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Reduced order dynamic models of reactive absorption processes
European Symposium on Computer Aided Process Engineering - 13 A. Kraslawski and I. Turunen (Editors) © 2003 Elsevier Science B.V. All rights reserved.
929
Reduced Order Dynamic Models of Reactive Absorption Processes S. Singare, C.S. Bildea and J. Grievink Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL, Delft, The Netherlands
Abstract This work investigates the use of reduced order models of reactive absorption processes. Orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE) are compared. All three methods are able to accurately describe the steady state behaviour, but they predict different dynamics. In particular, the OC dynamic models show large unrealistic oscillations. Balanced truncation, residualization and optimal Hankel singular value approximation are applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy.
1. Introduction Many important chemical and petrochemical industrial processes, such as manufacture of sulphuric acid and nitric acid, soda ash, purification of synthesis gases, are performed by the reactive absorption of gases in liquids, in large scale processing units. For example, the whole of fertilizer industry relies on absorption processes. Rising energy prices and more stringent requirements on pollution prevention impose a need to continuously update processing conditions, design and control of industrial absorption processes. Traditionally, the design of Reactive Absorption Processes (RAP) relies on equilibrium models, whose accuracy has been extensively criticised both by academic and industrial practitioners. In contrast, the rate-based approach (Kenig et al. 1999), accounting for both diffusion and reaction kinetics, provides very accurate description of RAP. Solving such models requires discretization of the spatial co-ordinates in the governing PDEs. This gives rise to a large set of non-linear ODEs that can be conveniently handled for the purpose of steady state simulation. However, the size of the model becomes critical in a series of applications. For example, real-time optimisation requires fast, easy-tosolve models, because of the repetitive use of the model by the iterative algorithm; in model based control applications, the simulation time should be 100 to 1000 times shorter than the time scale of the real event. This work investigates the use of reduced order RAP models for the purpose of dynamic simulation, controllability analysis and control system design. Three different discretization methods, namely orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE), are compared. All three methods are able to accurately describe the steady state behaviour. However, the predicted dynamic
930 behaviour is very different. In particular, the OC dynamic models show large unrealistic oscillations. In view of control applications, different reduction techniques, including balanced truncation and residualization and optimal Hankel singular value approximation, were applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy.
2. Model Description The reactive absorption column is modelled using the well-known two-film model (Fig. 1). In this model, the resistance to mass transfer is concentrated in a thin film adjacent to the phase interface and the mass transfer occurs within this film by steady-state molecular diffusion. Axial z co-ordinate represents the length of the column. Gaseous component, A diffuses through the film towards liquid bulk and in the process reacts with the liquid component, B. In the present model, assumptions of plug flow, constant temperature and pressure are made. Dynamic mass balance equations in the non-dimensional form for gas bulk, liquid bulk phase can be written as: 2.1. Bulk gas phase mass balance It is assumed that no reaction occurs in the bulk gas phase and the gas film. dY. er-
dr
dY.
^ - r a dz
(Y
-h
C )
B.c.atz = oy.,=y.,,
(1)
where. Da is Damkohler number, aj and hj Q = A, B components) are dimensionless mass transfer coefficients and Henry's constants, respectively, r is the ratio between gas and liquid residence times. GB
GF
LF
LB
Fig 1. Schematic of reactive absorption column model.
931 2.2. Bulk liquid phase The second order reaction occurs in the bulk of the liquid, as well as in the liquid film.
(l-s)
'-^ = dr
dz
^-0.-^1 - D « C , . - C , , ( l - 0 dx
B.C.atz=l,C.,=C,, I.C.atr = 0,C.,
(2)
=C°M(Z)
where, 0j are dimensionless diffusion coefficients. 2.3. Liquid film mass balance Neglecting the fast dynamics, application of Pick's law of diffusion gives rise to the following set of second order differential equations. d'C. ax B.C.atJc = 0
m.(Y.-h.'C\
)=
dC
(3)
dx atx = l
C , =C j.L
J
Here, Ha is Hatta number which represents the ratio of kinetic reaction rate to the mass transfer rate. As a test case, the data reported by Danckwerts and Sharma (1966) is chosen. The dimensionless parameters are Da = 1.87 10 ^, a^ = 37.92, /3A = 6.94, /?5 = 3.82, HaA = 15.48, Has = 20.88, m^ = 5.46, HA = 1.62, r = 0.325.
