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.
-------
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:
-------
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.
-------
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,
-------
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)
-------
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
-------
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
-------
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,
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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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