Critical consideration on the Freeman and Carroll method for evaluating global mass loss kinetics of polymer thermal degradation

Transcription

1 Thermochimica Acta 338 (1999) 85±94 Critical consieration on the Freeman an Carroll metho for evaluating global mass loss kinetics of polymer thermal egraation N.A. Liu *, W.C. Fan State Key Laboratory of Fire Science, University of Science an Technology of China, Hefei, Anhui , PR China Receive 27 January 1999; accepte 22 June 1999 Abstract By means of regression analysis, an insight into the limitations of the wiely use Freeman an Carroll metho is offere, an it is inicate that the Freeman an Carroll metho is essentially unstable for the etermination of the reaction orer. The reliabilities of the intercept an slope terms for any one regression equation are also investigate mathematically, suggesting that the slope term is in general easily to be etermine accurately, whereas the intercept term has possibility to be obtaine with great error. Base on these theoretical analyses, a new metho to calculate the kinetic parameters for polymer thermal egraation is propose, in which the three kinetic parameters are etermine from the slopes of three regression equations escribe in this paper. The experimental ata in the literature are use to verify the propose metho. By comparing the results from the Freeman an Carroll metho an the propose metho respectively with those from the Doyle metho, it is shown that the propose metho can be applie with a much better accuracy than the Freeman an Carroll metho. # 1999 Elsevier Science B.V. All rights reserve. Keywors: Dynamic thermogravimetry; Freeman an Carroll metho; Unstable regression equation; Thermal egraation; Kinetic parameter 1. Introuction During the past several ecaes, ynamic thermogravimetry, in which changes in weight are measure as a function of increasing temperature, has been wiely use to stuy the kinetics of thermal ecomposition reactions. Dynamic thermogravimetry has attracte wiesprea interest mainly ue to its explicit avantages over isothermal methos. As pointe out * Corresponing author. Tel.: ; fax: aress: liunai@mail.ustc.eu.cn (N.A. Liu) by Doyle [1], one ynamic mass-loss curve is equivalent to a large number of isothermal mass-loss curves. Other avantages of ynamic thermogravimetry have been summarize by Coats an Refern [2] an Wenlant [3]. With the evelopment of ifferent kinetic analysis methos, ynamic thermogravimetry has now been playing a very important role in the els such as fuel property, re research, fabric ammability, nuclear weapons effects, waste incineration, an aerospace technology, etc. It is known that a wie range of practical problems is associate with the polymer thermal egraation. For example, the pyrolysis of biomass is relate to the /99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserve. PII: S (99)

2 86 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 self-heating of cellulosic materials, the behavior of woo in re, the role of inorganic salts as re retarant for timber, etc. For the polymer thermal egraation, there has been an ongoing ebate in the literature concerning the global kinetics of polymer pyrolysis. Global kinetics are of interest in moeling polymer ecomposition in many applications in which trying to represent the full complexity of the polymer egraation process makes no sense. In general, the global kinetics of the pyrolysis are looke on as offering a clue to the key mechanistic steps in the overall weight loss process. In the case of polymer egraation, global kinetic analysis of thermogravimetric ata from a weight loss curve is base on the formal kinetic equation t ˆ k 1 n (1) where is the fraction of material ecompose at time t, n is the orer of reaction, an k the rate constant. The combination of this equation an the empiric Arrhenius expression gives the following relationship: t E ˆ Aexp RT 1 n (2) where A is the frequency factor, E is the activation energy of the reaction, an R the gas constant. Introucing the heating rate,, T ˆ A e E=RT 1 n (3) This is the funamental expression of analytical methos to calculate kinetic parameters on the basis