Optimal design of the inlet temperature based periodic operation of non-isothermal CSTR using nonlinear output frequency response functions

Proceedings, 10th IFAC International Symposium on Proceedings, 10th IFAC International Symposium on Proceedings, 10th of IFAC International Symposium on at www.sciencedirect.com Available online Advanced Control Chemical Processes Proceedings, 10th IFAC International Symposium on Advanced Control Chemical Processes Proceedings, 10th of IFAC International Symposium on Advanced Control of Chemical Processes Shenyang, Liaoning, July 25-27, 2018 Advanced Control of China, Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Advanced Control of China, Chemical Processes Shenyang, Shenyang, Liaoning, Liaoning, China, July July 25-27, 25-27, 2018 2018 Shenyang, Liaoning, China, July 25-27, 2018

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IFAC PapersOnLine 51-18 (2018) 620–625

Optimal design of the inlet temperature based periodic operation of non-isothermal Optimal design of the inlet temperature based periodic operation of non-isothermal Optimal design of the inlet temperature based periodic operation of non-isothermal Optimal design of the inlet temperature based periodic operation of non-isothermal CSTR using nonlinear output frequency response functions Optimal design of the inlet temperature based periodic operation of non-isothermal CSTR using nonlinear output frequency response functions CSTR using nonlinear output frequency response functions CSTR using nonlinear output frequency response functions CSTR using nonlinear output frequency response functions * ** Hongyan Shi , Z.Q. Lang**# ,Decheng Yuan**,Wei Wang## * **#,Yunpeng Zhu** * Hongyan Shi *, Z.Q. Lang**# **#,Yunpeng Zhu** **,Decheng Yuan* *,Wei Wang# # Hongyan Shi *, Z.Q. Lang**#,Yunpeng Zhu**,Decheng Yuan*,Wei Wang# Hongyan Shi *, Z.Q. Lang**#,Yunpeng Zhu**,Decheng Yuan*,Wei Wang# Hongyan Shi , Z.Q. Lang ,Yunpeng Zhu ,Decheng Yuan ,Wei Wang * *Department of Information Engineering, Shenyang University of Chemical and Technology, * *Department of Information Engineering, Shenyang University of Chemical and Technology, of Information Engineering, Shenyang University of Chemical and Technology, *Department Shenyang,110142, China(tel:13889890549;e-mail:[email protected]) of Engineering, *Department Department of Information Information Engineering, Shenyang Shenyang University University of of Chemical Chemical and and Technology, Technology, China(tel:13889890549;e-mail:[email protected]) ** Shenyang,110142, Shenyang,110142, China(tel:13889890549;e-mail:[email protected]) of Automatic and Control and Systems Engineering, University of Sheffield, Shenyang,110142, China(tel:13889890549;e-mail:[email protected]) ** Department ** ** Department Shenyang,110142, China(tel:13889890549;e-mail:[email protected]) of Automatic and Control and Systems Engineering, University of Sheffield, of Automatic and Control and Systems Engineering, University of Sheffield, ** Department Sheffield, S1 3JD,U.K.(e-mail:[email protected]) of Automatic and Control and Systems Engineering, University of Sheffield, ** Department Department of Automatic and Control and Systems Engineering, University of Sheffield, Sheffield, S1 3JD,U.K.(e-mail:[email protected]) # Sheffield, S1 3JD,U.K.(e-mail:[email protected]) Department of Control Science and Engineering, Dalian University of Technology, Sheffield, S1 3JD,U.K.(e-mail:[email protected]) # # Department of Control Science and Engineering, Dalian University of Technology, # Sheffield, S1 3JD,U.K.(e-mail:[email protected]) of Control Control Science and Engineering, Engineering, Dalian University University of Technology, Technology, # Department Dalian,116024, China(e-mail:[email protected]) of and # Department China(e-mail:[email protected]) DepartmentDalian,116024, of Control Science Science and Engineering, Dalian Dalian University of of Technology, Dalian,116024, China(e-mail:[email protected]) Dalian,116024, China(e-mail:[email protected]) Dalian,116024, China(e-mail:[email protected]) Abstract: The periodic operation of a non-isothermal continuous stirred tank reactor (CSTR) using inlet temperature modulation Abstract: The periodic operation of aa non-isothermal continuous stirred tank reactor (CSTR) using inlet temperature modulation Abstract: The periodic operation of reactor (CSTR) using inlet modulation is investigated in this paper. TheofDC component ofcontinuous the CSTR stirred outputtank concentration is optimized bytemperature tuning the modulation Abstract: The periodic operation aa non-isothermal non-isothermal continuous stirred tank reactor (CSTR) using inlet temperature Abstract: The periodic operation of non-isothermal continuous stirred tank reactor (CSTR) using inlet temperature is investigated in this paper. The DC component of the CSTR output concentration is optimized by tuning the modulation is investigated in this paper. The DC component of the CSTR output concentration is optimized by tuning the modulation parameters of the inlet temperature in order to achieve a maximum conversion using a isNonlinear Output Frequency Response is investigated in this The component of CSTR concentration optimized by the is investigated in inlet this paper. paper. The DC DC component of the the CSTR output output concentration optimized by tuning tuning the modulation modulation parameters of the temperature in order to achieve a maximum conversion using aa isNonlinear Output Frequency Response parameters of the inlet temperature in order to achieve maximum Nonlinear Output Frequency Functions (NOFRFs) based approach. The results show aathat the newconversion approach isusing fast and efficient in the analysis and Response design of parameters of the inlet temperature in order to achieve maximum conversion using a Nonlinear Output Frequency Response parameters of the inlet temperature in order to achieve a maximum conversion using a Nonlinear Output Frequency Response Functions (NOFRFs) based approach. The results show that the new approach is fast and efficient in the analysis and design of Functions (NOFRFs) approach. The results show the new approach fast and efficient the analysis and design of the periodic operationbased of CSTR and can potentially bethat applied to conduct theis optimal design ofin periodic operation of other Functions (NOFRFs) based approach. The results show that the new approach is fast and efficient in the analysis and design of Functions (NOFRFs) based approach. The results show that the new approach is fast and efficient in the analysis and design of the periodic operation of CSTR and can potentially be applied to conduct the optimal design of periodic operation of other the periodic operation of CSTR and can potentially be applied to conduct the optimal design of periodic operation of other chemical engineering processes. the periodic operation of CSTR and can potentially be applied to conduct the optimal design of periodic operation of other chemical engineering processes. the periodic operation of CSTR and can potentially be applied to conduct the optimal design of periodic operation of other chemical engineering engineering processes. processes. chemical chemical engineering processes. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Key words: optimal design, periodic operation, nonlinear output frequency response functions (NOFRFs), frequency domain Key words: optimal design, periodic operation, nonlinear output frequency response functions (NOFRFs), frequency domain Key words: optimal design, periodic operation, nonlinear output frequency response functions (NOFRFs), frequency frequency domain domain Key words: optimal design, periodic operation, nonlinear output frequency response functions (NOFRFs), Key words: optimal design, periodic operation, nonlinear output frequency response functions (NOFRFs), frequency domain