3. Solution Method 3.1. Steady state The complete model of reactive absorption column is solved using three different discretization methods: (1) Orthogonal collocation (OC), (2) Finite difference (FD) and (3) Orthogonal collocation on finite elements (OCFE). In case of OC and OCFE, roots of Jacobi orthogonal polynomial are used as collocation points. FD method requires two discretization schemes in the axial direction: backward finite difference method (BFDM, 2nd order accuracy) in up-axial direction for bulk gas phase; forward finite difference method (FFDM, 2nd order accuracy) in down-axial direction for bulk liquid phase. In FD scheme, liquid film equations are discretized using central finite difference method (CFDM) of 4th order accuracy. The whole set of equations are written in gPROMS, which solves the set of algebraic non-linear equations using NewtonRaphson method. The different numerical methods are summarized in Table 1. Results of steady state calculations are shown in Figure 2, where gas and liquid concentration profiles along the column are depicted.
932 Table 1. Discretization method and number of variables and equations involved. Discretization method OC FD OCFE
# discretization points Film Axial 7 17 21 51 13 21
# of variables and equations 289 2295 609
FD scheme with 51 and 21 discretization points in the axial and film co-ordinate respectively is taken as a basis for comparison with OC and OCFE method. In OC scheme, 15 and 5 internal collocation points (axial and film co-ordinate respectively) and in OCFE scheme, 5 and 3 finite elements with 3 internal collocation points in each finite element are needed. In SS simulation, OC scheme results in the lowest number of variables and provides good accurate results. But it is found that it can not be used beyond 22 discretization points due to ill-conditioning of the matrix calculations. In such situation, OCFE provides improved stability with slightly increased number of variables. As seen, FD requires the largest number of variables to get accurate results. It should be noted that the definition and use of the dimensionless variables allows a robust solution of the model equations, easy convergence being obtained for all three discretization methods and for very crude solution estimates (for example, all concentrations set at 0.5). 3.2. Dynamic The dynamic simulation of RAC model was carried out for the above three cases. The dynamic response of gas and liquid outlet concentrations, YAOouh CBLOUI-^ to changes in inlet flow rates FMG^ ^VL and concentrations YAGim CBUH was investigated. Figure 3 presents results for a 0.05 step change in YACin (similar results were obtained for the other inputs). The expected, realistic response is a gradual increase of outlet concentration occurring after a certain dead time. The computed response showed oscillations, which is attributed to numerical approximation of the convection term. This effect is discussed by Lefevre et. al (2000), in the context of tubular reactors. 2.50 -aSi—B—aAD/OE
2.00 h DC (15,5) - FD (50,20) r OCFE (5,3)
1.50
o"
- s - O C (15,5) — FD (50,20) ^OCFE(5,3)
1.00 0.50 0.00 0.2
0.4
0.6 Z
Fig 2. Steady state profile for different discretization methods.
0.8
933 200 180 160 E 140 a. €L 120 100 80 60 40 20 0
1
1
TTl M/ y^
-I
0.05
1/
-OCFE BFDM(2) - 50
OC ,
1
0.2
0.15
0.1
0.25
Time/[-]
Fig. 3. Dynamic response of gas-outlet concentration to a step change of gas-inlet concentration. OC scheme produces large oscillatory response right from the start, without any dead time. Thus, it is not suitable for dynamic simulation purposes. In the case of OCFE and FD, the oscillatory behaviour starts after some dead time. The amplitude is much smaller compared to OC. As expected, oscillations are reduced by increasing the number of discretization points. Taking into account the size of the model and the shape of dynamic response, the OCFE seems the preferred scheme. Further, we used the "Linearize" routine of gPROMS to obtain a linear model. Starting with the OCFE discretization, the linear model has 48 states. This might be too much for the purposes of controllability analysis and control system design. Therefore, we applied different model-reduction techniques (Skogestad and Postlethwaite, 1996). Fig. 4 compares the Bode diagrams of the full-order model and the models reduced to n = 10 states by different techniques. For the frequency range of interest in industrial application (10 rad / time unit, corresponding roughly to 5 rad/min), the balanced residualization offers the best approximation.
