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Modeling, Simulation and Control of a Nonlinear Distributed Parameter Isotope Separation Process
9th IFAC symposium on Control of Power and Energy Systems 9th IFAC symposium on Control of Power Energy 9th IFAC symposium on Control of Power and and Energy Systems Systems Indian Institute of Technology 9th IFAC symposium on Control of and Systems 9th IFAC symposium on Control of Power Power and Energy Energy Systems Indian Institute of Technology Available online at www.sciencedirect.com Indian Institute of Technology December 9-11, 2015. Delhi, India Indian Institute Institute of of Technology Technology Indian December 9-11, 2015. Delhi, India December 9-11, 2015. Delhi, India December 9-11, 2015. Delhi, India December 9-11, 2015. Delhi, India
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IFAC-PapersOnLine 48-30 (2015) 191–196
Modeling, Simulation and Control of a Nonlinear Distributed Parameter Isotope Modeling, Simulation and Control of a Nonlinear Distributed Parameter Isotope Modeling, of Modeling, Simulation Simulation and and Control Control of aa Nonlinear Nonlinear Distributed Parameter Parameter Isotope Isotope Separation Process Distributed Separation Process Separation Separation Process Process Vlad MUREŞAN*, Mihail ABRUDEAN* Vlad MUREŞAN*, Mihail ABRUDEAN* Vlad MUREŞAN*, MUREŞAN*, Mihail ABRUDEAN* Daniel Mihail MOGA* Vlad ABRUDEAN* Vlad MUREŞAN*, Mihail ABRUDEAN* Daniel MOGA* Daniel MOGA* Daniel Daniel MOGA* MOGA* *Technical University of Cluj-Napoca, Cluj-Napoca, CO 400114 ROMANIA (Tel: +40-744-420906; *Technical University of Cluj-Napoca, Cluj-Napoca, CO (Tel: +40-744-420906; *Technical University of Cluj-Napoca, Cluj-Napoca, CO 400114 400114 ROMANIA ROMANIA (Tel: +40-744-420906; ). e-mail: [email protected], [email protected], [email protected] *Technical University of Cluj-Napoca, Cluj-Napoca, (Tel: *Technical University of Cluj-Napoca, Cluj-Napoca, CO CO 400114 400114 ROMANIA ROMANIA (Tel: +40-744-420906; +40-744-420906; ). e-mail: [email protected], [email protected], [email protected] e-mail: [email protected], [email protected], [email protected] ). e-mail: e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] [email protected] ). ). Abstract: In this paper, an original solution of modeling and control of a distributed parameter 13 C 13 13 Abstract: In this this paper, paper, an is original solution ofmentioned modeling and and control of of process distributed parameter 13 C Abstract: In an original solution of modeling control aaa distributed parameter 13C isotope separation process presented. The technological takes place into C Abstract: In this paper, an original solution of modeling and control of distributed parameter Caa Abstract: In this paper, an is original solution ofmentioned modeling and control of process a distributed parameter 13 isotope separation process presented. The technological takes place into isotope separation process is presented. The mentioned technological process takes place into aa C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 isotope separation separation process is presented. The mentioned technological process takes place into isotope process is presented. The mentioned technological process takes place into a 13C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 exchange. A main difficulty in the modeling – control problem approach is the fact that, besides that the C enrichment enrichment through through carbon carbon dioxide dioxide – – carbamate carbamate chemical chemical separation column, column, whose whose work work ensures ensures the the C separation exchange. main in – problem approach is the besides that exchange. A main difficulty difficulty in the the modeling modeling – control control problem approach in is relation the fact fact that, that, besides that the the process is A a distributed parameter one, it presents strong nonlinearities to both independent exchange. A main in – problem approach is the fact besides that exchange. A main difficulty difficulty in the the modeling modeling – control control problem approach in is relation the 13 fact that, that, besides that the the process is a distributed parameter one, it presents strong nonlinearities to both independent process is is(the a distributed distributed parameter one, it column presentsheight). strong For nonlinearities in relation to both independent C isotope concentration variables time and the position in the the tuning of the process a parameter one, it presents strong nonlinearities in relation to both independent 13 process is a distributed parameter one, it presents strong nonlinearities in relation to both independent 13 Cisisotope isotope concentration variables (the time and the position in the column height). For the tuning of the 13C concentration variables the position the column height). For the tuning of the controller,(the the time relayand method is used. in Also, initially obtained controller structure modified in order to concentration variables (the time and the in the column height). For tuning the Cisisotope isotope concentration variables (the time and the position position in the the column height). For the the tuning of of the 13C controller, the relay method is used. Also, the initially obtained controller structure modified in order in to controller, the relay method is used. Also, the initially obtained controller structure is modified in order to ensure the same set of performances at the variation of the second independent variable (the position controller, the relay method is used. Also, the initially obtained controller structure is modified in to controller, the relay method is used. Also, initially of obtained controller structure is modified in order order in to 13the ensure the same set of performances at the variation the second independent variable (the position ensure the same set of performances at the variation of the second independent variable (the position in C isotope concentration value is controlled using an adaptive the column height). In this context, the ensure the same set of performances at the variation of the second independent variable (the position in 13 ensure the same set of performances at the variation of the second independent variable (the position in 13 C isotope concentration concentration value is controlled using an adaptive the column height). In this context, the 13C isotope controlled using an adaptive the column height). In this context, the 13 algorithm. the paper, some interesting are presentedvalue whichis the main advantages of C value is controlled using an the column height). In context, the C isotope isotope concentration concentration value isprove controlled using an adaptive adaptive the columnIn height). In this this context, the simulations 13 algorithm. In the paper, some interesting simulations are presented which prove the main advantages of algorithm. In the paper, some interesting simulations are presented which prove the main advantages of C isotope using a control structure with an adaptive controller: the possibility to control theadvantages algorithm. In the paper, some interesting simulations are presented which prove the main of 13 algorithm. In the paper, some interesting simulations are presented which prove the main advantages of 13C isotope using a control structure with an adaptive controller: the possibility to control the 13 C isotope using a control structure with an adaptive controller: the possibility to control the 13 concentration in any point from the column height; the possibility to reject the effect of the two types of C isotope isotope using aa control control structure structure with with an an adaptive adaptive controller: controller: the the possibility possibility to to control control the the C using concentration in from height; the to effect the types concentration which in any any point point from the the column height; the possibility possibility to reject reject the effecta of ofcertain the two tworestrictive types of of disturbances can occur in column the process work; the possibility to the satisfy concentration in from column height; the to the effect the types concentration in any any point point from the the column height; the possibility possibility to reject reject the effecta of ofcertain the two tworestrictive types of of disturbances which can occur in the process work; the possibility to satisfy disturbances set; which can occur in in the process work; the the possibility to satisfy satisfy a certain certain restrictive performance the possibility to adapt the controller parameters in order to obtain the same set of disturbances which can occur the process work; possibility to a restrictive disturbances which can occur in the process work; the possibility to satisfy a certain restrictive performance set; the possibility to adapt the controller parameters in order to obtain the same set of performance set; the possibility to adapt the controller parameters in order to obtain the same set of performances in any transversal section of the column. performance set; the possibility to adapt the controller parameters in order to obtain the same set performance set; thetransversal possibilitysection to adapt the column. controller parameters in order to obtain the same set of of performances in any of the performances in any transversal section of the column. performances in any transversal section of the column. 13 performances in any transversal section of the column. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear distributed parameter process, 13 C isotope, separation column, relay method, 13 isotope, separation column, relay method, Keywords: Nonlinear distributed parameter process, 13C isotope, separation column, relay method, Keywords: Nonlinear distributed parameter process, 13C concentration adaptive controller, chemical exchange process carbon dioxide – carbamate, C separation column, relay Keywords: Nonlinear distributed parameter process, Keywords: Nonlinear distributed chemical parameter process,process C isotope, isotope, separation column, ethanolamine. relay method, method, concentration adaptive controller, exchange carbon dioxide – carbamate, ethanolamine. concentration adaptive controller, chemical exchange process carbon dioxide – carbamate, ethanolamine. concentration concentration adaptive adaptive controller, controller, chemical chemical exchange exchange process process carbon carbon dioxide dioxide – – carbamate, carbamate, ethanolamine. ethanolamine. 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION C isotope The technological plant used for the 13 13 13C isotope The technological plant The technological used for the 13C presented in Fig. 1.plant C isotope isotope The technological plant used used for for the the 13 The technological plant used for the C isotope presented presented in in Fig. Fig. 1. 1. presented in Fig. 1. presented in Fig. 1.
