- Email: [email protected]
Sensitivity in optimization of a reactor system with deactivating catalyst
European Symposiumon Computer Aided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
517
Sensitivity in Optimization of a Reactor System with Deactivating Catalyst .Ingvild Lgtvika, Magne Hillestad b and Terje Hertzberg a* aDepartment of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. bStatoil R&D Center, N-7005 Trondheim, Norway. An optimal operating strategy for a fixed bed reactor system with a slowly deactivating catalyst is found. The process studied is Lurgi's methanol synthesis. A rigorous model of the reactor and the loop has been used and the actual control variables have been optimized. In this paper, we look specifically at sensitivity in the optimization. The parameters in the catalyst deactivation model are believed to be uncertain. The effect of variations in the deactivation parameters on the optimal operating variables and the objective function has been studied by a first order error propagation-approach.
1. Introduction Catalyst deactivation occurs in practically all fixed bed reactors. The two main questions in operation of fixed bed reactors with deactivation are when to change catalyst, and how to compensate for deactivation between the catalyst changes. This work looks at the last problem only, because in the methanol synthesis the time for catalyst change is decided by factors outside the process section. Much work has been done on optimal operation of fixed bed reactors undergoing catalyst deactivation. Some central references are [ 1-5]. Most of the earlier work [3-5] has focused on optimal distributed control, e.g. optimization of reactor temperature distributed in time and space. A more realistic approach is taken in this work. The actual time varying control variables in the reactor system, the recycle rate and coolant temperature, are optimized. This study also uses a detailed, realistic model of the total reactor system. Parts of this work are published earlier [6,7]. This paper looks specifically at sensitivity in the optimization with regards to the deactivation model. In the methanol synthesis, synthesis gas (CO, CO2 and H2) is converted to methanol over a Cu/ZnO/A1203 catalyst. The following exothermic reactions occurs [8]: CO 2 +3H 2 r CO + H 2 0 r
CH3OH+ H20 CO 2 + H 2
In the Lurgi reactor [9] the catalyst is packed in vertical tubes surrounded by boiling water. The reaction heat is transferred to the boiling water and steam is produced. Efficient heat transfer gives small temperature gradients along the reactor. Typical operating conditions are 523 K and 80 bar. The pressure of the boiling water is controlling the reactor temperature. Because o f the quasi-isothermal reaction conditions and high catalyst selectivity, only small amounts of byproducts are formed. Methanol conversion is limited by equilibrium. Unreacted synthesis gas is separated from crude methanol, compressed and recycled. The Lurgi reactor system consists of two parallel reactors with a common steam drum, a feed/effluent interchanger, a cooler, a methanol separator and a recycle
* Author to whom correspondence shoud be adressed, email: [email protected]
518
compressor. The control variables are the coolant temperature and the recycle rate. A flowsheet of the process is shown in Fig. 1. ....................................... S.'/n|hesk~ ~t"~
MP Steam
Cruc!e melhonol
Fig. 1: The meth~ol synthesisl..... The Cu/ZnO/A1203 catalyst can deactivate irreversibly because of chemical poisoning or thermal sintering [10-12]. Sintering is the cause of deactivation under normal operation. The catalyst poison sulfur is removed earlier in the process. Chlorine and heavy metals act as catalyst poisons but is not likely to occur in the process gas. Sintering is caused by high temperatures and increases when the catalyst is exposed to high partial pressure of water [ 10,11 ]. Copper is the active phase in the catalyst. During sintering, copper atoms migrate to larger agglomerates. This leads to increasing crystal size and decreasing active area. The sintering mechanism changes at higher temperatures; copper crystals migrate together, causing severe deactivation and loss of selectivity. Reported temperatures for when severe deactivation starts range from 543 K [9] to 670 K [11 ]. The lowest temperature is chosen as a constraint in the optimization. The catalyst deactivates slowly under normal operating conditions, and after 3 to 4 years, the activity is so low that the catalyst has to be replaced. A shut down of a part of the plant is necessary to change catalyst. The maintenance plan of the plant and the natural gas supply determine when to replace the catalyst. This is why the catalyst lifetime is not optimized in this study. A common opera7ion strategy is to increase the temperature at the end of the catalyst lifetime to compensate for decreased activity [9]. The decisions regarding temperature increase are based on the experience of the operators. Increased temperature gives higher reaction rates, but also higher deactivation rates. This makes coolant temperature an interesting optimization variable. Increased recycle rate leads to lower conversion per pass in the reactor, but higher overall conversion in the loop.
