360
Energy & Fuels 1990,4, 360-364
percritical fluid extraction, Western Canadian coals are promising candidates for pyrolytic preprocessing of coal destined for combustion and electrical energy generation.
Acknowledgment. Financial support for this work was provided by the Natural Sciences and Engineering Re-
search Council (NSERC) of Canada and is gratefully acknowledged. We are also grateful to the Canada-Japan Joint Academic Research Program on the Efficient Use of Coal for sponsoring M.S. Registry No. Toluene, 108-88-3.
Estimation of Ash Softening Temperatures Using Cross Terms and Partial Factor Analysis William G. Lloyd,* John T. Riley, Mark A. Risen, Scott R. Gilleland, and Rick L. Tibbitts Department of Chemistry and Center for Coal Science, Western Kentucky University, Bowling Green, Kentucky 42101 Received February 5, 1990. Revised Manuscript Received April 20, 1990
The relationship between elemental composition of coal ash and ash softening temperature has been studied, using a set of 70 ashes prepared from blends of seven source coals. Simple multiple linear regression analysis is of limited value. The inclusion of second-order cross terms, representing acid-base and other metathetic reactions, markedly improves the precision of predictive regressions. Excessive collinearity, notably among five major components of these coal ashes, limits the combinations of regressor terms usable in a predictive equation. A strategy for avoiding collinearity problems by means of partial factor analysis is illustrated. Best estimates of ash softening temperature show root mean square errors of 45 O F (25 K) or less.
Introduction Most coal-fired boilers are designed for the continuous removal of bottom ash as a dry solid. Others are engineered to burn at higher temperatures, with the removal of slagged ash as a viscous fluid. None, however, can operate in both modes. Since ash softening temperatures are commonly found over a range of 900 O F (500 K), it is of great importance to the power plant operator to know something about the ash softening temperature of his boiler feed before the coal is fed to the boiler. The experimental determination of ash fusion temperatures, for example, by ASTM Procedure D 1857 (IS0 540), provides a useful laboratory measurement closely related to fireside slagging potential.' This determination requires both skill and time. The ASTM procedure requires that the coal, after sampling, be prepared in accordance with Method D 20132and that it be ashed t~ two stages (in air and again under pure oxygen). The resulting ash is then compounded with a dextrin solution, the paste thus formed is pressed into cone molds, and the resulting ash cones are dried and then fired to remove the dextrin binder. All of these steps must be carried out before starting the actual determination of ash fusion temperat u r e ~ . In ~ the hands of experienced and careful workers (1)Gluskoter, H.J.; Shimp, N. F.; Ruch, R. R. Coal Analysis, Trace Elements, and Mineral Matter. In Chemistry of Coal Utilization;Elliott, M. A., Ed.; Why: New York, 1981;2nd suppl. vol., pp 394-395. (2) Method of Preparing Coal Samples for Analysis; Method D 2013; Annual Book of ASTM Standards, Vol. 5.05;American Society for Testing and Materials: Philadelphia, PA (published annually). (3) Test Method for Fusibility of Coal and Coke Ash; Method D 1857; in ref 2.
this procedure provides fusion temperatures which are repeatable and which correlate well with boiler'performance. From the viewpoint of the power plant operator, however, there are two big problems with reliance upon the determination of ash fusion temperatures: 1. If the above laboratory steps are followed faithfully, an ash fusion temperature determination will require a t least 3 days. The plant operator often needs to know the answer within hours, or even within tens of minutes. 2. Most power plants burn blended coals. The ash fusion temperatures of ashes from blended coals are not even roughly approximated by interpolations from the ash fusion temperatures of the unblended source coals.4* When a desired property, such as ash softening temperature, can be correlated with a more easily measured property, such as a specific ash component, regression analysis can provide a predictive equation that may be useful. For example, for the 70-ash database of this study Tso, is significantly correlated with the TiOp content of the ash:
?'son
(OF)
= 1652
+ 520.6[% Ti02]
(1)
For this regression the correlation coefficient R = 0.67. R2 (0.45) is the fraction of the variation in values of T S O ~ explained by the regression, the balance of variation being attributed to random error. (4)Riley, J. T.; Gilleland, S. R.; Forsythe, R. F.; Graham, H. D., Jr.; Hayes, F. J. Proc. Conj.-Znt. Coal Test. Conj. 1989, 7, 32-38. (5)Gray, V. R.Fuel 1987, 66, 1230. (6) Huffman, G. P.; Huggins, F. E.; Dunmyre, G. R. Fuel 1981,60,585.