separation is separation separation is separation is is separation is
C isotope separation plant. Fig. 1. The technological 13 13 Fig. 1. The The technological technological 13 13C Fig. 1. C isotope isotope separation separation plant. plant. Fig. C isotope isotope separation separation plant. plant. Fig. 1. 1. The The technological technological 13C
The carbon dioxide (CO2) absorption (2003;2009a) in ))) absorption (2003;2009a) in The carbon dioxide absorption (2003;2009a) in The carbon takes dioxide (CO ethanolamine place(CO in 222the absorber A (the two chemical (2003;2009a) in The dioxide (CO ) absorption absorption (2003;2009a) in The carbon carbon takes dioxide (CO 2the ethanolamine place in absorber A (the two chemical ethanolamine takes place in the absorber A (the two chemical elements circulating in counter-current), resulting the ethanolamine takes place in the absorber A (the two chemical ethanolamine takes place in the absorber A (the two chemical elements circulating counter-current), resulting the elements (pipe circulating insupplied counter-current), resulting the carbamate 3). A isin with ethanolamine through elements circulating in counter-current), resulting the elements circulating insupplied counter-current), resulting the carbamate (pipe 3). A is with ethanolamine through carbamate (pipe 3). Apump is supplied supplied with ethanolamine ethanolamine through the pipe 1 using the P, respectively with CO carbamate (pipe 3). A is with through 2 through carbamate (pipe 3). A is supplied with ethanolamine through the 1 the P, with through the pipe pipe 1 using using the pump pump 99.98% P, respectively respectively with CO CO 222Through pipe 5 (at approximately concentration). through the 1 the P, with through the pipe pipe 1 using using the pump pump 99.98% P, respectively respectively with CO CO 2Through pipe 5 (at approximately concentration). pipe 5 (at approximately 99.98% concentration). Through at a concentration 4, the gas phase, containing CO pipe 5 (at approximately 99.98% concentration). Through 2 pipe 5 (at approximately 99.98% concentration). Through at concentration pipe 4, the gas phase, containing CO 2 aaa concentration pipe the gas phase, containing CO 2 at C lower4, is evacuated from The 13 atsystem. pipe 4, the gas containing CO 2 13 a concentration concentration pipe 4,than the 0.1% gas phase, phase, containing COthe 13 2 atsystem. C lower than 0.1% is evacuated from the The 13C lower than 0.1% is evacuated from the system. The 13 – carbamate takes enrichment through isotopic exchange CO C lower than 0.1% is evacuated from the system. The 2 C lower than through 0.1% isisotopic evacuated from CO the2 –system. Thetakes carbamate enrichment exchange – carbamate takes enrichment through isotopic exchange CO 2 place in thethrough separation column SC (1994). The carbamate is carbamate takes enrichment isotopic exchange CO 2 – – carbamate takes enrichment through isotopic exchange CO 2 The carbamate is place in the separation column SC (1994). place in the separation column carbamate is introduced SC through pipe 3SC and(1994). the COThe place separation column SC (1994). The carbamate is 2 is introduced place in in the thein separation column SC (1994). The carbamate in is in introduced in SC pipe 3 CO is introduced introduced in introduced inpipe SC through through pipe elements 3 and and the the circulating CO222 is SC through 7 (the two in SC in is introduced introduced in SC through pipe 3 and the CO is introduced in introduced in SC through pipe 3 and the CO 2 SC through 7 two elements circulating SC in SC through pipe pipetoo). 7 (the (the two the elements circulating inthe SC13 in C counter-current, During separation process,in SC through 7 two elements circulating in SC in SC through pipe pipetoo). 7 (the (the two the elements circulating inthe SC13 in 13 C counter-current, During separation process, 13 C counter-current, too).inDuring During the separation process, the 13 isotope concentrates liquid phase in the lower part of the C counter-current, too). the separation process, the C counter-current, too).inDuring the separation process, the isotope concentrates liquid phase in the lower part of the isotope concentrates in liquid phase in the lower part of the which passes through column (pipe 2), respectively the CO isotope concentrates in liquid phase in the lower part of the 2 isotope concentrates in liquid phase in the lower part of the which passes column (pipe 2), respectively the CO which passes through column (pipe 2), respectively the CO SC is sent to A pipe 5. the222 reactor R, the through thermal which through column (pipe 2), respectively the CO which passes passes through column (pipe 2),through respectively theIn CO 2 reactor SC is sent to A through pipe 5. In the R, the thermal SC is sent to A through pipe 5. In the reactor R, the thermal decomposition of the carbamate is made and in the stripper S SC is sent to A through pipe 5. In the reactor R, the thermal SC is sent to Aofthrough pipe 5. In the reactor R,thethe thermal decomposition the carbamate is made and in stripper S decomposition of the carbamate is made and in the stripper S ) the is carbamate completely removed resulting the the gas (CO2of decomposition is made and in the stripper S decomposition of the carbamate is made and in the stripper S ) is completely removed resulting the the gas (CO 2) is completely removed resulting the the gas (CO 2 ethanolamine. The ethanolamine is reheated using the heater ) is is completely completely removed removed resulting resulting the the the gas (CO 2) the gas (CO 2 ethanolamine. The ethanolamine is using the ethanolamine. The ethanolamine is reheated reheated usingpipe the heater heater H and circulated to the absorber trough 1 and ethanolamine. The ethanolamine is using the ethanolamine. Theagain ethanolamine is reheated reheated usingpipe the heater heater H and circulated again to the absorber trough 1 H and and the circulated again to the thetheabsorber absorber trough pipe 1 and and resulting after the using pump P. Also, CO H circulated again to trough pipe 1 2 H and the circulated again to thetheabsorber trough pipe 1 and and resulting after using pump P. Also, CO 2 resulting afterto the the using the pump pump P. and Also, the CO22 resulting decomposition (in R) after stripping (in S) is sent after using the P. Also, the CO afterto the the using the pump P. and Also, the CO2 resulting decomposition (in R) after stripping (in S) is sent the decomposition (in R) and after stripping (in S) is sent to the SC through pipe 7. The pipe 6 used for supplying the plant decomposition (in R) and after stripping (in S) is sent to the decomposition (in R) and after stripping (in S) is sent to the SC through pipe 7. The pipe 6 used for supplying the plant SC through pipe 7. The pipe 6 used for supplying the plant and pipe 8 used for extracting the product in with CO SC through pipe 7. The pipe 6 used for supplying the plant 2 SC through pipe 7. The pipe 6 used for supplying the plant and pipe 8 used for extracting the product in with CO 2 pipe 8 used for extracting the product in with CO 2 and gaseous phase (both figured in Fig. 1 with dashed line) are and pipe 8 used for extracting the product in with CO 2 pipe figured 8 used inforFig. extracting the product in with COphase 2 and (both gaseous 1 with dashed line) are gaseous phase (both figured in Fig. 1 with dashed line) are used in production regime. The product is referring to CO gaseous phase (both figured in Fig. 1 with dashed line) are 2 gaseous phase (both figured in Fig. 1 with dashed line) are 13 used in production regime. The product is referring to CO 2 used production regime. The is referring to CO 2 C. In Fig. 1, it can be with in a certain concentration of product used in production regime. The product is referring to CO 13 used in production regime. The product is referring to CO 13C. In Fig. 1, it can be2 2 with a certain concentration of 13 C. In Fig. 1, it can be with a certain concentration of 13 remarked that the elements A, SC and S present steel pack of C. In In Fig. Fig. 1, 1, it it can can be be with aa certain certain concentration concentration of of C. with remarked that the elements A, SC and S present steel pack of remarked that that the elements elements A, SC SC and S S present steel steel pack of of remarked remarked that the the elements A, A, SC and and S present present steel pack pack of
Copyright 2015 IFAC 191Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 191 Copyright © 2015 IFAC 191 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 191Control. Copyright © 2015 IFAC 191 10.1016/j.ifacol.2015.12.376
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Halipack type (1994), figured with the hachured sections. The steel pack usage is determinant for the plant working as it makes possible the 13C isotope separation. In Fig. 1, the automation equipment associated to the separation plant are also presented, their significance and the working of the concentration control system being detailed later in the paper.
where T = T(p) represents the time constant of the process and uf(t) represents the final input signal in the process. The time constant of the process is determined using an experimental identification procedure based on the step response of the process. The signal uf is a function of Fin(t), hence uf(Fin(t)), dependency which in the particular case of the Fin(t) step type signal implies the fact that uf is a step type signal, too. Each experiment can be described as follows: a step type signal Fin(t) is applied at the input of the process and the evolution in time of the y(t,p) signal is measured in a certain predefined transversal section of SC given by the value of the independent variable (p). From the obtained experimental data, it results that the process is of first order one (as it was shown in (2)). The value of the time constant is determined using the tangent method, resulting the values Tp0 = 2h for p = 0+ (in the close neighbourhood of the origin 0 of the 0p axis) and Tpf =14h for p = pf. If the procedure is repeated for some intermediary values of (p) variable, a linear increasing evolution of T results from p = 0 to p = pf, the general relation of T(p) becoming:
The mathematical model proposed in the paper can be adapted, and after that, it will be used in order to model the work of the electric power systems. 2. 13C SEPARATION PROCESS MODELING The input signal in the 13C separation process is considered the flow of the 1M (molar concentration) ethanolamine solution Fin(t) introduced in the absorber A using the pump P, respectively the output signal from the process is considered the concentration of the 13C isotope y(t,p). The treated separation process is a distributed parameter one (2005;2011) due to the fact that the output signal depends on two independent variables: the time (t) and the position in the separation column in relation to its height, variable notated with (p). The concentration variation of the 13C isotope in the transversal section of the separation column is insignificant, not being considered in the process model (the transversal section of the column is referring to each section on which its height direction is a vertical line). In order to highlight the independent variable “length” (p), the 0p axis from Fig. 2 is defined.
