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State space formulation of ammonia reactor dynamics
Chpulcn md ChemicalE@weting Vol. 4, pp. 215-222 @IPergamonPress Ltd., 19&l. Printed in Great Britain
STATE SPACE FORMULATION OF AMMONIA REACTOR DYNAMICS L. M. PATNAIK,*N. VISWANADHAM and I. G. SARMA Schoolof Automation,Indian Institute of Science, Bangalore 560 012, India (Receiued 12
May 1980)
Abtract-The specific objective of this paper is to develop a state space model of a tubular ammonia reactor which is the heart of an ammonia plant in a fertiliser complex. A ninth order model with three control inputs and two disturbance inputs is generated from the nonlinear distributed model using linearization and lumping approximations. The lumped model is chosen such that the steady state temperature at the exit of the catalyst bed computed from the simplified state space model is close enough to the one computed from the nonlinear steady state model. The model developed in this paper is very useful for the design of continuous/discrete versions of single variable/multivariable control algorithms.
Scope-This paper presents the details on the development of a state space model for a tubular ammonia reactor. The reactor where the main reaction takes place critically affects the economic and safe operation of the ammonia synthesis process. Studies relating to its optimization and control therefore acquire utmost practical interest. In an earlier paper Patnaik ef a1.[8] developed an optimization program to determine the steady state flow rates and the optimum temperature profile that maximizes the yield of the reactor. The feed concentration and temperature disturbances and the plant parameter variations due to drift in the operating point tend to change the temperature profile inside the reactor and drive the reactor away from the maximal yield operating condition. Hence fast regulation of the temperature along the length of the reactor around its optimum profile is important and forms the dominant objective for the control system design. For this purpose, a dynamic model of the reactor-heat exchanger combination is needed. The dynamic model derived using mass and energy balance principles consists of a set of nonlinear coupled partial differential equations, the exact solution of which is not possible. Moreover, these equations cannot be used directly for the design of control laws/algorithms. Within the constraints provided by the nature of these problems and the specific objectives, this paper deals with the derivation of simpler dynamic models and the techniques and assumptions adopted to simplify them further. One particular representation of special interest is the state variable form which is basic to the design of various control laws/algorithms. The validity of the linear model is also established via simulation. The earlier studies proposing control system designs for ammonia reactors are all based on simplified models[2,4]. Brian et al.[l] have suggested P and PI type controllers whose parameters are to be tuned by trial and error. Gould et a/.[31 have obtained a linear constant coefficient differential equation model of the reacting portion using modal approach whereas this paper deals with the state space model of reacting portion and the heat exchanger which is the usual combination of industrial ammonia reactors. In the approach followed by Gould et a1.[3]the choice of the number of modes is not simple especially for diffusionless processes. Moreover there seems to be no check on the accuracy of the model developed by Gould et al. [3]. Conclusions and Sign&me+In this paper, a nonlinear distributed model is presented for the reactor-heat exchanger combination of an ammonia reactor. Since these equations present formidable difficulties in obtaining computational solutions, resorting to linearization And lumping techniques becomes mandatory, in order to simplify the model. The number of discretization intervals for the purpose of lumping has been chosen after conducting a number of simulation experiments. Linearized models of this nature provide information regarding the dynamics of the plant to small perturbations around the steady state operating point. The state variable equations obtained here are useful for the design of continuous as well as discrete single and multivariable controllers[6]. The multivariable discrete version of the controller based on feedforward or integral control principle ensures that the temperature profile along the reactor length can be maintained very close to the optimal one[7]. These controllers can be easily implemented on the digital processor used in the real time control system.
*Authorto whom correspondence should be addressed. 21s
L. M. PATNAIK et al.
216 REACFOR DESCmON
Ammonia is formed according to the exothermic reaction N1 t 3H2$2NH,. The synthesis gas, which consists of hydrogen and nitrogen in the stoichiometric proportions of 3: 1 is prepared in the reformer section of the plant. This gas mixture compressed to a pressure of 200-300 atmospheres enters the synthesis converter. Figure 1 shows the schematic of the ammonia synthesis converter under consideration. The data for this reactor is given in a paper by Patnaik et al.[8]. The converter consists of two parts: (a) the catalyst bed section shown in the upper part of the diagram, and (b) the heat-exchanger section in the lower part of the diagram. To ensure stable conditions with maximum yield, it is necessary to heat the feedgases to a temperature of about 420°C before they enter the catalyst bed. This is economically achieved by preheating the feedgases first in the bottom heat exchanger and subsequently in the tubes of the reacting section. The inlet gaseous mixture is split into three separate streams: (a) the main stream called the heat exchanger tlow (b) the second stream called the heat exchanger bypass flow and (c) the third stream, known as the direct bypass flow. The feedgases flowing down the catalyst bed react to produce ammonia. The outlet gases from the reacting portion enter the tubes of the heat exchanger and finally exit the converter. The three flow rates can be varied using valves. Also adequate instrumentation exists to measure the process variables of interest. The temperatures at five points along the length of the reactor are measured using thermocouples. Also the inlet flows are measured using differential pressure type flow meter. DYNAMIC MODELOF TEE REACTOR
In this section, the model equations representing the synthesis converter are derived. The method used is the same as that adopted by Brian et ul.[l]. However, since the reactor configuration under consideration differs
Hoot
I t
lxchangu
from the one considered in the above cited reference, appropriate modifications have been incorporated in the model equations. The proposed model consists of five partial differential equations and two algebraic equations with necessary boundary and initial conditions with respect to the space and time variables, respectively. The dynamic model is derived under the following assumptions: (1) No radial variation of temperature in the catalyst, cooling tube and walls. (2) No temperature difference between the catalyst particles and the gas phase. (3) Uniformly constant pressure throughout the converter. (4) Negligibly small values of heat capacities of the tube walls in the reacting and heat exchanger section. (5) No heat loss from the shell side of the heat exchanger to the environment. (6) Absence of longitudinal diffusion of the reactants in the reactor. (7) No transfer of enthalpy by conduction within the gas phase in the empty tube. (8) Temperature independence of the heat capacity of the gases in the reactor only. The larger variation of temperature across the heat exchanger warrants the calculation of these parameters from the exact expressions given in a paper by Patnaik et al. [8]. (9) It is assumed that changes in flow rate, pressure and composition propagate instantaneously throughout the reactor. Similar assumptions have been made by Brian et al.[l]. As a direct consequence of this assumption, this study is primarily concerned with transient analysis of the reactor for changes in the feed temperature. The dynamic model equations (Alt_(A8) are presented in Appendix A. The dynamic model consists of five coupled, nonlinear partial differential equations. Because of the complexity represented by this system of equations, it is not possible to obtain an analytical solution. It can be seen that even the steady state model equations have to be solved iteratively because of the boundary value nature of the problem. In the steady state case, the heat exchanger temperature at the shell entry end is matched with the feed temperature. But in the dynamic case, devising a fast converging iterative technique to match the timevarying boundary conditions is a formidable task. Pure digital simulation techniques based on finite difference schemes are also not suitable for these problems because of the two-point boundary value nature of the problem. Solving these equations using straight analog techniques is ruled out because of the excessive requirement of the computing modules. Partial differential equations can be solved efficiently using the hybrid computer by adopting either “Continuous Space Discrete Time” or “Discrete Space Continuous Time” techniques. Because of the time varying split boundary conditions associated with this problem, a convergent solution could not be obtained using hybrid simulation. Thus it became imperative to simplify the model equations so that the solution of this group of equations becomes possible within a reasonable amount of effort and computational time. Besides providing a first approximation regarding the dynamics of the reactor, these simplified model equations are also helpful in the design of on-line controllers and in stability studies.