0887-0624/90/ 2504-0360$02.5O/O 0 1990 American Chemical Society
Estimation of Ash Softening Temperatures
coal rank source
86027 LigA PRPSb
85091 LigA BDPSb
moisture % ash % VM %C % H % N
18.0 18.9 41.0 55.3 3.33 0.38 0.73 8770 20.4 35
18.4 15.1 39.5 59.4 3.57 0.96 0.34 9610 22.3 34 0
%
s
Btuflb MJ/kg HGI FSI
0
Energy & Fuels, Vol. 4, No. 4, 1990 361 Table I. Source Coals Used" 82045 86039 SubC SubB Belle Jacob's AyrC RanchC 16.0 5.8 43.2 68.0 4.22 0.93 0.36 11340 26.4 41
15.2 8.3 44.1 66.8 4.60 0.97 0.64 11340 26.4 40
0
0
85099 hvBb WKY no. 12d
86046 hvAb Poplar Licke
86026 mvb Consol Coal C d
5.9 15.2 35.7 64.6 4.03 1.37 3.78 11420 26.5 50 1
2.0 10.8 38.4 73.9 5.06 1.57 0.89 13190 30.7 40 4
1.1 6.0 25.3 84.4 4.55 1.08 0.85 14670 34.1 90 9
"Except for the moisture determination, analyses are on a dry basis. Apparent rank is estimated using analyzed moisture. Saskatchewan, Canada. e Wyoming. dMuhlenberg County, Kentucky. e Bell County, Kentucky. IPennsylvania.
oxide SiOz PZO6
86027 41.1 0.63 7.84 1.60 13.2 0.82 4.20 0.92 1.40 23.2
so3
KZ0 CaO TiO, Fez03 Na20 MgO A1203
Table 11. Elemental Composition of Ashes from Source Coals 85091 82045 86039 85099 44.9 35.5 45.5 33.1 0.45 1.50 0.20 1.31 9.26 0.72 10.54 5.66 1.02 0.16 2.63 0.48 13.2 26.9 0.55 18.5 1.07 1.25 1.00 1.12 24.1 3.49 5.57 6.25 0.51 0.72 1.04 1.20 1.07 3.27 1.20 4.59 19.2 21.4 14.7 16.7
Equation 1 has a root mean square error of estimate (rmse) of 112 O F (62 K), an uncertainty that is too large to make this regression useful. The goal of studies of the estimation of ash fusion temperatures is to find regression equations with high values of R-perhaps 0.90 or higher-and low values for the rmse-perhaps as low as 60 OF (33 K) or less. No such simple regression has been found. A number of ingenious efforts have been made, using as the predictive variable the sum of the basic oxides [FeZOB, CaO, MgO, Na20, and K20], the sum of the acidic oxides, the ratio of basic to acidic oxides, and more complex expressions such as slagging and fouling factors (products of the ratio of basic to acidic oxides and an additional concentration term such as [ 9% Na20]). These formulations have been extensively reviewed;*l0 they are interesting and indicative but insufficiently precise to be of great practical value. For the data in the present study, for example, the best of seven such formulations (the s u m of acidic oxides) affords an R2 value of 0.34 and a rmse of 123 OF (68 K), inferior to that of eq 1. Within the past decade, the rapid increase in the availability of small computers and of powerful statistical program packages has made possible the application of multiple linear regression (MLR) analysis, by means of which an ash fusion temperature such as Tsom is estimated by the simultaneous influence of several predictive variables: TSon = bo blX1 + b2X2 + b3X3 + ... (2) At the same time, the availability and improving accuracy of fast multielement analyzers has focused attention upon ways to make use of MLR to estimate ash fusion tem-
+
1, pp 1313-1387. (10)Reid, W. T. Coal Ash-Ita Effect on Combustion Systems;in ref 1, pp 1390-1445.
