SERA
           United States
           Environmental Protection
           Agency
            Industrial Environmental Research  EPA-600/2-78-195
            Laboratory           September 19,78
            Research Triangle Park NC 27711
           Research and Development
Statistical Analysis
of Fugitive Emission
Change Due to
Refinery Expansion

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                                   EPA-600/2-78-195
                                      September 1978
Statistical Analysis  of Fugitive
     Emission  Change Due  to
         Refinery  Expansion
                        by

              Hugh J. Williamson and Lloyd P. Provost
                   Radian Corporation
                     P. O. Box 9948
                   Austin, Texas 78766
                  Contract No. 68-02-2608
                     Task No. 27
                Program Element No. IABG04C
               EPA Project Officer: Irvin A. Jefcoat

             Industrial Environmental Research Laboratory
              Office of Energy, Minerals, and Industry
               Research Triangle Park, NC 27711
                     Prepared for

            U.S. ENVIRONMENTAL PROTECTION AGENCY
               Office of Research and Development
                  Washington, DC 20460

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                          DISCLAIMER
     This report has been reviewed by the Industrial Environmental
Research Laboratory - Research Triangle Park, U.S. Environmental
Protection Agency, and approved for publication.  Approval does
not signify that the contents necessarily reflect the views
and policies of the U.S. Environmental Protection Agency, nor
does mention of trade names or commercial products constitute
endorsement or recommendation for use.
                        ACKNOWLEDGMENT
     Consultations with Dr. Robert G. Wetherold of Radian
Corporation have been very helpful in this research.
                       DCN 78-200-187-27-09
                               ii

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                            CONTENTS
List of Tables	    iv
Executive Summary	     v

     1.   Introduction	     1
     2.   Average Emission Increment and Its
          Uncertainty (Given Type of Device,
          e.g. , Valves)	     3
     3.   Mean Emission Increment and Its
          Uncertainty (Entire Refinery)	    21
     4.   Combining Emission Rates	    30
     5 .   Summary	 .	    35

References	    37
Appendices
     A.   Properties of the Mean and Variance	    38
     B.   A Paired Measurement Scheme for Reducing
          Random E rrors	    44
                                iii

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                                  TABLES
Number                                                                  Page

3-1    Data for Example 1:  Hypothetical Test Care in Which
         A Catalytic Reformer is Added to An Existing
         Ref inery	 24

3-2    Data Required to Calculate Emission-Increment Statistics
         for New Facility for Flanges, Process Drains,
         and Relief Valves	 25
                                      IV

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                             EXECUTIVE SUMMARY
     The purpose of this report is to discuss a statistical approach for
testing whether a planned petroleum refinery expansion can be made without
increasing the fugitive emissions, when this is required by regulations.
Emission factors can be used to estimate the emission increase or decrease
due to the expansion.  Since emission rates are empirically determined,
however, they are subject to random sampling errors.  Thus, the effect of
the expansion on emissions cannot be computed exactly.

     For this reason, the problem has been treated statistically.  Analytical
methods are presented which can be used to compute the mean and standard
deviation of the emission change, whether positive or negative, due to the
expansion.  A method for computing the probability that the emissions will,
in fact, not be increased by the expansion is also discussed, along with
other related probability calculations.

     The methods presented here can be used as an aid in comparing several
options for reducing emissions to acceptable levels.  This involves simply
performing the statistical calculations in parallel for each option to be
considered.  In a given case, the options might include, for example:

     •    instigating an improved maintenance program
          for certain devices (e.g., valves),

     •    venting certain emissions to a flare, and

     •    shutting down a particular processing unit.

     The methods are designed to handle different control strategies in the
existing and in the planned facilities and different strategies for different
types of devices, if this is necessary.  The following situation, for example,
could be handled.

     •    A new unit, such as a catalytic reformer, is to
          be added to an existing refinery.

     •    To reduce emissions in the existing facilities,
          a maintenance program is instituted for valves,  .
          and the API separator is to be covered.

     •    To reduce emissions in the new facility, the same
          type of maintenance program used in the old facility
          will be instituted for valves, and double seals
          rather than single seals will be used on all pumps.
          Single seals remain on pumps in the old facility,
          however.

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     Additionally an approach is presented for obtaining an estimate of the
emission decrease for a particular type of device, such as valves, due to a
particular emissions control program.   The approach involves a paired-
measurement experimental design which eliminates certain sources of extra-
neous influence and, therefore, increases the precision of the estimates
obtained.

     Although the discussion and examples in this report are specific to
petroleum refineries, the methods developed and outlined here can be
generalized to similar situations in other plant expansion problems.

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                                 SECTION 1

                               INTRODUCTION
     The purpose of this report is to discuss a statistical approach for
testing whether a planned refinery expansion can be carried out without
increasing emissions, when this is required by regulations.  The problem has
been addressed statistically, since emission rates, being determined empiri-
cally, are subject to random errors.   Thus, the decision as to whether a
planned expansion is in compliance with regulations cannot be made with
absolute certainty but can be made with a certain level of confidence.
Probability calculations can be used to establish the level of confidence
with which the judgement can be made in a given case.

     The basic approach is first to determine, for each type of device in
the refinery which independently contributes to the emissions, the emission
rate from the new facility minus the emission reduction (if any) from the old
facility due to improved maintenance practices or equipment modifications.
This difference is the emissions increment for the device type being considered.
The device types would probably include pipeline valves and flanges, pressure
relief valves, pumps, compressors, cooling towers, etc.  Any system of cate-
gorization which is convenient and physically sensible can be used.

     Secondly, the emissions increments for all important device types are
summed to get the overall increment for the refinery.  If this increment is
less than or equal to zero, this means that the expansion will "probably" not
result in an emissions increase.  If the increment is positive, the expansion
probably will increase emissions.

     These qualitative statements, however, are unsatisfactory due to their
vagueness.  Thus, an approach for calculating the numerical probability that
emissions will not be increased is also presented in this report.  The confi-
dence with which a decision can be made regarding compliance with regulations,
then, can be assessed.

     An effort has been made to present the analysis in a self-contained form,
so that an extensive statistical background is not required to follow the
development.  Thoroughness and generality, however, have not been sacrificed
for the sake of simplicity.  Some of the situations which could arise in a
refinery expansion are complex.  The equations necessary to evaluate the
emission change due to the expansion in these cases, therefore, are also
complex.

     The statistical method presented here is illustrated with a number of
numerical examples.  The emission rates used in the examples are believed to

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be reasonable but are employed here for illustrative purposes only; they are
in no way being presented as emission factors for use other than in this
report.

     The analysis presented herein also provides certain guidelines for
experimental design for determination of emission factors.  This is because
the information needed to perform a refinery tradeoff study is listed, and
the manner in which the information would be used to do the study is pre-
sented.  These guidelines would be beneficial if existing estimates of
emission rates were not appropriate for a particular tradeoff study, and,
therefore, new estimates had to be developed.

     Radian is currently performing an extensive research study for the
Environmental Protection Agency, however, in which statistically valid
estimates of the emission rates and their uncertainties will be obtained.

     References 3 and 4 were very helpful in choosing a set of numerical test
cases.

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                                 SECTION 2

              AVERAGE EMISSION INCREMENT AND ITS UNCERTAINTY
                   (Given Type of Device, e.g., Valves)


     In this section, the calculation of the estimated emission increment for
a given device type is discussed.  The standard deviation of this increment,
which reflects random errors in the emission factors is also discussed.  Four
cases are discussed which include most trade-off situations anticipated.

     The purpose of this section is not to present a rigorous mathematical
treatment but rather to give* in.  an easily understandable way, the basic
equations and the circumstances under which they should be used.   The statis-
tical background for this section involves only the basic properties of mean
values, variances,and covariances.  These properties are discussed briefly in
Appendix A.  Also see Reference 5.

Case 1- Same Provisions  for Reducing Emissions  in New  and Old Facilities

     In this case,  it is assumed that control measures such as covering an
API separator or improving maintenance programs are applied to all sources of
a certain type.  Another example is venting emission gases from all sources
to a flare.

     Define:

     E1 = average emission rate per source without  improved
          maintenance,
    O
     E1 = standard deviation of E1,

    E11 = average emission rate per source with improved
          maintenance,
   c
    E11 = standard deviation of E11,

     N  = number of sources of type being considered in
          the old facility, and

     N  = number of sources in the new facility.
      n
     Then the emission reduction in the old facility corresponding to  this
particular type of source is:

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                            N (E1 - E11)
                             o
and the emission increment in the new facility is

                            N E11
                             n

Thus, the total emission increment I is

                            I = N E11 - N (E1 - E11)
                                 n       o
                              = (N  + N )E1: - N E1
                                  n
The variance, S*, of I is
     It is important to remember that the emission factors E1 and E   corres-
pond to a single device (e.g., a single valve), while I is the emission rate
from all devices of a given type.  A similar convention is used in the other
cases discussed, although additional variables are introduced.