2. Modeling and Optimization The catalyst deactivation was the only dynamic effect included in the model while steady state was assumed for the other states. This pseudo steady state assumption is reasonable because the dynamics in composition, temperature and pressure are much faster than the deactivation dynamics. A twodimensional, heterogeneous reactor model with the following assumptions was used: * Dispersion of mass in axial and radial directions is negligible. 9 Dispersion of heat in axial direction is negligible. 9 Isotherm catalyst pellets. 9 Inter-facial temperature gradients are negligible. 9 Viscous flow in catalyst pellets is negligible. The LHHW-type reaction kinetic by Bussche and Froment [8] was selected. The fifth order deactivation kinetic is based on Skrypek et al. [ 11] with deactivation energy Ed from Cybulski [ 13]. The activity is scaled to fit a temperature profile for a Lurgi reactor with deactivated catalyst [9].
519
1)/
a,t, dt
~Rg
- "~"0 "a(t)5
a(0) =a0 a ' ( t ) = 1 - ~'-atto)a ao
This model was selected because it predicts a reasonable long catalyst lifetime. Few deactivation models for this catalyst are published, and they predict quite different deactivation rates. Most of the models were developed in laboratory scale, and therefore predicts too fast deactivation [13-16]. One model [ 17] considers the reaction gas composition, but the mechanism that is assumed is in conflict with other literature. This model also predicts too fast deactivation. Lumped steady state models were used to describe the remaining unit operations in the reactor loop. Soave-Redlich-Kwong equation of states was used to find the phase equilibrium in the separator [ 18]. The task of finding an optimal operation strategy was formulated as a nonlinear dynamic optimization problem: tl
Max "Profit = ] (FMeOH " PMeOH + Fsteam " esteam )it
Tc(t),R(t)
to
s.t.:
Zrmeaa~ctor <_ 5 4 3
K
513K
2
5
4 ...............................................
I~
:
:
:
300
:
:
:
:
:
:
600
900
Time
[days]
:
:
:
1200
:
53~ t 525~
,
,.........
L .........
..
300
600
900
Time
[days]
Fig. 2: Optimal coolant temperature and recycle rate (- opt ' ref)
1200
520
3. Sensitivity Analysis We wanted to investigate the sensitivity in the optimization results with regards to the deactivation model. Two factors were studied, the rate constant Kd and the activation energy Ed. The effects and cross effects of the factors were found by varying the factors according to the 22 experimental design in Table l [23]. The effects and cross effects of varying the deactivation parameters on the optimal coolant temperatures and the scaled objective function are shown in Table 2. No effects were observed on the optimal recycle rate. The Kd effect on coolant temperature is positive. The temperatures are increased to compensate for the increased deactivation caused by increased Kd. The Ed effect on coolant temperatures is not significant in all intervals. It is negative and shrinking in the tree first intervals, resulting in a steeper profile that starts at a lower temperature. The temperatures are lowered in the start were most of the deactivation takes place, because the increase in Ed makes the deactivation rate more sensitive to temperature. Both effects on the objective function are negative as expected. A positive cross effect on the objective function exists, meaning that the Kd effect is lager when Ed is at the low level. Both the objective function and the coolant temperature were checked for nonlinear effects, with a negative result. The center point was close to the response surface for all responses.
Table 1" Experimental design. 0.8*Ed Ed 1.2*Ed 0.8*Kd
Kd 1.2*Kd
--
-'+
00 +--
Table 2: Effects of the deactivation parameters on the optimization results.