1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Periodic operations in chemical engineering processes Periodic operations in chemical engineering processes Periodic operations in chemical engineering processes Periodic operations in chemical engineering processes have received extensive attentions in the past decades have received extensive attentions in the past decades Periodic operations in chemical engineering processes have received extensive attentions in the past decades have received extensive attentions in the past decades (Bailey, 1973; Silveston et al., 1995; Petkovska et al., (Bailey, 1973; Silveston et al., 1995; Petkovska et al., have received extensive attentions in the past decades (Bailey, 1973; Silveston et al., 1995; Petkovska et al., 2010). The advantages of the periodic operations lie in the (Bailey, 1973; Silveston et al., 1995; Petkovska et al., 2010). The advantages of the periodic operations lie in the (Bailey, 1973; Silveston et al., 1995; Petkovska et al., 2010). The advantages of the periodic operations lie in the 2010). The advantages of the periodic operations lie in the fact that the average performance of a nonlinear chemical fact that the average performance of a nonlinear chemical 2010). The advantages of the periodic operations lie in the fact that the average performance of aa nonlinear chemical engineering system under a periodic operation is often fact that the average performance of nonlinear chemical engineering system under a periodic operation is often fact that the average performance of a nonlinear chemical engineering system under aa periodic operation is under often engineering operation often superior to the steady-state performance superior to tosystem the under steady-state performance under engineering system under a periodic periodic operation is is under often superior the steady-state performance superior to the steady-state performance under conventional constant input operations (Douglas, 1972). conventional constant input operations (Douglas, 1972). superior to the steady-state performance under conventional constant input operations (Douglas, 1972). Chemical reactors under the condition of periodic conventional constant input operations 1972). Chemical reactors under the condition(Douglas, of aaa periodic periodic conventional constant inputthe operations (Douglas, 1972). Chemical reactors under condition of Chemical reactors under the condition of a periodic modulation feed were studied both theoretically and modulation feed were were studied both theoretically theoretically and Chemical reactors understudied the condition of a periodic modulation feed both and modulation feed were studied both theoretically and experimentally (Silveston et al., 2012;Brzić et al. 2015). In experimentally (Silveston et al., 2012;Brzić et al. 2015). In modulation feed were studied both theoretically and experimentally (Silveston et al., 2012;Brzić et al. 2015). In these chemical reactors, the CSTR is a good candidate experimentally (Silveston et al., 2012;Brzić et al. 2015). these chemical reactors, the CSTR is a good candidate experimentally (Silveston et al., 2012;Brzić et al.candidate 2015). In In these chemical reactors, the CSTR is aa good these chemical reactors, the CSTR is good candidate because of its significant nonlinearity and has therefore because of its significant nonlinearity has therefore these chemical reactors, the CSTR is and a good candidate because of its significant nonlinearity and has therefore been widely to study the periodic operation of the because of significant nonlinearity and has been widely used to study the periodic operation of the because of its itsused significant nonlinearity and has therefore therefore been widely used to study the periodic operation of the been widely used to study the periodic operation of the flow-rate, concentration or temperature for flow-rate, concentration or temperature for been widely used to study the periodic operation of flow-rate, concentration concentration or temperature temperature for the the flow-rate, or for improvement of the process operational performance. improvement of process operational performance. flow-rate, concentration or temperature for the the improvement of the the the process operational performance. In recent years, periodic operation of the CSTR has improvement of the process operational performance. In recent years, the periodic operation of the CSTR has improvement of the process operational performance. Ininvestigated recent years, years, the periodic operation of the the CSTR has recent periodic operation of has beenIn using nonlinear frequency analysis. recent years, the the periodic operation of the CSTR CSTR has been using nonlinear frequency analysis. beenIninvestigated investigated using nonlinear frequency analysis. Particularly, the generalized frequency response functions been investigated using nonlinear frequency analysis. been investigated using nonlinear frequency analysis. Particularly, the generalized frequency response functions Particularly, the generalized generalized frequency response functions Particularly, the response functions (GFRFs) method has been frequency applied for the analysis of Particularly, the generalized frequency response functions (GFRFs) method has been applied for the analysis of (GFRFs) method has been applied for the analysis of (GFRFs) method has been applied for the analysis of different types ofhas CSTR for enhancement of process (GFRFs) method been applied for the analysis of different types of CSTR for enhancement of process different types of CSTR for enhancement of process performance through periodic modulation ofof single or different types of CSTR for enhancement process different types of CSTR for enhancement of process performance through periodic modulation of single or performance through periodic modulation of single or performance through periodic modulation of single or multiple inputs (Marković et al.,2008; Nikolić et al., 2014, performance through periodic modulation of single or multiple inputs (Marković et al.,2008; Nikolić et al., 2014, multiple inputs (Marković et al.,2008; Nikolić et al., 2014, multiple inputs (Marković et al.,2008; Nikolić et al., 2014, 2015). However, the GFRFs method requires derive the multiple inputs (Marković et al.,2008; Nikolić et al., 2014, 2015). However, the GFRFs method requires derive the 2015). However, the GFRFs method requires derive the higher frequency response functions (FRFs), which 2015). However, the method requires derive the 2015). order However, the GFRFs GFRFs method requires derive the higher order frequency response functions (FRFs), which higher order frequency response functions (FRFs), which higher order frequency response functions (FRFs), which is difficult to be implemented and widely applied in higher order frequency response functions (FRFs), which is difficult to be implemented and widely applied in is difficult to be implemented and widely applied in practice. is difficult to be implemented and widely applied in is difficult to be implemented and widely applied in practice. practice. practice. The NOFRFs is a novel concept for the analysis of practice. The NOFRFs is a novel concept for the analysis of The NOFRFs novel concept for the(Lang analysis of The is novel concept for analysis of nonlinear systems is in aaathe frequency et al., The NOFRFs NOFRFs is novel conceptdomain for the the(Lang analysis of nonlinear systems in the frequency domain et al., nonlinear systems in the frequency domain (Lang et al., 2007). The method allows the analysis of nonlinear nonlinear systems in the frequency domain (Lang et al., nonlinear systems in the frequency domain (Lang et al., 2007). The method allows the analysis of nonlinear 2007). The allows the analysis of nonlinear 2007). method allows the analysis of systems to bemethod implemented to the 2007). The The method allows in theaaa manner analysis similar of nonlinear nonlinear systems to be implemented in manner similar to the systems to be implemented in manner similar to the systems to be implemented in a manner similar to the analysis of linear systems and provides great insight into systems to be implemented in a manner similar to the analysis of linear systems and provides great insight into analysis of linear systems and provides great insight into analysis of linear systems and provides great insight into analysis of linear systems and provides great insight into