86026 38.6 0.01 7.99 1.34 9.26 1.18 12.3 0.79 1.61 24.7
peratures from ash composition analysis. Since the pioneering work of Rees," a number of such studies have been made, providing generally superior e s t i m a t e ~ . ~ * ~ J ~ - ' ~ In a recent comparison of seven predictive models with a base of 73 ashes, Slegeir and co-workers12 found that softening temperature was best predicted by the Attar model,15 with a mean error of 94 OF (52 K). Gray, using separate regressions for the fusion temperatures of ashes from each of seven coalfields, obtained mean errors of less than 72 OF (40 K).5 Riley and co-workers used a five-term MLR equation, after excluding four outliers from a 70-ash database, to estimate softening temperature with an average error of 44 OF (24 K).14 In the present study we use the same 70-ash database as Riley14 and evaluate two procedures for the analysis of these data. Experimental Section Seven source coals, of rank from lignite A to medium-volatile bituminous, were selected. Coal sources and characteristics are given in Table I. After reduction to -60 mesh (-0.25 mm) three blends were prepared, in the proportions 75:25,5050, and 25:75, for each of the 21 binary combinations of coals. Ash samples from each blend and from the seven source coals were prepared in accordance with ASTM Method D 1857.3 The ash softening temperature (reducing atmosphere) was determined on ash splits using a LECO Model AF-600ash fusibility system. Determinations were made in duplicate or triplicate for each of the 70 ashes. Softening temperatures were found in the range 2027-2869 O F (1108-1576 "C), with an overall (11) Rees, 0. W. Composition of the Ash of Illinois Coals; Circular No. 356; Illinois State Geological Survey: Urbana, IL; 1964. (12) Slegeir, W. A,; Singletary, J. H.; Kohut, J. F. J. Coal Qual. 1988, 7, 48.
(7) Bryers, R. W. J. Eng. Power 1976, 98, 528. (8) Vorres, K. W. J. Eng. Power 1979, 101, 497. (9) Ceely, F. C.; Daman, E.L. Combustion Process Technology;in ref
86046 50.4 2.36 1.08 1.91 2.71 1.66 5.32 0.63 1.06 24.5
(13) Sondreal, E.A.; Ellman, R. C. "Fusibility of Aah from Lignite and its Correlation with Ash Composition"; U S . Bureau of Mines Report GFERC/RI-75-1, Pittsburgh, PA, 1975. (14) Riley, J. T.;Lloyd, W. G.; Risen, M. A.; Gilleland, S. R.; Tibbitts, R. L. Proc. Conf.--lnt. Coal Test. Conf. 1989, 7, 58-63. (15) Attar, A. "Kinetic Studies Related to the LIMB (Limestone Injection Multistage Burner) Burner", Rep. PB84-20948.5,1984.
Lloyd et al.
362 Energy & Fuels, Vol. 4, No. 4, 1990 S
/
I3
I Figure 2. Multicollinearityamong five ash components. Numbers are values of Pearson’s correlation coefficient (R)between pairs of oxides.
0
25 50 75 Wt % Coal 85091 in Blend
100
Figure 1. SO3 contents of ashes from blends of coals 85091 and 85099. Open squares: interpolated. Solid squares: found. standard error of 9.0 OF (5.0 K),indicating excellent instrumental precision. Elemental analyses were carried out by energy-dispersive X-ray fluorescence (XRF)spectrometry, using an ORTEC Model 6141 spectrometer, and inductively coupled plasma (ICP)spectrometry, using a LECO Plasmarray ICP 500 spectrometer. The XRF spectrometer was calibrated with 13 reference standards (ashes from 5 NIST standard reference coals and 8 ashes which had been used in round-robin analyses), all uniformly prepared by fusion with lithium tetraborate. The analytical program entailed examination under two conditions: (1)10 kV, 200 PA, 200 s live time for Si, P, S, K, Ca, Ti, and Fe; and (2)6.0 kV, 500 PA, 400 s live time for Na, Mg, and Al. In each case the current was set to provide dead time of 40-60%. The raw counts were then adjusted for enhancement/attenuationeffects by Beer’s correction. Repeatability for these 10 elements was consistently better than that allowed by ASTM Method D 4326.16 Ash compositions for the seven source coals are given in Table 11. The samples for ICP analysis were prepared by fusing the ash with lithium tetraborate at lo00 O C for 15 min and dissolution of the melt in 2 % HC1.l’ The elements Al, Ca, Fe, Mg, Si, and Ti were analyzed by ICP spectrometry as a cross-check for the XRF data. Analyses by these two totally different methods agreed to within f10% relative. It is necessary to analyze the ashes of each coal blend, since the composition of blend ashes cannot be safely estimated by interpolation.’ For example, ashes from blends of a high-S low-Ca coal with a low-S high-Ca coal may contain considerably more sulfur than the ashes of either parent coal. Figure 1 illustrates this effect using the SO3 contents of ashes from blends of coals 85091 and 85099. Statistical analyses were conducted using the Statistical Analysis System.lS
Results For simple regressions such as that of eq 1,a high value of R between the predictive and dependent variables indicates a good predictive relationship. In MLR analysis high correlations among predictive variables present serious problems. When a substantial linear dependence exists between two predictive variables, they are said to be c~llinear.’~Multiple linear regression (MLR) analysis is based upon the assumption of orthogonality: each regressor is assumed to be independent of each of the other (16) Test Method for Major and Minor Elements in Coal and Coke Ash By X-RayFluorescence; Method D 4326; in ref 2. (17) Test Method for Major and Minor Elements in Coal and Coke Ash by Atomic Absorption; Method D 3682; in ref 2. (18) The Statistical Analysis System; SAS Institute: Cary, NC 27512. (19) Mason, R. L.; Gunst, R. F.; Hem, J. L. Statistical Design and Analysts of Expenments; Wiley: New York, 1989; p 490.
