Chapter 9 – Tennessee Eastman Plant-wide Industrial Process Challenge Problem

Chapter 9

Tennessee Eastman Plant-wide Industrial Process Challenge Problem* The objective of this case study is to analyse and solve the published models for Tennessee Eastman plant-wide industrial process challenge problem (Downs and Vogel, 1993). Two versions of the models are used: * *

simplified model by Ricker and Lee (1995); full model by Jockenhovel et al. (2004).

For both the cases, the model equations are presented and analysed, a solution strategy is determined and, based on it, numerical solutions are obtained and compared with the corresponding published results.

9.1. PROCESS DESCRIPTION Figure 1 shows the flowsheet of the Tennessee Eastman (TE) process (Downs and Vogel, 1993). There are four unit operations in this flowsheet: an exothermic twophase reactor, a flash separator, a reboiler stripper and a recycle compressor. The TE process produces two products (G and H) and one undesired byproduct (F) from four reactants (A, C, D and E), according to the following reduced reaction scheme: AðgÞ þ CðgÞ þ DðgÞ !Gðl Þ

ð1Þ

AðgÞ þ CðgÞ þ EðgÞ !H ðl Þ

ð2Þ

1=3AðgÞ þ DðgÞ þ 1=3EðgÞ !F ðl Þ

ð3Þ

9.2. SIMPLIFIED MODEL 9.2.1. Process Model The objective of this simplified version of the TE process model is to capture the essential characteristics of the process assuming perfect control of the *

Written in conjunction with Prof Mauricio Sales-Cruz

273

274

[(Figure_1)TD$IG]

Product and Process Modelling: A Case Study Approach

FIGURE 1 The Tennessee Eastman process (Downs and Vogel, 1993).

Chapter | 9

275

Tennessee Eastman Plant-wide Industrial Process

heating/cooling taking place in the process. That is, the energy balance equations are ignored and only the mass balance–related equations are solved. This means that the dynamics of the compressor and reboiler stripper are ignored but the accumulations of mass in the purge unit and the feed mixer (holding tank) are considered. More details of the model simplification are given in Ricker and Lee (1995).

Mass balance equations (holdup of component i in each unit operation) 3 X dN i;r ¼ yi;6 F 6 yi;7 F 7 þ nij Rj dt j¼1

