p.1 

OAR Box 1828 

Prepped by Ollie Stewart 

Document Number: 

299) IV-A-9 

Docket Number: 

A-83-27 

Printed 11/12/2008 11:12:59 AM Sheet 299 of 325 
17

 ~~ A-83-27 T 

IV-A-9 

DCN #85-231-020-12-31 

AN ANALYSIS OF THE EFFECTS OF 

COAL LOT SIZE ON THE VARIABILITY 

OF S02 EMISSIONS 

FINAL REPORT 

Prepared by: 

Radian Corporation 

Research Triangle Park, North Carolina 27709 

Under EPA Contract No. 68-02-3816 

Prepared for: 

U.S. Environmental Protection Agency 

Office of Air and Radiation 

Office of Air Quality Planning and Standards 

Emission Standards and Engineering Division 

Research Triangle Park, North Carolina 27711 

September 1986 
P.3 

1.0 BACKGROUND 

This technical note presents the results of an analysis of the 

relationship between coal lot size and the variability of sulfur dioxide 

(SOp) emissions resulting from coal combustion. The purpose of this 

analysis is to examine potential differences in SOp emissions variability as 

a function of coal lot size and to determine the effects of these 

differences on the long-term average S02 emission levels that must be 

achieved to comply with a given emission limit or standard. 

Coal lot size is defined as "the quantity of coal consumed by a boiler 

in one typical operating day." Although boiler size can be used as a rough 

indicator of coal lot size, boiler size and lot size are not necessarily 

equivalent because boilers of the same heat input capacity are often 

operated at different loads and fire coals with different heating values. 

Sulfur dioxide emissions resulting from the combustion of coal in a 

particular boiler can vary over time due to the non-homogeneous nature of 

coal and coal deposits. The amount of variation is influenced not only by 

variation in the distribution of sulfur throughout the seam from which the 

coal is mined, but can also be influenced by the manner in which the coal is 

mined. In addition, coal may be mined simultaneously from several locations 

within the same seam. The sulfur content of the coal from each location and 

the degree of mixing the coals undergo will influence overall variability in 

the sulfur content of the coal produced from the mine. 

The amount of variation is also influenced by the extent to which coal 

is cleaned prior to shipment. Physical coal cleaning removes a large 

portion of the impurities normally found in raw coal and dampens the 

variation in the sulfur content of the coal. It has been reported that coal 

washing, for example, reduces coal sulfur variability by approximately 50 

percent. Physical coal cleaning also provides opportunities for blending 

various coals in the loading, unloading, storage, and processing steps, 

thereby decreasing the variability in coal sulfur content. 

Finally, the amount of variation is influenced by coal handling 

practices at the mine, coal cleaning plant, and boiler site. Coal handling, 
for example, may involve blending coals with varying properties to produce a 

coal blend that is more uniform in sulfur content than the individual coals. 

Coal blending decreases the variability in coal sulfur content by physically 

averaging the sulfur contents of coals and by mixing. The degree of 

reduction in variability, however, depends on the properties of the coals 

blended and the specific blending method. 

The present analysis does not examine the effects of coal mining, 

preparation, and handling procedures on variability, however, because the 

effects of these procedures are difficult to quantify. Instead, this 

analysis examines the effect of coal lot size alone on coal sulfur content 

or SOx emissions variability. Some consideration is also given to the 

effects of averaging time on emissions variability relative to the effects 

of lot size. 

This technical note presents the results of both a theoretical analysis 

and an analysis of actual data relating coal lot size to emissions 

variability. The theoretical analysis, Section 3 of this technical note, is 

built on equations and discussion which are presented in Appendix A. The 

actual data analysis, Section 4, is based on several SOp emissions 

data sets from utility, industrial, and institutional boilers. Conclusions 

drawn from these two analyses are presented in Section 2. 

The statistical parameters that are used to describe coal sulfur 

variability are standard deviation, relative standard deviation (RSD), 

autocorrelation. The standard deviation is an indicator of the range of 

values measured and the deviation of these values from the sample mean. The 

RSD is defined as the ratio of the standard deviation to the sample mean, 

and is expressed as a percentage. The variability of SOp emissions is 

directly proportional to the standard deviation and RSD of the data set. 