Example 1 for Case 1

     The calculations for Case 1 will now be demonstrated with a numerical
example.  As is mentioned in the Introduction, the emission factors used in
this and other examples are intended for illustrative purposes only.  Statis-
tically valid estimates of emission rates and their uncertainties will be
calculated through another project currently being performed by Radian.

     In this example, the emission increment for valves will be calculated in
a hypothetical case in which a catalytic reformer is being added to an existing
refinery.  It is assumed that the reformer has 850 valves and that an improved
maintenance program is introduced which reduces the average emission rate from
each valve from 0.040 to 0.008 Ib/hr.  These and other necessary statistics
are presented below.

     E1 = 0.040 Ib/hr,

    SFI = 0.006 Ib/hr (estimated standard deviation of E *) ,

    E11 - 0.008 Ib/hr,

   S_n = 0.003 Ib/hr  (estimated standard deviation of E11),
    lit

     N  = 14,000 (number of valves in the refinery before the
          catalytic reformer was added), and

     Nn = 850.

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     The estimated emission increment can be obtained by direct substitution
of these values in the expression for I.

                            I = (N  + N )E1J - N E1
                                v n    o'       o

                              = (850 + 14000) 0.008 - 14000(0.040)

                              = -441

     Thus, the addition of the catalytic reformer and the instigation of
improved maintenance are estimated to decrease the total emissions from all
valves by 441 Ib/hr.

     If the values of E1 and E11 were exact, then the value of I given above
would also be exact.  Due to the uncertainty in the empirically determined
emission rates E1 and E11, however, the value of I is also uncertain; its
estimated standard deviation, ST , is calculated as follows.


                            SI2 = ^o + V'V^+VV2

                                = (850 + 14000) 2(0. 003) 2 + (14000) 2 (0. 006) 2

                                = 9041

                                       = 95 Ib/hr


     This concludes the calculations which would be necessary for valves
alone.  Similar calculations would also have to be performed for other device
types, such as flanges, pumps and compressors.  The values of I and Sj would
then be combined to estimate the emission increment for the entire refinery
and its uncertainty.  This set of calculations is presented in the examples
of this section and the next.  The final calculations are presented in Example
1 in Section 3.

Example 2 for Case 1

     In this example, we assume that the refinery's API separator is covered
and that this reduces the estimated emission rate from the separator from 6.2
to 0.31 lb/1000 gallons of wastewater.  This example is different from the
preceding one in that no units of the type being considered are added; thus,
N  is zero.  Covering the separator could be one of the steps to reduce
emissions from existing facilities, so that additional processing equipment
could be added without increasing emissions.

     The required inputs are as follows:

     E1 =6.2,
     11 = 0.31,

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   SEn = 0.15,

     N  = 1, and
      o

     N  = 0.
      n
     The quantities E1, Sgi, E11 and SgH  are in pounds per 1000 gallons of
wastewater.  Then:
                              = (0 + 1)  0.31 - (1)(6.2)

                              = -5.9

     Thus, covering the separator reduces the estimated  hourly emissions by
5.9 lb/1000 gallons of wastewater.  The standard deviation of this estimate,
which is due almost completely to the uncertainty in E1, is calculated as
follows.
                                                   (1)2(2.9)
Further Discussion of Case 1

     The case in which sources of a certain type are added in a new facility
but reduced in number in the old facility is discussed in this subsection.
This case would apply, for example, if a light ends/gas processing unit were
to be added to an existing refinery and a reformer feed pretreating unit were
to be shut down.  This would result in the addition of a number of valves,
for example, in the new facility and in the elimination of all valves which
were in the reformer feed pretreating unit.

     The formulation presented above must be modified slightly to handle this
type of situation.  Define:

     N  = number of sources of the type being discussed which are
          eliminated from the old facility.

     All other notation is exactly as defined above.  The emission reduction
in the old facility due to the improvements in hardware or maintenance pro-
grams is :

                            (NQ - N^CE1 - E11)

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and the reduction in the old facility due to the elimination of N  units is



                                     N E1
                                      c


     The added emissions in the new facility, as before, is:



                                     N E11
                                      n


     Thus, the total emission increment I is:



                            I = N E11 - (N  - N HE1 - E11) - N E1
                                 n        o    c               c


                              = (N  + N  - N )EJ1 - N E1
                                  n    o    c        o



and the variance Sy2 of I is:
Example 3 for Case 1



     The calculations for valves for the example mentioned above will now be

performed.  It will be assumed that putting in a light ends/gas processing

unit adds 1300 valves and that eliminating the reformer feed pretreating unit

removes 800 valves.  These and other needed inputs are listed below.



     E1 = 0.040 Ib/hr,



    S.,i = 0.006 Ib/hr,
     EJ


    E11 = 0.008 Ib/hr,



   SEH = 0.0012 Ib/hr,



     N0 = 14,000 valves,



     Nn = 1300 valves, and



     Nc = 800 valves.



     As in Example 1 for Case 1, it has been assumed that an improved mainte-

nance program is to be instituted which will reduce the average emissions from

valves from 0.040 to 0.008 Ib/hr.



                            I =  (N  + N  - N )EU - N E1
                                  n    o    c        o


                              =  (1300 + 14000 - 800) 0.008 - (14000)(0.040)



                              = -444

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     Thus, the maintenance program together with the addition and removal of
equipment results in a net decrease of 444 Ib/hr in the estimated fugitive
emissions from all valves.  The standard deviation is computed as follows:
                                = (1300 + 14000 - 800)2(0.0012)2 + (14000)2(0.006)2

                                = 7359

                                - 86
Example 4 for Case 1
     This example is exactly like the preceding one,  except that it is assumed
that a light gas oil hydrotreating unit is also to be shut down, resulting in
the elimination of an additional 800 valves.   The only change to the required
input quantities, then, is that N  is increased from 800 to 1600.

     Since the value of I obtained when only the reformer feed hydrotreating
unit was eliminated is negative, it is not necessary to eliminate another unit
to balance emissions from valves.  It is possible, however, that a large
decrease in emissions in valves would be necessary to balance an increase in
emissions from another type of device, such as flanges.

     The value of I, then, assuming elimination of the pretreatment and hydro-
treating unit is:

                            I = (N  + N  - N )En - N E1
                                  n    o    c        o

                              = (1300 + 14000 - 1600) (0.008) - (14000) (0.040)

                              = -450 Ib/hr

     Eliminating the additional hardware, then, changes the emission increment
by only 6 Ib/hr.

     The standard deviation S_ is computed as follows.
                                = (1300 + 14000 - 1600) (0.0012) 2 + (14000)2(0.006)2

                                = 7326

                             S  = 86

     This and the preceding example illustrate the use of the analysis methods
presented herein as an aid in evaluating the various options for reducing
fugitive emissions.  The emission reductions achieved by equipment shutdowns

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and different maintenance programs can be estimated.  The final decision, of
course, should also include economic and other considerations, as well as
reductions in emissions.

     The analysis below for Cases 2 and 3 is somewhat more complicated than
that for Case 1, and the number of required inputs is greater.  Cases 2 and
3 cover situations in which emissions monitoring is performed, and only the
high-leaking devices are repaired.  In these cases, if the mean and standard
deviation of the emissions from a collection of sources which are selectively
maintained can be estimated, then the analysis under Case 1 should be used.
The mean value would then play the role of E11, and the standard deviation
would play the role of S_n in the notation used in Case 1.
                        £•

     The analysis below, however, can be used when the information in this
exact form is not known.  Although more numerous, the required inputs for
Cases 2 and 3 may be more easily obtained than those for Case 1 in some
instances.

Case 2 - Emissions Monitoring and Subsequent Corrective Action Performed the
         Same Way in the Old and New Facilities

     An example of this case is the periodic checking of emissions from valves
and performing maintenance if required.  The possibility exists that sources
with different ranges of emission rates would be maintained in a different
way; for example, high leakers might be maintained immediately, but medium
leakers might be tagged, rechecked periodically, and ultimately maintained
only if the emissions exceeded a certain rate.  This situation, however,
reduces to the case in which units are maintained if they leak more than a
certain amount.

     Define:

     FI = average emission rate of maintainable leakers if
          maintenance is not performed,

    • Fz = average emission rate of maintainable leakers if
          maintenance is performed,

     F3 = average emission rate of other units,

    S-,  = standard deviation of F. , i = 1, 2, 3,
     £ .                          •^-
      i
      f = proportion of the units of the type being considered
          which would be high leakers at any given time on the
          average, if detailed maintenance were not performed,

     Sf = standard deviation of f,

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     N  = total number of units of the type being considered

          in the old facility, and



     N  = total number in the new facility.
      n


     The emission rate F2 is intended to apply to the more recently maintained

units, while F3 is intended to apply to the other units.  More  specifically,

the emission rate F2 applies to the fN  most recently maintained  units  in the

old facility and to the fN  most recently maintained units in the new facility.



