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Energy 233 (2021) 121194

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Energy

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Insight into master plots method for kinetic analysis of lignocellulosicbiomass pyrolysis

Laipeng Luo a, Zhiyi Zhang a, Chong Li a, Nishu a, Fang He b, Xingguang Zhang c,Junmeng Cai a, *, 1

a Biomass Energy Engineering Research Center, School of Agriculture and Biology, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240,People's Republic of Chinab School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo, Shandong, 255049, People's Republic of Chinac Department of Chemistry, School of Science, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai, 200093, People's Republic ofChina

a r t i c l e i n f o

Article history:Received 14 April 2021Received in revised form20 May 2021Accepted 6 June 2021Available online 14 June 2021

Keywords:Lignocellulosic biomass pyrolysisKineticsIsoconversional methodActivation energyMaster plots methodKinetic mechanism function

* Corresponding author.E-mail address: jmcai@sjtu.edu.cn (J. Cai).

1 http://biofuels.sjtu.edu.cn.

https://doi.org/10.1016/j.energy.2021.1211940360-5442/© 2021 Elsevier Ltd. All rights reserved.

a b s t r a c t

The determination of the kinetic triplet including the activation energy, frequency factor, and kineticmechanism function is a key objective in kinetic analysis of solid-state reactions like lignocellulosicbiomass pyrolysis. The master plots method is usually used to determine the kinetic mechanism functiononce the activation energies as a function of conversion are estimated from an isoconversional method. Acritical study of the master plots method has been performed by analyzing theoretically simulatedprocesses with varying activation energies and frequency factors. The accuracy of the resulting kineticmechanism function calculated by the master plots method is strongly dependent on the variation de-gree of the frequency factor with conversion and the conversion dependency of activation energy fromisoconversional methods. The utilization of the master plots method without considering the variationdegree of the frequency factor with conversion may result in misestimating kinetic mechanism function.

© 2021 Elsevier Ltd. All rights reserved.

1. Introduction

Energy is considered a keystone for the global development andeconomic growth of modern society. However, the increasingconsumption of energy during the last few years leads to theproblems of fossil energy resource depletion on their reservationand environmental pollution, and thus have made it vital to replacefossil fuels with alternative renewable resources of energy. Ligno-cellulosic biomass is found to be a promising alternative candidateand attained considerable attention in comparison to otherrenewable resources [1]. Crop residues are the main feedstocks oflignocellulosic biomass from agriculture [2]. China produces over890 million tons of crop residues annually [3], which can be con-verted into heat, biofuels, or bio-chemicals via various thermo-chemical conversion techniques [4,5]. Pyrolysis is the thermaldecomposition of biomass in the absence of oxygen at moderate

temperatures (usually 673e873 K), which can produce bio-oil,permanent gases, and charcoal [6]. Recently, China promised toachieve carbon emissions peak before 2030 and carbon neutralityby 2060, tightening its target to cut the emissions of greenhousegases [7]. Hence, to achieve this goal, pyrolysis conversion ofbiomass is considered a helpful technique as it is carbon neutral [8].

The kinetics of biomass pyrolysis is fundamental for the design,scale-up, optimization, and industrial application of biomass py-rolysis systems/processes [9,10]. One of the main objectives ofbiomass pyrolysis kinetics is to determine the kinetic triplet,including the activation energy, frequency factor, and kineticmechanism function [11e13].

The activation energy can be calculated by kinetic methods,which can be divided into model-fitting and isoconversionalmethods [14,15]. The use of model-fitting methods requires thepre-assumption of the kinetic mechanism function [16] and leadsto the ambiguity problem of the kinetic triplet [17]. The iso-conversional methods allow us to estimate the activation energyvarying with the degree of conversion without pre-assuming anyparticular form of the kinetic mechanism function [18,19]. Ofvarious isoconversional methods, the Flynn-Wall-Ozawa (FWO),

List of symbols

NomenclatureA frequency factor s�1 or min�1

E activation energy J mol�1 or kJ mol�1

T absolute temperature Kf(a) differential form of kinetic mechanism function

dimensionlessg(a) integral form of kinetic mechanism function

dimensionlessnd number of data points dimensionlessR universal gas constant 8.3145 J K�1 mol�1

RRMSE_A relative root mean square error of Aa/A0.5%RRMSE_f relative root mean square error of f(a)/f(0.5) %RRMSE_E relative root mean square error of Ea %R2 coefficient of determination dimensionlessa, b, c constantp1, p2, p3 constant

Greek lettersa degree of conversion dimensionlessda/dT conversion rate with respect to T K�1

da/dt conversion rate with respect to t s�1

b heating rate K s�1 or K min�1

AbbreviationsFWO Flynn-Wall-OzawaKAS Kissinger�Akahira�SunoseICTAC International Confederation for Thermal Analysis and

