EPA-650/4-74-033
SEPTEMBER 1974
Environmental Monitoring Series
ilUDY
\
UJ
O
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EPA-650/4-74-033
COLLABORATIVE STUDY
OF METHOD FOR DETERMINATION
OF STACK GAS VELOCITY
AND VOLUMETRIC FLOW RATE
IN CONJUNCTION
WITH EPA METHOD 5
by
H. F. Hamil and R. E. Thomas
Southwest Research Institute
8500 Culebra Road
San Antonio, Texas 78284
Contract No. 68-02-0626
ROAP No. 26AAG
Program Element No. 1HA327
EPA Project Officer: M. R. Midgett
Quality Assurance and Environmental Monitoring Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
September 1974
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This report has been reviewed by the Environmental Protection Agency
and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of tho Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
11
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SUMMARY AND CONCLUSIONS
This report presents the results of a statistical analysis on data collected in the use of EPA Method 2 in
conjunction with collaborative testing of Method 5 (Particulate Emissions). Method 2 is for the determination of
stack gas velocity and volumetric flow rate and specifies that the stack gas velocity be determined from the gas
density and from measurement of the velocity head using a Type S Pitot tube.
The collaborative tests of Method 5 were conducted at three sites: a Portland cement plant, a coal-fired
power plant, and a municipal incinerator. There were 15, 16 and 12 sampling runs, respectively, at the three sites
and four collaborating laboratories at each. The data from one laboratory at the power plant site were not used,
and some determinations were not made due to equipment failure during the sampling run. This resulted in a
total of 150 separate determinations of both velocity and flow rate being used in the analyses.
The runs at each site were grouped into blocks based upon the velocity heads. The precision components,
within-laboratory, between-laboratory and laboratory bias, are shown to be proportional to the mean of the deter-
minations and are expressed as percentages of the true mean, denoted by 6. The results are summarized below for
each factor.
Velocity-The between-laboratory standard deviation estimate is 5.0% of 6 with 8 degrees of freedom. The
within-laboratory standard deviation estimate is 3.9% of 5 with 113 degrees of freedom. From
these, a laboratory bias standard deviation of 3.2% of 5 may be estimated.
Volumetric Flow Rate—The estimated between-laboratory standard deviation is 5.6% of 8 with 8 degrees of
freedom. The estimated within-laboratory standard deviation is 5.5% of 5 with 113 degrees of
freedom. These give a laboratory bias standard deviation of 1.1% of 6.
The emission rate, denoted by r, is defined in the Federal Register as the product of the volumetric flow
rate and the pollutant concentration. Using the estimates for the precision of the flow rate determination and
estimates for the precision of Methods 5,6, and 7, the precision of r is estimated for each Method.
Based upon the results obtained, the precision of the volumetric flow rate seems adequate for use with other
test methods in determining the emission rate. The precision of r depends primarily upon the precision of the test
method used, which is the desirable result.
in
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TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS vi
LIST OF TABLES vi
I. INTRODUCTION 1
II. COLLABORATIVE TESTING 2
A. Collaborative Test Sites 2
B. Collaborators and Test Personnel 2
III. STATISTICAL DESIGN AND ANALYSIS 7
A. Statistical Terminology 7
B. Test Data 8
C. Test Design and Analysis 9
IV. VELOCITY DETERMINATION PRECISION ESTIMATES 12
V. VOLUMETRIC FLOW RATE PRECISION ESTIMATES 13
VI. EMISSION RATE VARIATION 14
APPENDIX A—Method 2. Determination of Stack Gas Velocity and Volumetric Flow Rate (Type S
PitotTube) 15
APPENDIX B-Statistical Methods 19
B.1 Proportional Relationship Between Mean and Standard Deviation in the
Velocity Determinations 21
B.2 Proportional Relationship Between Mean and Standard Deviation in the Flow
Rate Determination 22
B.3 Unbiased Estimation of Standard Deviation Components 25
B.4 Weighted Coefficient of Variation Estimates 26
B.5 Estimating Precision Components For Velocity Determination 29
B.6 Estimating Precision Components For Volumetric Flow Rate 30
B.7 Emission Rate Variability 33
LIST OF REFERENCES 35
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LIST OF ILLUSTRATIONS
Figure Page
1 Typical Velocity Profiles, Lone Star Portland Cement Plant 3
2 Typical Velocity Profiles, Allen King Power Plant 4
3 Typical Velocity Profiles, Holmes Road Incinerator 5
LIST OF TABLES
Table Page
1 Average \Mp's and Block Designations 10
2 Stack Gas Velocity Data, Arranged by Block 10
3 Volumetric Flow Rate Data, Arranged by Block 11
4 Precision Estimates for Emission Concentrations 14
B-l Velocity Transformation Results 21
B-2 Run Means and Standard Deviations (Velocity, ft/sec) 22
B-3 Collaborator-Block Means and Standard Deviations (Velocity, ft/sec) 23
B-4 How Rate Transformation Results 23
B-5 Run Means and Standard Deviations (Volumetric Flow Rate, ft3/hr X 10'4) . ... 24
B-6 Collaborator-Block Means and Standard Deviations (Volumetric Flow Rate,
ft3/hrX 10-4) 25
B-7 Run Beta Estimates and Weights (Velocity) 30
B-8 Collaborator-Block Beta Estimates and Weights (Velocity) 31
B-9 Run Beta Estimates and Weights (Volumetric Flow Rate) 32
B-10 Collaborator-Block Beta Estimates and Weights (Volumetric Flow Rate) 32
vi
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I. INTRODUCTION
This report describes the work performed on Contracts 68-02-0623 and 68-02-0626, and the results obtained
on Southwest Research Institute Project 01-3462-008, Contract 68-02-0626, which includes collaborative testing
of the method for determination of stack gas velocity and volumetric flow rate with use of Method 5 for particulate
emissions as given in "Standards of Performance for New Stationary Sources"' '.
This report describes the statistical analysis of data from collaborative tests conducted in a Portland cement
plant/2^ a coal-fired power plant/ * and a municipal incinerator/ *
The collaborative tests of the method for determination of stack gas velocity and volumetric flow rate were
not run as separate tests of Method 2 but as this method is used in conjunction with Method 5 for particulate
emissions/1*
The results of the data analyses are given in this report.
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II. COLLABORATIVE TESTING
A. Col laborative Test Sites
The site of the Portland cement plant test was the Lone Star Industries Portland Cement Plant in Houston,
Texas. This plant utilizes the wet feed process and operates three kilns. The flue gas from each kiln passes through
a separate electrostatic precipitator. The flue gases are then combined and fed into a 300-foot-high stack.( '
Samples were taken at the sample ports located on the stack 150 feet above grade. Inside diameter of the stack
at the sample ports is 13 feet.* The cross-sectional area of the stack at the sample ports is 132.73 ft2 .* The
average stack gas velocity ranged from about 50 to 60 ft/sec* during the test period. A typical velocity profile
is shown in Figure 1. The typical volumetric flow rate was about 12 X 106 fr /hi* dry gas basis at 70°F and
1 atmosphere.
The site of the coal-fired power plant was the Allen King Power Plant, The Northern States Power Company,
near St. Paul, Minnesota. The exhaust gas from the combustion chamber passes through the heat exchanger and
splits into two identical streams upstream of twin electrostatic precipitatofs. The twin emission gas streams are
fed into an 800-foot-high stack through two horizontal ducts/3* The sample ports were located in the south
horizontal duct upstream of the entrance to the stack. The inside duct dimensions are 12 feet wide by 27 feet
high. The duct cross sectional area is 324 ft2. The average gas velocity was about 50 ft/sec. A typical velocity
profile is shown in Figure 2. The typical total volumetric flow rate (flow rate in the duct times 2) was about
70 X 106ft3/hr.
The site for the municipal incinerator test was the Holmes Road Incinerator, City of Houston, Houston, Texas.
The facility consists of two independent parallel furnace trains. Refuse feeds continuously onto traveling grate
stokers in the furnaces. Gases leaving the furnaces are cooled in water spray chambers and then enter the flue gas
scrubbers to remove participates. The gases are then drawn through induced draft fans and exhaust into the
148-foot-high stacks. Samples were taken from the sample ports located on the stacks 102 feet above grade. The
inside diameter of both stacks is 6.5 ft. The cross-sectional area of each stack is 33.18 ft2. The typical stack
gas velocity for both stacks was about 50 ft/sec (Fig. 3). The typical volumetric flow rate for either unit was about
3.5 X 106 ft3/hr. Determinations were made on both stacks during the test. Only one furnace train was operating
at any time during the test.
