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by
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M. W
APRIL, l'«I
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DISCLAIMER
This report has been reviewed by the Municipal Environmental Research
Lab, 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.
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. -3
f
ABSTRACT
This research was undertaken to evaluate the adequacy of using a mass
balance technique with daily treatment plant data to determine combined
sewer runoff and overflow characteristics.
An hourly simulator was utilized to generate known runoff and overflow
concentrations as well as plant concentrations, similar to raw treatment
plant data. The daily balance technique was used to analyze the simulated
treatment plant data which provided comparisons of the calculated to the
known runoff and overflow concentrations.
The bias and variability associated with the mass balance technique
together with a theoretical analysis of the plant measurement error effects
are presented. The unit loads and average concentrations from the NYC 26th
Ward Treatment Plant area as well as the effect of rainfall characteristics
on combined sewer runoff concentrations are also presented.
This report was submitted in fulfillment of Grant No. R 806519-01 by
Manhattan College under the sponsorship of the U.. S. Environmental Protection
Agency and covers the project period June 1, 1979 to February 28, 1981.
iii
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TABLE OF CONTENTS
Abstract ill
Figures vi
Tables , ix
Acknowledgement xi
1. Introduction 1
2. Summary and Conclusions 3
3. Recommendations 6
4. Computational Framework 7
Hourly Simulator 7
Daily Mass Balance Technique 11
Flow Weighted Composite Sampling 12
Equal Volume Composite Sampling 13
5. Evaluation of Daily Mass Balance Method 14
Flow Weighted and Equal Volume Plant Sampling Analyses . 14
Effect of Runoff Coefficient 15
Diurnal Sewage Variation 15
6. Hourly Mass Balance Method 22
Derivation of Estimating Equations 22
Performance of Hourly Mass Balance Method 25
Measurement Error 29
Dry Weather Random Variability 33
Wet Weather Variability 33
Conclusion 38
7. Theoretical Analysis of Measurement Error 39
Variability of Overflow Concentrations 39
Variability of Runoff Concentrations 41
Comparisons of Predicted and Simulated Effects of
Measurement Errors 45
Conclusion 52
8. Analysis of Rainfall-Runoff Relationships 53
Simulator Data 53
NYC 26th Ward Data 63
Conclusion 63
9. Analysis of Hourly Sample Collection 67
Bibliography 72
Appendices. 73
I. Estimation of E{l/RD2} 73
II. Theoretical Analysis of Dry Weather Sewage Variability. ... 77
C_ - C Analysis 77
iv
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TABLE OF CONTENTS (cont'd)
C2 - CR Analysis 79
Q2 - CQ Analysis 80
Q2 - CR Analysis 80
Verification 81
III. Analysis of 26th Ward Plant Data 83
Flow Weighted and Equal Volume Analyses 83
Effect of Tidegate Leakage on Mass Balance 85
Hourly Mass Balance Method 89
Rainfall-Runoff Relationships 97
IV. Applicability 100
V. Definition of Symbols , 108
v
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FIGURES
Number Page
1. Hourly Simulator Schematic 10
2. Effect of Minimum Rainfall Analyzed and Interceptor
Capacity on Runoff and Overflow Values using Flow Weighted
Analysis 16
3. Effect of Minimum Rainfall Analyzed and Interceptor
Capacity on Runoff and Overflow Values using Equal Volume
Analysis 16
4. Effect of Assumed Runoff Coefficient on Calculated Runoff
and Overflow Loads - Equal Volume Analysis 17
5. Effect of Assumed Runoff Coefficient on Calculated Runoff
and Overflow Concentrations - Equal Volume Analysis . . 17
4
6. Diurnal Dry Weather Sewage Values for 26th Ward Plant ,
Tuesday, Oct. 12-13, 1976 18
7. Bias of Runoff and Overflow Concentrations for Diurnal and
Constant Sewage Characteristics. . . . 19
8. Variability of Runoff and Overflow Concentrations for
Diurnal and Constant Sewage Characteristics. ...... 19
9. Runoff and Overflow Histograms for Constant and Diurnal
Sewage Characteristics at a . = 4 Hr 21
mm
10. Runoff and Overflow Histograms for Diurnal Sewage Charac-
teristics Using Hourly Constant C Analysis 26
11. Runoff and Overflow Histograms for Diurnal Sewage Charac-
teristics Using Hourly Constant C Analysis for i . =
0.03 in/hr . . ?....... ?in . . 26
12. Runoff and Overflow Histograms for Constant C Analysis
with. Measurement Error for i . =0.03 in/hr 31
min
13. Runoff and Overflow Histograms for Constant C Analysis
with Measurement Error for i . = .03 in/hr. ...... 31
min
14. Runoff and Overflow Histograms for Variability on Dry
Weather Sewage Concentration 34
15. Runoff and Overflow Histograms for Variability on Dry
Weather Sewage Flow Rate 34
vi
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FIGURES (cont'd)
Number
16. Effect of Number of Samples and Rainfall Intensity on
Runoff and Overflow Variability Parameters ........ 44
17. Runoff, Variability Factors for Measurement Error
(STD = 10 mg/£) ...................... 46
18. Daily Averaging Error for Constant Rainfall and Variable
Sewage Characteristics During an Event .......... 47
19. Runoff Variability Factors for Combined Measurement
(STD = 10 mg/5.) and Daily Averaging Errors ........ 48
20. Overflow Variability Factors for Combined Measurement
(STD = 10 mg/&) and Daily Averaging Errors ........ 49
21. Runoff Variability for Varying Runoff Characteristics. . . 50
22. Overflow Variability for Varying Runoff Characteristics. . 51
23. Runoff Concentration - Storm Interval Regressions for
Strong Interval Effect .................. 55
24. Probability Distribution of Runoff Residuals for a 10 mg/&
Measurement Error for Strong Interval Effect ....... 56
25. Runoff Residuals Histograms for a 10 mg/£ Measurement
Error for Strong Interval Effect ............. 56
26. Effect of Length of Record on Regression Parameters for
a 15 mg/& Measurement Error for Strong Interval Effect . . 57
27 . Effect of Length of Record on Regression Parameters for
a 10 mg/£ Measurement Error for Weak Interval Effect ... 58
28. Effect of Measurement Error on Runoff Concentration -
Interval Correlation Coefficients .......... ... 59
29. Effect of Length of Record on Interval Regression Para-
meters for a Significant First Flush ......... . . 61
30. Effect on Length of Record on Duration Regression Para-
meters for a Significant First Flush ..... ...... 61
31. Effect of Minimum Duration on Runoff Concentration - Dura-
tion Regression Parameters for a Significant First Flush . 62
32. Effect of Minimum Duration on Runoff Concentration- Inter-
val Regression Parameters for a Significant First Flush. . 62
33. Effect of Minimum Duration Analyzed on Runoff-Duration
Regression Parameters for NYC 26th Ward Data ....... 64
34. Duration Regression Analysis for 26th Ward Suspended
Solids Data. ....................... 65
vii
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FIGURES (Cont'd)
Number Page
35. Interval Regression Analysis for 26th Ward Suspended
Solids Data 65
36. Effect of Minimum Storm Duration on Error Magnification
Factors for Hourly Plant Sampling 70
37. Duration Regression Parameters Using Hourly Plant Sampling
Technique 71
38. Interval Regression Parameters Using Hourly Plant Sampling
Technique 71
A-l. Effect of Minimum Storm Duration on Runoff Suspended Solids
Histograms for 1957-26th Ward Data Using Equal Volume
Analysis 87
A-2. Effect of Minimum Rainfall Intensities and Durations on
Runoff Suspended Solids Histograms for 1969 - 26th Ward
Hourly Analysis , . 90
A-3. Yearly Runoff Suspended Solids Concentrations for the
26th Ward Data Using the Hourly Analysis 94
A-4. Yearly Runoff BOD Concentrations for the 26th Ward Data
Using the Hourly Analysis 94
A-5. Yearly Sewage Concentrations for the 26th Ward Data .... 95
A-6. Percentage of Population in each State Served by Combined
Sewers 102
A-7. Number of Urban Areas in each State Having More than 30%
of the Population Served by Combined Sewers . 102
viii
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TABLES
Number Page
1. Daily and Hourly Mass Balance Results for Diurnal Sewage
2.
3.
4.
5.
6.
7.
8.
A-l.
A-2.
A-3.
A-4.
A-5.
A-6.
A-7.
A-8,
A-9.
Variability of Laboratory Analyses
Measurement Error Results
Dry Weather Random Variability Results
Runoff and Overflow Variabilities for Varying Runoff Character-
istics
Effect of Neglecting Short Storms on Runoff and Overflow Concen-
trations for Significant First Flush
Comparison of Theoretical and Actual Statistical Parameter for
Comparison of Theoretical and Actual Statistical Parameters for
Runoff Measurement Error. .... .......
Duration and Sampling Probabilities
Overall Probabilities for RD Values
Effect of RD . on E {-=T}
mm Z
Runoff and Overflow Variability Due to Dry Weather Random Varia-
bility
Flow Weighted and Equal Volume Analysis for 1957 - 26th Ward
Data. . .
Equal Volume Analysis for 1957 - 26th Ward Data, Alpha Minimum
=; 0
Effect of Tidegate Leakage on Volumes and Yearly Loads from
Effect of Tidegate Leakage on Runoff Concentrations from Equal
Comparison of Runoff and Overflow Concentrations Using One and
28
30
32
35
37
38
41
43
75
76
76
82
84
86
86
88
Two Rainfall Stations for 1969. - 26th Ward Data with Hourly
Analysis 91
A-10. Yearly Average Runoff and Overflow Concentrations Using the
Hourly Analysis 92
ix
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TABLES (cont'd)
Number Page
A-11. Comparison of Average Yearly Unit Loads by Daily and Hourly
Mass Balance Methods for 26th Ward Data 96
A-12. Weighted Average Concentrations for 26th Ward Data Over
Total Study Using Hourly Analysis 97
A-13. Rainfall-Runoff Relationships for Four Parameters Analyzed
at 26th Ward Using Separate Linear Regressions for Duration
and Interval 98
A-14. Rainfall-Runoff Relationship for Four Parameters Analyzed at
26th Ward Using Multiple Linear Regression for Duration and
Interval 99
'X.
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ACKNOWLEDGEMENTS
The capable participation of Edward J. Garland as Research Assistant
in the study is acknowledged as well as the initial work of Charles R.
Dabrusco. Special thanks are due to Mrs. Eileen Lutomski and Mrs. Margaret
Cafarella for their typing of the report manuscript. The interest and coop-
eration of Richard Field, Chief Storm and Combined Sewer Section, and
Douglas Ammon, Project Officer, are gratefully acknowledged.
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SECTION 1
INTRODUCTION
The assessment of the magnitude and characteristics of urban runoff
loads for specific regions is a difficult task due to the random nature of
storm events. Techniques typically used to evaluate urban runoff inputs in-
clude (1) direct sampling of storm overflow concentrations and flows or (2)
use of a stormwater quality model based on land use and rainfall characteris-
tics. Due to the highly variable nature of rainfall and associated runoff
phenomena, an extensive sampling program is generally necessary with the
first technique in order to provide accurate estimates of overflow loads, a
costly and time-consuming undertaking. The latter method, if based on de-
fault values incorporated in the models, may lead to significant errors. To
obtain reliability in the latter approach, the models must be calibrated for
specific areas, normally by direct sampling of stormwater overflows. A third
possibility is to use existing data bases namely, treatment plant influent
data, for determination of combined sewer overflow loads. This should pro-
vide municipalities with an alternate method to rapidly and economically
assess the importance of their combined sewer overflows when formulating
water quality management plans.
The objective of this research was to evaluate the adequacy of a mass
balance technique using treatment plant influent data to determine the magni-
tude of combined sewer runoff and overflow loads. The initial concept of
(1-3V
using treatment plant data to obtain these loads was developed to eval-
uate the relative importance of urban runoff inputs to New York Bight. The
mass balance method is a mathematical framework consisting of mass and flow
balances for the sewer system and regulators over the total drainage area
served by a treatment plant. Inputs to the sewer system include the dry
weather sewage flow, runoff into the combined sewer system during storm
events, and tidegate leakage. Outputs from the system include the wastewater
flow to the treatment plant and the combined sewer overflow from the' regula-
tors to the receiving waters. During the overflow event, the quality of the
overflow from the regulators is assumed equal to that of the treatment plant
influent. Since daily treatment plant data is normally available, hourly
mass and flow balance equations are integrated over the sampling day to pro-
vide estimates of the temporal and areal average daily overflow and runoff
concentrations. The runoff concentration includes the contribution from both
surface runoff and interceptor scour. The initial study utilized data from
the 26th Ward plant in New York City. A large degree of variability in the
daily runoff and overflow concentrations resulted, values over a number of
years used to characterize loads from the drainage area. Partial verifica-
tion was obtained by comparison to existing combined sewer sampling data from
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portions of the area.
The present study was conducted to determine the bias and variability
associated with the technique and evaluate modifications required to provide
maximum accuracy for the available data base. The approach taken was to
develop an hourly simulator in which all influent characteristics, both dry
weather sewage and runoff, were known. The daily composite simulator output
was analyzed by the mass balance technique and compared to the known inputs.
This comparison served as the basis for modifying the computational technique.
Two modifications were developed: one employing equal volume plant sampling
similar to the N.Y.C. sampling technique, and the other employing real time
which allows rainfall events to be. correlated with dr.y weather sewage diurnal
variability. The effects of errors in the estimation of dry weather sewage
characteristics and runoff volumes were evaluated along with the effect of
plant concentration measurement error. The ability of the technique to ex-
tract the effects of rainfall characteristics, interval between storms and
storm duration, on runoff loads was studied for both the New York City sam-
pling routine (every 4 hours skipping the 2 AM sample) and an hourly sampling
routine.
The modified computational techniques were utilized on the existing 26th
Ward data from N.Y.C. to evaluate the impact of the improved methodology on
the runoff and overflow load estimates. A literature review and letter sur-
vey were also conducted to evaluate the nationwide applicability of the
methodology.
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SECTION 2
SUMMARY AND CONCLUSIONS
The ability of a mass balance technique using treatment plant influent
data to accurately determine the overflow loads and runoff characteristics
from combined sewers was evaluated using an hourly simulator to generate
known runoff, overflow and plant influent concentrations. The plant influent
data generated by the simulator were analyzed by the daily mass balance tech-
nique to determine concentrations which were compared to the true values gen-
erated above. This provided the basis for analyzing the bias and variability
associated with the technique.
The initial results showed that a significant bias existed when inter-
ceptor capacity was greater than dry weather flow if a flow weighted analysis
of influent data was used on plant composite samples collected in equal vol-
umes. The bias was removed by modifying the technique to an equal volume
analysis of the plant composite samples. Lastly, variability due to the
averaging technique was minimized by using the hourly dry weather concentra-
tion coinciding with the time of the storm.
The variability in the calculated runoff and overflow concentrations due
to plant measurement error is significant. A theoretical analysis of the
error structure indicated that the variability of the runoff estimates was
greater than the overflow estimates. The variability in the individual con-
centrations could be reduced by deleting low average storm intensities (<0.03
in/hr) and low storm durations which provided only one wet sample at the
plant. However, excluding lower duration storms from the mass balance analy-
sis reduced the capability of extracting first flush effects from the data.
Random variability in hourly dry weather sewage concentrations using standard
deviations of 10 and 20% on the hourly values was found to be significant but
somewhat lower than that due to the measurement error. A summation of the
variance of each of the individual errors provided an excellent estimate of
the total variance of the estimated runoff and overflow concentrations.
The ability of the mass balance technique to analyze for the effect of
rainfall characteristics on runoff concentrations when both averaging and
measurement errors were present was evaluated for the New York City sampling
mode by linear regression analysis. The actual effects of both interval and
duration on the storm average runoff concentrations provided by the simulator
were successfully obtained from an analysis of the daily plant data. Approx-
imately 150 to 200 days of data are required to insure the confidence limits
on the interval effect, as measured by the slope of the regression curve, are
above zero when the runoff concentrations are significantly affected by a
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first flush. The correlation coefficients obtained from these regressions
are low, explaining only 3 to 14% of the observed variability. The remainder'
of the variability of these simulator data is due to the averaging and meas-
urement errors inherent in the analysis and not random variability of runoff
concentrations. Thus the mass balance technique is capable of accurately
predicting effects of duration and interval on storm weighted average runoff
concentrations.
In the simulated runoff data, the first flush effect was limited to the
first hours of the storm events with background levels attained after three
to four hours. Therefore, when short storms were neglected in the analysis,
lower runoff concentrations resulted with regression parameters similarly
reduced. Thus to properly evaluate the first flush effects on runoff char-
acteristics, short duration storms had to be included in the analysis.
Collecting samples every hour instead of the NYC sampling routine of
every 4 hours (skipping the 2 AM sample) caused a higher degree of variabil-
ity in the results especially when short duration storms were analyzed.
This is due to the fact that with durations of 1 or 2 hours, less than 10%
of the collected samples reflect wet weather conditions. Analyzing dura-
tions only equal to or greater than 4 hours for the hourly sampling routine
provided results similar to analysis of all plant data sampled by the NYC
routine as long as a runoff event occurred during a plant sampling time.
Thus a plant sampling routine based on hourly sampling reduces the capability
of evaluating runoff and overflow characteristics from plant data.
The actual daily data from the 26th Ward Plant in New York City were
then analyzed using the hourly mass balance technique. Unit loads were simi-
lar to those for the previous flow weighted daily mass balance analysis with
the exception of the soluble BOD. data, which was significantly lower than
previously estimated. For these estimates, the hourly variability in dry
weather concentrations for all four parameters: (SS, VSS, BOD,., soluble
BOD,.) was taken from the BOD- variability. Interval and duration signifi-
cantly affected runoff concentrations. For the 26th Ward data, similar first
flush effects were obtained when both 1 and 2 hour minimum duration storms
were analyzed, with a higher correlation coefficient for the latter. Plant
data analysis using a minimum storm duration of two hours and minimum average
intensity of 0.03 in/hr provided the best estimate of average runoff and
overflow concentrations as well as the effects of storm characteristics on
runoff concentration.
The following conclusions have been drawn from the study.
1. Average annual runoff and overflow loads and concentrations can be
obtained using long term influent data from treatment plants with
combined sewer systems by using a mass balance analysis.