flow
Eff
I I
Lwnt gels 1
I lnkt
LINEARUATION to reactor
Fii. 1. Schematic of synthesis converter.
In this section, the model equations (AI)are linearized around the steady state operating point corresponding to maximum yield. When the magnitudes of the disturbances are small, this linearized model
State space formulation of ammonia reactor dynamics
adequately describes the dynamics of the reactor in the neighbourhood of the steady state operating point. Let Y(W
217
where
0 = Ys+ Yt Making use of the corresponding steady state equation and using the following approximations
Tc(a,t) = Tcs+ T,, TT(%0 = TTS+ TT1 T;(a), ?) = T',, t r;,
(1)
f _ACYdh-Y* PO
1f Ys+ Yr
Ti(a’, f) = T:,t r:,. In the above equation ys, Tcs, TTs, T’Ts and Tis represent the steady state part and y,, T,,, TT1,Tk, and T:, represent the transient part of ammonia mole fraction, catalyst temperature, tube temperature in the reacting portion, tube temperature in the heat exchanger and shell temperature in the heat exchanger respectively. The steady state components are functions of a, the normalized distance. In order to carry out the linearization we substitute Eq. (1) in the dynamic equations (AI)and expand the latter around the operating point using Taylor series and neglect the higher order terms. The steady state equations which can be obtained from the dynamic model are used to further simplify these equations. During the simplification phase some terms have been neglected because their contribution is negligibly small. Equation (Al) can be rewritten in the form,
AH, - AC( T, - TB) = AH0 and 1+Ys+Yr=1+Ys,
Eq. (6) can be written as - K,&i~
t o.%)(T,, - T,,) - KsKz,AHo x(RI,,~‘t~I,~,y,)=~.
(7)
Similarly Eqs. (A4) and (AS) can be reduced to
2!&K5.gj=
KsU’Avs(T;, - Tit) 8 5 ECpi
(8)
I=,
z+-g =&(I ah
ah
+YS•t~t)*[r(T,s •tr,,,YS+ YJI. (2)
Let (V/F(l t y*))= K,. Expanding the right hand side of Eq. (2) by Taylor series and neglecting higher order terms, ah aYr_ ,IIt,,-K,[(1+ys)2+2y,(l+Ys)+y:l
rsV,s, ysI+-$lTcsT,, E
+$i,r;].
(3)
From steady state considerations, it is known that @y&a) = K1(l t ys)*rs. Making use of this relation, and neglecting the second order terms of the perturbed values, and rearranging, we get z
LUMPINGOFTHE LINEARPARTIAL DIFFEREIWU. EQUATIONS
= K,(l tys)+ ~l~~sT=,ty,[(lty,)2~l~~
I1
+2WysWWys~
YS
K,.
(4)
Similarly Eqs. (A2) and (A3) can be written as
2 =K2( T, -
T,,)
-K,
- K,lAHo - ACV,, + Tc, - T,)IK.,
where KS = (WCp,/h2S2). The terms (adaT,) and (aday) can be obtained by differentiating r[6] The values of (adaT,), (a&y) and r depend ‘on steady state temperature and concentration and thus are functions of distance. From a knowledge of the steady state values at various points, these terms are calculated at different points along the bed. These have been plotted in Fig. 2 as functions of normalized distance. The values of these partial derivatives at different points along the reactor length are later used, to develop the lumped model.
(5)
In this section, we further simplify the linear distributed model of the reactor described by Eqs. (4)-(S) and (7)-(9) using discrete space approximation. The linear partial differential equations describing the distributed model of the reactor are converted into ordinary differential equations by discretizing the length of the reactor and the heat exchanger into a number of segments. At each discretization point, the values of the quantities (adaT,), (J&y) and rs appearing in Eqs. (4) and (7) are obtained from Fig. 2 by computing the weighted average over the interval. The accuracy of the lumped model, obviously depends on the number of discretization steps used. The larger the number of intervals, the better is the accuracy. Also the number of differential equations increases linearly with the number of intervals chosen. On the other hand, coarser discretization may lead to inaccurate models. In this study we consider the effect of the fineness of discretization
L. M. PATNNK et al.
218
-2oo-? 2 2 O*O! -3oo-
i! y”
i
mLs
-4oo-
-5ooL
C
J
-04s
v
Normalized distance -
Fig. 2. Variationof (ar/aT,), (aday) and r with normalizeddistance. and conclude that the reacting portion can be divided into Eq. (10)and simplifyingwe get into five parts and the heat exchangerinto two parts. The results of this study can be found in a later section.
j=l,2,...,N Orcutt & Lamb[S] have used the linearized lumped y,(i)[~-o(i)]=~tb(i)T,O.), modelto investigatethe stabilityof the ammoniareactor. (13) They also have concluded that the reactor dynamicsfor this purpose can be satisfactorily described by dividing where Aa is the step size of normalized distance. The the reacting section into five parts and treating the heat above equation can be compactly written as exchanger as a single unit. In what follows, the discretization procedure is presented. Equation (4) can be Ay = Bt, t dye (14) arranged as, aYt a(l=
where y and t, are the N x 1 vectors representing the
(10) perturbed values of ammoniamole fraction and catalyst
4ah + b(a)T,,
temperature at different points, A and B are N x N matrices, d is N x 1 vector and y. = y,(O). Similarly Eq. (6) can be written as,
where K, +2WysW(l+ys~ II YS
(11) and (12
H is an N x N matrix, t, is an N x 1 vector representing the perturbed values of tube temperature at different points and tro= T,(O).Substituting the values of various constants in Eq. (7) we obtain
Substituting
0,
I
dai=
Yto')-Y&i-l) Aa
td(j)ye(j)=v
(16)
State spaceformulationof ammoniareactor dynamics where c(j) = -0.59 - KdKsAH& d(j) = -;
219
equations from Eqs. (2&o-(2). The fist set of five linear ordinary differential equations represent the dynamics of the reacting portion whereas the latter set of 4 equations represent the dynamics of the heat exchanger. If these nine temperatures form the elements of a state vector then one can obtain the state variable form of the dynamic equations which is presented in the following section.