B. Collaborators and Test Personnel
The collaborators for the Lone Star Industries Portland Cement Plant test were Mr. Charles Rodriguez and
Mr. Nollie Swynnerton of Southwest Research Institute, San Antonio Laboratory, San Antonio, Texas; Mr. Mike
Taylor and Mr. Ron Hawkins of Southwest Research Institute, Houston Laboratory, Houston, Texas; Mr. Quirino
Wong, Mr. Randy Creighton, and Mr. Vito Pacheco, Department of Public Health, City of Houston, Houston, Texas;
and Mr. Royce Alford, Mr. Ken Drummond, and Mr. Lynn Cochran of Southwestern Laboratories, Austin, Texas.
The collaborators for the Allen King Power Plant test were Mr. Mike Taylor and Mr. Hubert Thompson of
Southwest Research Institute, Houston Laboratory, Houston, Texas; Mr. Charles Rodriguez and Mr. Ron Hawkins of
Southwest Research Institute, San Antonio Laboratory, San Antonio, Texas; Mr. Gilmore Sem, Mr. Vern Goetsch,
and Mr. Jerry Brazelli of Thermo-Systems, Inc, St. Paul, Minn.; and Mr. Roger Johnson and Mr. Harry Patel of
Environmental Research Corporation, St. Paul, Minn.
The collaborators for the Holmes Road Incinerator test were Mr. Mike Taylor and Mr. Rick Hohmann of South-
west Research Institute, Houston Laboratory, Houston, Texas; Mr. Charles Rodriguez and Mr. Ron Hawkins of
*EPA policy is to express all measurements in Agency documents in metric units. When implementing this practice will result in undue
cost or difficulty in clarity, NERC/RTP is providing conversion factors for the particular non-metric units used in the document. For
this report, the factors are:
1 ft = 0.3048 meters 1.0 ft*= 0.0929 meters2
1 ft/sec = 0.3048 meters/sec 1 ft3/hr = 0.0283 meters3/hr
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70
65-
c
8
r. 60
c
c
55
50
45
Stack Diameter, feet
B-D Ports
10
~
45
6 8
Stack Diameter, feet
1C
A-C Ports
FIGURE 1. TYPICAL VELOCITY PROFILES,
LONE STAR PORTLAND CEMENT PLANT
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fin
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40
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Profile Across Lower Ports
FIGURE 2. TYPICAL VELOCITY PROFILES, ALLEN KING POWER PLANT
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c
•
B
50
4E
36 48
Stack Diameter, inches
Profile, East Stack
72 78
C
•:
36 48
Stack Diameter, inches
Profile, West Stack
D Axis through ports A, C
O Axis through ports B, D
FIGURE 3. TYPICAL VELOCITY PROFILES, HOLMES ROAD INCINERATOR
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Southwest Research Institute, San Antonio Laboratory, San Antonio, Texas; Mr. Quirino Wong, Mr. Randy Creighton,
and Mr. Steve Byrd, City of Houston, Department of Public Health; Mr. John Key, Mr. James Draper, Mr. Tom
McMickle, Mr. Tom Palmer, Mr. Michael Lee, and Mr. Charles Goerner, Air Pollution Control Services, Texas
State Department of Health.*
The Portland cement plant test was conducted under the supervision of Dr. Henry Hamil, and the power plant
and municipal incinerator tests were conducted under the supervision of Mr. Nollie Swynnerton, both of Southwest
Research Institute.
Collaborators for all three tests were selected by Dr. Hamil.
*Throughout the remainder of this report, the collaborative laboratories are referred to by randomly assigned code numbers. For the
cement plant test, code numbers 101,102,103, and 104 are used. For the power plant test, code numbers 201, 202, 203, and 204
are used. For the cement plant test, code numbers 301, 302,303, and 304 are used. These numbers do not correspond to the above
ordered listing of laboratories, and may differ from the code numbers assigned in the previous reports.^>^
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III. STATISTICAL DESIGN AND ANALYSIS
A. Statistical Terminology
To facilitate the understanding of this report and the utilization of its findings, this section explains the statis-
tical terms used in this report. The procedures for obtaining estimates of the pertinent values are developed and
justified in the subsequent sections.
A A
We say that an estimator, 9 , is unbiased for a parameter 6 if the expected value of 0 is 6 , or expressed in nota-
tional form, E(ff) = 6 . From a population of method determinations made at the same true level, /z, let Xi , .... xn
be a sample of n replicates. Then we define:
1 "
(1) 3c = — /_* Xj as the sample mean, an unbiased estimate of the true determination mean, 5, the center of
ni=l
the distribution of the determinations. For an accurate method, 5 is equal to M, the true level.
1 n
(2) s2 = - y")C*i ~ * )2 as the sample variance, an unbiased estimate of the true variance, o1 . This
«-i,tt
term gives a measure of the dispersion in the distribution of the determinations around 6.
(3) s = Vs as the sample standard deviation, an alternative measure of dispersion, which estimates a, the true
standard deviation.
The sample standard deviation, s, however, is not unbiased for o/5^ so a correction factor needs to be applied.
The correction factor for a sample of size n is an, and the product of an and s is unbiased for a. That is, E(ans) = a.
As n increases, the value of an decreases, going for example from a3 = 1.1284,04 = 1.0854 too^o = 1.0281. The for-
mula for <*„ is given in Appendix B.3.
We define
as the true coefficient of variation for a given distribution. To estimate this parameter, we use a sample coefficient
of variation, @, defined by
where 0 is the ratio of the unbiased estimates of a and 5 . The coefficient of variation measures the percentage
scatter in the observations about the mean and thus is a readily understandable way to express the precision of
the observations.
There were a total of 43 sampling runs for the three tests. Since the actual velocity, and hence the flow rate,
fluctuates, one can in general expect different true levels for each run. To permit a complete statistical analysis,
the individual runs are grouped into blocks, where each block has approximately the same true level.
We can apply the statistical terms of the preceding paragraphs both to the collaborators' values during a
given run and to each collaborator's values in a given block. In this report, statistical results from the first situa-
tion are referred to as run results. Those from the second situation are referred to as collaborator-block results.
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For example, a run mean is the average of all the determinations made in a run as obtained by Method 2. A col-
laborator-block coefficient of variation is the ratio of the unbiased standard deviation to the sample mean for all
the collaborator's runs grouped in the block.
The variability associated with a Method 2 determination is estimated in terms of the within-laboratory and
the between-laboratory precision components. In addition, a laboratory bias component can be estimated. The
following definitions of these terms are given with respect to a true level, M-
• Within-laboratory-Jhe within-laboratory standard deviation, a, measures the dispersion in replicate single
determinations made using Method 2 by one laboratory team (same field operators, laboratory analyst,
and equipment) sampling the same true level, /*. The value of a is estimated from within each col-
laborator-block combination.
• Between-laboratory '-The between-laboratory standard deviation, aj,, measures the total variability in
a determination due to simultaneous Method 2 determinations by different laboratories sampling
the same true stack level, ji. The between laboratory variance, a|, may be expressed as
and consists of a within-laboratory variance plus a laboratory bias variance, o| . The between-laboratory
standard deviation is estimated using the run results.
• Laboratory Mzs-The laboratory bias standard deviation, ff/, = ya \ - a2 , is that portion of the total
variability that can be ascribed to differences in the field operators, analysts and instrumentation, and
due to different manners of performance of procedural details left unspecified in the method. This
term measures that part of the total variability in a determination which results from the use of the method
by different laboratories, as well as from modifications in usage by a single laboratory over a period of time.
The laboratory bias standard deviation is estimated from the within- and between -laboratory estimates
previously obtained.
B. Test Data
This study is based upon velocities and volumetric flow rates obtained in the use of Method 5. The average
velocity, (Fs)aVg, i& calculated as
ft/sec
P^s
where
Kp = 85.48 for the units used,
Cp - the pitot tube coefficient
(\/^)avg — the average square root of the velocity head of the stack, inches H20
(5rs)avg — the average absolute stack gas temperature, °R
Ps — the absolute stack gas pressure, inches Hg
and
Ms - the molecular weight of the stack gas (wet basis), Ib/lb-mole.