the mechanisms underlying many nonlinear behaviors the underlying many behaviors the mechanisms mechanisms underlying many nonlinear nonlinear behaviors (Peng, et al., 2007). In the present study, the NOFRFs the mechanisms underlying many nonlinear behaviors (Peng, et al., 2007). In the present study, the NOFRFs the mechanisms underlying many nonlinear behaviors (Peng, et al., 2007). In the present study, the NOFRFs (Peng, et al., 2007). In the present study, the NOFRFs method is applied for the analysis and design of aa nonmethod is applied for the analysis and design of non(Peng, et al., 2007). In the present study, the NOFRFs method is applied for the analysis and design of nonmethod is applied for the analysis and design of aaa nonisothermal CSTR under a periodic operation of the inlet isothermal CSTR under a periodic operation of the inlet method is applied for the analysis and design of nonisothermal CSTR under a periodic operation of the inlet temperature. The results demonstrate the advantage of the isothermal CSTR under a periodic operation of the inlet temperature. The results demonstrate the advantage of the isothermal CSTR under a periodic operation of the inlet temperature. The results demonstrate the advantage of the temperature. The results demonstrate the advantage of new approach over existing GFRFs based analysis and the new approach over existing GFRFs based analysis and the temperature. The results demonstrate the advantage of new approach over existing GFRFs based analysis and the potential of the new analysis in the design of periodic new approach over existing GFRFs based analysis and potential of new analysis in design of new approach over existing GFRFs based analysis and the the potential offorthe the new analysis in the the design of periodic periodic potential of the new analysis in the design of periodic operations chemical engineering processes. operations chemical engineering processes. potential offor the new analysis in the design of periodic operations for chemical engineering processes. operations for engineering processes. operations for chemical chemical engineering processes. 2. ISSUES ASSOCIATED WITH ACHIEVING A 2. ISSUES ASSOCIATED WITH A 2. ISSUES ISSUES ASSOCIATED WITH ACHIEVING ACHIEVING A FAVORAVLE PERIODICAL OPERATION 2. ASSOCIATED WITH ACHIEVING A FAVORAVLE PERIODICAL OPERATION 2. ISSUES ASSOCIATED WITH ACHIEVING A FAVORAVLE PERIODICAL OPERATION FAVORAVLE PERIODICAL OPERATION PERIODICAL The FAVORAVLE output response response of nonlinear nonlinearOPERATION systems that that are are The output of systems The output response of nonlinear systems that are The output response of nonlinear systems that are stable at zero equilibrium can be represented by a Volterra stable at zero equilibrium represented by aa that Volterra The output response can of be nonlinear systems are stable at zero equilibrium can be represented by Volterra series (Rugh, 1981): stable at zero equilibrium can be represented by a Volterra series 1981): stable (Rugh, at zero equilibrium can be represented by a Volterra series (Rugh, 1981): series series (Rugh, (Rugh, 1981): 1981): y (t )  y   y (t ) (1)  ym (1) yy ((tt ))  yysss  tt )) m( (1) m m 1 y s  m( (1) yy ((tt ))   y  y ( t ) (1)  yss  mmm111 ymm (t ) m 1 y is the steady-state output corresponding to aa where m 1 s is the steady-state output y corresponding to where yyss is the steady-state output corresponding to aa where is the steady-state output corresponding to where yss is input, the steady-state output corresponding to a where y ( t ) is the system linear response, steady-state yy111 ((tt )) is the system linear response, steady-state input, is the system linear response, steady-state input, y ( t ) is the system linear response, steady-state input, y11 (the t ) issystem the system linear nonlinear response, steady-state input, yym ((tt ), m  22 is mth order and, ), m  is the system mth order nonlinear and, y ( t ), m  2 is the system mth order nonlinear and, m and, response. ymmm ((tt ), ), m m  22 is is the the system system mth mth order order nonlinear nonlinear and, y response. response. When the system is subject to a harmonic input response. response. When the system is subject to aa harmonic input When the system is subject to harmonic input When the system is subject to a harmonic input around a steady state with amplitude A and frequency x s with When the system is subject to a harmonic input amplitude A and frequency x around a steady state s with amplitude A and frequency around a steady state x s s with amplitude A and frequency around a steady state x around a steady state xs with amplitude A and frequency

