, 2, 5, 6, and 8, since the other reactions bring relatively small changes in material balance. T,(z) ma)- be obtained from furnace calculations which are described in the next section. If measurements of tube skin temperatures are available from plant data, they may be used to specify T w ( z ) . T h e heat transfer coefficient h( T ) for mixed ethane-propane feed is calculated by using the following expression :
mine b and c for Equation 33, it is necessary to solve Equation 34 to give Twand a value of T , at one more point along the pyrolysis tube. If the tube skin temperature at the pyrolysis section exit can be measured, information necessary for evaluating 6 and c is completed.
h = 0.023 (0.1438 d-O*’) X
n e t h a n e Cp*ethane
+
nsteam ~ p * s t e a m %thane
4
( n e t h a n e cp*ethane
+
+
npropane
f
npropane Cp*propane
nstesm
n s t e a m c p *steam) %thane
+npropane
1 . + + I +
n p r o p a n e Cp*propane
+
-0.4
10
&team
PRESSURE IN THE PYROLYSIS TUBES (32)
where I* and cp* are calculated a t the mean temperature,
EVALUATION OF T&) FROM CALCULATIONS
FURNACE
If neither measurements of tube skin temperature nor heat input Q ( t ) along the tube length is available, one must utilize furnace variables to estimate the tube skin temperature profile. An examination of the furnace design shown in Figure 1 indicates that the value of this temperature will depend on the furnace geometry, location of tubes and burners, air rate, excess air, flue gas recirculation, and many other factors. The method described here will take a few of these factors into account. I t is not possible for the purpose of this work to use sophisticated techniques of Hottel and Cohen (72). We will follow the method of Lobo and Evans (22) to calculate a mean temperature of the pyrolysis tube. T h e tube wall temperature will be expressed as a polynomial in tube length, and the mean temperature will be an integrated mean. Thus, a change in some of the controllable furnace conditions will bring about a change in the coefficients of the polynomial for T,(z). Let the mean temperature be expressed by
The reactions are carried out under low pressures (30-40 p.s.i.g.) because of the adverse effect of pressure on reaction equilibrium. Since the main Reactions 1, 2, 5 , and 8 increase the number of moles, pressure is increased during the progress of the pyrolysis. The pressure drop, then, will not be linear along the tube length as a simple fanning friction-type relation would indicate. T h e presence of steam, however, will reduce the nonlinear effect of the reaction to some extent, so that we simply express the pressure as a quadratic in z:
P(z)
=
Po
+ Bit + Bzz2
The data of Rase and Fair (24) indicate that this is a good approximation for ethane cracking. Equations 4 through 34 constitute our simulation model for cracking of mixed ethane-propane feed. If butane is also a significant part of feed, additional equations to take Reactions 9 to 11 must also be included in the material balance. The boundary conditions for the energy- balance equation require that the gas and tube skin temperature at the entrance of the radiation T ( 0 ) and ( Tw)z=a be specified. T(0) section-Le., may be obtained by energy balance in the convection section of the furnace:
WCP,*(T, - T,o)(l
+ o)(l - Y’) +
=
T ( o ) - Tc(0) (nethaneCp*ethane
+ bL7 +; L2 c
a
L
J
(33)
where, a is the tube skin temperature at the entrance of the radiation section (t = 0) . T , is also given by (22) 0.173 AAC+{[(’>I- 100
(&>I+
40.5 ( T , - Tw)
100
X
= equivalent cold plane (tube wall) surface = effective cold plane surface/actual tube wall surface;
4
=
W = H,
x
= = =
u
=
Y
&,, = TA = T, =
calculated from tube spacing and other furnace d a t a exchange factor, a function of gas emissivity, beam length and furnace geometry rate of fuel supplied calorific value of fuel per cent heat lost by furnace walls fraction of the flue gas recirculated amount of air/moment of fuel (dimensionless) flue gas specific heat at mean temperature ambient temperature of air fed to the furnace mean flue gas temperature
+
T h e calculation of A, A,, and from the furnace design is illustrated by Lobo and Evans (22). Thus, to deter-
npropanecp*propane
+
nsteamCp*steam) (36) where T,o and T,(O) are the flue gas temperature and feed temperature at the convection section inlet, c p , is the specific heat of flue gas at mean temperature ( T , -k T,o)/2 and similarly cp*ethane and so forth are evaluated at [T(O) Tc(0)]/2. If the value of T,(O) cannot be measured, but T,c(0)-i.e., the tube skin temperature at the inlet of the convection section is measured then a heat balance around the entire convec: tion section gives T,(O)
+
M’CpI*(T, - T,o)(l A,
(35)
+ u)(l
-
d L C h c ( T , ’ - T,’)
Y’)
=
+ /z,(T,’~- Tm’4) ( 3 7 )
where the temperatures T,’ and Tu’are geometric mean temperatures To’ = ( T , . and T,’ = [ T,(O). TZc X (0)]1’2. In Equations 36 and 37, Y’denotes the per cent heat lost or gained through the refractory wall (depending upon the type of furnace) in the convection section. Equations 4 through 37 comprise a set of coupled nonlinear ordinary differential equations which cannot be solved analytically. A computer program was Irritten for their numerical solution using a diyital sirnulation language DSL/90 (35). This language is quite VOL. 5 9
NO. 5
M A Y 1967
77
simple and requires little prior' knowledge of proqramming. Two integration techniques, four-order RungeKuttaand Milne, were found to be satisfactory for solution of our differential equations. A detailed description of the language and the integration routines used may be found (35). Coke Formation
Figure 3.
