1. Prediction of Coal Free-Swelling Index using Mathematical Modelling
Maedeh Tayebi-Khorami
PhD (Mineral Processing / Flotation)
M.Sc. (Mineral Processing)
B.Sc. (Mining Engineering)

2. Standard Laboratory Analysis of Coal
Proximate Analysis
Moisture, Ash, Volatile matter, and Fixed carbon (ASTM D-3172; to D-3175)
Ultimate Analysis (Organic coal substance)
Carbon, Hydrogen, Nitrogen, Sulphur, Oxygen, and Ash (ASTM D-3176 to D-3178)
Mineral Matter (Inorganic constituents of coal)
Clay minerals, sulphide and sulphate minerals, carbonate minerals, Silicate, etc.
Petrographic Analysis
Vitrinite, Liptinite, Inertinite (ASTM D-2798)
There are a large number of coal properties that are measured to help determine the quality or rank of the coal along with its characteristics. 
The most frequently coal analysis are Proximate Analysis, Ultimate Analysis, Mineral Matter, Petrographic Analysis.
These data will be used as the fundamental consideration for coal trading and shows if a particular coal is suitable for use in a particular process.

3. Other Analyses Grindability Plastic Properties (plasticity)
Relative amount of work needed to pulverize coal
Plastic Properties (plasticity)
Changes in a coal upon heating; caking properties of coal
Calorific value
Indication of energy content
Reflectivity
Free swelling index (FSI)
Measure of the increase in volume when a coal is heated
Beside the standard analysis there are various physical and electrical properties of coal that provide even more valuable information about the potential use for coal
Such as Grindability, plasticity, Calorific value, and Free swelling index

4. Free Swelling Index (FSI)
Measure of the volume increase of a coal when heated under specific conditions, and indicates the caking capacity of coal (ASTM D-720 or ISO 501).
FSI is determined by comparing the size and shape of the coke button with a chart of standard profiles and scaling a value from 0 to 9 at an interval of 0.5.
Free Swelling Index is a Measure of the volume increase of a coal when heated under specific conditions, and indicates the caking capacity of coal.
The outcome of the test is a coke button which by comparing the size and shape of the coke button with a series of standard profiles and scaling a value from 0 to 9 and the cokeability of samples is assigned.
Knowledge  of  the  swelling  property  of  a  coal  can  be  used  to  predict  or  explain  the  behaviour  of  the  coal  under various process conditions as well as assisting in the selection of process equipment. It is also an indication of the caking characteristics of the coal when it is burned as a fuel.
(Speight, 2005)

5. Problems associated with the method
Providing the proper heating rate in the furnace
Uneven heat distribution along the walls of the crucible may also cause erratic results.
Oxidation or weathering of the coal sample
A low free swelling index
An excess of fine coal in the analysis
An increase in the FSI
However, There are a few problems associated with the FSI measurement which lead to bias in the result of this test: such as providing the proper heating rate in the furnace, Oxidation or weathering of the sample and the size of samples.
Hence, Statistical models is an alternative approach which can be used to predict FSI and the caking property of the coal without any further experimental cost.
Statistical models is an alternative approach in prediction of FSI from coal analyses to better control the FSI results.

6. Statistical Modelling Artificial intelligence methods
Objective
To predict coke quality (Free Swelling Index) based on various coal analyses for a wide range of Kentucky coal samples by mathematical modelling.
Regression and Adaptive neuro-fuzzy inference systems (ANFIS) were used.
Mathematical Modelling
Statistical Modelling
Non-Linear Regression
Artificial intelligence methods
Adaptive Neuro Fuzzy Inference System (ANFIS)
Artificial Neural Network (ANN)
Genetic Algorithm (GA)
The aim of this work was to predict coke quality or (Free Swelling Index) based on various coal analyses for a wide range of Kentucky coal samples by mathematical modelling.
Regression and Adaptive neuro-fuzzy inference systems (ANFIS) were selected for this study.
Adaptive Neuro Fuzzy Inference System (ANFIS) is one of the most popular and well documented neural fuzzy systems.