   

  FFFF  FF

s

xx((tt ))  xxs  A cos(  )) Ft A cos(  Ft xx((tt ))   xxsss   A cos(  )) F Ft   A cos(  t s F Eq.(1) can be expressed x(t )  as xs (Douglas,  A cos(1972) Ft) Eq.(1) can be expressed as (Douglas, 1972)

Eq.(1) can be expressed as (Douglas, 1972) Eq.(1) as 1972) Eq.(1) can can be be expressed expressed as (Douglas, (Douglas, 1972) yy((tt ))  y B F tt   ) s  y DC  1 cos(  y y  B cos( DC  B11 cos( y(t )  yss  yDC FF t   111 ))  where Bn and where B and where Bnnn and where and where B Bn and

(2) (2) (2) (2) (2)

(3) F t  1 )  yy((tt ))   B11 cos(  (3) DC F t  1 )   Byy2sss cos(2  yyDC  (3) t   )  DC F B 1 cos( F 1 (3) 2 B cos(2  t   )  2 F 2 (3) B cos(2  t   )  2 F 2 2 F 2 B cos(2  t   )  2 F 2 are the amplitude and phase of B cos(2  t   )   , n  1, 2 n , n  1,2 2 are the F 2 and phase of  n , n  1, 2 are the amplitude and phase of  n n , n  1, 2 are the amplitude amplitude and phase  of n , n  1, 2 are the amplitude and phase of

n 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by nElsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 614Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 614 Copyright © 614 10.1016/j.ifacol.2018.09.355 Copyright © 2018 2018 IFAC IFAC 614 Copyright © 2018 IFAC 614

Shenyang, Liaoning, China, July 25-27, 2018

Hongyan Shi et al. / IFAC PapersOnLine 51-18 (2018) 620–625

the first and second order harmonic output of the system, respectively. yDC is the DC component of the output response produced by the system nonlinearity which can be expressed as follows (Weiner and Spina,1980) 2  A yDC  2   H 2 (F , F ) 2 (4)