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The data used in the calculation of the velocities and flow rates were obtained during the sampling runs and
not from the preliminary velocity traverses. These, then, represent 2-hour average velocities and flow rates across
the stack. The volumetric flow rate, Qs, is calculated as
Qs = (3600X1 -
where A is the cross-sectional area of the stack, and Bwo is the volume fraction of water vapor in the gas stream.
In conjunction with the testing of Method 5, the collaborators calculated average stack velocities but not
volumetric flow rates, since the concentration determinations in the previous studies were the final results used in
the analysis. The velocities were recalculated to ensure their accuracy, and the flow rates calculated using these
velocities and the other test data.
The results obtained by Lab 201 were excluded from the analysis. In a study on moisture fraction determination,
Lab 201 was eliminated due to the probable development of leakage during some runs and filter contamination due
to use of a low-melting ground-joint lubricant. Since this would adversely affect the volume of liquid collected due
to the introduction of ambient air into the train, their moisture fractions were not usable. Since the moisture
fraction is involved directly in the Qs determination and indirectly, throughMs, in the (Kj)avg determination,
these data were not judged acceptable.
C. Test Design and Analysis
The data were arranged in blocks where the true velocity was assumed to be essentially constant. The velocity
determination has been shown* ' to be principally dependent upon the value of (v^)aVg - Thus, this provides
a valid means of determining when there was a change in the stack gas velocity. The actual reading of the velocity
head is a function of the particular pitot tube that is used, but by comparing the values of all collaborators, the
increases and decreases in velocities can be determined. The average v^'s are shown in Table 1, along with the
blocks to which the runs were assigned. The determinations used in the analyses are shown arranged in blocks
in Table 2 and 3.
To determine the best method of analyzing the data, Bartlett's test for homogeneity of variance was used to
determine the appropriateness of an analysis of variance approach. The data were then transformed using the
logarithmic transformation and retested by Bartlett's test. The details are contained in Appendices B.I and B.2.
For the velocity data, significance levels under both transformations indicate suitability. If a logarithmic
transformation is accepted, the conclusion is that there is a proportional relationship between the true mean and
true standard deviation. If a linear transformation is accepted, then the indication is that the variance is
independent of the mean. An investigation of the proportional relationship was conducted on an empirical
basis to determine which of the two models should be used.
The correlation between the sample means and standard deviations is determined for both the run data and
the collaborator-block data. The model chosen is a no-intercept model, meaning that when the sample mean is
zero, the sample standard deviation must also be zero. The coefficient of determination, r2, is the measure of the
appropriateness of the model. For the run data, the value of r2 was 0.80, which gives a correlation coefficient,
r = VT% of 0.89 based on 43 pairs. This value is significant at the 5 percent level. For the collaborator-block
data, the value of r2 was 0.75, and the correlation coefficient, r, was 0.86, based on 37 pairs. This value also
exceeds the value for the 5 percent significance level.
Thus, there is evidence of a proportional relationship between the mean and the standard deviation for
the velocity data. This is equivalent to saying that the standard deviations, a/, and a, change as the mean, 6,
changes. That is,
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TABLE 1. AVERAGE v/A^'s AND BLOCK
DESIGNATIONS.
TABLE 2. STACK GAS VELOCITY DATA,
ARRANGED BY BLOCK.
Velocity, ft/sec
Run
Labs
Block
Site 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
101
0.83
0.83
0.83
0.75
0.75
0.78
0.68
0.72
0.74
0.74
0.72
0.73
0.72
_*
0.67
102
0.82
0.83
0.84
0.75
0.78
0.75
0.69
0.76
0.74
0.73
0.72
0.76
0.73
0.69
0.70
103
_*
0.91
1.02
0.77
0.76
0.76
0.69
0.73
0.73
0.74
0.72
0.73
0.71
0.67
0.69
104
0.83
0.83
0.83
0.79
0.75
0.79
0.74
0.72
0.77
0.73
0.75
0.76
0.75
0.70
0.70
1
1
1
2
2
2
4
3
2
3
3
3
3
4
4
Site 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
202 203
0.75 0.80
0.74 0.75
0.70 0.79
0.78 0.80
0.73 0.75
0.74 0.79
0.69 0.76
0.79 0.80
0.68 0.71
0.73 0.74
0.72 0.75
0.71 0.73
0.76 0.75
0.76 -*
0.71 0.73
0.75 0.80
204
0.76
0.76
0.77
0.78
0.73
0.77
0.74
0.76
0.74
0.77
0.75
0.67
0.75
0.74
0.75
0.72
1
2
1
1
2
1
2
1
3
2
2
3
2
2
2
2
Site 3
1
2
3
4
5
6
7
8
9
10
11
12
301
0.79
0.77
0.78
0.78
0.83
0.82
0.79
0.83
0.68
0.84
0.77
0.82
302
0.81
0.91
1.00
0.73
0.86
0.90
0.86
0.89
0.75
0.84
0.83
0.87
303
0.78
0.73
0.79
0.73
0.78
0.74
0.75
0.70
0.67
0.78
0.77
0.76
304
_*
0.88
0.78
0.86
0.83
0.81
0.84
0.74
0.72
0.87
0.84
0.94
3
2
1
3
2
2
3
3
4
1
3
1
*Run not made due to equipment failure.
Block
Run
Labs
Site 1
1
2
3
4
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
101
62.4
61.1
61.7
56.5
55.9
57.6
51.9
49.5
52.6
51.2
51.9
51.7
46.9
_*
52.0
102
62.3
63.2
63.1
56.5
60.3
56.0
52.8
53.9
53.4
54.1
55.3
53.2
48.7
52.0
51.5
103
_*
59.0
66.1
57.3
56.3
56.4
52.3
51.7
53.0
52.4
53.3
51.9
49.0
48.6
49.9
104
58.2
60.7
63.3
60.1
57.6
60.4
55.6
51.8
53.8
55.9
57.0
55.3
51.9
52.9
52.2
Site 2
1
2
3
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
202
51.2
47.3
53.1
50.2
53.7
50.1
49.8
46.5
48.8
48.1
51.9
51.3
48.2
50.6
46.0
48.6
203 204
52.3 47.6
51.8 51.3
52.5 49.6
52.0 49.3
51.3 47.5
48.7 49.9
49.3 48.1
48.8 45.4
48.3 44.8
46.6 48.8
47.6 48.8
-* 49.8
46.8 48.6
51.7 47.0
45.0 49.5
46.5 43.0
Site 3
1
2
3
4
3
10
12
2
5
6
1
4
7
8-
11
9
301
46.9
50.9
50.5
48.0
51.7
51.1
48.6
48.0
48.8
51.6
47.4
41.7
302
52.3
51.8
50.2
48.2
51.6
48.9
51.0
47.9
49.6
46.1
51.0
44.8
303
57.9
53.5
55.5
58.4
55.6
57.5
55.9
56.7
55.3
56.5
53.4
48.7
304
47.7
53.0
57.6
54.2
53.2
50.2
_*
54.3
52.0
46.0
51.5
48.5
*Run not made due to equipment failure.