     The emission reduction in the old facility then is:



                            fNQ(F! - F2)



and the emission increment in the new facility is:



                            fNQF2 + (l-f)NnF3




The total emission increment is:



                            I = fN F2 + (l-f)N F3 - fN  (Fi - F2)
                                  n           no


                              = [N F2 - N F3 - N Fi + N F2]f +  N  F3
                                  n      n      o      o        n



                              = [(N  + N )F2 - N F3 - N Fi]f +  N  F3
                                   no       no        n


                              = Af + N F3
                                      n


where A denotes the coefficient of f, which is enclosed in brackets in  the

expression  above.



               ST2 = A2S/ + f2S.2 + S/S,2 + (N )2S17 2 + 2C
                j-       £       A     Z  A      n   r 3


where S^2 is the variance of A, and C is the covariance between  Af and U F.

These two expressions are given below.                                  n
     If it were assumed that the more recently maintained sources had  the same

mean and variance as the other sources, then the following equations would hold:
     F2 = F3



      2 = q  2
     2    SF3
                                     10

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     The correlation between the estimated emission rates F2 and F3 would then
be one, and this fact should be taken into account in the analysis.  The
expressions which should be used for I and S,2 are as follows:

                            I=[(N  + N  - N )F2 - N Fi]f + N F2
                                   n    o    n       o        n

                              = N  (F2 - FOf + N F2
                                 o              n

                              = Bf + N F2
                                      n

where B denotes  the  coefficient of f.

               Sj2 = B2Sf2 + f2 SB2 + Sf2 SB2 + (Nn)2 Sp22 + 2C


where
 and
Example  for  Case  2
                             C = N N  fS
                                 on F2
      To  illustrate  Case 2, we  again assume that a catalytic reformer is
being added  and that  an improved maintenance program is being instigated for
valves,  for  which the values of I  and Si will be calculated.  In this example,
however,  the available information is assumed to conform to Case 2 rather
than  Case 1.

      Again,  it  is assumed that 850 valves are added.  Further we assume that
12% of the valves would leak above the  allowed rate if maintenance were not
performed.   These and other required inputs are as follows:

      F!  - 0.267,

    S_  - 0.035,
      FI
      F2  = 0.000,

    S,,  = 0.000,
      F2
      F3  = 0.0091,

    S-  = 0.0045,
      Fa

       f  = 0.120,
                                      11

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     Sf = 0.010,



     NQ = 14000, and



     N  = 850.
      n
Then,
where
                            I - Af + N  F3
                                      n
                          A =  [(NQ + NQ)F2  -  NnF3  - NQFi]
                            =  [(14850)(0)  -  (850)(0.0091)  - (14000)(0.267)]



                            =  -3745.74



Thus,

                          I =  (-3745.74)(0.12)  + (850)(0.0091)



                          I =  -441
The standard deviation of I can then be calculated  as  follows.



                              ' 2 4. f2^ 2 + c  2g  2 + jr  2

                              *f       A     f A    n
                           O  2 =  A2<5  2 4- f2Q  2  + C  2C 2 J. \f 2q  2 4.

                           S      A S   + t  b    + S   b   +    S
where
                           S.2  =  (N  4- N  )2S_ 2  + N 2S1, 2 + N 2ST7 2
                           A     n    o   Fa     n  F3     o  FI



                               -  (14850) 2(0)  + (850) 2(0. 0045) 2 + (14000) 2 (0. 035) :



                               =  240,115
                           C =  -\2fSF 2  =  -(850) 2(0. 120) (0. 0045)
                                     '3


                            =  -1.76



                          Sj2  =  (-3745.74)2(0.010)2 + (0.120)2(240,115)


                                 +  (0.010)2(240,115) + (850)2(0.0045)2 + 2(-1.76)



                               =  4896



                          c  — /G  ^ —  "7 r\





     This example was patterned  after  Example 1 for Case 1.  The emission  rates

used in the two cases are consistent,  and the resulting values of I are the
                                     12

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same, -441.  The values of S  are different, however, due to the very different
formulations and types of information used in calculating Si in the two cases.
The covariance structures are different in Cases 1 and 2, for example, and this
affects Sj.

Case 3 - Same Hardware Provisions for Reducing Emissions in New and Old
         Facilities, Along With Emissions Monitoring and Subsequent Corrective
         Action Performed in Both Facilities

     This  case is, in a sense, a composite of Cases 1 and 2.  The analysis
presented  here would apply in the case of pumps, for example, if the following
conditions held:

     •     in the old facilities, single mechanical pump seals
           were replaced by double seals, resulting in a
           reduction in the average emissions,

     •     double mechanical pump seals were  also used in the
           new facility, and

     •     in both  facilities, emissions monitoring and subse-
           quent corrective action were performed the same way.

The  following terms will be needed:

     E = average emission rate for hardware  modifications,
         without detailed maintenance, and

     Sg = standard  deviation of E.

     The Fj_ and f, defined below, apply to sources with hardware modifications:

     FJ =  average  emission rate of maintainable leakers if
           maintenance is not performed,

     Fa =  average  emission rate of maintainable leakers if
           maintenance is performed,

     F3 =  average  emission rate of other units,

       f =  fraction of sources of the  type being considered
           which would be high-leakers at any given time on
           the average if detailed maintenance were not
           performed,

     Sf =  standard deviation of f,

     N  =  total number of sources of  the type being
           considered in the old facility, and
                                     13

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     N  = total number in the new facility.

The emission reduction in the old facility is:

                          N0E - [fNQF2 + (1 - f)N0F3]

The emission increment in the new facility is:

                          fNnF2 + (1 - f)NnF3


     The emission increment for the entire refinery for the type of device
being considered is:

                          I = fNnF2 + (1 - f)NnF3 -  [N0E - fN0F2 - (1 - f)NQF3]

                            = f [(Nn + N0)F2 - (N0 + Nn)F3] + (Nn + NO)F3 - NOE

                            = Af -fc BF3 - N0E

where

                          A - (Nn + NQ)F2 - (N0 + Nn)F3 = (Nn + NQ) (F2 - F3)

and

                          B = (NQ + NQ)

The  variance of I,  then, is given by the following:

                          Sz2 = A2Sf2 + f2SA2 + Sf2SA2 + B2SF32 + 2C + NQ2SE2

where SA2  is the variance of A, and C is the  covariance of Af and BF3.  These
two  expressions are given below.
                          SA2 -  (Nn + N0)2SF22 +  (N0 + Nn)2SF32
                                            2

                                          22      2
                              -  (Nn + N0)(SF2  + SF32)

                            C = -B2fSF32
 Example  for Case  3
      In  this  example,  the  emission  increment  for pump  seals will  be  examined
 assuming that a  catalytic  reformer  is being added,  that single  seals are  being
 replaced by double  seals on existing pumps, and that all pumps  on the new
 equipment will have double seals.   In addition to  the  hardware  modifications,
 it  is assumed that  pump seals which are  found to leak  excessively (say, over
 one Ib/hr) are replaced.   The required inputs are  listed below:
                                     14

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     E - 0.308 Ib/hr,

    SE - 0.080 Ib/hr,

    Fi = 1.500 Ib/hr,

    F2 = 0.010 Ib/hr,

    F3 = 0.015 Ib/hr,

   SF  = 0.400 Ib/hr,

   SF  = 0.005 Ib/hr,

   SF  = 0.006 Ib/hr,

     f = 0.130,

    Sf = 0.020,

    NO = 264, and

    Nn = 17.

                          I = Af + BF3 - NQE

where

                          A - (Nn + N0)(F2 - F3)

                            = 281 (0.010 - 0.015) = -1.41

                          B = Nn + NQ - 281

Thus,

                          I = (-1.41)(0.130) + 281(0.015) - 264(0.308)

                            = -77

     Thus,  adding the catalytic reformer and replacing single seals  with
double seals results in a net decrease of 77 Ib/hr in the estimated  fugitive
emissions from all pumps.  The standard deviation of I can be calculated as
follows:

                          C 2 - A20 2,.p20 2 i  c 2n 2j_B2C  2 J. 9r 4.  M 2C  2
                          SI  * A Sf  + f SA  + Sf SA  + B SF3  + 2C +  NQ SE
                                     15

-------
whe re,
                          SA2
                              = (281)2[(0.005)2 + (0.006)2]

                              = 4.82

                          C = -BfSF 2 = 281(0.130)(0.006)2 = -0.001

Then,

                          Sz2 = (-1.41)2(0.020)2 + (0.130)2(4.82)2
                                +(0.020)2(4.82)2 + (281)2(0.006)2 + 2(-0.001)
                                + (264)2 (0.080)2

                              = 449

                                    "=21
Case 4 - Entirely Different Emission Factors Apply for the New and Old
         Facilities

     This case would apply if, for example, emissions from a particular type
of device were vented to a flare in the new facility, but it was infeasible
to introduce such a system in the old facility.  The results for this case
follow from the analyses for Cases 1, 2 and 3, with appropriate choices for
the parameters.

     Suppose, for example, that emissions monitoring with corrective action
as required is to be performed in the old facility.  Then the emissions incre-
ment I 1 , and its standard deviation Sj2 ., are obtained by using the analysis
for Case 2, with

     Nn = 0.