Calorimetry

Subscriptsa value related to the degree of conversion, acal calculated valuesim simulated valuei ith heating rate0.5 value corresponding to a ¼ 0.5

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

Kissinger-Akahira-Sunose (KAS), and Friedman methods are themost widely used methods because of their ease of use and theFriedman method can accurately determine the activation energy[20]. However, the FWO and KAS isoconversional methods werederived from integrating the Arrhenius equation by assumingconstant activation energy all over the reaction and approximatingthe temperature integral [21]. Hence, the use of FWO and KASisoconversional methods would lead to some substantial errors inthe estimation of the activation energies, especially for significantlyvarying activation energy processes [17,22]. On the other hand, theFriedman method uses instantaneous conversion rate data, whichare usually estimated by numerical differentiation of experimentaldata. Hence, the Friedman method may lead to numerical insta-bility and be very sensitive to experimental data noise [23]. In theavoidance of the above problems associated with the FWO, KAS andFriedmanmethods, a modified Friedmanmethodwas developed byLuo et al. [24]. The modified Friedman method can enable us tocalculate the activation energies of a chemical reaction processmore accurately than the FWO and KAS methods and reduce theeffect of the data noise effectively.

The kinetic mechanism function can be estimated by using themaster plots method coupled with the activation energies fromisoconversional methods [25e27]. The master plots method can beused for analyzing experimental data obtained under variousconditions [28]. It is performed by comparing the theoreticalmaster plots corresponding to various ideal kinetic mechanismfunctions with the experimental master plots derived from theexperimental kinetic data and kinetic parameters from isoconver-sional methods [29]. The comparison allows us to select the mostprobable kinetic mechanism function of the reaction process underinvestigation. Table 1 lists the common ideal kinetic mechanismfunctions including their mechanisms, symbols, and expressions. Inthe ICTAC (International Confederation for Thermal Analysis andCalorimetry) Kinetics Committee recommendations for performingkinetic computations on thermal analysis data from Vyazovkinet al. [14], the master plots method has been recommended for thedetermination of the kinetic mechanism function. The accuratedetermination of the kinetic mechanism function is very helpful forfurther kinetic and thermodynamic analyses [30e32].

In literature, there are many studies on the determination of themechanism function using the master plots method for kineticmodelling of solid waste thermal decomposition processes (see

2

Table 2). From Table 2, it can be obtained that (1) the FWO and KASmethods were the commonly used methods for the estimation ofactivation energies, and (2) the kinetic mechanism functions weredetermined from a simple graphical method coupled with the co-efficient of determination (R2) when applying the master plotsmethod.

Despite the popularity of the master plots method, the appli-cability and accuracy of the master plots method for the estimationof the kinetic mechanism function are not clear and require furtherinvestigation. This is the scope of the present paper. The structureof this paper is organized as follows: Section 1 gives the basic in-formation about the kinetics of biomass pyrolysis, the master plotsmethod and its application, and the aim of this study. Section 2 willprovide the general introduction of the isoconversional kineticmethods for the estimation of activation energies and the masterplots method for the determination of the kinetic mechanismfunctions. Section 3 will present the results of two type theoreti-cally simulated processes using the master plots method, anddiscuss the factors influencing the accuracy of the obtained kineticmechanism functions. Section 4 will show the examples of mis-estimating the kinetic mechanism function using master plotsmethod and mention a possible method to accurately estimate thekinetic mechanism function. Finally Section 5 will summarizeconcluding remarks.

2. Theory

2.1. Isoconversional kinetic methods

The activation energies as a function of conversion for a ther-mally activated solid-state reaction can be calculated from theisoconversional methods [43]. The FWO and KAS methods are themost commonly used conventional linear integral isoconversionalmethods [44]. With some assumptions and restrictions, the equa-tions of the FWO and KAS isoconversional methods for the esti-mation of the activation energies can be obtained [18]:

FWO method:

ln bi ¼ ln�Aa,EaR,gðaÞ

�� 5:331� 1:052

EaR,Ta;i

(1)

KAS method:

Table 1Common kinetic mechanism functions for thermally activated solid-state reactions.

