Fast evaluation of periodic operation of a heterogeneous reactor based on nonlinear frequency response analysis

ARTICLE IN PRESS Chemical Engineering Science 65 (2010) 3632–3637

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Fast evaluation of periodic operation of a heterogeneous reactor based on nonlinear frequency response analysis M. Petkovska a,, D. Nikolic´ a, A. Markovic´ b,1, A. Seidel-Morgenstern b,c a

University of Belgrade, Faculty of Technology and Metallurgy, Department of Chemical Engineering, Karnegijeva 4, 11000 Belgrade, Serbia Max-Planck Institute for Dynamics of Complex Technical Systems, Sandtorstasse 1, 39106 Magdeburg, Germany c Otto von Guericke University, Chair of Chemical Process Engineering, Universit¨ atsplatz 2, 39106 Magdeburg, Germany b

a r t i c l e in f o

a b s t r a c t

Article history: Received 12 November 2009 Received in revised form 24 February 2010 Accepted 2 March 2010 Available online 10 March 2010

The concept of higher-order frequency response functions (FRFs), which is based on Volterra series representation of nonlinear systems, is used to analyse the time-average performance of a perfectly mixed reactor subject to periodic modulation of the inlet concentration, for a simple n-th order heterogeneous catalytic reaction. The second order frequency response function G2(o,  o), which corresponds to the dominant term of the non-periodic (DC) component, essentially determines the average performance of the periodic process. Thus, in order to evaluate the potential of a periodic operation, it is sufficient to derive and analyse the G2(o,  o) function. The sign of this function defines the sign of the DC component and reveals whether the periodic operation is favourable compared to conventional steady state operation, or not. It will be shown that, for the case investigated, the sign of this function depends both on the reaction order and on the shape of the adsorption isotherm. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Forced periodic operation Frequency response functions Nonlinear dynamics Chemical reactors Adsorption Mathematical modelling

1. Introduction 1.1. Periodic operations for enchantment of reactor performance Periodic operations of different chemical engineering processes, especially of chemical reactors, have been attracting attention of a number of research groups in the last 20–30 years ¨ (Silveston, 1998; Aida and Silveston, 2005; Schadlich et al., 1983; Chanchlani et al., 1994; Nappi et al., 1985). The attractiveness of the periodic operations lies in the fact that the average process performance corresponding to the periodic operation can be superior to the optimal steady-state operation, i.e., the conversion or/and selectivity can be increased by cycling one or more inputs. Fig. 1 illustrates the difference between steady state and periodic operation of a chemical reactor for a simple reaction mechanism A-products. Let us assume that cAi,s and cA,s are the input and output concentrations of the reactant A, respectively, when the reaction is performed in a steady state operation. If the input concentration  Corresponding author. Tel.: + 381 11 3303 610; fax: + 381 11 3370 387.

E-mail addresses: [email protected] (M. Petkovska), [email protected] (D. Nikolic´), [email protected] (A. Markovic´), [email protected] (A. Seidel-Morgenstern). 1 Current address: University of Stuttgart, Institute for Chemical Engineering, ¨ Boblinger Str. 72, Geb. 78, 70199 Stuttgart, Germany. 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.03.011

is modulated periodically (e.g. in a co-sinusoidal way) around its steady-state value, the outlet concentration will eventually also oscillate. If the reactor is a nonlinear system, the mean value of the outlet concentration cAm will be different from cs. The difference D ¼ cAm cA,s can be negative, zero, or positive, depending on the type of nonlinearity. If D o0, the periodic operation can be considered as favourable, as it corresponds to increased conversion, in comparison to the steady-state operation. The questions of identifying candidate system for enhancement through periodic operation and estimating the magnitude of such enhancements have been occupying researchers for several decades (Douglas and Rippin, 1966; Douglas, 1972; Bailey, 1973; Watanabe et al., 1981; Farhadpour and Gibilaro, 1981; Sterman and Ydstie, 1990; Nowobilski and Takoudis, 1986), Thullie et al., 1986). Theoretical problems related to identification and estimation of enhancement were typically analysed based on suitable control criteria. Essentially four major approaches were developed (Watanabe et al., 1981): the Hamilton–Jacobi approach based on the Pontryagin’s maximum principle (Bailey and Horn, 1971; Bailey, 1973; Farhadpour and Gibilaro, 1981); the Hamilton–Jacobi approach based on relaxed steady-state analysis (Bailey and Horn, 1971; Horn and Lin, 1967) suitable for high forcing frequencies; p-criteria, suggested by Guardabassi et al. (1974), Noldus (1977), Sch¨adlich et al. (1983) which could cover broader frequency range; the singular control tests (Sch¨adlich, 1981; Sch¨adlich et al., 1983;

ARTICLE IN PRESS M. Petkovska et al. / Chemical Engineering Science 65 (2010) 3632–3637

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Fig. 1. Representation of a favourable periodic reactor operation.