of TG ata. Base on this expression, a consierable number of methos to erive kinetic parameters E, n, an A from thermogravimetric curves have been evelope. Accoring to the kin of ata use, the methos in general fall into two classes: integral metho an ifferential metho. Integral metho utilizes weight loss versus temperature ata irectly, while ifferential metho uses the rate of weight loss versus temperature. The most wiely use integral kinetic methos are those evelope by (a) Doyle [1], (b) Coats an Refern [4], (c) Zsako [5], () Broio [6], (e) Reich an Stivala [7], an (f) Ozawa et al. [8], an the ifferential methos in common use are those erive by (g) Freeman an Caroll [9], (h) Newkirk [10], (i) Achar et al. [11], (j) Vachuska an Voboril [12], an (k) Frieman et al. [13] From their mathematical erivations, it can be conclue that the methos (a),(b),(c),(e) an (i) can be applie only when the reaction orer is priorly known, while the methos () an (h) may be use only when the reaction is of rst orer. In other wors, for these methos the priori knowlege of the reaction mechanism is generally neee for calculating the kinetic ata. However, it is just such a limitation that usually leas to great trouble in ealing with the thermogravimetric curve, because the reaction mechanism is in general not able to be etermine in avance. In fact, the techniques to obtain the kinetic parameters when the reaction orer (or mechanism) is unknown have been a subject of much concern over the past several ecaes. The metho (j) evelope by Vachuska an Voboril [12] can be use to etermine the kinetic parameters E an n simultaneously. However, this metho involves the etermination of 2 /t 2, which cannot be obtaine reaily an accurately from the weight loss curve. Actually, such a isavantage strongly weakens the accuracy of this metho, an few applications of it can be foun in literature. The methos (f) an (k) evelope respectively by Ozawa an Frieman have been successfully use to stuy polymer thermal ecomposition kinetics, without any priori knowlege of the reaction mechanism. The common isavantage of the two methos is the requirement of several experiments with ifferent heating rates. Besies the Ozawa metho an the Frieman metho, the metho evelope by Freeman an Carroll (FC) [9] is another one wiely use without any priori knowlege of the reaction orer. The reaction orer n (or mechanism) an the activation energy E can be etermine from the equation E=R D 1=T Dln 1 ˆ Dln =t n (4) Dln 1 It is apparent that by plotting a graph of Dln (/t)/ Dln (1 ) versus DT 1 /Dln (1 ), the parameters E an n can be calculate respectively from the intercept an the slope of the regression line (in what follows we call it the FC regression line). However, in most practical instances it was foun that there is consierable if culty in obtaining a reliable value of the reaction orer n by the FC regression line [2]. By

3 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 87 means of the thermal ecomposition of calcium carbonate, Sharp an Ventworth [14] also prove the invaliation of the FC metho in etermining the reaction orer n. During their investigations it was foun that the obtaine reaction orer values sensitively uctuate with the number of the points selecte for tting the straight line. When the number of points was eight, the obtaine reaction orer was +0.55, whereas when 14 points were use, the reaction orer was obtaine to be In aition, the FC metho cannot be use irectly to etermine the pre-exponential factor A. These eviences hint that the FC metho can only be use to etermine the activation energy of the reaction accurately. With the above limitations of the FC metho in min, this stuy seeks to evelop a new metho for eriving the kinetic ata E, n, an A accurately, without any priori knowlege of the reaction mechanism. We begin with a theoretical attempt to unerstan the reason for the limitations of the FC metho in Section 2. Base on the theoretical analysis, in Section 3 we formally escribe three equations which make up the new metho. One of the equations is just the Eq. (4) erive by Freeman an Carroll. The parameters E, n, an A can be etermine accurately from the slopes, rather than the intercepts, of the regression lines accoring to the three equations respectively. The veri cation of the propose metho appears in Section 4, an Section 5 is a summary an conclusion section. 