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Coal breakage characterisation – Part 2: Multi-component breakage

modelling

Fengnian Shi⇑

The University of Queensland, Sustainable Minerals Institute, Julius Kruttschnitt Mineral Research Centre, 40 Isles Road, Indooroopilly, Brisbane, Qld 4068, Australia

h i g h l i g h t s

  A multi-component model to describe energy–size reduction for various coal particle sizes and densities.  Model parameters calibrated with JKFBC characterisation testing data.

 Using  t n-family of curves to determine product size distribution from the model predicted  t 10  parameter.

 Model was validated with over 100 sets experimental data using an Australian and a Chinese coal sample.

a r t i c l e i n f o

 Article history:

Received 14 May 2013

Received in revised form 7 July 2013

Accepted 9 July 2013

Available online 1 August 2013

Keywords:

Coal

Breakage characterisationHGI test

Multi-component breakage model

a b s t r a c t

A multi-component breakage model has been developed at the Julius Kruttschnitt Mineral Research

Centre (JKMRC) to describe the energy–size reduction in relation to particle size and density for coal

breakage characterisation. The model takes the following form:   t 10 ¼   M ðRD=RDmin Þc   f1 exp½ f mat   x  E g.

The model incorporates four parameters and fits 60 JKFBC (JK Fine-particle Breakage Characteriser) test

data for each of the Australian coal and Chinese coal samples, with R2 = 0.982 and 0.978 respectively. The

multi-component model can be switched into a single component model by setting c  = 0.

A set of  t n-family of curves for coals ground in the JKFBC are presented. It was found that the data from

various particle sizes and densities of the two coal samples, collected from the Australian and Chinesepower stations, all fall on similar   t n-curve trend lines. These   t n-family of curves can be used in the

multi-component model to estimate the product size distribution from the predicted  t 10 values. A proce-

dure has been developed to calibrate the multi-component model with seven tests based on a combina-

tion of various particle sizes, coal densities and grinding energy levels, using the JKFBC device. Over 100

sets of data have been used to validate the calibration procedure.

  2013 Elsevier Ltd. All rights reserved.

1. Introduction

Particle breakage characterisation aims to establish the rela-

tionship between a specific energy input and the resultant product,

through some type of laboratory test on a given sample  [1]. With

this energy–size reduction relationship characterised for a particu-lar coal sample, users can investigate how particles break when

subjected to a given input energy; or determine what energy is re-

quired in order to achieve a desired breakage degree. Obviously,

the traditional HGI test on a single size coal particle with a single

level of grinding energy cannot provide this level of detail.

An improved coal breakage characterisation method has been

developed at the Julius Kruttschnitt Mineral Research Centre

(JKMRC), which is presented in Part 1 of this paper  [2]. An experi-

ment was conducted on various narrowly-sized and density

franctionated groups of coal samples, using a JKFBC (JK Fine-parti-

cle Breakage Characteriser) consisting of a standard HGI grinding

device and a precision torque meter. Each size–density group sam-

ple was ground in the JKFBC with various input energies. The data

show a strong size effect and density effect on coal breakage, with

the larger and lower density particles being broken more easily un-der the same specific energy.

Part 2 of the paper presents a mathematical model which de-

scribes the multi-component breakage behaviour observed in the

 JKFBC tests. Part 3 will demonstrate the applications of the model

for HGI predictions and coal breakage simulations.

2. Particle breakage modelling 

 2.1. The JKMRC prior-art model

The JKMRC has been using a breakage model (Eq.   (1)) to

describe the energy–size reduction relationship for a long time:

0016-2361/$ - see front matter     2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.fuel.2013.07.026

⇑ Tel.: +61 7 3365 5913; fax: +61 7 3365 5999.

E-mail address: f.shi@uq.edu.au

Fuel 117 (2014) 1156–1162

Contents lists available at  ScienceDirect

Fuel

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t 10  ¼  Að1  eb:EcsÞ ð1Þ

where t 10 is a size distribution ‘fineness’ index (%), Ecs is the specific

comminution energy (kWh/t), and A and b are the ore impact break-

age parameters. The product form of the model parameters   Ab   is

used as an indicator of ore resistance to breakage, with a large

 Ab indicating less resistance to breakage  [1]. The  Ab indicator for

ore breakage is currently used by researchers and mining engineersworldwide.