¨ Grabmuller et al., 1985) which can be considered as special cases of the p-criterion. More details can be found in Parulekar (2003). However, although different and detailed optimization and control criteria for analysis of performance enhancement by cycling exist, they are not applied widely. The reason for this is in the complexity of their application and some lingering uncertainty about their reliability. In practice, testing whether a periodic operation is favourable, i.e., whether it leads to an increased productivity as compared to the corresponding conventional steady-state operation, is usually performed by long and tedious experimental and/or numerical work. Therefore, there is still a need for developing new simple and reliable methods which would enable to quantitatively evaluate the possibility of process improvements through periodic operations quickly and in early stages of investigation. In this paper we apply a relatively new, fast and easy method, which is based on the Volterra series approach (Volterra, 1959), nonlinear frequency response and the concept of higher order frequency response functions (Weiner and Spina, 1980). Here we will give only a brief overview of these tools. More details can be found in a book chapter by Petkovska (2006). 1.2. Nonlinear frequency response and the concept of higher order frequency response functions Without going into details, let us just remind that the frequency response (quasi-steady state response to a co-sinusoidal input change) of a weakly nonlinear system, in addition to the basic harmonic, contains a non-periodic (DC) term, and an indefinite number of higher harmonics (Weiner and Spina, 1980): x ¼ xs þ A cosðotÞ! y ¼ ys þyDC þ BI cosðot þ fI Þ t-1

þ BII cosð2ot þ fII Þ þ   

yDC  2ðA=2Þ2 G2 ðo,oÞ

ð2Þ

The dominant term of yDC is proportional to the asymmetrical second order function G2(o,  o). Based only on the second order

ð3Þ

In one of our previous papers (Markovic´ et al., 2008) we have illustrated the use of this function for evaluation of periodic operations of different types of reactors (ideal continuous stirred tank and tubular plug-flow reactors, as well as tubular reactor with axial dispersion) with feed concentration modulation, for a simple homogeneous n-th order reaction of type A- products. This analysis showed that for reaction orders no0 and n 41 the periodic operation would be favourable in comparison with the corresponding steady-state one, that for 0 on o1 the periodic operation is unfavourable, while for n ¼0 and 1 the periodic operation makes no difference. It is important to point out that these results were obtained for all three reactor types considered in the investigation. In this paper we use the same type of analysis to investigate periodic operations of a continuous stirred tank reactor (CSTR) with a simple heterogeneous catalytic n-th order reaction ads:eq:

k

A ! Aads !products:.

2. Model equations In the current investigation, we limit our analysis to the continuous stirred tank reactor model, which corresponds, from the mathematical point of view, to the simplest case. This model is applicable, for example, for heterogeneous catalytic fluidized bed reactors, in which the catalysts can be fully back-mixed. For a simple heterogeneous catalytic n-th order reaction ads:eq:

ð1Þ

On the other hand, any nonlinear model with polynomial nonlinearity (G), can, in the frequency domain, be replaced by an indefinite sequence of frequency response functions (FRFs) of different orders (G1(o), G2(o1,o2), G3(o1,o2,o3),y) (Weiner and Spina, 1980). These functions are directly related to the DC component and different harmonics of the response. The DC component, which is responsible for the average performance of the periodic process can be expressed as the following indefinite sum: yDC ¼ 2ðA=2Þ2 G2 ðo,oÞ þ 6ðA=2Þ4 G4 ðo,o,o,oÞ þ   

function G2(o,  o), an approximate value of the DC component can be easily estimated:

k

A ! Aads !products, the material balance for the reactant A can be written in the following form: Vf

dcA dqA þ Vs ¼ FðcAi cA ÞkVs qnA dt dt

ð4Þ

where t is time cA and qA are the concentrations of the reactant A in the fluid and in the solid phase, respectively, cAi is the concentration of the reactant in the feed stream, Vf and Vs are the volumes of the fluid and of the solid phase, respectively, F is the feed stream flow-rate, k is the reaction rate constant and n the reaction order. If the mass transfer resistances between the fluid and solid phases are negligible, the two phases are in equilibrium, defined by an adsorption isotherm relation, which is generally nonlinear: qA ¼ jðcA Þ