10
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TABLE 3. VOLUMETRIC FLOW RATE DATA,
ARRANGED BY BLOCK
Volumetric Flow Rate, ft3/hi X 10'4
°b =
Block
Run
Labs
Site 1
1
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
101
1267.6
1244.2
1238.5
1101.3
1123.3
1183.7
1257.4
1243.9
1240.1
1241.7
1208.4
1184.5
1078.2
_*
1136.8
102
1176.5
1243.7
1220.3
1141.6
1065.6
1110.5
1256.7
1335.0
1245.9
1167.2
1198.1
1160.7
1261.9
1086.1
1118.8
Site
1
2
3
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
202
7170.8
6617.2
7324.7
6995.2
7445.7
6895.3
6935.4
6555.7
6835.6
7364.3
7173.4
7119.1
6694.1
7093.1
5819.7
6835.7
103
_*
1384.0
1496.7
1250.2
1238.4
1248.5
1403.8
1337.1
1341.2
1255.8
1257.8
1266.2
1270.0
1152.5
1205.0
2
203
7145.4
7110.8
7162.2
7011.5
7199.8
6606.4
6741.9
6850.2
6861.3
6534.0
6887.8
_*
6586.4
7177.6
6582.5
6730.8
104
1254.2
1251.4
1410.4
1274.7
1158.1
1220.9
1330.8
1320.1
1255.2
1198.7
1192.6
1240.5
1431.5
1114.7
1135.8
204
6947.2
7240.3
6855.4
6880.2
6683.3
6958.7
6823.1
6551.8
6364.2
6839.9
7015.9
6928.9
6830.0
6602.6
7101.8
6107.2
Site 3
1
3
10
12
2
5
6
1
4
7
8
11
9
301
335.8
354.1
319.8
311.3
261.3
335.4
309.3
301.4
308.4
323.8
312.6
273.6
302
371.7
369.1
367.6
355.5
350.9
349.5
368.3
329.5
339.4
323.4
348.7
311.7
303
333.5
317
342
334
.3
.2
.6
310.5
325.6
357
.1
316.6
325.4
352.0
303
.7
276.7
304
311.8
353.9
374.5
345.1
284.5
320.2
_*
317.8
326.2
281.1
329.0
289.5
*Run not made due to equipment failure.
and
where fo and (1 are the true coefficients of variation for between-
laboratory and within-laboratory, respectively. The standard
deviations are estimated, then, as
and
where &b and 0 are the estimated coefficients of variation.
For the volumetric flow rates obtained, a similar
investigation is done. For these values, the only acceptable
transformation is the logarithmic, which implies, on a theoretical
basis, an underlying proportional relationship between the
population mean and the population standard deviation for
both the run data and the collaborator-block data.
To establish this empirically, the paired sample means
and standard deviations are fit to a no-intercept regression
model. The run data give an r2 of 0.73 and a correlation
coefficient of 0.85, based on 43 pairs. The collaborator-block
r2 is 0.63, with r = 0.79 based on 37 pairs. Both r values are
significant at the 5 percent significance level.
As a result, the volumetric flow rate within-laboratory
and between-laboratory standard deviations can be said to
be proportional to the mean level. The estimates of these
standard deviations will be expressed using coefficients of
variation times an unknown mean in the same manner as
the velocity data.
At each site, there were occasional missing values due to
equipment malfunctions and varying block sizes, so that not
all coefficients of variation are based on the same number of
observations. To account for this, for each site the individual
beta estimates are weighted so that a greater contribution to
the final estimate, is made by those values based on larger
samples. The weighting technique is based upon the number
of values in each run or block and is discussed in detail in
Appendix B.4. The beta values from all three sites form a
composite estimate of the coefficients of variation for
both velocity and flow rate.
11
-------
IV. VELOCITY DETERMINATION PRECISION ESTIMATES
The between-laboratory standard deviation, aj, and the within-laboratory standard deviation, a, for (Vs) avg
are estimated as
and
In Appendix B.5, the data from the three sites are used to obtain estimates of these terms using a linear combination
of the individual values.
The between-laboratory coefficient of variation is |3j = (0.050). This gives a between-laboratory standard
deviation of
= (0.050)5
or 5.0% of the mean. This estimate has 8 degrees of freedom associated with it.
The within-laboratory coefficient of variation is estimated as $ = (0.039). This gives an estimated within-
laboratory standard deviation of
0 = 08
= (0.039)6
or 3.9% of the mean. There are 1 13 degrees of freedom associated with this term.
From the formula in Section IIIA, the laboratory bias standard deviation, a/, , is given by
°L =
Substituting the estimates above into this formula gives
= V(0.050)282-(0.039)262
= >/[(0.050)2-(0.039)2]62
= >/(0.001)52
= (0.032)5
or 3.2% of the mean level.
12
-------
V. VOLUMETRIC FLOW RATE PRECISION ESTIMATES
The between-laboratory standard deviation, ffft, and the within-laboratory standard deviation, 6 , for Qs are
estimated as
and
where 0& and 0 are the estimated coefficients of variation. In Appendix B.6 the individual beta estimates are com-
bined to obtain estimates of these from the run data and collaborator-block data, respectively.
The estimated between-laboratory coefficient of variation is fa = (0.056). This gives an estimated between-
laboratory standard deviation of
= (0.056)5
or 5.6% of the mean. This estimate has 8 degrees of freedom associated with it.
The within-laboratory coefficient of variation is estimated by 3 = (0.055). This gives an estimated within-
laboratory standard deviation of
a=J36
= (0.055)6
or 5.5% of the mean. There are 113 degrees of freedom associated with this estimate.
The laboratory bias standard deviation is defined as
Substituting ab and a into this formula gives
5/.=V°J
= Jal-P
= x/(0.056)252 - (0.055)262
= N/[(0.056)2-(0.055)2]62
= (0.011)5
or 1.1% of the mean level.
13
-------
VI. EMISSION RATE VARIATION
The standards of performance*1* for certain sites (e.g., power plants, nitric acid plants, Portland cement plants)
specify that the product of the volumetric flow rate and the emission concentration obtained by the appropriate
method be used in determining compliance with the regulations. The rate is denoted in this study by r, where
It is of interest to determine the precision of this product based upon the precision of the individual components.
In Appendix B.7, the formula is developed for estimating a precision component for this product when both
the flow rate and the concentration determination follow the coefficient of variation hypothesis. The formulas for
the within-laboratory and between-laboratory variances are given by
and
where ^(G4) and P(QS) are the between and within-laboratory coefficients of variation for flow rate, $b(c) and
|3(c) are the coefficients of variation for emission concentrations, and 5r is the mean emission rate.
TABLE 4. PRECISION ESTI- In Table 4 are listed values of $b(c) and 0(c) for Methods 5, 6 and 7 based upon
. MATES FOR EMISSION previous collaborative studies. Using these and the coefficients of variation for Qs
CONCENTRATIONS developed in this study, estimates can be made of the precision associated with r.
Method
5<4>
6(7)
7(8)
0(0
0.253
0.040
0.066
*>
0.387
0.058
0.095
Method J-The between-laboratory standard deviation estimate is 39.2% of dr,
and the within-laboratory standard deviation estimate is 25.9% of 6r. This gives a
laboratory bias term of 29.4% of 6r.
Method 6—The estimated between-laboratory standard deviation is 8.1% of 5r,
with an estimated within-laboratory standard deviation of 6.8% of 5r. From these, the
laboratory bias standard deviation is estimated as 4.5% of 8r.
Method 7-The estimated between-laboratory standard deviation is 11.0% of 8r. The estimated within-laboratory
standard deviation is 8.6% of 8r. Using these, the laboratory bias standard deviation is estimated as 7.1% of 5,-.
As can be seen from these results, the precision in r depends primarily upon the precision of the emission con-
centration determination, and little variation is introduced by the volumetric flow rate determination.
14
-------
APPENDIX A
METHOD 2. DETERMINATION OF STACK GAS VELOCITY AND
VOLUMETRIC FLOW RATE {TYPE S PITOT TUBE)
15
-------
24884
RULES AND REGULATIONS
METHOD a—DETERMINATION Or STACK GAS
VELOCITY AND VOLUMETRIC HOW BATE (TYPE
S Pl'ruT TUBE)
1. Principle and applicability.
I.I Principle. Stack gas velocity Is deter-
mined from the gas density Bud from meas-
urement of the velocity head using a Type S
(SUuisohelbe or reverse type) pltot tube.
1.2 Applicability. This method should be
applied only when specified by the test pro-
cedures for determining compliance with the
New Source Performance Standards.
2. Apparatus.
2.1 Pttot tube—Type S (Figure 2-1), or
equivalent, with a coefficient within ±5%
over the working range.
2.2 Differential pressure gauge—Inclined
manometer, or equivalent, to measure velo-
city head to within 10% of the minimum
value.
2.3 Temperature gauge—Thermocouple or
equivalent attached to the pltot tube to
measure stack temperature to wrthln 1.5% of
the minimum absolute stack, temperature.
2.4 Pressure gauge—Mercury-filled U-tube
manometer, or equivalent, to measure stack
pressure to'within 0.1 In. Hg.
2.5 Barometer—To measure atmospheric
pressure to within 0.1 In. Hg.
2.6 Oas analyzer—To analyze gas composi-
tion for determining molecular weight.