ði ¼ A; B; . . . ; H Þ

dN i;s ¼ yi;7 F 7 yi;8 ðF 8 þ F 9 Þ xi;10 F 10 dt

ði ¼ A; B; . . . ; H Þ

3 dN i;m X ¼ yij F j þ yi;5 F 5 þ yi;8 F 8 þ F *i yi;6 F 6 dt j¼1

dN i;p ¼ ð1 fi Þxi;10 F 10 xi;11 F 11 dt

ð4Þ

ð5Þ

ði ¼ A; B; . . . ; H Þ

ð6Þ

ði ¼ G; H Þ

ð7Þ

Conditional equations (thermodynamic relations) for reactor Pr ¼

H X

ð8Þ

Pi;r

i¼A

where Pi;r ¼

N i;r RT r V Vr

ði ¼ A; B; C Þ

Pi;r ¼ g ir xir Psat i ðT r Þ V Lr ¼

ði ¼ D; E; . . . ; H Þ

H X N i;r i¼D

ri

ð9Þ ð10Þ ð11Þ

Conditional equations (thermodynamic relations) for separator Ps ¼

H X i¼A

Pi;s

ð12Þ

276


where Pi;s ¼

N i;s RT s V Vs

ði ¼ A; B; CÞ

Pi;s ¼ g is xi;10 Psat i ðT s Þ ði ¼ D; E; . . . ; H Þ V Ls ¼

H X N i;s i¼D

ri

ð13Þ ð14Þ ð15Þ

Constitutive (property) models Psat i ð T s Þ ¼ Ai Psat i ð T r Þ ¼ Ai

Bi ðC i þ T s Þ

Bi ðC i þ T r Þ

ði ¼ A; B; C; D; E; F; G; H Þ ði ¼ A; B; C; D; E; F; G; H Þ

ð16aÞ ð16bÞ

Conditional equations (volume) for reactor, separator and purge V Vr ¼ V r V Lr

ð17Þ

V Vs ¼ V s V Ls

ð18Þ

V Lp ¼

H X N i;p i¼G

ri

ð19Þ

Conditional equations (pressure) for mixer Pm ¼

H X i¼A

N i;m

RT m Vm

ð20Þ

Defined relations (mole fractions) xi;r ¼ 0 xi;r ¼

N i;r H P

ði ¼ A; B; CÞ

ð21Þ

ði ¼ D; E; . . . ; H Þ

ð22Þ

ði ¼ A; B; CÞ

ð23Þ

ði ¼ D; E; . . . ; H Þ

ð24Þ

N i;r

i¼D

xi;10 ¼ 0 xi;10 ¼

N i;s H P i¼D

N i;s

Chapter | 9


xi;11 ¼ x GH

N i;p N G;p þ N H;p

ði ¼ G; H Þ

Constitutive (kinetic models) equations 42600 1:08 0:311 0:874 R1 ¼ a1 V Vr exp 44:06 PA;r PC;r PD;r RT r

277

ð25Þ

ð26Þ

19500 1:15 0:370 1:00 R2 ¼ a2 V Vr exp 10:27 PA;r PC;r PE;r RT r

ð27Þ

59500 R3 ¼ a3 V Vr exp 59:50 PA;r 0:77PD;r þ PE;r RT r

ð28Þ

Defined relations (vapour mole fractions) yi;6 ¼

N i;m H P N i;m

ði ¼ A; B; . . . ; H Þ

ð29Þ

i¼A

yi;8 ¼ yi;9 ¼ yi;7 ¼

Pi;s Ps

Pi;r Pr

ði ¼ A; B; . . . ; H Þ

ði ¼ A; B; . . . ; H Þ

ð30Þ ð31Þ

Vapour (mole fractions) mass balance on the stripper yi;5 F 5 ¼ fi zi;4 F 4 yi;5 F 5 ¼ fi xi;10 F 10 yi;5 F 5 ¼ fi xi;10 F 10

ði ¼ A; B; C Þ

ð12Þ

ði ¼ D; E; F Þ

ð13Þ

ðfG ¼ 0:07; fH ¼ 0:04Þ; ði ¼ G; H Þ

ð14Þ

Conditional (flow) relations F 6 ¼ b6

2413:7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjPm Pr jj M6

ð35Þ

F 7 ¼ b7

5722:0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjPr Ps jj M7

ð36Þ

F 10 ¼ F p10 F *10

ð37Þ

278


9.2.2. Model Analysis The model represented by Eqs. 4–36 can be written in nonlinear state variable form as: dx ¼ f ðx; u; dÞ dt y ¼ gðx; u; dÞ

x_ ¼

ð38Þ ð39Þ

where x_ is the state vector, u is a vector of known inputs, d is a vector of unmeasured inputs (disturbances and time-varying parameters) and y is the output vector. There are 26 state variables represented by x: xT ¼ N A;r ;N B;r ;.. .;N H;r ;N A;s ; N B;s ;. ..;N H;s ;N A;m ; N B;m ;... ;N H;m ;N G;p ;N H;p ð40Þ where Ni,r, Ni,s and Ni,m are the molar holdups of compound i in the reactor, the separator and the feed mixing zone, respectively; NGp and NHp are the molar holdups of compounds G and H in the product reservoir (stripper base), respectively. The u vector has 10 variables: uT ¼ F 1 ; F 2 ; F 3 ; F 4 ; F 8 ; F 9 ; F p10 ; F 11 ; T cr ; T cs ð41Þ where Fj is the molar flow rate of stream j, while Tcr and Tcs are the reactor and separator temperatures, respectively. This model consists of 26 ODEs (Eqs. 4–7) and 100 explicit AEs (Eqs. 8– 36). The variables are classified as: 26 dependent variables (Eq. 40), 100 explicit unknown algebraic variables (Pr, Pi,r, VLr, Ps, Pi,s, VLs, VLp, VVr, R, Pm, xi,r, xi,10, yi,5, yi,7, yi,6, yi,8, yi,9, Psat, F6, F7) and 83 parameters (NC, A, B, C, Mw, r, v, w, g r, gs, VrT, VsT, VvT, Rg1, Rg2, b6, b7, a), and 10 known manipulated variables and/or process-design variables (Eq. 41).

9.2.3. Specified Data The operational scenarios involve four sets of operational conditions specified through the set of manipulated variables vector u. This gives the base case as well as three other steady states. The 83 parameters that need to be specified are divided into three groups: (a) pure component property parameters (see Table 1); (b) system parameters Rg1 = 8.314; Rg2 = 1.987; b6 = 1; b7 = 1; a1 = ; a2 = ; a3 = ; (c) process parameters NC = 8; VrT = 36.8; VsT = 99.1; VvT = 150.0. The initial values of the 26 state variables are given in Table 2 and the known values for the 10 input variables (vector u – see Eq. 41) are given in Table 3.

