Ind. Eng. Chem. Res. 2003, 42, 4919-4926
4919
Prediction of Coal Ash Thermal Properties Using Partial Least-Squares Regression Maurizia Seggiani* and Gabriele Pannocchia Department of Chemical Engineering, Industrial Chemistry and Science of Materials, University of Pisa, Via Diotisalvi, 2 - 56126 Pisa, Italy
The partial least-squares regression (PLSR) method is used to develop empirical correlations for predicting coal ash fusion temperatures (AFTs) under reducing conditions and the temperature of critical viscosity (Tcv) from the chemical composition. The database used in this work includes 433 ash samples from coals of different sources. PLSR is a powerful tool that is able to identify the most significant variables affecting coal ash thermal properties, suggesting ways of modifying the slag melting behavior. The final correlations show good predictive results with predictive standard deviations of less than 90 °C; they represent a simple and reliable tool for predicting the thermal behavior of coal ash and the effects of additives and blending of different coals. 1. Introduction The behavior of the inorganic matter in coal is frequently the limiting factor determining the choice of coal for use in power generation reactors. During combustion, the mineral matter undergoes complex chemical and physical transformations to produce ash that can deposit on the heat-transfer surfaces or other surfaces of the equipment. These deposit phenomena are known as slagging and fouling. Slagging is defined as the formation of fused or highly viscous ash deposits in zones directly subjected to radiant heat exchange (the hottest parts of the boiler). Fouling, instead, is defined as deposition of species that have vaporized and subsequently condensed on surfaces subject to convective heat exchange (the cooler parts of the boiler below the melting temperatures of the bulk coal ash). Although ash fusibility is not the only factor to be considered in choosing a coal for a given application (wet or dry bottom), ash fusion temperatures (AFTs) are the most common parameters used by furnace and boiler operators to predict the melting behavior of the coal ash in power generation reactors and to determine whether slagging and ash-deposition problems will be encountered during combustion. For these reasons, it is important to develop a simple and reliable tool for predicting AFTs from the ash chemical composition alone and for understanding the effects of various components on the ash thermal properties. The conventional method (ASTM D1857) of ash fusibility characterization consists of observing cones or pyramids of ash (prepared at 815 °C in a muffle furnace) in an oven in which the temperature is continuously increased under a reducing atmosphere. The four ash fusion temperatures recorded as the characteristic of various stages of ash melting are the IDT (initial deformation temperature), the temperature at which rounding of the specimen apex is first observed; ST (softening temperature), the temperature at which the height of the specimen is equal to its width; HT * To whom correspondence should be addressed. Tel.: +39050-511281. Fax: +39-050-511266. E-mail: m.seggiani@ ing.unipi.it.
(hemispherical temperature), the temperature at which the height becomes equal to half of the width; and FT (fluid temperature), the temperature at which the fused mass spreads out in a nearly flat layer with a maximum height of 1.5 mm. Another property of interest to engineers designing gasifiers, wet-bottom furnaces, and slag taps, as well as researchers modeling slag flow, is the temperature of critical viscosity, Tcv, that is, the temperature at which the viscosity properties of the molten slag change from those of a Newtonian fluid to those of a Bingham plastic. The determining the relationship between Tcv and the ash composition is much more complex than predicting the fluidity of the slag because the beginning of crystal formation (corresponding to Tcv) depends on phase equilibria within a complex multicomponent system, on the presence of catalysts for the formation of crystals, and on cooling rates of the slag system.1 Tcv is difficult and expensive to measure; for this reason, most fuel technologists use empirical correlations to estimate it. Sage2 proposed the calculation of Tcv by simply adding 200 °F (111 °C) to the ash softening temperature (ST) determined under reducing conditions, and some researchers of the British Coal Utilisation Research Association (BCURA)1 developed an empirical expression for Tcv in terms of the ash composition alone, based on measured values of 63 coal ash slags. However, Sage’s method cannot be generalized, and the BCURA correlation gives incorrect results when it is applied to coal ash samples from different sources and with different compositions. Many studies have reported empirical and statistical correlations between AFTs and coal ash compositions derived using multivariate regression analysis,1,3-6 neural networks,7 or ternary phase equilibria diagrams.8 In the latter case, the AFTs are correlated with the liquidus temperatures calculated by computer thermodynamic modeling of pseudo-ternary phase equilibria for the Al-Ca-Fe-O-Si system.8 Therefore, this approach is applicable when the ash composition can be expressed in terms of Al2O3, CaO, Fe2O3, and SiO2 alone and the concentrations of other components (MgO, K2O, Na2O, etc.) are negligible.
