A new Approach for the Reliable Estimation of Kinetic Parameters by Means of Dynamic DSC Experiments

Transcription

1 145 A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 36, 2014 Guest Edtors: Valero Cozzan, Eddy de Rademaeker Copyrght 2014, AIDIC Servz S.r.l., ISBN ; ISSN The Italan Assocaton of Chemcal Engneerng DOI: /CET A new Approach for the Relable Estmaton of Knetc Parameters by Means of Dynamc DSC Experments Roberto Sanchrco Isttuto d Rcerche sulla Combustone (IRC) Consglo Nazonale delle Rcerche (CNR)- P.le V. Teccho, Naples (Italy) r.sanchrco@rc.cnr.t A new method for the assessment of the knetc trplet (pre-exponental factor, Actvaton energy, knetc model) better descrbng a thermal decomposton process s presented. The approach s based on the processng of three DSC thermograms collected under dynamc condtons usng dfferent heatng rates. It has been shown that two of ths basc set of runs allows the assessment a pror of the knetc model sutable to descrbe the process under study (among reacton order and autocatalytc Sestak-Bergren model) whle a successve multvarate procedure carred out consderng these three set of data allows the assessment of the fnal estmate of the thermoknetc parameters. A number of examples are provded wth the am at showng that, when partcular hypothess are met, the extrapolaton of the results lead to relable prevsons. 1. Introducton In the context of a safety analyss nvolvng thermal unstable compounds, a crucal pont s concerned wth the relable prevson of the behavour of these substances under dfferent thermal condtons. Ths requres the knowledge of a knetc model sutable both to descrbe the expermental data from whch t has been dentfed (nterpolaton) and predct the behavour of the system outsde these condtons (Extrapolaton). To ths end two dfferent strateges are possble: the model free and the model fttng approach (Burnham et al., 2007). The frst class of technques ncludes all the methods that allow the determnaton of the dependence of the actvaton energy on the converson. The second class of methods are based on the strateges devoted to determne the best model that ft the expermental data. In both cases, the basc assumpton whch s consdered vald s the so called sngle step hypothess (Šmon, 2005). Ths assumpton allows us to wrte the reacton progress as the product of two ndependent terms: d K( T ) f ( ) dt (1) the frst dependng only on the temperature (K(T): Knetc constant, whch generally has an Arrhenus structure): E K( T ) A exp (2) RT and the second term f() (reacton model) dependng only on the converson. A lst of these models can be found n (Vyazovkn et al., 2011). It should be underlned that the sngle step hypothess s a strongest assumpton that should be verfed before the mplementaton of any successve dentfcaton procedure. A way by means ths can be accomplshed consder the plot E=E() evaluated adoptng a model-free method (Fredman method for example) and verfyng that these values are constant (or at least almost constant) over the whole converson nterval: E=E()=const for each belongng to [0,1]. In the classc model fttng approach we try to determne the model better descrbng the expermental (sothermal or dynamc) DSC data evaluatng the regresson coeffcent R 2 determned consderng a Please cte ths artcle as: Sanchrco R., 2014, A new approach for the relable estmaton of knetc parameters by means of dynamc dsc experments, Chemcal Engneerng Transactons, 36, DOI: /CET

