Approximation Expressions for the Temperature Integral
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1 20 7Π PROGRSS IN CHMISRY Vol. 20 No. 7Π8 Aug., ( ),,,, : O64311 ; O64213 : A : X(2008) 07Π Approimation pressions for the emperature Integral Chen Haiiang 3 3 Liu Naian 3 3 (State Key Laboratory of Fire Science, University of Science and echnology of China, Hefei , China) Abstract he recent ten2year research progress in the approimation epressions for the temperature integral in the field of thermal analysis kinetics is reviewed. Some new approaches to approimate the temperature integral are introduced, and the relationship between various approimation epressions is presented. he accuracy in calculating kinetic parameters by different integral methods with different approimation epressions is commented, and the methods to avoid the problem of the temperature integral are summarized. he research tendency about the temperature integral is also suggested. Key words temperature integral ; approimation epression ; thermal analysis kinetics, ( ), [1 ],,, [2, 3 ],, : Arrhenius,,, ( ICA) Flynn , [4 ] 1997,3 : [5 ],,,,, : , : (No ) 3 3 e2mail edu. cn ; edu. cn
2 ,, 1997 Flynn [5 ], 10,,,,, 1 10,,,,, : d = k ( ) f ( )Π (1) d,, f ( ), k ( ), Arrhenius : k ( ) = Aep ( - ΠR) (2),A, R (1) (2), 0 0 ( 0 ), : 0 f ( ) d = G( ) = A e - ΠR 0 d (3) e - ΠR d 0, 0,, 0 0, e - ΠR d = ΠR, 0 e - ΠR d = 0 2 p ( ), p ( ) = R ep ( - ) d h ( ), h ( ) = p ( ) 2 e p ( ), h ( ) 0 1, [6 ], p ( ) h ( ), k ( ) : k ( ) = A m ep ( - ΠR),, I ( m, ) = m ep ( - 0 p ( m, ) ( p ( m, ) = ) ( h ( m, ) = p ( m, ) ΠR) d ep ( - ) d) m +2 h ( m, ep ( - ) m+2 m 1Π2 1 2,0 Arrhenius 2, Scholmilch MKN [7 ], Coats2Redfern [8 ] Doyle [9 ], [10, 11 ],,Heal,,,,,,, Madhusudanan [12, 13 ] MKN : ln[ p ( ) ] = ln ln[ p ( ) ] = ln ln[ p ( ) ] = ln (4) MKN,MKN 2 3 ang [14 ], 9ln[ p ( ) ] ln[ p ( ) ] = c + b + aln, 9 = b + a 9ln[ p ( ) ] 9 1 b a ;
3 7Π a,ln[ p ( ) ] - aln = c + b,c b: ln[ p ( ) ] = ln (5),, b, (b ), [15 ] b,, Doyle [9 ] (ln[ p ( ) ] = (20 60) ),, MKN,Doyle 20 60, Gao [16 ] , h ( ) = [22 ] ln ln ,( ),,,, Chen [6], h ( ): h ( ) = h ( ) - 1 [23], h ( ), 0, h ( ), h ( ) = h ( ) - 1 Grobachev2Lee2Beck [24,25 ] h ( ) = Doyle, Π 1 + 2, p ( ) = ep ( ) Cai [17 ],,h ( ) = 3 2 [6 ] ( 215 %),, ( Starink [18 ] MKN Doyle Senum2Yang 4 [26 ] :, : ep ( - A + B) h ( ) = p ( ) =, k A B k ) Orfao [27 ] MKN Doyle (, ang Starink 2 [26 ]), ( A, B, k) : (1, ,1195) (110008, , 1192),,,,,Senum2Yang 4, [26 ],, ang [19 ],Orfao, h ( ) = :, ;,,Cai,, [20 ] h ( ) = [21 ] h ( ) =
