EPA-650/4-74-033 SEPTEMBER 1974 Environmental Monitoring Series ilUDY \ UJ O ------- 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 ------- 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 ------- 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 ------- 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 ------- 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 ------- 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. ------- 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 ------- 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 ------- 6bi fin 8 4- £r 55 g j._ t>u ,f. G Ea ;t : B _— •^- 5 2>- 1 4 . i - -i - ^ - £ •*-< i. g -» •\ 3 ,- '** - 5 - . 1 iA 2 est Duct Width, Feet Profile Across Upper Ports 0) DO 50 45 40 35 1 Ea g '^ •~> ^», a -, -, - » ,r ** ? / t 4 < P / i | J fee 2H 2 4 st ^ _^- gw*"t; ! ' • i t V s - 1 \ ^ 1 11 V, •^s — ^-' 6 8 10 12 West Profile Across Lower Ports FIGURE 2. TYPICAL VELOCITY PROFILES, ALLEN KING POWER PLANT ------- 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 ------- 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.^>^ ------- 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. ------- 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. ------- 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, ------- 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 ------- 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 *> ------- 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 |