Chapter | 9

Vapour pressure (Antoine equation)

Compound

Molecular weight

Liquid density (kg/m3)

gr

gs

w

A

B

C

A B C D E F G H

2.0 25.4 28.0 32.0 46.0 48.0 62.0 76.0

299 365 328 612 617

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1 1 1 1 1 1 0.07 0.04

20.81 21.24 21.24 21.32 22.10

-1444 -2114 -2144 -2748 -3318

259 266 266 233 250


TABLE 1 Component Physical Properties (at 100 C)

279

280


TABLE 2 Initial values of the state variables State Variables

Initial values

NA,m NB,m NC,m ND,m NE,m NF,m NG,m NH,m NA,r NB,r NC,r ND,r NE,r NF,r NG,r NH,r NA,s NB,s NC,s ND,s NE,s NF,s NG,s NH,s NF,p NG,p

51.7765845054 14.3100440019 42.4614755860 11.1168681570 30.1600733741 2.66082574139 5.68007202963 2.67883480622 4.71866674478 1.97858299828 3.43269304734 0.18201838615 10.3059014350 1.25329368022 66.0587540475 67.8869026640 28.8967272740 12.1166696783 21.0215212630 0.09658565923 5.90626944116 0.71825343663 20.4923456368 16.1950690323 21.7412698769 17.7363533518

TABLE 3 Specified Values of the Manipulated Variables, u, at Four Steady-state Conditions No.

Input

Units

Base case

Mode 1

Mode 2

Mode 3

1 2 3 4 5 6 7 8 9 10

F1 F2 F3 F4 F8 F9 F10 F11 Tcr Tcs

kmol/h kmol/h kmol/h kmol/h kmol/h kmol/h kmol/h kmol/h deg C deg C

11.2 114.5 98.0 417.5 1201.5 15.1 259.5 211.3 120.4 80.1

11.991 114.314 96.471 413.782 1441.021 9.497 253.563 210.885 123.074 92.078

13.848 22.948 174.679 383.109 1419.501 16.164 243.825 194.638 124.213 90.259

8.703 161.856 15.216 350.844 880.830 3.895 198.512 179.021 121.911 83.396

Chapter | 9

281


9.2.4. Numerical Solution Open-loop simulation results The DAE system of equations representing the simplified model [Eqs. 4–36] is solved with the BDF-method in ICAS-MoT (Sales-Cruz 2006) for the data given in Tables 1-3. Simulated values of the state variables and some output variables are given in Tables 4 and 5, respectively. These simulated values show a good match with those reported by Ricker and Lee (1995). Also, screenshots from ICAS-MoT highlighting the steady-state values of some of the state variable, x, and the right-hand sides of the ODEs (functions f) are seen in Figures 2 and 3. The model as implemented and solved in ICAS-MoT is given in Appendix A (see ch-9-1-te-dynamic-ricker.mot file). The simulated transient responses for four reactor outputs are shown in Figures 4 and 5 using the steady-state value as the initial condition to verify

TABLE 4 State Variables at Four Steady-state Conditions (all units are kmol) No.

State

Base case

Mode 1 (50/50)

Mode 2 (10/90)

Mode 3 (90/10)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

NAr NBr NCr NDr NEr NFr NGr NHr NAs NBS NCs NDs NEs NFs NGs NHs NAm NBm NCm NDm NEm NFm NGm NHm NGp NHp

4.722 1.9805 3.4354 0.18231 10.32 1.2572 66.06 67.87 28.895 12.119 21.022 0.09656 5.9052 0.71938 20.492 16.195 51.776 14.327 42.455 11.1 30.156 2.6668 5.684 2.6807 21.741 17.736

5.4536 3.6096 2.184 0.11589 7.7163 2.6844 59.961 58.48 28.565 18.906 11.44 0.0597 4.2311 1.4719 21.02 16.612 55.543 25.607 32.324 10.409 28.351 7.1577 8.5369 4.1394 21.741 17.736

6.0899 1.9562 2.4319 0.01556 9.9718 3.2853 10.927 95.896 32.039 10.291 12.794 0.00822 5.6803 1.8714 4.0319 28.679 59.874 14.1070 33.449 2.206 43.729 9.6449 1.7076 7.44 4.3633 32.037

4.4549 7.2066 1.47350 0.30814 2.1277 0.98381 123.15 14.052 25.365 41.032 8.3895 0.18371 1.3512 0.62479 41.458 3.6405 45.817 43.208 27.935 19.708 5.9845 2.0160 10.286 0.54295 38.999 3.535