10.1021/ie030074u CCC: $25.00 © 2003 American Chemical Society Published on Web 08/27/2003
4920 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Of the methods mentioned above, correlations that use linear combinations of parameters (deriving from the ash composition) based on a broad database with good predictive correlation coefficients represent the simplest and most flexible tools for predicting the ash thermal properties. Moreover, such correlations can be used to predict the effects of adding minerals (such as CaCO3 and K2CO3) that can be intentionally added to modify the slag behavior. In a previous work,6 empirical correlations were derived by multiple linear regression (MLR) employing the 49 variables suggested by Winegartner and Rhodes3 (including chemical properties such as weight and mole percentages of the main ash oxides; cross products, sums, and ratios of these values; and several commonly used parameters such as silica value, base, acid, etc.). The database included 295 samples: 260 samples of coal ash from different sources and 35 samples of biomass ash. In the present work, the previous approach is applied to a larger database (433 samples) including only coal ash samples with a wide range of composition, and the partial least-squares regression (PLSR) method is used to correlate the coal ash fusibility under reducing conditions with the chemical composition. (The database is omitted due to space limitations; however, it can be acquired by writing to the authors.) PLS regression (which also means projection to latent structures) is an extension of MLR that was developed a few decades ago.9,10 It is a powerful technique, frequently applied in inferential process monitoring11,12 and control,13-15 and it has become a standard tool in chemometrics16 in recent years. PLSR is able to analyze data of complex and highly correlated multivariate systems and to provide linear correlations that have a much higher prediction ability than those obtained with MLR. Moreover, PLSR can be used to identify the most significant parameters affecting the property investigated. This latter point is particularly useful in identifying a strategy for modifying slag melting or fluxing behavior, for example, by adding basic oxides or by blending coals. The main characteristics of PLSR are described in the next section. 2. Short Review of Partial Least-Squares Regression In this section, we provide a short review of the PLSR modeling method used to build a linear model for each AFT and for Tcv. In this work, each property is modeled separately for a number of reasons (this approach is often referred to as the PLS-1 approach). First, the AFTs and Tcv are not necessarily related to each other, and hence, separate PLS modeling is to be preferred.16 Also, the final objective of this work is to reduce the number of model parameters and to find, for each property, the most important predictors. This can only be done by modeling each property separately. 2.1. Introduction to Multivariate Regression. The objective of multivariate (or multiple) regression is to find a linear combination of the independent variables (X) that predicts the dependent variable (Y) as closely as possible
yˆ ) R0 + R1x1 + R2x2 + ... + Rmxm
(1)
where yˆ is the predicted value of one of the temperatures investigated, y (one of the AFTs or Tcv); x1, ..., xm are
the chemical parameters used in the model (see section 3); R0, R1, ..., Rm are the model coefficients. Given a training set of n calibration samples, both the Y variable and the X variables are centered with respect to their mean value and scaled to unit variance. Then, a column vector Y ∈ Rn and a matrix X ∈ Rn×m are built by stacking the corresponding (centered and scaled) y and xi values of each calibration run as rows. Thus, the corresponding linear model prediction can be written as
Y ˆ ) XKT
(2)
in which K ) [k1 k2 ‚‚‚ km] is a constant coefficient vector to be determined and Y ˆ ∈ Rn is the vector of the predicted (centered and scaled) Y variable. Note that a constant term does not appear in eq 2 because both yˆ and xi are centered. Also note that straightforward linear algebraic relations exist between the coefficients ki in eq 2 and Ri in eq 1. It is well-known that the MLR coefficient vector is given by the following least-squares solution (see, e.g., Golub and Van Loan17)
KMLR ) YTX(XTX)-1,
(3)
which minimizes the variance of Y - Y ˆ for the training data. A measure of the goodness of fit is given by the explained variance, R2 (sometimes referred to as the multiple correlation coefficient), which is defined as n
R2 ) 1 -