Emission measurements taken at equal time intervals constitute a time 

series in which values are not necessarily independent. A measure of the 

degree of association between observations in a time series is termed 

autocorrelation. The effect of higher autocorrelation is to increase SO,, 

emissions variability. 
P.5 

2.0 CONCLUSIONS 

Statistical theory indicates that, for the hypothetical case where both 

an industrial and a utility boiler obtain their coal supplies from the same 

mine and operate in an identical fashion, the difference in coal lot sizes 

typical for these boilers should result in a difference in daily average SOp 

emissions variability. Theoretical analysis predicts that the ratio of the 

daily average emission variances should be inversely related to the ratio of 

the respective lot sizes. However, when a 30-day rolling average technique 

is used to measure emissions and determine compliance, the additional 

averaging and autocorrelation effects of this technique act to reduce the 

difference in emissions variability. The result is that the allowance for 

variability that is required to determine the long-term average emission 

level consistent with a given standard is comparable for both the industrial 

and utility situation. The differences in required long-term average 

emission levels (and hence coal sulfur levels) are quite small and should 

not have a significant impact on the availability of coal supplies to meet a 

given emission limit or on operational procedures for either industrial or 

utility boilers. 

Furthermore, although statistical theory infers that emissions 

variability, or RSD, increases as lot size decreases, the analysis of 

available lot size and coal emissions variability data from industrial, 

institutional, and utility boilers does not support this inverse 

relationship. In fact, there does not appear to be any significant 

relationship between lot size and coal sulfur variability. 

Several reasons have been identified as possible explanations for this 

discrepancy between statistical theory and the analysis of actual data. One 

study has concluded that numerous physical attributes such as mining 

techniques, coal handling procedures (e.g., blending), and geologic factors 

are the primary factors affecting the distribution of coal sulfur. 

Different coal sampling methods and measurement error also contribute to 

variation in coal sulfur values. This study concluded that sulfur content 

varies widely among coal seams, geographical locations, and often even among 
P.6 

mines operating in the same coal seam in the same area. None of the above 

factors are a direct function of lot size. 

This analysis has shown that coal lot size is not a major factor 

affecting coal sulfur and emissions variability for industrial, 

institutional, and utility boiler operations. Other studies have shown, and 

this analysis supports the contention, that other physical and geological 

factors influence coal sulfur variability and tend to mask or confound any 

direct relationship between lot size and coal sulfur and emissions 

variability. 

Finally, a comparison of the effects of averaging time on emissions 

variability with the effects of lot size shows that averaging time 

differences have a similar, but lesser, impact on required long-term average 

S 0? emissions than lot size differences. This is due to the fact that the 

choice of averaging time influences the autocorrelation structure of 

emission averages to a lesser extent than lot size differences influence the 

standard deviation of 30-day emission averages. 
P.7 

3.0 THEORETICAL ANALYSIS RESULTS 

This section presents the results of a theoretical statistical analysis 

of the effects of lot size on coal sulfur and emissions variability. 

Appendix A presents the statistical theory and equations which are the basis 

of this analysis. The major assumptions in the analysis are that utility 

and industrial boilers obtain coal shipments from the same mine and that the 

only operational difference between these hypothetical boilers is their lot 

size. 

When samples are taken to estimate the value of a certain parameter, 

such as coal sulfur content, statistical theory indicates that larger sample 

sizes result in less variability in the measured values of that parameter. 

Following this reasoning, it has been suggested that industrial boilers 

might exhibit greater variability in sulfur dioxide emissions than utility 

boilers due to the smaller lot sizes typical of industrial boilers. The 

primary objective of this analysis, therefore, was to determine what 

influence, if any, lot or sample size has on coal sulfur and emissions 

variability from a purely theoretical point of view. A secondary objective 

was to compare the effects of averaging time on emissions variability 

relative to the effect of lot size. 