2. To remove bias from the analysis, the original flow weighted mass bal-
ance technique must be modified to reflect the type of composite sam-
pling being conducted at the plant. For the New York City plant sam-
pling routine, an equation based on equal sample volumes is required
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for the measured plant concentrations,
i
3. Individual estimates of daily runoff and overflow concentrations have a
high degree of variability due to subtractions inherent in the mass bal-
ance technique. Therefore long data bases are required to provide good
estimates of average loads.
4. An hourly mass balance technique using dry weather hourly sewage concen-
trations and flows with hourly rainfall intensities reduces averaging
errors inherent in the daily analysis. However, for evaluation of long
term BOD , suspended solids and volatile suspended solids loads for the
the 26th Ward Plant in New York City, the extra complexity of the hourly
analysis was not justified since average loads were not significantly
different. This was not the case for soluble BOD,, which was signifi-
cantly lower for the hourly analysis. In the hourly analysis of the
26th Ward data, the diurnal variability of all dry weather constituents
was assumed similar to that for BOD_ for which data was available.
5. Measurement error associated "with plant concentrations causes a major
portion of the variability in estimated runoff and overflow concentra-
tions using the hourly analysis. Other causes of this variability are
hourly dry weather sewage concentration variability and within storm
variable hourly runoff concentrations.
6. Runoff concentrations can be related to rainfall characteristics reli-
ably if a sufficient length of record is analyzed, the hourly analysis
providing greater reliability than the daily analysis. The manner of
sample collection and compositing significantly affects the length of
record required. For example, hourly sampling for the daily plant com-
posite requires approximately 400 days of data while sampling at 4 hour
intervals would require approximately 150 days data.
7. Use of the mass balance technique to obtain drainage area integrated
runoff and overflow concentrations from plant influent data should pro-
vide significant costs saving when laboratory analytical costs are high
as in the case of the toxics.
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SECTION 3
RECOMMENDATIONS
With the ability to obtain accurate estimates of combined sewer runoff
and overflow characteristics from plant data, plant influent sampling for
the toxics should be initiated to define the combined sewer contributions of
these loads to receiving waters and to treatment plants. The manner of sam-
ple collection, compositing and analysis should be optimized to minimize
costs of laboratory analyses. Utilization of suspended solids with a toxics-
suspended solids correlation may be the most cost effective approach.
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SECTION 4
COMPUTATIONAL FRAMEWORK
Two mathematical models were used in the study. The first was an hourly
simulator to develop daily composite treatment plant, runoff and overflow
loads using hourly rainfall input data. The second was the daily mass bal-
ance technique which analyzed the above treatment plant data. This section
describes the characteristics of both models along with the modifications to
the daily balance technique to reduce bias and variability associated with
the methodology.
HOURLY SIMULATOR
The adequacy of the daily balance technique used in the previous work
(1-3)
on the 26th Ward treatment plant was evaluated by comparing the esti-
mates to measured average overflow characteristics from field studies of
Jamaica Bay. The spread of the average field values from the two studies was
large, a factor of approximately 3 to 1. Thus no absolute runoff and over-
flow concentrations existed to accurately evaluate the accuracy of the daily
mass balance technique. The hourly simulator was developed to fill this gap.
To develop an efficient hourly simulator which did not require a signif-
icant amount of raw data handling, two modifications of the previous balance
programs were utilized. The first was that tidal inflow was not included in
the analysis so a chloride balance was not required. The second, and most
time saving, was the calculation of hourly rainfall volumes using internally
generated characteristics of storm average rainfall intensity, duration, and
interval between storms. It has been found , that the intensity, i, dura-
tion, d, and time between storms, &, are essentially independent, serially
uncorrelated and exponentially distributed. Thus the probability density
functions for these random variables are:
P±(t) - e
where I,D, and A, are the average intensity, duration, and time between
storms, respectively. In order to generate exponentially distributed random
variables, consider a uniformly distributed random variable, x:
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P(x)=l o i
Uniformly distributed random variables are directly available from internal
subroutines that are part of most programming languages (the RND function) ,
and the logarithmic transformation converts them to exponentially distributed
random variables with the appropriate mean. Similiar equations are used for
duration and time between storms. This provided storm average rainfall char-
acteristics which were exponentially distributed, similiar to actual distri-
butions. These are then converted to hourly sequences.
Hourly fluctuations of rainfall intensity within each event were obtained
using a random number generator with a zero mean and a specified variance, so
that
ijk = ^ + £ik' £ik = NCO> \2)
where k and j refer to the hour and day, respectively. Fluctuations on the
hourly intensities providing standard deviations (a) of 0 and 100% of the
hourly values were utilized with a constraint that the minimum value of i,, =
JK
0.01 in/hr. The hourly runoff flow rate (Q- ., ) was obtained from the hourly
rainfall values using the rational method:
Qljk = Cijk
where C combines the runoff coefficient, drainage area and unit conversion
factor. Using rainfall values in units of hundredths in/hr., a drainage
area of 5,000 acres and a runoff coefficient of 0.7 similar to the 26th Ward
data, a C value of 0.95 provides runoff flow rates in units of MG/hr.
The runoff concentration, C.. , was varied deterministically as a function
of both interval between storms and storm duration as follows:
Cljk - C10
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A linear increase of runoff concentration with interval between storms,
S, was utilized similar to the previous correlations obtained with the 26th
Ward data . An exponentially decreasing effect of storm duration, d, on
runoff concentration provided a strong first flush effect.
Hourly values of dry weather sewage concentration and flow rate from 26th
Ward plant data provided a significant diurnal fluctuation in the hourly
sewage characteristics. In addition to the known diurnal fluctuation, random
variability was assigned to the hourly values as follows:
C2jk = C2k + £C2k; £C2k
Q2jk - ^2k + £Q2k; £Q2k
Fluctuations on the hourly values which provided standard deviations (a)
of 0, 10 and 20% of the hourly values were utilized.
The above parameters served as input data to a regulator which propor-
tioned flow to the treatment plant influent and to the overflow as a function
of the plant interceptor capacity, QI, as shown in Figure 1. A constant value
of interceptor capacity typically equal to 2.5 times the average dry weather
flow rate was used in the simulator. During an event, the overflow concen-
tration was assumed equal to the plant concentration. The latter value was
calculated each hour from a mass balance on runoff and sewage loads. No
storage capacity of volume or load was assumed to exist in the interceptor
and regulator.
The plant influent concentrations were then sampled according to a
specified sampling routine and composited to provide the daily plant concen-
tration. Two sampling routines with equal volume compositing were used in
the analysis, the one historically used by NYC, every 4 hours but skipping
the 2:00 A.M. sample as well as an hourly sampling routine. To account for
plant measurement error, the daily plant concentration was varied randomly
to provide the measured plant concentration as follows:
Standard deviations of measurement error in the range of 0 to 20% of the
actual concentrations were utilized.
The daily loads and volumes at each location in the flow diagram were
calculated by taking the sum of the hourly values over the day and the flow
weighted average concentrations calculated as seen in Fig. 1. A summation
of the total wet hours (a) occurring each day was also made. Output then
consisted of the daily volumes and average flow weighted concentrations at
each location, the measured plant concentration, the hourly runoff flow rates,
along with the hours of rainfall and day of the year.
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0
_150
RAIN- d
«" &«J_
fl
a
' 0
450
RUNOFF
o>
SEWAGE
en
5
8 am 8 am
HOURLY INPUT
Q1+Q2>Q,
REGULATOR
Q2C2
CL
24
/ =1
24
/= 1
OVER-
FLOW
PLANT
SAMPLE
HOUR
CP
8 am 8 am
HOURLY OUTPUT
W
V
DAILY OUTPUT
DAY No.: a, Vv V2, V3, V4,
CP, C^, C2, Cg, €4
Q,/ values
Figure 1, Hourly Simulator Schematic
10
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All simulator runs were developed,in BASIC and conducted on a TRS-80
microcomputer. The simulator studies were initially conducted with constant
runoff and dry weather sewage characteristics. As the evaluation of the
daily mass balance technique progressed, the degree of complexity of the
input data was increased.
DAILY MASS BALANCE TECHNIQUE
The analysis of influent treatment plant data relies on flow and mass
balance equations. The derivation is given in detail since they form the
basis for all results in this report. The schematic is given in Fig. 1 which
defines surface runoff flow, Q. , and concentration, C.. ; sewage flow, 0~, and
concentration, C,.,; overflow flow Q , and concentration, C • and finally
treatment plant flow, Q,, and concentration, C,. The time scale of the analy-
sis is one day. Wet periods during the day (during rainfall) are subscripted
by "w" and the length of rainfall is t . Dry periods, subscripted "d", have
length t,.
The flow balance equations are
Dry:
Wet:
0+0=0 Q+Q«
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occurs. Then eqs. (5) and (6) become:
C3w(V)
and using the flow balance eqs. (2) and (3) yields:
independent of whether the system is overflowing or not. This equation forms
the basis of the daily average analysis. Two different models are obtained
depending on the sampling routine followed at the treatment plant.
Flow Weighted Composite Sampling
If the treatment plant sample is a flow-weighted daily composite, then
the reported plant concentration is:
n Q4dC4dtd + Q4wC4wtw ,.nN
CP " Q, .t. + Q, t C10)
X4d d 4w w
Define wet and dry period volumes as, V, = Q, t , V_, = QOJtj, V0 = Q0 t ,
* v Iw xlw w' 2d X2d d* 2w X2w w'
etc. Then using eqs. (1) and (4), and these definitions, eq. (10) becomes:
If the total daily volumes are defined as V0 = V0, + V0 , and V, = V, , + V,
2 Zd 2w 4 4d A-w
then:
V2d = V2 - V2w = V2C1 - a) (12)
where a = V_ /V~, the fraction of total sewage volume which corresponds to
wet periods. For constant within day sewage flow a is the wet fraction of
the day. Further,
V4w = V4 ~ V4d
= V - V
V4 V2d
= V4 - V2C1 - a) C13)
12
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, ' t Hence the overflow concentration, C^, is, from eq. (11):
i
cpv4 - d-g)v2c2d
0 3w V. - (l-a)V0
4.2
The runoff concentration, C , follows from the wet weather mass balance
K.
eq. (9) applied over the wet period, t :
Sw-lw'w * 'WWv = C3w(Qlw + Q2w}tw C15)
C., V, + C_ a V0 = C_ (Vn + aV0) C16)
Iw Iw 2w 2 3w Iw 2
so that:
aV
C_ = C- = C. + =-£ (C. - C0 ) (17)
R Iw 3w Vn 3w 2w
Iw
These equations (14 and 17) form the basis of the mass balance analysis
reported previously (1,2).
Equal Volume Composite Sampling
If the treatment plant sampling is not flow weighted, then the defining
equation for C is not equation (10), but:
Cp = a - a) C2d + a C3w CIS)
where a is the fraction of time corresponding to wet weather. Thus,
C - (1 - a) C»
c = c = _£ £l (19)
0 3w a
gives the overflow concentration and equation (17) , the runoff concentration
as before.
13
-------
SECTION 5
EVALUATION OF DAILY MASS BALANCE METHOD
FLOW WEIGHTED AND EQUAL VOLUME PLANT SAMPLING ANALYSES
The original flow weighted mass balance analysis was initially tested
using the simulator output. All analyses were made with constant values of
dry weather sewage flow and concentration (C? = 100 mg/fc, Q? = 2 MG/hr)
and constant runoff concentration and volumetric runoff coefficient (C- =
50 mg/£, C = 0.7). The New York City plant sampling procedure (10 AM, 2PM,
6PM, 10PM, 6AM) was also utilized.
Two comparisons were conducted: one with the interceptor capacity Q ,
equal to 5 MG/hr (Q = 2.SO-) and the other with a capacity of 2 MG/hr (Q
= Q-). For each comparison, the amount of wet weather data analyzed was
varied by specifying a . , and analyzing those data for which a >_ a . . The
results are summarized in Figure 2 which compares the ratio of the average
runoff and overflow concentrations to the known concentrations. When the
plant capacity is significantly greater than the dry weather flow, errors
of 30 to 40% result when the majority of the rainfall is analyzed (a . <
min ~
4 hours). This occurs due to the non-flow weighted plant sampling. During
high intensity storms the concentration at the plant would be lower during
the wet period than during the dry period while the flow is higher due to
the larger interceptor capacity.
Since New York City composites samples according to equal volume incre-
ments independent of flow, too great a weight is proportioned to the dry
weather conditions at the lower flows. When only these larger duration
storms are analyzed this proportioning error is reduced with no error re-
sulting for storms lasting 24 hours, since all concentrations are constant
over the day. Of course only a small proportion of the total rainfall is
analyzed at the larger a . thus neglecting the major portion of the data.
mm
When the interceptor capacity is equal to the dry weather flow, plant
flow is constant for both wet and dry days and no flow proportioning error
results. The overall error in the analysis is therefore significantly re-
duced to a maximum of about 12% at a . of 8 to 12 hours. When the majority
mm
of the data is analyzed, at a . <_ 4 hours, the error is small, + 3-5%.
14
-------
Underestimation of the runoff and overflow loads at the lower storm duration
^(some negative values result) help to balance out the overestimation of the
larger storms.
The results using the equal sample volume mass balance analysis to cal-
culate the runoff and overflow concentrations are shown in Figure 3. The
previous large error for the higher capacity plants has been removed with
interceptor capacity having no effect on runoff concentration and only slight
effect on overflow concentrations. This modification adds no increased com-
plexity to the daily mass balance technique and was therefore used for the
remainder of the analyses using the New York City plant sampling routine.
Depending on the actual method of plant compositing, either technique can be
employed to analyze plant data.
EFFECT OF RUNOFF COEFFICIENT
Using the equal sample volume mass balance technique, the effect of run-
off coefficient, C , on runoff and overflow loads and concentrations was
evaluated. The actual C value used in the hourly simulator was 0.7. It
was varied from 0.3 to 0.9 in the daily mass balance analysis. The loads are
seen to vary significantly with the C coefficient as indicated in Figure 4
while runoff and overflow concentrations (Figure 5) vary only slightly for
C from 0.5 to 0.9. The lowest C coefficient of 0.3 resulted in numerous
negative overflow volumes, an indication that the estimated runoff coeffi-
cient is unrealistically low for this test case. The above analysis confirms
the results previously obtained using the 26th Ward data: the daily mass bal-
ance technique provides good estimates of runoff and overflow quality inde-
pendent of the quantity estimation. The latter is still required for a good
estimate of the load.
DIURNAL SEWAGE VARIATION
The hourly sewage flow and BOD,, concentrations during dry weather from
the 26th Ward Plant (.Oct. 12 & 13, 1976, Figure 6) were used in the hourly
simulator. Random rainfall was again utilized and average daily treatment
plant sample concentrations were generated. To maintain consistency with
the previous analysis using a constant concentration of 100 mg/&, the con-
centrations were adjusted slightly (+5%) to yield a flow weighted average
concentration over the day of 100 mg/£. The daily runoff and overflow con-
centrations were calculated from the daily mass balance technique using a
constant average sewage concentration and flow. Figure 7 indicates that no
consistent bias in the runoff or overflow concentrations was introduced to
the analysis as a function of rainfall duration analyzed a . . Bias is the
average difference for the year of record analyzed between the actual and
estimated daily concentrations. For storms with durations >_ 2 and 4 hours,
the bias introduced in the daily mass balance analysis was greater than the
bias using a constant sewage input but lower when all storms (a . =0) and
15
-------
60
-i o 40
< LU
u. N
2 >-
<<2Q
<
I
J-
0
1.30
1.20
§ I 1.10
< £
0 8
o 1.00
o
OC
0.90
SEWAGE, C2= WOmg/L
RUNOFF, C, = 50 mg/L
RUNOFF"
OVERFLOW
DESIRED
\
I
0 4 8 12 16 20
ALPHA MINIMUM, hr
24
Figure 2. Effect of Minimum Rainfall Analyzed
and Interceptor Capacity on Runoff
and Overflow Values using Flow
Weighted Analysis
1.40
RUNOFF
OVERFLOW
DC
8 12 16 20
ALPHAMINIMUM.hr
24
Figure 3. Effect of Minimum Rainfall Analyzed
and Interceptor Capacity on Runoff
and Overflow Values using Equal Vol-
ume Analysis
-------
600
poo
ro
-Q
Q 400
O
LU
oc
III
_j
oc
300
200
100
TRUE LOAD.
TRUE Cv= 0.7
Qt = 2.5Q2
OVERFLOW
/ NUMEROUS NEGA TIVE
OVERFLOW VOLUMES
I I
0.2 0.4 0.6 0.8
RUNOFF COEFFICIENT (Cv)
1.0
60
o>
£
-=? 50
40
UJ
O
z
O
O
LU
O
LU
DC
<
LU
30
20
10
0
OVERFLOW
1
NUMEROUS NEGA 77 VE
OVERFLOW VOLUMES
I
JL
0.2 0.4 0.6 0.8
RUNOFF COEFFICIENT (C)
1.0
Figure U. Effect of Assumed Runoff Coefficient
on Calculated Runoff and Overflow
Loads - Equal Volume Analysis
Figure 5- Effect of Assumed Runoff Coefficient
on Calculated Runoff and Overflow
Concentrations - Equal Volume Analysis
-------
160
oo
Q
26th WARD DATA
TUES., WED., OCT 12-13, 1976
8:30 12:30 16:30 20:30 00:30
TIME, hr
4:30 8:30
Figure 6. Diurnal Dry Weather Sewage Values for 26th Ward Plant ,
Tuesday, Oct. 12-13, 1976
-------
VARIABLE SEWAGE
VARIABLE SEWAGE
0
4 6 8 10
ALPHA MINIMUM, hr
2.0
1.5 -
1.0
O
H
<0.5
DC
>
U-
O o
UJ
O
- \
u.
u.
UJ
o
o
1.5
1.0
0.5
0
RUNOFF
VARIABLE SEWAGE
CONSTANTA— -— .^^ "
SEWAGE ""
OVERFLOW
VARIABLE SEWAGE
CONSTANT"
SEWAGE
I
I
I
1
I
0
4 6 8 10
ALPHA MINIMUM, hr
12
Figure 7. Bias of Runoff and Overflow Concen-
trations for Diurnal and Constant
Sewage Characteristics
Figure 8. Variability of Runoff and Overflow
Concentrations for Diurnal and Con-
stant Sewage Characteristics
-------
those with durations _>_ 8 hours were analyzed.
Although no significant bias was introduced using diurnal dry weather
concentrations, the variability of the results was significantly increased
as shown in Figure 8. For both runoff and overflow concentrations the
diurnal sewage characteristics caused the coefficient of variation of cal-
culated versus actual values to be 40 to 50% greater than those using con-
stant sewage characteristics. The longer the storm duration analyzed, the
lower the variability.