(yi - yˆ i)2 ∑ i)1 n
(4)
(yi - yj) ∑ i)1
2
where yi and yˆ i are the ith values (in the training set) of the experimental Y variable and of the corresponding variable given by the linear model in eq 1, respectively, and yj is the mean value (in the training set) of the experimental Y variable. It is clear from eq 4 that a linear model fits the data well if R2 is close to 1; moreover, if R2 is negative, the linear model fits the data worse than a constant mean value does. Sometimes, it is also interesting to quantify the goodness of fit in terms of standard deviation, σ, defined as
σ)
x
1
n
(yi - yˆ i)2 ∑ n - 1i)1
(5)
It is straightforward to show that the higher the value of R2, the smaller σ. The main drawback of MLR is that, when the number of X variables is large (say, greater than 5-10) and the X variables are partially dependent on each other, the condition number of (XTX) is high, and the model coefficients are extremely sensitive to small perturbations in both the Y and X variables. Thus, in general, the prediction ability of MLR models can be poor, i.e., the explained variance computed as in eq 4 for data not used in the model calibration can be small (or even negative). This predictive explained variance is denoted as Q2.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4921
2.2. PLSR and Cross-Validation. PLSR generates a few X scores (a in number), called latent variables, that can capture most of information contained in the independent variables X, which is useful for predicting the dependent variable Y. These new variables are a linear combination of the original ones
T ) XW
(6)
in which T ∈ Rn×a and W ∈ Rm×a is an appropriate weight matrix, computed in such a way that the covariance between the Y variable and the X scores is maximized. When multiplied by an appropriate loading matrix P ∈ Rm×a, the X scores are good summaries of the original X variables
X ) TPT + E
(7)
in which E ∈ Rn×m contains the X residuals. Then, a linear regression model for Y is carried out, leading to
Y ) TCT + F ) XWCT + F
(8)
where C ∈ R1×a is an appropriate row vector and F ∈ Rn contains the Y residuals. Finally, the PLSR model can be written as
Y ˆ ) XKPLST in which the PLSR coefficient vector is given by
KPLS ) CWT
(9)
The choice of the number of latent variables a e m is particularly important in determining the goodness of fit and the prediction ability of the PLSR linear model, and it has been the subject of extensive research.16 It is easy to show that the explained variance R2 monotonically increases with the number of latent variables and is equal to that of MLR when a ) m. However, the prediction ability of the model is usually maximized when the number of latent variables a is smaller than the number of parameters m. It is well established that the best results can be obtained by means of the so called cross-validation (CV) method, which is briefly described here (see Wold et al.16 and references therein for more details on CV). Typically, CV is performed by dividing the training data set into a number of groups, G (usually from 5 to 9), and then developing G parallel models from reduced data with one of the groups deleted. For each model, the predictive explained variance Q2 is calculated for the group taken out, and a mean value is taken. This procedure is repeated for an increasing number of latent variables a ) 1, ..., m, and then the optimal number of latent variables, a*, is chosen so that the mean predictive explained variance Q2 is a maximum. Finally, the “optimal” model is obtained by performing PLSR with a* latent variables on the whole training data set. It is important to remark that, for a model to have good predictive ability, parameters significantly correlated with the property of interest should be used. This often requires one to preprocess the experimental data and generate an appropriate set of model parameters. 2.3. Reduction of Parameters. PLSR can also be used to select a smaller number from among the m
Table 1. Chemical Composition Ranges of Coal Ashes component
range (wt %)
SiO2 Al2O3 TiO2 Fe2O3 CaO MgO K2O P2O5 Na2O SO3
4.6-72.5 0.9-46.8 0.0-3.2 0.5-69.9 0.1-41.6 0.1-10.1 0.0-6.0 0.0-9.5 0.0-9.9 0.0-24.3