c I T,s’
KdKsAHo. I YS
Equation (16) can be rewritten as, !5TATE SPACE REPRJBENTATION
i, = Ct, + Dy t 0.59t,t 0.506dt,o where
tco= T,,(O). (17)
Solving Eqs. (14) and (15) for y and t, and substituting these into Eq. (17) we get
I 1 [I.
i, = [Ct DA-‘B-0.59x
3.1888H-‘It,
= 12&t:- t:)
(19)
53.2(r:- t;).
(20)
In arriving at the linear Eq. (20) from (9), the denominator of the latter equation has been obtained by taking the average of the steady state values of this term at various points in the heat exchanger. Since the heat exchanger is divided into two parts, Eqs. (19) and (20) can be written as 12.8tXj)-316rj(j)t
18.8t:(jt l), j=O, 1 (21)
T
= 53.2t:(j)- 147.2tKj)t 94rKj- l),
j = 1,2. (22)
The boundary conditions are, t:(2) = tf = perturbation in feed temperature and tX0)= &(a = 1). The mixing equation at the top of the reactor, Eq. (A6), can be perturbed around the steady state to yield t,,, = 0.955&o t O.O47t,- 0.0595~st 0.0029~~t 0.0029~~ (23) where ul, u2 and u3 are the perturbations in F,-R. Equation (A7) representing the mixing equation at the top of the heat exchanger can be linearized to result in t,(s) = 0.00288u,-
0.0&.4~
t O.S7t;(O)t O.l31t,. (24)
By repeated application of Eq. (5) in a backward difference scheme to obtain an expression for Ito and by using this expression along with Eqs. (23x24) we get a set of five coupled differential equations from Eq. (18). Similarly one can get a set of four coupled differential CACE
Vol. 4. No. 4-B
Xs,
X69 X7,
Xa,
X91
4
[u,,
u2,
us1
d== It,, YOI
Next let us consider Eqs. (8) and (9). Representing t: and t: as the perturbed values of shell side and tube side temperatures in the heat exchanger and substituting the values of various constants, we obtain
F=
x4,
A
t,o
47$=
x3,
= [G(l), &(2), t,(3), t,(4), t=(5), U), tX2), G(O),W)l UT
t CO
St
xr ’ (Xl, X2,
(18)
t [DA-‘d i0.506d:0.59H-‘d] y.
s-9.4$
Let
t,(l), t,(2), t,(3), t,(4) and t,(S) are the incremental catalyst temperatures at the five discrete points in the reactor. Then the linearized equations (18) and (21)-(22) after minor arrangements can be represented by i(t) = Ax(t) t h(t) t Dd(t)
(25)
where x(t) is a 9 x 1 state vector, u(t) is a 3 x 1 manipulated vector, d(t) is a 2 x 1 disturbance vector and A, B and Dare 9 X $9 X 3 and 9 x 2 matrices respectively (see Appendix B). SIMUWTED
RESPONSES
The nine ordinary differential equatjons given by Eq. (25) are simulated on the analog section of the AD511 hybrid computer using nine integrators, eleven inverters and forty-three potentiometers. The purpose of this simulation study is (i) to check whether the response obtained using the linear model equation agrees with those reported in the literature[l, 41, (ii) to determine the effect of the fineness of discretization on the response and from these results to fix the discretization intervals for both the reactor and heat exchanger. Apart from the above reasons, the knowledge of the open-loop behaviour of the system is important in determining the specifications for proceeding with the control system design. The system is subjected to a step disturbance of 5°C in the feed temperature which has a steady state value of 42°C. This corresponds to a 12.5% perturbation after this nominal value. Even though feed temperature and ammonia mole fraction are the components of the disturbance vector, we consider the former because the re<s obtained can be compared with those reported in the literature. Moreover, the effect of thermal perturbations are of interest to the plant operators. The response at different points along the length of the reactor is shown in Fig. 3. The steady state program is run with the new feed temperature (the nominal feed temperature incremented by the step disturbance), The perturbed temperature profile thus obtained should check with the profile obtained by the dynamic simulation. It has been established that the profiles match within reasonable accuracy. The error in the steady state temperature at
L. hf. PATNAtK eta/.
a-a2
a-0.4
a=O.6
a4.6
ad.0
no
0
1500
2250
limo (rconds) Fii. 3.Opcn4oop
responseto ST
the exit of the catalyst bed is 0.4%. In the absence of experimental data, the results obtained are compared with those reported by Brian et uL[ll. The trend of the response agrees with that shownin theii work. The pulse response of the system at the reactor exit is shown in
step in feedtemperahue (N = 5).
Fii. 4. The initial “inverse response*’is clearly reflected in both the above response curves. In order to gain a greater insight into plant dynamics, the system is subjected to a step change of 7.5kg mole&r in direct bypass flow rate. The response
5.0 -
temperature af aC=I.0
-
Catalysl
2.5I 0
.F !! I -I0 1L E cy
I
100
200
Tif!W (seconds)
Jo0
400
500
-
8-
5.0 -
‘-Pulse
2*5-
dtsturbance )emperature
0 lnne
(seconds) -
Fig. 4. Puke responseat catalyst bed exit end.
in feed
State space formulation of ammonia reactor dynamics
Time (seconds)
221
-
Fig. 5. Response to step change in direct bypass flowrate (step: 7.5 kg moleslhr). Table 1. Percentage error in catalyst bed exit temperature for different
discretizationstep sizes N
Percentageerror in t,(5)
3 4 5 7
Reactor model unstable 0.6% 0.4% 0.32%
curves are shown in Fig. 5. It is interesting to note that the “inverse response” is more pronounced as the bottom end of the reactor is approached. This can be explained by the presence of a right-half-plane zero in the corresponding transfer functions[6].