Symbol Reaction mechanism Kinetic mechanism function

Reaction orderF1 First-order ð1 � aÞF2 Second-order ð1� aÞ2F3 Third-order ð1� aÞ3AutocatalyticB1 Autocatalytic reaction að1 � aÞMampelP2/3 Power law ð2 =3Þa1=2P2 Power law 2a1=2

P3 Power law 3a2=3

P4 Power law 4a3=4

NucleationA1.5 Avrami-Erofeev ð1:5Þð1 � aÞ½�lnð1� aÞ�1=3A2 Avrami-Erofeev 2ð1 � aÞ½�lnð1� aÞ�1=2A3 Avrami-Erofeev 3ð1 � aÞ½�lnð1� aÞ�2=3A4 Avrami-Erofeev 4ð1 � aÞ½�lnð1� aÞ�3=4Contracting geometryR2 Contracting cylinder 2ð1� aÞ1=2R3 Contracting sphere 3ð1� aÞ2=3DiffusionalD1 One dimensional diffusion 1=ð2aÞD2 Two-dimensional diffusion 1

½�lnð1� aÞ�D3 Three-dimensional diffusion (Jander equation) 3ð1� aÞ2=3

2½1� ð1� aÞ1=3�D4 Three dimensional diffusion (Ginstling-Brounshtein equation) 3

2½ð1� aÞ�1=3 � 1�

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

ln

biT2a;i

!¼ ln

�Aa,R

Ea$gðaÞ�� EaR,Ta;i

(2)

where a is the degree of conversion (dimensionless), b is theheating rate (K s�1), A is the frequency factor (s�1), E is the acti-vation energy (J mol�1), T is the absolute temperature (K), R is theuniversal gas constant (8.3145 J K�1 mol�1), g(a) is the integral formof kinetic mechanism function (dimensionless), the subscript i

Table 2Some kinetic studies of biomass pyrolysis based on the master plots method.

Feedstock Estimation method of Ea Kinetic mec

Extracted oil from oil sludge FWO and KAS DevolatilizaDecomposit‘F3’Thermal crachars: ‘D1’

Coconut shell FWO, KAS and Friedman a<0.4: ‘F1’a > 0.7: ‘R6

Plum stone KAS and Friedman Empirical kCellulose Friedman ‘B1’Rice straw KAS and FWO 0.3 < a < 0.

a > 0.5: ‘D2Garlic husk KAS and FWO a < 0.2: F1

0.2 < a < 0.Waste tires KAS and FWO Empirical kPhragmites communis FWO Stage 1: ‘P4

Stage 2: ‘P3Stage 3: ‘R2

Peanut shell Friedman Stage 1: ‘F3Stage 2: ‘F1Stage 3: ‘F3

Green corn husk FWO ‘D3’Prosopis juliflora (PJ) and Lantana

camara (LC)Friedman, M-Friedman, FWOand KAS

‘F3’ for PJ‘D3’ for LC

3

represents the ith heating rate and the subscript a is the valuerelated to the degree of conversion, a.

For the FWO method, at a given a, the plot of ln bi vs. 1=Ta;i atvarious heating rates would be a straight line whose slope is�1:052 Ea

R . Similarly, for the KAS method, at a given a, the plot of

ln

biT2a;i

!vs. 1=Ta;i at various heating rates has slope of � Ea

R . The

schematic diagrams of the FWO and KAS isoconversional methods

hanism function Statistical measure forf(a)

Reference

tion of water and light hydrocarbons: ‘D1’ R2 [26]ion of medium molecular weight components:

cking of heavy carbonaceous substances and

R2 [33]’

inetic mechanism function: (1ea)3.11 R2 [34]e [35]

5: ‘R1’ R2 [36]’

e [37]7: ‘R2’inetic mechanism function: (1ea)3.065 R2 [38]’ e [39]’

’

’ R2 [40]’

’

e [41]R2 [42]

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

are shown in Fig. 1.Based on the Friedman method and the finite-difference theory,

the derivation of the modified Friedman isoconversional methodcan be found in our previous paper [24]. For the determination ofthe effective activation energies on the degree of conversion, themodified Friedman isoconversional method is more accurate thanthe FWO and KASmethods [24]. The computation complexity of themodified Friedman method is less than that of the advancednonlinear isoconversional method developed by Vyazovkin [45]. Inaddition, it can effectively reduce the effect of data noise on the useof the Friedman method (a widely used differential isoconversionalmethod) [23]. The equation of the modified Friedman isoconver-sional method is given below:

ln

bi ,

Da

TaþDa=2;i � Ta�Da=2;i

!¼ ln½Aa , f ðaÞ� � Ea

R,Ta;i(3)

where f(a) is the differential form of kinetic mechanism function.The determination procedure of the activation energies for themodified Friedman method is summarized as follows: (1) the Ta-△a/2,i, Ta,i, and Taþ△a/2,i values at various a and different b areinterpolated from the experimental kinetic conversion data; (2)

ln

bi

DaTaþDa=2;i�Ta�Da=2;i

!and � 1

RTa;ivalues are calculated; (3) the effec-

tive activation energy (Ea) values are obtained by the linear

regression of ln

bi

DaTaþDa=2;i�Ta�Da=2;i

!vs. �1000

RTa;ifor various a. The

schematic diagram of the modified Friedman isoconversionalmethod is shown in Fig. 2.