available parameters to build a simpler linear model with good predictive properties. The choice of the relevant parameters is of particular interest in inferential control,13,15,18 and in the present work, a technique similar to the one proposed by Mejdell and Skogestad13 is used and briefly summarized below. Once a PLSR model with m parameters has been found (as described in the previous paragraph), a coefficient vector, KPLS, is obtained for the meancentered covariance-scaled data. Because of this scaling, the magnitude of each coefficient is related to the correlation between the corresponding parameter and the investigated property. Hence, the m parameters can be ranked in descending order of the absolute value of the corresponding coefficient in KPLS. Next, a reduced set of parameters is chosen as the first r (with r e m) most significant parameters. Similarly to the approach used with the number of latent variables, the reduced number of parameters is chosen by means of CV, i.e., one computes different PLSR models with an increasing number of parameters r ) 1, ..., m, and then one chooses the optimal reduced number of parameters such that the mean predictive explained variance is a maximum. 3. Procedure In this work, 49 chemical parameters are used to predict the four ash fusion temperatures (IDT, ST, HT, FT) under reducing conditions and the temperature of critical viscosity, Tcv. The definitions and calculation procedures for the parameters and independent variables are given in Appendix A. Most of the parameters used are those suggested by Winegartner and Rhodes.3 The parameters include the mole percentages of the nine main ash oxides (i.e., SiO2, Al2O3, TiO2, Fe2O3, CaO, MgO, K2O, P2O5, and Na2O), normalized and SO3free, and squares, cross products, sums, and various combinations of these values. Widely used parameters such as silica value, base, acid, dolomite ratio, and R250 are also considered (see Appendix A). Information on these parameters can be found in ref 19. The database includes 433 samples of coal ash from different sources (American, Italian, Spanish, Polish, Australian, German, African, French, Albanian, etc.) with a wide range of compositions, as reported in Table 1. Different numbers of experimental data are available for each property: 312 for IDT, 226 for ST, 162 for HT, 260 for FT, and 202 for Tcv. The data available for each property are randomly separated into a set of training data (80% of the total data) used to build the models and a set of test data (20%) used to validate and compare the models. Using the training data set of each property, MLR and PLSR models are built using the 49 parameters previously discussed. Note that, for PLSR models, the number of
4922 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 2. Results for PLSR and MLR Models with 49 Parameters training PLSR
validation MLR
PLSR
MLR
property
R
σ (°C)
R
σ (°C)
Q
σ (°C)
Q
σ (°C)
IDT ST HT FT Tcv
0.77 0.85 0.81 0.76 0.48
86 65 80 87 88
0.82 0.88 0.92 0.82 0.76
77 59 54 76 65
0.83 0.81 0.79 0.78 0.52
78 74 83 86 87
0.71 0.71 0.46 0.23 a
99 88 121 134 123
a For the MLR model of T , the predictive explained variance cv Q2 is negative; hence, Q is not a real value.
Figure 2. FTs predicted for the validation data set by PLSR and MLR models with 49 parameters.
Figure 1. IDTs predicted for the validation data set by PLSR and MLR models with 49 parameters.
latent variables is chosen by means of CV with G ) 5 groups as discussed in section 2.2. Next, the number of parameters is reduced according to the procedure described in section 2.3, and finally, MLR and PLSR models for the reduced set of parameters are built. That is, for each property, four candidate models are built and finally compared in terms of predictive ability, i.e., in the terms of the explained variance and standard deviation computed on the validation data set (the set not used to build the models). These calculations were performed using a MATLAB program written by the authors. Note that several PLSR algorithms exist in the literature, and in the present work, the SIMPLS algorithm20 was used. 4. Results and Discussion The square root of the multiple correlation coefficient, R, and the standard deviation computed on the training data set for each of the PLSR and MLR models with all 49 parameters are reported in Table 2. In the same table, the corresponding square root of the predictive correlation coefficient, Q, and the predictive standard deviation computed on the validation data set are given. MLR gives better results (higher R and lower σ) than PLSR on the training data set but poorer predictions on the data set used in validation (lower Q and higher predictive σ). Comparing the Q values of MLR and PLSR, it is clear that the prediction ability of PLSR is significantly better than that of MLR. As examples, Figures 1 and 2 show the abilities to predict IDT and FT, respectively, by the PLSR and MLR models with 49 parameters. The Ri coefficients of the models obtained by PLSR with 49 parameters are reported in Table 3.