T Tp0 (Tpf Tp0 )
p . pf
In (3), the variations of the (p) independent variable occur at discrete time moments, being approximated as step type signals. The (p) variable variations are obtained by changing the transducer T position inside the SC along the 0p axis. Considering (2) and (3), it results that Ft is a function depending on both independent variables Ft(t,p). In order to determine the relation of uf signal, first, the dependency relation between the ethanolamine input flow Fin(t) and the height equivalent to a theoretical plate (HETP) is considered:
HETP(t ) HETP0 KH ·( Fin (t ) – Fin0 ) ,
In Fig. 2, the origin 0 represents the centre of the column section from the upper part of the column (the column has a cylindrical form). The column diameter is d = 2.5cm, respectively the column height is h = 300cm. In this case, the (p) independent variable takes values between p0 = 0cm (corresponding to the SC section which includes the origin 0) and pf = h = 300cm (corresponding to the lowest SC section). The approximating analytical solution (2010) which describes the separation process work in transitory regime is given by the relation: (1)
KH
where Ft(t) and Fp(p,Fin(t)) are two increasing functional terms and y0 = 1.108% represents the natural abundance of the 13C isotope. The analytical solution from (1) is valid for all working regimes, but only after the CO2 enters the first time in SC through the pipe 7. The Ft(t) function represents the solution of the following first order ordinary differential equation: dFt (t ) 1 1 Ft (t ) u f (t ) , dt T T
(4)
where HETP(t) is the instantaneous value of HETP, HETP0 is the steady state value of HETP for the ethanolamine input flow Fin0 = ct., KH is the proportionality constant which makes the connection between Fin and HETP signals, respectively Fin(t) is the instantaneous value of input signal. The value of the reference input flow Fin0 = 367ml/h is chosen, the corresponding HETP value being HETP0 = = 4.64cm. The KH constant is the gradient of the ramp resulted from the graphical representation of the function HETPst(Fin), where HETPst represents the steady state value of HETP corresponding to different Fin step signals. The computation relation of KH is:
Fig. 2. The 0p axis associated to the separation column.
yAN (t , p) y0 Ft (t )·Fp ( p, Fin (t )) ,
(3)
HETPst1 – HETP0 , Fin1 – Fin0
(5)
where the step signal Fin1 = 460ml/h and the corresponding steady state HETP value being HETPst1 = 5.43cm. The data HETP0 and HETPst1 are experimentally obtained applying at the process input the step signals Fin0, respectively Fin1 and waiting until the process reaches the steady state regime. The experiments can be made for any value of the input flow in the technologically allowed limits. After computation, it results the value KH = 0.0085(cm∙h)/ml.
(2) 192
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function influences only the values of the y(t,p) signal increase over y0. Using the experimental data and a mathematical procedure based on an interpolation method, respectively considering the exponential increasing evolution of the 13C concentration at the increase of (p) variable, the y(tf,p) function can be approximated by the FpA(p) function of the form:
The isotope separation can be computed using the relation:
S (t ) n(t ) ,
(6)
where α = 1.01 is the elementary separation factor of the 13C isotope for the carbamate – CO2 chemical exchange procedure and n(t) = h/HETP(t) is the instantaneous value of the number of the theoretical plates. Considering (4), (6) and the relation of n(t), it results that: S (t )
h HETP0 K H ·( Fin ( t )– Fin 0 )
,
p
FpA ( Fin (t ) ,p)=( y0 1) e 430 K P ( Fin (t ) Fin 0 ) ,
(7)
y (t f ,p f )( Fin (t )) ,
P 430 KP ·(Fin (t ) – Fin0 ) .
(8)
Considering the form of FpA(Fin(t), p) from (13), FpB can be obtained directly as FpB = FpA(Fin(t), pf). Having FpA and FpB, Fp(p,Fin(t)) results using (12). Also yAN results replacing the obtained forms of Ft(t,p) and Fp(p,Fin(t)) in (1).
where y(tf,pf)(Fin(t)) represents the steady state value of the y(t,p) signal for p = pf = h and for a certain value of the input step type signal which has the instantaneous value of the Fin(t) signal. From (8), it results that:
Analysing the equations presented at this Paragraph it results that the treated separation process is a strong nonlinear (2013) one. This remark is proved by the following aspects: both Ft and Fp functions depend on both independent variables; in (1) the two mentioned functions are multiplied; Fp results as the ration between FPA and FPB.
(9)
respectively, the 13C concentration increase over y0 for p = = pf = h and in steady state regime, is given by:
y(t f ,p f )( Fin (t )) y0 y0 (S (t ) 1) .
(10)
3. THE PROPOSED CONTROL STRUCTURE AND THE CONTROLLER TUNING
Equation (10) gives also the relation of the final input signal in the process, form presented in (11):
u f (t )( Fin (t )) y0 (S (t ) 1) .
The proposed control structure is presented in Fig. 3. The elements from the structure are: TP – technological separation process (being a distributed parameter process DPP); A – actuator (the pump P from Fig. 1); T – concentration transducer (the mass spectrometer from Fig. 1); AC – concentration controller (the same as in Fig. 1); RELAY – a relay type element with hysteresis. Also, the signals from the structure are: w(t) – set point signal used for the structure work in automatic control regime; w1(t) – set point signal used for the controller tuning; r(t) – feedback signal; c(t) – control signal generated by AC; b(t) – output signal of the relay element; Fin(t) – actuating signal (the input flow of ethanolamine); d1(t) – disturbance signal which affects directly the actuating signal; Finf(t) – disturbed actuating signal (the final value of the ethanolamine input flow); yi(t,p) – (initial) output signal (the 13C isotope concentration); d2(t) – disturbance signal which affects directly the output signal; y(t,p) – disturbed output signal due to the effect of d2(t). In Fig. 3, the elements A, TP and T connected in series represent the Fixed Part of the control system.
(11)
Considering (2), (3), (9), (11) and the remark according to Ft = Ft(t,p), the value changing of the (p) independent variable influences the value of Ft(t,p) only if the changing occurs in transitory regime. Also, considering (1), (2), (3), (9), (10) and (11) it results that at the value changing of the (p) independent variable, the speed evolution of the output signal is adapted, but its steady state value is not modified, remaining at y(tf,pf) and occurring the necessity to introduce in the analytical solution yan the Fp(p,Fin(t)) function. The Fp(p,Fin(t)) function is given by the following relation:
Fp ( p,Fin (t )) =
FpA ( Fin (t ) ,p) y0 . FpB y0
(14)
It can be remarked that the technological separation process presents only one “length” constant P. From (14), it results that P is a function of Fin(t) (P(Fin(t)) and implicitly P(t). Consequently, the notations FpA(Fin(t), p) and FpA(t,p) are both justified.
y0
y(t f ,p f )( Fin (t )) y0 S (t ) ,
(13)
where the constant C = 430cm is associated to SC being determined through interpolation and the proportionality constant KP = 0.7527(cm∙h)/ml is determined mathematically processing two consecutive sets of values {Fin, P}. P signifies the “length” constant of the process, having the relation:
S being a function of Fin(t) (S(Fin(t))). In (7), the input signal in the process Fin(t) occurs with positive sign at the denominator of the ratio from the exponent. Consequently, the increase of Fin(t) implies the decrease of S(t), implying also the decrease of the y(t,p) signal, respectively the decrease of Fin(t) implies the increase of S(t), implying also the increase of the y(t,p) signal. From the physical point of view, the 13C enrichment is a more efficient one at lower values of the input ethanolamine flows due to the fact that the contact duration between carbamate and CO2 in SC is longer. The separation S(t) results from the following relation, too: S (t )
193
(12)
In (12) the subtraction of the y0 value both from numerator and from denominator signifies the fact that the Fp(p,Fin(t)) 193
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Fig. 3. The proposed control structure. structure (characterized through undamped oscillations of the output, respectively feedback signals) and in the estimation of the obtained limit cycle parameters (the amplitude and the period of the sustained oscillations). In the case of the current application, the set point signal is maintained constant to the value w1(t) = 9.8mA (step type signal; this value is chosen considering the fact that, in many cases, the plant works for reference signals around this value; the considered value is interpreted as 5.8mA over the initial value of 4mA associated to the unified current signals), the three switches being set on the positions 2, 4 and 4’. Using the previously presented relay parameters, the evolution in time of the feedback signal r(t) in controller tuning regime is presented in Fig. 5.
Also, the elements S1, S2 and S3 are switches used in order to select the working regime of the structure: controller tuning regime or automatic control regime. The feedback from the value of the (p) independent variable to the AC controller is used in order to adapt its parameters at the values change of this variable. Due to this feedback, the AC controller can be considered an adaptive one. In Fig. 3, if d1(t) = 0 and d2(t) = 0, then Finf(t) = Fin(t), respectively y(t,p) = yi(t,p). Also the error signal a(t) is given by a(t) = w(t) – r(t) in automatic control regime, respectively a(t) = w1(t) – r(t) in controller tuning regime. The automation equipment from Fig. 3 work with unified current signal 4-20mA. In this case the signals w(t), w1(t), c(t), b(t) and r(t) are unified current ones. If the three switches are set on the positions 2, 4 and 4’, the w1(t) set point signal is active, respectively the relay element is connected in the control structure, working in controller tuning regime (2009b) (the connections figured with dashed line from the structure are active). If the three switches are set on the positions 1, 3 and 3’, the w(t) set point signal is active and the AC controller is connected in the control structure, working in control regime (the connections figured with continuous line from the structure, associated to S 1, S2 and S3 are active). The feedback in relation to the (p) signal is used only in automatic control regime. For the controller tuning the relay method is applied. The used two position relay is presented in Fig. 4.