2.2. Master plots method

There are several types of master plots methods, which arebased on the differential or integral forms of kinetic equation, or thecombination of the two forms (e.g., the z(a) master plots method)[46,47]. For the determination of the kinetic mechanism function,different types of master plots methods are equivalent. In thisstudy, themaster plotsmethod based on the differential form of thekinetic equation is considered.

For a solid-state reaction under non-isothermal conditions at aconstant heating rate, the kinetic equation can be written as [48]:

Fig. 1. Schematic diagram of (a) FWO an

4

dadT

¼Abe�E=ðR,TÞf ðaÞ (4)

According to Equation (4), the following equation can beobtained:

f ðaÞ¼ b

AeE=ðR,TÞ,

dadT

(5)

Based on the reference point at a ¼ 0.5, Equation (6) can bederived from Equation (5):

f ðaÞf ð0:5Þ¼

ðda=dTÞa,exp½Ea=ðR,TaÞ�,A0:5

ðda=dTÞ0:5,exp½E0:5=ðR,T0:5Þ�,Aa(6)

where (da/dT)0.5 is the conversion rate corresponding to a ¼ 0.5,f(0.5), E0.5 and T0.5 represent the kinetic mechanism function value,activation energy and temperature corresponding to a ¼ 0.5,respectively.

Assuming Aa/A0.5z1. Equation (6) can be simplified as follows:

f ðaÞf ð0:5Þ¼

ðda=dTÞa,exp½Ea=ðR,TaÞ�ðda=dTÞ0:5,exp½E0:5=ðR,T0:5Þ�

(7)

The conversion rate (da/dT)a can be calculated by finitedifference:

�dadT

�a

¼ Da

TaþDa=2 � Ta�Da=2(8)

The master plots method allows us to select the appropriatekinetic mechanism function for the chemical reaction underinvestigation [49].

The right-hand side of Equation (7) can be calculated from theexperimental data and the Ea values from the isoconversionalmethods. As such, the experimental data-derived values of f(a)/f(0.5) vs. a can be calculated according to Equation (7). Then, bysimply comparing the plot of f(a)/f(0.5) vs. a obtained from theexperimental data with that assumed by various theoretical kineticmechanism functions, the kinetic mechanism function can bedetermined [50].

Fig. 3 shows the flowchart and MATLAB pseudo-codes of kineticanalysis based on the master plots method.

d (b) KAS isoconversional methods.

Fig. 2. Schematic diagram of modified Friedman isoconversional method.

Fig. 3. Flowchart and MATLAB pseudo-codes of kinetic analysis based on master plots method.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

3. Results and discussion

3.1. Analysis of simulated process I

To validate the applicability of the master plots method for thedetermination of the kinetic mechanism function, some theoreti-cally simulated solid-state processes are analyzed. According toWuet al. [51], the activation energies of lignocellulosic biomass

5

pyrolysis significantly vary with the degree of conversion and theycan be described by the following equation [24]:

Ea ¼p1 þ p2,a

ln aþ p3,ln a (9)

where p1, p2, and p3 are constants. And the relationship betweenthe frequency factor and activation energy usually follows the

Fig. 4. Calculation diagram for obtaining kinetic data of theoretically simulatedprocesses.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

kinetic compensation effect [52,53]:

ln Aa ¼ aþ b,Ea (10)

where a and b are constants. According Eqs. (9) and (10), the

Fig. 5. (a) Kinetic curves of one simulated process: Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ 0:0a

ln aþ 5 ln a, ln Aa ¼ 14:5þ c,Ea ðc ¼ 0:00: 0:02: 0:10Þ.

6

theoretically simulated processes with the following parametersare considered: Ea ¼ 215� a

ln aþ 5 ln a (Ea is expressed in kJ

mol�1); ln Aa ¼ 14:5þ c,Ea (Aa is expressed in s�1, c is a constantvarying from 0.00 to 0.10); f(a) ¼ ‘F1’, ‘B1’, ‘P2’, ’A1.50, ’R20 or ’D3’;b ¼ 2.5, 5, 10 and 20 K min�1. The theoretically simulated data areobtained through numerical integration of the basic kinetic equa-tion. The classical Runge-Kutta fourth-order algorithm with arelative error lower than 10�6% was used for the numerical inte-gration. The detailed calculation procedures for obtaining kineticdata of theoretically simulated processes are shown in Fig. 4.