2 146 number of dfferent (reasonable) models. The expresson of f() that provde the hghest value of R 2 s consdered as the best choce for the system under study. The major drawback of ths approach s that ths crteron does not provde an objectve dscrmnaton procedure due to the fact that the R 2 values are generally comparable and close to one also consderng dfferent knetc models. Another possble dscrmnaton procedure try to select the best model descrbng the expermental data by means of the use of the so called y() and z() functons (Malek method) (Màlek, 1992). A good revew of all these methods can be found n (Vyazovkn et al., 2011). When the model fttng strategy s consdered, sem-emprcal models can be successfully taken nto account especally when the complexty of the system under study prevents any mechanstc concluson. The use of these models can be consdered as a possble strategy whch allows us to lump n an apparent knetc scheme the complex nature of the real process under study. Equaton 3 and 4 (Šmon, 2011) show the basc sem-emprcal models that are generally taken nto account. n RO ( n) : f ( ) (1 ) (3) SB ( p, q) : f ( ) ) p q (1 (4) Once determned the model whch s consdered vald for the descrpton of the system under study (dscrmnaton), the knetc trplet [A, E, f()] can be dentfed usng the approach suggested by (Mltký et al., 1992). Ths approach requres the mnmzaton of a sutable objectve functon. Due to the complex nature of the objectve functon, usually the sum of the squared errors, the success of ths approach s strongly affected by the proper choce of the ntal values of the parameters. If DSC dynamc technques are consdered, although a sngle dynamc DSC run contans all the nformaton, more than one scan performed at dfferent heatng rates are requred to dentfy the model correctly. The approach that wll be dscussed n the present work s based on the followng assumptons: a thermal process n a dynamcal DSC run s represented by a sngle peak; Equaton 1 s vald, the sem-emprcal models consdered are: RO(n) (Equaton 3) and SB(p,q) (Equaton 4). The approach proposed by the authors allows the dscrmnaton among RO(n) and SB(p,q) models wthout makng any smplfyng hypothess and consderng only the geometrc structure of two DSC dynamc runs. Once establshed the knetc model sutable at descrbng the system under study, a relable ntal estmate of the model s parameter s performed and the fnal estmate s determned by means of a sutable multvarate dentfcaton procedure. 2. Method Descrpton It has been shown (Sanchrco, 2012) that two DSC dynamc runs carred out at dfferent heatng rates are enough to the am at determnng the knetc model (among Reacton Order (RO(n) and Sestak-Berggren autocatalytc (SB(p,q) model) better descrbng the expermental data. The analyss of the rato q ( T ) for q j ( T ) the specfc heat powers performed usng the Equaton (5) q ( T ) K( T ) f ( ) ( H R ), 1, 2 (5) wrtten for two dfferent values of the heatng rate, and j, (H R s the heat of reacton) and consderng for f() the expressons gven by the Equatons (3) and (4) lead to the concluson that the smple fact that two dynamc DSC curves gathered n these condtons ntercept n a pont dfferent than the orgn allows us to conclude that RO(n) has to be ruled out and the applcaton of SB(p,q) should provde: 0<p<1 and q>0. The last crcumstance hghlghts the autocatalytc nature of the process under study. On the contrary, f two curves do not ntercept n a pont dfferent than the orgn, RO(n) can descrbe the expermental data provded that n>0 and SB(p,q) s not applcable. A frst estmate of the exponents p and q can be obtaned by means of the lnear nterpolaton of the followng varables (Relaton a): q ln 1 ln q j vs y 1 j (a), j x, j ln ln j j

3 (where the ndexes j refer to a couple of runs carred out at dfferent heatng rates). The slope of ths lne s equal to q and ntercept s equal to p: p q x (6) y, j, j If RO(n) holds, by applyng the Equaton 6 we should get an ntercept p=0 and a slope q=n>0; ths result could be derved also consderng (Relaton b) q ln vs 1 ln (b) q j 1 j obtanng a lne wth slope equal to the reacton order n. Once the model f() s estmated ts dervatve f () can be used for the estmaton of the actvaton energy and of the pre-exponental factor by means of the applcaton of the extended Kssnger s method. Ths requre at least three expermental curves gathered durng dfferent dynamcal runs carred out usng dfferent heatng rate 1 < 2 < 3. At ths pont, the dfferent curves correspondng to dfferent combnaton of ndexes ((, j)=(1, 2), (2, 3), (1, 3)), can be used also to verfy the results gathered about the model usng the frst two runs. The values of the Arrhenus and knetc model parameters assessed prevously can be adopted as ntal estmate n a successve optmzaton procedure carred out by means of multvarate Ordnary Least Square procedure that nvolves the search of the mnmum over the parameter space of the sum of the squared errors bult consderng all the data collected at the dfferent heatng rates. 3. Results and dscusson In order to show the method llustrated above, n the followng are reported the prncpal results obtaned for the thermal decomposton of Cumene Hydroperoxde (80 % n Cumene) [CHP] and Dcumyl Peroxde (98% pure crystallne powder) [DCP]. These substances are mportant ndustral ntermedates belongng to the class of organc peroxdes tendng to decompose exothermcally wth a huge heat release and gas evoluton. It s reported that CHP decomposes followng an autocatalytc behavor (Sanchrco, 2012). On the contrary, t has been confrmed that the thermal decomposton of DCP follow a reacton order knetc (Sanchrco, 2013). In both cases, DSC runs were performed usng a PerknElmer DSC 8000 nstrument equpped wth an Intracooler II coolng system. Both Dynamc and Isothermal runs were carred out usng Hgh Pressure capsules. Temperature ramps were performed usng three dfferent heatng rates for each compound. The prncpal results and the condtons adopted for these experments are reported n Table 1 whle n Fgure 1a and 1b are reported the correspondng DSC traces. Table 1: Expermental condtons and prncpal results gathered durng the Dynamc DSC runs carred out on DCP and CHP Substance Run Sample mass (mg) (K mn -1 ) T max (K) H R (J g -1 ) ,692 CHP , , DCP These curves have been consdered wth the am of dentfyng the knetc trplet concerned wth both CHP and DCP. The analyss of the Fgure 2b does not show evdent ntersecton ponts among all the possble combnatons of dstnct curves whle, at the opposte, n the case reported n Fgure 1a are evdent three dstnct ntersecton ponts j (hghlghted by arrows). These crcumstances allow us to conclude that n the case of DCP RO(n) holds and n the case of CHP the Sestack-Berggren has to be taken nto account. It has been demonstrated that a frst estmate of the exponent n can be obtaned by means of the lnear nterpolaton of sutable varable (Equaton 6) (Sanchrco, 2012). In Fgure 2 s reported an example of these nterpolatons both for CHP and DCP. In Table 2 are reported the results of these ntal estmates [(p 0, q 0 ) for CHP and n 0 for DCP)] along wth the correspondng evaluaton temperature ntervals. The evaluaton temperature ntervals are the temperature ntervals over whch the varables n Eq. 6 should to be evaluated due to the fact that, f two curves are too close (as t s evdent consderng the values reported for q vs T over the tals), the expermental errors could amplfy leadng to a poor evaluaton of these varables. Once determned a frst estmate of the reacton orders, Extended Kssnger s method has been appled consderng Eq