Figure 9 shows typical histograms for the above results at a . =4
nun
hours with the actual compared to the calculated values. The diurnal sewage
histograms show a significantly greater number of errant values at both the
negative and positive ends of the histogram than the constant sewage charac-
teristics. Using greater rainfall intensities and durations with diurnal
sewage characteristics gave similar variabilities as those above. This in-
creased variability is inherent in an analysis that ignores the actual time
of day and partitions the day into only wet and dry periods without regard
to the actual times that it rained. Since this information is directly
available from the rainfall record it can be incorporated within the mass
balance framework at the expense of some complexity of the resulting formu-
las. As shown in the next chapter, this refinement significantly improves
the performance of the mass balance analysis.
20
-------
o 40
2 30
HI
0 20
2
P 1°
GC
•^ n
Constan
RUNOFF t 57
t C2, V2
OVERFLOW
p ACTUAL
P CALCULATED
-
-
i r
-n i
I I I
T,
g -40 0 40 80120 -40 0 40 80120
8 40
I
w 30
£ 20
0
GC
S 10
z
n
RUNOFF t 57
-
-
—
-^TTf
™"
"|-
Diurnal
_
I
C2,V2
OVERFLOW
•
r~T
i
i
, , .rf
__
lr
40 0 40 80120 -40 0 40 80120
CONCENTRATION, mg/L
Figure 9. Runoff and Overflow Histograms for Constant and Diurnal
Sewage Characteristics at a . = 4 hr.
mm
21
-------
SECTION 6
HOURLY MASS BALANCE METHOD
DERIVATION OF ESTIMATING EQUATIONS
The estimating equations for runoff and overflow concentrations follow
from flow and mass balance equations applied to each hour during the day.
Letting i denote the hour of concern the flow balance and mass balance equa-
tions are:
Dry Hour:
Q2i - Q4i (20)
(21)
Wet Hour:
Qii + Q2i = V Qii + Q2i ^ Qi C22)
Cl Ai + C2iQ2i - C4 Ai Qii + Q2i ± Q! (23)
Qii + Q2i ' ^3i + Q! Qll + Q2i > Q! C24)
CliQli + C2iQ2i = C3iQ3i + C4iQI Qli + Q2i > Q! (25)
The assumption: C, . = C-. yields the wet hour mass balance equation, analo-
gous to equation (9); that is, equation (25) becomes:
cQ + GQ • c(Q + Q> (26)
The dry hour mass balance equation (21) similarly becomes:
C2iQ2. = C3iQ2. (27)
so that when Q is zero, equation (26) is equivalent to equation (27).
Thus it applies to all hours, wet or dry.
Constant Runoff Approximation
Since the application of these methods is for New York City data, only
equal volume treatment plant sampling will be investigated. Consider a
22
-------
variable, s., which is equal to one if a sample is taken at the i hour, and
'zero otherwise. The reported daily treatment plant concentration is:
C = E s.C / Z s, (28)
P i A 4i ± i
For NYC sampling, s. = 1 for the 5 hours that samples are taken (10 AM, 2 PM,
6 PM, 10 PM, 6 AM) and E± s± = 5.
Using the mass balance equation (26) yields:
SiCliQli SiC2iQ2i
CP = ( 2 Q +n + E o S > / E Si (29)
P i Qli+Q2i i Qli+Q2i i X
This equation cannot be solved for the hourly runoff concentrations since
there are 24 unknowns, C.. . , and one equation. The assumption that suggests
itself is that the runoff concentration is constant: C, . = C.. = C . For
J.i J. K
this situation, equation (29) can be solved for C :
K
z
i
The flow weighted overflow concentration is defined to be:
*. <>i<¥3i
c° ' t^T
where o = 1 if Q + Q_. > Q indicating an overflow occurs at hour i, and
zero otherwise. The overflow concentrations follow from the mass balance:
C3i
and the flows from the flow balance:
Q3i = Qli + Q2i ~ QI Q1± + Q2i > Q.J. (33)
Q3 . = 0 otherwise,
23
-------
The daily average interceptor capacity can be estimated from the difference
of sewage plus runoff flow and the recorded volume of influent to the treat-
ment plant. From equations (22) and (24), the wet weather flow balance equa
tions;
CQ
li
2i
' V
(34)
where V, is the reported daily volume treated at the plant. Expressing this
equation in terms of hours for which overflows occurred (o . =1) and did not
occur (o . = 0 and 1 - o. =1) yields:
Q2.)
- Q3i) =
(35)
and using equation (33) for overflow hours yields:
d - o) (Q
1±
Q2j.)
(36)
or:
QI = [V4 "
C37)
The solution technique for QT is iterative. An initial daily average
interceptor capacity is chosen (say Q = 0) . Then equation (33) is applied
to each hour, which established Q». and o. = 1 for Q.,. > 0 and o. = 0 for
Q_. = 0. These are used in equation (37) to compute a new estimate of X} .
The cycle is repeated until QT converges. A maximum of three iterations has
been required when analyzing simulator output.
Constant Overflow Approximation
An alternative to the assumption that runoff concentration is constant,
is that overflow concentration is constant: CL. = CQ. The treatment plant
sample concentration can be expressed as:
Cp =
(38)
.
where E is the sum over all wet hours (Q > 0) and E is the sum over all
i i
24
-------
dry hours (Q. . = 0). For the wet hours, the treatment plant concentration is
the overflow concentration C, . = C_.; for the dry hours it is the sewage flow:
C, . = C^.' Assuming that the overflow concentration is constant yields from
equation (38) :
C0 - (39)
This formula is valid only if at least one sampling time corresponds to a
wet hour and Z s . > 0 .
Since runoff concentration is also required, it is convenient to compute
the flow weighted average runoff concentration:
CR = f 'l:LCll'f Qli' (40)
This is available from the mass balance equation C26). Summing it over the
wet hours only yields:
f Qli Cli = C3f «li + Q2i) - f
•so that:
These equations complete the specification of the hourly mass balance tech-
niques .
PERFORMANCE OF HOURLY MASS BALANCE METHOD
The results of the hourly mass balance analysis which assumes constant
overflow concentration, C are shown in Figure 10 for the diurnal sewage
characteristics. Comparing the results to those in Figure 9, it is seen that
variability is significantly reduced by using the hourly analysis method.
The hourly mass balance analysis results indicated that the variability
did not significantly decrease when only longer duration storms were analyzed.
However, upon inspection of the individual daily values, it was determined
that low rainfall intensities produced the greatest errant values. Figure 11
25
-------
UJ
CD 40
to
cc
CD
30
20
O
jz 10
cc.
h-
o
•z.
8 40
x
h;
? 30
u. 20
O
DC
LU .,n
03 JU
0
( t Alfmin
RUNOFF '{tj 77
',
, ,-T
TJ62
S
£
"L,,
= 0hr
Ol/£/?/r£OM<'
i i i
n
"
1.
50
RUNOFF
-
-
i i r
;t
Alfmin
U-T-I 1
= 4hr
OV£/?/=-/.OM/
rq ACTUAL
P CALCULATED
1 1 1
"
•-
-40 0 40 80 120 -40 0 40 80 120
CONCENTRATION, mg/L
Figure 10. Runoff and Overflow Histograms for
Diurnal Sewage Characteristics
Using Hourly Constant C Analysis
2
O
OC
h-
O
O
O
OC
UJ
I\J
60
50
UJ
CD
< 40
> 30
~ 20
10
n
Alfmin = 0 hr
RUNOFF
~
-
—
-
-
-
1 1 r-
n
'min = C
1 I I
.03in/hr OVERFLOW
n ACTUAL
P CALCULATED
1 ! i
r~<
1 1
-40 0 40 80120 -40 0 40 80120
CONCENTRATION, mg/L
Figure 11. Runoff and Overflow Histograms for
Diurnal Sewage Characteristics
Using Hourly Constant C Analysis
for i . = 0.03 in/hr. °
-------
^indicates that deleting rainfalls less than 0.02 in/hr significantly reduced
the variability in runoff concentrations. No effect on overflow variability
occurred since the low intensity storms did not overflow.
The low rainfall intensities that showed the greatest error in the
above analysis were normally the longer rainfalls from 8 to 12 hr. duration
which occurred over the early morning periods. Since NYC sampling was every
4 hours except for 2 AM which was skipped, only 1 or 2 wet period samples
were obtained for these long duration storms giving too great a weight to the
dry weather data.
Table 1 summarizes the results for both runoff and overflow concentra-
tions in terms of bias and coefficient of variation. Define the error in
each estimated daily flow weighted runoff and overflow concentration as:
£rj = CR-true(j) " GR-estimated(j )
Vj = C0-true(j) ~ C0-estimated(j)
where j is the j day of record analyzed. Then the bias is defined as:
N
1 N
'£ =±Z e . C46)
o N ^ 03
for N days of record analyzed. They are the average of the difference between
true and estimated daily concentrations. The variability of the daily esti-
mates are represented by the coefficients of variation of runoff and over-
flow which are the ratios of the error standard deviation to the true concen-
trations:
\ '
-------
TABLE 1. DAILY AND HOURLY MASS BALANCE RESULTS
FOR DIURNAL SEWAGE CHARACTERISTICS
RAINFALL ANALYZED
Alpha Total Wet Overflow
Min. Inches Days Days
(a)
0
2
4
8
12
(b)
Daily Mass
42.4
40.3
33.6
21,2
10.5
Daily Mass
Balance
117
85
57
27
11
Balance
Method -
84
68
46
22
10
Method -
RUNOFF
Bias
mg/A
Equal
-2
+4
+2
-2
-1
Equal
Coeff.
of Var.
OVERFLOW
Bias Coeff.
mg/£ of Var.
Sampling Volumes
.9
.7
.0
.8
.9
1
1
0
0
0
.85
.06
.83
.55
.36
-3.
+9.
+6.
-3.
+2.
1
9
2
1
1
Sampling Volumes -
1
0
0
0
0
.34
.75
.61
.41
.28
Greater Inten-
sity and duration
0
2
Cc)
0
4
8
(d)
0
Ce)
0
85.5
83.5
Hourly Mass
37.6
29.5
20.2
Hourly Mass
36.1
126
101
95
76
Balance Method
77
50
25
62
42
21
Balance Method
61
61
Hourly Mass Balance Method
38.6
82
63
+11
+4
.0
.0
- Constant
+2
+3
+4
.3
.0
.4
- Constant
+0
,2
- Constant
0
1
0
co
0
0
0
co
0
CR
0.
.15
.89
.32
.34'
.50
- i >
.10
0004
+12.
+7.
+1.
+2.
+4.
0
4
6
0
2
0.02
+0.
+0.
7
4
0
0
0
0
0
in/hr
0
0.
.94
.72
.14
.17
.21
.10
0005
28
-------
2 1 N - 2
"CO - I.f, (£oj - eo>
For the daily analysis methods, Table 1 (a) and (b), the bias in both
the runoff and overflow concentration is small (< 10 mg/fc except in one case)
but the coefficient of variation is substantial, especially for a . < 4 where
it exceeds one, indicating that the standard deviation exceeds the mean con-
centration.
For the hourly analysis method with Cfi assumed constant, Table l(c),
the coefficient of variation for both runoff and overflow are significantly
reduced and the effect of a . is eliminated. This is a significant improve-
ment since short duration storms can make a significant contribution to over-
flow and they contain the information from which first flush effects are
extracted, as shown subsequently.
A significant reduction in variability (v and v ^ 0.1) can be achieved
K (j
if low intensity rainfalls are ignored Table l(d). This residual variation
is due to the assumption of constant overflow concentration which is not the
case if the sewage concentration has a diurnal variation, as it does for
these simulations. Note, however, that the small bias (< 1.0 mg/£) and coef-
ficient of variation (0.1) indicate that this method of analysis is quite
good and indeed extracts the runoff and overflow concentrations from the
composited treatment plant sampling information.
The hourly analysis method with assumed constant C , performs exactly
R
since it conforms to the assumptions of the simulation output used for this
case (constant C ) and it has exact knowledge of the diurnal sewage fluctu-
K
ations. The small errors and variation are due to numerical roundoff. This
result also serves as a check on the computer program implementation of the
method.
Measurement Error
In analysis of the plant samples, a certain amount of variability exists
in the laboratory technique. Table 2 summarizes the variability data from
Standard Methods for three of the parameters measured at 26th Ward.
29
-------
TABLE 2. VARIABILITY OF LABORATORY ANALYSES7
Average Standard Coefficient of
Concentration Deviation Variation
Parameter (mg/a) (mg/&) (%)
Suspended Solids 15 5.2 33
Suspended Solids 242 24 10
Volatile Susp.
Solids 170 11 6.5
BOD5 175 26 15
Since the mass balance analysis techniques depend upon differences of
measured concentrations it was suspected that measurement error .would in-
crease significantly the variability in computed runoff and overflow con-
centrations. In order to simulate the effect of measurement error, the simu-
lated treatment plant concentration was corrupted by adding Gaussian random
variables with zero mean and standard deviation = STD corresponding to rea-
sonable measurement precision.
A measurement error of 6 and 12% of the average plant concentration C85-
88 -mg/Ji) was used in the analysis which is in the range of the volatile and
suspended solids variabilities. This was equivalent to standard deviations
of 5 and 10 mg/£.
Figures 12 and 13 show that a significant variability was reintroduced
into both runoff and overflow concentration estimates due to measurement
error for both the constant overflow and constant runoff analyses. The
smaller 5 mg/£ standard deviation perturbation produces one-half the vari-
ability of the 10 mg/Jl perturbation.
Table 3 summarizes the measurement error results. An a . = 1 hr was
mm
used for all runs. Due to the measurement errors in the plant concentrations,
some bias was again introduced into the analysis for the approximately 60 days
analyzed. Duplicate runs using different random numbers on the perturbations
provided similar results with bias and variability slightly different.
Greater variability resulted when the low rainfall intensities, 0.01 and 0,02
in/hr, were included in the constant C analysis.
R.
The variability on the overflow is consistently lower than that of the
runoff concentrations in both analysis schemes. This results since the over-
flow concentration is directly related to the plant concentration while the
runoff concentration involves an additional subtraction due to the mass
balance analysis.
30
-------
m40
•z.
DC
z 30
1 1 1
UJ
5
2 20
"~
^^
INTRATIO
3 0
STD = 10mg/L
RUNOFF f 60
OVERFLOW
i — i
—
PJ ACTUAL
P CALCULATED \
•""
-//-frf "TKn
i
i
-^-rf
8 -40 0 40 80 120 -40 0 40 80
O
°40
X
H
w 30
>-
Q
u- 20
O
QC
111
i 10
0
STD = 5mg/L
RUNOFF t 61
-
-
r~ —
~n
r-r-l "T-i 1
OVER Ft
f—-
, , rT
""**'
-
-^
UJ /in
0 40
<
z 30
UJ
-
^ 20
^^
0
F 10
QC
H
•^ ^.
STD = 10mg/L
RUNOFF f fi?
-
-
m-n-
12" S " -40 6
1 -y
.ow
-u
o
O 40
™~^ "TV/
X
5 30
>-
Q
u. 20
O
DC
UJ
1 10
n
•—i
40 80
OVERFLOW
r_.
n ACTUAL
I I CALCULATED
lu.
, n-Tlf ' h
120 -40 0 40 80120
STD = 5mg/L
RUNOFF | 62
-
-
—
nrr
1 |
Ot/£/?FZ.CW
i
i
i
— |
, , J -1
-40 0 40 80120 -40 0 40 80120
CONCENTRATION, mg/L
40 0 40 80120 -40 0 40 80120
CONCENTRATION, mg/L
Fig. 12. Runoff and Overflow Histograms for
Constant C Analysis with Measure-
ment Error for i , =0.03 in/hr.
min
Fig. 13. Runoff and Overflow Histograms for
Constant C Analysis with Measure-
K
ment Error for i . = .03 in/hr.
mm
-------
TABLE 3. MEASUREMENT ERROR RESULTS
OJ
10
RAINFALL ANALYZED
S.D.
of Plant
Concentrat ion
Perturbation, mg/fc
Minimum
Intensity
in/hr
Total
Inches
Wet
Days
CONSTANT CQ
10
ioa
5
0.03
0.03
0.03
35.4
36.1
36.1
60
61
61
CONSTANT C_
K
(a)
(b)
10
10
ioa
5
same as above
0
0.03
0.03
0.03
with different
38.6
36.6
36.6
36.6
82b
62
62
62
random numbers for
RUNOFF
Bias
ANALYSIS
+ 4.
_ o
+ 4.
ANALYSIS
+ 4.
+ 5.
+13.
+ 2.
0
9
4
0
2
2
6
Coeff.
of Var.
1
1
0
1
1
1
0
.08
.20
.62
.54
.23
.22
.61
OVERFLOW
Bias
•mg/Jl
+ 3.3
- 3.0
+ 3.9
+ 1.5
+ 4.6
+10.9
+ 2.3
Coeff.
of Var.
0.
0.
0.
0.
0.
0.
0.
71
79
39
89
79
83
39
measurement error
-------
Measurement errors of 6 and 12% completely swamp any previous differ-
ences in output between the two analysis techniques (constant C or constant
R
Cn) with coefficients of variation of 40 to 120% reintroduced into the esti-
mated overflow and runoff concentrations. This effect of the measurement
error precludes use of any single estimated daily value for comparison to
observed overflow concentrations. Rather an average value must be utilized
in order to lessen the measurement error effect.
Dry Weather Random Variability
Errors in the estimation of dry weather sewage characteristics of both
10 and 20% on the hourly flows and concentrations were evaluated using the
constant C analysis.
Figure 14 shows that 10 and 20% perturbations on the dry weather con-
centrations causes significant variability in both runoff and overflow con-
centrations. Similar perturbations on the average sewage flows cause a much
lower variability to result on the overflow and runoff values (Figure 15).
This is to be expected since overflow concentration is a function of only
the plant and dry weather concentrations, not flow. A summary of the dry
weather random variability results is shown in Table 4. Once again bias is
not significantly affected by the dry weather variability. The dry weather
concentration perturbations of 10 and 20% cause coefficients of variation of
45 and 76% to result for the calculated runoff concentrations. Values of
only 15 to 17% resulted in the runoff concentrations when the same perturba-
tions were applied to the sewage flows. Much of this latter effect is due
to the inherent model error (^ 10%) since there is very little difference
in the results for the 10 and 20% perturbations. A number of runs have been
made with the flow perturbations which verified the similarity of results for
both the 10 and 20% perturbations.