0.1
1
10
100
1000
(o I [rad / dimensionless time]
Fig 4. Comparison of reduced-order models obtained by different techniques.
934 20 15
ICXFE
n=48, n=15v
OCFE Residualization
n=5 \
n=10
^ - ^ / ^
nonlinear
\f^\
0.1
1
10
100
CO I [rad / dimensionless time]
1000
0.05
0.1
0.15
0.25
Time/[']
Fig. 5. Comparison of linear models of different orders, a) Bode diagram, b) Step response. Figure 5 compares the effect of number of states that are retained in the reduced order model, using both the Bode diagram and step response. The linear model predicts well the dead time and the speed of response. For n= 15, the full and reduced-order models coincide. Reasonable results are obtained for n = 10. If the model is further reduced (n=5), it fails to predict the dynamic behaviour. From the time response presented in Figure 4, it seems that a second-order model including dead time should suffice. It is possible to identify such model, using for example, real plant data. However, we are interested to obtain the model starting with first-principles model. This is the subject of current research.
4. Conclusions - A dynamic model of reactive absorption column is developed in non-dimensional form. Three discretization methods; OC, FD, OCFE are used to solve the model equations. For steady state process synthesis and optimisation, orthogonal collocation based methods are found accurate and robust. - For dynamic simulation^ pure OC is unsuitable. OCFE is found to give realistic representation of column's behaviour, together with a small-size model. This presents a good option to FD scheme. Linear model reduction techniques are further applied to reduce the model for control design purpose. Balanced residualization with 15 states approximates satisfactorily the column dynamics. - In the future work, more complex reaction schemes, Maxwell-Stefan equations for diffusion, heat balance, axial dispersion term, hydrodynamics, thermodynamics, tray columns will be included in the model.
5. References Danckwerts and Sharma, 1966, Chem. Engrs. (London), CE 244. Kenig, E.Y., Schneider, R., Gorak, A., 1999, Chem. Eng. Sci. 54, 5195. Lefevre, L., Dochain, D., Magnus, A., 2000, Comp. & Chem. Engg. 24,2571. Skogestad, S. and Postlethwaite, L, 1996, Multivariable Feedback Control - Analysis and Design, John Wiley & Sons, Chichester.
929
Reduced Order Dynamic Models of Reactive Absorption Processes S. Singare, C.S. Bildea and J. Grievink Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL, Delft, The Netherlands
Abstract This work investigates the use of reduced order models of reactive absorption processes. Orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE) are compared. All three methods are able to accurately describe the steady state behaviour, but they predict different dynamics. In particular, the OC dynamic models show large unrealistic oscillations. Balanced truncation, residualization and optimal Hankel singular value approximation are applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy.
1. Introduction Many important chemical and petrochemical industrial processes, such as manufacture of sulphuric acid and nitric acid, soda ash, purification of synthesis gases, are performed by the reactive absorption of gases in liquids, in large scale processing units. For example, the whole of fertilizer industry relies on absorption processes. Rising energy prices and more stringent requirements on pollution prevention impose a need to continuously update processing conditions, design and control of industrial absorption processes. Traditionally, the design of Reactive Absorption Processes (RAP) relies on equilibrium models, whose accuracy has been extensively criticised both by academic and industrial practitioners. In contrast, the rate-based approach (Kenig et al. 1999), accounting for both diffusion and reaction kinetics, provides very accurate description of RAP. Solving such models requires discretization of the spatial co-ordinates in the governing PDEs. This gives rise to a large set of non-linear ODEs that can be conveniently handled for the purpose of steady state simulation. However, the size of the model becomes critical in a series of applications. For example, real-time optimisation requires fast, easy-tosolve models, because of the repetitive use of the model by the iterative algorithm; in model based control applications, the simulation time should be 100 to 1000 times shorter than the time scale of the real event. This work investigates the use of reduced order RAP models for the purpose of dynamic simulation, controllability analysis and control system design. Three different discretization methods, namely orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE), are compared. All three methods are able to accurately describe the steady state behaviour. However, the predicted dynamic
930 behaviour is very different. In particular, the OC dynamic models show large unrealistic oscillations. In view of control applications, different reduction techniques, including balanced truncation and residualization and optimal Hankel singular value approximation, were applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy.