Fig. 5(a) shows the a e T and da/dt e T curves for one of thesimulated processes: Ea ¼ 215� a

ln aþ 5 ln a, ln Aa ¼ 14:5þ 0:05Ea

and f(a) ¼ ’D3'. Fig. 5(b) shows the theoretical Aa/A0.5 values as afunction of a for the simulated processes with the following kineticparameters: Ea ¼ 215� a

ln aþ 5 ln a, ln Aa ¼ 14:5þ c,Ea ðc ¼

0:00: 0:02: 0:10Þ. These simulated processes are varying activa-tion energy processes. From Fig. 5(b), it can be obtained that (1) thefrequency factor is constant when c ¼ 0, and (2) with increasing cvalues, the Aa/A0.5 values vary with a more significantly, especiallywhen c reaches 0.10, the Aa/A0.5 values increase sharply at a > 0.8.

Figs. 6 and 7 show the Ea values as a function of a calculatedfrom the FWO, KAS, and modified Friedman isoconversionalmethods for the simulated processes: Ea ¼ 215� a

ln aþ 5 ln a,

ln Aa ¼ 14:5þ c,Ea ðc ¼ 0:00: 0:02: 0:10Þ, and f(a) ¼ ‘F1’, ‘B1’,‘P2’, ‘A1.5’, ‘R2’ and ‘D3’. From the results in Figs. 6 and 7, it can beobtained that (1) the modified Friedman method can provide ac-curate Ea values for the simulated processes with various c valuesand f(a), and (2) the FWO and KAS methods lead to significant er-rors in Ea. Similar results can be found in Refs. [18,21]. The reason isthat the FWO and KASmethods are derived from the assumption ofconstant activation energy for the chemical process and the use ofover-simplified approximations for the temperature integral.

Figs. 8e10 show the f(a)/f(0.5) values obtained from the masterplots method with the Ea values determined from the FWO, KAS,and modified Friedman isoconversional methods for the simulatedprocesses: Ea ¼ 215� a

ln aþ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00:

0.02: 0.10) and f(a)¼ ’F10, ‘B1’, ‘P2’, ’A1.50, ’R20 and ’D30, respectively.For the simulated processes with various kinetic mechanismfunctions, the calculated f(a)/f(0.5) values from the master plotsmethod are in good agreement with the simulated ones when c¼ 0(in that case Aa keeps constant), especially for the modified Fried-man method. Although Ea varies with a significantly for thosesimulated processes, indicating that the accuracy of the masterplots method for the estimation of the kinetic mechanism function

5Ea , and f(a) ¼ ’D3’. (b) Variation of Aa/A0.5 with a for simulated processes: Ea ¼ 215�

Fig. 6. Ea as a function of a calculated by FWO method for simulated processes: Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea ðc ¼ 0:00: 0:02: 0:10Þ, and (a) f(a) ¼ ’F10, (b)f(a) ¼ ’B10, (c) f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

is mainly dependent on the variation degree of Aa with a. When thevalue of c increases from 0.02 to 0.10, the differences between thecalculated f(a)/f(0.5) values from the master plots method and thesimulated ones sharply increase with Ea from all isoconversionalmethods considered in this study. The obtained f(a)/f(0.5) valuesare different with different a-dependencies of Ea by different iso-conversional methods, which indicates that the obtained f(a)/f(0.5)from the master plots method is affected by the a-dependency ofEa.

To quantitatively analyze the effect of the variation of Aa with a

on the accuracy of f(a) determined by themaster plots method, twostatistical measures are introduced:

7

RRMSE_f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nd

Pa

�½f ðaÞ=f ð0:5Þ�cal � ½f ðaÞ=f ð0:5Þ�sim�2r

1nd

Pa½f ðaÞ=f ð0:5Þ�sim

� 100%

(11)

RRMSE A¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nd

Xa

�Aa

A0:5� 1

�2vuut � 100% (12)

where RRMSE_f and RRMSE_A represent the relative root meansquare error of f(a)/f(0.5) and Aa/A0.5, respectively, the subscriptssim and cal represent the simulated value and the value calculated

Fig. 7. Ea as a function of a calculated by KAS method for simulated processes: Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea ðc ¼ 0:00: 0:02: 0:10Þ, and (a) f(a) ¼ ’F10, (b) f(a) ¼ ’B10,(c) f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

from the master plots method, and nd is the number of data points.A lower RRMSE_f value indicates that the calculated f(a)/f(0.5)curve is closer to the simulated one, while a higher RRMSE_A valueindicates the variation of Aa with a are more significant. In general,the accuracy of obtained f(a)/f(0.5) is considered excellent whenRRMSE_f � 10% and good if 10% < RRMSE_f � 20% according toRef. [54].