Wet Weather Variability
In addition to dry weather variability, runoff concentration and flow
variability within a storm are known to occur. Their effect was analyzed
by varying runoff characteristics in the hourly simulator. Runoff concen-
tration as a function of rainfall -duration, similar to a first flush effect
was specified as follows:
CRCti ' CR + CCR - CR > e"6Rt
00 max °°
The above expression provides the peak runoff concentration during the
first hour of the storm. The concentration then exponentially decreases to
a constant value, the time to attain the constant value a function of the
rate coefficient, g , For the hourly simulation data the following para-
ix
meters were used in the above equation:
33
-------
LO
a40
•z.
*30
LU
-20
•z.
NT RATIO
D 0
_
10% Variability C2
i •
RUNOFF jt 56
-
-
"lrff
3 -40 o
o
0 40
H
|30
Q
u- 20
O
cc
LU
H 10
0
RUNOFI
—
~~
—
r^f
_
OV£/?F/.OM/
P CALCULATED \
"h, ,
1 1 r-J
-•
i
40 80120 -40 0 40 80
20% Variability C2
C f 56
— I
"k,,
OVERF
i
1 i . .1 1
— j
...
Z
cc
z 30
LU
(5
—
"Z.
0
H 10
CC
•5» _
10% Variability Q2
— i i
RUNOFF 't'50
Fl46
}^ n x\c?
i i
OVERFLOW
'UAL
P CALCULATED
1 _ .
1 1
—
^j
l/;u o -40 0 4080120 -40 0 4080120
LOW
~\
•z.
o
*-* 4U
X
H
5 30
>-
^f
-20
cc
LU
i 10
n
20% Variability Q2
ff/y/VOFFitlBO
1
- jr
-
-
, ,r
T-i I 1
OVfflFLOM/
1 1
-
_
-|
-40 0 40 80 120 -40 0 40 80 120
CONCENTRATION, mg/L
-40 0 40 80 120 -40 0 40 80 120
CONCENTRATION, mg/L
Figure 14. Runoff and Overflow Histograms for
Variability on Dry Weather Sewage
Concentration
Figure 15. Runoff and Overflow Histograms
for Variability on Dry Weather
Sewage Flow Rate
-------
TABLE 4. DRY WEATHER RANDOM VARIABILITY RESULTS
10
Ul
S,D. of Perturbation on
Hourly Sewage Values, P
10
20
0
0
0
0
0
0
20
0
0
10
10a
10
20
20a
20a
20
RAINFALL ANALYZED
min
in/hr
0.03
0.03
0.03
0.03
0
0.03
0.03
0.03
0.03
Total
inches
45.4
35.9
37.3
36.6
38.3
34.3
36.5
36.6
37.2
Wet
Days
56
56
50
52
63b
50
50
52
59
RUNOFF
Bias
mg/fc
+ 0,4
+ 4.0
+ 1.6
+ 1.4
+ 0.7
- 0.4
+ 1.4
+ 1,4
+ 1.2
Coeff.
of Var,
0.45
0.76
0.17
0.15
0.29
0.19
0.17
0.15
0.88
OVERFLOW
Bias
+ 1.0
+ 1.4
+ 0.8
+ 1.7
+ 1.0
+ 0.4
+ 0.8
+ 1.3
- 0.4
Coeff.
of Var.
0.30
0.46
0.15
0.16
0.15
0.16
0.15
0.16
0.57
(a)
different rainfall than above
(b)
51 overflow days
-------
C = 40 mg/£
K
oo
a,, = 1000 mg/£
K
max
3R - 2/hr
which provided the following runoff concentrations during a storm:
time
(hr)
time CR(t)
1 170
2 58
3 42
4 40
» 40
To more closely simulate actual storm events, the rainfall intensity
was varied randomly during an event similar to the random generation of
measurement error and dry weather sewage variability. A high degree of
variability was given to the hourly rainfall intensity by using a standard
deviation of 100% of the mean intensity with the constraint of a minimum
intensity of 0.01 in/hr.
Table 5 indicates that the runoff and overflow variabilities due to
averaging errors were significantly increased from a previous standard devi-
ation of 5 to 15 mg/£ for constant rainfall characteristics to 30 to 40 mg/£
for variable characteristics. The major portion of the variability was due
to the varying runoff concentrations with varying hourly intensities provid-
ing a relatively slight effect. Using both the constant Cn and constant C_
U K.
computational techniques provided similar variabilities due to averaging
errors. Use of minimum rainfall intensities and minimum durations had neg-
ligible effect on averaging error.
With the strong first flush effect in the runoff concentrations, ne-
glecting low duration rainfalls significantly reduces the actual runoff and
overflow concentrations as seen in Table 6. As the minimum number of sam-
ples is increased, from 1 to 3, the storm duration is significantly increased
and thus more dilute samples obtained. The minimum duration of 4 hours also
has reduced runoff and overflow concentrations, but not as significantly as
a two sample minimum. Thus the technique of minimizing variability by ex-
cluding shorter duration events is unacceptable when a significant first
flush is present. Minimum rain intensities from 0.01 to 0.04 in/hr had no
effect on average runoff concentrations.
36
-------
TABLE 5. RUNOFF AND OVERFLOW VARIABILITIES
FOR VARYING RUNOFF CHARACTERISTICS
WET STD
DAYS (mg/£)
C,
1
78 0
67 0
60 0
78 10
C,
1
78 0
RD .
mm
(# samples)
VARIABILITY -
1
1
1
1
VARIABILITY -
1
i .
mm
CTCR
CTco
(0.01 in/hr) (mg/£) (mg/£)
CONSTANT Cn
0
1
2
3
1
CONSTANT C_
R
1
C± & Ql VARIABILITY - CONSTANT
90 0
67 0
69 0
90 10
90 10
69 10
1
1
Ha . =4)
mm
1
1
Ito . =4)
mm
1
3
1
1
1
1
ANALYSIS
37.0
34.2
33.2
81.4
ANALYSIS
33.9
CQ ANALYSIS
38.7
39.6
34.9
82.8
76.4
69.7
30.1
29.6
28.0
53.2
32.5
32.9
33.1
30.7
54.8
50.0
50.2
Cn & Ql VARIABILITY - CONSTANT CD ANALYSIS
X K
90 0 1 1 39.2 34.1
37
-------
TABLE 6. EFFECT OF NEGLECTING SHORT STORMS ON RUNOFF AND OVERFLOW
CONCENTRATIONS FOR SIGNIFICANT FIRST FLUSH
RD . a .
nun mm
(# samples) (hr.)
1 1
2
3
1 4
Flow Weighted Average
Concentration (mg/£)
Runoff, C^
63,0
50.4
44.7
60.3
Overflow,
66.9
56.0
50.8
64.6
C3
Conclusion
The conclusion is that measurement error causes significant variability
in the daily estimates and that it is affected by the characteristics of the
rainfall analyzed. Errors in dry weather sewage and wet weather concentra-
tions also cause significant variability but not to the extent of measure-
ment errors. These results suggest that a theoretical analysis of the re-
lationships that produce these variations would be useful in understanding
the results obtained using simulation techniques and may suggest strategies
for mitigating their impact such as ignoring small rainfall intensities in
the analysis.
38
-------
SECTION 7
THEORETICAL ANALYSIS OF MEASUREMENT ERROR
To evaluate the effect of the important parameters on measurement error,
a theoretical statistical analysis of the errors on overflow and runoff con-
centrations was conducted using the hourly-constant C analysis formulas.
VARIABILITY OF OVERFLOW CONCENTRATIONS
The statistical error analysis seeks to compute the mean and standard
deviation of the perturbations in overflow and runoff concentration due to
the random measurement errors. Let e be the random measurement error with
zero mean and standard deviation, cr ,p The overflow concentration, computed
using equation (39), is: **
Cn + £ =
0 o
Z+s.
i i
- ZiSiC2i
(51)
where £ is the perturbation in the overflow concentration, C , produced by
£ . Subtracting equation (39) from equation (51) yields:
C52)
Let Z.s. = N , the number of sampled hours (= 5 for NYC sampling). Let Z.s.
1 1 S J- J
= RD which is the number of wet hours sampled. Thus equation (52) becomes:
N
£o = RD
(53)
Note that RD is a random variable since its value depends on the timing of
the rainfall. Hence the statistics of £ depend not only on the statistics
of the measurement error, £ , but also on the rainfall characteristics as
they affect RD. p
The bias introduced by measurement error is the statistical average
(i.e. the expected value) of £ :
39
-------
E{eo} = Ns E} (54)
since N is constant. And since measurement error is independent of rain-
fall: s
E{£o} ' Ns E } E{V (55)
If the measurement error has zero mean as used in the simulator then E{e }
o
= 0 or no bias is introduced by plant measurement error into overflow con-
centration. This agrees with the simulation results.
The variance of a random variable, x, is defined by the equation:
V{x} = E{ (x-E{x})2} (56)
i.e. the variance is the average of the square of the deviations between x
and its average. By squaring and combining terms this equation becomes:
V{x} = E(x2} - E2{x} (57)
Applying this to e yields:
2
VU } = N2 £{-^5-} - E2{£ } (58)
0 s RD2 °
but the second term is zero and, by independence of e and ED, the result is:
VU } = N2 a2 E{-M (59)
S p RD2
22 2
where a =V{a}=E{e} since E{e } = 0. Hence the variance of e is lin-
P P P P 2 °
early related to the expected value of 1/RD .
2
In order to compute E{1/RD } it is necessary to examine the possible
values of RD and compute their probability. This is evaluated in Appendix I.
Comparing the above techniques to results of three sets of simulator data
2
shows that the theoretical value of E{1/RD } to be comparable to the simu-
lator results (Table 7) .
40
-------
TABLE 7. COMPARISON OF THEORETICAL AND ACTUAL STATISTICAL
PARAMETER FOR OVERFLOW MEASUREMENT ERROR
Number of
Days Analyzed
59
82
121
• Theoretical
0.64
0.64
0.64
Expected Value of
Actual (=95%
0.65
0.73
0.60
1/RD2
Confidence Limits)
(0.106)
(0.086)
2
The importance of E{1/RD } can be seen from equation (59) which express
the variance of the overflow estimation error in terms of the variance of the
plant measurement error. For NYC sampling, N =5, and the variance magnifi-
22 S
cation factor is N E{1/RD } = 25(0.6) = 15 so that if the standard deviation
s
of plant measurement error is 5 mg/£, the standard deviation of the errors
in overflow concentration estimates is predicted to be: a = /15~ x 5 mg/5, =
oU
19.4 mg/&. As shown subsequently these theoretical predictions agree with
simulator results.
VARIABILITY OF RUNOFF CONCENTRATIONS
A similar analysis is possible for runoff concentration. The runoff
concentration is determined from the overflow concentration, dry weather con-
centrations and flow ratios:
EiQ2i ZiQ2iC2i
C - CQ (1 + -^) - + 2l (60)
Z 0 Z 0
Lq g
For an overflow perturbation, e , the resulting runoff perturbation, e is:
£r = £o^ + W C61)
where Vn = Z.Q_.. the total sewage volume during wet hours and V- is the
ZW 1 /I JL
total runoff flow during wet hours. The variance is found using equation
(57):
C1 + V2w/Vl)2} - E2{eo(l + V^/V^}
2 2
41
-------
Since e is independent of rainfall characteristics the second term of this
equation is zero and the result is:
V{er} = NV E{-^j (1 + V^/V^2} • (63)
RD
Consider the ratio: V_ /V- . For constant sewage flow, Q_ . 9 and runoff
flow, Q . during a rainfall event of duration d:
V2w 4Q2i dQ2 Q2
vi
which is independent of duration and is only a function of rainfall intensity
i, through the runoff flow Q = f C^A± = Ci for drainage area, A, units con-
version factor, f, and runoff coefficient, GV< Thus:
RD
= V{eQ} E(C1 + (^/Q^ } (64)
and the runoff variation due to measurement error is increased over the over-
2
flow variation by the expression E{Cl + Q?/Q.,) K
To compute this expectation, let:
g = Q2/fCvA = Q2/C
2
so that the expectation E{ Cl + Q9/Q-,) } becomes:
<— J.
9 °° 7 °°
E{(1 + S/i)z} = f Cl + B/i) P4(i)di/f P±Ci)di C65)
+ S/i) } = f Cl + S/i) Pi(i)di/f
^i . •'i
'min min
where P. CO = v e > t*ie Probability density function of the intensity.
The integral in the numerator can be evaluated numerically. For example,
42
-------
CO
f (1 + g/i)2 P±(i)di = Z ( 1 + 3/in+1/2 )2 ( e ln+1/I - e ln/I ) (66)
' i . n=l
mm
where i L, ,„ = (i + i (1)/2 and i = 0, 0.015, 0.025, 0.035, etc. (in/hr).
n+1/2 n n+1 n
For the following values used in the simulator: Q_ = 2 MG/hr, C =
0.7 and A = 5000 acres, f = 2.715 x 10~ MG/acre-hundredths inch, then
g = 2.105, and:
E{1 + g/i)2} = 3.36
Two runs were used to compare the above theoretical values to the
simulator results and close agreement was obtained as shown in Table 8.
TABLE 8. COMPARISON OF THEORETICAL AND ACTUAL STATISTICAL PARAMETERS
FOR RUNOFF MEASUREMENT ERROR
No. of Days § E{ (1 + g/i)2}
Analyzed Theoretical Actual (95% C.L.) Theoretical Actual (95% C.L.)
82
80
2.105
2.105
2.107
2.109
(0.058)
(0.059) •
3.36
3.36
3.106
3.213
(0.632)
(0.624)
With the above theoretical framework, the effect of varying both the
number of samples taken during a rain event, RD, and the rainfall constant
intensity, i, can be evaluated.
Figure 16 indicates that significant reductions in the error magnifica-
2 2
tion factors, E{1/RD } and E{(1 + g/i) }, occur at i . > 0.02 in/hr and
RD . ^2 samples. Again good agreement between theoretical and simulated
values was obtained as shown. These results are important for two reasons.
2 2
They confirm that the methods used to compute E{1/RD } and E{(1 + g/i) }, as
described above are correct. In addition they suggest that significant re-
ductions in the variability of overflow and runoff concentration estimates
can be achieved by using an RD . ^2 and i . ^0.02 (in/hr). These
J ° mxn mm
results provide the explanation of the large increases in estimated concen-
tration variability that were observed to occur in the simulator investiga-
tions if short duration - low intensity storms were included in the analysis.
43
-------
CN
0
95% CONFIDENCE
LIMITS
I
0.8
0.6
Q
Qc
OA
0.2
0
I
01234
RDm\nt number of samples
0.00 0.01 0.02 0.03 0.04 0.05
/ MINIMUM, in/hr
Figure 16. Effect of Number of Samples and Rainfall Intensity
on Runoff and Overflow Variability Parameters
44
-------
COMPARISONS OF PREDICTED AND SIMULATED EFFECTS OF MEASUREMENT ERRORS
The theoretical expressions for the effect of measurement errors on
overflow, equation (59), and runoff, equation (63), concentration variances
can be compared directly to simulation results.
The effect of measurement error on runoff variability is given in Figure
17. The only source of variability in this figure is measurement error since
constant runoff and sewage characteristics were used. Good agreement between
theoretical and observed values is obtained with ,a significant degree of
scatter in the data. Again the major effect is the number of samples, RD . ,
rather than rainfall intensity analyzed.
Incorporation of variable sewage characteristics introduces additional
variability into the analysis as shown in Figure 18. Using a storm interval
of 1 day caused additional variability since the number of multiple rain
events occurring on one day was significantly increased over the three day
interval data. With the presence of significant model error, due to the
daily averaging of plant samples, as well as significant measurement errors,
it is necessary to combine these effects theoretically in order to compare
to simulator results. Assuming the measurement variance and averaging var-
iance are independent, the total variability can be obtained by summing the
variances.
v(cR} = VI{CR} + v2(cR}
where V {C } = Measurement Error Variance
1 R
V0{C_.} = Averaging Error Variance
Z K n
V {C,,} = Total Variance = an_
R UK
For the constant runoff characteristics used in the analysis, the magni-
tude of the averaging error is small compared to the measurement error as
seen in Figure 19. The data again show a significant amount of scatter but
in good agreement with the predicted values. Figure 20 shows the predicted
and observed variabilities on the overflow concentrations. As predicted,
rainfall intensity has no effect on overflow variability. To obtain over-
flow values at low rainfall intensities, for this data the interceptor capa-
city was set at the maximum dry weather flow rate of 2.32 MG/hr. instead of
5 MG/hr. used previously.
When measurement errors are combined with wet weather concentration
variation significantly higher variabilities were obtained. The effect of
both averaging and model errors on the runoff and overflow variabilities is
given in Figures 21 and 22. For runoff variability, the averaging error is
still less than the measurement error using a STD = 10 mg/&. However for
overflow the situation is reversed when more than two samples are taken
during a storm. Good agreement between predicted and observed variabilities
is obtained with a relatively large degree of scatter in the data due to the
one year data base (.only 5 to 10 values exist for an RD . of 3). The good
45
-------
Q
co
"5
8
4
0
STD = 10mg/L
o 3
• 3
I
Number Interval
of days (days)
121-82 1
192-131 3
62-40 1
88-60 3
RD
=3
Number Interval
of days (days)
30-19
39-27
1
3
I
1 234
/ MINIMUM, 0.01 in/hr
Figure 17. Runoff, Variability Factors for
Measurement Error (STD = 10
46
-------
16
en
I
8
4
0
20
16
12
8
RANGE OF VALUES
I I
= 2
01 234
/MINIMUM, 0.01 in/hr
Figure 18. Daily Averaging Error for Constant Rainfall and
Variable Sewage Characteristics During an Event
47
-------
8
6
4
I
STD = 10mg/L
O
O
MEASUREMENT AND
A VERAGING ERROR
0/0
MEASUREMENT
ERROR
A
RD,
'mm
A on Number Interval
num\n Of days (days)
O 1
• 1
A 2
A 2
93-58 3
194 1
37-18 3
99-67 1
I
= 3
on Number Interval
'"'min (day5)
Q 3
• 3
9-7
47-30
3
1
I
I
012345
/MINIMUM, 0.01 in/hr
Figure 19. Runoff Variability Factors for Combined Measurement
(STD = 10 mg/£) and Daily Averaging Errors
48
-------
0
RDm'm
O 1
* 2
a 3
MEASUREMENT AND
A VERAGING ERROR
= 2
a
a
= 3
I
I I
I
012345
/MINIMUM, 0.01 in/hr
Figure 20. Overflow Variability Factors for Combined Measurement
(.STD = 10 mg/A) and Daily Averaging Errors
49
-------
10.0
7.5
5.0
2.5
2.5
2.5
STD = 10mg/L
MEASUREMENT
AVERAGING
I I I I
= 1
I I
I I
o
_L
I
1234
/MINIMUM, 0.01 in/hr
Figure 21. Runoff Variability for Varying
Runoff Characteristics
50
-------
7.5
5.0
2.5
O
to
2.5
TOTAL
8 \
STD=10mg/L
MEASUREMENT
AVERAGING
RDm\n=1
= 2
_ o
1
1
1
1
1234
/MINIMUM, 0.01 in/hr
Figure 22. Overflow Variability for Varying
Runoff Characteristics
51
-------
agreement again verifies the technique of taking the sum of the variances of
the individual errors to obtain the total variance on the runoff and over-
flow concentrations.