2. Model Description The reactive absorption column is modelled using the well-known two-film model (Fig. 1). In this model, the resistance to mass transfer is concentrated in a thin film adjacent to the phase interface and the mass transfer occurs within this film by steady-state molecular diffusion. Axial z co-ordinate represents the length of the column. Gaseous component, A diffuses through the film towards liquid bulk and in the process reacts with the liquid component, B. In the present model, assumptions of plug flow, constant temperature and pressure are made. Dynamic mass balance equations in the non-dimensional form for gas bulk, liquid bulk phase can be written as: 2.1. Bulk gas phase mass balance It is assumed that no reaction occurs in the bulk gas phase and the gas film. dY. er-
dr
dY.
^ - r a dz
(Y
-h
C )
B.c.atz = oy.,=y.,,
(1)
where. Da is Damkohler number, aj and hj Q = A, B components) are dimensionless mass transfer coefficients and Henry's constants, respectively, r is the ratio between gas and liquid residence times. GB
GF
LF
LB
Fig 1. Schematic of reactive absorption column model.
931 2.2. Bulk liquid phase The second order reaction occurs in the bulk of the liquid, as well as in the liquid film.
(l-s)
'-^ = dr
dz
^-0.-^1 - D « C , . - C , , ( l - 0 dx
B.C.atz=l,C.,=C,, I.C.atr = 0,C.,
(2)
=C°M(Z)
where, 0j are dimensionless diffusion coefficients. 2.3. Liquid film mass balance Neglecting the fast dynamics, application of Pick's law of diffusion gives rise to the following set of second order differential equations. d'C. ax B.C.atJc = 0
m.(Y.-h.'C\
)=
dC
(3)
dx atx = l
C , =C j.L
J
Here, Ha is Hatta number which represents the ratio of kinetic reaction rate to the mass transfer rate. As a test case, the data reported by Danckwerts and Sharma (1966) is chosen. The dimensionless parameters are Da = 1.87 10 ^, a^ = 37.92, /3A = 6.94, /?5 = 3.82, HaA = 15.48, Has = 20.88, m^ = 5.46, HA = 1.62, r = 0.325.
3. Solution Method 3.1. Steady state The complete model of reactive absorption column is solved using three different discretization methods: (1) Orthogonal collocation (OC), (2) Finite difference (FD) and (3) Orthogonal collocation on finite elements (OCFE). In case of OC and OCFE, roots of Jacobi orthogonal polynomial are used as collocation points. FD method requires two discretization schemes in the axial direction: backward finite difference method (BFDM, 2nd order accuracy) in up-axial direction for bulk gas phase; forward finite difference method (FFDM, 2nd order accuracy) in down-axial direction for bulk liquid phase. In FD scheme, liquid film equations are discretized using central finite difference method (CFDM) of 4th order accuracy. The whole set of equations are written in gPROMS, which solves the set of algebraic non-linear equations using NewtonRaphson method. The different numerical methods are summarized in Table 1. Results of steady state calculations are shown in Figure 2, where gas and liquid concentration profiles along the column are depicted.
932 Table 1. Discretization method and number of variables and equations involved. Discretization method OC FD OCFE
# discretization points Film Axial 7 17 21 51 13 21
# of variables and equations 289 2295 609
FD scheme with 51 and 21 discretization points in the axial and film co-ordinate respectively is taken as a basis for comparison with OC and OCFE method. In OC scheme, 15 and 5 internal collocation points (axial and film co-ordinate respectively) and in OCFE scheme, 5 and 3 finite elements with 3 internal collocation points in each finite element are needed. In SS simulation, OC scheme results in the lowest number of variables and provides good accurate results. But it is found that it can not be used beyond 22 discretization points due to ill-conditioning of the matrix calculations. In such situation, OCFE provides improved stability with slightly increased number of variables. As seen, FD requires the largest number of variables to get accurate results. It should be noted that the definition and use of the dimensionless variables allows a robust solution of the model equations, easy convergence being obtained for all three discretization methods and for very crude solution estimates (for example, all concentrations set at 0.5). 3.2. Dynamic The dynamic simulation of RAC model was carried out for the above three cases. The dynamic response of gas and liquid outlet concentrations, YAOouh CBLOUI-^ to changes in inlet flow rates FMG^ ^VL and concentrations YAGim CBUH was investigated. Figure 3 presents results for a 0.05 step change in YACin (similar results were obtained for the other inputs). The expected, realistic response is a gradual increase of outlet concentration occurring after a certain dead time. The computed response showed oscillations, which is attributed to numerical approximation of the convection term. This effect is discussed by Lefevre et. al (2000), in the context of tubular reactors. 2.50 -aSi—B—aAD/OE
2.00 h DC (15,5) - FD (50,20) r OCFE (5,3)
1.50
o"
- s - O C (15,5) — FD (50,20) ^OCFE(5,3)
1.00 0.50 0.00 0.2
0.4
0.6 Z
Fig 2. Steady state profile for different discretization methods.