282


TABLE 5 Output Variables at Four Steady States No. Description

Units

Base case

Mode 1 (50/50)

Mode 2 (10/90)

Mode 3 (90/10)

1 2 3 4 5 6 7 8

kPa % kPa % % kPa kscmh mol%

2705.0 75.0 2633.7 50.0 50.0 3102.2 42.27 32.19

2800.0 65.0 2705.7 50.0 50.0 3325.7 47.27 32.28

2800.0 65.0 2705.4 50.0 50.0 3327.5 46.03 34.78

2800.0 65.0 2764.9 50.0 50.0 2995.6 32.04 29.47

mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol%

8.91 26.4 6.9 18.75 1.66 32.96 13.82 23.98 1.26 18.58 2.26 4.84 2.30 53.72 43.83

14.88 18.79 6.05 16.48 4.16 32.82 21.72 13.14 0.88 15.93 5.54 6.70 3.27 53.84 43.92

8.19 19.43 1.28 25.4 5.60 36.63 11.77 14.63 0.13 22.37 7.37 1.32 5.79 11.66 85.64

27.79 17.97 12.68 3.85 1.29 27.86 45.08 9.22 2.18 3.94 1.82 9.4 0.5 90.1 8.17

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Reactor pressure Reactor liquid level Separator pressure Separator liquid level Stripper bottoms level Stripper pressure Reactor feed flow rate A in the reactor feed (stream 6) B in the reactor feed C in the reactor feed D in the reactor feed E in the reactor feed F in the reactor feed A in purge (stream 9) B in purge C in purge D in purge E in purge F in purge G in purge H in purge G in product (stream 11) H in product

whether the process moves away from the steady states. It can be noted that the steady state is maintained (during the short time simulated, as shown in Figures 4–5); the small variation (note the scales) is probably due to the precision of the convergence criteria used. However, continuing the simulation for a longer time scale (not shown in the figures) the system moves away from the steady state, indicating an unstable state.

Closed-loop Simulation Results The model for a simple PI-controller is implemented, considering as measured (controlled) output y, XMEAS_15 = stripper liquid product flow (with a step change of + 15) and as manipulated (actuator) input u, F11 = stripper exit feed flow rate. The control law used is as follows (for set point, Setpt ¼ 65): Error and time derivative of the error e ¼ Setpt X meas ð15Þ

ð42Þ

Chapter | 9


283

[(Figure_2)TD$IG]

FIGURE 2 State variables at the base case steady state (screen shot from ICAS-MoT).

de 1 dN G;p 1 dN H;p ¼ 22:58 þ dt rG dt rH dt

ð43Þ

Tuning 1 0:3

ð45Þ

t I ¼ 0:05

ð45Þ

Kp ¼

Controller: Stripper liquid product flow % (XMV(8))

Z e F 11 ¼ F 11;ref erence K p e þ dt tI or

dF 11 de e þ ¼ K p dt t I dt

ð46Þ

ð47Þ

Adding Eqs. 42–47 to the model allows a closed-loop simulation to be performed. The BDF-method from ICAS-MoT is used as the DAE-solver. The obtained simulation results are shown in Figures 6-8, where it can be seen that

284


[(Figure_3)TD$IG]

FIGURE 3 Function f (right-hand side of Eq 38) values at base case steady state (screenshot from ICAS-MoT).

the actuator F11, reaches a steady-state value, while the measured (controlled) output XMEAS_15 reaches the given set point = 65. The other outputs (XMEAS_6, XMEAS_8 and XMEAS_12) also reach almost constant values (note the small scale), thereby achieving the desired control objectives.

9.2.5. Discussion Check the eigenvalues of the Jacobian matrices at the steady states to verify which of the steady states are unstable. Perform a sensitivity analysis to identify the most sensitive actuators (manipulated variables).

Nomenclature (for Section 9.2) d F F* F10p F10* Nij

unmeasured input vector molar flow rate, kmol/h molar ‘pseudo-feed’, kmol/h (added at the feed mixing point) apparent stream 10 flow rate, as indicated by the valve position, kmol/h bias adjustment for stream 10, kmol/h total molar holdup of i-th component in the j-th unit, kmol

Chapter | 9

FIGURE 4 Dynamic behaviour of reactor outputs: (a) XMEAS_6 = reactor feed flow rate; (b) XMEAS_7 = reactor pressure.