2.7 Pltot tube—Standard type, to cali-
brate Type S pltot tube.
3. Procedure.
3.1 Set up the apparatus as shown In Fig-
ure 2-1. Make sure all connections are-tight
and leak free. Measure the velocity head and
temperature at the traverse points specified
by Method 1.
3.2 Measure the static pressure in the
stack.
3.3 Determine the stack gas molecular
weight by gas analysis and appropriate cal-
culations as Indicated In Method 3.
4. Calibration.
4.1 To calibrate the pitot tube, measure
the velocity head at some point In a flowing
gas stream with both a Type S pltot tube and
a standard type pltot tube with known co-
efficient. Calibration should be done In the
laboratory and the velocity of the flowing gas
stream should be varied over the normal
working range. It is recommended that the
calibration be repeated after use at each field
site.
4.2 Calculate the pitot tube coefficient
using equation 2-1.
equation 2-1
where:
Cp,,.,=Pltot tube coefficient of Type S
pitot tube.
Cp,tJ=Pitot tube coefficient of standard
type pltot tube (if unknown, use
0.99) .
Apnd= Velocity head measured by stand-
ard type pitot tube.
Apt. .t=: Velocity head measured by Type S
pitot tube.
4.3 Compare the coefficients of the Type S
plbot tube determined first with one leg and
then the other pointed downstream. Use the
pitot tube only if the two coefficients differ by
no more than 0.01.
6. Calculations.
Use equation 2-2 to calculate the stack gas
velocity.
PIPE COUPLING
TUBING ADAPTER
Equation 2-2
where :
(V»)«c.= Stack gas velocity, feet per second (f.p.s.).
Cp=Pltot tube coefficient, dimensionl'1ss
(T.)«£.=A7erage absolute stack gas temperature
R.
( VAP) >T,.= Average velocity head of stack gas, inches
HiO (see Fig. 2-2).
P.=Absohite stack gas pressure, Inclips \JK.
M.= Molecular weight of stack gaa (wet basis).
Ib./Ib.-mole.
Mj(l-Bwo)+18Bwo
Md=Dry molecular weight of stack gas (from
Methods).
B»0=Proportion by volume of water vapor in
the gas stream (from Method 4).
Figure 2-2 shows a sample recording sheet
for velocity traverse data. Use the averages
In the last two columns of Figure 2-2 to de-
termine the average stack gas velocity from
Equation 2-2.
Use Equation 2-3 to calculate the stack
gas volumetric flow rate.
.=3600 a-
Equation 2-3
where:
Q.=Volumetric flow rate, dry basis, standard condi-
tions. ft.'/hr.
A =Cross-secttonal area of stack, ft."
T,td= Absolute temperature at standard conditions,
Figure 2-1. Pttot tube-manomotar assembly.
P.id= Absolut* pressure at standard conditions, 29.92
inches Hg.
fEOHAL UGISRR, VOL M. NO. 247—THURSDAY, DECEMBER 23, 1971
17
-------
RULES AND REGULATIONS
24885
6. References.
Mark. L. 3., Mechanical Engineers' Hand-
book. McGraw-Hill Book Co., Inc.. New York,
N.T., 1951.
Ferry, J. H., Chemical Engineers' Hand-
book, McGraw-Hill Book Co., Inc., New York,
N.Y., 1960.
Shigehara, R. T., W. P. Todd, and W. 8.
Smith, Significance of Errors In Stack Sam-
pling Measurements. Paper presented at the
Annual Meeting of the Air Pollution Control
Association, St. Louis, Mo., June 14-18, 1970.
Standard Method for Sampling Stacks for
Paniculate Matter, In: 1971 Book of ASTM
Standards, Pan 23. Philadelphia, Pa, 1971,
ASTM Designation D-2S29-71.
Vennard, J. K., Elementary Fluid Mechan-
ics, John Wiley & Sons, Inc., New York, N.Y,
1947.
PIANT_
DATE
RUN NO.
STACK DIAMETER, in..
BAROMETRIC PRESSURE, in. Hg._
STATIC PRESSURE IN STACK (Pg). in. Hg._
OPERATORS
SCHEMATIC OF STACK
CROSS SECTION
Traverse point
number
Velocity head.
in. HjO
Stack Temperature
AVERAGE:
.Figure 2-2. Velocity traverse data.
FEDERAl REGISTER, VOL. 36, NO. 247—THURSDAY, DECEMBER 23, 1971
18
-------
APPENDIX B
STATISTICAL METHODS
19
-------
APPENDIX B. STATISTICAL METHODS
This appendix consists of various sections which contain detailed statistical procedures carried out in the
analyses of the Method 2 data. Reference to these sections has been made at various junctures in the body of this
report. Each Appendix B section is an independent ad hoc statistical analysis pertinent to a particular problem
addressed in the body of the report.
B.1 Proportional Relationship Between Mean and Standard Deviation in the Velocity
Determinations
The velocities shown in Table 2 are tested to determine if the variance is independent of the mean level in
their original form (linear) and after having undergone a logarithmic transformation. Bartlett's test for homogeneity
of variance^ is used to determine the suitability of each transformation. The obtained values of the statistic with
degrees of freedom and significance levels are shown in Table B-l. The significance levels are obtained from a
chi-square distribution with the degrees of freedom shown.
TABLE B-l. VELOCITY TRANSFORMATION RESULTS
Data
Run
Collaborator-Block
Transformation
Linear
Logarithmic
Linear
Logarithmic
Test
Statistic
44.391
46.219
47.932
48.084
DF
42
42
36
36
Significance
Level
0.371
0.302
0.088
0.086
For both the run data and the collaborator-
block data, either form is acceptable. The
acceptance of the linear form of the data implies
that the variance is independent of the mean,
that is, constant regardless of the mean value.
The acceptance of the logarithmic transforma-
tion implies a proportionality between the pop-
ulation mean and the population standard
deviation, or that as the mean level rises (falls),
the standard deviation rises (falls) in a propor-
tional manner.
Both transformations are acceptable at nearly equal significance levels. To determine if there is further
evidence of a proportional relationship between the mean and standard deviation, a regression model is fit to the
data. The model chosen is a no-intercept model,
y = bx
so that a sample mean of zero implies a sample standard deviation of zero. Define
Xjjk as the determination by collaborator i on run k in block/.
1
x . = —
k as the mean of run k in block / for p collaborators
and
»/* =
as the run standard deviation.
The paired means and standard deviations, (jc ./#, Sjk), shown in Table B-2 are fit to the model, and the degree of fit
determined by the coefficient of determination, r2. For this model, r2 is calculated as^
21
-------
TABLE B-2. RUN MEANS AND STANDARD
DEVIATIONS (Velocity, ft/sec)
Run
Mean
Velocity
Standard
Deviation
Site 1
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
61.0
61.0
63.5
57.6
57.5
57.6
53.1
51.7
53.2
53.4
54.4
53.0
49.1
51.2
51.4
2.4
1.7
1.8
1.7
2.0
2.0
1.7
1.8
0.5
2.0
2.2
1.7
2.1
2.3
1.0
Site 2
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
50.4
50.1
51.7
50.5
50.8
49.6
49.1
46.9
47.3
47.8
49.4
50.5
47.9
49.8
46.8
46.0
2.5
2.5
1.9
1.4
3.1
0.8
0.9
1.7
2.2
1.1
2.2
1.1
0.9
2.5
2.4
2.8
Site 3
3
10
12
2
5
6
1
4
7
8
11
9
51.2
52.3
53.4
52.2
53.0
51.9
51.8
51.7
51.4
50.0
50.8
45.9
5.1
1.2
3.7
5.0
1.9
3.8
3.7
4.5
2.9
5.0
2.5
3.3
For the run data, r2 = 0.80, which indicates that 80 percent of the
variation in the magnitude of the standard deviation is attributed to
variation in the magnitude of the mean. The correlation coefficient,
r = V^3", is 0.89 based on 43 pairs of observations, which is significant
at the 5 percent level.
For the collaborator-block data, we define
_J_ "V
determinations.
as the mean of collaborator-block ij, for
and
- x ,y.)2 as the collaborator-block standard
deviation.
The values obtained are shown in Table B-3. Fitting these to a no-
intercept model, we have a coefficient of determination of r2 = 0.75
and a correlation coefficient of 0.86. This value is also significant at
the 0.05 level, based upon 37 pairs of observations.