EFTWTOl’FlNENESOF DISCRETIZATION In an earlier section the linear state space model is derived after dividing the reacting section into 5 parts and heat exchanger into 2 parts. The division of the heat exchanger into 2 equal parts is justified because its dynamics is very fast and this is not as significant as the reacting portion where the main reaction takes place. The partial derivatives being functions of the length of the reactor, weighted average values of these are used in the model. To study the sensitivity of the dynamic model of the system to the number of discretization steps, the reacting section is considered separately. The state variable representation for this is obtained for different cases of discretization: (1) N = 3; (2) N = 4; (3) N = 5; (4) N = 7, where N is the number of steps into which the reactor is divided. The reactor shows unstable behaviour when N c 3 even though this corresponds to a stable operating point. The dynamic behaviour of the reactor for N = 4,5 and 7 is considered. The criterion used for comparing these cases is the percentage error in the steady state tem-
perature at the exit of the catalyst bed. This information is summarized in Table 1. From this table, it is clear that no significant improvement can be achieved by choosing N >5 thus justifying our earlier choice of five discretization intervals for the reactor length. NOMENCLATURE specific heat of catalyst, kcal/kg mole “K specific heat of ith component, kcal/kgmole”K (i= 1 for hydrogen, 2 for nitrogen, 3 for ammonia and 4 for inerts) average heat capacity of feed gas, kcallkg mole “K molar Row rate of feed, kg moleslhr flow in the jth branch (j= 1 for heat exchanger shell, 2 for heat exchanger bypass, 3 for direct bypass), kg moleslhr molar flowrate of ith component in the feed gas, kg moleslhr heat transfer area in catalyst bed, m2 heat transfer area in heat exchanger, m2 total surface available for heat transfer between tube wall-catalyst section, m* reference temperature, 25°C catalyst temperature, “C feed temperature, “C temperature in the lube of catalyst portion, “C temperature in the tub-eof heat exchanger, “C temperature in the shell of heat-exchanger, “C overall heat transfer coefficient in catalyst bed, kcal/hr m2“C overall heat transfer coefficient in heat exchanger, kcallhr m2“C volume of the catalyst bed, m3 weight of the catalyst, kg disturbance vector average heat transfer coefficient from wall to catalyst, kcal/hr m*“C length of the heat exchanger, m reaction rate, kg mole NHJhr m3 catalyst state vector mole fraction of ammonia normalized distance, catalyst portion
L. M. PATNAIK et al.
222
a’ normalized distance, heat exchanger B ratio of heat exchanger bypass to heat exchanger flow Y ratio of direct bypass flow to heat exchanger bypass s ratio of heat exchanger flow to feed flow AC decrease in specific heat due to formation of one mole of ammonia, kcal/kg mole “K AHo enthalpy of formation of ammonia at 298”K, kcal/kg mole e VS
t,(i), i = 15 ty), i = 1,2 t:(i), i = 0,l ui i=l i=2 i=3 u(t) tr Yo
Energy balance in the catalyst: O.B(T, - Tc)
j$po-AC~)-$t 22
643) Energy balance in the shell of the heat exchanger:
(O=@)
nondimensionalunitoftime
velocity of gas mixture in the shell ?ide of heat exchanger, m/set velocity of the gas mixture in the tube of heat exchanger, m/set incremental catalyst temperature at the ith location, “C incremental tube temperature in heat exchanger at ith location, “C incremental shell temperature in heat exchanger at ith location, “C incremental flow rates (manipulated variables), kg moleslhr main stream flow in heat exchanger heat exchanger bypass flow direct bypass flow control vector incremental feed temperature, “C incremental feed composition
Superscripts * inlet condition
Energy balance in the tube of the heat exchanger:
Mixing equations and boundary conditions: At the top of reactor-Energy balance s(lt /3)(T& =O, e))t&?ByU'fite))=(Tcb =O,eN.
M
At the top of heat exchanger-Energy balance 6(T:(d ; 0, e)) t sB(TF(e)) = T&x = 1, e).
(-47)
Boundary conditions: Y(c~= 0, e) = y*(e);
REFERENCES 1. P. L. T. Brian, R. F. Baddour & J. P. Eymery, C/rem. Engng Sci. 24,297 (l%S). 2. S. S. Gluzman & V. N. Krainov, ht. Chem. Engng 7, 289 (1%7). 3. L. A. Gould & F. M. Schlaepfer, PreprintsIACC, 81 (1%7). 4. B. W. Iljin, D. Balzer, G. Reinig & V. Scholz, Preprints of IFAC Symp. on Digital Simulation of Continuous Processes. 11 (1971). 5. J. C. Grcutt & D. E. Lamb, Proc. 1st Znt.Cona. of the ZFAC, p. 274 (1960). 6. L. M. Patnaik, Optimization and multivariable computer control of an ammonia reactor. Ph.D. Thesis, Indian Institute of Science (1977). 7. L. M. Patnaik, N. Viswanadham & I. G. Sarma, Proc. LE.E.E. Conf. Decision and Control (1978). 8. L. G. Patnaik, I. G. Sarma & N: Viswanadham, Int. J. Sys. Sci. 10,225 (1979).
T& = 1, 0) = T;(a’ =0, 0): T&Y’= 1, e) = T&B).
(A@
Equations (AD-o-(B) describe the dynamics of the reactorexchanger combination. The details of the derivation of these equations can be found elsewhere[6]. APPENDMB State space model for the reactor: In Eq. (25) the elements of A(9 x 9), B(9 X3) and D(9 X2) are given by, al, = -4.019, aI2 = 5.12, au = -2.082, aI9 =0.87, a2, = -0.346, azz= 0.986,ax = -2.34, ar, = 0.97, a), = -7.909, a,* = 15.407, a,, = -4.069, aJ5 = -6.45, aj9 = 2.68, adI = -21.816, au = 35.606, ad,= -0.339,aa=-3.870,a,,=-17.8,
ae=7.39,aS1 =60.1%,
as2= 98.188,as3= -7.907, as4= 0.340,a,, = -53.008, aS9= 20.4, ab5= 94, a6 = -147.2, aa = 53.2,aT6= 94, a,, = -147.2, au,=12.8,ass=-31.6,a9~=12.8,a,=18.8,a99=-31.6. b,,=O.Ol, b,*=-0.011, bu=-0.051, bz, =0.003, bz,=-0.021,
APPENDIX A Material balance in the catalyst section: $Y
-cm-
vu+Y)*
clcy F(1 ty*)
bj, = O.OO!I, bD = -0.059, b4, = 0.024,bh2= -0.162, b,, = 0.068, bS2= -0.445.
00
Y).
(AU
d,, = 0.251,d,z = -1438.916,dz, = 0.147,dz2= -323.846, d,, = 0.405,ds2= -82.771, d4, = 1.13,d.,?= -22.637,
Energy balance in the empty tube section:
d5, = 3.87,d52= -5.456, d6, = 53.2,d,, = 18.8. z=$T,-T,). PO
642)
(All other elements of the above matrices are zero).