Figure 3. IDTs predicted for the validation data set by PLSR and MLR models with a reduced number of parameters.
Reducing the number of parameters does not produce negative effects on the predictive ability of PLSR models; in fact, it improves the predictive abilities for the HT and Tcv (see Tables 2 and 4). The MLR models with the reduced set of parameters suggested by PLSR show predictive abilities much better than those of the MLR models with 49 parameters for all properties (compare the Q values for MLR in Tables 2 and 4); this indicates that not all initial parameters are good descriptors for the model, so that the removal of some of them is beneficial. As reported in Figures 3 and 4 for IDT and FT, respectively, the PLSR and MLR models with the reduced set of parameters show similar predictive properties. The final reduced models involve from 11 to 17 parameters. The Ri coefficients of the models obtained by PLSR with the reduced set of parameters are reported in Table 5. A Q value of 0.7 is generally acceptable, 0.8 is good, and 0.9 or higher is excellent.3 As can be seen in Table 4, all of the reduced models, except those for Tcv, give Q values above 0.75 with predictive σ values below 90 °C. It must be remembered that the reproducibility of the AFTs determined by the same operator and using the same apparatus is about 30-50 °C. Between different laboratories (different operators and apparatuses), the AFT reproducibility is about 50-80 °C, and that for the
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4923 Table 3. Coefficients of PLSR Models for the Four AFTs and Tcv no.
term
IDT
1 2 3 4 5
e0.1SV2 [P2O5] [SiO2] [Fe2O3] [Al2O3]
1540 88.7 -0.725 -1.14 4.18
6 7 8 9 10
[TiO2] [CaO] [MgO] [K2O] [Na2O]
38.0 0.531 -4.47 0.235 -6.72
11 12 13 14 15
[P2O5]2 [SiO2]2 [Fe2O3]2 [Al2O3]2 [TiO2]2
16 17 18 19 20
ST
HT
1890 287 -0.531 -0.005 27 4.02
930 18.3 0.345 -0.495 0.0730
-41.3 0.377 -2.56 8.24 -3.69
FT
Tcv
1320 -21.4 -0.424 0.838 4.85
92.2 -82.8 -0.153 -0.392 1.89 -2.67 -0.0313 -0.373 -1.54 -0.865
50.3 0.118 -1.47 -14.8 -12.6
41.6 0.120 -8.23 -2.12 3.40 26.9 0.006 14 0.245 0.233 10.9
-27.7 -0.001 53 0.341 0.142 -5.91
-516 0.000 281 0.244 0.172 26.5
-27.4 0.009 20 0.0728 0.0505 21.1
[CaO]2 [MgO]2 [K2O]2 [Na2O]2 [SiO2][Fe2O3]
0.0444 0.230 -2.61 -0.414 -0.106
0.0630 0.617 -4.58 -0.548 -0.0656
0.0351 0.0405 -0.554 -0.0922 -0.0440