     Suppose further that hardware capabilities have been introduced to reduce
fugitive emissions in the new facility.  The emissions increment Inew and its
standard deviation Si2    can be obtained from the analysis of Case 1, with
                      ZiSvv

     NO = 0.

     The total emissions increment corresponding to this particular type of
device is:

                          I = I ,, + I
                               old    new
                                     16

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and


                           I     •*• old  ' "-1- new

Example 1 for Case 4
                                  2    ,  0 2
                                         SI
     In this example, it is once again assumed that a catalytic reformer is
to be added to an existing refinery.  The emission statistics for compressors
are computed below.

     Four compressors are assumed to be required for the new reformer, and
these are all vented to a flare.  No provisions for reducing emissions from
existing compressors, which are not presently vented to a flare, are planned.
Thus, different emission factors entirely apply in the old and new facilities,
and this situation is handled under Case 4.

     This is a very special case in which the emission increment and its
standard deviation are both zero, as is shown below.

Old Facility—
     It is clear that nothing has been done to change the compressor emissions
in the existing facility.  Thus,

     Iold = 0, and

    ST ., = 0.
     -"•old

New Facility—
     Since emissions are vented to a flare, the fugitive emissions are
essentially reduced to zero.  Thus,

     Znew " °» and

    ST    = 0.
     -"-new

Both Facilities—
     The emission increment for the entire facility and its uncertainty are
readily seen to be zero:

                          1 = Zold + 'new = °

                          ST2 . ST2    + ST2    = 0
                           1     L old    I new

Example 2 for Case 4

     In this case, we assume that a light ends/gas processing unit is to be
added, resulting in the addition of 15 pumps.  The single seals on the existing
pumps are replaced by double seals.  Hardware provisions, including venting
to a flare, again essentially reduce the emissions from the new pumps to zero.
The required calculations for pumps, then, are as follows.
                                     17

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New Facility—
     As in the preceding example,

     I    =0, and
      new
    S     = 0.
     •4iew

Old Facility—
     The calculations for the old facility can be made by  using the Case 1
analysis and setting Nn equal to zero.  The required  inputs,  then,  are as
follows:

     E1 - 0.308,

    SEi = 0.080,

    E11 = 0.060,

   SEii = 0.015,

     N0 = 264, and

     Nn = 0.

Then,

                           old     no        o

                               = (264)(0.060) -  (264)(0.308)

                               = -65

and,

                          SI2 .. = 
-------
Method for Reducing Error  in Emissions  Increments

     A method designed  to  reduce  the  uncertainty in  the  emissions estimates
is presented in Appendix B.  The  resulting  analysis  required  to  calculate
I and Sx is more complex than  in  the  cases  presented above.   The circumstances
under which this method can be used advantageously and the  analysis itself are
discussed in detail in  Appendix B.

     The method does not lend  itself  to brief summarization without loss of
accuracy.  It should be considered, however,  if the  conditions discussed below
hold.

     Define:

     x = measured  emission rate for  a particular source  of
         the type  being considered before a planned  program
         for reducing  emissions is put into effect,  and

     y = measured  emission rate for  the same source
         after  the program is  put into effect.

     Then,  if x and y  are  correlated  (see Reference  5),  the method discussed
 in  Appendix B  can  be  used  to  reduce  the variance in  the  estimated value of
 I.

 Other  Cases

     While  every effort has been made to cover the most  likely scenarios, it
 is  almost  inevitable  that  situations  will arise which do not  clearly  fall under
 any of  the cases discussed above.  In these cases, the analysis  necessary to
 estimate  the  emission increment and its uncertainty  for  the device being
 considered should be  worked out.   The analysis under "Further Discussion of
 Case 1,"  earlier in  this  section, is  an example of how the  basic equations
 can be  extended to handle  other cases.

      It  is  probable,  however,  that the cases presented here will cover most,
 if  not  all,  device types  in a given refinery.  The  approach  discussed in
 Section 3  for combining the I's and Sj's for all device  types in the  refinery,
 moreover,  can be used in  any case.

 Systematic Errors

      It is correctly pointed out in Reference 4 that, if present, biases in
 measurements  could affect the estimated emission rates.   If a known bias
 exists in measuring emissions, then the data will  not be representative of
 the true emissions and therefore must be corrected before the analysis is
 performed   A quality-assurance program using specified  calibration tech-
 niques and utilizing known standards for verification will  be essential in
 determining the magnitude of the bias and the resulting  correction  factors
 needed   If quality-assurance procedures are not  incorporated,  then unknown
 systematic errors are best handled as random errors, as  is  discussed  below.


                                      19

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     If an emission factor were estimated from measurements at several
refineries, and if each data collection involved an independent equipment
setup and calibration, then one would expect that the data set would include
a random collection of systematic or bias errors (if, in fact, biases were
present).  These errors would then be averaged, along with the other random
variations, in obtaining the final estimated emission rate.  If present,
bias errors would increase the standard deviation of the emission rate.

     Under these conditions, the analysis should be carried out exactly as
described above; one should assume that the standard deviations of the emission
rates reflect all measurement errors.

     Suppose, on the other hand,  that data were only available from a
single equipment setup and calibration.   If the variance from setup to setup
for the particular type of measuring equipment used were known from experience,
this variance should be added to the variance of the mean calculated directly
from the data.  The results, then,  would represent the uncertainty due both
to measurement-to-measurement variations and the random error due to setup
and calibration.

     If in collecting emissions data an unknown bias error existed which
could not be measured or accounted for,  then the resulting estimates would
be erroneous.  The standard deviations calculated would underestimate the
actual uncertainty of the emission rates.   There is always the potential for
this type of occurrence in any measurement project.   Proper calibration and
quality-assurance standards will minimize this problem.   If no attempt is
made to measure these biases, then statistical procedures are of no help in
deriving estimates of emissions.
                                     20

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                                  SECTION 3

                MEAN EMISSION  INCREMENT AND ITS  UNCERTAINTY
                              (Entire  Refinery)


     Now, suppose, that  for  each  device type,  the mean (or  expected)  emission
increment, I, and standard deviation,  S-j-,  have been computed.   The  emission
increment IT for the entire  refinery  is:

                          IT = II

and the variance of If is

                          SZ 2 =  ZST2
                           •""£       I

In both cases, the summation is over  all device  types.

     Now, !„ is a linear combination  of the various  emission factors.
Assuming that the most important  emission factors were estimated from a large
number of observations,  these  factors  can reasonably be assumed approximately
normally distributed.  This  statement  does not imply that the original
emissions data are normally  distributed.

     The quantity 1-j- then is a linear sum whose  largest terms are approximately
normal; thus, I-j is also approximately normal.   The  inclusion of the multipli-
cative random factor f , however,  weakens the argument  for normality somewhat.
This objection would be  insignificant,  however,  if f were estimated from a
large sample and S^2 were small.

     If I_ is assumed to be  normal, and if  the emission  factors which con-
tribute most to the standard deviation  estimate  S    are  calculated from samples
of size .30 or greater, then  the following  vari-    T  able Z can be assumed to
have the standard normal distribution  (i.e., Z is  normally distributed and
has mean zero and variance one)
                                T

where \i is the true and unknown mean of the emission increment.
                                     21

-------
     As is mentioned in the Introduction  of  this  report, Radian is
currently performing an extensive  research study  in which statistically
valid estimates of emission rates  and their  uncertainties will be obtained.
Due to the magnitude of the data collection  being performed  in this project,
it is felt that the emission factors  will be calculated from sample sizes
large enough to satisfy the requirements  for Z  to have approximately  the
standard normal distribution.   The standard  normality assumption, moreover,
greatly simplifies the probability calculations.

     A few comments about cases in which  this assumption is  not valid,
however, are in order.  If I_ is approximately  normal, but some of the
important variance estimates are calculated  from  samples of  size less than
30, then Z as defined is not normally distributed.  Since S   2 is the sum of
a set of estimates of different variances, moreover, Z       T  does not
have exactly the t - distribution, either.   It  is possible that the distri-
bution of Z could be approximated  by  a t  - distribution, however.  A
similar application of the t - distribution  is  discussed in  Section 4.14 of
Reference 7.  Cases in which IT is not approximately normally distributed
should probably be treated individually.  Further analytical work regarding
the distribution of Z will be beneficial  if  such  cases arise in the future.

     Assuming that Z has approximately the standard normal distribution,
then, well-known statistical methods  can  be  used  to make probabilistic
statements about the value of u, as is discussed  below.  Of  particular
interest is the probability that ]i is less than or equal to  zero; that is,
that the plant expansion can be made  with no increase in emissions.

     The probability that y is less than  some value a can be obtained as
follows:

                          P(y
-------
using the Z-statistic instead  of  the t-statistic.   The use of  the Z-statistic
would be valid as long as  the  emissions  factors  for the greatest contributors
to emissions, such as valves,  were  computed from large samples.

     It should be noted  that the  estimated emission increment  IT for  the
entire refinery  can be calculated even if only the emission factors,  but not
their variances, are available.   The variances are required only for  the
calculation of Sj  and for the probability calculations.