STATE SPACE FORMULATION OF AMMONIA REACTOR DYNAMICS L. M. PATNAIK,*N. VISWANADHAM and I. G. SARMA Schoolof Automation,Indian Institute of Science, Bangalore 560 012, India (Receiued 12
May 1980)
Abtract-The specific objective of this paper is to develop a state space model of a tubular ammonia reactor which is the heart of an ammonia plant in a fertiliser complex. A ninth order model with three control inputs and two disturbance inputs is generated from the nonlinear distributed model using linearization and lumping approximations. The lumped model is chosen such that the steady state temperature at the exit of the catalyst bed computed from the simplified state space model is close enough to the one computed from the nonlinear steady state model. The model developed in this paper is very useful for the design of continuous/discrete versions of single variable/multivariable control algorithms.
Scope-This paper presents the details on the development of a state space model for a tubular ammonia reactor. The reactor where the main reaction takes place critically affects the economic and safe operation of the ammonia synthesis process. Studies relating to its optimization and control therefore acquire utmost practical interest. In an earlier paper Patnaik ef a1.[8] developed an optimization program to determine the steady state flow rates and the optimum temperature profile that maximizes the yield of the reactor. The feed concentration and temperature disturbances and the plant parameter variations due to drift in the operating point tend to change the temperature profile inside the reactor and drive the reactor away from the maximal yield operating condition. Hence fast regulation of the temperature along the length of the reactor around its optimum profile is important and forms the dominant objective for the control system design. For this purpose, a dynamic model of the reactor-heat exchanger combination is needed. The dynamic model derived using mass and energy balance principles consists of a set of nonlinear coupled partial differential equations, the exact solution of which is not possible. Moreover, these equations cannot be used directly for the design of control laws/algorithms. Within the constraints provided by the nature of these problems and the specific objectives, this paper deals with the derivation of simpler dynamic models and the techniques and assumptions adopted to simplify them further. One particular representation of special interest is the state variable form which is basic to the design of various control laws/algorithms. The validity of the linear model is also established via simulation. The earlier studies proposing control system designs for ammonia reactors are all based on simplified models[2,4]. Brian et al.[l] have suggested P and PI type controllers whose parameters are to be tuned by trial and error. Gould et a/.[31 have obtained a linear constant coefficient differential equation model of the reacting portion using modal approach whereas this paper deals with the state space model of reacting portion and the heat exchanger which is the usual combination of industrial ammonia reactors. In the approach followed by Gould et a1.[3]the choice of the number of modes is not simple especially for diffusionless processes. Moreover there seems to be no check on the accuracy of the model developed by Gould et al. [3]. Conclusions and Sign&me+In this paper, a nonlinear distributed model is presented for the reactor-heat exchanger combination of an ammonia reactor. Since these equations present formidable difficulties in obtaining computational solutions, resorting to linearization And lumping techniques becomes mandatory, in order to simplify the model. The number of discretization intervals for the purpose of lumping has been chosen after conducting a number of simulation experiments. Linearized models of this nature provide information regarding the dynamics of the plant to small perturbations around the steady state operating point. The state variable equations obtained here are useful for the design of continuous as well as discrete single and multivariable controllers[6]. The multivariable discrete version of the controller based on feedforward or integral control principle ensures that the temperature profile along the reactor length can be maintained very close to the optimal one[7]. These controllers can be easily implemented on the digital processor used in the real time control system.
*Authorto whom correspondence should be addressed. 21s
L. M. PATNAIK et al.
216 REACFOR DESCmON
Ammonia is formed according to the exothermic reaction N1 t 3H2$2NH,. The synthesis gas, which consists of hydrogen and nitrogen in the stoichiometric proportions of 3: 1 is prepared in the reformer section of the plant. This gas mixture compressed to a pressure of 200-300 atmospheres enters the synthesis converter. Figure 1 shows the schematic of the ammonia synthesis converter under consideration. The data for this reactor is given in a paper by Patnaik et al.[8]. The converter consists of two parts: (a) the catalyst bed section shown in the upper part of the diagram, and (b) the heat-exchanger section in the lower part of the diagram. To ensure stable conditions with maximum yield, it is necessary to heat the feedgases to a temperature of about 420°C before they enter the catalyst bed. This is economically achieved by preheating the feedgases first in the bottom heat exchanger and subsequently in the tubes of the reacting section. The inlet gaseous mixture is split into three separate streams: (a) the main stream called the heat exchanger tlow (b) the second stream called the heat exchanger bypass flow and (c) the third stream, known as the direct bypass flow. The feedgases flowing down the catalyst bed react to produce ammonia. The outlet gases from the reacting portion enter the tubes of the heat exchanger and finally exit the converter. The three flow rates can be varied using valves. Also adequate instrumentation exists to measure the process variables of interest. The temperatures at five points along the length of the reactor are measured using thermocouples. Also the inlet flows are measured using differential pressure type flow meter. DYNAMIC MODELOF TEE REACTOR
In this section, the model equations representing the synthesis converter are derived. The method used is the same as that adopted by Brian et ul.[l]. However, since the reactor configuration under consideration differs
Hoot
I t
lxchangu
from the one considered in the above cited reference, appropriate modifications have been incorporated in the model equations. The proposed model consists of five partial differential equations and two algebraic equations with necessary boundary and initial conditions with respect to the space and time variables, respectively. The dynamic model is derived under the following assumptions: (1) No radial variation of temperature in the catalyst, cooling tube and walls. (2) No temperature difference between the catalyst particles and the gas phase. (3) Uniformly constant pressure throughout the converter. (4) Negligibly small values of heat capacities of the tube walls in the reacting and heat exchanger section. (5) No heat loss from the shell side of the heat exchanger to the environment. (6) Absence of longitudinal diffusion of the reactants in the reactor. (7) No transfer of enthalpy by conduction within the gas phase in the empty tube. (8) Temperature independence of the heat capacity of the gases in the reactor only. The larger variation of temperature across the heat exchanger warrants the calculation of these parameters from the exact expressions given in a paper by Patnaik et al. [8]. (9) It is assumed that changes in flow rate, pressure and composition propagate instantaneously throughout the reactor. Similar assumptions have been made by Brian et al.[l]. As a direct consequence of this assumption, this study is primarily concerned with transient analysis of the reactor for changes in the feed temperature. The dynamic model equations (Alt_(A8) are presented in Appendix A. The dynamic model consists of five coupled, nonlinear partial differential equations. Because of the complexity represented by this system of equations, it is not possible to obtain an analytical solution. It can be seen that even the steady state model equations have to be solved iteratively because of the boundary value nature of the problem. In the steady state case, the heat exchanger temperature at the shell entry end is matched with the feed temperature. But in the dynamic case, devising a fast converging iterative technique to match the timevarying boundary conditions is a formidable task. Pure digital simulation techniques based on finite difference schemes are also not suitable for these problems because of the two-point boundary value nature of the problem. Solving these equations using straight analog techniques is ruled out because of the excessive requirement of the computing modules. Partial differential equations can be solved efficiently using the hybrid computer by adopting either “Continuous Space Discrete Time” or “Discrete Space Continuous Time” techniques. Because of the time varying split boundary conditions associated with this problem, a convergent solution could not be obtained using hybrid simulation. Thus it became imperative to simplify the model equations so that the solution of this group of equations becomes possible within a reasonable amount of effort and computational time. Besides providing a first approximation regarding the dynamics of the reactor, these simplified model equations are also helpful in the design of on-line controllers and in stability studies.