Temperature projiles f o r ethane cracking
In practice, the rate of coke deposition on the interior of the tube wall is important, as periodically the furnace requires a shutdowm to clean the pyrolysis tubes. The rate of coke biiildup with passage of time is directly proportional to the rate of excess coke (decomposition of hydrocarbons less steam-coke reaction) formation in the pyrolysis tube. T h e effect of the coke buildup on heat transfer coefficients can be evaluated from the rate of coke formation using relationships given by Lichtenstein (27). Useful tube life for pyrolysis can be related by the coke formation via heat transfer relationships.
DISCUSSION OF RESULTS
Figure 4. Temperature projles f o r ethane cracking
To obtain significant results from a simulation study, comparison of the computer prediction with actual plant observations is a first essential step. It must also be realized that an exact matching of the two is difficult i n view of the errors in plant measurements and the uncertainties in the values of the kinetic parameters, A I and E,. It was possible to obtain data from two furnaces: one cracked feed containing mostly ethane; the other \\as supplied with gas primarily containing propane. I n both cases, adjustment in the values of '4d for the various reactions gave a vary satisfactory agreement of computer predictions of composition and temperatiire a t the pyrolysis tube exit, with the average measured \ alues. Figures 3 and 4 show the gas and tube wall temperature profiles in the pyrolysis tubes for ethane feed. Fieures 5, 6, and 7 show the profiles of ethylene and methane formation and the formation of C? compounds. Th-se figures indicate that in the first 20% of the radiation section, the feed gas is merely heated to the reaction
Figure 5. Ethane and ethylene conversion
Figure 6. Methane and acetylene formation 78
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Figure 7. Formation of C4 hyh'ocarbons
temperature, and no significant reactions occur. T h e gas temperature profile changes the slope further along the tube and becomes flat, and the major reactions begin. Toward the end of the tube the ethylene formation appears to reach equilibrium. Figures 6 and 7 indicate, however, that the side reactions proceed much faster in the last section of the pyrolysis tube, and degradation products in cracking increase almost exponentially. Similarly, for feed containing about 90% propane and 595 ethane, Figures 8 through 10 indicate the temperature and composition profiles. The results follow the same trend as in ethane cracking. For both furnaces, the tube wall teinperaturc measurements were available. In one instance the measurements fit very well to a polynomial. Once the acciiracy of the simulation model is established by reasonable agreement of model results with plant measurements, the next step of interest is the effect of various input parameters on the furnace output. T h e major purpose of our investigation is to obtain the output-input relationships. A series of runs 1% as made with our program, varyine; inlet pressure, feed-to-steam ratio, feed rate, inlet gas temperature, and tube wall temperature profile. Table IV shows the values of these parameters and the resulting ethylene yields and gas temperature at tube exit. All variations were made from the base case (Run 1) which represented the normal furnace operating conditions. These runs are shown for a furnace with feed containine about 95% ethane. the rest being propane, ethylene, and propylene. The results of propane feed were found to be very similar, $0 that in what follo\vs Ive will discuss only the results of simulation runs for feed containing primarily ethane.
Figure 8.
Temperature projles in tube.r for propane feed
Figuie 9. Ethane and propane concmtratzon proJles f m propane f e d
Pressure
Runs I to 4 show the effect of variation in inlet pressure on ethane conversion and per cent ethylene yield.