0.0205 0.396 3.53 -0.218 -0.0507
-0.002 08 -0.0624 -0.410 -0.142 -0.009 79
21 22 23 24 25
[SiO2][Al2O3] [SiO2][CaO] [Fe2O3][Al2O3] [Fe2O3][CaO] [Fe2O3][MgO]
0.0607 -0.0205 -0.200 0.516 -0.0450
0.0790 -0.0507 -0.314 0.630 -1.50
0.0111 -0.0244 -0.0869 0.108 -0.131
0.0379 -0.0281 -0.0350 0.554 -0.793
0.0193 -0.000 317 0.004 01 -0.000 886 -0.0514
26 27 28 29 30
[Al2O3][CaO] [CaO][MgO] [SiO2] + [Al2O3] [CaO] + [MgO] [Fe2O3]/[CaO]
0.002 97 -0.0158 0.106 0.0952 8.05
-0.0503 0.0526 -0.0134 0.0951 11.9
-0.0499 0.0709 0.269 -0.133 11.4
0.0224 0.142 0.341 -0.577 3.90
0.0121 -0.0453 0.206 -0.0468 -0.525
31 32 33 34 35
[SiO2]/[Al2O3] ([SiO2]/[Al2O3])2 ([CaO] + [MgO])/[Fe2O3] B B2
6.22 1.94 -0.234 -0.215 0.0229
5.60 1.79 0.110 0.0341 0.0325
5.49 0.399 2.10 -0.338 0.0140
6.43 1.97 0.906 -0.384 0.0206
-5.80 -0.644 -0.164 -0.185 -0.004 51
36 37 38 39 40
e0.1(B/A-1)2 DR2 B/A (B/A)2 (B/A - 1)2
122 9.09 32.8 34.9 7.12
41 42 43 44 45
[Na2O](B/A) [Fe2O3]/B R250 |B/A - 1| SV
-4.52 -91.5 -20.7 -64.3 96.0
46 47 48 49
SV2 DR e10-4[SiO2][Al2O3] e10-2([SiO2] + [Al2O3]) constant
-239 -38.8 41.8 26.2 -33.5 -3.30 -51.0 -20.5 -93.8 97.5
155 -31.6 609 29.0 -1540
standard deviation σ (°C) R predictive standard deviation σ (°C) Q
665 9.00 30.1 47.3 66.5 0.428 12.2 -1.69 20.3 57.5
187 -124 793 31.3 -1610
86 0.77 78 0.83
93.8 -26.3 126 31.3 -845
65 0.84 74 0.81
-132 -0.001 81 -0.0176 0.0556 -1.63
351 -4.09 36.8 39.6 32.1
68.5 5.02 -12.2 -19.5 7.28
-10.4 -30.0 -35.2 -46.7 60.6
-2.23 -10.9 14.7 12.2 12.6
133 -34.6 395 42.2 -1270
9.64 6.33 175 9.14 932
80 0.81 83 0.79
87 0.76 86 0.78
88 0.48 87 0.52
Table 4. Results for PLSR and MLR Models with Reduced Number of Parameters training
validation
PLSR
MLR
PLSR
MLR
property
number of parameters
R
σ (°C)
R
σ (°C)
Q
σ (°C)
Q
σ (°C)
IDT ST HT FT Tcv
13 11 11 12 17
0.75 0.82 0.80 0.75 0.59
88 70 81 88 81
0.77 0.83 0.82 0.75 0.64
86 68 77 87 77
0.80 0.76 0.84 0.78 0.62
84 82 73 86 80
0.82 0.77 0.80 0.79 0.66
80 80 82 85 76
IDT is 70-150 °C.21 With this in mind, it appears that the predictive σ values calculated are more than acceptable.
The difficulty of predicting Tcv (lower Q values) in terms of the ash composition alone was to be expected given the complexity of the crystallization phenomenon
4924 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 5. Coefficients of the Reduced PLSR Models for the Four AFTs and Tcv no.