Previous work has found that the crack density of larger parti-

cles is much greater than that of smaller particles  [3,4]. In view

of this, larger particles tend to be weaker and therefore easier to

break than smaller particles. In the standard data reduction proce-

dures, however, Eq. (1) is used to fit the Drop Weight Tester (DWT)

data with one set of  A  and  b  parameters for all particle sizes. This

‘average’ set of  A  and  b  parameters assume that particles of differ-

ent sizes would be broken in the same way when subjected to the

same impact energy. However, this assumption is questionable.

A new breakage model incorporating particle size effect has

been reported [5]  to describe the degree of breakage. It was mod-

ified from Vogel and Peukert’s work  [6] describing the probability

of breakage. The breakage degree model takes the following form:

t 10  ¼  M f1 exp½ f mat :  x  kðE   E minÞg ð2Þ

where M  (%) represents the maximum  t 10  for a material subject to

breakage,   f mat   (kg J1 m1) is the material breakage property,   x

(m) the initial particle size,  k  is the successive number of impacts

with a single impact energy,  E  (J kg1) the mass-specific impact en-

ergy, and E min  (J kg1) is the threshold energy. Analysis of the fitted

 f mat  from many DWT and JKRBT test data enables the development

of a sub-model of   f mat, which allows determination of the particle

size effect on the impact breakage result.  Fig. 1  demonstrates the

model fitting quality by Eqs. (1) and (2) respectively, using the same

set of DWT data. It is apparent that the size-dependent breakage

model (Eq. (2)) can describe the data better.

 2.2. A multi-component model for coal breakage

 2.2.1. Analysis of the JKFBC data

Eq.  (2) was developed to treat impact breakage data, and has

been tested with hundreds of sets of DWT and JKRBT data to prove

its validity. A question was raised regarding whether or not Eq. (2)

can be applied for coal breakage characterisation, knowing the fol-

lowing differences exist between the JKFBC and the DWT/JKRBT

test procedures:

(1) The breakage mode utilised in the JKFBC is not by impact as

in the DWT/JKRBT, but rather by compression grinding.

(2) The cumulative energy controlled by the grinding time in

the JKFBC tests is much larger than the energy threshold

(E min  in Eq.  (2)) of coal particles; while in the JKRBT incre-

mental low energy breakage, each impact energy is in simi-

lar order to the energy threshold of rock particles.

(3) Eq. (2)  was developed as a size-dependent model for DWT/ JKRBT ore data reduction. The JKFBC tests were conducted

on a multi-component basis, and the data suggest that both

particle size and density affect breakage results  [2].

Regarding the first difference, the breakage modes will certainly

affect the degree of breakage (t 10), resulting in different breakage

parameters when the model is fitted to the experimental data.

The same model structure can be applied, allowing the model

parameters to take care of the difference in breakage mode (either

impact or grinding).

In the case of the second difference, the energy threshold E min in

Eq. (2)  can be ignored for coal. For a 60-revolution grinding test,

similar to the standard HGI test, the total specific energy is approx-

imately 7000 J/kg. The energy threshold, or the minimum energy

required to cause breakage for a coal particle, is in the order of 

10 J/kg that was estimated from single-particle dropping on an an-

vil from a known height, using large amount of coal particles for

the test. This energy threshold is minor compared with the

7000 J/kg input energy. Therefore the term  E min   in Eq.  (2)  can be

dropped without any significant influence on the model fitting

results.

The major challenge in developing the coal breakage model is to

describe the effects of both particle size and density. The simplest

way is to treat the multi-component data with a single component

model, i.e. to fit one set of breakage parameters for each density

group, using the size-dependent model as presented in Eq.  (2). In

an application for a particular density, the set of parameters for

the given density can be used. However, as the individual sets of 

parameters are independent of the density effect, they may predictan inconsistent trend in the density effect on breakage. In addition,

there is a difficulty in selecting the right set of parameters for a

density that was not measured.

Analysis of the data presented in Fig. 6 in Part 1 of this paper [2]

found that the JKFBC data show a regular trend in the size effect,

i.e. the large particle size is consistently on top of the small particle

sizes on the   t 10–Ecs   plots. This indicates that the size-dependent

model (Eq. (2)) will be able to handle the data; working from the

expectation that all points of various sizes will fall on one trend

line for each density group.