regressors. Multiple regression is a “forgiving” procedure: it tolerates a moderate amount of collinearity among regressors, though it becomes unreliable in the presence of high collinearity. The collinearity among major components of coal ashes can be extremely high, especially among five of the oxides: CaO, K20,Na20, SO3,and SOz. Figure 2 illustrates the multiple collinearities among these five components for this 70-ash data set. When multicollinearity (excessive collinearity) is encountered, regression equations lose their predictive value and commonly display several serious signs of abnormality. One convenient “flag” for multicollinearity is the size for the standard error of the calculated intercept term (SEI). In a good MLR analysis SEI is of the same general magnitude as rmse. When multicollinearity is present, the SEI will be substantially larger than the rmse. Riley14 has reported a case of extreme multicollinearity for which R2 and rmse looked very good but the SEI was found to be more than 10-fold greater than the rmse. For the following analyses the regressions are selected on the basis of three criteria: 1. A regression is included only if the SEI < 90 O F (50
W.
2. Subject to rule 1,the regression providing the lowest value of the root mean square error (rmse) is selected. 3. An expansion of the number of regressors, subject to rules 1 and 2, is permissible only if it results in an improvement in R2of at least 0.01. Simple MLR using the compositional data of the ten principal oxides yields a good four-term expression (regression 1 in Table 111) containing the concentrations of K20,Ti02, Fe203,and MgO, with a rmse of 76.6 O F . Addition of further terms (e.g., regression 2 of Table 111) provides higher values for R2and lower values for rmse, but at the cost of violating rule 1. Compared with regression 1 of Table 111, regression 2 is apparently better (based upon values for R2and rmse), but the large value of SEI for regression 2-more than 4-fold that of its rmse-signals that this regression is worthless, owing to excessive collinearity. It is intuitively evident that ash properties, including fusion properties, cannot be adequately described by assuming a mixture of 10 separate basic and acidic oxides: these components interact, in acid-base reactions and in other metathetic reactions. T o identify and make use of those interaction products which are important to the estimation of softening temperature, a set of second-order terms is generated.5J3J4 When MLR analysis is carried out using the 10 original concentration terms plus the 55 generated second-order terms, the best four-term regression (regression 3 in Table 111) increases R2 from 0.76 to 0.85 and reduces the rmse from 76.6 to 59.5 O F . Since the SEI is also low, regression 3 is a valid predictive expression.
Estimation of Ash Softening Temperatures
Energy & Fuels, Vol. 4, No. 4, 1990 363
Table 111. Regressions on Ash Softening Temperaturea SEI Simple MLR 1. 1561 + 2.39E4(K) 4.13E4(Ti) - 1.94E3(Fe) + 2.30E3(Mg) 2. 2015 2.79E4(K) + 4.50E4(Ti) - 2.01E3(Fe) - 1.51E3(Si) - 2.79E3(S) + 3.63E4(Na)
+
+
R2 rmse E,,
65.2 0.755 76.6 61.1 324.6 0.807 69.1 55.1
With Cross Terms 3. 2004 + 7.30E4(Al)(Na) + 3.15E6(Na)(P)+ 6.92E5(K)(P)- 3.76E5(P)(S) 4. 1815 + 1.35E5(Ti) - 2.89E5(Si)(Ti) - 7.54E5(S)(P)- 1.60E4(Fe)(S) - 3.54E4(Fe)(K) + 1.03E5(Al)(K) + 6.54E6(Na)(P)
27.5 0.852 59.5 47.5 55.7 0.919 45.0 35.9
With Partial Factor Analysis 5. 2362 - 2.34E5(P)(S)+ 50.2(Facl) + 190(Fac3) + 28.1(Fac14) 6. 2379 + 2.50E3(Fe)2+ 3.53E5(Mg)(Ti) - 5.69E4(Fe)(Mg) - 2.87E5(P)(S) + 73.2(Facl) + 191(Fac3) 18.8(Fac13) + 31.O(Fac14)
12.9 0.872 55.3 44.1 14.0 0.925 43.8 34.9
+
OTemperatures and error estimates in OF; element symbols represent weight fractions of corresponding oxides, e.g., Mg for [MgO];2.9934 represents 2.99 X lo‘, etc.; Facl represents factor 1, etc.