++
Effects
Response Obj TJ Tc2 TJ Tc4 Tc5 Tc6 Tc7 TJ
Mean 21.00 520.3 523.9 526.1 527.6 529.0 530.9 530.9 531.7
Kd -1.39 0.7_-/-0.4 0.9_+0.4 0.9_+0.4 1.1_+0.4 0.8_+0.4 0.8_+0.4 1.0-+0.4 0.9_+0.4
Ed KaxEd -0.73 0.19 -1.4_--t-0.4 -1.0-+0.4 -0.7_+0.4
The propagation of statistical uncertainty from the deactivation parameters to the optimization results were calculated from the equation [23]"
Var(f)=l~i2Var(Kd)+ ~~f-~l)~21Z. Var(Ed)+2I_~f d Y_)f) Var(Kd ) =
Var(Ed ) =CYEd
Ea)
CoV(Kd ,Ed ) = P Kd,Edt~Kd~ Ed
The partial derivatives were estimated from the effects in Table 2. Different cases with combinations of uncertainties and correlation factors in the deactivation parameters were used, see Table 3. The propagated uncertainties in objective function and coolant temperatures are shown in Table 4. The uncertainties in the scaled objective function is relative large. Converted to percent increased profit from optimization, the uncertainties are +0.2 percent in case 4 and +0.1 percent in case 2. The uncertainties in the optimal coolant temperatures are relative small. Varying the correlation coefficient had small effects on the uncertainties in both the objective function and the coolant temperatures. It is interesting that the uncertainties in coolant temperatures in case 3 and 4 are of the same size as the optimization accuracy of +0.4 K. This point is illustrated in Fig. 3. From these results, it can be concluded that 20 percent standard deviation in the deactivation parameters is
521
sufficient for optimization purposes. More accurate deactivation parameters will not lead to a more accurate optimal operation strategy. This can be used as a target for uncertainty in a future model development. In adition to uncertanty in calculation of the optimal coolant temperatures, there is also uncertainty in implementation of the optimal coolant temperatures. The implementation uncetanty is estimated to +0.5 K. This is another reason not to use a more accurate deactivation model for optimization. Table 3: Standard deviation in the deactivation parameters Case (YKd (YEd PKd,Ed 10 10 20 20 40 40
1
2 3 4 5 6
10 10 20 20 40 40
Table 4: Standard deviation in optimization results. Center O'respo.se Response Point Case Case Case Case 20.86 520.03 523.47 525.86 525.86 527.34 528.67 529.94 530.89
Obj
0.5 0.95 0.5 0.95 0.5 0.95
Tc1 Tc2
Tc3 Tc4 Tcs Tc6 Tc 7
Tc8
532
'
3
4
Case 6 +2.10 +0.77 +0.32 +0.32 _+0.73 +0.61 +0.80 +1.00 +1.29
........ i|
528
r.
524
-
.
.
.
.
-
.
- : _- I: "
.....
.
~-" - = "
522 520
2
'
53O ~'
Case 5 +0.47 +.52 +0.93 • +1.86 +0.30 +0.19 +0.61 +0.38 + 1 . 2 1 +0.24 +0.08 +0.48 +0.16 +0.95 _+0.21 _+0.08 +0.41 +0.16 +0.82 _+0.24 _+0.18 _+0.48 +0.37 _+0.96 +0.18 +0.15 +0.36 +0.31 +0.72 _+0.20 _+0.20 +0.40 +0.40 +0.80 +0.25 +0.25 +0.50 +0.50 +1.00 +0.29 _+0.32 _+0.58 _+0.64 +1.15 1
......
4
0
I
500
!
.........
T i m e [ d a y s ] 1000
1500
Figi3" Standard deviation in optimal coolant temperature in case 4 ( m prop. err. ' opt. acc.).
4. Conclusions The operation of the methanol synthesis with catalyst decay has been optimized, and the sensitivity in the optimization results with regards to the deactivation model has been investigated by a first order error propagation - approach. The optimization results have been published earlier [7]. The effects of variations in the two parameters in the deactivation model on optimal coolant temperature profile and the objective function was found. Both the rate constant and the activation energy had a negative effect on the objective function, and a cross effect between the to factors was observed. The rate constant had a positive effect on the optimal coolant temperature profile. The activation energy had a negative effect on the first intervals of the coolant temperature profile, resulting in a steeper profile. The propagated uncertainties was relative large in the scaled objective function and relative small in the optimal coolant temperatures. It was found that the uncertainties in coolant temperatures with 20 percent standard deviation in the deactivation parameters are of the same size as the optimization accuracy. From these results, it can be concluded that 20 percent deviation in the deactivation
522
parameters is sufficient for optimization purposes. This can be used as a target for uncertainty in a future model development.