3.1 Impact of Lot Size 

The analysis in Appendix A indicates that the lot-to-lot variability of 

average sulfur content (or SOp emissions potential in lb SOp/million Btu) is 

directly related to the size of the sample or lot selected by the expression 

a

 x n

X U *,I U 

a

 x n

X I £,U

 l 

2 

where a - is the lot-to-lot variability (with subscripts "I" and "U" 

x * 

(3-1) 
representing industrial and utility boiler operations, respectively) and n 

is a measure of lot size. The Appendix A discussion points out that the 

units used to measure lot size (e.g., number of coal lumps, tons, rail 

cars) is not important as long as consistency is maintained. Equation 3-1 

does imply, however, that SOp emissions variability increases with 

decreasing lot size. 

Moreover, when using a multiple-day averaging technique to measure 

compliance with a given emissions limit, it is also necessary to consider 

the further averaging effects of this averaging technique as well as the 

day-to-day autocorrelation effects. As shown in Appendix A, 

2 

a x 

m m (1-Pd) 

(3-2) 

2 

where m is the number of days in the averaging period, a - is the variance 

xm 

of the m-day averages, and p . is the lag-one autocorrelation among daily 

average SOp emission levels. Using a 30-day rolling average and a 1 in 10 

years exceedance frequency, which approximates a "never to be exceeded" type 

of standard, it is shown that 

2 

0

 x 30 30 (1-P(J) 

and 

(3-3) 

C = 3.46 ' a- (3-4) 

x30 

Here C is the difference between an emissions limit and the long-term 

average, or target, emissions level required to comply with the limit; it 

represents the allowance required for emissions variability. 
A numerical example helps to put the theoretical impact of coal lot 

size on SOp emissions variability into perspective. For the case of a 

typical utility boiler rated at 500 MW electrical output and operating at 

60 percent of capacity, the daily coal lot size would be approximately 3,100 

tons. A typical 150 million Btu/hour industrial boiler, on the other hand, 

operating at 60 percent of capacity would consume about 90 tons of coal per 

day. The bases for these lot size estimates are presented in Appendix B. 

Based on actual boiler emissions data presented in Table l (in 

Section 4.1), typical values for a- , which is the standard deviation of 

daily average SOp emissions, range between 0.06 and 0.23 lb/million Btu. 

For example purposes let a equal 0.20 lb S09/million Btu, which implies 

I 2 2 that a - equals 0.040 (lb SOp/million Btu) . Based on Equation 3-1 and 

xa,I

 c 

the relative lot sizes, the variance of the daily emission averages for a 

2 

utility boiler operating on the same coal supply, a - , is expected to be 

% U 

(90/3,100 x 0.040 =) 0.00116 (lb S02/million Btu)2 . This corresponds to a 

utility boiler daily average standard deviation of 0.034 lb SOp/million Btu. 

The values for daily average autocorrelation in Table 2 range from 0.49 

to 0.79. A value of 0.70 is assumed for example purposes for both the 

industrial and utility boiler. The variances of the 30-day averages are 

2 

computed using Equation 3-3 as 0.0444 (lb SOp/million Btu) for the 

industrial boiler and as 0.000129 (lb SOp/million Btu)2 for the utility 

boiler. Finally, the allowance for variability, C, contained in 

Equation 3-4 can be calculated as 0.231 and 0.039 lb SOp/million Btu, 

respectively, for the industrial and utility operations. 

In order to meet an emission limit of 1.2 lb SOp/million Btu, for 

example, an industrial boiler operator would have to fire coal with a 

long-term average emission potential of (1.200 - 0.231 =) 0.969 lb 

SOp/million Btu. Under the same conditions, a utility boiler operator could 

fire coal with a long-term average emission potential of (1.200 - 0.039 =) 

1.161 lb S02/million Btu. 

For the case where the a- for industrial boiler operation equals 0.10 

instead of 0.20 lb SOp/million Btu, the corresponding long-term average coal 
P.10 

emissions potentials for industrial and utility boilers are estimated in the 

same fashion to be 1.085 lb S02/million Btu and 1.180 lb SOp/million Btu, 

respectively. 