flow
Eff
I I
Lwnt gels 1
I lnkt
LINEARUATION to reactor
Fii. 1. Schematic of synthesis converter.
In this section, the model equations (AI)are linearized around the steady state operating point corresponding to maximum yield. When the magnitudes of the disturbances are small, this linearized model
State space formulation of ammonia reactor dynamics
adequately describes the dynamics of the reactor in the neighbourhood of the steady state operating point. Let Y(W
217
where
0 = Ys+ Yt Making use of the corresponding steady state equation and using the following approximations
Tc(a,t) = Tcs+ T,, TT(%0 = TTS+ TT1 T;(a), ?) = T',, t r;,
(1)
f _ACYdh-Y* PO
1f Ys+ Yr
Ti(a’, f) = T:,t r:,. In the above equation ys, Tcs, TTs, T’Ts and Tis represent the steady state part and y,, T,,, TT1,Tk, and T:, represent the transient part of ammonia mole fraction, catalyst temperature, tube temperature in the reacting portion, tube temperature in the heat exchanger and shell temperature in the heat exchanger respectively. The steady state components are functions of a, the normalized distance. In order to carry out the linearization we substitute Eq. (1) in the dynamic equations (AI)and expand the latter around the operating point using Taylor series and neglect the higher order terms. The steady state equations which can be obtained from the dynamic model are used to further simplify these equations. During the simplification phase some terms have been neglected because their contribution is negligibly small. Equation (Al) can be rewritten in the form,
AH, - AC( T, - TB) = AH0 and 1+Ys+Yr=1+Ys,
Eq. (6) can be written as - K,&i~
t o.%)(T,, - T,,) - KsKz,AHo x(RI,,~‘t~I,~,y,)=~.
(7)
Similarly Eqs. (A4) and (AS) can be reduced to
2!&K5.gj=
KsU’Avs(T;, - Tit) 8 5 ECpi
(8)
I=,
z+-g =&(I ah
ah
+YS•t~t)*[r(T,s •tr,,,YS+ YJI. (2)
Let (V/F(l t y*))= K,. Expanding the right hand side of Eq. (2) by Taylor series and neglecting higher order terms, ah aYr_ ,IIt,,-K,[(1+ys)2+2y,(l+Ys)+y:l
rsV,s, ysI+-$lTcsT,, E
+$i,r;].
(3)
From steady state considerations, it is known that @y&a) = K1(l t ys)*rs. Making use of this relation, and neglecting the second order terms of the perturbed values, and rearranging, we get z
LUMPINGOFTHE LINEARPARTIAL DIFFEREIWU. EQUATIONS
= K,(l tys)+ ~l~~sT=,ty,[(lty,)2~l~~
I1
+2WysWWys~
YS
K,.
(4)
Similarly Eqs. (A2) and (A3) can be written as
2 =K2( T, -
T,,)
-K,
- K,lAHo - ACV,, + Tc, - T,)IK.,
where KS = (WCp,/h2S2). The terms (adaT,) and (aday) can be obtained by differentiating r[6] The values of (adaT,), (a&y) and r depend ‘on steady state temperature and concentration and thus are functions of distance. From a knowledge of the steady state values at various points, these terms are calculated at different points along the bed. These have been plotted in Fig. 2 as functions of normalized distance. The values of these partial derivatives at different points along the reactor length are later used, to develop the lumped model.
(5)
In this section, we further simplify the linear distributed model of the reactor described by Eqs. (4)-(S) and (7)-(9) using discrete space approximation. The linear partial differential equations describing the distributed model of the reactor are converted into ordinary differential equations by discretizing the length of the reactor and the heat exchanger into a number of segments. At each discretization point, the values of the quantities (adaT,), (J&y) and rs appearing in Eqs. (4) and (7) are obtained from Fig. 2 by computing the weighted average over the interval. The accuracy of the lumped model, obviously depends on the number of discretization steps used. The larger the number of intervals, the better is the accuracy. Also the number of differential equations increases linearly with the number of intervals chosen. On the other hand, coarser discretization may lead to inaccurate models. In this study we consider the effect of the fineness of discretization
L. M. PATNNK et al.
218
-2oo-? 2 2 O*O! -3oo-
i! y”
i
mLs
-4oo-
-5ooL
C
J
-04s
v
Normalized distance -
Fig. 2. Variationof (ar/aT,), (aday) and r with normalizeddistance. and conclude that the reacting portion can be divided into Eq. (10)and simplifyingwe get into five parts and the heat exchangerinto two parts. The results of this study can be found in a later section.
j=l,2,...,N Orcutt & Lamb[S] have used the linearized lumped y,(i)[~-o(i)]=~tb(i)T,O.), modelto investigatethe stabilityof the ammoniareactor. (13) They also have concluded that the reactor dynamicsfor this purpose can be satisfactorily described by dividing where Aa is the step size of normalized distance. The the reacting section into five parts and treating the heat above equation can be compactly written as exchanger as a single unit. In what follows, the discretization procedure is presented. Equation (4) can be Ay = Bt, t dye (14) arranged as, aYt a(l=
where y and t, are the N x 1 vectors representing the
(10) perturbed values of ammoniamole fraction and catalyst
4ah + b(a)T,,
temperature at different points, A and B are N x N matrices, d is N x 1 vector and y. = y,(O). Similarly Eq. (6) can be written as,
where K, +2WysW(l+ys~ II YS
(11) and (12
H is an N x N matrix, t, is an N x 1 vector representing the perturbed values of tube temperature at different points and tro= T,(O).Substituting the values of various constants in Eq. (7) we obtain
Substituting
0,
I
dai=
Yto')-Y&i-l) Aa
td(j)ye(j)=v
(16)
State spaceformulationof ammoniareactor dynamics where c(j) = -0.59 - KdKsAH& d(j) = -;
219
equations from Eqs. (2&o-(2). The fist set of five linear ordinary differential equations represent the dynamics of the reacting portion whereas the latter set of 4 equations represent the dynamics of the heat exchanger. If these nine temperatures form the elements of a state vector then one can obtain the state variable form of the dynamic equations which is presented in the following section.
c I T,s’
KdKsAHo. I YS
Equation (16) can be rewritten as, !5TATE SPACE REPRJBENTATION
i, = Ct, + Dy t 0.59t,t 0.506dt,o where
tco= T,,(O). (17)
Solving Eqs. (14) and (15) for y and t, and substituting these into Eq. (17) we get
I 1 [I.
i, = [Ct DA-‘B-0.59x
3.1888H-‘It,
= 12&t:- t:)
(19)
53.2(r:- t;).