2. Insight into the limitations of the FC metho As pointe out in the last section, there exists lack of consistency in calculating the reaction orer n from the intercept of the FC regression line. In the literature, this isavantage has been consiere to arise mainly ue to the ranom error in the slope evaluation uctuation [15]. However, no insight was offere into the sensitivity of the resulte reaction orer value to the slope evaluation uctuation until now, an it has been presume that the FC metho can be accurate on conition that the slope is etermine with satisfactory accuracy. By contrast, this section seeks to inicate in terms of theoretical analysis that the regression Eq. (4) is in nature unstable for etermining its intercept value. Due to this instability, a small error of the slope estimator may lea to a great error of the intercept evaluation, so that the reaction orer cannot be etermine accurately by the FC metho in essence, so long as any experimental uctuation exists. For convenience in the present iscussion, we begin with a brief erivation of the FC metho. Taking logarithms of both sies of Eq. (3), we have ln T ˆ ln A E nln 1 (5) RT Suppose another temperature, referre to as T 0,is ranomly selecte from the thermogravimetric curve. By analogy to Eq. (5) we obtain ln 0 T ˆ ln A E RT 0 nln 1 0 (6) where 0 is the fraction of material ecompose at temperature T 0, an here for the convenience of representation, (/T) T=T 0 has been expresse simply as 0 /T. Subtracting Eq. (6) from Eq. (5) yiels Dln T ˆ E R D 1 ndln 1 (7) T Diviing both sies by Dln(1 a), we have Dln =T A Dln 1 ˆ E D 1=T RDln 1 n (8) This expression is just the one use in the FC metho (see Eq. (4)), but the erivation here is more concise than that of Freeman an Carroll. This equation is use in the propose new metho to etermine the activation energy E. Hence, we mark this equation by (A) for sake of emphasis. As has been explaine, ealing with the weight loss curve by the FC metho comes own to tting a straight line in terms of this equation. In a strictly mathematical sense, the FC regression line may be expresse as follows: Y ˆ ^a ^bx (9) where X an Y are two variables relate to each other by the least squares metho, while aã an ^b are two estimators. Corresponing to Eq. (8), we have the following relations: X ˆ D 1=T (10) RDln 1

4 88 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 Y ˆ Dln =T Dln 1 (11) E ^a ˆn (12) E ^b ˆE (13) Here the notation E( ) enotes the preicte value of the brackete term. The terms aã an ^b are respectively, the estimators of n an E. For a speci c thermogravimetric experiment, samples of the ata point (X, Y) are measure an calculate ranomly from the weight loss curve. In this sense, X an Y can be looke on as two observe variables, an it can be assume that the two variables follow normal istribution. In general, ifferent samples of (X, Y) lea to ifferent values of aã an ^b. In other wors, the estimators aã an ^b will uctuate when the use sample changes from one to another. If the uctuations of aã an ^b both result to be small compare with their respective targets, i.e. n an E, the Eq. (9) may be referre to as a stable regression equation. If not, we call Eq. (9) an unstable regression equation, an the estimator with great uctuation will be unreliable. In mathematics, the uctuations of aã an ^b can be formulate by their stanar eviations [16]: s P N iˆ1 X2 i ^a ˆ N P N iˆ1 X i X 2 (14) ^b ˆ q P (15) N iˆ1 X i X 2 where is the stanar eviation of Y aroun the regression line. We e ne the relative errors of aã an ^b as the ratios of aã an ^b respectively to E(aÃ) an E(^b), an expresse them as ~^a an ~^b ~^a ˆ n ~^b ˆ s P N iˆ1 X2 i N P N iˆ1 X i X 2 P N iˆ1 X i X 2 (16) q (17) E The orers of magnitue of these two errors cannot be evaluate theoretically in a general sense. However, for the thermal ecomposition of soli materials, the orer of ~^b can be inferre from the literature. So