ð5Þ

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For easier analysis in the frequency domain, it is convenient to transform the model equation into dimensionless form:   qn dC dQ c þf ¼ Ai,s ð1 þCi Þð1 þCÞkts A,s ð1 þQ Þn ð6Þ cA,s cA,s dy dy The dimensionless variables used in this equation are defined in Table 1. In Table 1 and Eq. (6) the subscript s denotes the concentrations corresponding to the steady-state. By solving the material balance equation for the steady-state conditions, the following relation among the steady-state concentrations cA,s, qA,s and cAi,s is obtained:   qn cAi,s ¼ cA,s 1 þ kts A,s ð7Þ cA,s Two auxiliary parameters have also been introduced in Eq. (6): f¼

Vs qA,s Vf cA,s

and

ts ¼

Vs : F

In order to apply the Volterra theory and the concept of higher order FRFs, all nonlinearities in the model equation need to have polynomial forms. For that reason, the nonlinear term (1+ Q)n in Eq. (6) is replaced by its Taylor series expansion: ð1þ Q Þn ¼ 1þ nQ þ 12nðn1ÞQ 2 þ . . .

ð9Þ The dimensionless nonlinear adsorption isotherm is also replaced by its Taylor series expansion around the steady state: Q ¼ FðCÞ ¼ aC þ bC 2 þ . . .

ð10Þ

where a and b are proportional to the local isotherm derivatives: 2 cA,s @F @j cA,s 1 @2 F 1 @2 j ¼ , b¼ ¼ a¼ 2 2 @C C ¼ 0 @cA c ¼ c qA,s 2 @C C ¼ 0 2 @cA c ¼ c qA,s A,s

(1) defining the input concentration Ci(y) in the form of a cosinusoidal function; (2) expressing the output concentration C(y) in the Volterra series form; (3) substituting the expressions for Ci(y) and C(y) into the model equations; (4) applying the method of harmonic probing to the equations obtained in step 3 (collecting the terms with the same power of the input amplitude and same frequency, and equating them to zero); (5) solving the equations obtained in Step 4. This procedure is recursive, i.e., the first order FRF has to be derived first, than the second order FRFs, and so on. For our application, it is enough to derive only the first order FRF G1(o) and the asymmetrical second order FRF G2(o,  o). Some details of the derivation can be found in the Appendix A. 3.2. The first and asymmetrical second order FRFs

ð8Þ

The resulting model equation is   qn qn dC dQ 1 qnA,s þf ¼ 1 þ kts A,s Ci Ckts A,s nQ  nðn1Þkts Q 2  . . . cA,s cA,s dy dy 2 cA,s

A

The basic steps of this procedure, applied to our case, are as follows:

A

When the procedure defined in the previous Section was applied to our model equations in the dimensionless form (Eqs. (9) and (10)), the following final expressions were obtained:

 For the first order FRF: G1 ðoÞ ¼

3. Frequency response functions for the heterogeneous catalytic CSTR

ð12Þ

 For the asymmetrical second order FRF: G2 ðo,oÞ ¼

nð12ð1nÞa2 bÞ bð1þ bÞ2 ð1þ nbaÞ ð1 þnbaÞ2 þ o2 ð1 þ faÞ2

ð13Þ

where

A,s

ð11Þ

1þb 1 þ bna þjoð1 þ faÞ

b ¼ kts

qnA,s cA,s

ð14Þ

is again an auxiliary parameter. It should be noticed that in Eqs. (12) and (13) the frequency o is also dimensionless.

3.1. The procedure for derivation of the frequency response functions The next step in our procedure is deriving the necessary FRFs which correlate the outlet and the inlet dimensionless concentration changes. For evaluating the average reactor performance, it is necessary and enough to estimate the DC component. As explained in the Introduction, the sign of the DC component is determined by the asymmetrical second order FRF G2(o,  o). The procedure for deriving FRFs of different orders is standard and can be found in our previous publications (Petkovska, 2006; Petkovska and Markovic´, 2006).