12 Amax 11 Ao
Aav
r(t) [mA]
10 9
Ao
8
Amin
7 6
To
The feedback signal
To
5 t1 4 0
50
100
t3
t2 150
200
t4 250
TIME [h]
Fig. 5. The evolution in time of the feedback signal in controller tuning regime. The Fig. 5 is presented due to the fact that the oscillations parameters of the feedback signal are used for the controller tuning. From Fig. 5, it results that the average value of the r(t) signal oscillations is Aav = 9.816mA, the maximum value is Amax = 11.134mA, respectively the minimum value is Amin = 8.498mA. The oscillations amplitude A0 can be calculated using the relation:
Fig. 4. The used relay with hysteresis. The output signal from the relay element is given by the relation:
b,if a(t ) as , b(t ) b, if a(t ) as
A0 Amax Aav Aav Amin 1.318mA .
(15)
(16)
Also, the period of the oscillations T0 results from Fig. 5 as the period of time between two consecutive oscillations maximum values or minimum values. Taking the case of the maximum values, T0 is given by the relation:
where b = 8mA and the hysteresis is determined by the error value as = 1.3mA. The relay method consists in using the relay to obtain the stability limit regime for the control 194
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T0 t2 t1 168.086 149.946 18.14h .
maximum, respectively the minimum values of the unified current signals (4-20)mA. In (21), Fin(s) and C(s) represent the Laplace transformation of the Fin(t) and c(t) signals. From (21) it results that the technological plant works in starting regime at the maximum flow Finmax = 1000ml/h. The negative value of KA signifies that the second term from the right member of (21) (the term corresponding to C(s)) represents the ethanolamine flow value which has to be subtracted from Finmax in order to obtain the actuating signal Fin(s). The proportionality constant of the transducer T, KT, is computed using the relation KT = (cmax – cmin)/(ymax – y0)= 8.45mA/%, where cmax and cmin have the same significance as in the case of KA, y0 has the same significance as in (1), respectively the ymax (3%) represents the maximum value of the produced 13C isotope concentration.
(17)
The value of T0 obtained in (17) is verified for the case of the t3 and t4 values, too. Using the A0 value, the equivalent amplifying coefficient can be calculated using the relation:
k0
4b 7.7283 . A0
(18)
Using the obtained values of T0 and k0, the PI and the PID type controllers parameters can be computed using the Ziegler-Nichols formulae (1999). The following notations are used: KC represents the proportionality constant of the controller, TI represents the integral time constant of the controller, respectively TD represents the derivative time constant of the controller. Using the relations associated to the Ziegler-Nichols tuning method, for the value p = pf = h of the (p) independent variable, the transfer function of the PI controller is presented in the equation:
1 , H PI ( s) KC (1 ) TI s
The mathematical models presented in (21) and (22) are used in the case of the simulation from Fig. 5 and in the case of the simulations from Paragraph 4, too. 4. SIMULATIONS RESULTS
(19)
The simulations from this paper MATLAB/Simulink (User Guide).
where KC = 3.47 and TI = 14.51h. The PID controller is not presented in the paper due to the fact that in the simulations, it generated very high values of the control signal that cannot be used in practice. If a saturation element is used in order to limit the control signal values or the controller parameters are adjusted as value (decrease of the KC and TD values, increase of TI value, respectively the increase of the value of the filter time constant used for obtaining the feasibility of the controller derivative component), the generated set of performances is a weak one, the complexity of the controller not being justified.
KA 1000 C ( s) ; (TA s 1) s HT ( s )
KT . TT s 1
made
in
1.8 1.7
(20) y(t,p) [%]
1.6
where TI0 = 14.51h is the value of TI from (19) and Tp0 has the same significance as in (3). The A and T elements from Fig. 3 are both linear elements. Their work is expressed using the following relations ((21) for A, and (22) for T):
Fin ( s)
are
In Fig. 6, the comparative graph between the step response of the open loop process and the control system step response (in the case when the three switches from Fig. 3 are set on the positions 1, 3 and 3’) is presented. The value of the reference signal in concentration is set to 1.7%, respectively the simulation of the open loop process is made setting for the Fin(t) signal (of the step type) the value previous computed (641.7ml/h) necessary to obtain for the y(t,p) signal the value 1.7%. Also, the two responses from Fig. 6 are obtained for p = pf = ct.
The adaptive control consists in the automatic changing of the value of the TI constant at the value changing of the (p) independent variable. The corresponding relation is:
p , TI ( p) Tp0 (TI 0 Tp0 ) pf
195
The control system response The open loop process response
1.5 1.4
t7
1.3
(21)
1.2 1.1 0 t6
(22)
t5 50
100
150
200
250
TIME [h]
Fig. 6. The comparative graph between the open loop process response and the control system response.
In (21) and (22) TA = 20s and TT = 3min are the time constants of the A and T elements. The proportionality constant of the actuator A, KA, is computed using the relation KA = (Finmin – Finmax)/(cmax – cmin) = (100 – 1000)/(20 – 4) = = –56.25ml/(h∙mA). In this relation Finmin = 100ml/h and Finmax = 1000ml/h represent the minimum, respectively the maximum technological values of the actuating signal and cmax = 20mA, respectively cmin = 4mA represent the
The obtained set of the control system performances are: the steady state error – 0% (the imposed value), the overshoot – – 0% (the imposed value, being also a critical restriction) and the settling time t6 = 20.4h (much smaller than the imposed value – 35h). Also, the value of the obtained settling time in the case of using the control system is much smaller than the 195
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value of the settling time associated to the open loop process: t5 = 48.45h (the difference between them being t7 = 28.05h). Consequently, the usage of the control system is justified.
In Fig. 8, with dashed line, the case of using the unmodified form of the PI controller from (19) is presented, respectively with continuous line, the case of using the adaptive PI controller (applying (20), too) is presented. The reference signal is maintained constant during the simulation. In Fig. 8, the fact that the adaptive controller generates a much faster stabilization of the y(t,p) signal than the unmodified PI controller, after the p variable variation, is highlighted.
In Fig. 7, the simulations of the open loop process and of the control system are made if the step type disturbance d1(t) = = 200 ml/h occurs in the system after 100h after the simulation start, respectively the negative step type disturbance d2(t) = -0.2% occurs in the system after 200h from the simulation start. The positive value of d1(t) represents a disturbance, it implying the decrease of y(t,p).
5. CONCLUSIONS In this paper, an original solution for modeling and for the adaptive control of a strong nonlinear distributed parameter isotope separation process is presented. In order to verify the efficiency of using a control system, some simulations are presented in Paragraph 4. In Paragraph 4, the proposed control structure is tested both in the case when the disturbance signals occur the system and in the case when the p independent variable presents variations. The simulations results proved the necessity of using the proposed control system, it generating a restrictive set of control performances (much better than the imposed one). Also, the controller rejects the effect of the two possible disturbance signals and it has the possibility (being an adaptive one) to preserve the system performances at the p variable value changing. In all the simulations from Paragraph 4, the intermediary signals c(t) and Fin(t) do not exceed their maximum, respectively minimum imposed limits (cmax, cmin, Finmax, Finmin).
1.8 1.7
y(t,p) [%]
1.6 1.5 1.4 1.3 1.2 1.1 0
The control system response The open loop process response 50
100
150
200
250
300
TIME [h]
Fig. 7. The open loop step process response and the control system step response, if the disturbances occur in the system.
REFERENCES Axente, D., Abrudean, M., Bâldea, A. (1994). 15N, 18O, 10B, 13 C Isotopes Separation trough Isotopic Exchange. Science Book House. Dobra, P. (1999). Nonlinear Systems. U.T.PRES Publishing House. Dang, H., Rochelle, G.T. (2003). CO2 absorption rate and solubility in monoethanolamine/ piperazine/ water. Separation Sci. & Tech., 38 (2), pp. 337–357. Smyshlyaev, A., Krstic, M. (2005). Control design for PDEs with space-dependent diffusivity and time-dependent reactivity. Automatica, Vol. 41, pp. 1601-1608. Love, J. (2007). Process Automation Handbook. 1 edition, Springer Publishing House. Dugas, R., Rochelle, G. (2009). Absorption and desorption rates of carbon dioxide with monoethanolamine and piperazine. Energy Procedia, Vol. 1 (1), pp. 1163–1169. Golnaraghi, F., Kuo, B. C. (2009). Automatic Control Systems. 9th edition, Wiley Publishing House. Mureşan, V., Abrudean, M. (2010). Temperature Modelling and Simulation in the Furnace with Rotary Hearth. Proc. of IEEE AQTR–17th ed., Cluj-Napoca, Romania, pp. 147-152. Li, H.-X., Qi, C. (2011). Spatio-Temporal Modeling of Nonlinear Distributed Parameter Systems: A Time/Space Separation Based Approach. 1st Edition, Springer Publishing House. Coloşi, T., Abrudean, M., Ungureşan, M.-L., Mureşan, V. (2013). Numerical Simulation of Distributed Parameter Processes. Springer Publishing House. User Guide, Matlab 7.5.0 (R2007b).