term
1 2 3 4 5
e0.1SV2
6 7 8 9 10
[TiO2] [CaO] [MgO] [K2O] [Na2O]
11 12 13 14 15
[P2O5]2 [SiO2]2 [Fe2O3]2 [Al2O3]2 [TiO2]2
16 17 18 19 20
[CaO]2 [MgO]2 [K2O]2 [Na2O]2 [SiO2][Fe2O3]
21 22 23 24 25
[SiO2][Al2O3] [SiO2][CaO] [Fe2O3][Al2O3] [Fe2O3][CaO] [Fe2O3][MgO]
26 27 28 29 30
[Al2O3][CaO] [CaO][MgO] [SiO2] + [Al2O3] [CaO] + [MgO] [Fe2O3]/[CaO]
31 32 33 34 35
[SiO2]/[Al2O3] ([SiO2]/[Al2O3])2 ([CaO] + [MgO])/[Fe2O3] B B2
36 37 38 39 40
e0.1(B/A-1)2 DR2 B/A (B/A)2 (B/A -1)2
41 42 43 44 45
[Na2O](B/A) [Fe2O3]/B R250 |B/A - 1| SV
46 47 48 49
SV2 DR e10-4[SiO2][Al2O3] e10-2([SiO2] + [Al2O3]) constant
[P2O5] [SiO2] [Fe2O3] [Al2O3]
standard deviation σ (°C) R predictive standard deviation σ (°C) Q
IDT 2040 83.4
ST 5360 91.3
HT 2150
2.12
FT 2240
6.13
39.3
53.1
Tcv -935
4.11
58.0 -13.8
-25.3 2580 0.335 0.118
0.282
0.259 0.278
0.254
16.0 0.135
0.178 0.939
0.0877 0.736 -0.139 0.108
-0.116 0.0768 0.533
0.0377 0.630 -1.03
0.259 -0.730
19.3 2.42
2.34
0.285
2.03
14.0 3.05 0.006 91
910 41.9 86.4
92.0
7.40 -113 -5.48 -164 -7.40
205 780 -2170 88 0.75 84 0.80
that characterizes it. Nevertheless, the Tcv correlation obtained by PLSR with the reduced set of parameters shows a moderate predictive standard deviation (σ ) 80 °C). The numbers and types of parameters that give the best predictive results in PLSR modeling are different for the various properties, as shown in Tables 4 and 5. Owing to interactions between the various parameters, care must be exercised in interpreting these results. The parameters that arise more frequently in the reduced correlations for the AFTs are e0.1SV2, [TiO2], [Fe2O3]2, [CaO]2, [Fe2O3]‚[CaO], ([SiO2]/[Al2O3])2, and SV2.
-140 -85.9 3120 -7820 70 0.82 82 0.76
216
231 409
-2120 81 0.80 73 0.84
-1340 88 0.75 86 0.78
675 81 0.59 80 0.62
[SiO2] and [SiO2]2 do not appear in the reduced models; rather, the combinations of silica with other components represented by ([SiO2]/[Al2O3])2 and SV2 appear to be most significant. The reduced PLSR models can be used to predict the effects of basic additives such as CaCO3 (CaO) purposely added to modify and control the ash fusion behavior. As example, Figure 5 shows the effects of the addition of different amounts of CaO to a coal ash sample with 5 wt % CaO (15 mol % base) on IDT and FT as predicted by the corresponding reduced PLSR models. The calculated IDTs and FTs of ash-CaO mixtures gradually decrease with increasing CaO content until the CaO
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4925
Figure 4. FTs predicted for the validation data set by PLSR and MLR models with a reduced number of parameters.
Figure 5. Effects of CaO content in ash-CaO mixtures on IDTs and FTs calculated by reduced PLSR models.
reaches about 20-25 wt % for IDT and 30-40 wt % for FT (corresponding to about 50% base on a mole basis), and then they increase as the CaO content of the mixture is further increased. The predicted trends are consistent with the typical U-shaped curves obtained experimentally for the AFTs plotted against the percent base, in which a minimum typically occurs at about 50 mol % base.1,3 Another important chemical factor frequently used is the [SiO2]/[Al2O3] ratio. Huggins et al.22 observed that the AFTs increase as this ratio decreases, that is, as the ash becomes richer in kaolinite. To predict the individual effect of this factor on the ash fusibility, a coal ash sample ([SiO2]/[Al2O3] ) 3.3 on a mole basis) was considered in which the sum of the SiO2 and Al2O3 contents (on a weight basis) was kept constant and the ratio between the two acid oxides was varied. The effects of the [SiO2]/[Al2O3] ratio on IDT and FT predicted by reduced PLSR models are reported in Figure 6. As shown, the results are in accordance with the findings of others,22,23 that is, as this ratio increases, the AFTs increase as well. 5. Conclusions In this paper, empirical correlations were derived to predict coal ash fusibility temperatures (AFTs) and the temperature of critical viscosity (Tcv) from the ash
Figure 6. Effects of SiO2/Al2O3 mole ratio on IDTs and FTs calculated by reduced PLSR models.