     The analysis method presented  here  can be used as an  aid  to assess several
options for reducing the emissions  to acceptable levels.   The  following two
numerical examples illustrate  this  process.  In Example 1,  the  calculations are
performed assuming certain emission control procedures will be used.  In Example
2, then, the control procedures are  altered in several  ways,  and the calculations
are updated accordingly.  The  final selection of control strategies,  of course,
should also include economic and  other considerations,  as  well as the emission
calculations.

Example 1

     The calculations will now be worked out for a hypothetical example.  It
is assumed  that  a catalytic reformer is  being added to an  existing refinery
and that various provisions to reduce emissions  are being  taken.  The provisions
and the values of I and  S^ are listed in Table 3-1.  The value of I and Sj
shown  for valves are  calculated in  Example 1 for Case  1; for the API  separator
in Example  2,  Case  1;  for  pumps in  the example for Case 3;  and for compressors
in Example  1  for Case  4.

     The value of I and  S-r for the  API separator are -6.0  and  3.0, respectively.
Both values are  in  pounds  of emissions per thousand gallons  of wastewater and
must be converted to  Ib/hr. It is  assume'd that  300,000 gallons of wastewater
are processed per hour;  therefore,  the conversion can  be made  by multiplying
I and  Sj by 300.  The  resulting values are:

                           I =  (-6)(300)  = -1800  Ib/hr

                           SZ = 3(300)  ^ 900 Ib/hr


     The values  of  I  and Sj for flanges,  drains,  and relief valves must also be
calculated.   Since no  change is made in  the old  facility for these three sources,
the values  of I     and SI    are  both zero (see  the discussion of Example 1 for
Case 4).  The statistics °  regarding the emission increment  in the  new
facility, however, must  be computed. The required inputs are shown in Table
3-2.

     The new-facility  calculations  will  be done  as a Case  1 problem,  with
N0 = 0.  Then in the case  of flanges:
                                     23

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   TABLE 3-1.   DATA FOR EXAMPLE 1:   HYPOTHETICAL  TEST CASE IN WHICH A
               CATALYTIC REFORMER IS ADDED TO  AN  EXISTING REFINERY
Device
Valves
API Separator
Pumps
Steps to Reduce Emissions
Maintain high leakers
Cover
Replace single seals
I
-441
-6*
-77
Si
95
3*
21
                          with double  seals,  then
                          maintain high leakers
Compressors               Vent emissions from new        0              0
                          compressors  to a flare
Flanges                   None                          2              1
Process Drains            None                          2              2
Relief Valves             None                          2              1
*In pounds per thousand gallons of wastewater.   All  other  values  of  I  and
 are pounds per hour.
                                    24

-------
  TABLE 3-2.   DATA REQUIRED TO CALCULATE EMISSION-INCREMENT STATISTICS  FOR
              NEW FACILITY FOR FLANGES, PROCESS DRAINS AND RELIEF VALVES
Quantity
E11
SEn
N
o
N
Flanges
0.00076
0.00050
0

2800
Process
Drains
0.034
0.025
0

60
Relief
Valves
0.50
0.20
0

4
E11 and Sgn
NOTE:  The values of E  and SEi are not needed, since No = 0.
       are the emission rate and its standard deviation in Ib/hr/unit with
       the programmed maintenance  (none in these cases).  All variable names
       are as defined under Case 1.
                                    25

-------
                          I = N^11 - NoCE1 - E11)

                            =NnE11

                            = (2800)(0.00076)

                            = 2.13

and

                          Sz2 = (Hn + N0)2 SE:i2 + N02SEi2

                              = \2sEii2

                              = (2800)2(0.0005)2

                              = 1.96

                          SI = V^I5 = 1'40

     The calculations for drains and relief valves are very similar.  The
values of I and Sj for all source types are given in Table 3-1, as is mentioned
above.

     The mean and standard deviation of the emission increment will now be
calculated.

                          IT = El = -441 - 1800 -77+0+2+2+2

                                  = -2312


                          S  2 = (95)2 + (900)2 + (21)2 + (O)2 + (I)2 + (2)2 + (I)2
                            T
                               - 819472

                          S   - 905
                            T

     Now we are interested in computing the probability that the true emission
increment is less than zero:
                                    26

-------
                          P(U<0) =

                          P(0<-y) =
                          P(-2.55 < Z) = 0.9946


     The probability 0.9946 was obtained from a normal probability table.
The probability that the emissions are not increased, then, is extremely
high.

     This example requires one further comment.  Any indirect effect of
maintenance or hardware changes should be taken into account in establishing
the emission rate estimtes to be used.  Suppose, for example, that an equip-
ment expansion (such as adding a catalytic reformer) resulted in a significant
increase in fugitive emissions from either the API separator or the cooling
towers due to an increased volume of water processed.  Then the emission
increase should be reflected in the calculations.

Example 2

     As a further illustration, we will now alter the preceding example
somewhat.  Suppose the maintenance program for valves is not planned; that
high leaking pumps are maintained, but that single seals are used in the new
and old facility; and that the API separator is not covered.

     The values of:

     I = 34 Ib/hr, and

     Si - 68 Ib/hr
                                     27

-------
for valves are obtained exactly as were the values for flanges in the
prececing example; in this case

     E11 = 0.040,

    SEM = 0.080, and

      Nn = 850.

     The calculation of I and Sj for pumps falls under Case 1, with the
following required inputs:

     E1 = 0.308 Ib/hr,

    S£i = 0.080 Ib/hr,

    E11 = 0.060 Ib/hr,

   SEn = 0.015 Ib/hr,

     N0 = 264, and

     Nn = 17.

Thus,

                          I - (Nn + No)Eu - NoE1 = -64

                          Si2 = (Nn + N0)2SE112 + N02SEi2 - 464

                          Sj = 22

     Since no maintenance or hardware modifications are to be made regarding
the API separator, the values of I and S-j- for it become zero.

     The estimated emission increment for the entire refinery is:

                          IT= II =34 +0-64 +0+2+2+2

                                  = -24

and

                          ST 2 = EST2 = (68)2 + (O)2 + (22)2 + (O)2 + (I)2
                           J-np      JL
                                        + (2)2 + (I)2

                                      = 5114

                                  ST  = 72
                                      28

-------
     The situation here is much less clear cut than it was in the preceding
example.  The estimated emission increment is negative, which means that the
planned expansion probably would not increase the fugitive emissions.  The
probability that the true mean u is actually negative can be calculated as
follows:

                          P(y<0) -

                          P(0<-u) =
                           P(-0.33  <  Z) = 0.63  (from the tabulated normal
                                               distribution)

      If it is desired,  the probability that the true mean is less than a
 specified positive value can be  computed by essentially the same method.
 The probability that y  is less than  50 Ib/hr,  for example, is:

                           P(U<50)  =

                           P(-50<-u)  =
                           P(-1.03  <  Z) =  0.85


      Thus,  there is only a 15%  chance  that  the  true emission increment is
 greater than 50 Ib/hr.
                                     29

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                                 SECTION 4

          COMBINING DIFFERENT ESTIMATES OF THE SAME EMISSION RATE
     In this section, the situation is discussed in which independent estimates
of the same emission factor are available from:

     (1)  data collected at the refinery being investigated and

     (2)  other refineries (from which general emission factors
          have been computed).

     The estimate from Source (1) may be more accurate in that random differences
among refineries do not contribute to the error.  Source (2), on the other hand,
would be less subject to random measurement errors and random device-to—device
differences if it were based on a larger data set, which would usually be the
case.

     The objective, then, is to determine whether: (a) the estimate from Source
(1) only should be used, or (b) the estimates from the two sources should be
combined.  In the second case, the question is how to combine the estimates
most effectively. Alternative (a) or (b) is selected by performing a statisti-
cal hypothesis test.  Hypothesis tests are discussed in References 3 and 5 and
in many other statistics works.

     The analysis to be discussed in this section applies equally to any of the
emission factors required for any of the cases discussed above.

     Define the following variables:

     EI = emission factor estimated from data collected at
          the refinery in question,

    SE  = standard deviation of EI,

     Ez = emission factor estimated from other refineries,
          and

    Sg  = standard deviation of £2-

     Note that S_  reflects only device-to-device variations and random
measurement errors, while Sg2 reflects these variations and whatever refinery-
to-refinery variations may exist.
                                     30

-------
     Now unless a measurement malfunction  invalidates  EI,  it  can  be used to
obtain the final estimate.  Both Ej  and £2 should  be used  if  they can  be
considered independent estimates of  the same  quantity=   If, on  the other
hand, the particular refinery being  studied deviates significantly from  the
average, so that E2 is actually a biased estimate  of the emission factor of
interest, then only EI should be used.

     To decide which course to take, we will  test whether Ej and  E2 are  equal
within random variation.  The statistical  hypothesis test should  be made using
the Z - statistic, which is defined below,  if Ej. and E2 were both  computed
from reasonably large samples, say with size  over 30.  Otherwise,  the t  -
statistic should be used.  The t - statistic  is discussed in many  introductory
statistics textbooks, such as References 5  and 7.