Fig. 1.   Comparison of model fitting results to the same DWT data of a quarry material (after Shi and Kojovic  [5]).

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The analysis also pointed out that particle density affects the

positions of the trend lines.   Fig. 2   depicts the density effect on

the energy–size reduction relationship for the 2.8–4.0 mm Chinese

coal ground in the JKFBC. Obviously, as density increases (indicat-

ing more mineral matter associated with the coal particles), the

breakage degree (t 10) decreases.The physical meaning of parameter M  in Eq. (2) is the maximum

t 10 that a sample can reach as the other variables (specific energy E 

or particle size  x) change. Mathematically, changing  M   in Eq.   (2)

will change the position of the exponential line. Based on this anal-

ysis, therefore, the density effect was modelled in the  M -term, and

the size effect was modelled separately in the exponential term.

However, as the same test data were used to fit the density-related

and size-related parameters simultaneously, these parameters

would represent the joint effects of coal density and particle size.

 2.2.2. A multi-component breakage model

A multi-component breakage model was developed to describe

the JKFBC data. The model takes the following form:

t 10  ¼  M 

ðRD=RDminÞc   f1  exp½ f mat   x  E g ð3Þ

where  t 10,  M  and  x  have been defined in Eq.  (2),  RD  is the relative

density of the particle,   RDmin   is the minimum relative density of 

the sample (RDmin = 1.25 for coal),   c   is a parameter determining

the trend line position,  f mat   is a model parameter that is described

by a size-dependent equation with parameters   p  and  q ,   E  (J kg1)

is the mass-specific energy in the JKFBC that is determined by Eq.

(4):

E  ¼ 2pNT 

m  ð4Þ

where N  is the mill revolution number,  T  (Nm) is the mean net tor-que, and m (kg) is the coal mass being ground.

The model incorporates four parameters,  M ,  c ,  p  and  q, which

can be calibrated by fitting the model to the measured data.  Figs. 3

and 4 show the fitting results for the Australian coal and the Chi-

nese coal respectively. There are nominal 64 data points (4

Fig. 2.  Coal density effect on the energy–size reduction relationship for the 2.8–

4.0 mm Chinese coal ground in the JKFBC.

Fig. 3.  The multi-component breakage model fitted to the JKFBC data for a nominal 4 sizes   4 densities  4 energies (61 data points sharing 4 parameters) for the Australian

coal.

Fig. 4.  The multi-component breakage model fitted to JKFBC data for a nominal 4 sizes   4 densities  4 energies (59 data points sharing 4 parameters) for the Chinese coal.

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sizes   4 densities  4 energies), minus a couple of data points

which are missing due to insufficient samples for those tests. To

avoid over-crowded plots in one graph, only the two extreme den-

sities are presented on the   t 10–E   graph (left), with the measured

data in dots and the model fitting in lines. The overall fitting qual-

ity can be seen from the predicted versus the measured plots

(right).

The model fitting results suggest that the multi-componentbreakage model (Eq. (3)) can well describe the particle size effect

and the density effect on the energy–size reduction relationship.

With four model parameters fitted, the model is able to represent

the coal breakage behaviour of the 60 tests from each of the two

coal samples, covering a wide range of coal sizes and densities typ-

ically found in coal-fired power station grinding facilities. The

model has been validated with an Australian coal (HGI = 53) and

a Chinese coal (HGI = 80). The HGI range of the two coal samples

is typical in the marketplace of thermal coals used in power sta-

tions worldwide.

A question may be raised regarding the flexibility of the model

structure. Fig. 7 in Part 1 of the paper [2] presents the density effect

on breakage, in which the coarse particles show a very pronounced

density effect while the fine particles do not (with the four sets of 

density data falling on one similar line for the 0.6–1.18 mm size

particles). Apparently there is a strong interaction between the

density effect and the particle size effect on coal breakage, and

these effects are not presented in a fixed pattern. Can the model

present such variations correctly?  Fig. 5 shows the model fitting

results (in line) and the measured data (in dots) for the four density

groups in the 0.6–1.18 mm and 2.36–3.35 mm size fractions

respectively. With one set of four parameters shared by the 61 Aus-

tralian coal tests, the model demonstrates its robustness to repli-

cate the complex patterns of coal density and size effects on

breakage.