[(Figure_4)TD$IG]

285

286

[(Figure_5)TD$IG] Product and Process Modelling: A Case Study Approach

FIGURE 5 Dynamic behaviour of reactor outputs: (c) XMEAS_8 = reactor liquid level; (d) XMEAS_12 = separator liquid level.

Chapter | 9


287

[(Figure_6)TD$IG]

FIGURE 6 Dynamic behaviour of input F11 = stripper exit feed flow-rate.

Psat p P Rnet R t T u V x x y y z

vapour pressure partial pressure, kPa total pressure, kPa molar rate of reaction, kmol/h gas constant = 1.987 cal/gmol-K in Eqs. 26–28; otherwise it is equal to 8.314 kJ/kmol-K time, h absolute temperature, K known input vector total volume, m3 state vector mole liquid fraction output vector mole vapour fraction mole feed fraction

Subscript r reactor s separator m mixing zone p product L liquid V vapour i component j stream

288

[(Figure_7)TD$IG] Product and Process Modelling: A Case Study Approach

FIGURE 7 Dynamic behaviour of reactor outputs: (a) XMEAS_15 = stripper level; (b) XMEAS_6 = reactor feed flow-rate.

Chapter | 9

FIGURE 8 Dynamic behaviour of reactor outputs: (c) XMEAS_8 = reactor liquid level; (d) XMEAS_12 = separator liquid level.


[(Figure_8)TD$IG]

289

290


Greek symbols a adjustable parameter used in reaction rate equations (Eqs. 26–28); dimensionless with nominal value of unity b used to adjust flow/pressure drop relation in Eqs. 35–36 for streams 6–7 (nominal values are unity) g activity coefficient w stripping factor y stoichiometric coefficient r molar density, mol/m3 x GH purity of G and H in the product (as a fraction)

9.3. FULL (PROCESS) MODEL 9.3.1. Process Model This model considers the mass as well as the energy balance. The model derivation and assumptions can be found in Jockenhovel et al. (2004).

Mass balances for the mixer, reactor, separator and purge (unit operations operating in the dynamic mode) X dN i;m ¼ yij F j yi;6 F 6 dt j¼1;2;3;5;8

ði ¼ A; B; . . . ; H Þ

3 X dN i;r ¼ yi;6 F 6 yi;7 F 7 þ nij Rj dt j¼1

dN i;s ¼ yi;7 F 7 yi;8 ðF 8 þ F 9 Þ xi;10 F 10 dt

ði ¼ A; B; . . . ; H Þ

ði ¼ A; B; . . . ; H Þ

dN i;p ¼ ð1 fi Þ xi;10 F 10 þ yi;4 F 4 xi;11 F 11 dt

ði ¼ G; H Þ

ð48Þ

ð49Þ

ð50Þ

ð51Þ

Energy balances for the mixer, reactor, separator and purge (unit operations operating in the dynamic mode) H X i¼A

! N i;m cp;vap;i

! H X X dT m ¼ Fj yi;j cp;vap;i T j T m dt i¼A j¼1;2;3;5;8

ð52Þ

Chapter | 9


H X N i;r cp;i

!

i¼A

dT r dt

H X

¼ F6

291

! yi;6 cp;vap;i ðT 6 T r Þ

i¼A 3 X Q_ r DH rk Rk

ð53Þ

k¼1 H X

! N i;s Cpi

i¼A

dT s dt

H X yi;7 Cpvi

¼ F7

! ð54Þ

i¼A

ðT r T s Þ H 0 V s Q_ s H X

! N i;p Cpi

i¼G

! H X dT p ¼ F 10 xi;10 Cpi T s T p dt i¼A ! H X v þF 4 yi;4 Cpi T 4 T p H 0 V p þ Q_ p i¼A

F4

H X

! yi;4 cp;vap;i

T 4 T p H 0 V p þ Q_ p

ð55Þ

i¼A

Conditional equations (mixer) Pm ¼

H X

N i;m

i¼A

yi;6 ¼

N i;m H P

RT m Vm

ð56Þ

ði ¼ A; B; . . . ; H Þ

ð57Þ

N i;m

i¼A

Conditional equations (reactor) Pr ¼

H X

ð58Þ

Pi;r

i¼A

where Pi;r ¼

N i;r RT r V Vr

Pi;r ¼ g ir xir Psat i ðT r Þ 3 ð T Þ ¼ 1x10 exp Ai þ Psat i

ði ¼ A; B; C Þ

ð59Þ

ði ¼ D; E; . . . ; H Þ

ð60Þ

Bi Ci þ T r T *

ði ¼ D; E; . . . ; H Þ

ð61Þ

292


Flow rates as a function of pressure drop kmol pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 6 ¼ 0:8334 pffiffiffiffiffiffiffiffiffiffi jjPm Pr jj s MPa kmol pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 7 ¼ 1:5344 pffiffiffiffiffiffiffiffiffiffi jjPr Ps jj s MPa