     As is seen in the numerical examples  below, if one  sample size is much
larger than the other (e.g., by a factor of 16) then very little  is gained by
including the estimate calculated from the  smaller sample size.    If both sample
sizes are very small, it is unlikely that  an  adequate emission increment will
be obtained even by using both data  sets.   For these reasons, the discussion
below is centered around the Z - statistic; application of the t - statistic,
in accordance with References 5 and  7, however, is very similar.

     Define:

                                 E2 - Ei
                          Z =

      If Ei  and E2  are normally distributed, then Z is normally distributed with
variance  one.  If  EI and E2  estimate the same emission factor, then Z has mean
zero.  Standard  statistical  methods then can be used to perform the statistical
hypothesis  test, as is  discussed below.

      If  |z|  > Zcrit, where Zcrit is a value chosen from a normal variable
table, then the  difference between EI and E2 is too great to be explained by
random errors.   In this case, EI should be used as the estimate.

      If the test is to  be performed at the 0.05 confidence level, for example,
then  Zcr-j_t  = 1.96.  The confidence level is the probability of concluding that
there is  a  true  difference between EI and E2 when there is not.

      If  |z   < Zcrit» then the difference between EI and E2 can reasonably
be explained~~in  terms of random sampling errors alone.  In this case, EI and
E2 should both be  used  to derive the emission estimate.

      The  minimum variance estimate of the emission factor, then, is as
follows:

                                   + a2E2
                                      31

-------
where
and
                         a.2
     That is, EI and Ea are weighted by the inverses of their variances.  The
variance of the estimate E, then, is:

                          a2 _ 0 20  2 ,    2C  2
                          S  = ai Sp   +32 Sp
                                   EI        E2

Since:

     0 < ai < 1 and 0 < 3.1 < 1 when both variances are greater than zero, the
variance S2 is less than either Sg 2 or Sg22.  Thus, the uncertainty has been
reduced in using both EI and E2 to compute the final emission factor.

     To reiterate, it has been assumed that the sample sizes used to compute
Ei and £2 are large enough (at least 30) to justify using the Z- rather than
the t-statistic.  At least this sample size would be required in most cases
to obtain acceptable accuracy, anyway.

Example

     Suppose the analysis described above is to be used to test the emission
factor for valves and that the following data are available:

     Ej = 0.0400 Ib/hr,

    SE1 = 0.0080 Ib/hr,

     E2 = 0.0600 Ib/hr, and

    SEz = 0.0320 Ib/hr.

     The value of Sg  is four times as large as the value of SEl.  This  is
about what one would expect if EI were determined from a sample size  16  times
as large as that used to determine £2-
                                     32

-------
     Then:

                          7 - 0-060 - 0.040
                          Li — ——————^————



     Since:

                           Z| =  0.61 < 1.96

the difference between EI and E2 can reasonably be explained in terms of
random errors.  It is important  to notice that this conclusion is reached
despite the rather large difference between EI and E2.  This is due to the
relatively large uncertainty in  E2.

     The  two  emission rate estimates  then could be combined as follows:

                          ,. .   i/(o.QQ8)2         _ A „,
                                      rr +
                                (O.OOSr    (0.032)2

Similarly:

                           a2  =  0.06

NOTE:  ai + a2 must be one.

     Then the updated emission  rate  is:

                           E = 0.94(0.040) +  (0.06)(0.060)

                           E = 0.041

and
                          S =  /(0.94)2(0.008)2 +  (0.06)2(0.032)2

                            =  0.0078

     The emission rate estimate then has been changed by only 0.001, from
0.040 to 0.041, and  the standard deviation of the emission rate has also been
changed by a very small account.  Thus, since E2 was estimated from a small
sample size, the improvement in the emission increment is very small.

     Suppose, then,  that E2 had been estimated from a sample one-fourth as
large as that used to estimate EI, and that:

     Sp  = 0.016.
      E2
                                      33

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Note that
Then:

     3! = 0.80,

     a 2 = 0.20, and
                          S - /(0.80)2(0.008)2 + (0.20)2(0.016)2 - 0.0072
     In this case, the uncertainty in the emission estimate is reduced by 10%
from 0.0080 to 0.0072, by pooling the data.

     One might ask, how large a sample of data would be required for the
estimation of EZ for the data pooling to result in a significant improvement
in the emission rate estimate.  The type of calculations demonstrated here
can be used to address this question; the fact that the standard deviation of
a sample mean value decreases as the square root of the sample size increases
was used in choosing the values of Sg2; e.g., increasing the sample size by a
factor of four would be expected to reduce the uncertainty in the mean by a
factor of two.

     If the sample sizes used to estimate EI and £2 were the same, and if

                          Sw  = Sp  = 0.0080,
                           Ei    E2
then,

                          ai = a2 = 0.5

and

                          SE = /(0.5)2(0.008)2 + (0.5)2(0.008)2 = 0.0057

     In this case, the uncertainty in the emission estimate is reduced from
0.0080 to 0.0057, or by a factor of 0.707.
                                     34

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                                 SECTION  5

                                  SUMMARY


     This report presents a  statistical approach for testing whether a planned
refinery expansion can be made without increasing the fugitive emissions, when
this is required by regulations.  Emission factors can be used to estimate the
emission increase or decrease due to  the  expansion.  Being empirically deter-
mined, however, the emission factors  are  subject to random errors.  Thus, the
effect of the expansion on emissions  cannot be computed exactly.

     For this reason, the problem has been treated statistically.  Anaytical
methods are presented which  can be used to estimate the emission increment
as a function of:

     •    the facilities which are to be  added,

     •    existing facilities which will  be shut down
          (if any), and

     •    hardware changes or improved maintenance programs
          designed to reduce emissions in the old facilities
          and to limit the new emissions  in the planned
          facilities.

     The emission increment, I, can be thought of as:

                          I  = A - D

where

     A = emissions from the  new facility  in, say, Ib/hr,
         and

     D = emission decrease in the existing facility due
         to hardware changes or improved  maintenance
         practices.

     If the value of I is negative, the expansion "probably" can be made with-
out increasing emissions.  If I is positive, emissions  P^bably  will be
increased   As is indicated  above, however, I is affected by the random errors
in the emission factors.  Thus, the standard deviation of I is also computed,
                              less than zero is obtained; this is tne
                                  Planned .ill not increase fugltive emissions.
                                     35

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Other probabilities, e.g. that I is less than 50 Ib/hr, can also be computed,
if this is desired.

     The methods presented here can be used as an aid in comparing several
options for reducing emissions to acceptable levels.  This involves simply
performing the statistical calculations in parallel for each option to be
considered.  The final decision, of course, should also involve economic
and other considerations.

     The methods are designed to handle different control strategies in the
existing and in the planned facilities and different strategies for different
types of device, if this is necessary.  The following situation, for example,
could be handled.

     •    A catalytic reformer is to be added to an
          existing refinery.

     •    To reduce emissions in the existing facilities,
          an improved maintenance program is instituted for
          valves, and the API separator is to be covered.

     •    To limit the added emissions in the new facility,
          the same type of maintenance program used in the
          old facility will be instituted for valves,  and
          double seals will be used rather than single seals
          for all pumps.  Single seals remain on all pumps
          in the old facility, however.
                                     36

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                                REFERENCES
1.    Aitchison, John, "On the Distribution of a Positive Randon Variable Having
     a Discrete Probability Mass at  the Origin", Journal of the American
     Statistical Association. 50, 9  (1955), pp. 901-908.

2.    Finney, D. J., "On the Distribution of a Variate Whose Logarithm is
     Normally Distributed", Journal  of the Royal Statistical Society Series  B,
     7, pp. 155-161.

3.    Jefcoat,'!. A., Leigh Short, R. G. Wetherold, "Fugitive Emission Control
     Strategy for Petroleum Refineries", Paper Presented at the Refinery
     Emissions Symposium, Jekyll Island, Georgia, April 26-28, 1978.

4.    Jones, Harold R. , Pollution Control in the Petroleum Industry, Noyes Data
     Corporation, Park Ridge, New Jersey, 1973.

5.    Mood, Alexander M., Franklin A. Graybill, and Duane C. Boes, Introduction
     to the Theory of Statistics, McGraw-Hill Book Company, New York, 1963.

6.    Serth, R. W. and T. W. Hughes,  "Error Analysis for Plant Expansion
     Problem", Monsanto Research Corporation, Dayton, Ohio, March 18, 1977.