Additional terms can be added. The best second-order regression, subject to the above rules, is regression 4 in Table 111:
Tsom ( O F ) =
1815 + (1.35 X 105)[Ti02]- (2.89 X 105)[Si02][Ti02](7.54 X 105)[S03][P205] - (1.60 X 1O4)[Fe2O3][SO3] (3.54 X 104)[Fe203J[K20] (1.03 X 1O5)[Al2O3J[K20] (6.54 X 106)[Na20][P205](3)
+
+
for which the rmse is 45.0 OF (25 K). Principal component analysis, a particularly useful form of factor analysis, was developed by Karl Pearson.m This mathematical process extracts a relatively small number of principal components from a larger number of raw variables, each principal component being a linear combination of the original variables. Principal component analysis has been shown to be an important tool in the analysis of geochemical databases21 and has been used effectively by Glick and Davis in identifying common factors underlying the distribution of trace elements in
U.S. Principal component analysis is of particular utility in regression analysis when excessive collinearity is a problem. The extracted factors are mutually orthogonal and therefore completely free of collinearity among themselves. Thus these factors can be used even when severe collinearity prevents use of the original data. We have used the SAS Factor Procedurel8 for this approach, selecting the principal component analysis option and then optimizing the distribution of factor weights by factor rotation, using the standard Varimax rotation (so named because it maximizes the variance of the squared loadings of each column of the factor matrix). In the present study some of the analytical data, namely, those terms involving only A1203,Fe2O3, MgO, P2OS,and TiO,, can be used in their original form, since severe collinearity is only observed in those terms involving CaO, K20, Na20, SO3,and Si02. We have found it advantageous to carry out a partial factor analysis and then to estimate the fusion temperature by MLR using as regressors both the raw data (from components showing little collinearity) and the derived factors (from the components showing high mutual collinearity). There are various ways to separate the 65 terms into those to be retained and those to be pooled and subjected (20) Pearson, K. Philos. Mag. 1901, 6 (2), 559. (21) Joreskog, K. G.; Klovan, J. E.; Reyment, R. A. Methods in Geomathematics. I . Geological Factor Analysis;Elsevier: New York, 1976. (22) Glick, D. C. M.S. Thesis, Pennsylvania State University, 1984. (23) Glick, D. C.; Davis, A. Part 10, Final Report, US.Department of Energy Contract No. DE-AC22-80PC30013,1984. (24) Glick, D. C.; Davis, A. Org. Geochem. 1987, 11, 331.
to factor analysis. We have excluded from factoring (a) all terms that do not contain any of the “multicollinear oxides” of Figure 2 and (b) one cross term each composed of Al,Fe, Mg, P, or Ti oxides with a “multicollinear oxide”. Five terms are discarded and the remaining 40 terms are factored to 20 principal components. The best four-term expression using partial factor analysis (regression 5 in Table 111) yields R2 = 0.87 and rmse of 55.3 OF, better than either of the other four-term regressions. The best overall expression (regression 6 in Table 111) yields R2 of 0.925 and rmse of 43.8 OF (24.3 K). The regressions in Table I11 show a wide range of values of SEI, the measure we have used as a flag for multicollinearity. For regression 2 there are three very strong collinear interactions [Na-S, Na-Si, and S-Si] and the average value of Pearson’s R between terms is 0.52. This strong multicollinearity, as noted above, is signaled by a SEI that is much larger than the rmse. Regression 4 is the only other expression in Table I11 for which SEI exceeds rmse. Here two of the 21 interactions [Na-P and P-SI are strong, and the average value of R between terms is 0.40. This equation shows some collinearity, but at a probably acceptable level (SEI < 2(rmse)). Regression 1 has no strong interactions among regressors; Regression 3 has just one such interaction [(Na)(P)-(P)(S)]. For both of these regressions SEI < rmse and these are, we believe, good clean regressions. Regressions 5 and 6 have no strong interactions among regressors. The average value of R between terms is 0.2. The SEI for both regressions is less than one-third the rmse. Not only are these regressions superior estimators of softening temperature in terms of mean errors of estimate, but also-owing to the use of factor analysis-they are free of significant collinearity problems. The auerage error estimated for large data sets is E,, = E , , , , [ ~ / T ] ~ = ~0.7983,,,, ~ For the best regression, therefore (regression 6 of Table 1111, the average error of estimate of 2 ’ 8 0 ~ is 34.9 O F (19.4 K). Figure 3 illustrates the good fit of this estimate to the experimental fusion data. No outliers have been rejected for the above calculations. Trial rejections of three outliers reduces each of these error estimates by several degrees but does not alter relative rankings. Discussion Huffman,6 hug gin^,^^ and co-workers have conducted detailed phase studies of ashes and ash-oxide blends, (25) Huggins, F. E.; Kosmack, D. A.; Huffman, G. P. Fuel 1981, 60, 577.