A theoretical analysis of the effect of dry weather random variability,
similar to the measurement error analysis, is presented in Appendix II. The
results of this analysis show that random variability on hourly dry weather
concentrations can produce significant variability on overflow and runoff
concentrations. To accurately assess the magnitude of this error, data on
the hourly variability in dry weather sewage concentrations is required.
CONCLUSION
From the above analysis, it is seen that the variability due to treat-
ment plant measurement error is magnified 4 to 7 times for estimates of
overflow and runoff concentrations respectively by the mass balance tech-
nique. By analyzing longer duration storms in which two out of the five
samples taken for the composite occur during the runoff event, the vari-
ability due to measurement error can be reduced to factors of about 2 and
3.5 to 4 respectively for overflow and runoff. The average rainfall inten-
sity has no effect on overflow variability and some effect on runoff vari-
ability, especially for 0.01 and 0.02 in/hr. average intensities. The
theoretical analysis substantiates these results and explains the source
of the magnification of measurement errors at the plant.
These results suggest that the variability of the individual daily
estimates of overflow and runoff concentrations are an inherent part of
the mass balance method and are large, relative to measurement errors, be-
cause of the magnification factors. These are an unavoidable consequence
of the method employed, which attempts to extract the runoff and overflow
concentrations from differences of measured concentrations. However, the
analysis also confirms that there are no. biases present in the resulting
estimates. This suggests that although the estimates are noisy they may
be still useful for analyzing the properties of runoff concentrations ob-
tained from an analysis of actual treatment plant data. This is investi-
gated in the next section.
52
-------
SECTION 8
ANALYSIS OF RAINFALL - RUNOFF RELATIONSHIPS
An important topic in the modeling and analysis of runoff generation
mechanisms is the relationship between rainfall properties and resulting
runoff concentrations. For example, if a strong first flush effect exists,
then storm-averaged runoff concentrations should show a significant inverse
relationship to storm duration. Also if dry deposition of pollutants is
the principle mechanism by which they accumulate on the drainage basin, or
if in combined sewers solids are accumulating during dry periods, then a
positive correlation is expected between storm interval and runoff concentra-
tration.
The purpose of this section is to investigate the degree to which the
hourly mass balance methods, using daily (equal volume) composite treatment
plant data, can be used to uncover these relationships. The methodology
used involves building into the simulator a known relationship between rain-
fall properties and runoff concentrations. This simulator output is then
sampled and composited as before, measurement error is introduced at the
plant, and these observations are analyzed. Since runoff concentrations
will be varying, the constant overflow method is employed to estimate the
runoff for each day, C . These concentrations are then analyzed using re-
K
gression analysis to estimate the relationships between runoff concentra-
tions and rainfall properties.
SIMULATOR DATA
The hourly simulator was modified to incorporate the effect of interval
between storms on runoff concentration. Two interval correlations were
utilized. The first uses the same effect as that previously found in the
26th Ward data, Mueller and Anderson, 1979:
(^ = 0.542 * 6 + 146
-------
obtained by the daily balance analysis. Figure 23 shows the regression plots
for the strong interval effect for the total length of record of 260 days.
The data are plotted in groups of 20 to reduce some of the scatter. The
2
degree of variability (r ) explained by the interval effect is relatively
low, 12% for the strong interval effect and only 2% for the weaker interval
effect. The reason for these low correlation coefficients is the errors in-
herent in the daily balance analysis due to the averaging technique and
measurement errors. As can be seen from Figure 23, the results are quite
good. The estimated slope of the relationship (0.584) is very close to the
actual slope (0.542) as are the intercepts: 136 and 146 mg/& respectively.
In order to make a quantitative statement of the goodness of the estimates
it is necessary to know how close is close enough. This information is
available from the regression analysis since the 95% confidence limits for
slope and intercept are available using standard regression theory. However
it is necessary to check that indeed the assumptions implicit in linear re-
gression theory are met. Most important is that the residuals are normally
distributed. A normal probability plot and histogram of the residuals are
shown in Figs. -24 and 25. These confirm the assumption of normally distrib-
uted residuals and allow the use of the confidence intervals for slope and
intercept as correct indications of the extent to which slope and intercept
are known. These limits are then compared to the true values used in the
simulator.
It is clear that the length of record analyzed, and therefore the num-
ber of C_ data used in the regression analysis, will affect the confidence
K
limits associated with the slope and intercept. These indicate the quantity
of data required in order to use the mass balance method for the investiga-
tion of rainfall-runoff relationships.
The effect of length of record on the slope and intercept of the re-
gression analyses is given in Figs. 26 and 27. For the strong interval cor-
relation, a nonzero intercept is ruled out (95% confidence) but not a non-
zero slope for a record length of 65 days. A data base of 150 to 200 days
provides tighter confidence limits. For the weak interval effect, the 95%
confidence limits on the slope are relatively wide and the lower limit still
approaches zero at a data base as high as 330 days. Thus the weaker the
effect, the longer the data base required.
The effect of the magnitude of the measurement error on the percent of
2
variability (r ) explained by the interval correlation is shown in Fig. 28.
2
The lower the measurement error, the greater the r , The stronger interval
2
effects on runoff concentration have significantly greater r values than the
weaker effects. Averaging error, although relatively small since constant CL,
2
values were used over an event, result in a maximum r value of 29 and 78%
respectively, for the weak and strong interval effects. Thus as measurement
2
error increases, r decreases since more of the total variability is due to
the measurement error and less to the interval effects.
54
-------
O)
oc
°
400
350
300
250
200
150
100
50
260 Events
STD=15mg/L
J_
J_
MEAN±S.E.
REGRESSION LINE
y =136 + 0.584 X
2= 0.117
_L
J_
_L
J_
_L
20 40 60 80 100 120 140 160 180 200 220 240
INTERVAL BETWEEN STORMS, hr
Figure 23. Runoff Concentration - Storm Interval Regressions
for Strong Interval Effect
55
-------
200 i-
150
100
50
o>
0
o
5 -50
-100
-150
X
\
\
I
1
1
1 5 20 50 80 95
100 X PROBABILITY OF A VALUE
THE STATED VALUE
99
Figure 24. Probability Distribution of Runoff Residuals for a 10 mg/&
Measurement Error for Strong Interval Effect
IU
S3 14
0
w 12
oc
3 10
0
0 8
u.
0 6
cc
ffl 4
i 2
0
Strong Interval
- Effect
—
•—
-
-
-
nnT
—
—
—
™—
—
STD = 10mg/L
o
I I
-160 -80 0 80 160 •
-200 -120 -40 40 120
C/?-C, (mg/L)
Figure 25. Runoff Residuals Histograms for a 10 mg/£
Measurement Error for Strong Interval Effect
56
-------
£.\J\J
-1
^>150
h-"
2] 100
o
cc
LU
£ 50
0
-C
^ 1.0
£
uT
-0.5
CO
n
<
STD = 15mg/L
Interval
T T T
» 1 \ T T
T \ 1 J-
\
L •"• TRUE VALUE
\
\
95% CONFIDENCE LIMITS
1 l I I I I
r"2- =0.07-0.12
_
1 -
TRUE VALUE
I I, /I
i I i
l 1 1 1 1
50 100 150 200 250 300 350
DAYS OF RAINFALL ANALYZED
Figure 26. Effect on Length of Record on Regression Parameters for
A 15 mg/& Measurement Error for Strong Interval Effect
57
-------
100
75
Si 50
o
cc
11!
2 25
_i
OJ
ii
a,
0.50
0.25
0
-0.25
r2 = 0.02-0.05 STD = 10
9&
(
I •
i
S CONFIDE
xx
1
1
I (
WC£ Z./M7S
' V '
• TRUE VALUE
1 1 1
i
TRUE VALUE
I/I
i i I
1 1 1 1 1
mg/L
T
1
1
1
0 50 100 150 200 250 300 350
DAYS OF RAINFALL ANALYZED
Figure 27. Effect of Length of Record on Regression Parameters for
A 10 mg/£ Measurement Error for Weak Interval Effect
58
-------
to 78%
LU
CJ
tr.
in
OL
CMX
W
STRONG INTERVAL EFFECT
=119mg/L
RANGE OF
VALUES
5 10 15
STD, mg/L
Figure 28. Effect of Measurement Error on Runoff Concentration-
Interval Correlation Coefficients
59
-------
The ability of the daily balance technique to determine the runoff prop-
erties when both first flush and dry interval between storms were incorpo-
rated in the model was next evaluated. The following strong interval and
first flush effects were used for the analysis:
C_(t) = [40 + (1000 - 40)e~2t(hrs)][0.0067 S + 1.8]
K
Typical hourly values for an interval of 68 hours (2.8 days) are as follows:
t
hr
1
2
3
4
5
6
The effect of first flush is negligible after 4 hours for this assumed
runoff relationship. The concentrations are much higher than previously used
due to the stronger interval effect.
Figures 29 and 30 show the slopes and intercepts of the regression equa-
tions as a function of length of record analyzed. For both interval and
duration, close to the true values of both parameters are obtained. Between
150 and 200 days data are required to insure the 95% confidence limit on the
slopes are different than zero in both cases.
If the significant first flush exists in the rainfall properties then
analyzing data at greater minimum durations reduces the average runoff con-
centration significantly since the highest values occur in the first hour of
the storm. This is shown in Figs. 31 and 32, using the full record length
with 15 mg/£ measurement error. For the duration regressions the intercept
decreases as the average concentration analyzed decreases, while the slope
approaches zero. The amount of variance explained by the duration effect is
also markedly reduced. As the duration effect becomes weakened, the interval
effect becomes somewhat greater as seen from the increasing slope, but con-
centration is still reduced. The correlation coefficient also increases, but
the large degree of measurement and averaging error inherent in the data base
2
and daily analysis technique keeps the r values below 10%. However in all
cases the regression slopes and intercepts obtained from the calculated C
JX
data are remarkably consistent with the regression slopes and intercepts ob-
tained from the true runoffs, C-. Further, these analyses indicate that to
obtain the true runoff effects, all duration data must be analyzed. Addi-
tional regressions conducted at an i of 0.03 in/hr had negligible effect
on the duration and interval correlations.
60
-------
ZOU
_i 20°
CD
E
H-150
Q.
LU
DC 100
LU
~ 50
0
•= 1.0
_j
en
E. 0.5
LU
Q_
0
ti °
-ns
_ <
-95%CC
1
r2 = 0
<
i
>
NFIDENCl
1
.03-0.04
> <
STD = 15mg/L
I I I
I l\i
TRUE VALUE
F LIMITS
till
TRUE VALUE
I T /I
rr^
1 | 1 1 1
o
cc
LU
LU
ouu
250
200
150
100
50
0|0
on
I I
r \
i \
-L r/?6/F W\ /.(/f
~ 95% CONFIDENCE LIMITS
X
STD = 15mg/L
Duration
r2 =0.11-0.14
i i t i i
-r TRUE VALUE
\ /
I
II | 1 1
1
I
I
1
0 50 100 150 200 250 300 350
DAYS OF RAINFALL ANALYZED
0 50 100 150 200 250 300 350
DAYS OF RAINFALL ANALYZED
Figure 29. Effect of Length of Record on
Interval Regression Parameters
for a Significant First Flush
Figure 30. Effect on Length of Record on Duration
Regression Parameters for a Significant
First Flush
-------
80
30
20
&
cr>
to
10
400 1 0
Q.
UJ .
O -|
DC ^,200
0|0
LU
a.
g
to
10
-20
Figure 31.
Duration
STD = 1 5 mg / L
BASED ON Cj
BASED ON CR
y
i
i
j
40
STD=15mg/L
Interval
8
10
200 1 0
t
UJ _,
cc~5>100
UJE
0.810
ai
en
1234
ALPHA MINIMUM, hr
Effect of Minimum Duration on Runoff
Concentration - Duration Regression
Parameters for a Significant First Flush
0
O/V C
BASED ON CR
1
1
1
01234
ALPHA MINIMUM, hr
Figure 32. Effect of Minimum Duration on Runoff
Concentration - Interval Regression
Parameters for a Signifiant First Flush
-------
NYC - 26th WARD DATA
Numerous analyses were conducted on the New York City 26th Ward data
with the various model changes as described in detail in Appendix III. Ana-
lyses of rainfall-runoff characteristics were conducted with the hourly mass
balance analysis at 0.03 in/hr. minimum average intensity to reduce the
variability in the runoff estimates for the low rainfall storms. Also storms
that lasted more than one day were combined into one event.
The regression parameters describing the effect of minimum duration ana-
lyzed are shown in Fig. 33 for concentration vs. duration. The 95% confi-
dence limits on the slope and intercept show that the analysis can predict
the effect of duration, even when small storms are included. The plot of
slope vs. alpha minimum shows that a first flush does exist, and is most
pronounced in storms of at least 2 hours in duration and 0,03 in/hr intensity.
With the exception of the 1 hour rainfall, these results are similar to those
from the hourly simulator with the first flush.
They also agree well with the NYC 208 results which showed a signifi-
cant first flush to exist in combined sewer overflows for BOD and suspended
solids over the first two hours of storm events. The linear regression plots
for suspended solids versus storm duration and interval at a . of 2 hours
mxn
and a minimum rainfall intensity of 0.02 in/hr are given in Figs. 34 and 35.
The first flush effect as measured by the duration regression is signif-
2
icantly greater than that obtained previously (y = 257 - 4.7x, r = 0.033)
using the flow weighted balance technique at an a . of 4 hours . The inter-
val regression has a similar slope but with a greater intercept and lower
2
correlation coefficient than obtained previously (y = 13x + 146, r = 0.078)
as predicted by the simulator results (Fig. 32), Rainfall characteristics
have less of an effect on the three remaining parameters analyzed at the 26th
Ward plant as summarized in Appendix III,
CONCLUSION
These results strongly suggest that the mass balance estimates of runoff
concentrations, although noisy, can be successfully used to obtain the rela-
tionship between runoff concentration and rainfall properties and that the
proper method of analysis is linear regression. The confidence limits for
slope and intercept decrease as record length increases as expected. The
surprising result is that the regression estimates are quite close to the
actual values even for the cases where the confidence limits are quite large.
This suggests that the 95% confidence limits are a conservative estimate of
the probable range of the true values.
2
The low value of r obtained from the regression is not to be inter-
preted as an indication that the estimates of slope and intercept from the
regression analysis are not useful. Their utility should be judged from
63
-------
Duration, 26th Ward Data
95% CONFIDENCE LIMITS
Number of events =
199-64
2468
ALPHA MlNIMUM,hr
Figure 33. Effect of Minimum Duration Analyzed on Runoff-Duration
Regression Parameters for NYC 26th Ward Data
64
-------
26th Ward Data
REGRESSION LINE
= 307-8.32*
= 0.063
12 16 20 24
DURATION, hr
28
Figure 34. Duration Regression Analysis for 26th Ward
Suspended Solids Data
26th Ward Data
uuu
Q 500
_j :H
O o>
S 6-«0
i-
II300
3z
"• o 200 •
i :
QC 100
0
181 Events
20
i MEAN±S.E.
" 20 2Q V^f--^"
Y | REGRESSION LINE
J- K=179+13.2x
r2 = 0.046
i I I i i
22
T
0 2 4 6 8 10 12
INTERVAL BETWEEN STORMS, days
Figure 35. Interval Regression Analysis for 26th Ward
Suspended Solids Data
65
-------
2
their confidence limits. The small r values are a result of the large vari-
ability in the estimates of C_, which is inherent in the mass balance method.
2
There is no reason to expect that a large fraction (r close to. one) of this
variability, which is due to measurement error, should be explained by the
2
rainfall property correlations. As shown above, r decreases sharply as
measurement error increases. However, the slope and intercept are still
reasonably well estimated, as judged by the confidence limits.
Analysis of the actual 26th Ward data shows a significant first flush
to exist at this location for suspended solids concentration. The proper
magnitude of this effect could not be obtained when only storm durations
greater than 4 hours were analyzed as required by the flow weighted balance
technique used previously.
66
-------
SECTION 9
ANALYSIS OF HOURLY SAMPLE COLLECTION
The impact of the sample collection procedure was investigated. The New
York City regime composites five equal volume samples over the day and mea-
sures the concentration of the composite. An alternate method of collection
is to obtain a sample at each hour and composite the resulting twenty-four
samples. The composite sample is then analyzed. As shown in Appendix IV, a
significant number of plants (24 out of 54 surveyed) composite samples at a
one hour interval or less. This regime does not increase the quantity of
measurements obtained, rather it samples the influent more frequently. It
was expected that this regime may improve the behavior of the mass balance
estimates.
For the constant overflow assumption, the treatment plant concentration
is, (equation 38):
o
/N
Z C_. + £ C0.
. 33. .21
s
where now s. = 1 for each hour since a sample is removed at each hour, so
that N =24.
s
The overflow concentration estimate is (.equation 39):
VP - E°C2i Z+C3i + Z°C3i - Z°C2i
^*
v+ v+
L S. 2S.
i X i X
yr — y r
zc3i z c2
E+Si
i
i NsCp-JC2i
a
where a = £.s. = numbers of hours of rainfall in the day.
The last term in the above equation is the dry weather contribution for
the daily sample. The runoff concentration is calculated as previously dis-
cussed from a mass balance on the collection system.
67
-------
Two data bases were used in the analysis, one with a weak interval ef-
fect and no first flush, the other with a strong interval effect with first
flush.
When no measurement error was present, the linear regression analysis on
runoff concentration versus interval successfully predicted the input data
2
with r values of 70 and 12% for no first flush and first flush, respectively.
However, when measurement error was introduced into the analysis the degree
of variability was greatly increased over that observed previously with the
NYC sampling technique. This surprising result requires an explanation.