7.    Snedecor, George W. and William G. Cochran, Statistical Methods, The
     Iowa State University Press, Ames, Iowa, 1967.
                                     37

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            APPENDIX A




PROPERTIES OF THE MEAN AND VARIANCE
                 38

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     This Appendix includes  a  brief  discussion of  the properties of the mean,
variance, and covariance.   These  statistical  measures have been used extensively
in the study reported herein.  Also,  Reference 5 includes an excellent discus-
sion of these statistical  measures.   Some properties of  the covariance that
are not readily available  in statistical texts are derived in this appendix.
Mean
     The mean or  arithmetic average of a random variable X can be estimated
from a set of N values,  X^, i = 1 to N,  as  follows:
     If Ux  is  the mean of X and K is a constant,  then  the mean of KX is
If Y has mean  Vy, then the mean of X + Y is  vx +  Uy-   From these properties
the mean of any  linear combination of random variables can be obtained.  The
mean of:

                           3 + 4X + 2Y

for example is:
     If  X and Y are independently distributed,  that  is, if the value of one
 is not influenced by the value of the other,  then  the mean of XY is
Variance
      The variance a2 is a measure of the amount of  scatter or dispersion a
 quantity has;  the more widely it varies, the greater  the variance is.  Thus,
 a quantity with a large variance is considered to have  a large uncertainty.

      The variance a 2 of a variable X can be estimated  from a set of N values
 as follows:
                                 N
                            a
                           S
                                   N -
      The standard deviation, a, is the positive square  root of the variance.

      If X and Y are independently distributed,  the variance of X + Y is:
                           ax2 + aY2
                                     39

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where CT 2 is the variance of Y.  The variance of Kv is
       I                                          A
where, as before, K is a constant.  The variance of any constant is  zero.
From these properties, the variance of a linear combination of independent
variables can be derived.  The variance of:

                          3 + 4X + 2Y

for example is
     If X and Y are independent, the variance of XY is:
     If X and Y are not independent, the variance of X + Y is

                          crx2 + aY2 + 2cov(X,Y)

where cov(X,Y) , the covariance between X and Y, can be estimated from a set
of N values of X and Y as follows :

                          N        _       _
                          £  (X± - X) (Y± - Y)
                                 N - 1
     The more closely X and Y are (linearily) related, the larger the covariance
is in magnitude.  If Y is a perfect linear function of X with positive  slope,
then the covariance equals its maximum possible value, ^x^Y-  When X and Y
are independent, the covariance is zero.

     The variance of X + XY where X and Y are independent is

                          ax2 + cr^2 + 2 cov(X, XY) =


                          a 2 + u 2a/ + uY2aY2 + aY2aY2 + 2 cov(X, XY).
                           A     1  •"•     •"•  •*•     A,  *•

     The covariance between X and XY is
This relationship, which is not ordinarily given in textbooks,  is proved
below.
                                     40

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     The same expression for the variance of X + XY can be obtained by
writing X + XY  as  X(l + Y)  and using the formula given above for the variance
of a product of two  independent random variables.

Derivation of the  Covariance Between X and XY

     Suppose X  and Y are independent random variables with respective means
Ux and VY anc^ variances 
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Derivation of the Covariance Between XY and Z

     Suppose X, Y,and Z are random variables with respective means yx> |Jy and
y ,  and X is independent of Y and Z.  Then:
COV(XY, z)  = E[(XY -

           = E[XYZ -

           = UE[YZ] -
                                                       - yz)]
                                                yxcov(Y, Z) -
                                     = Uxcov(Y, Z)
Derivation of the Covariance Between X and Y

     An addition property of covariance is needed for use in Appendix B.
Suppose the covariance cov(X, Y) between two random variables X and Y is
known, and the covariance between the sample means X and Y is needed.  The
quantities X and Y are the means of samples of size N of values of X and Y,
respectively.

     Then, if l-i  and VL. denote the means of X and Y, respectively.
                          cov(X, Y) = cov
                                                N
                                  I (X  - y )|l  E (Y
                                 1-1  x    X/\i=l  ]
                                                      E[(x, - y.
                             [N cov(x, Y)]
                          cov(X, Y)
                             N
                                    42

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Estimation of Mean and Variance  for Skewed Distributions

     The expressions  given  above for estimating  the mean and variance are
unbiased estimators,  that is,  their expected values  equal  the population
values they estimate.  This is  true regardless of  the underlying probability
distribution.

     If the distribution is highly skewed (asymmetric), however, these esti-
mators are not  efficient; other estimators exist which have smaller uncertainties.
The lognormal distribution is  an example of such a distribution.

     Reference  2 presents a discussion of the efficient estimation of the mean
and variance of a lognormally  distributed random variable.
                                      43

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                     APPENDIX B




A PAIRED MEASUREMENT SCHEME FOR REDUCING RANDOM ERRORS
                           44

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     In this appendix,  a method designed to reduce the variance of I is
described.  This  is  beneficial in that if I is known more accurately, then a.
more confident  decision can be made as to whether a refinery can be expanded
without increasing emissions.   The extent of the reduction in uncertainty,  or
whether any reduction is achieved, depends on factors discussed below.

     The  basic approach involves making emissions measurements before and
after  a program to reduce emissions is instituted.  The program could include
hardware  modification or improved maintenance practices.  Then define

     X. = measured emission rate from the i   tested unit before the program
           is  started

     Y. = measured rate after the program is started for the same unit, and

     r. = X.-Y , the decrease in emissions for the i   unit due to the program.

     The  average emission reduction, then, is
          N
               N

 where N is the number of units tested and the variance a* is estimated by
                                                         j\
      Sf
               (N-l)

      In the expression above, the quantity in parentheses is the variance of
 the individual values of r, and this must be divided by N to obtain the
 variance of the mean of N values of r.  (More complicated estimates exist for
 the mean and variance which are more efficient when the distrxbutxon is highly
 skewed, as is discussed in Appendix A.)

      Now, it is clear that the average  emission reduction can also be expressed

            N
        R =
          = X - Y
       —     —                 *  t-v,^ Y'O  and  the Y's, respectively.
 where X and Y are the means  of  the X  s  and  cue
                                        45

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     The final expression for R indicates that the variance of R can also be
written
                          a £  +  a ^  -  2 cov (X,Y)
                            X       Y

where cov (X,Y) is the covariance between X and Y.  The following brief dis-
cussion of covariance relates importantly to the physical problem:

     The covariance, which is defined formally and discussed in Appendix A,
is a measure of the extent to which two variables are linearly related.  If
there is no relationship, then the covariance is zero.  If one variable
tends to increase as the other increases, then the covariance is positive.

     Thus, if the emission rates before and after the program is started are
not related, then the variance of R becomes


                          a 2  +  a 2
and the effort to ensure that the same set of sources are measured before and
after achieves nothing; any set of sources could have been tested before and
any other set after, and the result would have been the same (excluding
differences in random errors).

     If X. and Y. are linearly related, however, cov (X,Y) is positive, and
using a paired measuring scheme reduces the uncertainty in R; the amount of
the variance reduction is 2 cov (X,Y).  Whether the paired approach should
be used in a given case depends on the extent to which X and Y are related
and the expense and inconvenience of using a paired scheme for estimating R.

     Additionally, estimated emission rates may be  available which were not
obtained by using the paired approach.  If a paired experiment were performed,
then either (1) the existing factors would not be used or (2) a much more
complicated analysis approach would be employed to  combine emission rates
estimated by different methods.  The second option  is not considered to be
desirable, since it is not at all certain that a significant reduction in
uncertainty would be achieved over using the simpler, unpaired approach.

     In the analysis which follows, the covariances between related emission
rates (such as X and Y) and between emission rates  and emission reductions
(such as X and R) are needed to compute the uncertainty of the estimated
emission increment.  The equations needed to calculate these covariances are
given below.
                                     46

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     The covariance between X and Y is estimated as follows

                                      N      _     _
                           cov(X,Y) = Z  (X.-X)(Y.-Y)
                                     1-1   1     x
                                             N-l

The covariance between X! and Y,  then,  is


                           cov(X,Y)  = cov(X.Y)
                                          N


r is Stlmatef bynSh±P " '^ " APPendlX A'   The covariance bet— * and

                                     N      _
                           co-v(X,r)  = 2   (X.-X)(r.-R)
                                              N-l

and the estimated covariance between X  and R  is


                           cov(X,R) = cov(X,r)
                                         N

     In Section 2, the calculation of the mean and variance of the emissions
increment for a given type of device is discussed.  Several scenarios
representing different types of effort  to reduce emissions are analyzed.
The same scenarios are discussed below, except that the emissions reductions
are assumed to have been estimated by using the paired-measurement scheme.

Case I1-  Same Provisions  for  Reducing  Emissions  in  the New and Old Refineries

     Define:

     R    = average reduction in emissions per source due to maintenance,

     S    - standard deviation of R,
      R
     E1   = average emission rate per -source  without maintenance,

     S *  = standard deviation of E1,

     E11  = average emission rate per  source  with maintenance,

     S-.11  = standard deviation of E11,
      u
     N    - number of sources of  the  type being  considered in the
      0     old facility,  and
    Nn   =  number of sources  in the new facility.

                                     47

-------
     This case corresponds to Case 1 discussed in Section 2.  The covariance

between R and E11 will also be needed, as is discussed below.