These results illustrate the point that the long-term average S0? 

emission levels for industrial boilers must be slightly lower than for 

comparable utility boilers in order to meet a given standard, based on the 

theoretical impact of lot size differences. However, the differences are 

quite small and should not have a significant impact on the availability of 

coal supplies to meet a given emission limit or on operational procedures 

for either industrial or utility boilers. Moreover, if the exceedance 

frequency is relaxed from the 1 in 10 years level examined in the example, 

the required allowance for variability declines. This would allow both 

industrial and utility boiler operators to fire coals with long-term average 

SOp emissions potential closer to the limit and further reduce the 

differences between them. 

3.2 Averaging Time Effects Versus Lot Size Effects 

Comparisons are sometimes made between the effects of averaging time on 

emissions variability and the effects of lot size, since in both cases there 

is a difference in the magnitude of coal combusted in the boiler. In the 

example used above, the ratio of coal combusted in a typical 500 MW utility 

boiler to that combusted in a 150 million Btu/hr industrial boiler is about 

34 to one. A similar ratio would apply to the coal combusted in a given 

boiler (boiler size is not a factor here) during a 30-day averaging period 

versus a 24-hour averaging period (i.e., the ratio is 30 to one). As shown 

above, the impact of this lot size difference on the long-term average 

emissions required to meet a 1.2 lb SOp/million Btu standard is small; for a 

standard deviation of daily averages equal to 0.20 lb SOp/million Btu, the 

smaller lot size reduces the required long-term average emissions level from 

1.16 to 0.97 lb SOp/million Btu. These values correspond to RSD levels of 

2.9 and 20.6 percent for the utility and industrial boiler, respectively. 
P.11 

The required long-term average emission level can also be calculated as 

a function of averaging time based on the equations presented in Appendix A. 

For a 1.2 lb SOp/million Btu emission limit, one-in-10-years exceedance 

frequency, 3 percent daily RSD, and a 30-day rolling average period, the 

required long-term average is given by: 

L + C 

L = 

1 + (Za) (RSDd )Vf7 2 o(p)/f 24(p) 

In this case, 

1.2 

L = 

1 + (3.46) (0.03)i/0.1180 

V n0.836 SIK9 Q 

L = 1.16 lb S02/million Btu 

For the same conditions but using a 24-hour averaging period: 

L + C 

L = 

1 + (Za) (RSDd) Vf^fp ) 

1.2 

L = 

1 + (3.46) (0.03)1/0.8369 

L = 1.10 lb S02/million Btu 

Thus for similar order of magnitude changes, averaging time differences 

have a similar, but lesser, impact (1.16 vs. 1.10 lb/million Btu) on 

required long-term average S02 emissions than lot size differences (1.16 vs. 

0.97 lb/million Btu). The reason for this smaller impact is that the choice 

of averaging time influences the autocorrelation structure of the emission 
P.12 

averages to a lesser extent than lot size differences influence the standard 

deviation of 30-day emission averages. 

10 
P.13 

4.0 ANALYSIS OF S0p_ EMISSIONS DATA 

This section presents the results of an analysis of SOp emissions 

data from utility, industrial, and institutional boilers. Twelve data sets 

were examined in this analysis; each data set was collected using either a 

certified S02 continuous emission monitor (CEM) or EPA Method 6B (40 CFR 60, 

Appendix A) . Daily coal lot size estimates for all data sets are based on 

several assumptions: steam enthalpy, steam flow rate, boiler efficiency, and 

coal heating value; a sample calculation of coal lot size is presented in 

Appendix B. 

For each data set, the mean, the standard deviation, and the relative 

standard deviation (RSD) of the daily average S02 emissions were calculated. 

In cases where sufficient data were available, the daily autocorrelation was 

also calculated. The methodology used to calculate these statistics and 

descriptions of the various data sets may be found in References 3 

through 7. 