Lloyd et al.
364 Energy & Fuels, Vol. 4,No. 4, 1990 3000
0
2800
P
2
$ 2600 .-P C
8
8 -
2400
.-
tj 2200-
W
2000Y 2000
2200
2400
2600
2800
3( I
Actual Softening Temp (deg F)
Figure 3. Estimation of softening temperature using partial factor analysis. Average error is 34.9 O F (19.4 K).
concluding that even the initial deformation temperature is preceded by extensive formation of liquid phase. The progression from initial deformation through softening and subsequent fusion temperatures is “due principally to changes of viscosity and flow properties of the This finding may account of the relatively strong predictive powers of elemental compositional data as compared with mineralogical data. Coefficients for Na, K, Mg, and Ti oxides are consistently positive, while those for Fe, s, and Si oxides are negative. Cross terms of P205with the alkali-metal oxides are large and positive, while the important P205-SO, cross term is large and negative, as are all of the cross terms involving Fe20,. Factors obtained from principal component analysis cannot be usefully defined in compositional terms; each factor is a linear combination of 40 different terms. On the basis of correlation analysis, however, it is possible to identify certain compositional terms that are strongly associated with certain factors. Factor 1 is strongly associated with CaO and its interactions with Si02 and K20. Factor 3 is strongly associated with the interactions of P205 with Si02,K20, and Na20. Coefficients of both factors are positive. The components of factors 13 and 14 are widely distributed among the 40 input variables. Interpretation of predictive relationships requires cau-
tion, owing to the many strong correlations noted above. For example, since Si02 and SO3 have a very strong negative correlation ( R = -0.93),a positive coefficient for a term including Si02 may imply that high silica content tends to increase Tsom, or that sulfur oxides tend to decrease Tsom, or that one of the other strongly correlated oxides is implicated. For this database the best partial factor regressions are moderately better than the best cross-term regressions in terms of R2 and rmse and are markedly free of multicollinearity as gauged by SEI. The coefficients recorded in Table I11 account well for the behavior of the blends of the seven source coals used in this study. In principle, if the above analysis identifies the chemical bases of ash softening temperature, it should be applicable to any coal ash. In practice we would like to examine the ashes of a number of different coals and coal blends before making that claim. Boiler operators with a collection of high-quality ash analyses and ash fusion data and with access to any one of the scores of commercial statistical programs written for microcomputers can generate their compositional cross terms on a spreadsheet, run a correlation analysis to identify the combinations of terms which cannot be used together in the same regression, and then run a large number of multiple linear regressions, screening out those with unacceptably high values of SEI. This should yield valid MLR equations estimating softening temperature with average errors as low as 50 OF (28 K). With any of several advanced statistical programs that now are available in versions running on personal computers equipped with hard disks, it is possible to submit selected variables to factor analysis and to obtain slightly better predictive regressions with somewhat less effort. There are several valid instrumental approaches to ash analysis, including X- and y-ray fluorescence methods, atomic absorption, ICP emission, and electrochemical methods, as well as the classical methods. Virtually every method has its own systematic bias. It is therefore important to use the same analytical procedure for the original ashes upon which the regression is based as for those ashes for which the regression will be used to estimate ash softening temperatures. Acknowledgment. We gratefully acknowledge financial support of this work through the Robinson Professorship of The Ogden Foundation and a faculty research grant from Western Kentucky University.