5. Notation a a
Ed [J/mol] Kd [da y-i] Tc(t) [K] Tre actor max
[K]
R [mol/mol] Profit [USD]
Catalyst activity Scaled catalyst activity Activation energy for deactivation Rate constant for deactivation Coolant temperature Maximum reactor temperature Recycle rate Profit over catalyst lifetime
FMeOH[tons/day] [;'Steam[tons/day] Pi [USD/ton] Qcomp[ k W ] tl [days]
Var09 Cov~ g) (yf pf, g
Production rate of methanol Net production rate of steam Price of product i Compressor power Catalyst lifetime Variance Co variance Standard deviation Correlation coefficien
References 1. A.F. Ognue and W.H. Ray, AIChE J., 17(1), (1971) 43. 2. G. Buzzi-Ferraris, E. Facchi, P. Forzatti and E.Tronconi, Ind. Eng. Chem. Proc. Des. Devel. 23 (1984) 126. 3. J.R. Gonzalez-Velasco, M.A. Gutierrez-Ortiz, J.I. Guiterrez-Ortiz and J.A Gonzales-Marcos, Canad. J. Chem. Eng., 7 (1987) 65. 4. M.T. Asrar and A.S. Moharir, Comp.Chem.Eng., 15 (1991) 7. 5. R.S Dixit and N. Grant., Canad. J. Chem. Eng., 74 (5) (1996) 652. 6. I. Lcvik, M. Hillestad and T. Hertzberg, Comp. Chem. Eng. Suppl., 22 (1998) 707. 7. I. Lcvik, M. Hillestad and T. Hertzberg, Comp. Chem. Eng. Suppl., 23 (1999) 839. 8. K.M. Vanden Bussche and G.F. Froment, J. Catal., 161 (1996)1. 9. E. Supp, Hydrocarbon Proc., 3 (1981) 71. 10. H.H. Kung, Catal. Today, 11 (1992) 443. 11. J. Skrzypek, J. Sloczynski and S. Ledakowicz, Methanol synthesis, Polish Scientific Publishing, Warszawa, 1994, Chap.II.2.2.4. 12. J. Ladebeck, Natural Gas Conversion W-Stud. Surf. Sci.Catal., 107 (1997) 73. 13. A Cybulski, Catal. Rev.-Sci.Eng., 36 (4) (1994) 557. 14. C. Keuchen and U. Hoffmann, Chem. Eng. Sci., 48 (22), (1993). 3767. 15. M. Sahibzada, D. Cadwick and I.S. Metcalfe, Natural Gas Conversion 1V - Stud. Surf. Sci. Catal.., 107 (1997) 29. 16. G.W. Roberts, D.M. Brown, T.H. Hsiung and J.J. Lewnard, Ind. Eng. Chem. Res. 32 (1993) 1610. 17. M.R. Rahimpour, J. Fathikalajahi and A. Jahanmiri, Canad. J. Chem. Eng., 76, August, (1998) 153. 18. R.C.Reid, J.M Prausnitz and B.E Poling, Properties of gases and liquids, McGraw-Hill, 1988 19. Methanex, Current methanol price, http://www.methanex.com. 1998. 20. T.F. Edgard and and D. H. Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1989. 21. PSE, gPROMS User's Guide 0.2 ed., London, 1997. 22. PSE, gPROMS Advanced User's Guide, 1.1 ed., London, 1998. 23. G.E.P Box, W. G. Hunter and J. S. Hunter, Statistics for experimenters, John Wiley, New York, 1975.
517
Sensitivity in Optimization of a Reactor System with Deactivating Catalyst .Ingvild Lgtvika, Magne Hillestad b and Terje Hertzberg a* aDepartment of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. bStatoil R&D Center, N-7005 Trondheim, Norway. An optimal operating strategy for a fixed bed reactor system with a slowly deactivating catalyst is found. The process studied is Lurgi's methanol synthesis. A rigorous model of the reactor and the loop has been used and the actual control variables have been optimized. In this paper, we look specifically at sensitivity in the optimization. The parameters in the catalyst deactivation model are believed to be uncertain. The effect of variations in the deactivation parameters on the optimal operating variables and the objective function has been studied by a first order error propagation-approach.