Thus, we have that on a theoretical basis, from the acceptability
of the logarithmic transformation, and an empirical basis, from the regres-
sion model, there is strong evidence that a proportional relationship exists
between the mean and standard deviation for the velocity data. This
is equivalent to saying that the coefficients of variation for both between-
and within-laboratory components remain constant. This gives the
equations
and
Then we estimate the standard deviations by estimating the
coefficients of variation and defining new estimators 8& and d,
and
where fe and |3 are the estimated coefficients of variation for between-
laboratory and within-laboratory, respectively. Thus, the standard deviations
are estimated as percentages of an unknown mean, 5.
B.2 Proportional Relationship Between Mean and Standard Deviation in the Flow Rate Determination
The calculated volumetric flow rates in Table 3 are tested for equality of variance in two forms: their original
form (linear) and after having been passed through a logarithmic transformation. Bartlett's tesr ' for homogeneity
of variance is used to determine the adequacy of each transformation, and the test statistic is compared to a chi-square
22
-------
TABLE B-3. COLLABORATOR-BLOCK MEANS
AND STANDARD DEVIATIONS
(Velocity, ft/sec)
TABLE B-4. FLOW RATE TRANSFORMATION RESULTS
Block
Collaborator
Mean
Velocity
Standard
Deviation
Site 1
1
2
3
4
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
61.73
62.87
62.55
60.73
55.47
56.40
55-57
58.42
51.38
53.98
52.46
54.76
49.45
50.73
49.17
52.33
0.65
0.49
5.02
2.55
2.49
3.07
2.23
2.26
1.16
0.82
0.69
2.02
3.61
1.78
0.67
0.51
Site 2
1
2
3
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
51.10
51.98
49.06
49.48
48.47
47.91
47.30
45.75
46.25
Site 3
1
2
3
4*
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
49.43
51.43
55.63
52.77
50.27
49.57
57.17
52.53
48.88
49.12
55.56
50.95
41.70
44.80
48.70
48.50
2.55
0.47
1.58
1.72
1.62
1.82
1.84
1.06
4.60
2.20
1.10
2.20
4.95
1.99
1.80
1.43
2.08
1.62
2.12
1.33
3.52
_
-
-
-
*No standard deviations since block contains
only one run.
Data
Run
Collaborator-Block
Transformation
Linear
Logarithmic
Lineal
Logarithmic
Test
Statistic
192.451
48.401
192.416
62.844
DF
42
42
36
36
Significance
Levei
0.000
0.230
0.000
0.004
distribution with the appropriate degrees of freedom. The values
for both the run and collaborator-block data are shown in Table B-4.
Clearly, for the run data the logarithmic transformation is
acceptable, while the linear form of the data is not. The reason for
this is apparent from the formula for Qs. The principal factor upon
which Qs depends is (f^avg, but the use of the multipliers of (3600)
and the cross-sectional area of the stack increases the magnitude of
the velocity variation. However, the relative variation, as expressed
by the coefficient of variation, tends to remain constant from site
to site. For the collaborator-block data, the logarithmic transformation
would not be considered acceptable but is an improvement over the
linear form. The acceptance of the logarithmic transformation implies,
on a theoretical basis, a proportional relationship between the mean
and standard deviation of the distribution.
To further investigate the proportional relationship, a least
squares model is fit to the paired sample means and sample standard
deviations. For the run data, define
Xjjk — the flow rate determined by collaborator /on run k
in block/.
V - _ V
*•/'*" L,
collaborators
as the run mean, where p is the number of
and
P ~ 1 i = 1
the run standard deviation.
A no-intercept model is fit to the pairs (x./jt, s/jt), since a mean of zero
automatically implies a standard deviation of zero. The paired means
and standard deviations are shown in Table B-5. The fit to this model
23
-------
TABLE B-5. RUN MEANS AND STANDARD
DEVIATIONS (Volumetric Flow Rate,
ft*/hrX lO'4)
is measured by the coefficient of determination, r2. For the no-intercept
model, r2 is calculated as* ^
Run
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
Mean
Flow Rate
Standard
Deviation
Site 1
1232.8
1280.8
1341.5
1191.9
1146.3
1190.9
1312.2
1309.0
1270.6
1215.8
1214.2
1213.0
1260.4
1117.8
1149.1
49.2
68.9
134.3
83.7
72.3
59.8
70.3
44.1
47.5
40.5
29.8
48.8
144.4
33.3
38.2
Site 2
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
3
10
12
2
5
6
1
4
7
8
11
9
7087.8
6989.4
7114.1
6962.3
7109.6
6820.1
6833.5
6652.6
6687.0
6912.7
7025.7
7024.0
6703.5
6957.8
6501.3
6557.9
Site 3
338.2
348.6
351.0
336.6
301.8
332.7
344.9
316.3
324.8
320.1
323.5
287.9
122.4
328.8
238.3
71.6
389.1
187.8
97.2
171.2
279.9
419.9
143.1
134.5
122.1
310.5
644.9
393.8
24.8
22.0
25.0
18.9
38.4
12.9
31.3
11.5
12.7
29.2
19.8
17.3
For the run data, r2 = 0.73 based on 43 pairs of observations. This
indicates that 73% of the variation in the magnitude of the standard deviation
is attributed to variation in the magnitude of the mean. The correlation
coefficient, r = v^3", is 0.85 which is a significant value at the 5 percent
level.
Similarly, for the collaborator block data define
1 "9
x ff = — V Xijk as the collaborator-block mean for collaborator
"'/" * = i
i in block/, and n,y determinations in the collaborator-block
and
as the collaborator-block standard
deviation.
The paired values (*#., s/y) are shown in Table B-6. For the no-intercept
model,/-2 = 0.63 and the correlation coefficient, r, is 0.79 based on
37 pairs. As in the case of the transformations, the proportional relation-
ship does not appear as strong for the collaborator-block data. However,
a correlation coefficient of 0.79 is significant at the 5 percent level.
As a result, then, we have the model for the between-laboratory
and within-laboratory standard deviations of
and
= 05
where fa and |3 are the true between-laboratory and within-laboratory
coefficients of variation, and 6 is an unknown mean. The coefficients
of variation remain constant, and the standard deviation may be expressed
as a percentage of the mean value. Thus, the standard deviations are
estimated by obtaining estimates of the coefficients of variation, fo and 0,
and expressing the estimators as
»&=!
and
5 = 1
24
-------
TABLE B-6. COLLABORATOR-BLOCK MEANS AND
STANDARD DEVIATIONS
(Volumetric Flow Rate, ft'/hr X lO'4)
Block
Collaborator
Mean
Flow Rate
Standard
Deviation
Site 1
1
2
3
4
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
1250.10
1213.50
1440.35
1305.33
1166.42
1143.57
1285.22
1246.12
1223.72
1221.38
1291.62
1241.42
1107.50
1155.60
1209.17
1227.33
15.42
34.11
79.69
91.00
69.94
81.63
79.22
73.87
26.32
71.89
43.59
51.45
41.44
93.50
58.86
177.13
Site 2
1
2
3
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
7110.72
7125.94
6921.28
6962.89
6780.70
6768.34
6327.70
6656.65
6604.50
323.46
71.54
203.17
251.59
210.72
215.90
718.42
104.86
703.29
Site 3
1
2
3
4*
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
336.57
369.47
331.00
346.73
302.67
351.97
323.57
316.60
311.10
341.86
330.96
313.52
273.60
311.70
276.70
289.50
17.16
2.07
12.64
31.96
37.80
3.14
12.18
30.46
8.19
17.65
22.95
22.13
-
*No standard deviations since block contains
only one run.
B.3 Unbiased Estimation of Standard Deviation
Components
In Appendices B.I and B.2, the theoretical and empirical
arguments from the collaborator-block data indicate that a
suitable model for the within-lab standad deviations of both
variables is
o =
ship
To estimate this standard deviation, we use the relation-
where Cis a constant, representing the proportionality. As
previously discussed, s,y is a biased estimator for the true
standard deviation, a. The correction factor for removing
the bias is dependent on the sample size n, and is given by
Ziegler
(5)
as
where T represents the standard gamma function. Thus, we
can say that
or
a =
= anCE(xi/.)