The theoretical analysis developed previously was applied to predict the
variability of the results with the hourly sample collection. Applying a
perturbation on the above equation for overflow concentration and subtracting
the overflow concentration yields:
o a p
where: £ = the plant concentration perturbation, a = the hours of rainfall,
and N = 24.
s
The variance of the overflow concentration due to measurement error is
then:
Since E{e } = 0, the last term in the above equation is zero leaving:
P
V{e } = N 2 E{e 2} = N 2 STD2 E{-±}
o s p s a2
The standard deviation of the overflow concentration is then:
STD E{-}
As previously determined for the NYC sampling routine, the standard deviation
on the overflow concentration when all rainfall is analyzed, i . = 1, is
mm
2
The expected value of I/a can be determined at any minimum duration,
RD . = a . , as:
mm mm '
68
-------
,
d=RD . d
E{l/a2}
d=RD .
mm
The resulting error magnification factors are compared to those from the NYC
sampling routine in Figure 36. Good agreement between theoretical and calcu-
lated values is again observed. The magnitude of the error magnification
factor is much larger than that for the NYC sampling routine at a similar
KD . . A minimum of 4 wet samples provides magnification factors similar to
those for 1 wet sample using the NYC routine. This is equivalent to 17% of
the plant samples being wet for the hourly sampling routine compared to 20%
for the NYC sampling routine. With the large measurement error at low rain-
fall durations, the ability to obtain information at these low durations is
impaired. The reason is that the sample collected during the short storm is
mixed with many samples collected during dry weather thus reducing the effect
of the runoff contribution. A small measurement error effectively masks this
small concentration impact.
Figure 37 indicates that the daily mass balance analysis can still de-
termine the correct values of slope and intercept from a regression analysis
of runoff concentration versus duration analyzing all storm durations for 476
days of data. However, the confidence limits are wide due to the large meas-
urement error effect. Increasing the minimum duration analyzed reduces the
confidence limits, however lower values of runoff concentration result due to
missing the first flush effect. Figure 38 shows that the true values of the
effect of interval on runoff concentration cannot be obtained with 388 days
data at KD . = 1. The predicted slope is lower and the intercept higher than
the true values. The confidence limits on the slope are also large and show
the slope to be not significantly different than zero. Analyzing storm dura-
tions greater than 4 hours, provides higher correlation coefficients than
the lower durations, since the first flush effect is diminished. Again, this
results in low estimated runoff concentrations due to missing the first flush.
The above analysis indicates that hourly sample collection at a treatment
facility has significant drawbacks when analyzing for runoff and overflow
characteristics. To obtain reliable estimates, greater duration storms have
to be analyzed which miss a first flush effect if present. Therefore this
type of sample collection procedure is not recommended. The New York City
sampling collection method is superior since a larger proportion of the col-
lected sample is affected by runoff. These results suggest that separate
sampling during wet and dry periods may be beneficial.
69
-------
NYC SAMPLING FOR RDm-m = 1
234
' number of samples
Figure 36. Effect of'Minimum Storm Duration on Error Magnification
Factors for Hourly Plant Sampling
70
-------
£ 15
8
te 10
Q.
•5L 5
400 1 0
c! 300
O)
H
uj 200
o
cc
LU
H
2 100
0(0.
v_
JZ
i
ci -10
O)
E
UJ
0 ~20
V)
-30
Length of Record = 476-276 days
STD = 15mg/L
- Duration
_____ _.__
—
V 555K CONFIDENCE L IMITS
"^ — — •?
_L.
"•
1 1 1
>^-H
— -,
— ~
r sf
s\
/ L
""\
r/?ty£ wiz.6/£
I i i
012345
RDm!n, number of samples
Figure 37. Duration Regression Parameters Using
Hourly Plant Sampling Technique
20
£
§ 15
c? 10
5
300 1 0
_, 250
• —
E 200
S 15°
o
S 100
H
- 50
1.5|0
5 1.0
O)
E 0.5
UJ
o 0
g 0
w OR
Length of Record
~ STD=15mg/L
Interval
"^
—
—
^_ x^j
— TRUE
= 388-208 Days
0
/
/
/
/
I I
VALUE 95% CONFIDENCE LIMITS
~ \.ps. /
~ ^Ss*^J
I*
- •L
—
I
__ T ..TRUE
f T
I "
-^
/
>«»*^>i>^
-L
I I
VALUE
-5
I I I
01234
fiDm-m, number of samples
Figure 38. Interval Regression Parameters Using
Hourly Plant Sampling Technique
-------
BIBLIOGRAPHY
1. Mueller, J.A. and Anderson, A.R., "Contaminant Inputs to New York
Bight, Phase II, Urban Runoff", NOAA, MESA Report. October 1977.
2. Mueller, J.A. and Anderson, A.R., "A Mass Balance Method for Estimating
Combined Sewer Runoff and Overflow Quality from Sewage Treatment Plant
Data", Prog. Wat. Tech. 10, pp. 727-739. 1978.
3. Mueller, J.A. and Anderson, A.R., "Combined Sewer Overflow Quality
from Treatment Plant Data", JWPCF, 51, pp. 958-973. 1979.
4. Hydroscience, Inc., "Available Storm/CSO Sampling", NYC 208 Task Report,
PGP TASK 222. March 1978.
5. Di Toro, D.M., and Small, M.J., "Stormwater Interception and Storage",
Journal of the Environmental Engineering Division, ASCE, 105, pp. 43-54.
1979.
6. Davenport, W.B. and Root, W.L., "Random Signals and Noise", McGraw-Hill
Book Co., Inc., New York, p. 34. 1958.
7. Hydroscience, Inc., "Rainfall, Runoff and Statistical Receiving Water
Models", NYC 208 Task Report, PGP Task 225. March 1978.
72
-------
APPENDIX I
ESTIMATION OF E U/RD2}
The number of samples obtained during storm events, KD, is a function
of a number of factors; the time at which the storm begins, storm duration
and interval between storms. The first two factors have a major effect since
sampling at the treatment plant occurs at specified times with the greater
the duration the greater the likelihood of obtaining samples during the wet
period of the day. The following equation was used to describe this effect:
N N
E {R?2} = ll (R¥2) PRD > ll PRD CA1)
tuJ . KD .
mm mm
where ?„_. = probability of RD occurring over all durations,
KD
and N = 5 for the New York City sampling regime.
s
The overall probability of a sample being wet is a function of both the prob-
ability of a specified duration occurring during a day and the probability of
that duration being wet as follows:
PRD '/A X Pwd CA2)
d—J.
where P = probability of occurrence of a specified duration from 1 to 24
hours
- d2 -d/D -d./D -d,/D
= ± / e dd = ei - e CA3}
dl
and P , = probability of duration, d, having RD wet samples.
wd.
The above equation ignores the effect of multiple events occurring in a
day, the smaller the storm interval the greater the probability of two or
more events occurring in 1 day. It also ignores the impact of starting time
on spreading the total duration over 2 days instead of a single day. These
two effects are somewhat self compensating in that the first effect tends to
give longer storm durations in a day and the latter effect shorter durations.
Storm durations greater than 24 hours which occur relatively infrequently
(< 1.5% of the time), are neglected since these would necessarily occur over
2 sampling days.
73
-------
Evaluation of the first probability, P., is straightforward using the
average duration of the simulator, D = 6 hr, and increments of 0-1.5, 1.5-2.5,
2.5-3.5, etc.
To evaluate the latter probability, P ,, an evaluation procedure was
utilized to properly account for the New York City Sampling regime which
skipped the 2 AM sample. A given duration continuous storm was assumed to
start at a specified hour. The number of samples which would be taken over
the storm duration were counted. The storm was then assumed to start on the
next hour and the same procedure followed. This was continued until all pos-
sible 24 starting times were evaluated. For example, a 13 hour storm first
appearing at the plant at 10 AM would be sampled at 10 AM, 2 PM, 6 PM, and
10 PM resulting in an RD of 4. However when started an hour later it would
only be sampled at 2, 6, and 10 PM resulting in an RD of 3. Continuing this
procedure over 24 hours provides 9 possibilities for RD = 2, 13 for RD = 3,
and 2 for RD = 4. Dividing each of the total number of possible RD values by
24 yields P , for each RD, so P , = 0.375 for RD = 2, 0.542 for RD = 3, and
wd wd
0,083 for RD = 4. The results for all durations and RD values is given in
Table A-l.
74
-------
TABLE A-l. DURATION AND SAMPLING PROBABILITIES
Storm Duration Probability of d having RD wet samples,
Duration Probability P , x 100, %, for specified number of
Pd x 100 samples
d,hr %
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
22.1
12.0
10.1
8.6
7.3
6.1
5.2
4.4
3.7
3.2
2.7
2.3
1.9
1.6
1.4
1,2
0.98
0.83
0.70
0.60
0.50
0.43
0,36
0.31
1
20.8
41.6
62.5
83.3
70.8
58.3
45.8
33.3
25.0
16.7
8.3
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
16.7
33.3
50.0
66.7
62.5
58.3
54.2
50.0
37.5
25.0
12.5
0
0
0
0
0
0
0
0
0
RD
3
0
0
0
0
0
0
0
0
12.5
25.0
37.5
50.0
54.2
58.3
62.5
66.7
50.0
33.3
16.7
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
8.3
16.7
25. .0
33.3
45.8
58.3
70.8
83.3
62.5
41.6
20.8
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4.2
8.3
12.5
16.7
37.5
58.3
79.2
100.0
«T*/^T% *
For each RD value
n^\ T T T +• IT TT«s1 11 a o r.Vt^ "i r
in Table A-l,
»r» ^ >• a 4* n QT> r* r\v
the summation of P .
*P . is
1 wd
o-rt/3 "DT* t
made over all
Tol^i^e* o e*
RD
shown in Table A-2. The last column in Table A-2 is used in the next section
for determining E{RD}.
75
-------
TABLE A-2. OVERALL POSSIBILITIES FOR RD VALUES
RD
1
2
3
4
5
Sum
Overall
probability
PRD X 10°'
%
37.3
16.8
7.88
3.65
1.33
66.96
PRD X ~^2 X 100>
RD
%
37.3
4.2
0.88
0.23
0.05
42.7
P-._ x RD x 100,
KJJ
%
37.3
33.6
23.6
14.6
6.7
115.8
The above evaluation accounts for only 66.96% of the number of wet sam-
ples since the effects of multiple storm events occurring in a day and a
storm event being spread over 2 days were ignored. In using the above re-
sults, the assumption is implicit that the remaining 33.04% of the wet sam-
ples will be distributed in the same fashion as that given in Table A-2.
Comparison of the above distributions to a number of computer simulations
showed the results to be acceptable.
When all data are analyzed (RD . =1), then E {^2} = 42.7/66.96
min KD
= 0.638. The summations from specified RD . values can then be made to
.. mm
yield the E {—r-} as shown in Table A-3.-
RD
TABLE A-3. EFFECT OF RD . ON E {-^}
RD2
RD .
mm
1
2
3
4
5
5
I
RD .
mm
T> 1
^ RD2
42.7
5.4
1.16
0.28
0.05
5
min
66.96
29.66
12.86
4.98
1.33
t? f \
E I— j>
RD
0.638
0.182
0.090
0.056
0.038
76
-------
APPENDIX II
THEORETICAL ANALYSIS OF DRY WEATHER SEWAGE VARIABILITY
The constant C equations for the hourly mass balance technique were
used for the theoretical analysis of dry weather variability on the runoff
and overflow variabilities.
C2 - CQ ANALYSIS
The overflow concentration with the sewage concentration perturbation
Ce±) is:
CPSiSiS°Si(C2i+ei)
C0 + SQ -- -i -
1 Si
i
Subtracting the overflow concentration from the above yields:
CA4)
Is. i
Applying the definition of variance yields:
v{£0} ' E{(R
The latter term is zero since E{e.} = 0, giving:
55? z CA5)
Q} = £{^2} E{(S s^)} CA7)
If one wet sample occurs during the day for the New York City sampling
regime of 5 samples total over a day, then 4 dry samples result, 2 wet sam-
ples require 3 dry, etc. The latter term in equation A7 can thus be ex-
pressed as follows:
77
-------
E{(E°Si£i)2} = E{(£1 + £2 + £3 + £4)2}PrRD=l + E{(£1 + £2
PrRD=2 + E{(£1 + £2)2pr= + E((£)2}Pr= (A8)
where 5
PrD_. - = P__ ../E P__ = 37.3%/66.96% = 0.557 from Table A-2
KU— J_ KJJ=X KU
= Probability that RD wet sample occurs over a day-
Expanding the above and realizing that all cross products are equal to
zero since E{e.} = 0, yields:
.
(A9)
2
An analysis of the different sampling intervals showed that E {s } was
not a function of RD but a constant, thus Equation A9 becomes:
E{(E0S.£.)2} . E{£±2}(4 PrffljBl + 3 PrRD=2 + 2 Pr^^ + Pr^^) (MO)
i
The latter term in the above can be simplified as follows:
4 PrRD=l + 3 PrRD=2 + 2 PrEID=3 + PrRD=4
C5-4)PrR])=4 + C5-5)PrRD=5
= 5 E Pr ' - E RD Pr__ = 5 - E {RD} = N - E {RD} (All)
.. i\D .. KD S
For the hourly simulator, the dry weather sewage variability was input
as a fraction (F) of the dry weather concentration as:
a . = F x C0.
ex 2i
78
-------
thus:
E{£ 2} = V{e.} = a 2 = F2 x C 2 . (A12)
j- JL £J_ ^w S
_ N
21s 2
where: C~ = r;— £ C_ . = average of the dry sewage concentrations
s i=l squared for the sampling hours.
Substituting Equations A-12, A-ll and A-10 into A-7, yields:
V{s } = F2 C 2 E{-~ } (N - E{RD» (A-13)
0 Zs RD
C_ - C_ ANALYSIS
/ K
The runoff perturbation due to sewage variability is given:
CR + ER = CC0 + £o) Cl
Subtracting the runoff concentration yields:
V{£R} = E{(£o(l + g/i) - |-
2
No E value results in the above since E{S } = 0, E{£.} = 0. Equation A15
can be expanded and the cross product deleted for the above reason to yield:
V{£_} = V{e
_ -..
K O L LI. 1
when: E{ (Z+Q .e.)2} = Q2 ECZ+£,)2
= Q 2 E{£.2}
D Q22 V{£i} (A-17)
since D = E{Z } = average storm duration.
79.
-------
The approach used to obtain Equation A-17 is similar to that used to obtain
A-ll. The V{s.} in the above is somewhat different than Equation A-12 since
it is taken over the total day not only over the sampled hours, thus:
= F2
24
where C 2 = \r £ C 2 • (A18)
1=1 Zl
Letting V.. = Cid where i is the mean storm intensity for the event, then:
or
Substituting A19-A17 into A16 yields:
(A19)
D Q C
22
^
(A20)
Q2 - CQ ANALYSIS
Variability in dry weather sewage flow rate has no effect on overflow
concentration since Q- does not appear in the equation.
Q0 - C_ ANALYSIS
Z K
The runoff concentration is given by;
£i)C2i
°R
which after subtracting C_, yields:
K
' V^ f £i - ^ f C2i£i
80
-------
and:
C - c7 2
V{e } = E{(-9- ?-) } E{(£+e,)2} (A23)
1 i x
-JT-.I.-J O
where C~ = average dry weather sewage concentration and no E term exists
since E{s±} = 0. From the results in Equations A17 and A19, Equation A23
becomes:
(C —C~~)
V{e } » ° 2— E{~-} E{\} D V{e.} (A24)
R C2 d2 i2 x
22 21 2
since V{e.} = F Q where Q = — Z Q , the above equation becomes:
i £* z, £,i\ •• «x
7 9 _ o
D FV
-------
D = 6 hr
Q2 =4.09 (MG/hr)2
C2. = 11020 (mg/£)2
C^ = 100 mg/A
C = 58 mg/£ (average of 6 runs)
Table A4 shows excellent agreement between predicted and observed stand-
ard deviations on overflow and runoff concentrations. The effect of dry
weather flow variability is relatively small, normally less than the aver-
aging error associated with the daily data base. Neglecting small storm
average rainfall intensities (<0.03' in/hr) significantly reduces the runoff
variability similar to the effect of measurement error.
TABLE A-4. RUNOFF AND OVERFLOW VARIABILITY DUE TO DRY WEATHER
RANDOM VARIABILITY
F,
C2
10
10
20
20
0
0
0
0
20
%
Q2
0
0
0
0
10
10
20
20
20
min
in/hr
0
0.03
0
0.03
0
0.03
0
0.03
0
0.03
C0, rng/A
Predicted Observed
15.6
15.6 17.8
32.5
32.5 27.8
0 M}9
0 7.1
0 .
0 7.1Z
32.5
32.5 35.1
CTCR,
Predicted
30.2
21.1
60.4
42.2
5.4
2.0
10.7
4.0
43.5
mg/fc
Observed
21.9
37.5
MD2
6.2
7.02
43.7
Variance of Averaging error removed from results
2
Averaging error same magnitude or greater than sewage variability error.
82
-------
APPENDIX III
ANALYSIS OF 26th WARD PLANT DATA
FLOW WEIGHTED AND EQUAL VOLUME ANALYSES
The New York City 26th Ward data was analyzed initially with the daily
mass balance technique using flow weighted composite sampling. A chlorides
balance was used to include tidegate leakage effects, the equations and re-
1-3
suits given in previous publications . To compare the results of the
equal volume composite sampling analysis as developed in this study to the
above flow weighted results required development of equations incorporating
the chlorides balance into the equal volume technique. The resulting equa-
tion for overflow concentration is:
CP l-« V2C2 + V5C5
[V4 - (.l-a)(V2 + V5)][<£ - (1ft ( \2 + ^ 5)] - aV5
CQ = c^ = V4 - (l-a)V2 - V5
which reduces to Equation 19 when V,. = 0 and C_ is constant over the day.
V_, the tidegate leakage volume over the day, is given by:
CA26)
+ / B2 - 4C
where :
B = D/F, C = E/F
D = CCp(2V2(l-a) - V4) - CC2V2(l-c02 + CC3(V4 - V2(2-a))
CC5(V4(l-a) - V2(l-2a))
E = V2(V4 - V2(l-a))(CCp - CC2(l-a) -
F = aCC3 + (l-2a)CC5 - (l-a)CCp
and CC refers to the concentration of chlorides at the various locations.