     Then the emission reduction in the old facility corresponding to this

particular type of source is



                          N R
                           o


and the added emissions in the new facility is



                          N E11
                           n


     Thus, the total emission increment I is



                          I = N E11 - N R
                               n
     The variance S 2 of I is



                          S2 = N 2 S_n2 + N 2 S 2 - 2N N
                           I     n   E       oR      no




     It will now be shown that the analysis here is consistent with that

presented in Section 2.  To do this, we will replace R by E : - E 11  in the

expression for I, to obtain:



                          N E11 - N R =
                           n       o


                          N E11 - N (E1 - E11 ) =
                           n       o


                          (N  + N ) E11 - N E1
                            no         o




     This is exactly the expression for I given in Case 1, Section 2.



     If E1 and E11 are assumed to be independent, the variances given here

and in Section 2 can also be shown to be consistent.  Under this  assumption,


                            S2> _. C1  2 j  ri   2
                              " S_i  + S 11
                           K.     £>      Ci



and



                          cov(R,En ) = - S^u2
                                     48

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             2
     Thus, ST  becomes
                            2NnNo + V> V'2
and this is exactly the expression given  for S  2 in Section 2.


Example  for Case I1



     As in Example 1 for  Case  1,  suppose  a  catalytic reformer is being added

to an existing refinery and  that  this  adds  850  valves.  An improved maintenance

program is introduced which  reduces  the average emission rate from a given

valve from 0.040  to 0.008 pounds  per hour.  These and other necessary statistics

are presented below.



     R    =  0.040 - 0.008 = 0.032 lb/hr,


     S*   =  0.004 lb/hr,
      R


     E1 •  =  0.040 lb/hr,



     S_i  =  0.006 lb/hr,
      £i


     E11  =  0.008 lb/hr,



     S_ii =  0.003 lb/hr,
      £u


     N    =  14000 ,
      o


     N    =  850,  and
      n


     cov  (R.E11)  = 3.6 x  10'6  (lb/hr)2


     covCR.E11)  and SR would have the values  given above if  the emission

factors E1  and E11 had a  correlation of 0.7.



     The value of I is the  same as in Example 1 for Case 1:



     I = -441 lb/hr
                                     49

-------
 but  Sj2  is  reduced:


        ST2  - N 2 S^n2 + N 2 S 2 - 2 N  N  cov (R,E11)
         I      n   E       o   R       no

             -  (850)2(0.003)2 + (14000)2 (0.004)2 -

                (850) (14000) (3.6 x 10~6)

             =  3100

          S  -  56  Ib/hr.

     Thus, a reduction from S  = 95 pounds per hour, which is the result
given in Example 1 for Case 1, to S  = 56 pounds per hour is achieved by
using the pairing scheme.

     If the emission factors E1 and E11 had a correlation of 0.3,  instead of
 0.7 as in the case above, S_ and cov (RjE11) would have the following values:
                          R
          S_ = 0.006
           R
         cov (R.E^1) = -3.6 x 10

     The resulting value of S  is 84 pounds per hour.  In this case a smaller
reduction (84 versus 95) in S  is achieved by using the pairing scheme.

     As is discussed in Section 2, the analysis of Case 1 (Case I1 here) can
be used to handle the situations covered under Cases 2 and 3 (Cases 2l and
31 here).  Cases 21 and 31 treat scenarios in which screening for maintenance
is performed, and high-leaking devices are repaired.   In these cases, if the
mean (E11) and standard deviation (S_n) of the emissions from a collection
of units which are selectively maintained are known, then the analysis dis-
cussed under Case I1 should be used.  The analysis below, however, can be
used when the information in this exact form is not known.  Although more
numerous, the required inputs for Cases 21 and 31 may be more easily obtained
than those for Case I1 in some instances.

Case 2l - Emissions Monitoring and Subsequent Corrective Action. Performed the
          Same in the Old and New Facilities

     Define

        RI = reduction in emissions per unit due to maintenance,

      S2   = variance of R!,
        K.1
        F2 = average emission rate of maintainable leakers if maintenance is
             performed,
                                     50

-------
        F 3  =  average emission rate of other units,

        S    =  standard deviation of F., 1*2, 3,
          i                          x
         f  =  proportion of the units of the type being considered which would
              be high- leakers at any given time on the average,  if detailed
              maintenance were not performed,

         S, =  standard deviation of f,

         NQ =  total number of units of the type being considered in the old
              facility, and

        N  =  total number in the new facility.

     This  case corresponds to Case 2 in Section 2.

     The emission reduction in the old faclity, then, is

                          f N  R
                             o

and the emission increment in the new facility is

                          f N F2 +  (1-f) N F3
                             n             n

     The total emission increment is

          I - f N F2 + (1-f) N F3 - f N R
                 n            no
                            -NR]f
            •  Af
where A denotes the coefficient of f, which is enclosed in brackets  in the
expression above.  The variance S 2 of 1 is given by
                                     51

-------
                     S 2 = A2S 2 + f2S 2 -I- S 2S,2 + N 2S  2
                                               ,
                      I       f       A     A  f     n  Fa




                       2f  [N 2 cov (F2,F3) - N 2S  2 - N N  cov (R,F3)1
                             n                 n  r 3     o n           _)
                                                  F3





where



                     SA2 = N 2S  2 + N 2S  2 + N  2S_
                      A     n  Fa     n  F3     o  R
Case 31 - Same Hardware Provisions for Reducing Emissions in New and Old

          Facilities, Along with Emissions Monitoring and Subsequent

          Corrective Action Performed in Both Facilities
     Define



     R    = average reduction in emissions per source due to the hardware

            modification,



     S 2  = variance of R,
      R


     RI   = average reduction in emissions per source due to maintenance,



     S_ 2 = variance of R , and
      RI


     F2, FS, S^, S^, f, Sf, No,




     and N  are as defined in the preceding section.



     This case corresponds to Case 3 in Section 2.



     The emission reduction in the old facility is



                     N R + f N  R!
                      o       o


and the emission increment in the new facility is




                     f N F2 + (1-f) N F3
                        n            n
                                     52

-------
     I = f NnF2 +  (1-f)  NnF3  - NQR - f i
       = Af + N F3 - N  R
               n      o




where A denotes the coefficient of f which appears  in  brackets above.





           S 2 = A2S  2  + f2S  2  + S. 2S2+ N 2SV  2 +  N 2S  2
            1       f       A     AfnFa     oR




         + 2f [ Nn2 cov  (F2,F3)  - Nn2Sp 2 - N^ cov (RlfF3)





           - NN  cov (R,.F2)+NN  cov (R,F3) + N
           + 2N N   cov  (R,F3)
               on
 Case  41  -  Different Emission Factors Entirely Apply for the New and Old

           Facilities



     The discussion  presented  under  Case  4  in  Section 2 applies directly in

this case whether the paired-measurement  approach  is used or not.
                                     53

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                                 TECHNICAL REPORT DATA
                          (Please read Instructions on the reverse be/ore completing)
1. REPORT NO.
EPA-600/2-78-195
                                3. RECIPIENT'S ACCESSION-NO.
                                    78-200-187-27-09
4. TITLE AND SUBTITLE
Statistical Analysis of Fugitive Emission Change Due
 to Refinery Expansion
                                5. REPORT DATE
                                 September 1978
                                6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Hugh J. Williamson and Lloyd P. Provost
                                                        8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Radian Corporation
P.O. Box 9948
Austin, Texas  78766
                                10. PROGRAM ELEMENT NO.
                                1AB604C
                                11. CONTRACT/GRANT NO.

                                68-02-2608, Task 27
12. SPONSORING AGENCY NAME AND ADDRESS
 EPA, Office of Research and Development
 Industrial Environmental Research Laboratory
 Research Triangle Park, NC 27711
                                13. TYPE OF REPORT AND PERIOD COVERED
                                Task Final: 2-7/78	
                                14. SPONSORING AGENCY CODE
                                  EPA/600/13
15.SUPPLEMENTARY NOTES jERL-RTP project officer is Irvin A. Jefcoat, Mail Drop 62,  919/
541-2547.
16. ABSTRACT
           The report discusses a statistical approach for determining if a planned
 petroleum refinery expansion can be carried out without increasing fugitive emissions.
 The random uncertainty of the empirically determined emission factors is taken into
 account during the determination.  The method presented is designed to handle dif-
 ferent control strategies in the existing and planned facilities and different strategies
 for different types of devices (e.g. ,  pumps and valves) if necessary. It is also pos-
 sible to evaluate different options for reducing emissions to acceptable levels.
17.
                              KEY WORDS AND DOCUMENT ANALYSIS
                 DESCRIPTORS
                                           t>.IDENTIFIERS/OPEN ENDED TERMS
                                             C. COSATI Field/Group
 Pollution
 Processing
 Leakage
 Refineries
 Petroleum Refining
 Petroleum Industry
 Expansion
Random Error
Measurement
Probability Theory
Pollution Control
Stationary Sources
Fugitive Emissions
13 B
13H
14B
131

05C
12A
13. DISTRIBUTION STATEMENT
 Unlimited
                                            19. SECURITY CLASS (This Report)
                                            Unclassified
                                             21. NO. OF PAGES
                                                  60
                    20. SECURITY CLASS (This page;
                    Unclassified
                                             22. PRICE
EPA Form 2220-1 (3-73)

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