Yotice that column 11 of Table V shoivs per cent ethane decomposed, M hereas column 1 2 represents per cent of ethane con\ erted to eth\-lene. Column 1 3 subsequently shou s per cent yield of ethylene from ethane decomposed, the rest of the ethane coni.ersion > ielding b>-products or further degradation products of cracking. Table L' indicates that the increase in inlet pressure
Figure 70. Ethylene and propylene formation with p7opane feed VOL. 5 9
NO. 5
MAY 1967
79
leads to increased decomposition of ethane ; however, the per cent of ethane yielding ethylene actually decreases, owing to a higher increase in the rate of side reactions than in the conversion rate of ethane to ethylene. T h e coke buildup in the pyrolysis tubes leads to increased pressure drop and, in practice, the inlet pressure is increased to maintain a constant pressure of gases at the tube exit. Our results indicate that this practice will lead to an even faster rate of coke and tar deposition. Steam to Feed Ratio
Runs 1, 5, 6, a n d 7 indicate the effect of variation in steam/feed ratio on ethylene yield. There is a slight increase in ethane decomposition as steam is increased u p to twofold. T h e ethylene yield also increases slightly as a result of suppression of side reactions. However, we have assumed here that the tube wall temperature is maintained constant even with increased total gas. In practice, increase in steam will bring about a decrease in tube wall temperature unless the head load is in-
creased as well. As we shall see later, the tube wall temperature has a much stronger effect on ethylene yield than the effect of steam, so that it is preferable to keep steam to a minimum-essential for reduction of coke formation and suppression of polymerization reactions.
Feed Rate
T h e effect of variation in feed rates, with fixed steam/ feed ratio, is shown in Runs 1, 8, 9, and 10 and in Figure 11. Conversion of ethane drops with increased feed; however, the total moles of ethylene produced increases and, furthermore, the per cent of ethane yielding ethylene increases. T h e latter is a result of suppression of side reactions due to a decrease in gas residence time in the tube with increased feed rate. Figure 11 shows that the ethylene production increases almost linearly with feed rate. \Ve have assumed, again, that the tube wall temperature is held constant by adjusting the furnace heat load. In practice, the tube wall temperature will drop
TABLE V.
Run No.
1"
b
Steam (Moles/Sec.)
Inlet Gas Temp. To [ ' K.)
7.8994 7.8994 7.8994 7.8994
0.363 0.363 0.363 0.363
818.3 818.3 818.3 818.3
Tube W a l l Temp. at Inlet Two, 'K.
Corr.
Factor Inlet Press., for h POAtm. hwrr
Outlet Ethane, Moles/Sec.
Outlet Ethylene, Moles /Sec
Ethane Decomposed, c*
iC
957.2 957.2
4.158 3.3 5.0 5.6
1. o 1. o 1. o 1.0
9.88 10.07 9.71 9.503
10.114 IO. 31 9.92 9.396
52.5 51.6 53.3 54.3
3 4
21.554 21.554 21.554 21.554
5 6 7
21.554 21.554 21,554
6.0 13.0 15.0
0.279 0.604 0.697
818.3 818.3 818.3
957.2 957.2 957.2
4.158 4.158 4.158
1 .o 1. o 1. o
9.923 9.75 9.696
10.005 10.38 10.47
52.3 53.25 53.4
8 9 10
17.23 25.85 32.25
6.32 9.479 11.83
0.363 0.363 0.363
818.3 818.3 818.3
957.2 957.2 957.2
4.158 4.158 4.158
1. o 1. o 1.0
7.34 12.515 16.62
8.404 11.702 13.9
55.9 49.8 46.7
11
21.554 21.554
7.8994 7.8994
0.363 0.363
818.3 818.3
957.2 957.2
4.158 4.158
0.9
12
1.1
10.583 9.244
9.555 10.598
49.1 55.6
13 14 15
21.554 21.554 21.554
7.8994 7.8994 7.8994
0.363 0.363 0.363
600.0 700.0 900.0
957. 2 957.2 957.2
4.158 4.158 4.158
1.0 1 .o 1. o
10.204 10.09 9.624
9.891 9.972 10.282
51 . O 51.5 53.7
16 17 18 19
21.554 21.554 21.554 21.554
7.8994 7.8994 7.8994 7.8994
0.363 0.363 0.363 0.363
818.33 818.33 818.33 818.33
900.0 942.0 972.0 1000.0
4.158 4.158 4.158 4.158
1. o 1. o 1. o 1. o
14.38 11.113 8.65 6.212
6.254 9.13 11.01 12.404
30.9 46.6 58.5 70.0
2
5
Total Feed (Moles/Sec.)