far, some stuies have been evote to the polymer thermal egraation, in which the FC metho an several other kinetic methos were use simultaneously in orer to obtain relatively reliable kinetic ata. It seems that for a speci c experiment all these methos always resulte in approximately the same activation energy values. This hints that, for most cases, the activation energy E can be estimate by the FC metho accurately, with a quite small error ~^b obtaine. A recent reference of this kin is that by Huang an Li [17]. By means of four kinetic methos, Huang an Li have obtaine the kinetic parameters for cellulose an cellulose esters at 108C/min in nitrogen, an it was foun that the E values by the FC metho iffere little from those by other three methos. Another reference of this kin is that of Sharp an Ventworth [14], as has been state in Section 1. During their investigation, the great uctuation of the reaction orer value by the FC metho hinte at a very big orer of ~^a. However, with ifferent numbers of points use, the obtaine E values by the FC metho were also foun to iffer not too much from those by other two methos. In fact, there has been a substantial agreement that the FC metho can be use in the etermination of the E values with satisfactory accuracy, an an investigation of the literature reveale that the obtaine E value is generally in error by 5±15%, that is to say, ~^b 5± 15%. Here an in the following iscussion we use the symbol `' to mean `the orer of magnitue is equal to'. Combining Eqs. (16) an (17) yiels the following relation ~^a ~^b ˆ E n s P N iˆ1 X2 i N (18) Taking Eq. (10) into account, Eq. (18) can be rewritten as the following expression: ~^a ED 1=T (19) ~^b nrdln 1 For simplicity, let =1 y. Here y is the fraction of material not yet ecompose at time t. To evaluate the orer of magnitue of ~^a =~^b, we e ne two functions z 1 (T, T 0 ) an z 2 (y, y 0 )as z 1 ˆ 1 1 R T 1 T 0 300T; T (20)

5 z 2 ˆ ln y ln y 0 "y; y 0 1:0 (21) Eq. (19) can then be rewritten as N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 89 ~^a ~^b Ez 1 nz 2 (22) In general, the thermal ecomposition is carrie out in an inert atmosphere from room temperature to about 8008C, meanwhile the fraction of material not ecompose ecreases from 1.0 to " continuously. Here " is the fraction not yet ecompose when the temperature has been increase up to 8008C. In ealing with the weight loss curve by the FC metho, the temperatures T an T 0 are selecte ranomly an inepenently, an the corresponing fractions y an y 0 are etermine from the curve. Due to this proceure, the fractions y an y 0 can be looke on as two quantities inepenent of each other, with possible values ranging from " to 1.0 for each of them. Fig. 1 shows the graph of z 1 versus T for various values of T 0. Inspection of Fig. 1 inicates that for any xe value of T 0, z 1 (T) is on the orer of 10 5 ±10 4 over most of the interval of T. State in another way, for any two temperatures T an T 0 selecte ranomly from the weight loss curve, the following relation hols in almost all cases: z 1 T; T (23) In an analogous manner, the plot of z 2 versus y for various values of y 0 shown in Fig. 2 reveals that z 2 y; y (24) Fig. 2. Plot of z 2 (y) vs. y for various values of y 0. is vali uner general circumstances. Combining Eqs. (23) an (24) the orer of magnitue of z 1 /z 2 can be obtaine as the following expression: z (25) z 2 It has been shown in the literature that the activation energy E is generally on the orer of 10 4 ±10 5 J/mol, an the reaction orer n 1. An investigation of the literature reveale that almost all the stuies with few exceptions have supporte these orers of magnitue. Some papers of this kin in recent years are those by Huang an Li [17], Jimenez et al. [18], Chang [19], Antal an Varhegyi [20], an Milosavljevic an Suuberg [21]. Taking this fact into account, substituting Eq. (25) into Eq. (22) yiels: ~^a ~^b (26) Recalling the fact that ~^b 5±15%, we can then obtain the following evaluation of the error term ~^a : ~^a 0:5 1500% (27) We note in passing that since y is epenent on T, Eq. (27) is just an evaluation of the orer of magnitue of z 1 /z 2 in a relatively rough sense. It can be inferre from the literature that uner real situations Z 1 /Z 2 has generally an orer of magnitue of 10 5 ±10 4 (see, e.g. [14,18]), an therefore a more accurate evaluation of ~^a can be obtaine: Fig. 1. Plot of z 1 (T) vs. T for various values of T 0. ~^a 0:5 150% (28)