It is worth emphasising that although the model structure is de-

signed for multi-component data, the model can be used easily for

single component data of a bulk sample (without float-sink tests to

fractionate the size-by-size feed into size-by-density bases). Sim-ply setting the parameter  c  = 0 will make the denominator in Eq.

(3) equal to one, thereby converting the multi-component model

to a single component model with three parameters  M , p  and q  fit-

ted to the size-by-energy test data. This feature will give users flex-

ibility to apply the model for coal breakage characterisation

according to their available data.

 2.3. Modelling of breakage distribution

The breakage model (Eq. (3)) predicts a single variable t 10 which

represents the cumulative per cent of the product passing 1/10th of 

the feed size. This t 10 parameter indicates the fineness of a product

when subjected to breakage with a certain input energy. Some-

times it is not sufficient knowing the fineness of a product. It

may be necessary to predict the product size distribution. For

example, for optimum combustion efficiency, the pulverised fuel

(PF) has a certain size distribution requirement, such as 75% pass-

ing 75 lm and 99.5% passing 300 lm.

Narayanan and Whiten [7]  found that the  t 10  parameter is un-

iquely related to other points on a family of size distribution

curves, with t n, defined as the cumulative percentage passing a gi-

ven fraction of the initial size, x/n. The t 10 can then be used to gen-

erate a size distribution from relationships between   t 10   and   t n-

family of curves established from the drop weight test database.

The   t n-family of curves have been independently confirmed by

Pauw and Maré [8]  and by King and Bourgeois [9]  using different

materials over a wide range of fracture energies.  Fig. 6  depicts

the t n-family of curves for a range of ore types [9]. Spline regression

analysis can be carried out to describe each of the relationships

t 10–t 2, t 10–t 4, t 10–t 25, t 10–t 50  and  t 10–t 75.

Using the   t n-family of curves to produce a size distribution is

straightforward. For a given  t 10  (e.g.  t 10 = 20),  Fig. 6 indicates that

t 2 = 75,   t 4 = 42,   t 25 = 10,   t 50 = 8,   t 75 = 6. For a feed geometric mean

size of 50 mm,   t 2  = 75 means 75% passing 25 mm (=50/2 mm),

t4 = 42 means 42% passing 12.5 mm (=50/4 mm), and so on. The

size distributions at the required size fractions can be determined

from the   t n   knots using spline regression. Thus the whole size

Fig. 5.   Modelling the coal density effect for various feed sizes, with one set of four parameters shared by 61 Australian coal tests.

t75t50

t25

t10

t4

t2

0

20

40

60

80

100

0 10 20 30 40 50

   t

   (   %    P

  a  s  s   i  n  g   )

  n

Breakage Index, t (%)10

Fig. 6.  Determination of size distribution parameter t n  from the breakage index  t 10

(Redraw after Narayanan [10]).

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distribution of the progeny can be determined once the breakageindex t 10  is known.

Considering that the t n-family of curves presented in Fig. 6 were

derived from impact tests on rock particles, and the JKFBC data

were generated from grinding coals, an investigation was carried

out using the JKFBC data to generate the   t n-family of curves.

Fig. 7 shows the plots of   t n   versus   t 10  using the JKFBC data from

grinding the Australian coal and the Chinese coal samples together.

Each t n-curve consists of 120 data points.

The data illustrate that the Australian coal and the Chinese coal

fall on similar trend lines, despite the significant differences in

their HGI values, ash content, and ash compositions (refer to Part

1, Section 2.3.1). Note that each  t n-curve comprises different den-

sity and different feed size data, appearing to all fall on similar

trend lines.   Fig. 8 is an enlarged plot of  Fig. 7, with only one   t 25

curve formed on a density by density basis. The graph supports

the notion that both the coal density effect and the particle size ef-

fect on breakage have been eliminated in the plot of  t n  versus  t 10.

Although the plots look promising, only two coals have been used.

Further data is desirable, particularly for a low HGI coal, to confirm

the approach of  t n-family of curves. More coal test data across the

commercial HGI range are required.