Table 1 Definitions of the dimensionless variables. Outlet concentration Input concentration Concentration in the solid phase Time

cA cA,s cA,s c c Ci ¼ A,i Ai,s cAi,s qA qA,s Q¼ qA,s F y¼ t Vf C¼

4. Analysis and discussion 4.1. Analysis of the sign of G2(o,  o) As it has already been explained in the Introduction, the sign of the G2(o,  o) function determines whether a periodic operation would give better results that the corresponding steady-state one, or not. For the case of a periodic operation of a reactor, where the output is the outlet concentration of the reactant, the periodic operation will be favourable if G2(o,  o)o0 (the average concentration of reactant cA will be smaller than cA,s, i.e conversion will be higher) and unfavourable for G2(o,  o)40. Therefore, we will analyse the sign of the G2(o,  o) function defined by Eq. (13) in order to define in which cases the periodic operation of a heterogeneous catalytic CSTR would be favourable. The sign of the G2(o,  o) function is determined by the sign of the constant term nð1=2ð1nÞa2 bÞ=ð1þ nbaÞ, which depends on the reaction order n, as well as on the values defining the isotherm derivatives a and b. Analysis of this term shows that when the reaction order n changes, there are three possible changes of the sign of the analysed function. The three values of n for which

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4.2. Simulation results

these changes occur are n1 ¼ 0 n2 ¼ 1

n3 ¼ 

2b a2

1

ð15Þ

ba

While n1 equals zero and n3 is always negative, the sign of n3 depends on the values of b and a. Therefore, regarding the isotherm derivatives a and b, we can consider two cases: Case 1, for boa2/2 and Case 2, for b 4a2/2. Case 1. boa2/2. This case corresponds to all favourable and most unfavourable isotherms. In this case n2 is positive (see Eq. (15)), and the sign of G2(o,  o) changes with n in the following way: n o n3 ) G2 ðo,oÞ 4 0 n3 on o0 ) G2 ðo,oÞ o 0 n ¼ 0 ) G2 ðo,oÞ ¼ 0 0 o n o n2 ) G2 ðo,oÞ 4 0 n ¼ n2 ) G2 ðo,oÞ ¼ 0 n 4 n2 ) G2 ðo,oÞ o0

ð16Þ

The value n2 depends on the adsorption isotherm derivatives a and b, while the value n3 depends on a and the auxiliary parameter b, which, in turn, depends on the reaction order (see Eq. (14)). Accordingly, the value n3 should be determined by solving the following set of nonlinear algebraic equations: n3 kts

qnA,s3 cA,s

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In order to illustrate the obtained results, we simulated the asymmetrical second order FRFs for a set of arbitrary model parameters, given in Table 2, and for different values of the reaction order n. The simulations were performed for a favourable (Langmuir) type of isotherm. For the model parameters used for simulation, the value n2, defined in Eq. (15) is n2 ¼1.235. The simulated G2(o,  o) functions corresponding to 6 different values of n are shown in Fig. 2. The values of n were chosen from all three regions defined in Eq. (18). As expected, the value of G2(o,  o) is negative for negative values of n (  3,  1 and  0.5) and for n 41.235 (3), while it is positive for 0 on o1.235 (0.5 and 1). It can also be observed that for all values of n the values of the G2(o,  o) function become negligibly small at high frequencies (approximately higher than 1) and tend to constant values at low frequencies (approximately lower than 0.01). In Fig. 3 we show the simulated values of the low-frequency asymptotic values of G2(o, o), as a function of the reaction order n, in a wide range between  10 and 10. As it can be observed from this figure, the limiting value of G2(o, o) is negative for all negative values of n, but it becomes very close to zero for no 5. There is a local minimum in this range of n, corresponding to nE  0.9 and limo-0 G2 ðo,oÞ  0:1. In the range 0ono1.235 G2(o, o) is positive, with a maximum in this range corresponding to nE0.6 and limo-0 G2 ðo,oÞ  0:1. Finally, for n41.235, G2(o, o) is again negative, and there is a minimum in this interval, corresponding to