6 90 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 Eqs. (27) an (28) both imply that when ifferent sets of ata points (T, ) an (T 0, 0 ) are selecte stochastically from the weight loss curve, the values of ~^a woul uctuate from less than 1% to over 100%. It has been state that aã is the estimator of the reaction orer n. Therefore, it has been now clear that the FC metho may lea to a great error for the n value, even if the error of the E value remains small. In mathematics, this result reveals that the regression Eq. (9) is in itself unstable for the etermination of its intercept term. Just for the instability, the obtaine value of the reaction orer generally uctuates greatly with the number of points use, as state previously. Due to the inescapability of the experimental an measure uctuations, the error of the E value cannot be reuce to extremely small. Hence, it can be conclue that the FC metho cannot be use to etermine the reaction orer accurately in essence. Only by improving the FC metho itself can the reaction orer be expecte to be etermine with satisfactory accuracy. 3. Propose metho to obtain the kinetic ata Investigating the literature, it is interesting to note that although the intercept term of the FC regression equation showe weak reliability, almost all the regressions ha acceptable correlation coef cient values. Before proceeing further into the etaile escription of the new metho, let us pause here to examine the regression reliabilities of the slope an intercept terms for any one general regression equation, unerlying the interpretation of the principle of the new metho. For a set of observe ata (X 1, Y 1 )(X 2, Y 2 ),..., (X N, Y N ), suppose the true relation between X an Y is Y = a + bx, while the regression line is Y = aã + ^bx.by comparing the e nition of the correlation coef cient r XY with the expression of ^b, the following relation can be achieve [22]: ( P ) Yi Y ^b 2 1=2 ˆ P Xi X 2 r XY (29) where an in what follows summations are all over i = 1,2,...,N. Combining Eq. (15) an (29) allows us to obtain the following expression: ^b ^b ˆ r XY q P (30) Yi Y 2 This expression is the approximate relative error of the estimator ^b. In practice we cannot know the value of 2, which is the variance of Y about the true line. Nevertheless, it can be estimate by the following unbiase eviation about the tte line [23]: s 2 ˆ 1 X Yi ^Y i 2 (31) N 2 where ^Y i is the fitte value of Y on the fitte line: ^Y i ˆ ^a ^bx i (32) The summation term in Eq. (30) can be expresse by the following equality [22]: X Yi Y 2 ˆ X ^Y i Y 2 X Y i ^Y i 2 (33) Finally, we can rewrite the approximate relative error of ^b by substituting Eqs. (31) an (33) into Eq. (30). Thus q P ^b ˆ 1 Yi ^Y i 2 p q ^b N 2 P ^Y i Y 2 P Y i ^Y i 2 r XY (34) It is clear that for a r XY close to unity, the error of the slope term ^b woul besmall on conition that thenumber N is big enough. Even for a small value of N, there is still possibilitythat a value of^b be achieve with minor error, ue to the reuction effect of the later fractional term in the above equation. This equation provies a goo interpretation of the reason for which the activation energy value resulte from the FC metho usually has a reasonable accuracy. As state previously, in the existing applications of the FC metho, the values of r XY were usually foun to be close to unity, implying that the variables X an Y for the FC regression line ha a closely positive relation. In view of this fact, reasonable accuracy of the activation energy E by the FC metho is not surprising. The more important conclusion rawn from the above equation is that, if the parameter being examine appears in the slope term of any one general regression equation, this parameter woul be expecte likely to be etermine accurately. This conclusion unerlies the theory of the new metho.