4 ,, ( A ),, (3) : ln G( ) = ln A R ln A R + ln[ p ( ) ] = + ln[ h ( ) ] - 2ln - (6) ln[ h ( ) ], : A ln[ h ( ) ] ln A R,, (6) ln G( ) = ln AR h 2 R - R, ;B ln[ h ( ) ] - 2ln -,ln[ h ( ) ] ln[ h ( ) ] = a ln + b + c, ln[ p ( ) ] = aln + b + c, (6) ln G( ) - a = c + ln A R R a + b R, A Coats2Redfern [8 ] Agrawal [28 ],B Doyle [9 ] MKN [12, 13 ],ang [14 ] B, A, Ortega [29 ] Coats2Redfern, 10 2 %,Coats2Redfern Liu [23 ] A B, A, [23,30 ],, [31 ] Liu B, MKN (ang [14 ] ),Doyle Perez2Maqueda [32,33 ],,A, ;B ln[ p ( ) ] = aln + b + c,,,, 4,? p ( ) = ep ( - A + B) k, [34 ] h ( ) = p ( ) 2 e, : ln[ h ( ) ] = (2 - k) ln + (1 - A) + B, h ( ) h ( ) = 2 - k h ( ) = h ( ) = + (1 - A) h ( ) h ( ) - 1 [23 ], A + k : y = A + k ( y = h ( ) ) ( A, k) [15,60 ], ( A, k) = ( , ), p ( ) = ep ( - A + B) k B, B,B, B, ang [14 ],c,,,
5 7Π8 1019, B,B A [23 ] 5, Singh [35 ],,ang [36 ] [19 ],, : I ( m, ) = I ( m, ) = ep - R R m+2 + ( m + 2) R ep - R R m+2 + ( m + 2) ( R) 1976 [37 ] Chen [38 ] h ( ) = h ( ) - 1 [6 ] h ( m, ) = 1 + m + 2 [47 h ( m, ) - 1, Serra 50 ] NPK ( non2 parametric kinetics), k ( ),, h ( m, ) = Arrhenius 1 + m m + 2 ( m + 1) ( m + 2) + 2, Cai [39 ],h ( m, ) a + bm + c + dm + e,,, [40 ],,,,,,, k ( ) k ( ) = A m ep ( - ΠR),,, k ( ) = A m ep ( - 6 ΠR),? Arrhenius,3 Arrhenius,,,, [2, 3, ] Arrhenius, [44, 45 ] k ( ) Dollimore Harcourt2sson (H2) : k ( ) = B m Simon [46 ] Berthelot2Hood (B2H) [5 ] : k ( ) = Ce D, B m CD,, A k ( ) d, (1) : = dt k ( ) f ( ), 7,,,,,,,,,, Flynn [5 ] :
6 Indeed, in this age of vast computational capabilities, there is no valid reason not to use precise values for the temperature integral when calculating kinetics parameters. [ 1 ] (Lu Z R). (Chinese Journal of Inorganic Chemistry), 1998, 14 : [ 2 ] Chen H X, Liu N A, Shu L F, et al. Journal of hermal Analysis and Calorimetry, 2004, 78 : [ 3 ] Caballero J, Conesa J. Journal of hermal Analysis and Calorimetry, 2005, 73 : [ 4 ] Flynn J H. hermochimica Acta, 1992, 203 : [ 5 ] Flynn J H. hermochimica Acta, 1997, 300 : [ 6 ] Chen H, Liu N. AICh J., 2006, 52 : [ 7 ] (Hu R Z), (Shi Q Z). (hermal Analysis Kinetics ). Press), 2001 : ( Beijing : Science [ 8 ] Coats A W, Redfern J P. Nature, 1964, 201 : [ 9 ] Doyle C D. J. Appl. Polymer Sci., 1961, 5 : 285 [10] Heal G R. hermochimica Acta, 1999, 341 : [11] Heal G R. Instrumentation Science & echnology, 1999, 27 : [12] Madhusudanan P M, Krishnan K, Ninan K N. hermochimica Acta, 1986, 97 : [13] Madhusudanan P M, Krishnan K, Ninan K N. hermochimica Acta, 1993, 221 : [14] ang W J, Liu Y W, Zhang H, et al. hermochimica Acta, 2003, 408 : [15] Liu N, Fan W. hermochimica Acta, 1999, 338 : [16] Gao Z M, Nakada M, Amasaki I. hermochimica Acta, 2001, 369 : [17] Cai J M, Liu R H, Shen F. Journal of Mathematical Chemistry, 2008, 43(3) : [18] Starink M J. hermochimica Acta, 2003, 404 : [19] Wanjun, Yuwen L, Hen Z, et al. Journal of hermal Analysis and Calorimetry, 2003, 74 : [20] Cai J M, Yao F S, Yi W M, et al. AICh J., 2006, 52 : [21] (Cai J M), ( Yi W M), ( He F). (Chemistry), 2006, 4 : [22] Cai J M, He F, Yi W M, et al. Chemical ngineering Journal, 2006, 124 : [23] Liu N, Chen H, Shu L, et al. Journal of hermal Analysis and Calorimetry, 2005, 81 : [24] Gorbachev V M. Journal of hermal Analysis, 1975, 8 : [25] Lee V, Beck S R. AICh J., 1984, 30 : [26] Senum J I, Yang R. Journal of hermal Analysis, 1977, 11 : [27] Orfao J J M. AICh J., 2007, 53 : [28] Agrawal R K. AICh J. 1987, 33 : [29] Ortega A, PerezMaqueda L A, Criado J M. hermochimica Acta, 1996, 283 : [30] Chen H X, Liu N, Shu L F. Polymer Degradation and Stability, 2005, 90 : [31] Gao Z M, Wang H X, Nakada M. Polymer, 2006, 47 : [32] Perez2Maqueda L A, Sanchez2Jimenez P, Criado J M. International Journal of Chemical Kinetics, 2005, 37 : [33] Perez2Maqueda L A, Sanchez2Jimenez P, Criado J M. Polymer, 2005, 46 : [34] Chen H X, Liu N A. Journal of hermal Analysis and Calorimetry, 2008, 92 : [35] Singh S D, Devi W G, Singh A K M, et al. Journal of hermal Analysis and Calorimetry, 2000, 61 : [36] Wanjun, Yuwen L, Xil Y, et al. Journal of hermal Analysis and Calorimetry, 2005, 81 : [37] Gorbachev V M. Journal of hermal Analysis, 1976, 10 : [38] Chen H X, Liu N A. Journal of hermal Analysis and Calorimetry, 2007, 90 : [39] Cai J M, Liu R H. Journal of hermal Analysis and Calorimetry, 2007, 90(2) : [40] Cai J M, Liu R H, Wang Y Y. Solid State Sciences, 2007, 9 : [41] Liu N, Chen H X, Shu L F, et al. Industrial & ngineering Chemistry Research, 2004, 43 : [42] Sbirrazzuoli N. hermochimica Acta, 1996, 273 : [43] Varhegyi G, ill F. hermochimica Acta, 1999, 329 : [44] Dollimore D, ong P, Aleander K S. hermochimica Acta, 1996, 283 : [45] Dollimore D, Lerdkanchanaporn S, Aleander K S. hermochimica Acta, 1997, 290 : [46] Simon P. Journal of hermal Analysis and Calorimetry, 2005, 79 : [47] Sempere J, Nomen R, Serra, et al. AFINIDAD, 2005, 62 : [48] Serra R, Sempere J, Nomen R. hermochimica Acta, 1998, 316 : [49] Sempere J, Nomen R, Serra R, et al. hermochimica Acta, 2002, 388 : [50] Sempere J, Nomen R, Serra R. Journal of hermal Analysis and Calorimetry, 1999, 56 :
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