Table 2 Model parameters used for simulations. Parameter

Value

Reaction rate constant, k Feed concentration of the reactant, cAi Total volume of the CSTR, Vf + Vs Volume of the solid phase Vs Volumetric flow rate of CSTR, F Adsorption isotherm

0.001 s  1 mol1  n 1 mol/dm3 10 dm3 7 dm3 0.1 dm3/s 0:2cA qA ¼ 20 1 þ 0:2cA

aþ1 ¼ 0

  qn cA,s 1 þ kts A,s ¼ cAi,s cA,s qA,s ¼ jðcA,s Þ

ð17Þ

It can be shown that this set of equations has no solutions. Taking all this into account, three intervals of the values of the reaction order n can be defined in which the sign of the function G2(o,  o) needs to be considered: n o0, 0 onon2 and n 4n2. The following final results regarding the sign of G2(o,  o) are obtained: G2 ðo,oÞ o 0

for

no0

G2 ðo,oÞ 4 0

for

0 o n o 1

2b a2

G2 ðo,oÞ ¼ 0

for

n¼0

n ¼ 1

or

or

n 41

2b a2

2b a2

ð18Þ

Case 2. b 4a2/2 (possible only for unfavourable isotherms). For this case the value n2 is negative, and the situation is reversed, so it holds 2b on o0 a2

G2 ðo,oÞ o 0

for

1

G2 ðo,oÞ 4 0

for

n o 1

2b a2

G2 ðo,oÞ ¼ 0

for

n¼0

or

or

n40

n ¼ 1

2b a2

ð19Þ

Fig. 2. Simulated G2(o, o) functions for different values of the reaction order n, for Case 1 (bo a2/2).

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5. Conclusions

Fig. 3. The low-frequency asymptotic value of G2(o,  o) as a function of the reaction order n, for Case 1 (b o a2/2).

nE2.8 and limo-0 G2 ðo,oÞ  0:15. These results were obtained for a favourable isotherm and with the simulation parameters defined in Table 2. For another set of parameter values, the values corresponding to the local minimums and maximum will be changed, but the overall behaviour will qualitatively remain the same.

The method for fast evaluation of periodic processes, based on nonlinear frequency response analysis, first presented in (Markovic´ et al., 2008), was used here for analysis of a nonlinear CSTR with a simple isothermal n-th order heterogeneous catalytic reaction of the ads:eq: k type A !Aads !products. For this purpose, the asymmetrical second order frequency response function G2(o, o), which is mainly responsible for the time-average performance of the periodic operation, was derived and analysed. The analysis showed that the conditions for the periodic operation of the CSTR to be favourable in comparison to the corresponding steady-state operation depend both on the reaction order and on the shape and parameters of the adsorption isotherm, more precisely, on the values of its local first and second derivatives. It was shown that the ratio between the second and the square of the first isotherm derivative plays a very important role in this sense. In our future work, we are planning to apply the nonlinear frequency response method for evaluation of periodic operations to evaluate some real reaction systems, including non-isothermal ones, as well as for the analysis of periodic operations with modulation of more than one input.

Notation a A

4.3. Comparison with the homogeneous case

b

As already mentioned in the Introduction (Section 1.2) similar analysis has been applied to investigation of periodic operation of homogeneous reactors with simple reaction mechanism ¨ A- products (Markovic´ et al., 2008). In agreement with Schadlich et al. (1983) the results of that analysis were G2 ðo,oÞ o 0

for

no0

G2 ðo,oÞ 4 0

for

0ono1

G2 ðo,oÞ ¼ 0

for

n¼0

or

or

n 41

n¼1

ð20Þ

Comparison with Eqs. (18) and (19) (Section 4.2) leads to the following conclusions: 1. The results obtained for the heterogeneous reaction corresponding to Case 1 (boa2/2) are similar to the results obtained for the homogeneous reaction: the periodic operation would be favourable for negative reaction orders and for reaction orders higher than a certain positive value n2, and it would be unfavourable for reaction orders between 0 and n2. The only difference is that for the homogeneous reaction this value is n2 ¼ 1, while for the heterogeneous reactor it depends on the local isotherm derivatives. 2. On the other hand, the result obtained for Case 2 of the heterogeneous reaction is just the opposite from the homogeneous case: the periodic operation would be favourable for 0on on2 and unfavourable for n o0 and n4n2. Also, the shapes of the G2(o,  o) functions presented in Fig. 2 are similar to the shapes obtained for the CSTR with a homogeneous reaction (Markovic´ et al., 2008).