7 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 91 Fig. 3. Illustration of the unreliability of the intercept term for a general regression equation. It shoul be pointe out, however, the intercept term for a general regression equation has not such a goo feature. Fig. 3 is use here to make the point. This gure shows a general regression. Suppose the true regression E(Y) = a+bx is the ashe line shown in the gure. This is unknown to the statistician, who must estimate it as accurately as possible by observing X an Y. Suppose the estimate regression is Y ˆ ^a ^bx. The correlation coef cient r XY is assume to be close to unity, hinting at a closely positive relation between X an Y. Our emphasis here is on the reliability of the tte intercept term aã. In Fig. 3, two coorinate systems are use to illustrate that a ata set (X 0, Y 0 ) with unreliable tte intercept value can be reaily constructe, even if the sample correlation coef cient between X 0 an Y 0 is close to unity. It's evient that the relation of the two coorinate systems is X 0 ˆ X L (35) Y 0 ˆ Y (36) In the space (X 0,Y 0 ), the true an the estimate regression lines can be expresse by X 0 an Y 0, respectively as an Y 0 ˆ a bl bx 0 (37) Y 0 ˆ ^a ^bl ^bx 0 (38) Combining these two equations, the error of the estimate intercept in this space can be obtaine: ^a 0 a 0 ˆ ^a a b ^b L (39) It can be seen from Fig. 3 that the quantity aã a hasthe same sign as b ^b. Hence, the error of the estimate intercept in the space (X 0,Y 0 ) is enlarge compare with that in the space (X,Y). It's obvious that if L is taken to be bigenough,thequantityaã 0 woul nallybecometoiffer from a 0 so much that the quantity aã 0 a 0 / a 0 woul excee unity, hinting at a great error of the estimate intercept. State in another way, the ata set (X 0,Y 0 ), i.e. (X + L,Y), woul lea to a regression line with an unreliable intercept term. During this proceure, the correlation coef cient r XY remains xe, which can be justi e by referring to the e nition of r XY. Hence, a conclusion can be rawn from this analysis, that is, with the correlation coef cient approximating unity, the intercept term is still likely to be greatly unreliable. Base on the above arguments, we now procee to escribe the metho for the eterminations of the reaction orer an the pre-exponential factor. In view of the above features of the slope an intercept terms for a general regression equation, the FC metho may be improve upon by eriving two regression equations whose slope terms respectively involve the reaction orer n an the pre-exponential factor A. The values of n an A can then be etermine from the slope terms of the two equations. Base on this iea, we rst refer back to Eq. (7) an erive the equation for the reaction orer n. Rearrangement of Eq. (7) yiels the following equation: Dln 1 B Dln =T ˆ E D 1=T n RDln =T 1 n (40) The use of its slope term, i.e. E/n, is therefore an essentially better way to calculate the reaction orer value, on conition that the activation energy E has been obtaine using the FC regression equation. The above equation is marke by (B). Although the erivation is trivial, this regression equation is of major importance for the accurate etermination of the reaction orer n. In fact, for the soli material thermal egraation, the following orer of magnitue is generally achieve (see, e.g. [14,18]): Dln T 1 10 (41) Due to the fact that D(1/T)/R10 5 ±10 4 an E 10 4 ±10 5, the intercept term of Eq. (40) can be

8 92 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 shown, by the same analysis metho for the FC regression equation, to have a relative error of 1± 100 times the slope term. This result suggests that Eq. (40) is also unstable for the etermination of its intercept term. However, the use of its slope term is expecte to lea to an accurate n value, for the reason state before. The equation for the pre-exponential factor A can be erive base on the same iea. In oing so, we woul refer back to Eqs. (5) an (6). Diviing both sies of them respectively by ln (1 ) an ln (1 0 ), an then subtracting one from the other, we obtain ln =T D ˆ ln A ln 1 D 1 ln 1 E R D 1 (42) Tln 1 Diviing both sies by D(1/T ln (1 )) yiels the following equation: D ln =T Š=ln 1 Š C D 1= Tln 1 Š ˆ ln A D 1=ln 1 D 1= Tln 1 Š E R (43) Obviously the pre-exponential factor A can be obtaine from the slope term of this equation. We mark this equation by (C) for the sake of emphasis. This regression equation is worth a few other