Table A-5 compares the equal volume to the flow weighted analysis for an
alpha minimum of 4 hours using the 1957 data. In both analyses, the dry
83
-------
TABLE A-5. FLOW WEIGHTED AND EQUAL VOLUME ANALYSIS FOR 1957 - 26th WARD DATA
ALPHA MINIMUM = 4 HOURS
FLOW WEIGHTED
SEWAGE
Avg Cone. , mg/A
Suspended
Solids
140
Volatile
Suspended
Solids
113
ANALYSIS
BOD5
122
Soluble
BOD
51
EQUAL VOLUME ANALYSIS
Volatile
Suspended Suspended
Solids Solids 5
Soluble
BOD5
SAME AS FLOW WEIGHTED ANALYSIS
Coeff. Variation
RUNOFF
o? Avg Cone., mg/A
Yearly load /volume
Arithmetic mean
Coeff. Variation
OVERFLOW
Avg. Cone. , mg/&
Yearly load/volume
Arithmetic mean
Coeff. Variation
WET DAYS
RAINFALL, in.
0.26
167
238
1.04
157
205
0.77
25
16.9
0.26
105
167
1.05
'
100
145
0.78
25
0.17
103
138
.0.88
103
128
0.66
24
0.29
43
51
0.79
44
49
0.65
21
173 95 93
261 173 134
1.22 1.30 1.08
161 89 94
220 147 123
0.93 1,00 0.78
25 25 24
16.9
35
44
1.03
37
43
0.83
21
-------
weather sewage data is analyzed similarly, thus no differences result. In
the table, the average concentration is calculated by 2 techniques, (1) by
dividing the total load over the year by the total volume and (2) by taking
an arithmetic mean of the daily concentrations.
Since sewage flows are relatively constant, arithmetic means are approx-
imately equal to the yearly load/volume. However both runoff and overflow
concentrations have significant differences due to the large flow variations.
Comparing average concentrations between flow weighted and equal volume ana-
lyses shows differences between 2 and 23% depending on the parameter analyzed.
From the analysis conducted with the hourly simulator, this magnitude of dif-
ference would be expected for an interceptor capacity of 1.5 to 2.0 Q2, typi-
al for New York City. The variability of the equal volume daily concentra-
tions about the arithmetic means is somewhat greater than the flow weighted
values.
Table A-6 presents the results of the equal volume analysis when all
rainfall durations were analyzed for the 1957 data. A significantly greater
variability results as well as some negative arithmetic mean concentrations.
Thus it is obvious that the individual concentrations from the short duration
storms with the resulting low rainfall volumes cannot be analyzed with the
daily mass balance technique. The overall load is not significantly affected
due to the low rainfall volumes of the shorter duration storms.
Figure A-l, the histogram for the runoff suspended solids concentration
for the 26th Ward data, indicates the large number of values existing at the
low and high runoff concentrations when all data are analyzed using the equal
volume analysis. These results are similar to those obtained previously with
the flow weighted analysis, resulting in a minimum duration storm analyzed of
4 hours.
EFFECT OF TIDEGATE LEAKAGE ON MASS BALANCE
If the effect of tidegate leakage is the same in wet and dry periods,
incorporating the tidegate leakage into dry weather sewage should result in
negligible changes in wet weather concentrations. This effect on the mass
balance calculation was studied by setting the tidegate volume equal to zero,
thereby incorporating the effect of tidegate leakage into the dry weather
sewage. Table A-7 indicates that the sewage volumes calculated without
chlorides are higher, but the change in overflow volumes is negligible. The
loads per unit area differ very little with a maximum difference of 3%. The
mean runoff concentrations (Table A-8) calculated without tidegate leakage
vary little from those calculated with tidegate leakage, and in most cases
the standard deviation of those calculated without was less. This shows that
the effect of tidegate leakage is approximately the same during wet and dry
periods, and therefore, the tidegate leakage was incorporated into the dry
weather sewage in the following analyses.
85
-------
TABLE A-6. EQUAL VOLUME ANALYIS FOR 1957 - 26th WARD DATA. ALPHA MINIMUM = 0
RUNOFF
Avg Cone. , mg/&
Yearly load/volume
Arithmetic mean
Coeff. Variation
OVERFLOW
Avg Cone. , mg/£
Yearly load/volume
Arithmetic mean
Coeff. Variation
WET DAYS
RAINFALL, in.
Suspended
Solids
169
-231
-8.83
159
10
95.1
77
24.0
Volatile
Suspended
Solids
98
-231
-9.15
99
-0.7
-1220
77
BOD5
109
155
7.54
104
148
4.47
76
Soluble
BOD5
51
46
18.5
56
62
7.7
65
TABLE A-7. EFFECT OF TIDEGATE LEAKAGE ON VOLUMES AND YEARLY
LOADS FROM EQUAL VOLUME ANALYSIS OF 26th WARD DATA
Parameter
RUNOFF SEWAGE OVERFLOW
with w/o with w/o with w/o
Tidegate Tidegate Tidegate Tidegate Tidegate Tidegate
Volume
LOAD Clb/acre)
Susp. Solids
Volatile S.S.
BOD
Soluble BOD
2220
2220
8600
9230
1990
2000
609
383
241
32
598
375
239
33
1971
1521
1643
517
1982
1526
1646
518
522
341
230
43
508
332
227
44
86
-------
>
a
u.
O
QC
LU
CQ
13
10
12
8
4
0
26th Ward 1957 Data
-
—
n
„, rffdllU
ALFMIN ' = 4 HOURS
-I. i //n
CO
Q
u.
O
oc
111
CQ
16
12
0
<-
yi//-
ALFMIN = 0 HOURS
,Jft
*-//-
450' -300 0 400 800 1200 >1450
RUNOFF SUSPENDED SOLIDS
CONCENTRATION (mg/L)
Figure A-l. Effect of Minimum Storm Duration on Runoff
Suspended Solids Histograms for 1957 - 26th
Ward Data Using Equal Volume Analysis
87
-------
TABLE A-8. EFFECT OF TIDEGATE LEAKAGE ON RUNOFF CONCENTRATIONS FROM EQUAL VOLUME
ANALYSIS OF 6 YEARS OF 26th WARD DATA
00
00
YEAR
54
57
60
63
66
69
PARAMETER
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
ARITHMETIC
TIDEGATE
LEAKAGE
150
72
34
261
173
134
44
145
80
55
11
273
184
171
- 4
221
139
164
34
162
114
93
29
MEAN, mg/£
w/o TIDEGATE
LEAKAGE
144
69
33
259
172
134
44
141
78
53
10
271
183
174
1
213
134
160
11
158
111
90
27
STANDARD
TIDEGATE
LEAKAGE
247
181
180
319
225
146
45
213 .
145
143
59
283
232
427
68
158
130
140
110
162
119
92
74
DEVIATION, mg/H
w/o TIDEGATE
LEAKAGE
244
177
175
315
222
144
45
209
143
141
58
272
221
412
69
151
128
139
113
165
119
89
73
-------
HOURLY MASS BALANCE METHOD
To analyze the 26th Ward WPCP data by the Hourly method the "Daily Bal-
ance" program was expanded from one to four parameters and changed to run on
a Tecktronix 4051 computer. The hourly rainfall data used by the program was
taken from the Avenue V, Brooklyn rainfall 'station, since this is the closest
rainfall station with an adequate data base. The equal volume variation of
the program, without chlorides, was run to obtain the wet weather data re-
quired by the hourly analysis program. The wet weather data required were:
alpha (the wet fraction of the day); total daily rainfall; daily volumes
for runoff, dry weather sewage, overflow, and the plant; as well as plant
and dry weather sewage concentrations for each parameter.
The first year analyzed was 1969. Various runs were made to determine
the sensitivity of the analysis to different minimum rainfall durations and
intensities. Figure A-2 shows histograms of runoff suspended solids concen-
trations calculated by: analyzing all data; using a minimum duration of 1
hour and intensity 0.03 in/hr; and a minimum duration of 2 hours and inten-
sity of 0.03 in/hr, respectively. Twenty-eight days were analyzed with a
minimum duration of 2 hours and intensity of 0.03 in/hr, with a standard
deviation of 131 mg/£. By lowering the minimum duration to 1 hour, 2 more
days were analyzed while the standard deviation increased to 170 mg/£. If
all the data were analyzed (minimum duration of 1 hour and intensity of 0.01
in/hr) concentrations are calculated for 41 days, with a standard deviation
of 329 mg/£. For these 3 analyses the range of the calculated mean runoff
suspended solids concentration is 67 mg/£, while the flow weighted average
concentration varies little, with a range of 19 mg/£. This is explained by
the large variability in the runoff concentration resulting from short storms
C<1 hr duration, <0.03 in/hr intensity) which have small volumes associated
with them.
Since the Avenue V rainfall station is southwest of the drainage area,
there was concern that the rainfall data recorded at Avenue V would not be
the same as the rainfall occurring over the drainage area. Using hourly
rainfall data from the La Guardia Airport weather station, a new hourly rain-
fall record was developed by combining the two rainfall records. This com-
bined hourly data was used by the hourly analysis to test for the number of
samples taken at the plant during wet periods of the day. Alpha (the wet
fraction of the day) was still input from the Daily Balance program based on
Avenue V, La Guardia, and Central Park rainfall data. As can be seen in
Table A-9, there is little difference between the analyses, thus for all
additional analyses, the Avenue V data was used.
Beginning in April 1959 the ten samples taken on Saturday and Sunday
were analyzed in one composite, as if the weekend was one 48 hour day.
Table A-10 shows the yearly average concentrations and standard deviations
for runoff and. overflow calculated with and without weekends. Since the
calculated concentrations vary little, and the standard deviations are gen-
erally higher without weekends with less data analyzed, the analysis to fol-
low included weekends.
The hourly analysis program obtains hourly plant and dry weather sewage
89
-------
26th Ward 1969 Data
£2
o
go
£6
CJ
O 2
CC
LU
OQ 0
i 6
- -4i£ 0/4 TA ANALYZED
. MINIMUM INTENSITY = 0.03 in/hr
nr
n n
. MINIMUM INTENSITY = 0.03 in/hr
MINIMUM DURA TION = 2 hr
, or
n
-1400 -1000 -600 -200 0 200 600
RUNOFF SUSPENDED SOLIDS CONCENTRATION
(mg/L)
Figure A-2. Effect of Minimum Rainfall Intensities
and Durations on Runoff Suspended Solids
Histograms for 1969 - 26th Ward Data
Using Hourly Analysis
90
-------
TABLE A-9. COMPARISON OF RUNOFF AND OVERFLOW CONCENTRATIONS USING ONE AND TWO RAINFALL
STATIONS FOR 1969 - 26th WARD DATA WITH HOURLY ANALYSIS1
RUNOFF VALUES, mg/Jl
PARAMETER
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Days Analyzed
ONE
Mean
Cone.
110
76
39
-6
STATION
Standard
Deviation
122
85
122
70
23
TWO
Mean
Cone.
109
74
44
-7
STATIONS
Standard
Deviation
131
86
142
63
25
OVERFLOW VALUES, mg/Jl
ONE STATION
Mean
Cone.
117
81
56
6
Standard
Deviation
91
63
94
54
23
TWO
Mean
Cone.
116
80
61
5
STATIONS
Standard
Deviation
95
64
103
51
25
1 a . = 2 hr
mm
min
=0.03 in/hr
-------
TABLE A-10. YEARLY AVERAGE RUNOFF AND OVERFLOW CONCENTRATIONS USING THE HOURLY ANALYSIS
to
RUNOFF VALUES, mg/£
YEAR
1960
1963
1966
1969
PARAMETER
Suspended Solids
Volatile SS
BOD
Soluble BOD
Days Analyzed
Suspended Solids
Volatile SS
BOD
Soluble BOD
Days Analyzed
Suspended Solids
Volatile SS
BOD
Soluble BOD
Days Analyzed
Suspended Solids
Volatile SS
BOD
Soluble BOD
Days Analyzed
WITH
Mean
Cone.
54
12
-3
263
191
33
-29
182
123
79
-3
109
82
43
-6
WEEKENDS
Standard
Deviation
206
174
173
56
28
306
249
404
74
26
163
146
150
79
20
146
102
133
63
28
WITHOUT
Mean
Cone.
55
21
2
3
303
209
63
-15
188
131
68
-4
110
76
39
-6
WEEKENDS
Standard
Deviation
213
180
179
62
26
313
262
439
73
21
170
136
130
95
14
122
85
122
70
23
WITH
"Mean
Cone.
63
22
9
3
232
172
35
16
178
127
79
1
116
86
61
5
OVERFLOW VALUES, mg/S,
WEEKENDS
Standard
Deviation
159
133
131
42
246
200
233
56
110
98
95
64
93
73
96
49
WITHOUT
Mean
WEEKENDS
Standard
Cone. Deviation
65
31
13
4
277
193
58
-4
182
132
74
1
117
81
56
6
164
136
135
45
254
214
250
54
115
99
95
76
91
63
94
54
1
a .
mm
= 2 hr
i . =0.03 in/hr
mm
-------
concentrations and volumes by assuming them to vary according to the varia-
tion of BOD and flow obtained by the Hydroscience Inc. 208 study, Oct. 12-13,
1976 at the 26th Ward WPCP. When the concentrations calculated by the hourly
analysis program (min. duration = 2 hr., min intensity = 0.03 in/hr) were
compared to those calculated previously by the "Daily Balance" program (min
duration = 4 hr) the hourly analysis results were significantly lower. The
runoff suspended solids, for example, were more than 25% lower. To determine
if the assumed variable sewage characteristics were the cause of the lower
concentrations, the analysis was run for 1960 with constant sewage character-
istics. The concentrations calculated with constant characteristics were an
average of 50% higher than those calculated with variable characteristics.
Assuming the sewage variation to be correct, the effects of the New York City
sampling schedule were studied. New York City samples at 6 AM, 10 AM, 2 PM,
6 PM, and 10 PM, using equal volume composites. The dry weather sewage con-
centration, Cpd, obtained from analysis of the dry weather data is greater
than the arithmetic average concentration as follows:
5Cpd = 0.99+1.59-KL.36+1.36+0.25
Cpd = 1.11
where C , = REPORTED AVERAGE DRY WEATHER SEWAGE CONCENTRATION
= ACTUAL AVERAGE DRY WEATHER SEWAGE CONCENTRATION
When this was incorporated into the Hourly Analysis the resulting runoff
suspended solids concentrations were an average of 23% higher. The bias re-
sulting from the NYC sampling scheme would also affect the equal volume con-
centrations, where the same dry weather sewage error exists, but the error in
the plant concentration would be a function of the number of hours of rain-
fall.
A comparison of the average unit loads calculated by the flow weighted
and equal volume variations of "Volbal" and by the hourly analysis is shown
in Table A-ll. All unit loads except for soluble BOD5 are similar. The
hourly analysis shows that runof-f and overflow soluble BOD_ are significantly
lower than previously estimated. The above analyses assume that all para-
meters have diurnal fluctuations similar to BOD-. Dry weather hourly data
for all parameters would be required to verify this assumption.
The yearly runoff concentrations for all 4 parameters using the hourly
analysis technique are shown in Figures A-3 and A-4 and in Figure A-5 for
sewage concentrations. A significant degree of variability occurs from year
to year while the values for soluble BOD,, are close to zero. Table A-12 pre-
sents the weighted average concentrations for runoff, sewage and overflow
from the hourly analysis results. The suspended solids concentrations are
higher in the runoff and overflow than in the sewage while the other para-
meters are lower except for the volatile suspended solids which is similar
for the three locations.
93
-------
RUNOFF CONCENTRATIONS
CO
0
SPENDEDS
mg/L
D
CO
4UU
300
200
100
0
KO
M
~: 1
i i
OU
51 54
>EAN±
I
1
57
«^I
T
}
i i
60 63
I
1
66
J
1
69
Q
111
Q
z -J
£ g
co E
if
58
^j
o
^
4UU
300
200
100
0
en
•—
—
T
~- J 5 * 5
JL
1 t 1 1 I I 1
51 54 57 60 63 66 69
YEAR
Figure A-3. Yearly Runoff Suspended Solids Con-
centrations for the 26th Ward Data
Using the Hourly Analysis
RUNOFF CONCENTRATIONS
fUU
_j 30°
05
^ 200
o
O
00 100
0
KO
__
—
_
•r^MEAN - S.E.
- ' ' ' I
***
; i
I ! | i i i
i
i
51 54 57 60 63 66 69
*»uu
_l
^300
§200
LU
_J
M 100
co 0
RO
-
-
-
-
5 51 I n
5 i s
I i i I ¥ i i
51 54 57 60 63 66 69
YEAR
Figure A-4, Yearly Runoff BOD Concentrations
for the 26th Ward Data Using the
Hourly Analysis
-------
SEWAGE CONCENTRATIONS
VOLATILE
SOLUBLE SUSPENDED SUSPENDED
BOD, mg/L BOD, mg/L SOLIDS, mg/L SOLIDS, mg/L
o§8g8oS8§8 og8S8o§8g|
\
MEAN ± S.D.
54 57
-5 I
i i
54 57
:* I
54 57
i i
54 57
60 63 66 69
I $ $ ;
i i i i
60 63 66 69
i i 5 i
i i i I
60 63 66 69
5 I 5 5
1 1 1 1
60 63 66 69
YEAR
Figure A-5. Yearly Sewage Concentrations
for the 26th Ward Data
95
-------
TABLE A-ll. COMPARISON OF AVERAGE YEARLY UNIT LOADS BY DAILY AND HOURLY MASS BALANCE
METHODS FOR 26th WARD DATA
RUNOFF LOAD (Ib/ac-in)
1 2
Daily Balance Hourly Analysis
Parameter
Suspended Solids
Volatile S.S.
BOD
Soluble BOD
Days Analyzed
Flow
Weighted
28.5
18.2
13.0
2.47
221
Equal
Volume
29.9 29.0
17.7 17.4
14.6 12.0
2.41 0.42
221 226
OVERFLOW LOAD (Ib/ac-in)
1 2
Daily Balance Hourly Analysis
Flow
Weighted
24.4
16.1
12.1
2.80
Equal
Volume
25.2
13.2
15.7
3.21
24.2
15.0
12.9
1.23
a . = 4 hr.
min
Skip Intensities <0.03 in/hr, a'. =2 hr.
mm
-------
' -i
t
TABLE A-12. WEIGHTED AVERAGE CONCENTRATIONS FOR 26th WARD
DATA OVER TOTAL STUDY USING HOURLY ANALYSIS
Weighted Average
Concentration (mg/£)
Flow
Point
Runoff
Sewage
Overflow
SS
183
143
174
VSS
109
109
107
BOD
82
121
87
Soluble
5
3
55
9
RAINFALL - RUNOFF RELATIONSHIPS
Linear regression was performed on the runoff concentrations to deter-
mine the relationship between concentration and duration of storms, and be-
tween concentration and interval between storms. For storms that lasted more
than one day the two days were combined into 1 event.