ð62Þ ð63Þ

Enthalpy calculation DH Rj ¼

H X

H i ni;j þ H 0 F j ;

H i ¼ cp;i T r T * ðj ¼ 1; 2; 3Þ ð64Þ

with

i¼A

Heat exchanger

Q_ r ¼ mCW;r cp;CW T CW;r;out T CW;r;in # DT DT 1;r 2;r Q_ r ¼ UAr ln DT 1;r =DT 2;r

ð65Þ

"

ð66Þ

where DT 1;r ¼ T r T CW;r;in ;

DT 2;r ¼ T r T CW;r;out

ð67Þ

Defined relations (reactor) xi;r ¼ 0 xi;r ¼

ði ¼ A; B; CÞ

ð68Þ

ði ¼ D; E; . . . ; H Þ

ð69Þ

N i;r H P

N i;r

i¼D

yi;7 ¼

Pi;r Pr

ði ¼ A; B; . . . ; H Þ

V Lr ¼

H X N i;r i¼D

ri

V Vr ¼ V r V Lr


Defined relations (separator) Ps ¼

H X

Pi;s

ð73Þ

i¼A

where Pi;s ¼

N i;s RT s V Vs

ði ¼ A; B; CÞ

ð74Þ

Chapter | 9


Pi;s ¼ gis xi;10 Psat i ðT s Þ ði ¼ D; E; . . . ; H Þ

293

ð75Þ

Heat transfer T8 ¼ Ts

H0V s ¼

H X

Pm Ps

1k k

ð76Þ

xi;10 F 10 H vap;i

ð77Þ

i¼D

Q_ s ¼ mCW;s cp;CW T CW;s;out T CW;s;in "

DT 1;s DT 2;s Q_ s ¼ UAs ln DT 1;s =DT 2;s

ð78Þ

# ð79Þ

where DT 1;s ¼ T s T CW;s;in ; yi;8 ¼ yi;9 ¼

Pi;s Ps

xi;10 ¼ 0 xi;10 ¼

N i;s H P N i;s

DT 2;s ¼ T s T CW;s;out ði ¼ A; B; . . . ; H Þ

ð80Þ ð81Þ

ði ¼ A; B; CÞ

ð82Þ

ði ¼ D; E; . . . ; H Þ

ð83Þ

i¼D

V Ls ¼

H X N i;s i¼D

ri

V Vs ¼ V s V Ls H0V p ¼

H X

yi;5 F 5 yi;4 F 4 H vap;i


i¼D

kJ _ steam Q_ p ¼ 2258:717 m kg

ð87Þ

Defined relation (purge) V Lp ¼

H X N i;p i¼D

ri

ð88Þ

294


Other defined relations fi ¼

3 X

ai;j ðT s 273Þ

ði ¼ D; E; . . . ; H Þ

ð89Þ

j¼0

fi ¼ 1

ði ¼ A; B; C Þ

F 5 ¼ F 10 þ F 4 F 11

H X dN i;p i¼G

yi;5

fi yi;4 F 4 þ xi;10 F 10 ¼ F5

xi;11 ¼

xi;11 ¼

1

j¼D

! xj;11

N i;p H P

dt

ði ¼ A; B; . . . ; H Þ

yi;4 F 4 þ xi;10 F 10 yi;5 F 5 F 11 F X

ð90Þ ð91Þ

ð92Þ

ði ¼ D; E; F Þ

ð93Þ

ði ¼ G; H Þ

ð94Þ

N j;p

j¼D

Constitutive equations

42600 1:08 0:311 0:874 R1 ¼ a1 V Vr exp 44:06 PA;r PC;r PD;r RT r

ð95Þ

19500 1:15 0:370 1:00 R2 ¼ a2 V Vr exp 10:27 PA;r PC;r PE;r RT r

ð96Þ

59500 R3 ¼ a3 V Vr exp 59:50 PA;r 0:77PD;r þ PE;r RT r

ð97Þ

9.3.2. Model Analysis The above full-model can be written, in nonlinear state variable form as, x_ ¼

dx ¼ f ðx; u; dÞ dt

y ¼ gðx; u; dÞ

ð98Þ ð99Þ

Chapter | 9


295

where x is the state vector, u is a vector of known (manipulated) inputs, d is a vector of unmeasured inputs (disturbances and time-varying parameters) and y is the output vector. There are 30 state variables arranged as follows: xT ¼½N A;r ; N B;r ; . . . ; N H;r ; N A;s ; N B;s ; . . . ; N H;s ; N A;m ; N B;m ; . . . ; N H;m ; N G;p ; N H;p ; T m ; T r ; T s ; T p