4 148 a) b) Fgure 1: Calculated (sold) and expermental (dashed lnes) results gathered durng the DSC dynamc experments on CHP (panel a) and DCP (panel b) a) b) Fgure 2: Examples of Regresson lnes used for the ntal estmate of (p,q) [Panel (a): CHP] and n [panel (b): DCP]. (Indexes refer to some of the runs reported n Table 2) Table 2: Frst estmates of the exponents evaluated consderng the method proposed by the authors both for DCP (RO(n)) and CHP (SB(p,q)) Substance Data Set at Evaluaton Temperature Range (K) P q n CHP DCP ln E AR ln (, max) ( 1, 2,3) 2 T O (7) T,max R,max E where ) D[ f ( )]. O (, max, max Consderng the temperature at the maxmum T max, (see Table 1) for the dfferent heatng rates, the lnear nterpolaton carred out consderng the Equaton 7, allowed the assessment of the Arrhenus parameters reported n Table 3.

5 149 Table 3: Intal and fnal estmates evaluated for the parameters concerned wth the RO(n) and SB(p,q) models used to descrbe the thermal decomposton of DCP and CHP Substance Parameter Intal estmate Fnal estmate A (s -1 ) DCP E (J mol -1 ) 125, ,892 n A (s -1 ) CHP E (J mol -1 ) 95,895 91,220 p q The fnal estmate of the parameters was determned by means of a multvarate ordnary least square (OLS) procedure carred out consderng all the curves shown n Fgure 1a and 1b and consderng as ntal guess for the vector parameter the values prevously determned. It should be stressed that, f we consder the SB(p,q) model, the Equaton 1 provdes the trval soluton (T)=0 f ths equaton s ntegrated consderng the ntal condton(t 0 )=0. Ths fact, along wth the extreme senstvty of the soluton n respect to the ntal condton adopted to carry out the ntegraton (Rodut et al., 2013) led to the ntegraton of the equaton (1) for the dfferent heatng rates (=1, 2, 3) fxng for CHP a value of 0 =10-3 and evaluatng the ntal temperatures T o, for whch the expermental converson assume ths value. Thus, the expermental values consdered for the OLS procedure were: T =[T o,,t fnal, ], =[10-3 1] and q =[q(t o, ) q(t fnal, )], (=1,2,3) 4. Extrapolatons Wth the am at showng that the dentfcaton procedure descrbed above lead to estmates whch can be used also for extrapolaton purposes, a seres of sothermal experments were performed both on DCP and CHP and the results compared wth the correspondng theoretcal predctons. These results are reported n Fgure 3 and, as t s evdent from ts analyss, clearly ponts out a good agreement between the expermental and the calculated curves both n the case of CHP and DCP. a) b) Fgure 3: Expermental DSC Isothermal conversons (dashed) and calculated (sold) curves for CHP (panel a) and DCP (panel b). Temperatures are reported near the curves 5. Conclusons A method for the dscrmnaton between two dfferent knd of sem-emprcal models (reacton order and autocatalytc Sestak-Berggren) has been proposed. It has been shown that two curves are enough to determne the knetc model that better descrbe the expermental data and t has been ponted out that three dynamc DSC curves could provde a complete set of expermental data sutable to assess the complete knetc trplet [A, E, f()]. It has been shown that f two curves gathered usng dfferent heatng rates ntercept n a pont dfferent than the orgn the model sutable at descrbng the system under study s the autocatalytc Sestak-Berggren model. At the opposte, f these ntersecton ponts doesn t exst, the RO(n) model should be taken nto account. A procedure provdng the ntal guess of the knetc parameters has been llustrated along wth the successve multvarate least square algorthm for the fnal dentfcaton of the knetc trplet [A E f()] better descrbng the system under study

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