For both duration and interval the regression was performed on the re-
sults of three analyses: all data analyzed, alpha minimum of 1 hour and min-
imum intensity of 0.03 in/hr, and alpha minimum of 2 hours and minimum inten-
sity of 0.03 in/hr. For both duration and interval the best correlation
occurred with an alpha minimum of 2 hours and minimum intensity of 0.03 in/hr.
The regression plots for suspended solids and the effect of minimum
duration analyzed have been shown previously in Section 8. The regression
parameters for suspended solids, BOD-, volatile suspended solids, and soluble
BOD are shown in Table A-13 for both duration and interval between events,
obtained using an alpha minimum of 2 hr. and minimum intensity of 0.03 in/hr.
Only the suspended solids data, duration data for BOD,, and interval data for
volatile suspended solids have slopes that are not zero within the 95% con-
fidence limits. The regression parameters from a multiple regression of run-
off concentration vs. duration of events and interval between events is shown
in Table A-14. Higher correlation coefficients result when both parameters
are analyzed together as anticipated.
97
-------
TABLE A-13. RAINFALL-RUNOFF RELATIONSHIPS FOR FOUR PARAMETERS
ANALYZED AT 26th WARD USING SEPARATE LINEAR REGRESSIONS1
FOR DURATION AND INTERVAL
SLOPE
PARAMETER
Suspended
Volatile
Solids
BOD5
Solids
Suspended
Soluble BOD
-8
-1
-4
-0
.34(±
.44(±
.49(±
.50(±
4.
1.
4.
2.
8)
8)
4)
9)
Y- INTERCEPT ,
PARAMETER
307.
162.
156.
33.
2
mg/Jl r , %
N, days
VS. DURATION
2 (±56
3 (±36
4 (±50
5 (±33
PARAMETER VS.
Suspended
Volatile
Solids
BOD5
Solids
Suspended
Soluble BOD5
13
9
4
2
.15(±
.88(±
.23(±
.25(±
9.
7.
8.
4.
0)
7)
1)
9)
178.
108.
98.
20.
8 (±48
1(±43
6 (±43
1(±20
.0)
.4)
-2)
• 0)
INTERVAL
.4)
.0)
.7)
•0)
6.
1.
2.
0.
4.
4.
0.
1.
3
82
32
22
6
6
62
22
181
138
179
74 •
181
138
179
74
a . = 2 hr., i . = 0.03 in/hr.
mm mm
(. ) refers to 95% confidence limits
r2 not different than zero at 5% level of significance.
98
-------
TABLE A-14. RAINFALL-RUNOFF RELATIONSHIP FOR FOUR PARAMETERS
ANALYZED AT 26th WARD USING MULTIPLE LINEAR REGRESSION1
FOR DURATION AND INTERVAL
PARAMETER
Suspended Solids
Volatile S.S.
BOD5
Soluble BOD
EQUATION
y =
y =
y =
y =
256-7.41 ^ +
123-1.12 j^ +
142-4.24 X;L +
22.6-0.25 XT
11.0 x2
9.16 x2
3.07 x2
+ 2.17 x9
r2, %
9.4
5.7
2.72
1.22
N, days
181
138
179
74
where y = Concentration of Parameter (mg/&)
x1 = Duration of Storm (hours)
x- = Interval Between Storms (days)
a . = 2 hr, i . =0.03 in/hr.
mxn mm
O
2
r not different than zero at 5% level of significance.
99
-------
APPENDIX IV
APPLICABILITY
The applicability of the mass balance technique throughout the U.S. was
investigated. The first step taken was to determine the extent of combined
sewers in the U.S. It was found that 37.6 million people living in urban
areas in the U.S. are served by combined sewers (Sullivan, Richard H. et al.,
1977) 1. This represents 25.3% of the urban population.
Using the above data on population served by type of sewerage system,
Figure A-6 was developed. This map of the U.S. shows the range of combined
sewerage service in the U.S. Within each EPA region there is a wide range of
percentage of population served by combined sewers. States with over 30% of
the population served by combined sewers are mostly in the Northeast and Mid-
west, with Washington and Oregon also included.
Since the Balance technique is applicable to specific urban areas con-
taining CSOs and treatment facilities, an analysis of these areas was con-
ducted. Table A-15 lists 71 urban area's in 27 states with over 30% of the
population served by combined sewers. This table contains 72% of the total
population served by combined sewers and gives a much better picture of the
extent of combined sewerage. For example, the state of Georgia has 21.3% of
the population served by combined sewers, seemingly low, but Albany, Savannah
and Augusta, Ga. have 100, 61, and 49% served, respectively. California, as
a state, has only 9.2%, but San Francisco has 47% of the urban population
served by combined sewers. This represents 84.8% of the combined sewered
population in the state. In some states, rather than having a state-wide
application, the "Balance" technique would be limited to one or two urban
areas, but still include most of the combined sewerage in the state. This is
the case for California, as well as Nebraska, Nevada, Oregon, Rhode Island,
Texas, Virginia, and Washington.
Three states, Maine, New Hampshire, and Vermont, are sewered exclusively
by combined sewers. Some states, Illinois, Indiana, Michigan, Missouri, New
York, and West Virginia, have several urban areas, each with high percentages
of combined sewers, accounting for most of the combined sewerage in the state
and a high percentage of the total sewerage in the state. The remaining 11
states from Table II, Arkansas, Connecticut, Georgia, Iowa, Kansas, Kentucky,
Massachusetts, Ohio, Pennsylvania and Tennessee, have percentages of combined
sewerage in specific urban areas ranging from 100-34%, but a small percentage
of the total sewerage of the state.
Sullivan, Richard H., et al. "Nationwide Evaluation of Combined Sewer
Overflows and Urban Stormwater Discharges, Volume I: Executive Summary".
EPA-600/2-77-064a. 'Sept. 1977.
100
-------
Figure A-7 shows the number of urban areas in each state. It is clear
that the concentration of combined sewerage is greatest in the Northeast and
Midwest, with a few additional urban areas scattered in other parts of the
country.
2
In reviewing Canadian literature Gore and Storrie Limited, 1978 , deter-
mined that 20.4% of the population is served by combined sewers. Twenty
urban areas in Ontario have more than 35% of their population served by com-
bined sewers (Sullivan, Richard H., et al., 1978)3. Similar statistics might
be expected in Quebec and Manitoba since the percentage of combined sewerage
is higher in these provinces than in Ontario.
Once the extent of combined sewerage was determined, a questionnaire
was prepared to be used in gathering information on the method of sampling
used by treatment plants across the country. Initially, attempts were made
to obtain information over the telephone, this however proved to be ineffec-
tive. Questionnaires were mailed to EPA regional offices with a request for
addresses of state offices if the information was not available at the region-
al office. While some state offices were able to supply the requested infor-
mation others forwarded the questionnaire to local governments. Twenty-six
cities from eleven states returned the questionnaires with information from
fifty-four sewage treatment plants. All plants reported using composite sam-
ples, 35 of which were equal volume, and 19 flow weighted. Of the 54 plants,
21 take samples for the composite four hours apart, 9 plants take samples
every two hours, 12 plants take samples every hour, and 12 plants sample be-
tween every ten and twenty minutes. The mass balance hourly analysis would
be adaptable to the above plants with greater variability expected from the
24 plants sampling hourly or relatively continuously.
2
Gore and Storrie Limited. "Review of Canadian Municipal Urban Drainage
Policies and Practices." Ontario Ministry of the Environment. 1978.
Sullivan, Richard H., et al. "Evaluation of the Magnitude and Significance
of Pollution Loadings from Urban Stormwater Runoff in Ontario." Ontario
Ministry of the Environment. 1978.
101
-------
EPA regions
30<%<55
10<%<30
Figure A-6. Percentage of Population in each State
Served by Combined Sewers
Figure A-7. Number of Urban Areas in Each State Having More Than
30% of the Population Served by Combined Sewers
102
-------
SEWAGE TREATMENT PLANT DATA SURVEY
State New York
City New York City
Number of Plants in City 12
Plant Name Summary for City
Drainage Area
Population Served
Population Served by Combined Sewers
FLOW:
Peak Hourly Hydraulic Capacity 2x Plant Design Capacity if Primary
ByPass Available - 1.5x Plant Design
Capacity if Primary Bypass Not Available
Daily Plant Flow CYearly Average)
SAMPLING:
Type of Sampling: Grab or Composite X
If Composite
1) Number of Samples Comprising Composite 6
2) Times of Sampling for Composite 10AM, 2PM, 6PM. 10PM, 2AM,
SAM CPrior to ^ 1975 no 2 AM sample).
3) Method of Composite:
a) Equal Volume X
b) Flow Weighted
Number of Composite Samples per Week 6 No Friday Samples
103
-------
ANALYSIS
Number of years for which data is available Approximately 35 years
Parameters
Analyzed
BOD
Soluble BOD
Suspended Solids
Volatile S.S.
NH3-N
Org-N
Total P
Total Coliform
Fecal Coliform
C
u
CR
Daily
X
X
X
X
Frequency of Analysis
Weekly Monthly
X
X
X
Other
2/Month
2/Month
2/Month
2/Month
2/Month
H
g
Pb
ZN
104
-------
TABLE A-15. URBAN AREAS WITH HIGH PERCENTAGES OF POPULATION SERVED BY COMBINED SEWERS
o
Ln
Urban Area
Albany
Anderson
Evansville
Huntington
Lima
Lynchberg
St. Joseph
Steubenville
Steubenville Metro
Philadelphia Metro
Lafayette
Binghampton
Chicago
Springfield
New Bedford
Manchester
Lewis ton
Owensboro
Portland
Scranton
Wheeling
Spokane
Cincinnati
Saginaw
Fall River
Galveston
New York City
State
Ga.
In.
In.
WV
Oh.
Va.
Mo.
Oh.
WV
NJ
In.
NY
11.
Oh.
Ma.
NH
Me.
Ky.
Me.
Pa.
WV
Wa.
Oh.
Mi.
Ma.
Tx.
NY
% of Total Pop.
Served by Com-
bined Sewer
100
100
100
100
100
100
100
100
100
99.5
92.4
86.8
77.3
76.6
75.4
74.7
73.8
73.6
72.6
72.5
72.0
71.7
70.1
69.6
t, 66.2
64.5
64.3
Pop. Served by
Combined Sewers
(1000 Persons)
76
81
142
121
70
71
77
45
40
201
73
145
4416
72
101
71
48
39
77
148
67
165
778
103
92
40
6764
Area Served by
Combined Sewers
(1000 Acres)
12.0
20.0
15.8
14.8
10.1
10,4
14.4
4.2
6.6
20.3
5.0
15.2
204.9
7.3
8.9
7.0
4.7
3.5
10.3
17.1
6.7
19,4
73.4
11.5
7.2
2.5
108.3
% of Total
Area
56.9
72.7
60.3
63.8
58.4
43.9
70.2
53.2
38.6 .
64.6
41.0
45.6
32.7
45.6
40.8
28.0
10.8
45.5
28.8
27.3
37.4
38.9
10.3
40.8
26.2
17.0
44.5
-------
TABLE A-15 (cont'd)
Urban Area
Detroit
Lawrence
Savannah
South Bend
Omaha
Buffalo
Hartford
Nashua
St. Louis
Bay City
Albany
Indianapolis
Decatur
Charleston
Washington B.C.
New Haven
Norwalk
Fort Wayne
Davenport Metro
Jackson
Springfield
Augusta
Richmond
Muncie
Lowell
Peoria
San Francisco
Des Moines
State
Mi.
Ma.
Ga.
In.
Ne.
NY
Ct.
NH
Mo.
Mi.
NY
In.
11.
WV
—
Ct.
Ct.
In.
11.
Mi.
Ma.
Ga.
Va.
In.
Ma.
11.
Ca.
la.
% of Total Pop.
Served by Com-
bined Sewer
62.3
61.5
61.0
60.8
60.2
59.1
59.1
59.0
58.2
57.7
55.8
55.6
54.0
53.2
52.8
51.4
51.4
50.7
50.0
50.0
49.4
49.0
48.1
47.8
47.5
47.4
47.0
46.3
Pop. Served by
Combined Sewers
(1000 Persons)
2475
123
100
175
296
624
275
36
1096
45
271
456
54
84
400
179
55
114
56
39
254
73
200
43
87
117
1410
118
Area Served by
Combined Sewers
(1000 Acres)
166.2
12.0
6.5
20.2
20.9
38.3
20.8
4.9
112.7
5.2
19.3
34.0
6.5
6.8
12.7
14.9
1.5
8.1
5.7
4.4
32.9
9.0
15.9
4.3
6.2
14.8
54.1
4.0
% of Total
Area
29.8
22.3
15.9
30.7
21.6
28.0
24.8
22.5
38.2
31.3
20.0
13.9
27.4
17.1
32.3
21.8
5.5
18.3
25.3
19.1
21.6
24.3
17.1
26.9
15.6
51.9
12.4
5.7
-------
TABLE A-15 (cont'd)
<< -
•4*
I-1
o
Urban Area
Chicago Metro
Youngs town
Syracuse
Providence
Toledo
Nashville
Rochester
Fort Smith
Seattle
Hamilton
Portland
Kansas City
Pittsburgh
Reno
Topeka
Beaumont
State
In.
Oh.
NY
RI
Oh.
Tn.
NY
Ar.
Wa.
Oh.
Or.
Mo.
Pa.
Nv.
Ks.
Tx.
% of Total Pop.
Served by Com-
bined Sewer
45.3
43.5
42.3
41.9
41.8
40.2
39.9
39.5
39.0
38.5
38.0
37.2
36.1
35.0
34.1
32.8
Pop. Served by
Combined Sewers
(1000 Persons)
453
172
159
333
204
180
240
30
483
35
316
308
667
35
45
38
Area Served by
Combined Sewers
(1000 Acres)
32.3
13.7
13.2
21.0
15.9
14.7
14.3
4.3
37.9
2.4
24.2
21.0
31.5
2.4
3.7
2.8
% of Total
Area
16.9
16.6
21.5
13.4
15.0
6.7
15.3
11.2
14.3
9.9
14.2
8.1
8.3
9.9
10.9
5.8
TOTAL
27,076
1,559.6
SOURCE: Heany, James P. et al., "Nationwide Evaluation of Combined Sewer Overflows
and Urban Stormwater Discharges Volume II: Cost Assessment and Impacts" EPA-600/2-77-064
March 1977.
-------
APPENDIX V
DEFINITION OF SYMBOLS
Symbol Subscript Superscript Description.
A drainage area
C overall runoff coefficient = fC A
v
1,2,3,4,5 actual concentrations at specified loca-
tion:
1 = Runoff
2 = Dry weather sewage
3 = Overflow
4 = Plant
5 = Tidegate leakage
i,k concentration during specified hour
j concentration during specified day
- average concentration over day
d,w concentration during dry and wet portions
of day, respectively
0,P,R estimated concentration at specified loca-
tion:
0 = Overflow
P = Plant
R = Runoff
R peak hourly runoff concentration
TflflTf * *
R runoff concentration at infinite time
CO
after beginning of event
s - average concentration during sampling hours
v volumetric runoff coefficient
CC 2,3,4,5,P chloride concentration at specified loca-
tion (.see "C")
d duration of storm
i duration of storm during specified hour
D mean duration of storm for total record
E{ } expected value
f units conversion factor for rainfall-runoff
relationship
108
-------
DEFINITION OF SYMBOLS (continued)
Symbol Subscript Superscript _ Description _
F ratio of the hourly random sewage varia-
bility (standard deviation) to the
hourly sewage concentration
i average rainfall intensity for event
j average rainfall intensity for event on
specified day
k rainfall intensity for specified hour
I mean event rainfall intensity for total
record
N number of days analyzed
s number of sampling hours used to obtain
daily composite sample for plant vari-
ance
o i variable used to indicate hours in which
overflows occurred (o . =1) or did not
occur (o . =0)
P( ) i,d,6 probability density function for specified
variable, rainfall, duration, interval,
respectively
P d probability of occurrence of a specified
duration
wd probability of occurrence of a specified
duration being wet
ED probability that a specified number (ED)
of samples are taken during wet hours
over all durations from 1 to 24 hours
Pr RD probability that a specified number (RD)
of samples are taken during wet hours
over a day
Q 1,2,3,4,5 flow rate at specified location (see "C")
d,w flow rate during dry and wet hours, re-
spectively
i,k flow rate during specified hour
j flow rate during specified day
I maximum hourly flow rate to treatment
plant during storm event. (interceptor
capacity)
109
-------
i , '
DEFINITIONS OF SYMBOLS (continued)
Symbol Subscript Superscript Description
2
r correlation coefficient squared = fraction
of variability explained by correlation
RD number of samples taken during wet hours
for daily plant influent composite .sam-
ple
s i variable used to indicate hours in which
a plant influent sample is taken (s.=l)
or not taken (s.=0) for the daily com-
posite
STD standard deviation of measurement error on
daily composite plant influent sample
t d,w length of dry and wet periods respectively
during a day
V 1,2,3,4,5 total daily volume at specified location
Csee "C")
d,w total daily dry and wet volume, respec-
tively
V{ } variance of specified parameter
1 variance of measurement error
2 variance of averaging error
W total load over day
a fraction of day over which rainfall
occurred
min minimum a analyzed in record
6 ratio of sewage flow rate (.0-) to overall
runoff coefficient (C)
R rate coefficient for exponential decay of
runoff concentration during first flush
6 interval between storm events
A mean interval between storm events for
total record
e ik,C?k,Q9k fluctuation about hourly values for inten-
sity, dry weather sewage concentration,
and flow respectively
110
-------
DEFINITIONS OF SYMBOLS (continued)
Symbol Subscript Superscript Description
£ C fluctuation about daily plant value due to
•* measurement error
oj,rj error in daily flow weighted and runoff
overflow concentrations, respectively
o,r - runoff overflow and bias, respectively,
for period of record analyzed
o,p,r perturbation on overflow, plant and runoff
concentrations, respectively
i hourly perturbation on dry weather sewage
concentration or flow rate
a CR,CO standard deviation of the errors (true -
estimated) of the daily runoff and over-
flow concentrations for the period of
record analyzed
p see "STD"
CR,C0,p 2 variance of specified values
Z ' sum between specified limits
v R,0 coefficient of variation of the errors
(true - estimated) of the daily runoff
and overflow concentrations for the
period of record analyzed
111
-------
|