(20)
In arriving at the linear Eq. (20) from (9), the denominator of the latter equation has been obtained by taking the average of the steady state values of this term at various points in the heat exchanger. Since the heat exchanger is divided into two parts, Eqs. (19) and (20) can be written as 12.8tXj)-316rj(j)t
18.8t:(jt l), j=O, 1 (21)
T
= 53.2t:(j)- 147.2tKj)t 94rKj- l),
j = 1,2. (22)
The boundary conditions are, t:(2) = tf = perturbation in feed temperature and tX0)= &(a = 1). The mixing equation at the top of the reactor, Eq. (A6), can be perturbed around the steady state to yield t,,, = 0.955&o t O.O47t,- 0.0595~st 0.0029~~t 0.0029~~ (23) where ul, u2 and u3 are the perturbations in F,-R. Equation (A7) representing the mixing equation at the top of the heat exchanger can be linearized to result in t,(s) = 0.00288u,-
0.0&.4~
t O.S7t;(O)t O.l31t,. (24)
By repeated application of Eq. (5) in a backward difference scheme to obtain an expression for Ito and by using this expression along with Eqs. (23x24) we get a set of five coupled differential equations from Eq. (18). Similarly one can get a set of four coupled differential CACE
Vol. 4. No. 4-B
Xs,
X69 X7,
Xa,
X91
4
[u,,
u2,
us1
d== It,, YOI
Next let us consider Eqs. (8) and (9). Representing t: and t: as the perturbed values of shell side and tube side temperatures in the heat exchanger and substituting the values of various constants, we obtain
F=
x4,
A
t,o
47$=
x3,
= [G(l), &(2), t,(3), t,(4), t=(5), U), tX2), G(O),W)l UT
t CO
St
xr ’ (Xl, X2,
(18)
t [DA-‘d i0.506d:0.59H-‘d] y.
s-9.4$
Let
t,(l), t,(2), t,(3), t,(4) and t,(S) are the incremental catalyst temperatures at the five discrete points in the reactor. Then the linearized equations (18) and (21)-(22) after minor arrangements can be represented by i(t) = Ax(t) t h(t) t Dd(t)
(25)
where x(t) is a 9 x 1 state vector, u(t) is a 3 x 1 manipulated vector, d(t) is a 2 x 1 disturbance vector and A, B and Dare 9 X $9 X 3 and 9 x 2 matrices respectively (see Appendix B). SIMUWTED
RESPONSES
The nine ordinary differential equatjons given by Eq. (25) are simulated on the analog section of the AD511 hybrid computer using nine integrators, eleven inverters and forty-three potentiometers. The purpose of this simulation study is (i) to check whether the response obtained using the linear model equation agrees with those reported in the literature[l, 41, (ii) to determine the effect of the fineness of discretization on the response and from these results to fix the discretization intervals for both the reactor and heat exchanger. Apart from the above reasons, the knowledge of the open-loop behaviour of the system is important in determining the specifications for proceeding with the control system design. The system is subjected to a step disturbance of 5°C in the feed temperature which has a steady state value of 42°C. This corresponds to a 12.5% perturbation after this nominal value. Even though feed temperature and ammonia mole fraction are the components of the disturbance vector, we consider the former because the re<s obtained can be compared with those reported in the literature. Moreover, the effect of thermal perturbations are of interest to the plant operators. The response at different points along the length of the reactor is shown in Fig. 3. The steady state program is run with the new feed temperature (the nominal feed temperature incremented by the step disturbance), The perturbed temperature profile thus obtained should check with the profile obtained by the dynamic simulation. It has been established that the profiles match within reasonable accuracy. The error in the steady state temperature at
L. hf. PATNAtK eta/.
a-a2
a-0.4
a=O.6
a4.6
ad.0
no
0
1500
2250
limo (rconds) Fii. 3.Opcn4oop
responseto ST
the exit of the catalyst bed is 0.4%. In the absence of experimental data, the results obtained are compared with those reported by Brian et uL[ll. The trend of the response agrees with that shownin theii work. The pulse response of the system at the reactor exit is shown in
step in feedtemperahue (N = 5).
Fii. 4. The initial “inverse response*’is clearly reflected in both the above response curves. In order to gain a greater insight into plant dynamics, the system is subjected to a step change of 7.5kg mole&r in direct bypass flow rate. The response
5.0 -
temperature af aC=I.0
-
Catalysl
2.5I 0
.F !! I -I0 1L E cy
I
100
200
Tif!W (seconds)
Jo0
400
500
-
8-
5.0 -
‘-Pulse
2*5-
dtsturbance )emperature
0 lnne
(seconds) -
Fig. 4. Puke responseat catalyst bed exit end.
in feed
State space formulation of ammonia reactor dynamics
Time (seconds)
221
-
Fig. 5. Response to step change in direct bypass flowrate (step: 7.5 kg moleslhr). Table 1. Percentage error in catalyst bed exit temperature for different
discretizationstep sizes N
Percentageerror in t,(5)
3 4 5 7
Reactor model unstable 0.6% 0.4% 0.32%
curves are shown in Fig. 5. It is interesting to note that the “inverse response” is more pronounced as the bottom end of the reactor is approached. This can be explained by the presence of a right-half-plane zero in the corresponding transfer functions[6].