621

3.1 The NOFRFS concept For system (1), Lang and Billings (Lang et al., 2007) have derived an expression for the output frequency response N  Y ( j )   Yn ( j ) for  n 1   (5) 1/ n  H n ( j1 , , jn ) Yn ( j )  n 1     1 n (2 )  n    U ( ji )d n   i 1  where N is the maximum order of system nonlinearity, Y ( j ) and U ( j ) are the system input and output

4

 A  6   H 4 (F , F , F , F ) 2 where H 2 n (), n  1, 2,... represents the 2nth order GFRFs

of the system. Considering a chemical reaction A→product(s) where one or more inputs can be modulated periodically around an established steady-state. If this type of chemical reaction process is subject to the harmonic input (2) and the outlet concentration of the reaction can be represented by Eq (3). Marković et al. (Marković et al., 2008) have shown that the mean of the outlet concentration denoted by

spectrum, Yn ( j ) represents the frequency response and

H n ( j1 ,

, jn )  



n th order output









hn ( 1 ,

, n )

 e (11  n n ) j d 1 is the definition of the nth order GFRFs with

cAm  ys  yDC is different from the corresponding steady state outlet concentration denoted by cA, s  ys and the



1

difference   c  cA,s  yDC is the indicator of the m A

n 

H n ( j1 ,

n

, jn )U ( ji )d n (7) i 1

denoting the integration of H n ( j1 ,

process improvement that can be achieved by the introduction of a period operation. If   yDC  0 , the introduction of the periodic operation is favorable, as it increase the conversion in comparison to the steady state operation. Otherwise, i.e. when   yDC  0 , the periodic operation is unfavorable. Consequently, in order to introduce a favorable periodic operation, it is necessary to ensure that   yDC  0 . Theoretically, this can be achieved using Eq.(4) but the multi-dimensional nature of the GFRFs H 2 n (), n  1, 2,... in Eq.(4) and associated complexities imply that this is difficult in practice. Currently, the solution is to assume that Eq.(4) can be approximated well by the first term and analytically evaluate the sign of H 2 (,  ) using the physical model of the CSTR to assess whether a favorable periodic operation is achievable. There are two fundamental problems with this available solution. First, it is impossible or difficult to analytically derive and study H 2 (,  ) if the physical model of CSTR is not available or complicated. Secondly, this approach cannot be used to optimally determine the periodic operation parameters A and  to minimize   yDC so as to maximize the conversion. In the present study, this challenge will be addressed by the development of a new approach based on the concept of Nonlinear Output Frequency Response Functions (NOFRFs).

(6)

d n

, jn )i 1U ( ji ) n

over the n-dimensional hyper-plane 1   n   . Based on Eq.(5), the concept of the NOFRFs of nonlinear system (1) was introduced by Lang and Billings (Lang and Billings,2005) as

Gn ( j ) 

Yn ( j ) , n  1, , N U n ( j )

(8)

where U n ( j ) 

1/ n (2 ) n 1



1

n

n 

U ( j )d  (9)  i

n

i 1

Clearly, using the NOFRFs concept, Eq.(5) can be written as N

N

n 1

n 1

Y ( j )   Yn ( j )   Gn ( j )U n ( j )

(10)

providing a representation for the system output frequency response similar to the representation in the linear case. When subject to the harmonic input (2), it can be shown that the output response of system (1) contributed by system nonlinearity at zero frequency can be represented using (10) as yDC  Y (0)  ys

In (11),

2 4 (11)  A  A  2   G2F (0)  6   G4F (0)  2 2 G2 nF  0  , n  1, 2, denotes the NOFRFs

G2 n   , n  1, 2,

of system (1) evaluated at   0 in the case where the system is subject to harmonic input (2)

3. THE NOFRFS METHOD 615

Shenyang, Liaoning, China, July 25-27, 2018 Hongyan Shi et al. / IFAC PapersOnLine 51-18 (2018) 620–625

622

As G2 nF  0  , n  1, 2,

with frequency  F . A comparison of Eq.(11) and Eq.(4) implies that the problems with the application of the existing GFRFs based analysis for the favorability of introducing a periodic operation could be addressed by using a NOFRFs based approach. This is because the NOFRFs can be numerically evaluated up to an arbitrary order only using the system input and output data (Lang et al., 2007).

above for all relevant  F of interest, Eq. (11) and G2 nF  0  , n  1, 2,

4. SIMULATION STUDY OF THE EFFECT OF MODULATION PARAMETERS ON PERIODIC OPERATION

The idea of the NOFRFs based analysis is to numerically determine the NOFRFs G2 nF  0  , n  1, 2,

4.1 The mathematic model of a non-isothermal CSTR

in Eq.(11) using the system input and output data so as to facilitate the evaluation and optimal design of the favorable effects of a periodic operation. This can be achieved by using the method for the evaluation of the NOFRFs proposed in (Lang et al., 2007) and the system input output data which can be obtained from either model simulation or experimental test. Mathematically, this idea can be described as follows. Assume that the component yDC of the response of system (1) to harmonic input (2) under M different amplitudes is all available. Denote the M different amplitudes as