ð100Þ

where Ni,r is the molar holdup of chemical i in the reactor, and Ni,s, Ni,m and Ni,p are in the separator, feed mixing zone and product reservoir (stripper base), respectively. The u vector contains 14 variables < mml : mathaltimg ¼ 105:gif} uT ¼½F 1 ; F 2 ; F 3 ; F 4 ; F 8 ; F 9 ; F 10 ; F 11 ; T CW;r;in ; >

T CW;r;out ; T CW;s;in ; T CW;s;out ; mCW;r ; mCW;s

ð101Þ

where Fj is the molar flow rate of stream j, (TCW,s,in, TCW,r,in) and (TCW,s,out, TCW, r,out) are the cooling water inlet and outlet temperatures in the reactor and separator, respectively, and mCW,s and mCW,r are the cooling water flow rates in the reactor and separator, respectively.

9.3.3. Specified Data The operational scenarios involve four sets of operational conditions specified through the set of manipulated variables vector u. This gives the base case as well as three other steady states. The 127 parameters that need to be specified are divided into three groups (a) pure component property parameters (see Table 6); (b) system parameters Rg1 = 8.314; Rg2 = 1.987; b6 = 1; b7 = 1; a1 = 1 ; a2 = 1; a3 = 1 ; (c) process parameters NC = 8; VrT = 36.8; VsT = 99.1; VvT = 150.0. The initial values of the 30 state variables are given in Table 7 and the known values for the 14 input variables (vector u – see Eq. 101) are given in Table 3 and values of the additional variables are: TCW,r,in = 308.00 K; TCW,r,out =367.59 K; TCW,s,in = 313.00 K; TCW,s,out = 350.45 K; mCW,s = 93.37 mol/s; mCW,s = 49.37 mol/s. The known measured values for a selection of the output variables are given in Table 8. The simulated values for these 23 variables are compared to these measured values to validate the model.

296

TABLE 6 Component Physical Properties (at 100 C) Liquid density (kg/m3)

Liquid heat capacity (kJ/kg. C)

Vapour heat capacity (kJ/kg. C)

Heat of vaporisation (kJ/kg)

Vapour pressure (Antoine equation) A

B

C

A B C D E F G H

2.0 25.4 28.0 32.0 46.0 48.0 62.0 76.0

299 365 328 612 617

7.66 4.17 4.45 2.55 2.45

14.60 2.04 1.05 1.85 1.87 2.02 0.712 0.628

202 372 372 523 486

20.81 21.24 21.24 21.32 22.10

-1444 -2114 -2144 -2748 -3318

259 266 266 233 250


Compound

Molecular weight

Chapter | 9

297


TABLE 7 Initial values of the state variables State variables

Initial values

NA,m NB,m NC,m ND,m NE,m NF,m NG,m NH,m Tm NA,r NB,r NC,r ND,r NE,r NF,r NG,r NH,r Tr NA,s NB,s NC,s ND,s NE,s NF,s NG,s NH,s Ts NG,p NH,p Tp

48.83 13.49 40.03 10.44 28.48 2.51 5.40 2.51 359.25 5.20 2.29 4.65 0.12 7.45 1.15 56.10 59.80 393.55 27.50 12.10 24.60 0.0836 5.86 0.901 24.10 19.80 353.25 20.40 17.30 338.85

TABLE 8 Elements of the Output Vector (y)

No.

Description

Units

Downs and Vogel ( XMEAS)

1 2 3 4 5 6

Reactor pressure Reactor liquid level Separator pressure Separator liquid level Stripper bottoms level Stripper pressure

kPa % kPa % % kPa

7 8 13 12 15 16 (continued)

298


TABLE 8

(continued)

No.

Description

Units

Downs and Vogel ( XMEAS)

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Reactor feed flow rate A in the reactor feed (stream 6) B in the reactor feed C in the reactor feed D in the reactor feed E in the reactor feed F in the reactor feed A in purge (stream 9) B in purge C in purge D in purge E in purge F in purge G in purge H in purge G in product (stream 11) H in product

kscmh mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol% mol%

6 23 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41

[(Figure_9)TD$IG]

FIGURE 9 Simulated steady-state values of some state variables (screenshot from ICASMoT).