1. Introduction Catalyst deactivation occurs in practically all fixed bed reactors. The two main questions in operation of fixed bed reactors with deactivation are when to change catalyst, and how to compensate for deactivation between the catalyst changes. This work looks at the last problem only, because in the methanol synthesis the time for catalyst change is decided by factors outside the process section. Much work has been done on optimal operation of fixed bed reactors undergoing catalyst deactivation. Some central references are [ 1-5]. Most of the earlier work [3-5] has focused on optimal distributed control, e.g. optimization of reactor temperature distributed in time and space. A more realistic approach is taken in this work. The actual time varying control variables in the reactor system, the recycle rate and coolant temperature, are optimized. This study also uses a detailed, realistic model of the total reactor system. Parts of this work are published earlier [6,7]. This paper looks specifically at sensitivity in the optimization with regards to the deactivation model. In the methanol synthesis, synthesis gas (CO, CO2 and H2) is converted to methanol over a Cu/ZnO/A1203 catalyst. The following exothermic reactions occurs [8]: CO 2 +3H 2 r CO + H 2 0 r
CH3OH+ H20 CO 2 + H 2
In the Lurgi reactor [9] the catalyst is packed in vertical tubes surrounded by boiling water. The reaction heat is transferred to the boiling water and steam is produced. Efficient heat transfer gives small temperature gradients along the reactor. Typical operating conditions are 523 K and 80 bar. The pressure of the boiling water is controlling the reactor temperature. Because o f the quasi-isothermal reaction conditions and high catalyst selectivity, only small amounts of byproducts are formed. Methanol conversion is limited by equilibrium. Unreacted synthesis gas is separated from crude methanol, compressed and recycled. The Lurgi reactor system consists of two parallel reactors with a common steam drum, a feed/effluent interchanger, a cooler, a methanol separator and a recycle
* Author to whom correspondence shoud be adressed, email: [email protected]
518
compressor. The control variables are the coolant temperature and the recycle rate. A flowsheet of the process is shown in Fig. 1. ....................................... S.'/n|hesk~ ~t"~
MP Steam
Cruc!e melhonol
Fig. 1: The meth~ol synthesisl..... The Cu/ZnO/A1203 catalyst can deactivate irreversibly because of chemical poisoning or thermal sintering [10-12]. Sintering is the cause of deactivation under normal operation. The catalyst poison sulfur is removed earlier in the process. Chlorine and heavy metals act as catalyst poisons but is not likely to occur in the process gas. Sintering is caused by high temperatures and increases when the catalyst is exposed to high partial pressure of water [ 10,11 ]. Copper is the active phase in the catalyst. During sintering, copper atoms migrate to larger agglomerates. This leads to increasing crystal size and decreasing active area. The sintering mechanism changes at higher temperatures; copper crystals migrate together, causing severe deactivation and loss of selectivity. Reported temperatures for when severe deactivation starts range from 543 K [9] to 670 K [11 ]. The lowest temperature is chosen as a constraint in the optimization. The catalyst deactivates slowly under normal operating conditions, and after 3 to 4 years, the activity is so low that the catalyst has to be replaced. A shut down of a part of the plant is necessary to change catalyst. The maintenance plan of the plant and the natural gas supply determine when to replace the catalyst. This is why the catalyst lifetime is not optimized in this study. A common opera7ion strategy is to increase the temperature at the end of the catalyst lifetime to compensate for decreased activity [9]. The decisions regarding temperature increase are based on the experience of the operators. Increased temperature gives higher reaction rates, but also higher deactivation rates. This makes coolant temperature an interesting optimization variable. Increased recycle rate leads to lower conversion per pass in the reactor, but higher overall conversion in the loop.
2. Modeling and Optimization The catalyst deactivation was the only dynamic effect included in the model while steady state was assumed for the other states. This pseudo steady state assumption is reasonable because the dynamics in composition, temperature and pressure are much faster than the deactivation dynamics. A twodimensional, heterogeneous reactor model with the following assumptions was used: * Dispersion of mass in axial and radial directions is negligible. 9 Dispersion of heat in axial direction is negligible. 9 Isotherm catalyst pellets. 9 Inter-facial temperature gradients are negligible. 9 Viscous flow in catalyst pellets is negligible. The LHHW-type reaction kinetic by Bussche and Froment [8] was selected. The fifth order deactivation kinetic is based on Skrypek et al. [ 11] with deactivation energy Ed from Cybulski [ 13]. The activity is scaled to fit a temperature profile for a Lurgi reactor with deactivated catalyst [9].