The simulations from Fig. 7 are made for p = pf = ct. From Fig. 7, it can be remarked that the effects of the two disturbances is rejected by the AC controller (in the case of the control regime), proving again the necessity of implementing the control structure from Fig. 3. In the case of the open loop process the disturbances effects generate the decrease of the y(t,p) signal value. In Fig. 8, the case of the p independent variable variation is treated, considering the following values: before the moment 150h p = pf , between the moments 150h and 225h p = 230mm, respectively after the moment 225h p = 280mm (the variation between two consecutive values are considered of step type). 1.9 1.8
y(t,p) [%]
1.7 1.6 1.5 1.4 1.3 1.2 1.1 0
The case of using the unmodified initial PI controller The case of using the adaptive PI controller
50
100
150
200
250
300
TIME [h]
Fig. 8. The control system simulations, if the p independent variable presents variations. 196
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Modeling, Simulation and Control of a Nonlinear Distributed Parameter Isotope Modeling, Simulation and Control of a Nonlinear Distributed Parameter Isotope Modeling, of Modeling, Simulation Simulation and and Control Control of aa Nonlinear Nonlinear Distributed Parameter Parameter Isotope Isotope Separation Process Distributed Separation Process Separation Separation Process Process Vlad MUREŞAN*, Mihail ABRUDEAN* Vlad MUREŞAN*, Mihail ABRUDEAN* Vlad MUREŞAN*, MUREŞAN*, Mihail ABRUDEAN* Daniel Mihail MOGA* Vlad ABRUDEAN* Vlad MUREŞAN*, Mihail ABRUDEAN* Daniel MOGA* Daniel MOGA* Daniel Daniel MOGA* MOGA* *Technical University of Cluj-Napoca, Cluj-Napoca, CO 400114 ROMANIA (Tel: +40-744-420906; *Technical University of Cluj-Napoca, Cluj-Napoca, CO (Tel: +40-744-420906; *Technical University of Cluj-Napoca, Cluj-Napoca, CO 400114 400114 ROMANIA ROMANIA (Tel: +40-744-420906; ). e-mail: [email protected], [email protected], [email protected] *Technical University of Cluj-Napoca, Cluj-Napoca, (Tel: *Technical University of Cluj-Napoca, Cluj-Napoca, CO CO 400114 400114 ROMANIA ROMANIA (Tel: +40-744-420906; +40-744-420906; ). e-mail: [email protected], [email protected], [email protected] e-mail: [email protected], [email protected], [email protected] ). e-mail: e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] [email protected] ). ). Abstract: In this paper, an original solution of modeling and control of a distributed parameter 13 C 13 13 Abstract: In this this paper, paper, an is original solution ofmentioned modeling and and control of of process distributed parameter 13 C Abstract: In an original solution of modeling control aaa distributed parameter 13C isotope separation process presented. The technological takes place into C Abstract: In this paper, an original solution of modeling and control of distributed parameter Caa Abstract: In this paper, an is original solution ofmentioned modeling and control of process a distributed parameter 13 isotope separation process presented. The technological takes place into isotope separation process is presented. The mentioned technological process takes place into aa C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 isotope separation separation process is presented. The mentioned technological process takes place into isotope process is presented. The mentioned technological process takes place into a 13C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 C enrichment through carbon dioxide – carbamate chemical separation column, whose work ensures the 13 exchange. A main difficulty in the modeling – control problem approach is the fact that, besides that the C enrichment enrichment through through carbon carbon dioxide dioxide – – carbamate carbamate chemical chemical separation column, column, whose whose work work ensures ensures the the C separation exchange. main in – problem approach is the besides that exchange. A main difficulty difficulty in the the modeling modeling – control control problem approach in is relation the fact fact that, that, besides that the the process is A a distributed parameter one, it presents strong nonlinearities to both independent exchange. A main in – problem approach is the fact besides that exchange. A main difficulty difficulty in the the modeling modeling – control control problem approach in is relation the 13 fact that, that, besides that the the process is a distributed parameter one, it presents strong nonlinearities to both independent process is is(the a distributed distributed parameter one, it column presentsheight). strong For nonlinearities in relation to both independent C isotope concentration variables time and the position in the the tuning of the process a parameter one, it presents strong nonlinearities in relation to both independent 13 process is a distributed parameter one, it presents strong nonlinearities in relation to both independent 13 Cisisotope isotope concentration variables (the time and the position in the column height). For the tuning of the 13C concentration variables the position the column height). For the tuning of the controller,(the the time relayand method is used. in Also, initially obtained controller structure modified in order to concentration variables (the time and the in the column height). For tuning the Cisisotope isotope concentration variables (the time and the position position in the the column height). For the the tuning of of the 13C controller, the relay method is used. Also, the initially obtained controller structure modified in order in to controller, the relay method is used. Also, the initially obtained controller structure is modified in order to ensure the same set of performances at the variation of the second independent variable (the position controller, the relay method is used. Also, the initially obtained controller structure is modified in to controller, the relay method is used. Also, initially of obtained controller structure is modified in order order in to 13the ensure the same set of performances at the variation the second independent variable (the position ensure the same set of performances at the variation of the second independent variable (the position in C isotope concentration value is controlled using an adaptive the column height). In this context, the ensure the same set of performances at the variation of the second independent variable (the position in 13 ensure the same set of performances at the variation of the second independent variable (the position in 13 C isotope concentration concentration value is controlled using an adaptive the column height). In this context, the 13C isotope controlled using an adaptive the column height). In this context, the 13 algorithm. the paper, some interesting are presentedvalue whichis the main advantages of C value is controlled using an the column height). In context, the C isotope isotope concentration concentration value isprove controlled using an adaptive adaptive the columnIn height). In this this context, the simulations 13 algorithm. In the paper, some interesting simulations are presented which prove the main advantages of algorithm. In the paper, some interesting simulations are presented which prove the main advantages of C isotope using a control structure with an adaptive controller: the possibility to control theadvantages algorithm. In the paper, some interesting simulations are presented which prove the main of 13 algorithm. In the paper, some interesting simulations are presented which prove the main advantages of 13C isotope using a control structure with an adaptive controller: the possibility to control the 13 C isotope using a control structure with an adaptive controller: the possibility to control the 13 concentration in any point from the column height; the possibility to reject the effect of the two types of C isotope isotope using aa control control structure structure with with an an adaptive adaptive controller: controller: the the possibility possibility to to control control the the C using concentration in from height; the to effect the types concentration which in any any point point from the the column height; the possibility possibility to reject reject the effecta of ofcertain the two tworestrictive types of of disturbances can occur in column the process work; the possibility to the satisfy concentration in from column height; the to the effect the types concentration in any any point point from the the column height; the possibility possibility to reject reject the effecta of ofcertain the two tworestrictive types of of disturbances which can occur in the process work; the possibility to satisfy disturbances set; which can occur in in the process work; the the possibility to satisfy satisfy a certain certain restrictive performance the possibility to adapt the controller parameters in order to obtain the same set of disturbances which can occur the process work; possibility to a restrictive disturbances which can occur in the process work; the possibility to satisfy a certain restrictive performance set; the possibility to adapt the controller parameters in order to obtain the same set of performance set; the possibility to adapt the controller parameters in order to obtain the same set of performances in any transversal section of the column. performance set; the possibility to adapt the controller parameters in order to obtain the same set performance set; thetransversal possibilitysection to adapt the column. controller parameters in order to obtain the same set of of performances in any of the performances in any transversal section of the column. performances in any transversal section of the column. 13 performances in any transversal section of the column. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear distributed parameter process, 13 C isotope, separation column, relay method, 13 isotope, separation column, relay method, Keywords: Nonlinear distributed parameter process, 13C isotope, separation column, relay method, Keywords: Nonlinear distributed parameter process, 13C concentration adaptive controller, chemical exchange process carbon dioxide – carbamate, C separation column, relay Keywords: Nonlinear distributed parameter process, Keywords: Nonlinear distributed chemical parameter process,process C isotope, isotope, separation column, ethanolamine. relay method, method, concentration adaptive controller, exchange carbon dioxide – carbamate, ethanolamine. concentration adaptive controller, chemical exchange process carbon dioxide – carbamate, ethanolamine. concentration concentration adaptive adaptive controller, controller, chemical chemical exchange exchange process process carbon carbon dioxide dioxide – – carbamate, carbamate, ethanolamine. ethanolamine. 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION C isotope The technological plant used for the 13 13 13C isotope The technological plant The technological used for the 13C presented in Fig. 1.plant C isotope isotope The technological plant used used for for the the 13 The technological plant used for the C isotope presented presented in in Fig. Fig. 1. 1. presented in Fig. 1. presented in Fig. 1.