Ratio Steam ( t o Feed, y )
COMPUTER RUNS FOR VARIOUS VALUES OF
957.2 957.2
R u n No. 7 was the base at which the furnace was operating. All disturbances are variations from this point. For these runs, the wall temperature for the entire pyrolysis section was reduced or increased uniformly.
80
INDUSTRIAL A N D ENGINEERING CHEMISTRY
owing to design limitations of the furnace and, hence, the production curve will show a decrease beyond a certain feed rate. A plant profit function which normally includes separation and recycling cost together with capacity limitation on distillation train will thus drop off beyond a certain feed rate. Feed Temperature at Tube Inlet
Runs 1, 13, 14, and 15 show the effect of variation in feed temperature T ( 0 ) on mole fraction ethylene yield and T ( L ) . T h e ethylene yield and T ( L ) are virtually unaffected by a decrease in T ( 0 ) provided the wall temperature T,(t) is maintained at the same level by adjusting furnace heat load. More interesting are the resultant temperature profiles from variation of T ( 0 ) shown in Figure 12. It is seen that the temperature profile within the pyrolysis tube is virtually unaltered from 2000 cm. to 12,000cm., even when T ( 0 ) is varied by as much as 300" C. Thus, for the first 2000 cm., the feed stream is merely heated to the reaction temperature by convection, and there is little
PARAMETERS Yield of Ethane Con- Ethylene verted to from Ethane Ethylene. Decomposed, ~.
Figure 7 7 . Temp. at Tube Exit: T L ( K.1
70
9%
47.6 48.7 45.7 44.2
90.8 94.3 85.7 81.5
1034.0 1025.0 1043.0 1048.0
Change in P a
47.2 48.9 49.3
90.3 91.8 92.3
1034.0 1032.9 1032.7
Change in steam/feed ratio
49.6 45.9 43.7
88.7 92.2 93.5
1038.8 1030.0 1026.1
Change in feed/rate
Effect of feed rate variation on ethane c r a c k i q
Remarks
45.0 50.0
91.7 89.9
1025.6 1041.1
Change i n heat-transfer coefficient
46.6 47.1 48.6
91.4 91.5 90.5
1030.6 1031.7 1036.0
Change in T o
29.2 42.9 52.5 58.8
94.5 92.0 89.7 84.0
988.4 1020.1 1048.0 1078.7
Change in tube
Figure 12. values
Temperature projles in cracking tubes f o r various To
Figure 73. values
Temperature profiles in cracking tubes for carious
wall temperature
VOL. 5 9
NO. 5 M A Y 1967
Tu 81
A large ethylene plant may have up t o 100 controllable variables and a similar number of dependent variables
reaction in this zone. After the point z = 2000 CIII., the process is heat-transfer controlled, and regardless of the value of T(O), the ethylene production remains unaffected as long as the tube wall temperature in the zone 0 < Z < 2000 cm. is maintained constant. Heat Transfer Coefficient
I t is known that the buildup of tar and coke on the interior of the pyrolysis tubes is detrimental to furnace operation. Results in the previous section indicate that heat transfer rate controls the ethylene formation in the reaction zone. Therefore, three runs were made to observe the effect of changes in the value of /z on ethylene yield. R u n 11 shows that a 10% decrease in heat transfer rate reduces the product temperature at tube exit T ( L ) only about 1yc; however. the ethane decomposition decreases by 3 to 47,. Similarly, a 10% increase in heat transfer rate increases the ethane conversion by 3.170. T h e lower heat transfer coefficient gives a slightly better percentage of ethylene yield. Thus, the effect of even a small reduction in heat transfer either by tar or coke or scale buildup is quite severe on the furnace performance. As the heat transfer coefficient is not easily measured, it can hardly be counted upon as a control variable. In practice, the decrease in h will be manifested in an increase in tube skin temperature and will give rise to the so-called “hot spots.” The need for tube skin temperature measurements because of their critical role in the optimum performance of a cracking furnace is therefore evident. Tube Wall Temperature
If the tubte skin temperature is accessible, then it is of interest to [observe the effect of change in T , on the pyrolysis reaction. T h e effect of change in tube wall temperature, profile in the first 20% of the tube length is shown on the gas temperature profile in Figure 13. I n spite of the rather severe reduction in tube temperatures for the initial pyrol)-sis tube section, the gas temperature profile and, hence, ethane decomposition is not affected seriously. O n the other hand, as Runs 1, 16, 17, 18, and 19 indicate, if the tube temperature is increased or decreased uniformly over the entire pyrolysis section, both per cent ethane conversion and ethylene yield are aKected drastically by a change of 100” C. 82
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