EFTWTOl’FlNENESOF DISCRETIZATION In an earlier section the linear state space model is derived after dividing the reacting section into 5 parts and heat exchanger into 2 parts. The division of the heat exchanger into 2 equal parts is justified because its dynamics is very fast and this is not as significant as the reacting portion where the main reaction takes place. The partial derivatives being functions of the length of the reactor, weighted average values of these are used in the model. To study the sensitivity of the dynamic model of the system to the number of discretization steps, the reacting section is considered separately. The state variable representation for this is obtained for different cases of discretization: (1) N = 3; (2) N = 4; (3) N = 5; (4) N = 7, where N is the number of steps into which the reactor is divided. The reactor shows unstable behaviour when N c 3 even though this corresponds to a stable operating point. The dynamic behaviour of the reactor for N = 4,5 and 7 is considered. The criterion used for comparing these cases is the percentage error in the steady state tem-
perature at the exit of the catalyst bed. This information is summarized in Table 1. From this table, it is clear that no significant improvement can be achieved by choosing N >5 thus justifying our earlier choice of five discretization intervals for the reactor length. NOMENCLATURE specific heat of catalyst, kcal/kg mole “K specific heat of ith component, kcal/kgmole”K (i= 1 for hydrogen, 2 for nitrogen, 3 for ammonia and 4 for inerts) average heat capacity of feed gas, kcallkg mole “K molar Row rate of feed, kg moleslhr flow in the jth branch (j= 1 for heat exchanger shell, 2 for heat exchanger bypass, 3 for direct bypass), kg moleslhr molar flowrate of ith component in the feed gas, kg moleslhr heat transfer area in catalyst bed, m2 heat transfer area in heat exchanger, m2 total surface available for heat transfer between tube wall-catalyst section, m* reference temperature, 25°C catalyst temperature, “C feed temperature, “C temperature in the lube of catalyst portion, “C temperature in the tub-eof heat exchanger, “C temperature in the shell of heat-exchanger, “C overall heat transfer coefficient in catalyst bed, kcal/hr m2“C overall heat transfer coefficient in heat exchanger, kcallhr m2“C volume of the catalyst bed, m3 weight of the catalyst, kg disturbance vector average heat transfer coefficient from wall to catalyst, kcal/hr m*“C length of the heat exchanger, m reaction rate, kg mole NHJhr m3 catalyst state vector mole fraction of ammonia normalized distance, catalyst portion
L. M. PATNAIK et al.
222
a’ normalized distance, heat exchanger B ratio of heat exchanger bypass to heat exchanger flow Y ratio of direct bypass flow to heat exchanger bypass s ratio of heat exchanger flow to feed flow AC decrease in specific heat due to formation of one mole of ammonia, kcal/kg mole “K AHo enthalpy of formation of ammonia at 298”K, kcal/kg mole e VS
t,(i), i = 15 ty), i = 1,2 t:(i), i = 0,l ui i=l i=2 i=3 u(t) tr Yo
Energy balance in the catalyst: O.B(T, - Tc)
j$po-AC~)-$t 22
643) Energy balance in the shell of the heat exchanger:
(O=@)
nondimensionalunitoftime
velocity of gas mixture in the shell ?ide of heat exchanger, m/set velocity of the gas mixture in the tube of heat exchanger, m/set incremental catalyst temperature at the ith location, “C incremental tube temperature in heat exchanger at ith location, “C incremental shell temperature in heat exchanger at ith location, “C incremental flow rates (manipulated variables), kg moleslhr main stream flow in heat exchanger heat exchanger bypass flow direct bypass flow control vector incremental feed temperature, “C incremental feed composition
Superscripts * inlet condition
Energy balance in the tube of the heat exchanger:
Mixing equations and boundary conditions: At the top of reactor-Energy balance s(lt /3)(T& =O, e))t&?ByU'fite))=(Tcb =O,eN.
M
At the top of heat exchanger-Energy balance 6(T:(d ; 0, e)) t sB(TF(e)) = T&x = 1, e).
(-47)
Boundary conditions: Y(c~= 0, e) = y*(e);
REFERENCES 1. P. L. T. Brian, R. F. Baddour & J. P. Eymery, C/rem. Engng Sci. 24,297 (l%S). 2. S. S. Gluzman & V. N. Krainov, ht. Chem. Engng 7, 289 (1%7). 3. L. A. Gould & F. M. Schlaepfer, PreprintsIACC, 81 (1%7). 4. B. W. Iljin, D. Balzer, G. Reinig & V. Scholz, Preprints of IFAC Symp. on Digital Simulation of Continuous Processes. 11 (1971). 5. J. C. Grcutt & D. E. Lamb, Proc. 1st Znt.Cona. of the ZFAC, p. 274 (1960). 6. L. M. Patnaik, Optimization and multivariable computer control of an ammonia reactor. Ph.D. Thesis, Indian Institute of Science (1977). 7. L. M. Patnaik, N. Viswanadham & I. G. Sarma, Proc. LE.E.E. Conf. Decision and Control (1978). 8. L. G. Patnaik, I. G. Sarma & N: Viswanadham, Int. J. Sys. Sci. 10,225 (1979).
T& = 1, 0) = T;(a’ =0, 0): T&Y’= 1, e) = T&B).
(A@
Equations (AD-o-(B) describe the dynamics of the reactorexchanger combination. The details of the derivation of these equations can be found elsewhere[6]. APPENDMB State space model for the reactor: In Eq. (25) the elements of A(9 x 9), B(9 X3) and D(9 X2) are given by, al, = -4.019, aI2 = 5.12, au = -2.082, aI9 =0.87, a2, = -0.346, azz= 0.986,ax = -2.34, ar, = 0.97, a), = -7.909, a,* = 15.407, a,, = -4.069, aJ5 = -6.45, aj9 = 2.68, adI = -21.816, au = 35.606, ad,= -0.339,aa=-3.870,a,,=-17.8,
ae=7.39,aS1 =60.1%,
as2= 98.188,as3= -7.907, as4= 0.340,a,, = -53.008, aS9= 20.4, ab5= 94, a6 = -147.2, aa = 53.2,aT6= 94, a,, = -147.2, au,=12.8,ass=-31.6,a9~=12.8,a,=18.8,a99=-31.6. b,,=O.Ol, b,*=-0.011, bu=-0.051, bz, =0.003, bz,=-0.021,
APPENDIX A Material balance in the catalyst section: $Y
-cm-
vu+Y)*
clcy F(1 ty*)
bj, = O.OO!I, bD = -0.059, b4, = 0.024,bh2= -0.162, b,, = 0.068, bS2= -0.445.
00
Y).
(AU
d,, = 0.251,d,z = -1438.916,dz, = 0.147,dz2= -323.846, d,, = 0.405,ds2= -82.771, d4, = 1.13,d.,?= -22.637,
Energy balance in the empty tube section:
d5, = 3.87,d52= -5.456, d6, = 53.2,d,, = 18.8. z=$T,-T,). PO
642)
(All other elements of the above matrices are zero).