= 06
so that in obtaining an unbiased estimate of 0, we can obtain
an unbiased estimate of a as well. Thus, we define an estimator
for a, a, where
From Appendices B.I and B.2, we determine that a suitable model for the run data from both variables is
ab =
25
-------
where oj, = VOL +o2 is the between-lab standard deviation. Empirically, we have
Sjk = CbX.jk
and Sjk is a biased estimator for o/,. Thus, for p collaborators,
E(apSjk) = o
and we have
a = E(apsjk)
= apE(Cbxjk)
= 0*6-
Obtaining an estimate of ft,, we have a new estimator, <% , of <;& given by
But a^ = Vffi + °2 implies
and substituting our estimates of
-------
for a sample of size n. This estimator is shown in B.5 to be unbiased for the true coefficient of variation. However,
since we are dealing with small samples to obtain our individual estimates, weighting is more desirable in that it pro-
vides for more contribution from those values derived from larger samples. There is more variability in the beta
values obtained from the smaller samples, as can be seen by inspecting the variance of the estimator. We have that
for normally distributed samples/ 1 1 * and true coefficient of variation, j3. Rewriting this expression, we have
and all terms are constant except for c?n and n. Thus, the magnitude of the variance changes with respect to the
factor <&ln. Now, since an decreases as n increases, the factor o£/n must decrease as n increases, and the variance
is reduced.
The weights, w,-, are determined according to the technique used in weighted least squares analysis(6), which
gives a minimum variance estimate of the parameter. The individual weight, w,-, is computed as the inverse of the
variance of the estimate, ft, and then standardized. Weights are said to be standardized when
1 *
To standardize, the weights are divided by the average of the inverse variances for all the estimates. Thus, we can
write
"i
r
u
where
Vaitf/)
and
1 * 1
k fa Vaitf/)
Now, from the above expressions we can determine «,-, u and w,- for the beta estimates. For any estimate, ft,
27
-------
for sample size «/, and
Thus, the ith weight, w,-, is
1 * 1
; =
* «/ r 2
I 2
The estimated coefficient of variation is
k ^
*/-1
* A f
/ -1
28
-------
B.5 Estimating Precision Components For Velocity Determination
In Appendix B.I the models are given for the between-laboratory and within-laboratory standard deviations,
a}, and a, respectively,
<>b =
and
where jjj and|3 are the true coefficients of variation for between-laboratory and within-laboratory, respectively.
The coefficients of variation remain constant for changing mean levels.
Estimates of o^ and a are obtained using the technique of Appendices B.3 and B.4. The coefficients of
variation are estimated as a linear combination of beta values obtained from each run or collaborator-block. The
estimator is of the form
where k is the number of individual estimates and w; is the weight applied to the Mi beta estimate.
From the run data, the estimated between-laboratory coefficient of variation is estimated as
-— V
fe~43 ^ Wjfc
The individual beta estimate and the weights applied are shown in Table B-7. Substituting these into the above
formula gives
ft, =(0.050).
The estimated between-laboratory standard deviation, then, is
= (0.050)6.
The degrees of freedom associated with this estimate are determined by taking the number of collaborators
at a site less one, summed for all three sites. This gives (4 — 1) + (3 - 1) + (4 - 1) = 8 degrees of freedom for this
estimate.
29
-------
TABLE B-7. RUN BETA ESTIMATES AND WEIGHTS
(Velocity)
Run
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
3
10
12
2
5
6
1
4
7
8
11
9
Beta Hat
Site 1
0.0444
0.0307
0.0315
0.0322
0.0375
0.0374
0.0342
0.0377
0.0105
0.0416
0.0477
0.0339
0.0457
0.0500
0.0220
Site 2
0.0551
0.0555
0.0408
0.0307
0.0694
0.0172
0.0201
0.0417
0.0520
0.0265
0.0506
0.0263
0.0223
0.0557
0.0569
0.0693
Sir e 3
0.1073
0.0244
0.0748
0.1047
0.0382
0.0800
0.0810
0.0938
0.0616
0.1092
0.0535
0.0789
Weight
0.723
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
0.723
1.043
1.030
1.030
1.030
1.030
1.030
1.030
.030
.030
.030
.030
.030
0.556
.030
.030
.030
.030
1.026
1.026
1.026
1.026
1.026
1.026
0.712
1.026
1.026
1.026
1.026
1.026
The within-laboratory coefficient of
variation, j3, is estimated from the collaborator-
block data as
1 37
37 ~j
The collaborator block beta estimates and their
corresponding weights are shown in Table B-8.
Substituting into the above equation gives
0 = 0.039
and the within-laboratory standard deviation estimate
is
= (0.039)6.
There are are (n,y - 1) degrees of freedom for
this estimate from each collaborator-block, where
n,y is the number of determinations in the col-
laborator block. Summing over the 37 blocks gives
113 degrees of freedom.
B.6 Estimating Precision Components For
Volumetric Flow Rate
In Appendix B.2, the models are developed
for the standard deviation components, a/, and a,
ob =
and
= 06,
where |3j and (3 are the between-laboratory and
within-laboratory coefficients of variation, and 8
is the mean method determination. The coefficients
of variation are shown to remain constant as the
mean changes. To estimate a^ and a, then, estimators
% and a are defined as
and
Estimated coefficients of variation are used to estimate the standard deviations as percentages of the mean value, 6.
The technique used in estimating fa and ft is discussed in Appendices B.3 and B.4. The estimator is a linear
combination of beta values,
30
-------
TABLE B-8. COLLABORATOR-BLOCK BETA
ESTIMATES AND WEIGHTS (Velocity)
Block
Collaborator
Beta Hat
Weight
Site 1
1
2
3
4
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
0.0119
0.0089
0.1006
0.0474
0.0486
0.0591
0-0435
0.0420
0.0241
0.0162
0.0139
0.0392
0.0914
0.0396
0.0153
0.0111
0.786
0.786
0.425
0.786
1.133
1.133
1.133
1.133
1.475
1.475
1.475
1.475
0.425
0.786
0.786
0.786
Site 2
1
2
3
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
0.0531
0.0095
0.0342
0.0360
0.0346
0.0391
0.0487
0.0291
0.1245
0.960
0.960
0.960
1.837
1.618
1.837
0.277
0.277
0.277
Site 5
1
2
3
4*
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
0.0503
0.0241
0.0447
0.1059
0.0446
0.0409
0.0282
0.0447
0.0352
0.0458
0.0254
0.0749
-
0.796
0.796
0.796
0.796
0.796
0.796
0.796
0.796
1.494
1.494
1.494
1.148
-
*No estimates possible since this block
contains only one run.
/-!
where $,- is the ith beta estimate from a run or collaborator-block, w/
is the weight assigned, and k is the number of estimates.
From the run data, there are 43 separate estimates of 0j, which gives
43
The values and their weights are shown in Table B-9. Substituting
these into the above formula gives
& = 0.056.
The between-laboratory standard deviation is estimated as
= (0.056)5
or 5.6% of its mean value.
The degrees of freedom for this estimate are determined by
taking the number of collaborators less one at each site, and summing
over the three sites. This gives (4 - 1) + (3 - 1) + (4 - 1) = 8 degrees
of freedom for this estimate.
The collaborator-block data gives an estimate of the within-lab-
oratory precision components. The within-laboratory coefficient of
variation is estimated as
1 37
*B Z "A
The individual beta estimates and weights are shown in Table B-10.
Substituting into this equation gives
$ = 0.055
and a standard deviation estimate of
&=|35
= (0.055)5.
Thus, the within-laboratory standard deviation is estimated as 5.5% of the mean level. Letting n/y be the number
of determinations in a collaborator-block, there are («/y - 1) degrees of freedom for this estimate from each. Sum-
ming, there are a total of 113 degrees of freedom from the 37 collaborator-blocks for this estimate.