LVith an increase in T,, although ethane conversion increases, per cent yield of ethylene decreases because of side reactions. T h e profit function which includes a penalty for ncreased separation costs with increased byproducts, and for increased maintenance cost due to faster tar buildup, will therefore indicate an optimum tube wall temperature profile. An examination of these results, together with the observation that a change in inlet gas temperature has little effect on product yield (Figure 12), suggests that the most t xnperature sensitive part of the ethylene furnace is the last section of pyrolysis tubes. Figure 3 indicates that not only is the gas temperature profile fairly flat in this region (1500 to 9000 cm.), but the effects of small changes in T , will show up as variations in ethylene production and in by-product concentrations. Thus, a good control strategy for the ethylene furnace will start with monitoring tube wall temperature in the last half of the pyrolysis tubes in the radiation zone, as well as measurement of exit gas temperature. As indicated in the section on ethylene process, the furnace simulation together with the distillation train calculations are used next in an optimization scheme for optimum operation of the ethylene plant by means of a digital control computer. Since the optimization will handle the separation section variables merely as dependent constrained variables, the distillation section may be simulated by means of simplified relationships of Fenske-Underwood (together with Gilliland correlations) Lvith provision for correction of minimum reflux values during the operation as in ( 5 ) . T h e furnace Optimization will presume that the separation section can handle the optimum output from the furnaces within given constraints determined by compressor and distillation column capacities. These constraints may vary depending on the frequency of plant disturbances, so that the points of optimum furnace operation may vary u i t h the constraints. In general, a large ethylene plant (200-500 million pounds per year) may have as many as 50 to 100 independent (controllable) variables and about as many or more dependent variables. A recently available program (73, 38) uses a nonlinear optimization technique for handling such a large system of variables. Basically, it linearizes sectionally the nonlinear relationships between independent and dependent
t
variables, uses linear programming to find an optimum under specified constraints, corrects for nonlinearity and moves in the optimum gradient direction to the next section, In an operating plant not all the constraints are rigid. Furthermore, unless the plant is continually disturbed b)- large step changes, one finds the plant to be always near optimum. T h e technique described in (73) takes these plant operational considerations into account and provides a computer program for the so-called “on-line control optimization.” Once the optimum values of the independent or control variables are calculated, the implementation of the changes in the control variables must consider the dynamics of the control loops which are affected by these variables. LVhile certain sophisticated techniques for determining an optimum method of changing the control variables have been published (78), in practice, owing to insufficient information available on dynamic interactions of the various control loops, one chooses a conservative “poor man’s dvnamic control’’ approach suggested in a recent publication (8). Essentially, this method specifies the total chanqe in the control variable, a given time interval during which the total change is to be made and the number of sections of this time interval. At each time interval section, a portion of the change is carried out. For example, in the first-quarter time interval, one half of the step change is made, in the next quarter a three-quarters additional step change is carried out, thus overshooting the total change; in the third quarter a one-half negative step change is made, and in the final quarter a one-quarter positive step change is made to complete the total step change. Any combination of the step changes mal- be used dependent upon the dynamic characteristics and stability of the control loop associated with the controlled variable. It is not the intent of this paper to present a case study of Optimization of the entire ethylene plant. SVe will show in a subsequent publication that there are many steps involved between simulation of furnace and implementation of an on-line optimizer on an ethylene plant. However, there are certain interesting deductions to be made from the simulation stud)-. The results presented here are carried out by perturbing one independent variable while holding all other independent variables constant. In the optimization technique, the independent variables are perturbed siniultaneously so that the total effect of conflicting variables on the overall profit function can be seen more clearly. T h e optimization program calls for solution of the furnace simulation equations, many times a t calculation of each gradient direction matrix. In fact, the simulation equations must be solved (2 n 1) times for each sectional optimization loop, n being the number of independent variables. O n an average, offline optimization may take about 5 to 6 such loops, so that even Mith large computers the computation time begins to get appreciable. I t is clear then, that for on-line optimization with control computers in real time, the computation time must be reduced considerably by simplification of simulation models.