wors. It has been clear that using Eqs. (A) an (B) the activation energy E an the reaction orer n can be obtaine with permissible errors. However, when E an n have been obtaine, the pre-exponential factor A shoul not be estimate by selecting only one ata point (T,) from the thermogravimetric curve an substituting the ata an the values of E an n into Eq. (5). The reason is ue to the fact that the measure ata point (T,) an the obtaine estimators of E an n all contain errors, an using them to estimate A woul certainly lea to aitional error of the result. Another possible strategy for etermining A is using Eq. (5) irectly as the regression equation, in which the parameters of E an n are taken as the values obtaine by the methos escribe earlier. However, it's obvious that the errors of E an n will still affect the reliability of the obtaine value of A. For this reason, this strategy is also not recommene. By contrast, Eq. (43) is not irectly affecte by the errors of the values of E an n, an thus it woul be expecte to result in the value of A with less error. So far, we have complete the erivations of the three regression equations which consist of the new metho to obtain the kinetic information for thermal egraation reactions. Eqs. (A)±(C) are respectively use to etermine the activation energy E, the reaction orer n, an the pre-exponential factor A. This metho overcomes the isavantage of the FC metho in the theoretical sense, an therefore it is expecte to result in reliable values of the kinetic parameters. 4. Experimental verification of the propose metho In our earlier paper [24], the thermal ecomposition of six ifferent samples of woo an leaves in nitrogen has been stuie by using ynamic thermogravimetry. We give now one example for the veri cation of the above-explaine metho, using the experimental ata publishe in the paper. The sample we take here is the leaves of oil-tea tree, whose botanical name is Camellia oleifera, an the initial weight of this sample is 5.73 mg. The sample was collecte from Qimen forest zone in Anhui province of China. It was groun an the fraction passing 20 mesh was use for analysis. Thermal ecomposition was observe in terms of the overall weight loss by using a WRT-3 thermobalance mae by Shanghai Balance Instrument Plant. An atmosphere of ry nitrogen, oxygen-free, was passe into the furnace at a ow rate of 80 ml/min. The sample was use in a wie-mouth porcelain crucible lightly packe so as not to impee loss of volatile proucts. The temperature was increase from room temperature to 5008C at a rate of 158C/min. The buoyancy effect was consiere negligible, which was justi e by some other researchers, e.g., Simons et al. [25]. Using the Doyle metho, we have foun the following kinetic ata uring the temperature interval from 154 to 2758C: n ˆ 2; E ˆ 63:68 KJ=mol; A ˆ 1: min 1 : To verify the above-suggeste metho, we selecte several ata points ranomly from the weight loss curve, an tte three lines respectively accoring to

9 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 93 Fig. 4. Data calculate from the suggeste metho for (a) the activation energy, (b) the reation orer, an (c) the pre-exponential factor. The unit of T is K, an the unit of R is J/K mol. Note that plot (a) is just the one use in the FC metho. Eqs. (A)±(C). The regression lines are shown in Fig. 4. A irect test establishe that these plots showe very goo linearity. For sake of comparison, the kinetic parameters have been calculate respectively by the FC metho an the suggeste metho, an the results are shown in Table 1. It is apparent that a singular value of n, i.e. 1.11, has been achieve by the FC metho. Contrarily, the comparison of the values of E, n, an A by the propose metho with those by the Doyle metho has shown goo agreement. Table 2 is a Table 1 Kinetic parameters by ifferent methos E (KJ/mol) n A (min 1 ) FC metho ± Doyle metho Propose metho Table 2 Regression results for the FC metho an the propose metho FC metho Propose metho a (A) (B) (C) I b 1.11 I 0.64 S c S r a (A), (B) an (C) correspon respectively to Eqs. (A), (B) an (C). b Intercept term. c Slope term. summary of the regression results for the FC metho an the propose metho. With reference to the table we can see that all the correlation coef cients approximate unity, implying that the points in each plot of