519
1)/
a,t, dt
~Rg
- "~"0 "a(t)5
a(0) =a0 a ' ( t ) = 1 - ~'-atto)a ao
This model was selected because it predicts a reasonable long catalyst lifetime. Few deactivation models for this catalyst are published, and they predict quite different deactivation rates. Most of the models were developed in laboratory scale, and therefore predicts too fast deactivation [13-16]. One model [ 17] considers the reaction gas composition, but the mechanism that is assumed is in conflict with other literature. This model also predicts too fast deactivation. Lumped steady state models were used to describe the remaining unit operations in the reactor loop. Soave-Redlich-Kwong equation of states was used to find the phase equilibrium in the separator [ 18]. The task of finding an optimal operation strategy was formulated as a nonlinear dynamic optimization problem: tl
Max "Profit = ] (FMeOH " PMeOH + Fsteam " esteam )it
Tc(t),R(t)
to
s.t.:
Zrmeaa~ctor <_ 5 4 3
K
513K
2
5
4 ...............................................
I~
:
:
:
300
:
:
:
:
:
:
600
900
Time
[days]
:
:
:
1200
:
53~ t 525~
,
,.........
L .........
..
300
600
900
Time
[days]
Fig. 2: Optimal coolant temperature and recycle rate (- opt ' ref)
1200
520
3. Sensitivity Analysis We wanted to investigate the sensitivity in the optimization results with regards to the deactivation model. Two factors were studied, the rate constant Kd and the activation energy Ed. The effects and cross effects of the factors were found by varying the factors according to the 22 experimental design in Table l [23]. The effects and cross effects of varying the deactivation parameters on the optimal coolant temperatures and the scaled objective function are shown in Table 2. No effects were observed on the optimal recycle rate. The Kd effect on coolant temperature is positive. The temperatures are increased to compensate for the increased deactivation caused by increased Kd. The Ed effect on coolant temperatures is not significant in all intervals. It is negative and shrinking in the tree first intervals, resulting in a steeper profile that starts at a lower temperature. The temperatures are lowered in the start were most of the deactivation takes place, because the increase in Ed makes the deactivation rate more sensitive to temperature. Both effects on the objective function are negative as expected. A positive cross effect on the objective function exists, meaning that the Kd effect is lager when Ed is at the low level. Both the objective function and the coolant temperature were checked for nonlinear effects, with a negative result. The center point was close to the response surface for all responses.
Table 1" Experimental design. 0.8*Ed Ed 1.2*Ed 0.8*Kd
Kd 1.2*Kd
--
-'+
00 +--
Table 2: Effects of the deactivation parameters on the optimization results.