+
MODEL SlMPLl FICATION O n e of the simplest schemes for reducing the model complexity is to perform a number of simulation runs varying the independent variables of the furnace in the operating range and observing their effect on the dependent variables, such as ethylene yield, exit temperature, and coking rate, LYhile the simulation runs shown here are for variation of a single variable at a time, holding the rest of them constant, more realistic runs can be made through the optimizer program itself, which via the gradient matrix gives a direct relationship between the objective or profit function and the furnace independent variables. T h e matrix values determined at various points of plant operation can be directly supplied to a multiple step-wise regression program, and a n algebraic relationship between the profit function and independent variables may be obtained for use in on-line optimizer programs. If the regressed model approximates the simulation model well, then in the operating range it is as good as the more complex simulation model. T h e trouble with this method is that if a linear approximation is not satisfactory one must a priori know the form of nonlinearities. The form of nonlinear equations can be arrived at by careful perusal of our simulation results. O u r results suggest that the ethylene yield (total) increases almost linearly with the feed rate and is affected only slightly by changes in steam rate, and feed temperature (to radiation zone). T h e ethylene yield shows an optimum with respect to pressure variation and is quite sensitive to the heat transfer coefficient and to T,. It is possible to construct a simple model based on these results of the form: Ethylene yield = CI
+ C2F + C ~ P -O C3P02 -
Since the tar formation may also be a function of steam rate, the profit function may also include a linear function of steam rate and cost of separation of the byproducts which are increased by increased PO and T,. T h e effect of furnace conditions such as burner fuel rate, ambient temperature, and flue gas recirculation will be manifested inf(T,) o r f [ Q ( z ) ]as the case may be. T h e regressed model will be useful for short-period (every 30 to 40 min.) optimization with an on-line control computer, presuming that during this period the plant does not drift too far from the optimum operation point. For a long-period (every 1 2 to 24 hr.) optimization, however, a more accurate representation of the furnace than Equation 38 is needed. The simulation results point out that (1) the tube wall temperature in the latter half of the radiation section and the heat transfer coefficient, h, are important controlling parameters, and (2) as far as the enerzy balance is concerned, the reactions producing ethylene and propylene are the only ones contributing significantly to the heat of reaction term. For the purpose vot.
59
NO. 5
MAY
1967
a3
would be very nearly complete and xl(L) = 1, so that Equation 44 simplifies to :
of this discussion, let us assume that the feed is either pure ethane or pure propane. Let F be the moles of feed and X I , x2 be the fractional conversions due to the two major reactions for ethane. T h e first reaction produces ethylene, whereas the second reaction yields by-products such as acetylene. T h e rate equation then simplifies (28, 29) to :
1 - xl(L)h+mE exp
[ -(1
+)z;
(45)
with the result
4 Fdxl F(l - X I - ~ 2 ) P ( z ) A1 exp ( -E l / R T ) ad2 dz F(l XI) NsR T ( z )
+ +
(39) and the energy balance reduces to:
(F
T h e asymptotic condition h ---t 0 signifies adiabatic conditions, and hence the energy balance Equation 31 reduces to
T dx + L2rs)cprn(T)d- F A H i ( T ) 4dz dz 1 -
ndh[T,(z) - T ] = 0
(40)
(47)
where cum is an average specific heat. Similar equations can also be written for propane feed. Note that there are two asymptotic conditions possible in the furnace, viz., (1) h + 0, heat transfer rate controlling, and (2) h +. a ,reaction-rate controlling. For the second case, to have finite energy transfer, T , must equal T . If T,(z) is specified as a polynomial in z, and if we define 4 F ~-
xd2L
E1/RTUo= M ;
- Z;
-
@; P X
Since under adiabatic conditions only small ethane conversion will take place, little change in temperature will occur, and hence both cum and AH1 will not vary significantly. Thus we get
where z1 = 12'
=
(1
+ y) AH1 CU?R
-
Equation 43 then reduces to and
exp [ E I I R ( T o- x1/:V)]dxl = where T,, a n d Po are a reference temperature a n d pressure, respectively. Equation 41 reduces to
(1
+ Y) In [1 -
1
d2 A1 [exp ( - E l / R T o ) 4 FTo
3
- { (2
+y +
x1)
1-x1-x2
dxl =
1
'
0
wl)exp [-M/OW(zl)]dzl
~
(43)
Since the P(z1) and O , ( Z ~ ) are polynomials in 21, the right-hand side of Equation 43 is independent of xl. As for the left side, x z is so small that it can either be neglected, or, as our simulation results suggest, it can be taken as a linear function of X I . For x2 = 0, it follows that
84
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
+
1
=
which, after expanding the logarithmic term, gives
+ K1 2 L2 + 5 L 3 )
(POL
(51)
3
As we indicated in a previous publication (28), the value of x l ( L ) for 0 < /i < co can be close1)- approximated from Equations 46 and 51 with an interpolation formula Xl(L) =
where is the value of the integral on the right-hand side in Equation 43. Since h is large, the reaction
P(z)dz ( 4 9 )
Note, however, that x 1 N 0 when h -+ 0 so that one may assume T = To and hence no change in reaction rate constant along the pyrolysis tube. Equation 49 will then further simplify to:
and hence
s,
SOi.