Regression was performed on the data displayed in Fig. 7 to pro-

duce the matrices for plotting the   t n-family of curves for coals

ground in the JKFBC, which are presented in Fig. 9.

The multi-component breakage model and the   t n-family of 

curves provide a useful tool for coal breakage characterisation.

Once the model is calibrated with the JKFBC test data, the model

will be able to predict the t 10  for a given particle size and density.The corresponding t n values to this predicted t 10 can be found from

the t n-family of curves (Fig. 9), regardless of particle size and den-

sity; thus the whole product size distribution can be determined

from a single breakage index  t 10.

Using this approach to determine the product size distribution

is helpful when there is no sizing data available, as in the case of 

numerical simulations. However, it is noticed from  Fig. 7 that the

data on each trend line are rather scattered, similar to the t n-curves

plotted from the rock impact tests [10]. This implies that errors are

associated with the generalised t n-family of curves. Table 1 gives R2

values for each   t n   curve, when plotting the measured   t -values

against those calculated from the  t n-family matrices. R2 is the sta-

tistical term of coefficient of determination, which provides a mea-

sure of how well observed outcomes are replicated by the

regression model.   R2 varies from zero to 1.   R2 = 1 indicates that

the two sets of data are exactly matched, with a perfect diagonal

line on the plot of the predicted versus the measured t-values, on

the same x-axis and  y-axis scales.

For an understanding of the  R2 values, readers are referred to

Fig. 8 (the t 25 plot). In this plot R2 = 0.95. The t 2 (R2 = 0.94) has sim-

ilar results to the t 25, the t 4 (R2 = 0.97) is slightly better than the t 25,

but both the  t 50 and the t 75 are not as good as the  t 25. This implies

that for a feed size of 10 mm, predictions of the cumulative per-centage passing 5 mm (t 2), 2.5 mm (t 4) and 0.4 mm (t 25) would

be accurate if the predicted  t 10   is accurate, however larger errors

may be associated with the cumulative percentage passing

0.2 mm (t 50) and 0.133 mm (t 75). To minimise the errors associated

with product size distributions, using the sizing data to establish a

set of coal-specific  t n-family of curves for the tested coal sample is

recommended, rather than using the generalised   t n-family of 

curves, particularly if experimental data is available, as was the

case in the JKFBC tests.

3. Model calibration and validation

To study coal breakage behaviour, the experiment, asreported in Part 1 of the paper, was designed for 4 particle sizes  4

Fig. 7.   The plots of   t n-family of curves using the JKFBC data from grinding the

Australian and the Chinese coal samples.

Fig. 8.   The plot of  t 25  versus t 10   for various densities of the Australian and Chinese

coals, consisting of 120 measurement points.

Fig. 9.   The t n-family of curves used for coal grinding in the JKFBC.

 Table 1

R2 values for various regression lines between the measured and the predicted values

using the generalised  t n-family of curves presented in Fig. 9.

t 2   t 4   t 25   t 50   t 75

0.94 0.97 0.95 0.90 0.88

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densities  4 specific energies; generating a nominal number of 64

tests for each coal sample being conducted. Based on the size-by-

density experimental results, the effects of particle size and density

on the energy–size reduction relationship have been established,

and a multi-component breakage model has been developed. The

model incorporates four parameters to define these effects. To gen-

erate a set of unique parameters, the minimum number of tests re-

quired is four, but if more test data were available the calibration

would likely generate more robust parameters.However, using a complete set of 64 tests to calibrate the four

model parameters is more than adequate. It is hypothesised that

the multi-component model can be calibrated with a reduced

number of tests on a given coal, and the calibrated model will still

be able to predict the coal breakage behaviour on a size–density–

energy basis. Many different combinations of the particle size, den-

sity and grinding energy were investigated. Eventually, seven tests

were selected by trial-and-error, covering the whole range of par-

ticle size, density and grinding energy.  Fig. 10a shows the model

fitting results using the data of the seven tests on the Australian

coal. The four model parameters fit the seven data points well.