4.1 Results 

Sulfur dioxide emissions measurement data are presented in Table 1. Of 

the twelve data sets presented, half are from utility boilers, and half are 

from industrial or institutional boilers. A plot of daily coal lot size (in 

tons) vs. daily coal sulfur RSD (in percent) is shown in Figure 1. As 

stated previously in the theoretical discussion in Section 3, the S02 

emissions standard deviation, or RSD, is expected to decrease as lot size 

increases. However, the plot of available data does not confirm this 

expectation. The scatter of the data indicates that coal lot size is not 

the sole major factor in determining coal sulfur RSO. 

It is interesting to note that the standard deviations shown in Table 1 

for daily average emission rates for industrial/institutional boilers do 

fall in the general range of the standard deviations assumed in the 

numerical examples in Section 3.1 (i.e., 0.20 and 0.10 lb SOp/million Btu). 

However, the daily average standard deviations for utility boilers are also 

11 
in this same range and are well above the standard deviations calculated in 

the numerical examples (i.e., 0.034 and 0.017 lb S02/mi11 ion Btu) that would 

be expected for the larger lot sizes. Although the data in Table 1 

represent only 12 data sets, which is a relatively small sample size, these 

trends in daily average emission standard deviations do indicate that other 

factors are involved which influence emissions variability to a greater 

extent than lot size. 

12 

J 
TABLE 1. S0? EMISSIONS OATA 

Plant (Source) Type of Unit 

No. of 

Hourly 

Data 

Points 

1896 

1848 

1152 

792 

a 

Raw or 

Cleaned 

Coal 

-

Raw 

Raw 

Cleaned 

Cleaned 

Type of Coal 

Bituminous 

Bituminous 

Bituminous 

Bituminous 

_ 

Daily Coal 

Lot Size 

(tons) 

550 

500 

330 

150 

150 

Mean , 

(lb S02/106 Btu) 

Daily 

Standard 

Daily RSO Deviation Daily 

(%) (lb S02/10° Btu) Autocorrelation 

Plant A (Ref. 3) 

Plant B (Ref. 3) 

Plant C (Ref. 3) 

Plant D (Ref. 3) 

Plant E (Ref. 4) 

Plant F (Ref. 7) 

Plant G, Unit 3 

(Ref. 5) 

Plant H, Unit 1 

(Ref. 5) 

Plant I Unit 1 

(Ref. 5) 

Plant J, Unit 4 

(Ref. 5) 

Plant K (Ref. 5) 

Plant I, Unit 4 

(Ref. 6) 

Industrial 

Industrial 

Industrial 

Institutional 

Institutional 

Industrial 

Utility 

Utility 

Utility 

Utility 

Utility 

Utility 

1368 Cleaned Bituminous 

1896 Raw 

2712 Raw 

2040 Raw 

1200 Cleaned Bituminous 

4500 

6600 

Subbituminous 7500 

5100 

2000 Raw - 6200 

612 - Subbituminous 900 

0.92 

0.64 

0.79 

1.44 

>1.46 

0.99 

0.92 

0.45 

0.78 

0.83 

1.06 

9.9 

32.3 

29.1 

9.5 

10.7 

10.5 

8.8 

17.3 

17.3 

7.6 

10.6 

0.091 

0.207 

0.230 

0.137 

>0.156 

0.103 

0.081 

0.078 

0.135 

0.063 

0.117 

0.49 

0.66 

0.63 

0.67 

0.67 

0.79 

0.63 

0.72 

a0nly daily data available. 
1—• 

3 2 -

3 0 -

28-i 

2 6 -

2 4 -

2 2 -

2 0 -

1 8 -

CO 

a 14 -

^ 12 -

1 0 -

8 -

6 -

4 — 

2 -

• 

• 

• • 

I 

5 

Figure 

• 

• 

• • 

• 

lllllllllll l 

10 16 20 26 30 36 40 46 60 65 60 65 

LOT SIZE 

(ton* X 100) 

1. Daily coal lo t size versus S02 emissions variabilit y for industrial 

and utilit y boilers. 

1 

70 

• 

1 

75 
5.0 REFERENCES 

1. Warholic, G. R., et al. (Foster Associates, Inc.). A Statistical 

Study of Coal Sulfur Variability and Related Factors. Prepared for 

U.S. Environmental Protection Agency. Research Triangle Park, N.C. 