B c C F Gn k n q Q t V x y

first order coefficient of the Taylor series expansion of the dimensionless adsorption isotherm input amplitude, general and of the dimensionless inlet concentration second order coefficient of the Taylor series expansion of the dimensionless adsorption isotherm output amplitude concentration in the fluid phase, mol/cm3 dimensionless concentration in the fluid phase flow-rate, cm3/s n-th order frequency response function reaction rate constant, s  1 mol1  n reaction order concentration in the solid phase, mol/dm3 dimensionless concentration in the solid phase time, s volume, cm3 input, general output, general

Greek symbols auxiliary parameter, Eq. (14) b D difference between the mean and the steady-state concentration dimensionless time y ts solids contact time, s phase f F dimensionless isotherm relation j adsorption isotherm relation o frequency, general and dimensionless Subscripts I first harmonic II second harmonic A reactant A DC non-periodic term f fluid i inlet

ARTICLE IN PRESS M. Petkovska et al. / Chemical Engineering Science 65 (2010) 3632–3637

s

steady-state, solid

3637

Step 5. Solving the equations obtained in Step 4 Eqs. (A.4) and (A.5) are linear algebraic equations, so their solution is rather trivial. Eq. (A.4) is first solved, resulting with the expression for the first order FRF G1(o), which is given by Eq. (14) in the main text. This solution is introduced into Eq. (A.5), using the fact that G1(  o) and G1(o) are conjugate-complex functions, and the resulting equation is solved, leading to the expression for the asymmetrical second order FRF G2(o,  o). This expression was given by Eq. (15) in the main text.

Superscripts m mean Abbreviations CSTR continuous stirred tank reactor FRF frequency response function

Acknowledgements This work was partly supported by the Serbian Ministry of Science in the frame of Project no. 142014G.

Appendix A. Derivation of the first and asymmetrical second order frequency response functions Step 1. Defining the input (inlet concentration) in the form of a co-sinusoidal function A joy ðA:1Þ ðe þ ejoy Þ 2 Step 2. Expressing the output (outlet concentration C(y)) in the form of a Volterra series

Ci ðyÞ ¼ A cosðotÞ ¼

CðyÞ ¼

A ðG1 ðoÞejoy þ G1 ðoÞejoy Þ 2  2 A þ ðG2 ðo,oÞe2joy þ 2G2 ðo,oÞe0 þG2 ðo,oÞe2joy Þ þ . . . 2

ðA:2Þ where G1(o), G2(o,o), G2(o,  o) are the first and second order FRFs defining the heterogeneous CSTR. Using this expression and the local equilibrium relation (Eq. (10)), the concentration in the solid phase can also be expressed in the series form:   A ðG1 ðoÞejoy þ G1 ðoÞejoy Þ Q ðyÞ ¼ a 2  2 A ðG2 ðo,oÞe2joy þ 2G2 ðo,oÞe0 þ G2 ðo,oÞe2joy Þ þa 2  2  2 A A G21 ðoÞe2joy þ 2b G1 ðoÞG1 ðoÞe0 þ . . . ðA:3Þ þb 2 2 Step 3. Substituting of expressions (A.1), (A.2) and (A.3) into the model Eq. (9) The resulting equation is rather cumbersome, but straightforward. It is not presented here. Step 4. Collecting all terms in the equation obtained in Step 3 with the same frequency and same power of the input amplitude, and equating them to zero In order to obtain the asymmetrical second order function, it is necessary to obtain the equations defining G1(o) and G2(o,  o). They are obtained collecting all terms with Aejoy : joG1 ðoÞ þ jofaG1 ðoÞ ¼ ð1 þ bÞG1 ðoÞbnaG1 ðoÞ
2 and A2 e0 0 ¼ G2 ðo,oÞbn½aG2 ðo,oÞ þ bG1 ðoÞG1 ðoÞ 1  bnðn1Þa2 G1 ðoÞG1 ðoÞ 2