separation is separation separation is separation is is separation is
C isotope separation plant. Fig. 1. The technological 13 13 Fig. 1. The The technological technological 13 13C Fig. 1. C isotope isotope separation separation plant. plant. Fig. C isotope isotope separation separation plant. plant. Fig. 1. 1. The The technological technological 13C
The carbon dioxide (CO2) absorption (2003;2009a) in ))) absorption (2003;2009a) in The carbon dioxide absorption (2003;2009a) in The carbon takes dioxide (CO ethanolamine place(CO in 222the absorber A (the two chemical (2003;2009a) in The dioxide (CO ) absorption absorption (2003;2009a) in The carbon carbon takes dioxide (CO 2the ethanolamine place in absorber A (the two chemical ethanolamine takes place in the absorber A (the two chemical elements circulating in counter-current), resulting the ethanolamine takes place in the absorber A (the two chemical ethanolamine takes place in the absorber A (the two chemical elements circulating counter-current), resulting the elements (pipe circulating insupplied counter-current), resulting the carbamate 3). A isin with ethanolamine through elements circulating in counter-current), resulting the elements circulating insupplied counter-current), resulting the carbamate (pipe 3). A is with ethanolamine through carbamate (pipe 3). Apump is supplied supplied with ethanolamine ethanolamine through the pipe 1 using the P, respectively with CO carbamate (pipe 3). A is with through 2 through carbamate (pipe 3). A is supplied with ethanolamine through the 1 the P, with through the pipe pipe 1 using using the pump pump 99.98% P, respectively respectively with CO CO 222Through pipe 5 (at approximately concentration). through the 1 the P, with through the pipe pipe 1 using using the pump pump 99.98% P, respectively respectively with CO CO 2Through pipe 5 (at approximately concentration). pipe 5 (at approximately 99.98% concentration). Through at a concentration 4, the gas phase, containing CO pipe 5 (at approximately 99.98% concentration). Through 2 pipe 5 (at approximately 99.98% concentration). Through at concentration pipe 4, the gas phase, containing CO 2 aaa concentration pipe the gas phase, containing CO 2 at C lower4, is evacuated from The 13 atsystem. pipe 4, the gas containing CO 2 13 a concentration concentration pipe 4,than the 0.1% gas phase, phase, containing COthe 13 2 atsystem. C lower than 0.1% is evacuated from the The 13C lower than 0.1% is evacuated from the system. The 13 – carbamate takes enrichment through isotopic exchange CO C lower than 0.1% is evacuated from the system. The 2 C lower than through 0.1% isisotopic evacuated from CO the2 –system. Thetakes carbamate enrichment exchange – carbamate takes enrichment through isotopic exchange CO 2 place in thethrough separation column SC (1994). The carbamate is carbamate takes enrichment isotopic exchange CO 2 – – carbamate takes enrichment through isotopic exchange CO 2 The carbamate is place in the separation column SC (1994). place in the separation column carbamate is introduced SC through pipe 3SC and(1994). the COThe place separation column SC (1994). The carbamate is 2 is introduced place in in the thein separation column SC (1994). The carbamate in is in introduced in SC pipe 3 CO is introduced introduced in introduced inpipe SC through through pipe elements 3 and and the the circulating CO222 is SC through 7 (the two in SC in is introduced introduced in SC through pipe 3 and the CO is introduced in introduced in SC through pipe 3 and the CO 2 SC through 7 two elements circulating SC in SC through pipe pipetoo). 7 (the (the two the elements circulating inthe SC13 in C counter-current, During separation process,in SC through 7 two elements circulating in SC in SC through pipe pipetoo). 7 (the (the two the elements circulating inthe SC13 in 13 C counter-current, During separation process, 13 C counter-current, too).inDuring During the separation process, the 13 isotope concentrates liquid phase in the lower part of the C counter-current, too). the separation process, the C counter-current, too).inDuring the separation process, the isotope concentrates liquid phase in the lower part of the isotope concentrates in liquid phase in the lower part of the which passes through column (pipe 2), respectively the CO isotope concentrates in liquid phase in the lower part of the 2 isotope concentrates in liquid phase in the lower part of the which passes column (pipe 2), respectively the CO which passes through column (pipe 2), respectively the CO SC is sent to A pipe 5. the222 reactor R, the through thermal which through column (pipe 2), respectively the CO which passes passes through column (pipe 2),through respectively theIn CO 2 reactor SC is sent to A through pipe 5. In the R, the thermal SC is sent to A through pipe 5. In the reactor R, the thermal decomposition of the carbamate is made and in the stripper S SC is sent to A through pipe 5. In the reactor R, the thermal SC is sent to Aofthrough pipe 5. In the reactor R,thethe thermal decomposition the carbamate is made and in stripper S decomposition of the carbamate is made and in the stripper S ) the is carbamate completely removed resulting the the gas (CO2of decomposition is made and in the stripper S decomposition of the carbamate is made and in the stripper S ) is completely removed resulting the the gas (CO 2) is completely removed resulting the the gas (CO 2 ethanolamine. The ethanolamine is reheated using the heater ) is is completely completely removed removed resulting resulting the the the gas (CO 2) the gas (CO 2 ethanolamine. The ethanolamine is using the ethanolamine. The ethanolamine is reheated reheated usingpipe the heater heater H and circulated to the absorber trough 1 and ethanolamine. The ethanolamine is using the ethanolamine. Theagain ethanolamine is reheated reheated usingpipe the heater heater H and circulated again to the absorber trough 1 H and and the circulated again to the thetheabsorber absorber trough pipe 1 and and resulting after the using pump P. Also, CO H circulated again to trough pipe 1 2 H and the circulated again to thetheabsorber trough pipe 1 and and resulting after using pump P. Also, CO 2 resulting afterto the the using the pump pump P. and Also, the CO22 resulting decomposition (in R) after stripping (in S) is sent after using the P. Also, the CO afterto the the using the pump P. and Also, the CO2 resulting decomposition (in R) after stripping (in S) is sent the decomposition (in R) and after stripping (in S) is sent to the SC through pipe 7. The pipe 6 used for supplying the plant decomposition (in R) and after stripping (in S) is sent to the decomposition (in R) and after stripping (in S) is sent to the SC through pipe 7. The pipe 6 used for supplying the plant SC through pipe 7. The pipe 6 used for supplying the plant and pipe 8 used for extracting the product in with CO SC through pipe 7. The pipe 6 used for supplying the plant 2 SC through pipe 7. The pipe 6 used for supplying the plant and pipe 8 used for extracting the product in with CO 2 pipe 8 used for extracting the product in with CO 2 and gaseous phase (both figured in Fig. 1 with dashed line) are and pipe 8 used for extracting the product in with CO 2 pipe figured 8 used inforFig. extracting the product in with COphase 2 and (both gaseous 1 with dashed line) are gaseous phase (both figured in Fig. 1 with dashed line) are used in production regime. The product is referring to CO gaseous phase (both figured in Fig. 1 with dashed line) are 2 gaseous phase (both figured in Fig. 1 with dashed line) are 13 used in production regime. The product is referring to CO 2 used production regime. The is referring to CO 2 C. In Fig. 1, it can be with in a certain concentration of product used in production regime. The product is referring to CO 13 used in production regime. The product is referring to CO 13C. In Fig. 1, it can be2 2 with a certain concentration of 13 C. In Fig. 1, it can be with a certain concentration of 13 remarked that the elements A, SC and S present steel pack of C. In In Fig. Fig. 1, 1, it it can can be be with aa certain certain concentration concentration of of C. with remarked that the elements A, SC and S present steel pack of remarked that that the elements elements A, SC SC and S S present steel steel pack of of remarked remarked that the the elements A, A, SC and and S present present steel pack pack of
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Halipack type (1994), figured with the hachured sections. The steel pack usage is determinant for the plant working as it makes possible the 13C isotope separation. In Fig. 1, the automation equipment associated to the separation plant are also presented, their significance and the working of the concentration control system being detailed later in the paper.
where T = T(p) represents the time constant of the process and uf(t) represents the final input signal in the process. The time constant of the process is determined using an experimental identification procedure based on the step response of the process. The signal uf is a function of Fin(t), hence uf(Fin(t)), dependency which in the particular case of the Fin(t) step type signal implies the fact that uf is a step type signal, too. Each experiment can be described as follows: a step type signal Fin(t) is applied at the input of the process and the evolution in time of the y(t,p) signal is measured in a certain predefined transversal section of SC given by the value of the independent variable (p). From the obtained experimental data, it results that the process is of first order one (as it was shown in (2)). The value of the time constant is determined using the tangent method, resulting the values Tp0 = 2h for p = 0+ (in the close neighbourhood of the origin 0 of the 0p axis) and Tpf =14h for p = pf. If the procedure is repeated for some intermediary values of (p) variable, a linear increasing evolution of T results from p = 0 to p = pf, the general relation of T(p) becoming:
The mathematical model proposed in the paper can be adapted, and after that, it will be used in order to model the work of the electric power systems. 2. 13C SEPARATION PROCESS MODELING The input signal in the 13C separation process is considered the flow of the 1M (molar concentration) ethanolamine solution Fin(t) introduced in the absorber A using the pump P, respectively the output signal from the process is considered the concentration of the 13C isotope y(t,p). The treated separation process is a distributed parameter one (2005;2011) due to the fact that the output signal depends on two independent variables: the time (t) and the position in the separation column in relation to its height, variable notated with (p). The concentration variation of the 13C isotope in the transversal section of the separation column is insignificant, not being considered in the process model (the transversal section of the column is referring to each section on which its height direction is a vertical line). In order to highlight the independent variable “length” (p), the 0p axis from Fig. 2 is defined.