EPA Publication No. EPA-450/5-80-008a. May 1980. 

2. Giguere, G. C. and S. C. Margerum. (Radian Corporation). 

Determination of Mean S02 Emission Levels Required to Meet a 1.2 lb 

SOp/million Btu Emission Standard for Various Averaging Times and 

Compliance Policies. Prepared for U.S. Environmental Protection 

Agency. Research Triangle Park, N.C. March 1985. 

3. Margerum, S. C. and G. Giguere. (Radian Corporation). Variability of 

SOp Emissions from Low Sulfur Coal-Fired Industrial Boilers. Draft 

Report. Prepared for U.S. Environmental Protection Agency. Research 

Triangle Park, N.C. February 1985. 

4. Whittle, Richard and Peter Westlin. Air Pollution Emission Test 

Report: Development and Evaluation of an Intermittent Integrated 

S02 /C02 Emission Sampling Procedure. U.S. Environmental Protection 

Agency. Research Triangle Park, N.C. December 19, 1979. 

5. PEI Associates, Inc. Analysis of Sulfur Dioxide Emission Data from 

Five NSPS Power Plants Firing Compliance Coal. Prepared for U.S. 

Environmental Protection Agency. Contract No. 68-02-3512. Research 

Triangle Park, N.C. January 1985. 

6. Kelly, W. E., et al. Air Pollution Emission Test. Second Interim 

Report: Continuous Sulfur Dioxide Monitoring at Steam Generators. 

Volume I - Summary of Results. Prepared for U.S. Environmental 

Protection Agency. Research Triangle Park, N.C. EMB Report No. 

77SPP23C. March, 1979. 

7. Peduto, E. F., Jr., et al. (RTI). AirPollution Emission Test. Third 

Interim Report: Continuous Sulfur Dioxide Monitoring at Steam 

Generators. Volume II - Data Listings. Prepared for U.S. 

Environmental Protection Agency. Research Triangle Park, N.C. EMB 

Report No. 77SPP23C. EPA Contract No. 62-02-2688, T. D. 106. April, 

1979. 

8. Lowry, H. H. (Editor). Chemistry of Coal Utilization, Supplementary 

Volume. John Wiley & Sons, Inc. New York. 1963. p. 1. 

9. Edgar, T. F. Coal Processing and Pollution Control. Gulf Publishing 

Co. Houston. 1983. p. 31. 

15 
P.18 

10. Dubose, D. A., et al. Statistical Analysis of Wet Flue Gas 

Desulfurization Systems and Coal Sulfur Content, Volume I: Statistical 

Analysis. Prepared for U.S. Environmental Protection Agency. Contract 

No. 68-02-3816. Research Triangle Park, N. C. August 1983. 

pp. A-5 - A-9. 

11. Andersen, T. W. The Statistical Analysis of Time Series. John Wiley & 

Sons, Inc. New York. p. 444. 

12. Reference 11, p. 478. 

9. Kendall, M. G. and A. Stuart. The Advanced Theory of Statistics, 

Volume I (3rd Edition). Griffin, London. 1969. 

16 
P.19 

APPENDIX A 

THEORETICAL STATISTICAL ANALYSIS 

This appendix discusses the application of statistical theory to the 

issue of S02 emissions variability as a function of coal lot size. To do 

so, it is helpful to consider the physical situation at hand. Coal is mined 

from seams formed from plant substances which accumulated thousands of years 

ago, were preserved from complete decay by a favorable environment, and were 

Q 

later altered by various chemical and physical processes. Sulfur is 

present in coal as a result of impurities laid down with the original plant 

substances. Sulfur occurs in two general forms: organic sulfur, which is 

distributed throughout the hydrocarbon structure of the coal, and inorganic 

g 

sulfur, which appears mainly in combination with iron, calcium, and barium. 

Thus, sulfur is distributed in "bands" throughout the coal seam, as opposed 

to a random distribution; this distribution has important statistical 

implications. 