Fig. 11 shows the RRMSE _f value as a function of RRMSE _A forthe simulated processes: Ea ¼ 215� a

ln aþ 5 ln a, ln Aa ¼ 14:5þ c,

Ea (c¼ 0.00: 0.02: 0.10) and f(a) ¼ ‘F1’, ‘B1’, ‘P2’, ‘A1.5’, ‘R2’ and ‘D3’.The analysis results for the simulated processes with the same Eaand Aa, and other theoretical f(a) are shown in SupplementaryMaterial Figs. S1eS12. The RRMSE_f values are calculated from

8

the master plots method coupled with Ea from the FWO, KAS, andmodified Friedman methods. Hence, the effect of the variationdegree of Aa with a and different isoconversional methods forestimating Ea on the resulting f(a)/f(0.5) values can be analyzed.From Fig. 11, it can be obtained that (1) in general, RRMSE_f in-creases with the increasing of RRMSE_A for most of the simulatedprocesses considered in this section, for the simulated process withf(a) ¼ ‘P2’, RRMSE_f with Ea from the FWO and KAS slightly varieswith RRMSE_A; (2) the variation trend of RRMSE_fwith RRMSE_A isdifferent with different simulated f(a), for example, RRMSE_f in-creases more sharply for ‘D3’ than other kinetic mechanism func-tions; (3) when RRMSE_A equals zero (that is, A keeps constant), theRRMSE_f values with the modified Friedman method are close to

Fig. 8. Validation of master plots method for simulated processes (Ea from FWO method): Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00: 0.02: 0.10) and (a) f(a) ¼ ’F10, (b)f(a) ¼ ’B10, (c) f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’. .

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

zero, although all simulated processes studied in this section arevarying activation energy processes; and (4) RRMSE_ f withdifferent isoconversional methods shows complex trends: althoughthe modified Friedman isoconversional method can provide accu-rate Ea values, it does not always give more accurate f(a)/f(0.5)values than the FWO and KAS methods. The accuracy of f(a) fromthe master plots method strongly depends on the variation degreeof Aa with a and the isoconversional method for estimating Ea, asshown in Fig. 11.

As analyzed above, RRMSE_f would increase with increasingRRMSE_A for various simulated processes. Therefore, there exists avalue RRMSE_Aa, when RRMSE_A � RRMSE_Aa, RRMSE_f � 20%,where the accuracy of the master plots method is considered good.

9

In this study, RRMSE_Aa is defined as the maximum acceptableRRMSE_A values. Fig. 12 shows the maximum acceptable RRMSE_Avalues as RRMSE_ f � 20% (Ea is estimated by the modified Fried-man method) for the simulated processes with Ea ¼ 215� a

ln aþ

5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00: 0.02: 0.10) and all theoreticalf(a). It can be obtained that (1) the acceptable RRMSE_A values arerelatively low for the simulated process with diffusional kineticmechanism functions (e.g., ‘D1’, ‘D2’, ‘D3’ and ‘D4’), which is causedby the significant variation of the theoretical f(a)/f(0.5) master plotwith a; (2) the acceptable RRMSE_A values are relatively high(ranging from 26.8% to 38.5%) for the simulated processes withautocatalytic, nucleation and contracting geometry kinetic mech-anism functions. To some extent, findings in Fig. 12 provide

Fig. 9. Validation of master plots method for simulated processes (Ea from KAS method): Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00: 0.02: 0.10) and (a) f(a) ¼ ’F10, (b)f(a) ¼ ’B10, (c) f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

reference applicability conditions for the master plots method. Forthe real kinetics of lignocellulosic biomass pyrolysis, the variationdegree of Aa with a is generally large. For example, Aa varies be-tween 105 and 1012 s�1 in the a range of 0.05e0.70 for the pyrolysisof poplar wood [55], and Aa changes from 109 and 1017 s�1 in the a

range of 0.1e0.8 for the pyrolysis of garlic husk [37] (see Fig. 13).According to Equation (12), the RRMSE_A values are 88.5% and93.5% for the pyrolysis of poplar wood and garlic husk.

According to ICTAC Kinetics Committee recommendations forperforming kinetic computations on thermal analysis data byVyazovkin et al. [14], the use of the master plots method requires

10

that “Ea does not vary significantly with a”. Based on the aboveanalyses, the accuracy of the master plots method for the deter-mination of f(a) significantly depends on the variation of Aa with a.Even if Ea varies significantly with a, the use of the master plotsmethod coupled with Ea from the modified Friedman method canprovide accurate f(a) when Aa keeps constant.