31
-------
TABLE B-9. RUN BETA ESTIMATES AND
WEIGHTS (Volumetric Flow Rate)
Run
Beta Hat
Weight
Site 1
1
2
3
4
5
6
9
8
10
11
12
13
7
14
15
0.0450
0.0584
0.1087
0.0762
0.0685
0.0545
0.0581
0.0365
0.0406
0.0362
0.0266
0.0436
0.1244
0.0336
0.0361
0.723
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
1.043
0.723
1.043
Site 2
1
3
4
6
8
2
5
7
10
11
13
14
15
16
9
12
0.0195
0.0531
0.0378
0.0116
0.0618
0.0311
0.0160
0.0290
0.0472
0.0685
0.0230
0.0240
0.0205
0.0504
0.1119
0.0678
1.030
1.030
1.030
1.030
1.030
1.030
1.030
1.030
1.030
1.030
1.030
0.556
1.030
1.030
1.030
1.030
Site 3
3
10
12
2
5
6
1
4
7
8
11
9
0.0796
0.0686
0.0774
0.0610
0.1381
0.0420
0.1025
0.0396
0.0425
0.0991
0.0664
0.0653
1.026
1.026
1.026
1.026
1.026
1.026
0.712
1.026
1.026
1.026
1.026
1.026
TABLE B-10. COLLABORATOR-BLOCK BETA
ESTIMATES AND WEIGHTS
(Volumetric Flow Rate)
Block
Collaborator
Beta Hat
Weight
Site 1
1
2
3
4
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
0.0139
0.0317
0.0693
0.0787
0.0651
0.0775
0.0669
0.0643
0.0229
0.0626
0.0359
0.0441
0.0469
0.0913
0.0549
0.1628
0.786
0.786
0.425
0.786
1.133
1.133
1.133
1.133
1.475
1.475
1.475
1.475
0.425
0.786
0.786
0.786
Site 2
1
2
3
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
Lab 202
Lab 203
Lab 204
0.0484
0.0107
0.0312
0.0373
0.0322
0.0329
0.1423
0.0197
0.1335
0.960
0.960
0.960
1.837
1.618
1.837
0.277
0.277
0.277
Site 3
1
2
3
4*
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
Lab 301
Lab 302
Lab 303
Lab 304
0.0575
0.0063
0.0431
0.1040
0.1409
0.0101
0.0425
0.1086
0.0280
0.0549
0.0738
0.0766
-
—
-
0.796
0.796
0.796
0.796
0.796
0.796
0.796
0.796
1.494
1.494
1.494
1.148
_
_
-
-
*No estimates possible since block contains
only one run.
32
-------
B.7 Emission Rate Variability
The emission rate, r, is given by
where
and
Qs is the volumetric flow rate of the stack, ft3/hr
c is the concentration of pollutant, determined by the applicable test method, appropriate weight units/scf.
The flow rate calculation does not involve the mass or volume of pollutant obtained, and the concentration
of pollutant is determined separately from the velocity calculation. Thus, it is reasonable to say that Qs and c are
independent variables. Under this assumption, we can estimate a variance component for r, V(r), from the estimated
terms for Qs and c.
In this section, these relationships will be used.
[1 ] The variance of any random variable, x , is defined as
[2] For independent variables jc and y
and
for any two functions /i and/2 .
[3] For any variables x and y
(jc)] -E\f2(y)}
~ E(y2)[E(x)]2 + [E(x)]2 [E(y)]
This can be derived easily from [1] .
For the variable r, from [1]
•
-------
Rearranging terms gives
and from 1
where
6| — the mean pollutant concentration, and
6gs — the mean flow rate
When both the flow rate and pollutant concentration have constant coefficients of variation, &(QS) and 0(c
respectively, the variances are written as
and substituting these into the equation for K(r) gives
where
Br is the mean emission rate, 5r =
34
-------
LIST OF REFERENCES
1. Environmental Protection Agency "Standards of Performance for New Stationary Sources," Federal Register,
Vol. 36, No. 247, December 23, 1971, pp 24876-24893.
2. Hamil, H.F. and Camann, D.E., "Collaborative Study of Method for the Determination of Paniculate Matter
Emissions from Stationary Sources (Portland Cement Plants)," Southwest Research Institute report for
Environmental Protection Agency, May, 1974.
3. Hamil, H.F. and Thomas, R.E., "Collaborative Study of Method for the Determination of Particulate Matter
Emissions from Stationary Sources (Fossil Fuel-Fired Steam Generators)," Southwest Research Institute
report for Environmental Protection Agency, June 30,1974.
4. Hamil, H.F. and Thomas, R.E., "Collaborative Study of Method for the Determination of Particulate Matter
Emissions from Stationary Sources," (Municipal Incinerators). Southwest Research Institute report for
Environmental Protection Agency, July 1, 1974.
5. Ziegler, R.K., "Estimators of Coefficients of Variation Using k Samples," Technometrics, Vol. 15, No. 2,
May 1973, pp 409414.
6. Hamil, H.F., "Laboratory and Field Evaluations of EPA Methods 2, 6 and 7." Southwest Research Institute
report for Environmental Protection Agency, October, 1973.
7. Hamil, H.F. and Camann, D.E., "Collaborative Study of Method for the Determination of Nitrogen Oxide
Emissions from Stationary Sources," (Fossil Fuel-Fired Steam Generators) Southwest Research Institute
report for Environmental Protection Agency, October 5, 1973.
8. Hamil, H.F. and Camann, D.E., "Collaborative Study of Method for the Determination of Sulfur Dioxide
Emissions From Stationary Sources (Fossil Fuel-Fired Steam Generators)." Southwest Research Institute
report for Environmental Protection Agency, December 10, 1973.
9. Dixon, W.J. and Massey, F.J., Jr., Introduction to Statistical Analysis, 3rd Edition. McGraw-Hill, New York,
1969.
10. Searle, S.R., Linear Models. Wiley, New York, 1971.
11. Cramer, H., Mathematical Methods of Statistics. Princeton University Press, New Jersey, 1946.
12. Hogg, R.V. and Craig, A.T., Introduction to Mathematical Statistics, 3rd Edition. The Macmillan Company,
New York, 1970.
35
-------
TECHNICAL REPORT DATA
(Please read Ituouctions on the reverse before completing)
1. REPORT NO. 2.
EPA-650/4-74-033
4. TITLE AND SUBTITLE
Collaborative Study of Method for Determination of Stack Gas Velocity and
Volumetric Flow Rate in Conjunction with EPA Method 5
7. AUTHOR(S)
H.F. Hamil, R.E. Thomas
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Southwest Research Institute
8500 Culebra Road
San Antonio, Texas 78284
12. SPONSORING AGENCY NAME AND ADDRESS
Quality Assurance and Environmental Monitoring Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 277 1 1
3. RECIPIENT'S ACCESSION-NO.
5. REPORT DATE
September, 1974 (date of issue)
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
1HA327
11. CONTRACT/GRANT NO.
68-02-0626
13. TYPE OF REPORT AND PERIOD COVERED
Task Order
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
!
I
16. ABSTRACT
This study concerns itself with the determination of stack gas velocity and volumetric flow rate (EPA Method 2) as
used with EPA Method 5 (Particulates). The determinations were calculated from data obtained in collaborative testing of
Method 5 at three sites: a Portland cement plant, a coal-fired power plarit and a municipal incinerator. These data were
submitted to statistical analysis to obtain estimates of the precision that can be expected with the use of Method 2. The
standard deviations for both velocity and flow rate are shown to be proportional to the mean value.
The between-laboratory standard deviations are estimated as 5.0% of the mean and 5.6% of the mean for velocity and
volumetric flow rate, respectively. The volumetric flow rate is used to calculate the emission rate for compliance testing.
The precision of the emission rate is shown to be primarily a function of the precision of the pollutant test method used. The
conclusion, then, is that the volumetric flow rate determination is sufficiently precise as it appears in Method 2.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS c. COSATI Field/Group
Air Pollution 1302
Flue Gases
Collaborative Testing
Method Standardization
13-B
07-B
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40
2O. SECURITY CLASS (This page)
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EPA Form 2220-1 (9-73)
-------
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the primary posting(s).
18. DISTRIBUTION STATEMENT
Denote releasability to the public or limitation for reasons other than security for example "Release Unlimited." Cite any availability to
the public, with address and price.
19. & 20. SECURITY CLASSIFICATION
DO NOT submit classified reports to the National Technical Information service,
21. NUMBER OF PAGES
Insert the total number of pages, including this one and unnumbered pages, but exclude distribution list, if any.
22. PRICE
Insert the price set by the National Technical Information Service or the Government Printing Office, if known.
EPA Form 2220-1 (9-73) (Revert*)
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