When coal is mined, the seam is broken into lumps and transported out 

of the mine and to industrial and utility boilers where they are combusted 

to generate flue gas. The sulfur present in the coal is combusted to form 

primarily sulfur dioxide (SOp). Thus S02 emissions are directly related to 

the sulfur content of mined coal. In this theoretical model, the only 

difference between industrial and utility boilers is the quantity of coal 

combusted in a given time period. For this analysis, lot size is defined as 

the quantity of coal combusted is a given boiler during one typical day of 

operation. 

Coal lumps are a convenient unit for this analysis because they are • 

analogous to samples or observations taken from a large population. In this 

case the population, or mine, is made up of many, many lumps, each with a 

sulfur content and heating value which correspond to potential S02 emissions 

(e.g., in units of lb SOp/million Btu). 

Because of the manner in which sulfur is distributed in the coal seams, 

the sulfur (or SOp) contents of the lumps are not independent of each other. 

The SOp contents of successive lumps will be autocorrelated, which suggests 

A-l 
P.20 

that a time series model be used for modeling and analysis. Without data, 

it is impossible to identify a time series model for the coal lumps, but an 

understanding of the effects of autocorrelation can be illustrated by 

assuming a first-order autoregressive time series, i.e., an AR(1) process. 

This assumption is supported by the finding that hourly S09 emissions data 

10 from coal-fired boilers tend to follow an AR(1) process. 

Assuming that the SOp contents of the coal lumps, in the order of 

2 combustion, are an AR(1) process with mean p, variance a , and lag-one 

i 

autocorrelation of p, then the lag-k autocorrelation is p . Each successive 

batch, or lot, of coal lumps will have an overall average SOp content, 

termed "X . The variance of the lot-average S02 contents, X , is given by 

2 ' !_ I I A |t-sl o? "I1 n k , Jc a z =± y Z Z o'pT

 3| = §- E (1 - J ) p* (A-l) 

\ r\L t=l s=l

 n k=0 n 

where n is the number of lumps in a given lot. 

For large n, 

a v 

2 n-1 k s a2 

a I pR " (A-2) Xa " n k=0 n (1-p) 

Since a and p are population parameters and will be the same for a 

utility boiler and an industrial boiler which obtain their coal from the 

same mine, the lot-to-lot S02 content variances for these boilers are 

related by 

2_ 

V 

1,1 n 

0 X 

U (A-3) 

2 

where the subscripts "I" and "U" refer to the industrial and utility 

situations, respectively. Note that the ratio of the variances is dependent 

A-2 
P.21 

only on the relative size of the lots -- the actual units used to measure 

lot size (e.g., number of coal lumps, tons, rail cars) are immaterial as 

long as consistency is maintained. 

Since a lot is defined as the amount of coal combusted in a single 

operating day, X is equivalent to the daily average S09 emission level (in 

A* ty La, 

lb SOp/million Btu) and a -z is the variance of the daily average SOp 

emission levels. For an emissions standard that measures compliance on the 

basis of a 30-day rolling average, the average S02 emission levels 

associated with 30 consecutive lots are averaged to generate a 30-day 

rolling average. The average daily SOp emission levels, IL's, will 

themselves be a random variable with mean, L, and variance a -s . These 

values will also be autocorrelated with a lag-one autocorrelation of p .. 

Following the same analysis outlined above for lump-to-lump autocorrelation, 

it is possible to show that 

2 

0

 X 

2 

(A-4) 

a

 X 

m m (1-Pd) 

2 where m is the number of days in the averaging period and a z is the 

m 

variance of the m-day averages. For a 30-day rolling average, 

2 

f 

l 

0

 X 

a X 30 30 (l-pd) 

(A-5) 

Moreover, by the central limit theorem for autoregressive processes, X-n is 

12 approximately normally distributed for large m. 