Consider a simple nonlinear homogeneous n-th order reaction A→product(s), the rate of reaction is given by

r  k0e ( EA / RT ) cAn

E A is the activation energy, k0 is the pre-exponential factor in the Arrhenius equation and R is the gas constant, respectively. The material balance and the energy balance for the reactant A can be written as

V V cp

4

(12)

          

dT  F  c pTi  F  c pT  (H R )k0e dt



EA RT

(16)

cAnV  UAW (T  TJ )

is the heat of reaction, AW is the surface area of the heat exchanger, U is the overall heat transfer coefficient,  is the density and c p is the specific heat capacity. Subscript i

can be

determined by using a Least Squares approach as follows,  yDC ,1  G2F (0)      G4F (0)   ( X T X ) 1 X T  yDC ,2  (13)          yDC , M    where 4   A1  2  A1  2   6    2   2 4  A 2  2  2  6  A2  X   2   2      A 2  A 4 2  M  6  M    2   2 

E  A dcA  FcA,i  FcA  k0e RT cAnV dt

(17) where t is the time, F is the volumetric flow rate of the reaction stream, V is the volume of the CSTR reactor, H R

Then, it is known from Eq.(11) that

Consequently, the NOFRFs G2 nF  0  , n  1, 2,

(15)

where c A is the reactant concentration, T is the temperature,

Am , m  1, , M and the corresponding yDC as yDC m , m  1, , M 2

thus determined provide a numerical

relationship that can be used to analyze and design the effect of modulation parameters A and  F on the favorability and performance of a chemical reaction process periodic operation.

3.2 The NOFRFs based analysis of the periodically operated chemical reaction process

A  A  yDC m  2  m  G2F (0)  6  m  G4F (0)   2   2  m  1, 2, , M

can be determined as described

for the inlet and Subscript J for the heating/cooling fluid in the reactor jacket, respectively (Nikolić et al.,2014, 2015). All the parameters of the model equations used in the simulation are listed in Table 1 (Marlin, 2000). Assume that Eq. (11) can, in this case, be represented as 2

yDC

4

 A  A  2   G2,F (0)  6   G4,F (0) 2 2

(18)

Then, the values of G2,F (0) , G4,F (0) in (18) can be (14)

determined using the method in Section 3.2 from the system output data yDC under the harmonic input (2) of amplitudes A1 and A2 , respectively.

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623

in the cases of A1  10% , A2  20% and frequency  F

Table 1. Parameters and operating conditions of CSTR

varying from 0 to 100 rad/min, the values of

G2,F (0) ,

Parameter

Value

Reaction order,n

1

Preexponential factor of the reaction rate constant,k0

1010

1/min

of

Activation energy,EA

69256

kJ/kmol

Fig.2 gives the diagram of the yDC as the function of log

Heat of reaction,ΔHR

-543920

kJ/kmol

when the input amplitude is

Heat capacity,Cp

4.184×103

kJ/K/m3

Units

G4,F (0) in (18) were obtained. Fig.1 shows the diagram

scaled frequency  F

A1  10% . The obtained G4,F (0)  0 over all the frequency ranges of concern. From Fig.1-2, it can be concluded that in the low frequency range of F  5.76 rad / min and in the high

3

Volume of the reactor,V

1

Steady-state flow-rate,Fs

1

m /min

Steady-state inlet concentration,cAi,s

2

kmol/m3

Steady-state inlet temperature,Ti,s

323

K

Steady-state temperature of the coolant,TJ,s

365

K

Overall heat transfer cofficient multiplied by the heat transfer area,UAw

27337

kJ/K/min

m

3

frequency range of F  30 rad / min , G2,F (0)  0 and yDC  0 , indicating that the improvement of reactant conversion is impossible. While, when 5.76 rad / min  F  30 rad / min , G2,F (0)  0, yDC  0 , showing that performance improvement can be expected by the introduction of periodic operation. Particularly, both G2, (0) and yDC have minima near F  7 rad / min ,

4.2 The NOFRFs based analysis From the input output data of system (16)(17) produced by the periodic modulation of inlet temperature, i.e., (19) Ti (t )  Tis  A cos(F t ) 3

x 10

G2,F (0) as the function of log scaled frequency F .

F

showing, a best periodic operation performance can be reached.