Chapter | 9


299

[(Figure_0)TD$IG]

FIGURE 10 Function f [right-hand side of Eq 98] values at steady state (screenshot from ICAS-MoT).

9.3.4. Numerical Solution Open-loop Simulation Results The DAE system of equations representing the simplified model [Eqs. 48–97] is solved with the BDF-method in ICAS-MoT (Sales-Cruz, 2006), for the data given in Tables 6-8. The simulated results as screenshots from ICAS-MoT highlighting the steady-state values of some of the state variable, x, values and

300

[(Figure_1)TD$IG]

Dynamic behaviour of reactor outputs: (a) XMEAS_6 = reactor feed flow rate; (b) XMEAS_7 = reactor pressure.


FIGURE 11

Chapter | 9

FIGURE 12 Dynamic behaviour of reactor outputs: (c) XMEAS_8 = reactor liquid level; (d) XMEAS_9 = reactor temperature.


[(Figure_2)TD$IG]

301

302


the right-hand sides of the ODEs (functions f Eq. 98) are seen in Figures 9 and 10. The model as implemented and solved in ICAS-MoT is given in Appendix 1 (see ch-9—2-te-dynamic-complete.mot file). The dynamic behaviour for four reactor outputs is seen in Figures 11 and 12, where the simulation results were generated using the steady-state value as initial condition. As can be seen from these figures the steady state almost remains constant (for a short time); the small variation (note the scales on the y-axis) is probably due to numerical accuracy of the computer.

9.3.5. Discussion Check the eigenvalues of the Jacobian matrices at the steady states to verify whether steady state is stable or unstable. Try to develop a control scheme to keep the operation stable.

Nomenclature (for 9.3) Cp,CW Cp,i Cp,vap,i d Fj Hi H0 DHRj mCW,r mCW,s Nik Pi,j Pisat Pk Qk Ri R t Tcw,k,in Tcw,k,out Tk T* UA u Vk VL,k

specific heat capacity of cooling water, kJ kg1 K1 specific heat capacity of component i in liquid phase, kJ kg1 K1 specific heat capacity of component i in vapour phase, kJ kg1 K1 unmeasured input vector molar flow rate of stream j, kmol h1 enthalpy of component i, kJ reference enthalpy, kJ exothermic heat, kJ kmol1 cooling water flow rate reactor, kg h1 cooling water flow rate separator, kg h1 total molar holdup of component i in the unit k (k = m, r, s, p), kmol partial pressure of component i in the unit k (k = m, r, s, p), kPa saturation pressure of component i, kPa total pressure in the unit k (k = m, r, s, p), kPa energy removed from the unit k (k = r, s, p), kW reaction conversion component i, kmol h1 gas constant time, h cooling water inlet temperature in the unit k (k = r, s), K cooling water outlet temperature in the unit k (k = r, s), K temperature of the unit k (k = m, r, s, p), K absolute temperature, K specific heat transfer rate, kW K1 known input vector total volume of the unit k (k = m, r, s, p), m3 liquid volume in the unit k (k = m, r, s, p), m3

Chapter | 9

VV,k xi,k x yi,k y


303

vapour volume in the unit k (k = m, r, s, p), m3 mole liquid fraction of component i in the unit k (k = m, r, s, p) state vector mole vapour fraction of component i in the unit k (k = m, r, s, p) output vector

Subscript r reactor s separator m mixing zone p product (stripper) L liquid V vapour i component j stream Greek symbols a adjustable parameter used in reaction rate equations. g activity coefficient w stripping factor y stoichiometric coefficient r molar density, mol m3

REFERENCES Downs, J.J., Vogel, E.F., 1993. A Plant-Wide Industrial Process Control Problem. Computers and Chemical Engineering. 17, 245–255. Jockenhovel, T., Biegler, L.T., Wachter, A., 2004. Dynamic Optimization of the Tennessee Eastman Process Using OptControlCentre. Computers and Chemical Engineering. 27, 1513–1531. Ricker, N.L., Lee, J.H., 1995. Nonlinear Modelling and State Estimation for the Tennessee Eastman Challenge Process. Computers and Chemical Engineering. 19, 983–1005. Sales-Cruz, M., 2006. Development of a computer aided modelling system for bio and chemical process and product design, PhD-thesis. Technical University of Denmark: Lyngby, Denmark.

Chapter 9 – Tennessee Eastman Plant-wide Industrial Process Challenge Problem

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