ðA:4Þ

ðA:5Þ

References Aida, T., Silveston, P.L., 2005. Cyclic Separation Reactors. Blackwell Publishing Ltd, Oxford. Bailey, J.E., 1973. Periodic operation of chemical reactors: a review. Chemical Engineering Communications 1, 111. Bailey, J.E., Horn, F.J.M., 1971. Comparison between two sufficient conditions for improvement of an optimal steady state process by periodic operation. Journal of Optimization Theory & Application 7, 378. Chanchlani, K.G., Hudgins, R.R., Silveston, P.L., 1994. Methanol synthesis under periodic operation: an experimental investigation. The Canadian Journal of Chemical Engineering 72, 657. Douglas, J.M., Rippin, D.W.T., 1966. Unsteady state process operation. Chemical Engineering Science 21, 305. Douglas, J.M., 1972. Process Dynamics and Control. Prentice-Hall, Englewood Cliffs, New Jersey. Farhadpour, F.A., Gibilaro, L.G., 1981. Analysis of concentration forcing in heterogeneous catalysis. Chemical Engineering Science 36, 143. Guardabassi, G., Locatelli, A., Rinaldi, S., 1974. Status of periodic optimization of dynamical systems. Journal of Optimization Theory & Applications 14, 1–20. ¨ ¨ Grabmuller, H., Hoffmann, U., Schadlich, K., 1985. Prediction of conversion improvements by periodic operation for isothermal plug-flow reactors. Chemical Engineering Science 40, 951–960. Horn, F.J.M., Lin, R.C., 1967. Periodic processes: a variation approach. Industrial & Engineering Chemistry Process Design & Development 6, 21. Markovic´, A., Seidel-Morgenstern, A., Petkovska, M., 2008. Evaluation of the potential of periodically operated reactors based on the second order frequency response functions. Chemical Engineering Research and Design 86, 682–691. Nappi, A., Fabbricino, L., Hudgins, R.R., Silveston, P.L., 1985. Influence of forced feed composition cycling on catalytic methanol synthesis. The Canadian Journal of Chemical Engineering 63, 963. Noldus, E.J., 1977. Periodic optimization of a chemical reactor system using perturbation methods. Journal of Engineering Mathematics 11, 49. Nowobilski, P.J., Takoudis, C.G., 1986. Periodic operation of chemical reactor systems: are global improvements attainable? Chemical Engineering Communications 40 249. Parulekar, S.J., 2003. Systematic performance analysis of continuous processes subject to multiple input cycling. Chemical Engineering Science 58, 5173. Petkovska, M., 2006. Nonlinear frequency response method for investigation of equilibria and kinetics in adsorption systems. In: Spasic, A.M., Hsu, J.P. (Eds.), Finely Dispersed Particles: Micro, Nano- and Atto-Engineering. CRC Taylor & Francis, Boca Raton, pp. 283. Petkovska, M., Markovic´, A., 2006. Fast estimation of quasi-steady states of cyclic nonlinear processes based on higher-order frequency response functions. Case study: cyclic operation of an adsorption column. Industrial & Engineering Chemistry Research 45, 266–327. ¨ Schadlich, K., 1981. Periodischer Ablauf Isothermer Heterogen-katalytischer ¨ Oberflachen-reaktionen in Systemen Mit Konzentrierten Parametern, Disser¨ tation, University of Erlangen-Nurnberg. ¨ Schadlich, K., Hoffman, U., Hofmann, H., 1983. Periodical operation of chemical processes and evaluation of conversion improvements. Chemical Engineering Science 38, 1375. Silveston, P.L., 1998. Composition Modulation of Catalytic Reactors. Gordon and Breach Science Publishers, India. Sterman, L.E., Ydstie, B.E., 1990. The steady-state process with periodic perturbations. Chemical Engineering Science 45, 721. Thullie, J., Chiao, L.R.G., Rinker, 1986. Analysis of concentration forcing in heterogeneous catalysis. Chemical Engineering Communications 48, 191. Volterra, V., 1959. Theory of Functionals and Integral and Integrodifferential Equations. Dover, New York. Watanabe, N., Onogi, K., Matsubara, M., 1981. Periodic control of continuous stirred tank reactors-I. The p criterion and its application to isothermal cases. Chemical Engineering Science 36, 809. Weiner, D.D., Spina, J.F., 1980. Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits. Van Nostrand Reinhold Company, New York.