-5

G2 discrete dot spline algorithm

2

G2,ωF(0)

1

0

-1

-2

-3 -2 10

10

-1

Fig.1 The NOFRF G2,

0

10 Logarithmic input frequency ω F

F

10

1

10

2

(0) vs. logarithmic input frequency  F

0.015 yDC discrete dot spline algorithm 0.01

yDC

0.005

0

-0.005

-0.01

-0.015 -2 10

10

-1

0

10 Logarithmic input frequency ω F

10

1

10

2

Fig.2 DC component yDC when amplitude A1=10% vs. logarithmic input frequency  F

periodic operation unfavourable (as yDC    0 ). In Fig.4, the fluctuation ranges of output concentration and temperature are cA (t ) [0.25 0.4] kmol / m3 and T (t ) [380 400] K when A  15% and F  10 rad / min . cAm  0.3368 kmol / m3

Figs.3-4 shows the numerical simulation results of the outlet concentration and temperature. It can be observed from Fig.3 that, the fluctuation ranges of output concentration and temperature are cA (t ) [0.1 0.6] kmol / m3 and T (t ) [370 430] K when

A  15% and F  5.53 rad / min . The mean value of output concentration cAm  0.3575 kmol / m3 is higher than

is lower than

cA,s  0.3468 kmol / m3 , indicating the

periodic operation is favorable. These results are obviously consistent with the NOFRFs based analysis above.

the steady state value cA, s  0.3468 kmol / m3 , making the 617

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624

simulation, the existing GFRFs method, and the new NOFRFs method are compared in Table 2.

In addition, the output temperatures values in Figs.34 both are close to the steady state Ts  388K , showing that the output temperature response is not be affected by periodic modulation of input temperatures.

Modulation of input temperature,A=15%,ωF=10 rad/min 0.45

c (t) [kmol/m3]

c A (t)

c A,s cm A

0.4

cm A

0.3

A

c A (t)

0.6

0.25

0

5

10

15

Time [min]

0.2 0

400

0

5

10

15

20

25

440

Tm

385

375

Ts

420

Ts

390

380

T(t) Tm

0

5

10

15

Time [min]

400

Fig.4 The output concentration and temperature when modulation of input temperature with amplitude A=15% and frequency ω F=10 rad/min

380 360

T(t)

395

Time [min]

T(t) [Ｋ ]

c A,s

0.35

T(t) [Ｋ ]

A

c (t) [kmol/m3]

Modulation of input temperature,A=15%,ωF=5.53 rad/min

0.4

0

5

10

15

20

25

It can be observed in Table 2 that yDC , obtained by the NOFRFs based method has a better agreement with the values obtained by numerical solution in almost all the cases, demonstrating the effectiveness and advantage of the NOFRFs based new method.

Time [min]

Fig.3 The output concentration and temperature when modulation of input temperature with amplitude A=15% and frequency ωF=5.53rad/min

The values of

yDC

calculated by numerical

Table 2. The values of

yDC calculated by numerical simulation, the existing GFRFs and the new NOFRFs method Modulation of inlet temperature

15% input amplitude Frequency ωF

Δnum

y,GFRFs y,NOFRFs

10% input amplitude

Δnum

5% input amplitude

y,GFRFs y,NOFRFs

3% input amplitude

Δnum y,GFRFs y,NOFRFs

Δnum y,GFRFs y,NOFRFs

1

0.0186

0.0181

0.0185

0.0082

0.0080 0.0082

0.0020

0.0020

0.0020

0.0007

0.0007

0.0007

2

0.0192

0.0196

0.0196

0.0088

0.0087 0.0088

0.0022

0.0022

0.0022

0.0008

0.0008

0.0008

3

0.0210

0.0224

0.0208

0.0095

0.0100 0.0095

0.0025

0.0025

0.0024

0.0009

0.0009

0.0009

5

0.0200

0.0166

0.0201

0.0105

0.0074 0.0105

0.0026

0.0018

0.0028

0.0009

0.0007

0.0010

5.53

0.0119

-0.0325 0.0111

0.0053

-0.0144 0.0053

0.0003

-0.0036

6

0.0022

-0.0600

-0.0023 -0.0267 -0.0023

-0.0038 -0.0067

-0.0008

-0.0024 -0.0024 -0.0003

7

-0.0154 -0.0424 -0.0184

-0.0112

-0.0169 -0.0112

-0.0042 -0.0039

-0.0033

-0.0016 -0.0014 -0.0012

10

-0.0099 -0.0096 -0.0099

-0.0046 -0.0042 -0.0046

-0.0013 -0.0011

-0.0012

-0.0006 -0.0004 -0.0004

-0.0013

4.3 Optimal design using the NOFRFs In order to achieve a minimum value for yDC so as to reach a maximal conversion, it is desirable to find an optimal value for both the periodic operation amplitude A and frequency  F . From Tab.2, F [5.53,10] rad / min is selected as the frequency boundary for the effective frequency range associated with a negative value of yDC . The maximum input amplitude is determined as 15% ( Ti  48.5K ) to ensure the temperature not to exceed 100 . Based on the discrete data generated by the NOFRFs based analysis, a data fitted relationship between yDC , A and F , denoted as yDC  f (F , A) , was

0.0014

-0.0004 -0.0013 0.0005

obtained as follows and illustrated in Fig. 5. fitted surf data

DC component y

DC

0.01 0 -0.01 -0.02

-0.03 40 Am

30 plit ud e

20 A

10

6

8

7 Fre

ωF cy en qu

Fig.5 The data fitted surface of 618

9

yDC

10

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Hongyan Shi et al. / IFAC PapersOnLine 51-18 (2018) 620–625

50

Amplitude A

35

-0.004 -0.006

40

-0.018

-0.01 6

-0 .01

.01 -0

4

ACKNOWLEDGEMENTS

08 -0 .0

30

The first author gratefully acknowledges the support of the Doctoral Science Research Start Fund Program of Liaoning Provincial (201501072), China.