0.8
933 200 180 160 E 140 a. €L 120 100 80 60 40 20 0
1
1
TTl M/ y^
-I
0.05
1/
-OCFE BFDM(2) - 50
OC ,
1
0.2
0.15
0.1
0.25
Time/[-]
Fig. 3. Dynamic response of gas-outlet concentration to a step change of gas-inlet concentration. OC scheme produces large oscillatory response right from the start, without any dead time. Thus, it is not suitable for dynamic simulation purposes. In the case of OCFE and FD, the oscillatory behaviour starts after some dead time. The amplitude is much smaller compared to OC. As expected, oscillations are reduced by increasing the number of discretization points. Taking into account the size of the model and the shape of dynamic response, the OCFE seems the preferred scheme. Further, we used the "Linearize" routine of gPROMS to obtain a linear model. Starting with the OCFE discretization, the linear model has 48 states. This might be too much for the purposes of controllability analysis and control system design. Therefore, we applied different model-reduction techniques (Skogestad and Postlethwaite, 1996). Fig. 4 compares the Bode diagrams of the full-order model and the models reduced to n = 10 states by different techniques. For the frequency range of interest in industrial application (10 rad / time unit, corresponding roughly to 5 rad/min), the balanced residualization offers the best approximation.
0.1
1
10
100
1000
(o I [rad / dimensionless time]
Fig 4. Comparison of reduced-order models obtained by different techniques.
934 20 15
ICXFE
n=48, n=15v
OCFE Residualization
n=5 \
n=10
^ - ^ / ^
nonlinear
\f^\
0.1
1
10
100
CO I [rad / dimensionless time]
1000
0.05
0.1
0.15
0.25
Time/[']
Fig. 5. Comparison of linear models of different orders, a) Bode diagram, b) Step response. Figure 5 compares the effect of number of states that are retained in the reduced order model, using both the Bode diagram and step response. The linear model predicts well the dead time and the speed of response. For n= 15, the full and reduced-order models coincide. Reasonable results are obtained for n = 10. If the model is further reduced (n=5), it fails to predict the dynamic behaviour. From the time response presented in Figure 4, it seems that a second-order model including dead time should suffice. It is possible to identify such model, using for example, real plant data. However, we are interested to obtain the model starting with first-principles model. This is the subject of current research.
4. Conclusions - A dynamic model of reactive absorption column is developed in non-dimensional form. Three discretization methods; OC, FD, OCFE are used to solve the model equations. For steady state process synthesis and optimisation, orthogonal collocation based methods are found accurate and robust. - For dynamic simulation^ pure OC is unsuitable. OCFE is found to give realistic representation of column's behaviour, together with a small-size model. This presents a good option to FD scheme. Linear model reduction techniques are further applied to reduce the model for control design purpose. Balanced residualization with 15 states approximates satisfactorily the column dynamics. - In the future work, more complex reaction schemes, Maxwell-Stefan equations for diffusion, heat balance, axial dispersion term, hydrodynamics, thermodynamics, tray columns will be included in the model.
5. References Danckwerts and Sharma, 1966, Chem. Engrs. (London), CE 244. Kenig, E.Y., Schneider, R., Gorak, A., 1999, Chem. Eng. Sci. 54, 5195. Lefevre, L., Dochain, D., Magnus, A., 2000, Comp. & Chem. Engg. 24,2571. Skogestad, S. and Postlethwaite, L, 1996, Multivariable Feedback Control - Analysis and Design, John Wiley & Sons, Chichester.