T Tp0 (Tpf Tp0 )
p . pf
In (3), the variations of the (p) independent variable occur at discrete time moments, being approximated as step type signals. The (p) variable variations are obtained by changing the transducer T position inside the SC along the 0p axis. Considering (2) and (3), it results that Ft is a function depending on both independent variables Ft(t,p). In order to determine the relation of uf signal, first, the dependency relation between the ethanolamine input flow Fin(t) and the height equivalent to a theoretical plate (HETP) is considered:
HETP(t ) HETP0 KH ·( Fin (t ) – Fin0 ) ,
In Fig. 2, the origin 0 represents the centre of the column section from the upper part of the column (the column has a cylindrical form). The column diameter is d = 2.5cm, respectively the column height is h = 300cm. In this case, the (p) independent variable takes values between p0 = 0cm (corresponding to the SC section which includes the origin 0) and pf = h = 300cm (corresponding to the lowest SC section). The approximating analytical solution (2010) which describes the separation process work in transitory regime is given by the relation: (1)
KH
where Ft(t) and Fp(p,Fin(t)) are two increasing functional terms and y0 = 1.108% represents the natural abundance of the 13C isotope. The analytical solution from (1) is valid for all working regimes, but only after the CO2 enters the first time in SC through the pipe 7. The Ft(t) function represents the solution of the following first order ordinary differential equation: dFt (t ) 1 1 Ft (t ) u f (t ) , dt T T
(4)
where HETP(t) is the instantaneous value of HETP, HETP0 is the steady state value of HETP for the ethanolamine input flow Fin0 = ct., KH is the proportionality constant which makes the connection between Fin and HETP signals, respectively Fin(t) is the instantaneous value of input signal. The value of the reference input flow Fin0 = 367ml/h is chosen, the corresponding HETP value being HETP0 = = 4.64cm. The KH constant is the gradient of the ramp resulted from the graphical representation of the function HETPst(Fin), where HETPst represents the steady state value of HETP corresponding to different Fin step signals. The computation relation of KH is:
Fig. 2. The 0p axis associated to the separation column.
yAN (t , p) y0 Ft (t )·Fp ( p, Fin (t )) ,
(3)
HETPst1 – HETP0 , Fin1 – Fin0
(5)
where the step signal Fin1 = 460ml/h and the corresponding steady state HETP value being HETPst1 = 5.43cm. The data HETP0 and HETPst1 are experimentally obtained applying at the process input the step signals Fin0, respectively Fin1 and waiting until the process reaches the steady state regime. The experiments can be made for any value of the input flow in the technologically allowed limits. After computation, it results the value KH = 0.0085(cm∙h)/ml.
(2) 192
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function influences only the values of the y(t,p) signal increase over y0. Using the experimental data and a mathematical procedure based on an interpolation method, respectively considering the exponential increasing evolution of the 13C concentration at the increase of (p) variable, the y(tf,p) function can be approximated by the FpA(p) function of the form:
The isotope separation can be computed using the relation:
S (t ) n(t ) ,
(6)
where α = 1.01 is the elementary separation factor of the 13C isotope for the carbamate – CO2 chemical exchange procedure and n(t) = h/HETP(t) is the instantaneous value of the number of the theoretical plates. Considering (4), (6) and the relation of n(t), it results that: S (t )
h HETP0 K H ·( Fin ( t )– Fin 0 )
,
p
FpA ( Fin (t ) ,p)=( y0 1) e 430 K P ( Fin (t ) Fin 0 ) ,
(7)
y (t f ,p f )( Fin (t )) ,
P 430 KP ·(Fin (t ) – Fin0 ) .
(8)
Considering the form of FpA(Fin(t), p) from (13), FpB can be obtained directly as FpB = FpA(Fin(t), pf). Having FpA and FpB, Fp(p,Fin(t)) results using (12). Also yAN results replacing the obtained forms of Ft(t,p) and Fp(p,Fin(t)) in (1).
where y(tf,pf)(Fin(t)) represents the steady state value of the y(t,p) signal for p = pf = h and for a certain value of the input step type signal which has the instantaneous value of the Fin(t) signal. From (8), it results that:
Analysing the equations presented at this Paragraph it results that the treated separation process is a strong nonlinear (2013) one. This remark is proved by the following aspects: both Ft and Fp functions depend on both independent variables; in (1) the two mentioned functions are multiplied; Fp results as the ration between FPA and FPB.
(9)
respectively, the 13C concentration increase over y0 for p = = pf = h and in steady state regime, is given by:
y(t f ,p f )( Fin (t )) y0 y0 (S (t ) 1) .
(10)
3. THE PROPOSED CONTROL STRUCTURE AND THE CONTROLLER TUNING
Equation (10) gives also the relation of the final input signal in the process, form presented in (11):
u f (t )( Fin (t )) y0 (S (t ) 1) .
The proposed control structure is presented in Fig. 3. The elements from the structure are: TP – technological separation process (being a distributed parameter process DPP); A – actuator (the pump P from Fig. 1); T – concentration transducer (the mass spectrometer from Fig. 1); AC – concentration controller (the same as in Fig. 1); RELAY – a relay type element with hysteresis. Also, the signals from the structure are: w(t) – set point signal used for the structure work in automatic control regime; w1(t) – set point signal used for the controller tuning; r(t) – feedback signal; c(t) – control signal generated by AC; b(t) – output signal of the relay element; Fin(t) – actuating signal (the input flow of ethanolamine); d1(t) – disturbance signal which affects directly the actuating signal; Finf(t) – disturbed actuating signal (the final value of the ethanolamine input flow); yi(t,p) – (initial) output signal (the 13C isotope concentration); d2(t) – disturbance signal which affects directly the output signal; y(t,p) – disturbed output signal due to the effect of d2(t). In Fig. 3, the elements A, TP and T connected in series represent the Fixed Part of the control system.
(11)
Considering (2), (3), (9), (11) and the remark according to Ft = Ft(t,p), the value changing of the (p) independent variable influences the value of Ft(t,p) only if the changing occurs in transitory regime. Also, considering (1), (2), (3), (9), (10) and (11) it results that at the value changing of the (p) independent variable, the speed evolution of the output signal is adapted, but its steady state value is not modified, remaining at y(tf,pf) and occurring the necessity to introduce in the analytical solution yan the Fp(p,Fin(t)) function. The Fp(p,Fin(t)) function is given by the following relation:
Fp ( p,Fin (t )) =
FpA ( Fin (t ) ,p) y0 . FpB y0
(14)
It can be remarked that the technological separation process presents only one “length” constant P. From (14), it results that P is a function of Fin(t) (P(Fin(t)) and implicitly P(t). Consequently, the notations FpA(Fin(t), p) and FpA(t,p) are both justified.
y0
y(t f ,p f )( Fin (t )) y0 S (t ) ,
(13)
where the constant C = 430cm is associated to SC being determined through interpolation and the proportionality constant KP = 0.7527(cm∙h)/ml is determined mathematically processing two consecutive sets of values {Fin, P}. P signifies the “length” constant of the process, having the relation:
S being a function of Fin(t) (S(Fin(t))). In (7), the input signal in the process Fin(t) occurs with positive sign at the denominator of the ratio from the exponent. Consequently, the increase of Fin(t) implies the decrease of S(t), implying also the decrease of the y(t,p) signal, respectively the decrease of Fin(t) implies the increase of S(t), implying also the increase of the y(t,p) signal. From the physical point of view, the 13C enrichment is a more efficient one at lower values of the input ethanolamine flows due to the fact that the contact duration between carbamate and CO2 in SC is longer. The separation S(t) results from the following relation, too: S (t )
193
(12)
In (12) the subtraction of the y0 value both from numerator and from denominator signifies the fact that the Fp(p,Fin(t)) 193
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Fig. 3. The proposed control structure. structure (characterized through undamped oscillations of the output, respectively feedback signals) and in the estimation of the obtained limit cycle parameters (the amplitude and the period of the sustained oscillations). In the case of the current application, the set point signal is maintained constant to the value w1(t) = 9.8mA (step type signal; this value is chosen considering the fact that, in many cases, the plant works for reference signals around this value; the considered value is interpreted as 5.8mA over the initial value of 4mA associated to the unified current signals), the three switches being set on the positions 2, 4 and 4’. Using the previously presented relay parameters, the evolution in time of the feedback signal r(t) in controller tuning regime is presented in Fig. 5.
Also, the elements S1, S2 and S3 are switches used in order to select the working regime of the structure: controller tuning regime or automatic control regime. The feedback from the value of the (p) independent variable to the AC controller is used in order to adapt its parameters at the values change of this variable. Due to this feedback, the AC controller can be considered an adaptive one. In Fig. 3, if d1(t) = 0 and d2(t) = 0, then Finf(t) = Fin(t), respectively y(t,p) = yi(t,p). Also the error signal a(t) is given by a(t) = w(t) – r(t) in automatic control regime, respectively a(t) = w1(t) – r(t) in controller tuning regime. The automation equipment from Fig. 3 work with unified current signal 4-20mA. In this case the signals w(t), w1(t), c(t), b(t) and r(t) are unified current ones. If the three switches are set on the positions 2, 4 and 4’, the w1(t) set point signal is active, respectively the relay element is connected in the control structure, working in controller tuning regime (2009b) (the connections figured with dashed line from the structure are active). If the three switches are set on the positions 1, 3 and 3’, the w(t) set point signal is active and the AC controller is connected in the control structure, working in control regime (the connections figured with continuous line from the structure, associated to S 1, S2 and S3 are active). The feedback in relation to the (p) signal is used only in automatic control regime. For the controller tuning the relay method is applied. The used two position relay is presented in Fig. 4.
12 Amax 11 Ao
Aav
r(t) [mA]
10 9
Ao
8
Amin
7 6
To
The feedback signal
To
5 t1 4 0
50
100
t3
t2 150
200
t4 250
TIME [h]
Fig. 5. The evolution in time of the feedback signal in controller tuning regime. The Fig. 5 is presented due to the fact that the oscillations parameters of the feedback signal are used for the controller tuning. From Fig. 5, it results that the average value of the r(t) signal oscillations is Aav = 9.816mA, the maximum value is Amax = 11.134mA, respectively the minimum value is Amin = 8.498mA. The oscillations amplitude A0 can be calculated using the relation:
Fig. 4. The used relay with hysteresis. The output signal from the relay element is given by the relation:
b,if a(t ) as , b(t ) b, if a(t ) as
A0 Amax Aav Aav Amin 1.318mA .