6 00 -0 .

25

-0.00 4

2 00 -0 .

20

approach has potential to perform an optimal design to reach a maximum conversion of the reactant, which cannot be achieved by existing techniques.

-0 .0 12

12 -0 .0 -0 .01 -0.00 8

45

625

REFERENCES

15 10

Bailey J.E. (1973). Periodic operation of chemical reactors: a review. Chemical Engineering Communications, 1(3), 111-124. Brzić D., Petkovska M. (2015). Nonlinear Frequency Response Fig.6 Contour map of the data fitted surface yDC measurements of gas adsorption equilibrium and kinetics: New apparatus and experimental verification. Chemical Engineering 2 yDC  f (F , A)  0.273  0.1187F  0.0058 A  0.0167F  0.00157F A Science, 132, 9-21. Douglas, J.M. (1972). Process dynamics and control. Prentice-Hall,  3 106 A2  0.00076F 3  0.0001F 2 A  2 106 F A2  1107 A3 Englewood Cliffs, New Jersey. (20) Lang Z.Q, Billings S.A. (2005). Energy transfer properties of nonlinear systems in the frequency domain. International Journal Fig.5 shows that yDC has a global minimum within of Control, 78(5), 345-362. the effective boundaries of amplitude A and frequency  F , Lang Z.Q., Billings S.A., Yue R., et al. (2007). Output frequency response function of nonlinear Volterra systems. Automatica, which is the solution to the following optimization 43(5), 805-816. problem, Marković A., Seidel M.A., Petkovska M. (2008). Evaluation of the min f (F , A) potential of periodically operated reactors based on the second order frequency response function. Chemical Engineering s.t. 5.53 rad / min  F  10 rad / min; （21） Research & Design, 86(7), 682-691. Marlin, E.T. (2000). Process control: designing processes and 9 K  A  48.45 K control systems for dynamics performance, 2nd edition and can be obtained as McGraw Hill, New York. F  7.8 rad / min Nikolić D., Seidel-Morgenstern A., Petkovska M. (2014). Nonlinear frequency response analysis of forced periodic operation of A  48.45 K (15% input amplitude) non-isothermal CSTR using single input modulations. Part II: producing Modulation of inlet temperature or temperature of the cooling/heating fluid. Chemical Engineering Science, 117(9), yDC ,min  0.02 kmol / m3 31-44. The contour map of the surface yDC  f (F , A) is Nikolić D., Seidel-Morgenstern A., Petkovska M. (2015). Nonlinear frequency response analysis of forced periodic operation of nonshown in Fig. 6 which is the relationship between isothermal CSTR with simultaneous modulation of inlet amplitude A and frequency  F for a given yDC . For concentration and inlet temperature. Chemical Engineering Science, 137, 40-58. example, given A  32K (10% amplitude), the input Peng Z.K., Lang Z.Q., Billings S.A., et al. (2007). Analysis of frequency  F can be found to be 7rad / min if an bilinear oscillators under harmonic loading using nonlinear output frequency response functions. International Journal of optimal conversion with yDC  0.012kmol / m3 is to be Mechanical Sciences, 49(11), 1213-1225. reached. These demonstrate the significance of the Petkovska M., Nikolic D., Markovic A.et al. (2010). Fast evaluation NOFRFs based design for a desired periodic operation. of periodic operation of a heterogeneous reactor based on nonlinear frequency response analysis. Chemical Engineering 5. CONCLUSIONS Science, 65, 3632-3637. Rugh, W.J. (1981). Nonlinear system theory: The volterra/wiener In the present study, a new NOFRFs based approach is approach. Johns Hopkins University Press. proposed to analyse and design the periodic operation of a Silveston P.L., Hudgins R.R., Renken A. (1995). Periodic operation of catalytic reactors-Introduction and overview, Catalysis non-isothermal CSTR with a periodically modulated inlet Today, 25(2), 95-112. temperature. A comparison between the existing GFRFs Silveston P.L., Hudgins R.R. (2012). Periodic operation of reactors, based method and the new NOFRFs approach has 1st ed., Elsevier, Amsterdam. demonstrated the effectiveness and advantage of the new Weiner D.D., Spina J.F. (1980). Sinusoidal analysis and modeling of method. In addition, an optimal design of the periodic weakly nonlinear circuits: with application to nonlinear operation parameters using the NOFRFs approach has interference effects. Van Nostrand Reinhold Com. 5.5

6

6.5

7

7.5 8 Frequency ωF

8.5

9

9.5

10

been conducted. The results have shown that the NOFRFs

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