10 94 N.A. Liu, W.C. Fan / Thermochimica Acta 338 (1999) 85±94 Fig. 4 are nearly in a line. Inspection of the stanar eviations reveale that only the FC regression line resulte in an error of its intercept being of the same orer as the intercept value itself, hinting at a great unreliability of the reaction orer obtaine by the FC metho. This result illustrate the invaliation of the FC metho in etermination of the reaction orer. On the other han, as is clear from Table 2, the stanar eviations for the suggeste metho are small compare with the corresponing estimator values, inicating that the suggeste metho has goo accuracy in the etermination of all the kinetic ata. 5. Concluing remarks Previous thermogravimetric stuies showe that the FC kinetic metho, for which no assumption about the reaction orer is neee, has great instability in the etermination of the reaction orer n. In this paper, an insight into this limitation has been offere by means of the regression theory, which suggests that the regression line use in the FC metho is in essence unstable for the calculation of the n value from the intercept term. Just for this instability, the n values by the FC metho often uctuate greatly with the number of points use for tting. For any one general regression equation, the reliabilities of the intercept an slope terms have also been investigate mathematically. It is shown that the slope term can be expecte to be etermine accurately on conition that the correlation coef cient is close to unity. Whereas even with the correlation coef cient approximating unity, the intercept term is still likely to be obtaine with great error. Base on this analysis, a new metho to etermine the three kinetic parameters E, A, an n accurately has been suggeste. In the propose metho, the three kinetic parameters are etermine respectively from the slopes of three regression lines. The comparison of the results respectively from the Doyle metho, the FC metho, an the propose metho has shown that the propose metho can be applie with a much better accuracy than the FC metho. The propose metho can be use without a priori knowlege of the reaction orer, which is neee for most of the other existing kinetic methos. Meanwhile, the propose metho removes the isavantages of the FC metho, an its statistical way of working up experimentalata insures the accuracy in estimating thekinetic parameters. The propose metho is therefore a reasonable improvement upon the FC metho, an it can be applie in the stuy of global mass loss kinetics of polymer thermal egraation. As a nal remark, this work also reveals that when any one new kinetic metho is evelope, the reliabilities of its resulte parameters shoul be investigate in etail, by either experimental or theoretical means. Only with reliable kinetic parameters can the suggeste metho be recommene in use for the kinetic ata of the thermal egraation reactions. Acknowlegements This work was sponsore by National Natural Science Founation of China uner Grant References [1] C.D. Doyle, J. Appl. Polym. Sci. 5 (1961) 285. [2] A.W. Coats, J.P. Refern, Analyst 88 (1963) 906. [3] W.W. Wenlant, Thermal Analysis, 3r e., Wiley, New York, 1986, p. 58. [4] A.W. Coats, J.P. Refern, Nature 201 (1964) 68. [5] J. Zsako, J. Phys. Chem. 72 (1968) [6] A. Broio, J. Polym. Sci. A-2(7) (1969) [7] L. Reich, S.S. Stivala, Thermochim. Acta 24 (1978) 9. [8] T. Ozawa, Bull. Chem. Soc. Jpn. 38 (1965) [9] E.S. Freeman, B. Carroll, J. Phys. Chem. 62 (1958) 394. [10] A.E. Newkirk, Anal. Chem. 32 (1960) [11] B.N.N. Achar, G.W. Brinley, J.H. Sharp, Proc. Int. Clay. Conf. Jerusalem 1 (1966) 67. [12] J. Vachuska, M. Voboril, Thermochim. Acta 2 (1971) 379. [13] H.L. Frieman, J. Polym. Sci. 6 (1963) 183. [14] J.H. Sharp, S.A. Wentworth, Anal. Chem. 41 (1969) [15] M.J. Antal Jr., H.L. Frieman, F.E. Rogers, Combust. Sci. Technol. 21 (1980) 141. [16] T.H. Wonnacott, R.J. Wonnacott, Regression: A Secon Course in Statistics, Wiley, New York, 1981, p. 29. [17] M. Huang, X. Li, J. Appl. Polym. Sci. 68 (1998) 293. [18] A. Jimenez, V. Berenguer, J. Lopez, A. Sanchez, J. Appl. Polym. Sci. 50 (1993) [19] W.L. Chang, J. Appl. Polym. Sci. 53 (1994) [20] M.J. Antal, G. Varhegyi, In. Eng. Chem. Res. 34 (1995) 703. [21] I. Milosavljevic, E.M. Suuberg, In. Eng. Chem. Res. 34 (1995) [22] N.R. Draper, H. Smith, Applie Regression Analysis, 2n e., Wiley, New York, 1981, p. 45. [23] T.H. Wonnacott, R. J. Wonnacott, Regression: A Secon Course in Statistics, Wiley, New York, 1981, p. 74. [24] N.A. Liu, W.C. Fan, Fire Mater. 22 (1998) 103. [25] E.L. Simons, A.E. Newkirk, I. Aliferis, Anal. Chem. 29 (1957) 48.