{
[Xl(L)lL
+ ~xl(L)lL,ps
(j2)
where s is of the order of 2 / 3 to 3 / 4 . We note that since [x,(L)]~,,is quite small, it is more or less a correction
moles of a component in the reaction mixture
n
=
P
= pressure
Q R
=
heat input to pyrolysis tube
= gas constant
rate of reaction temperature of gas in pyrolysis tubes TA ambient air temperature T, tube skin temperature T, flue gas temperature V = reactor volume W = rate of fuel to furnace x = mole fraction conversion = per cent lost by furnace walls Y z = variable tube length
r T
= = = = =
Greek Letters A a , AB, A? = constraints in heat capacity equations y = defined by Equation 41 =
Figure 14. G a s temperature projles i n cracking tubes for varying tube wall temperatures
X = effective cord plane surface/actual tube wall surface, Equa-
0
term in Equation 52. For plant optimization one only needs to know the functional dependence of x l ( L ) on Tu,, T o , F, Y,and so forth which is described well by the above expression. A comparison for XI dependence on z between Equation 52 and the simulation result in Figure 14 indicated that at least the functional dependence is predicted reasonably well bb- the expression for h + a . I t has not been possible to examine the accuracy of Equation 52 in predicting the ethylene yield so far. Qualitatively, however, the interpolation formula shows trends of x1 similar to those observed in the simulation results. Simplified expressions for x 2 , x3, . . . etc., can be The derived in the same manner as in the case of XI. resultant expressions may be further simplified by expanding the logarithmic and exponential functions, since the xi’s for i > 1 are quite small. For low h values only the first reaction is significant. Thus, again a profit function for a particular plant can be constructed from the simplified models. Needless to say, some of the coefficients in Equation 52 will be revised based on plant observations. Equation 52, however, is more realistic than a simple linear repression model or Equation 38, since it is based on physical phenomena.
NOMENCLATURE
A (I
= =
B I , Bz b =
C cp d
E F
= =
= =
=
AF = AH =
H,
=
h
=
I K L A4
= = = = =
rn
Arrhenius constant constant in Equation 33 = constants in Equation 35 constant in Equation 33 concentration of a component heat capacity tube diameter activation energy total moles feed to furnace free energy of formation heat of reaction calorific value of fuel heat transfer coefficient as defined by Equation 41 equilibrium constant pyrolysis tube length defined by Equation 41 order of reaction as defined in Equation 12
viscosity
$ = radiation heat exchange factor, Equation 34 @J = defined by Equation 41
= u = =
x
tion 34 defined by Equation 41 amount of air/amount of fuel in the furnace fraction of flue gas recirculated