(15)
(16)
Also, the period of the oscillations T0 results from Fig. 5 as the period of time between two consecutive oscillations maximum values or minimum values. Taking the case of the maximum values, T0 is given by the relation:
where b = 8mA and the hysteresis is determined by the error value as = 1.3mA. The relay method consists in using the relay to obtain the stability limit regime for the control 194
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T0 t2 t1 168.086 149.946 18.14h .
maximum, respectively the minimum values of the unified current signals (4-20)mA. In (21), Fin(s) and C(s) represent the Laplace transformation of the Fin(t) and c(t) signals. From (21) it results that the technological plant works in starting regime at the maximum flow Finmax = 1000ml/h. The negative value of KA signifies that the second term from the right member of (21) (the term corresponding to C(s)) represents the ethanolamine flow value which has to be subtracted from Finmax in order to obtain the actuating signal Fin(s). The proportionality constant of the transducer T, KT, is computed using the relation KT = (cmax – cmin)/(ymax – y0)= 8.45mA/%, where cmax and cmin have the same significance as in the case of KA, y0 has the same significance as in (1), respectively the ymax (3%) represents the maximum value of the produced 13C isotope concentration.
(17)
The value of T0 obtained in (17) is verified for the case of the t3 and t4 values, too. Using the A0 value, the equivalent amplifying coefficient can be calculated using the relation:
k0
4b 7.7283 . A0
(18)
Using the obtained values of T0 and k0, the PI and the PID type controllers parameters can be computed using the Ziegler-Nichols formulae (1999). The following notations are used: KC represents the proportionality constant of the controller, TI represents the integral time constant of the controller, respectively TD represents the derivative time constant of the controller. Using the relations associated to the Ziegler-Nichols tuning method, for the value p = pf = h of the (p) independent variable, the transfer function of the PI controller is presented in the equation:
1 , H PI ( s) KC (1 ) TI s
The mathematical models presented in (21) and (22) are used in the case of the simulation from Fig. 5 and in the case of the simulations from Paragraph 4, too. 4. SIMULATIONS RESULTS
(19)
The simulations from this paper MATLAB/Simulink (User Guide).
where KC = 3.47 and TI = 14.51h. The PID controller is not presented in the paper due to the fact that in the simulations, it generated very high values of the control signal that cannot be used in practice. If a saturation element is used in order to limit the control signal values or the controller parameters are adjusted as value (decrease of the KC and TD values, increase of TI value, respectively the increase of the value of the filter time constant used for obtaining the feasibility of the controller derivative component), the generated set of performances is a weak one, the complexity of the controller not being justified.
KA 1000 C ( s) ; (TA s 1) s HT ( s )
KT . TT s 1
made
in
1.8 1.7
(20) y(t,p) [%]
1.6
where TI0 = 14.51h is the value of TI from (19) and Tp0 has the same significance as in (3). The A and T elements from Fig. 3 are both linear elements. Their work is expressed using the following relations ((21) for A, and (22) for T):
Fin ( s)
are
In Fig. 6, the comparative graph between the step response of the open loop process and the control system step response (in the case when the three switches from Fig. 3 are set on the positions 1, 3 and 3’) is presented. The value of the reference signal in concentration is set to 1.7%, respectively the simulation of the open loop process is made setting for the Fin(t) signal (of the step type) the value previous computed (641.7ml/h) necessary to obtain for the y(t,p) signal the value 1.7%. Also, the two responses from Fig. 6 are obtained for p = pf = ct.
The adaptive control consists in the automatic changing of the value of the TI constant at the value changing of the (p) independent variable. The corresponding relation is:
p , TI ( p) Tp0 (TI 0 Tp0 ) pf
195
The control system response The open loop process response
1.5 1.4
t7
1.3
(21)
1.2 1.1 0 t6
(22)
t5 50
100
150
200
250
TIME [h]
Fig. 6. The comparative graph between the open loop process response and the control system response.
In (21) and (22) TA = 20s and TT = 3min are the time constants of the A and T elements. The proportionality constant of the actuator A, KA, is computed using the relation KA = (Finmin – Finmax)/(cmax – cmin) = (100 – 1000)/(20 – 4) = = –56.25ml/(h∙mA). In this relation Finmin = 100ml/h and Finmax = 1000ml/h represent the minimum, respectively the maximum technological values of the actuating signal and cmax = 20mA, respectively cmin = 4mA represent the
The obtained set of the control system performances are: the steady state error – 0% (the imposed value), the overshoot – – 0% (the imposed value, being also a critical restriction) and the settling time t6 = 20.4h (much smaller than the imposed value – 35h). Also, the value of the obtained settling time in the case of using the control system is much smaller than the 195
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value of the settling time associated to the open loop process: t5 = 48.45h (the difference between them being t7 = 28.05h). Consequently, the usage of the control system is justified.
In Fig. 8, with dashed line, the case of using the unmodified form of the PI controller from (19) is presented, respectively with continuous line, the case of using the adaptive PI controller (applying (20), too) is presented. The reference signal is maintained constant during the simulation. In Fig. 8, the fact that the adaptive controller generates a much faster stabilization of the y(t,p) signal than the unmodified PI controller, after the p variable variation, is highlighted.
In Fig. 7, the simulations of the open loop process and of the control system are made if the step type disturbance d1(t) = = 200 ml/h occurs in the system after 100h after the simulation start, respectively the negative step type disturbance d2(t) = -0.2% occurs in the system after 200h from the simulation start. The positive value of d1(t) represents a disturbance, it implying the decrease of y(t,p).
5. CONCLUSIONS In this paper, an original solution for modeling and for the adaptive control of a strong nonlinear distributed parameter isotope separation process is presented. In order to verify the efficiency of using a control system, some simulations are presented in Paragraph 4. In Paragraph 4, the proposed control structure is tested both in the case when the disturbance signals occur the system and in the case when the p independent variable presents variations. The simulations results proved the necessity of using the proposed control system, it generating a restrictive set of control performances (much better than the imposed one). Also, the controller rejects the effect of the two possible disturbance signals and it has the possibility (being an adaptive one) to preserve the system performances at the p variable value changing. In all the simulations from Paragraph 4, the intermediary signals c(t) and Fin(t) do not exceed their maximum, respectively minimum imposed limits (cmax, cmin, Finmax, Finmin).
1.8 1.7
y(t,p) [%]
1.6 1.5 1.4 1.3 1.2 1.1 0
The control system response The open loop process response 50
100
150
200
250
300
TIME [h]
Fig. 7. The open loop step process response and the control system step response, if the disturbances occur in the system.
REFERENCES Axente, D., Abrudean, M., Bâldea, A. (1994). 15N, 18O, 10B, 13 C Isotopes Separation trough Isotopic Exchange. Science Book House. Dobra, P. (1999). Nonlinear Systems. U.T.PRES Publishing House. Dang, H., Rochelle, G.T. (2003). CO2 absorption rate and solubility in monoethanolamine/ piperazine/ water. Separation Sci. & Tech., 38 (2), pp. 337–357. Smyshlyaev, A., Krstic, M. (2005). Control design for PDEs with space-dependent diffusivity and time-dependent reactivity. Automatica, Vol. 41, pp. 1601-1608. Love, J. (2007). Process Automation Handbook. 1 edition, Springer Publishing House. Dugas, R., Rochelle, G. (2009). Absorption and desorption rates of carbon dioxide with monoethanolamine and piperazine. Energy Procedia, Vol. 1 (1), pp. 1163–1169. Golnaraghi, F., Kuo, B. C. (2009). Automatic Control Systems. 9th edition, Wiley Publishing House. Mureşan, V., Abrudean, M. (2010). Temperature Modelling and Simulation in the Furnace with Rotary Hearth. Proc. of IEEE AQTR–17th ed., Cluj-Napoca, Romania, pp. 147-152. Li, H.-X., Qi, C. (2011). Spatio-Temporal Modeling of Nonlinear Distributed Parameter Systems: A Time/Space Separation Based Approach. 1st Edition, Springer Publishing House. Coloşi, T., Abrudean, M., Ungureşan, M.-L., Mureşan, V. (2013). Numerical Simulation of Distributed Parameter Processes. Springer Publishing House. User Guide, Matlab 7.5.0 (R2007b).
The simulations from Fig. 7 are made for p = pf = ct. From Fig. 7, it can be remarked that the effects of the two disturbances is rejected by the AC controller (in the case of the control regime), proving again the necessity of implementing the control structure from Fig. 3. In the case of the open loop process the disturbances effects generate the decrease of the y(t,p) signal value. In Fig. 8, the case of the p independent variable variation is treated, considering the following values: before the moment 150h p = pf , between the moments 150h and 225h p = 230mm, respectively after the moment 225h p = 280mm (the variation between two consecutive values are considered of step type). 1.9 1.8
y(t,p) [%]
1.7 1.6 1.5 1.4 1.3 1.2 1.1 0
The case of using the unmodified initial PI controller The case of using the adaptive PI controller
50
100
150
200
250
300
TIME [h]
Fig. 8. The control system simulations, if the p independent variable presents variations. 196