For the Australian coal sample, there are 61 sets of available

data. Seven tests were used for calibration, and the remaining 54

tests were used to validate the model. The model was employed

to predict the  t 10  values from the tested size, density and energy,

using the calibrated four model parameters. Fig. 10b presents the

predicted  t 10   versus the measured   t 10. Comparing   Fig. 10b with

Fig. 3, where all 61 sets of data were used to fit the four model

parameters, the new characterisation procedure using seven tests

to calibrate the model parameters predicts almost the identical re-

sults as those using the 61 sets of data, with  R2 of 0.976 for the 7

tests and 0.982 for the 61 tests.

The same seven test conditions were applied to the Chinese coal

sample, with minor variations in the feed particle sizes tested. Thesimilar results are presented in Fig. 11. Note that the prediction re-

sults for the Chinese coal, using the same seven testing conditions

as for the Australian coal, appear more scattered (Figs. 11b com-

pared with 10b). This indicates that more work can be done to

optimise the set of 3D test conditions (size–density–energy) used

for the model calibration.

A statistical   t -test was performed on paired comparisons be-

tween the measured and the predicted t 10, using the model param-

eters calibrated by the seven tests.   Table 2   summarises the

statistical test results. The null hypothesis is that each pair of re-

sults is equal, or that each difference is zero. The null hypothesis

is assumed true unless proved otherwise. The result given in

Table 2 indicates that the differences in  t 10  values determined by

the two methods for both the Australian and Chinese coal

samples do not reach the 95% significant level threshold. The null

Fig. 10.   The multi-component model predictions for the Australian coal, using seven tests to calibrate the model parameters.

Fig. 11.   The multi-component model predictions for the Chinese coal, using the seven Australian coal testing conditions to calibrate the model parameters.

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hypothesis is therefore accepted. This confirms that the multi-

component coal breakage model calibrated with seven reduced

tests can represent the coal breakage behaviour on a size-by–den-

sity-by–energy basis.

Future work remains to develop standardised testing conditions

for calibration when more data are available. Nevertheless this pio-

neering work demonstrates that using reduced coal breakage tests

with the JKFBC to calibrate the model parameters is a viable ap-

proach, and the multi-component model will predict the full

breakage characteristic response of the coal.

4. Conclusions

Based on the multi-component JKFBC test results and the anal-

ysis of the effects of particle size and density on the energy–size

reduction relationship, a multi-component breakage model for coal

has been developed. The model fits the approximately 60 sets of 

 JKFBC tests on each of the Australian coal and Chinese coal samples

well, with only four parameters needing to be calibrated. The mod-

el demonstrates its robustness to replicate the complex patterns of 

coal density and size effects on breakage. The multi-component

model can be switched into a single component model.

A set of   t n-family of curves have been established for coals

ground in the JKFBC. It was found that the data of various particle

sizes and densities, from the two coal samples collected at the Aus-tralian and Chinese power stations, all fall on similar t n-curve trend

lines. These   t n-family of curves can be employed to estimate the

product size distribution from the predicted t 10 values by the mul-

ti-component model.

The multi-component model can be calibrated with seven

 JKFBC tests on a combination of various feed sizes, densities and

specific energies. Predictions for the other testing conditions using

the calibrated model were compared with the measured data to

validate the model. Statistical analysis confirms that the model

predictions are similar to the measured data for both the Austra-

lian and Chinese coal samples.

 Acknowledgements

The coal breakage modelling work was funded by the Australian

Government Department of Resources, Energy and Tourism as part

of the Asia-Pacific Partnership on Clean Development and Climate

program (APP). The Chinese coal sample was provided by the APP

Project collaborative research team from the China University of 

Mining and Technology led by Prof. Yaqun He. The contributions

made by Dr Stephen Larbi-Bram and Mr Weiran Zuo in the exper-

imental work to provide the JKFBC data for the model development

were gratefully acknowledged. The support from Prof. Emmy Man-

lapig of JKMRC in the APP project was much appreciated.

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 Table 2

Summary of   t -test results on a paired comparison between the measured and the

predicted t 10  from the model parameters calibrated by the seven tests.

Coal source Australia China

Mean difference   0.35   0.05

Standard deviation 2.14 3.69

Degree of freedom 60 58

t    1.26   0.096

Distribution 2-Tailed 2-TailedSignificance level (%) 79 8

1162   F. Shi/ Fuel 117 (2014) 1156–1162