7. Database
A total of  808 set of coal samples including the Proximate, Ultimate, Petrography, Mineral Matter, Rmax, and FSI analyses in as determined basis were used.
More than 800 coal sample analyses from the University of Kentucky were used for this study.
The database included the proximate, ultimate, petrography, mineral matter, Rmax analysis and also Free Swelling Index.

8. Correlation Coefficient (R)
By increasing the Moisture and Oxygen content of coal, the FSI decreased.
Higher amount of Carbon, Vitrinite, and Rmax contents of coal results in higher FSI.
The Correlation Coefficient between input variables and FSI were calculated
It showed by increasing the moisture and oxygen content of coal, the FSI decreased. However Higher amount of carbon, vitrinite, and Rmax contents of coal results in higher FSI.

9. Input Parameters
Three different input sets of coal analyses were applied as FSI predictors:
(a) Proximate analysis (moisture, ash, and volatile matter);
(b) Ultimate analysis (carbon, hydrogen, nitrogen, oxygen), forms of sulphur, and mineral matter;
(c) Group-macerals analysis (vitrinite, inertinite, and liptinite), mineral matter, moisture, Rmax, and forms of sulphur.
Three different input sets of coal analyses were applied as FSI predictors:
proximate analysis,
ultimate analysis and mineral matter,
group-macerals analysis, mineral matter, moisture, Rmax, and forms of sulfur, were used to predict FSI by using stepwise regression method and ANFIS procedure.

10. Group-macerals analysis, MM, M, Rmax, and forms of sulphur
Non Linear Regression
Input set
Variables
Regression (R2) a
Proximate analysis
0.38 b
Ultimate analysis forms of sulphur, and MM
0.49 c
Group-macerals analysis, MM, M, Rmax, and forms of sulphur
0.70
The non-linear equations can predict FSI with correlation coefficients (R2) of 0.38, 0.49, and 0.70, for models (a), (b), and (c), respectively.
The results show that input set C can produce better results than Other input sets. The graph shows comparative plots of the FSIs determined experimentally and estimated by multivariate regression from input set C.

11. Group-macerals analysis, MM, M, Rmax, and forms of sulphur
ANFIS
Input set
Variables
Regression (R2) a
Proximate analysis
0.47 b
Ultimate analysis forms of sulphur, and MM
0.82 c
Group-macerals analysis, MM, M, Rmax, and forms of sulphur
0.95
To increase the accuracy of regression results, ANFIS procedure was applied.
The input sets (a) b and (c) were used. And 608 data sets were selected randomly for training and 200 data sets for testing.
The ANFIS procedure can predict FSI with correlation coefficients of 0.47, 0.82, and 0.95 for the input models of (a), (b), and (c), respectively.
The normal distributions of difference between estimated FSI and actual values in for input set (c) is shown in right graph
The left graph shows graphical comparison of experimental FSIs with those estimated by ANFIS model (c) in testing stage.

12. Non-linear Regression vs ANFIS
In this table the results of non-linear regression is compared with ANFIS for different FSI range. It can be seen that the best prediction belong to the strongly caking range.

13. Conclusions
The Correlation Coefficient between input variables and FSI showed by increasing the moisture and oxygen content of coal, the FSI decreased. However Higher amount of carbon, vitrinite, and Rmax contents of coal results in higher FSI.
ANFIS can predict the FSI with more accuracy than regression.
The input set (c) comprised of group-macerals, mineral matter, moisture, organic sulfur, and Rmax was the best model to predict FSI.
The results indicated that ANFIS can be applied as a reliable method to predict FSI.

14. Implication of this method
This method have been used successfully for other coal analyses such as Hardgrove Grindability Index (HGI), Gross Calorific Value (GCV), and Coal Flotation.
The mathematical modelling can significantly reduces the cost of coal analysis by decreasing the number of experiment.

15. Article