When Ea keeps constant and Aa varies with a, the master plotsmethod may lead to inaccurate f(a). To validate the assumption, thefollowing simulated processes were considered: Ea ¼ 215,

ln Aa ¼ 14:5þ c,215� a

ln aþ5 ln a

(c ¼ 0.00: 0.02: 0.10) and

Fig. 10. Validation of master plots method for simulated processes (Ea from modified Friedman method): Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00: 0.02: 0.10) and (a)f(a) ¼ ’F10, (b) f(a) ¼ ’B10, (c) f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

f(a) ¼ ‘F1’ and ‘R2’, where Ea and Aa are expressed in kJ mol�1 ands�1, respectively. The corresponding results by the master plotsmethod coupled with Ea from the modified Friedman method areshown in Fig.14. It can be observed that (1) the calculated f(a)/f(0.5)values deviate from the simulated f(a)/f(0.5) data except for thecase with c ¼ 0 (Aa keeps constant), (2) the deviations between thecalculated f(a)/f(0.5) values and the simulated ones increase withincreasing c for all simulated processes although Ea keeps un-changed with a. The results indicate that the variation of Aa with a

is the key factor for the accurate estimation of f(a) via the masterplots method.

As mentioned above, the variation of Aa with Ea generally fol-lows the kinetic compensation effect for some reaction processesincluding lignocellulosic biomass pyrolysis processes. Therefore, a

11

significant variation of Eawith a would lead to Aa varying sub-stantially with a, affecting the accuracy of the master plots methodfor the determination of f(a).

3.2. Analysis of simulated process II

Another type of theoretically simulated processes are consid-ered: Ea¼ 200þ5ln(1-a) (Ea in kJ mol�1), lnAa¼ c$Ea-2 (Aa inmin�1,c ¼ 0.00: 0.05: 0.25) and f(a) ¼ ‘F2’, which was also analyzed byCriado et al. [21] and Cai and Chen [18]. Fig. 15(a) shows the vari-ation of Aa/A0.5 with a. It can be obtained that larger values of cresult in a more significant variation of Aa. Fig. 15(b) shows the Eavalues calculated by the FWO, KAS andmodified Friedmanmethodsfor the simulated process with the same parameters except c¼ 0. It

Fig. 11. Relationship between RRMSE_f and RRMSE_A for simulated processes: Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00: 0.02: 0.10) and (a) f(a) ¼ ’F10, (b) f(a) ¼ ’B10, (c)f(a) ¼ ’P20 , (d) f(a) ¼ ’A1.50 , (e) f(a) ¼ ’ R20 , (f) f(a) ¼ ’D3’.

Fig. 12. Maximum acceptable RRMSE_A when RRMSE_f RMSE (Ea from modified Friedman method) for simulated process: Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ c,Ea (c ¼ 0.00:0.02: 0.10).

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

12

Fig. 13. Variation of Aa with a for pyrolysis of poplar wood [55] and garlic husk [37].

Fig. 14. Validation of master plots method for simulated processes (Ea from modified Friedman method): Ea ¼ 215, ln Aa ¼ 14:5þ c,215� a

ln aþ5 ln a

(c ¼ 0.00: 0.02: 0.10), and

(a) f(a) ¼ ‘F1’ and (b) f(a) ¼ ‘R2’.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

can be deduced that the FWO and KAS methods lead to significanterrors in Ea (RRMSE_E values for the FWO and KAS methods reachup to 50.0% and 10.7%, respectively, where RRMSE_E is calculatedby Equation (13)). Fig. 15(c) shows the analysis results from themaster plots method for the above theoretically simulated pro-cesses. The errors in the calculated f(a)/f(0.5) values from thesimulated ones are more significant with larger values of c, aspresented in Fig. 15(d). The results indicate that the variation of Aa

with a is the key factor for the accuracy of the master plots methodin determining f(a).

RRMSE E¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nd

Pa

�Ea;cal � Ea;sim

�2r1nd

PaEa;sim

� 100% (13)

where RRMSE_E represents the relative root mean square error ofEa.