REF E RENCES (1) Andrews, A. J., Pollock. I,. W.,IND.ENC. CHEM.51, 125 (1959). (2) Buell, C. K., Weber, L. J., Petroi. Process 5 , 266 (I 950). (3) Davidson, B., Shah, M. J., “Simulation of the Catalytic Cracking Process for Styrene Production,’’ IB,ZI J . Res. Deueiop. (July 1965). (4) Delleugenio, B., Fossati, A , , Slauri, M., Piovan, M., Zechini, F., “Optimization of Furnace Operation in a Cracking Plant for Production of Olefins by hleans of a n Online Dieital Computer,’’ presented at the Automation and Instrumentation Seminar, Milan, Nov. 19-25, 1966. (5) Dimpel, F., “Experience in Online Control of a n Ethylene Plant with a Digital Computer,” Ibid. (6) Din, F., “Thermodynamic Functions of Gases,” Vols. 1-3, Butterworth’s, London (1962). ( 7 ) Eisenhardt, R . D., Williams,T. J., Contr. Eng. 7 ( I l ) , 103 (1960). (8) Ewing, R . W., Glahn, G. L., Larkins, R. P., Zartman, W.N., Chem. Eng. Progr. 63 (2), 104 (1967). (9) Fedor, W.S., Chem. Eng. Ntwr 4 0 (37), 149 (1962). (10) Feigin, E . A., Platonov, V. M., Mukhina, T. N., Girsanov, I. V., Int. Chem. Eng. 4, 245 (1964). (11) “Homogeneous Reaction Kinetics,” Xatl. Bur. of Stand., publication 571,411 ( M a y 1960). (12) Hottel, H . C., Cohen, E. (13) I B M 1800 Control O p t ication Description, Form So. H20-0208-1,I B M Corp., IVhite Plains, N. Y . (14) I B M Program Information Dept., “A Process Optimization Program for Nonlinear Optimization,” program No. IBM-0021-7090-1965, Hawthorne, N. Y . (15) I B h i 1800 Time-sharing Executive System, Form S o . C26-5990-02, I B M Corp., White Plains, S. Y .
(16) Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley (1955). (17) Kirk, R. E., O t h m e r , D. F., “Encyclopedia of Chem. Tech.,” Intersci. Encycl. Inc., Vol. 5 (1950). (18) Latour, P. R., Koppel, L. B., Coughanowr, P. R., “ T i m e Optimal Control of Chemical Processes for Set Point Changes,” 60th A.1.Ch.E. National hleetinq, Atlantic City, September 1966. (19) Lee, E. S., I N D .ENC. CHEM.5 5 ( 8 ) , 30 (1963). (20) Lewis, G. K., Randall, hi., “Thermodynamics,” SlcGraw-Hill, New York (1 961 ). (21) Lichtenstein, I., Ciiern. Eng. Progr. 60 (12), 64 (1964). (22) Lobo, \V. FV., Evans, J . E., Tranr. A.Z.Ch.E. 31,743 (1939). (23) Myers, P. F., [\’atson, K. M., X d l . Petroi. .?Jewr38 (18), 388 (1940). (24) Rase, H. F., Fair, J. R . , private communication. ( 2 5 ) Roberts, S. hi., I N D .ENG.CHEM.P R O C E S S D E S IDEVELOP. CN 3, 14 (1964). (26) Schutt, H . C., Z. Elektrochem. 6 5 , 245 (1961). (27) Scott, G . S . , I U D .ENC. CHEM.33, 1279 (1951). (28) Shah, hl. J.: “ A n Approach t o Model Simplification in Process Optimization and Control,” presented a t the A I C h E - I C h E >itg., J u n ? 1965, London. (29) Shah, M.J.: “Simulation of a Thermal AIChE 56th Sationnl MLlretinS, San Franci ( 3 0 ) Shah, M . J., James, Chester, Duffin, J. thesis Reactor,” paper presented a t the 1966 IF.1C Meeting. London. (31 j Smith, J. hi., “Chemical Engineering Kinetics.” SicGraw-Hill, New York (1956). (32) Snow, R . H., Peck, R . E., Von Fredersdorff, C. G., .I.I.Ch.E. J . 5 , 304 (10591. (33) Steacie, E . \V. R., Puddington, I. E., Can. J. Res. 16B, 176. 260, 303, 411 (1938). (34) Steacie, E . \Y. R., Shane, R . G., Ibtd., 18B, 203 (1940). (35) Syn, \V. M . , IVyman. D. G., “DSL/90 Disital Simulation Languagz Cscr’s Guide,” I B h l , San Jose, T R 02.355, Julv 1965. (36) Towell, G . D., Ph.D. Thesis, University of hlichigan, Ann Arbor (1960). (37) Towell, G.D., hlartin, J. J., :l.I.Ch.E.J. 7, 693 (1961). (38) Lfeisenfeldrr. A . J., Fritz, J . C., Thompson, M’.G., ZS.1 Preprint 33-3-66.
VOL.59
NO. 5
MAY 1967
85