++
Effects
Response Obj TJ Tc2 TJ Tc4 Tc5 Tc6 Tc7 TJ
Mean 21.00 520.3 523.9 526.1 527.6 529.0 530.9 530.9 531.7
Kd -1.39 0.7_-/-0.4 0.9_+0.4 0.9_+0.4 1.1_+0.4 0.8_+0.4 0.8_+0.4 1.0-+0.4 0.9_+0.4
Ed KaxEd -0.73 0.19 -1.4_--t-0.4 -1.0-+0.4 -0.7_+0.4
The propagation of statistical uncertainty from the deactivation parameters to the optimization results were calculated from the equation [23]"
Var(f)=l~i2Var(Kd)+ ~~f-~l)~21Z. Var(Ed)+2I_~f d Y_)f) Var(Kd ) =
Var(Ed ) =CYEd
Ea)
CoV(Kd ,Ed ) = P Kd,Edt~Kd~ Ed
The partial derivatives were estimated from the effects in Table 2. Different cases with combinations of uncertainties and correlation factors in the deactivation parameters were used, see Table 3. The propagated uncertainties in objective function and coolant temperatures are shown in Table 4. The uncertainties in the scaled objective function is relative large. Converted to percent increased profit from optimization, the uncertainties are +0.2 percent in case 4 and +0.1 percent in case 2. The uncertainties in the optimal coolant temperatures are relative small. Varying the correlation coefficient had small effects on the uncertainties in both the objective function and the coolant temperatures. It is interesting that the uncertainties in coolant temperatures in case 3 and 4 are of the same size as the optimization accuracy of +0.4 K. This point is illustrated in Fig. 3. From these results, it can be concluded that 20 percent standard deviation in the deactivation parameters is
521
sufficient for optimization purposes. More accurate deactivation parameters will not lead to a more accurate optimal operation strategy. This can be used as a target for uncertainty in a future model development. In adition to uncertanty in calculation of the optimal coolant temperatures, there is also uncertainty in implementation of the optimal coolant temperatures. The implementation uncetanty is estimated to +0.5 K. This is another reason not to use a more accurate deactivation model for optimization. Table 3: Standard deviation in the deactivation parameters Case (YKd (YEd PKd,Ed 10 10 20 20 40 40
1
2 3 4 5 6
10 10 20 20 40 40
Table 4: Standard deviation in optimization results. Center O'respo.se Response Point Case Case Case Case 20.86 520.03 523.47 525.86 525.86 527.34 528.67 529.94 530.89
Obj
0.5 0.95 0.5 0.95 0.5 0.95
Tc1 Tc2
Tc3 Tc4 Tcs Tc6 Tc 7
Tc8
532
'
3
4
Case 6 +2.10 +0.77 +0.32 +0.32 _+0.73 +0.61 +0.80 +1.00 +1.29
........ i|
528
r.
524
-
.
.
.
.
-
.
- : _- I: "
.....
.
~-" - = "
522 520
2
'
53O ~'
Case 5 +0.47 +.52 +0.93 • +1.86 +0.30 +0.19 +0.61 +0.38 + 1 . 2 1 +0.24 +0.08 +0.48 +0.16 +0.95 _+0.21 _+0.08 +0.41 +0.16 +0.82 _+0.24 _+0.18 _+0.48 +0.37 _+0.96 +0.18 +0.15 +0.36 +0.31 +0.72 _+0.20 _+0.20 +0.40 +0.40 +0.80 +0.25 +0.25 +0.50 +0.50 +1.00 +0.29 _+0.32 _+0.58 _+0.64 +1.15 1
......
4
0
I
500
!
.........
T i m e [ d a y s ] 1000
1500
Figi3" Standard deviation in optimal coolant temperature in case 4 ( m prop. err. ' opt. acc.).
4. Conclusions The operation of the methanol synthesis with catalyst decay has been optimized, and the sensitivity in the optimization results with regards to the deactivation model has been investigated by a first order error propagation - approach. The optimization results have been published earlier [7]. The effects of variations in the two parameters in the deactivation model on optimal coolant temperature profile and the objective function was found. Both the rate constant and the activation energy had a negative effect on the objective function, and a cross effect between the to factors was observed. The rate constant had a positive effect on the optimal coolant temperature profile. The activation energy had a negative effect on the first intervals of the coolant temperature profile, resulting in a steeper profile. The propagated uncertainties was relative large in the scaled objective function and relative small in the optimal coolant temperatures. It was found that the uncertainties in coolant temperatures with 20 percent standard deviation in the deactivation parameters are of the same size as the optimization accuracy. From these results, it can be concluded that 20 percent deviation in the deactivation
522
parameters is sufficient for optimization purposes. This can be used as a target for uncertainty in a future model development.
5. Notation a a
Ed [J/mol] Kd [da y-i] Tc(t) [K] Tre actor max
[K]
R [mol/mol] Profit [USD]
Catalyst activity Scaled catalyst activity Activation energy for deactivation Rate constant for deactivation Coolant temperature Maximum reactor temperature Recycle rate Profit over catalyst lifetime
FMeOH[tons/day] [;'Steam[tons/day] Pi [USD/ton] Qcomp[ k W ] tl [days]
Var09 Cov~ g) (yf pf, g
Production rate of methanol Net production rate of steam Price of product i Compressor power Catalyst lifetime Variance Co variance Standard deviation Correlation coefficien
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