13

4. Implications

In the implementation of the master plots method for theestimation of f(a), the experimental master plot is compared withthe various theoretical master plots, and the closest theoretical f(a)to the experiment master plot is considered as the optimal kineticmechanism function. Based on the above analyses, the master plotsmethod would lead to some errors in the estimation of f(a), whichdepend on the variation degree of Aa with a and isoconversionalmethods for estimating Ea. Four examples are shown here. Fig.16(a)and (b) show the results from the master plots method coupledwith the Ea values from the FWO and KAS methods for differentsimulated processes, respectively. Fig. 16(c) and (d) show the re-sults from themaster plotsmethod coupled with the Ea values fromthe modified Friedman method. From Fig. 16, the resulting optimalkinetic mechanism functions are different from the pre-assumedones, which may be associated with the following reasons thatthe FWO and KAS method provide inaccurate Ea values, and Aa

strongly varies with a contradicting with the assumption Aa/A0.5z1. From the practical point of view, the current master plots

Fig. 15. Analysis results of simulated processes: Ea ¼ 200þ5ln(1-a) (Ea in kJ mol�1), lnAa ¼ c$Ea-2 (Aa in min�1, c ¼ 0.00: 0.05: 0.25) and f(a) ¼ ‘F2’. (a) Variation of Aa/A0.5 with a; (b)Ea e a from FWO, KAS and modified Friedman methods for simulated process with c ¼ 0; (c) Validation of master plots method (Ea from modified Friedman method); (d)Relationship between RRMSE_f (Ea from modified Friedman method) and RRMSE_A.

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

method needs more consideration, especially for the reactionprocesses with the significant variation of Aa with a.

To improve the accuracy of the resulting f(a), the current masterplots method should bemodified or an alterative method should bedeveloped. In this study, a possible idea for the determination of thekinetic mechanism function is presented here. Cai and Liu [56]proposed an empirical kinetic mechanism function:

f ðaÞ¼am,ð1� q,aÞn (14)

wherem, n, and q are constants. The empirical function can fit mostof the ideal kinetic mechanism functions very well. According toEquation (14) and the kinetic compensation effect, the followingequation can be obtained:

ln½Aa , f ðaÞ�¼ aþ b , Ea þ ln am , ð1� q,aÞn� (15)

The values of Ea and ln[Aa$f(a)] at various a can be obtainedwhen analyzing the kinetic data using the Friedman isoconver-sional method or its modification [24]. Equation (15) can be used tofit the ln[Aa$f(a)] values from the experimental data and iso-conversional methods. Then, the parameters a, b,m, n and qmay be

14

optimized by nonlinear regression analyses. Detailed informationabout this idea will be presented in our future work.

5. Conclusions

The master plots method coupled with Ea from an isoconver-sional method is usually used to determine f(a). This study focuseson the critical investigation of the master plots method by pro-cessing theoretically simulated processes with different parametersand mechanisms. The accuracy of f(a) calculated from the masterplots method depends on the variation degree of Aa with a and a-dependency of Ea from isoconversional methods. In general, a sig-nificant variation of Aa with a may result in large deviations of f(a)when the master plots method is implemented. At the same time,the use of the masters plots method probability lead to mis-estimating the kinetic mechanism function. In general, Aa signifi-cantly varies with a for lignocellulosic biomass pyrolysis, therefore,the applicability of the master plots method for the determinationof f(a) needs deliberation in kinetic analysis of lignocellulosicbiomass pyrolysis.

Fig. 16. Misestimating f(a) by master plots method: (a) Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼ 14:5þ 0:06,Ea , and f(a) ¼ ‘R2’ (Ea from FWO method); (b) Ea ¼ 215� aln a

þ 5 ln a, ln Aa ¼14:5þ 0:04,Ea , and f(a) ¼ ‘A1.5’ (Ea from KAS method); (c) Ea ¼ 215� a

ln aþ 5 ln a, ln Aa ¼ 14:5þ 0:04,Ea , and f(a) ¼ ‘A2’ (Ea from modified Friedman method); (d) Ea ¼ 215�

aln a

þ 5 ln a, ln Aa ¼ 14:5þ 0:06,Ea , and f(a) ¼ ‘F1’ (Ea from modified Friedman method).

L. Luo, Z. Zhang, C. Li et al. Energy 233 (2021) 121194

Declaration of competing interest

The authors declare that they have no known competingfinancial interests or personal relationships that could haveappeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge financial support fromSino-German Center for Research Promotion (Grant No.: M-0183)and Guangdong Provincial Key Laboratory of Distributed EnergySystems (Grant No.:2020B1212060075). Dominic Yellezuome, aPh.D. candidate from the School of Agriculture and Biology,Shanghai Jiao Tong University, is acknowledged for his help inproofreading.

Appendix A. Supplementary data

Supplementary data to this article can be found online athttps://doi.org/10.1016/j.energy.2021.121194.

Credit author statement

Laipeng Luo: Data curation, Software, Writing - original draft,Investigation; Zhiyi Zhang: Writing - original draft, Data curation;Chong Li: Writing - review & editing, Validation; Nishu: Writing -review & editing; Fang He: Writing - review & editing, Fundingacquisition; Xingguang Zhang: Writing - review & editing; Jun-meng Cai: Methodology, Conceptualization, Writing - review &editing, Software, Supervision, Funding acquisition.

15

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