When assessing compliance impacts for a given standard, a 1 in 10 year 

exceedance frequency is often assumed to approximate a "never to be 

exceeded" limit. Let C be the allowance for variation in emissions, or the 

A-3 
P.22 

difference between the long-term mean, L, and an emissions limit, S, For a 

1 in 10 year exceedance frequency, 

P ()Ln > S = L + C) =

 X 

30

 J w = 0.00027 (A-6) 

(10)(365) 

_ 2 

Since X~n is normally distributed with mean L and variance a -s , 

JU A30 

K - L C 

P | — > | = 0.00027 (A-7) 

OZ OZ 

A30 A30 

and 

P (Z > -£ — ) = 0.00027 (A-8) 

A30 

P (Z > 3.46) = 0.00027 (A-9) 

where Z is the standard normal variate. 

Thus 

- £ — = 3.46 (A-10) 

aX *30 

and 

C = 3.46 ' or 

x30 

C = 3.46 • aj (A-ll) 

Averaging Time Effects on Variability 

Comparisons are sometimes made between the effects of averaging time on 

emissions variability and the effects of lot size, since in both cases there 

is a difference in the magnitude of coal combusted in the boiler. 

A-4 
P.23 

In the technical note entitled "Determination of Mean S02 Emission 

Levels Required to Meet a 1.2 LB SOp/Million Btu Emission Standard for 

Various Averaging Times and Compliance Policies," the mathematics of 

2 averaging time effects on variability are discussed. The equations derived 

to calculate the long-term average emission level or target level, L, 

required to meet a given emission limit (assuming a normal distribution of 

averages) are: 

L + C 

L = , , for n >25 (A-12) 

1 + (Z ) (RS0d )Vfn (p)/f2 4 (p) 

and 

L + C 

L = m , for n <25 (A-13) 

1 + (Z ) (RSDd)yf7pl 

where 

L = long-term average emission level 

C = difference between long-term average emission level and emission 

limit 

Z = standard normal variate 

a 

RSD = relative standard deviation based on 24-hour, or daily, averages 

n = number of hours in averaging period. 

p = lag 1 or first-order hourly autocorrelation coefficient 

The expression for f_(p) is given by 

1 

UP)'-

n (1-P ) 

(A-14) 

The standard normal deviate, Z , is a function of the averaging period 

and the assumed emission limit exceedance frequency. For a one-in-10-years 

exceedance frequency, the value of Z is 3.98 for a 3-hour averaging period. 

For a 24-hour averaging period, or any larger averaging period where 

A-5 
P.24 

compliance is measured on a daily basis (including a 30-day averaging 

period), the value of Z is 3.46. 

The expression f_(p) is a function of both p and n. Based on other 

data bases, a typical value of p is 0.9769. Using this p value and the 

expression for f.(p)t the following values can be calculated. 

Averaging Period Up) 

3 hours f3(p) = 0.9796 

24 hours f2 4(p) = 0.8369 

30 days f72()(p) = 0.1118 

It is clear then that when calculating the required long-term average 

emission level, or L, the choice of averaging time affects Z to a small 

extent and f.(p) to a greater extent. For averaging times greater than 24 

hours where compliance is measured on a daily basis (e.g., a 7-day rolling 

average or 30-day rolling average), the Z value does not change. 

A-6 
P.25 

APPENDIX B 

CALCULATION OF COAL LOT SIZE 

The following are sample calculations of coal lot sizes for utility and 

industrial boilers. These calculations were used in developing Tables 1 and 

2 in the main body of this report. 

Utility Boiler Case 

o Boiler size = 500 MW electrical output. 

o Overall thermal-to-electric energy efficiency = 33 percent. 

0 Capacity factor = 0.60 

0 Coal heating value = 12,000 Btu/lb. 

500 MW x 10Q0 KW x 341 4 _Btu_ x 0>6Q x 24JJT x lb Coal 

0.33 Efficiency MW KW-hr day 12,000 Btu 

x — ^ = 3104 M 

2,000 lb day 

0 Round to 3100 ton/day lot size. 

Industrial Boiler Case 

0 Boiler size = 150 million Btu/hr heat input 

0 Capacity factor =0.60 

0 Coal heating value = 12,000 Btu/lb 

B-1 
P.26 

'-•I.J . 

150 million ^ x 0.60 x 24 ^- x }±TML ton 90 

hr day 12,000 Btu 2,000 lb 

ton 

day 

Use 90 ton/day as lot size. 

B-2