composition using partial least-squares regression (PLSR), which represents a reliable tool for analyzing highly correlated multivariate systems.16 The models used 49 parameters suggested by Winegartner and Rhodes,3 which include mole percentages of the main oxides and their linear and nonlinear combinations. The model coefficients were found by performing regression on a database of 433 samples corresponding to a wide coal ash composition range. A randomly chosen subset of 80% of the samples was used to calibrate the models, and the remaining subset was used to validate the predictive abilities of the models. To maximize the predictive abilities of the models, the number of latent variables used in the PLSR was chosen by means of cross-validation.10,16 PLSR models were compared with the corresponding multiple linear regression (MLR) models, and the PLSR models showed significantly higher predictive capabilities. Moreover, PLSR was used to identify the most important parameters among the 49 considered, giving models with a reduced number of parameters (ranging from 11 to 17) for AFTs and Tcv. These reduced models showed predictive abilities similar or even superior to those of the 49-parameter models (predictive standard deviations of less than 90 °C), indicating that not all of the original parameters are good descriptors of the AFTs and Tcv. Given the broad composition range of the database used in training and validation, the final reduced PLSR models represent a simple and reliable tool for predicting the AFTs of coal blends and/or the effects of basic oxides added to modify coal ash melting behavior. Appendix A. Explanation and Calculation of the Parameters Used in the Correlations (a) The ash composition is expressed as mole percentages of the oxides, assuming that iron is present as Fe2O3 and normalizing to an SO3-free basis. (b) The silica value, SV, is calculated as
SV )
SiO2 SiO2 + Fe2O3 + CaO + MgO
in which the compositions are expressed as weight percentages;
4926 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
(c) The percent base, B, is determined as the sum of the mole percentages of the basic components, i.e.
B ) Fe2O3 + CaO + MgO + K2O + Na2O (d) The percent acid, A, equals 100 minus the percent base or the sum of the mole percentages of SiO2, Al2O3,TiO2, and P2O5. (e) The dolomite ratio, DR, is the weight fraction of base that is present as CaO and MgO, i.e.
DR )
CaO + MgO Fe2O3 + CaO + MgO + K2O + Na2O
(f) R250 is calculated as
R250 )
SiO2 + Al2O3 SiO2 + Al2O3 + Fe2O3 + CaO
in which the compositions are expressed as mole percentages. Literature Cited (1) Bryers, R. W. Fireside slagging, fouling, and high-temperature corrosion of heat-transfer surface due to impurities in steamraising fuels. Prog. Energy Combust. Sci. 1996, 22, 29-120. (2) Sage, W. L.; McIlroy, J. B. Relationship of Coal-Ash Viscosity to Chemical Composition. Trans. ASME Ser. A: J. Eng. Power 1960, 82, 145-155. (3) Winegartner, E. C.; Rhodes, B. T. An EmpiricalStudy of the Relation of Chemical Properties to Ash Fusion Temperatures. Trans. ASME, Ser. A: J. Eng. Power 1975, 97, 395-406. (4) Kucukbayrak, S.; Ersoy, M. A.; Haykiri, A. H.; Guner, H.; Urkan, K. Investigation of the relation between chemical composition and ash fusion temperatures for some Turkish lignites. Fuel Sci. Technol. Int. 1993, 11 (9), 1231-1249. (5) Llyod, W. G.; Riley, J. T.; Zhon, S.; Risen, M. A.; Tibbitts, R. L. Ash fusion temperatures under oxidizing conditions. Energy Fuels 1993, 7 (4), 490-494. (6) Seggiani, M. Empirical Correlations of the Ash Fusion Temperatures and Temperature of Critical Viscosity for Coal and Biomass Ashes. Fuel 1999, 78 (9), 1121-1125. (7) Yin, C.; Luo, Z.; Ni, M.; Cen, K. Predicting coal ash fusion temperature with a back-propagation neural network model. Fuel 1998, 77, 1777-1782.
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Received for review January 27, 2003 Revised manuscript received July 14, 2003 Accepted July 15, 2003 IE030074U
