water and wastewater calculation manuals (1)

The author has performed anextensive survey of literature on surface water and groundwater per-taining to environmental engineering and compiled them in this book.Rules and regulations a

Water and Wastewater Calculations Manual

ABOUT THE AUTHOR

Shun Dar Lin is an Emeritus Faculty of University of

Illinois and in Taiwan A registered professional engineer

in Illinois, he has published nearly 100 papers, articles, and

reports related to water and wastewater engineering

Dr Lin brings to the book a background in teaching, research,

and practical field experience spanning nearly 50 years

Dr Lin received his Ph.D in Sanitary Engineering from

Syracuse University, an M.S in Sanitary Engineering from

the University of Cincinnati, and a B.S in Civil Engineering

from National Taiwan University He has taught and

conducted research since 1960 at the Institute of Public

Health of National Taiwan University In 1986, Dr Lin

received the Water Quality Division Best Paper Award for

“Giardia lamblia and Water Supply” from the American

Water Works Association He developed the

enrichment-temperature acclimation method for recovery enhancement

of stressed fecal coliform The method has been adopted in

the Standard Methods for the Examination of Water and

Wastewater since the 18th edition (1990) Dr Lin is a life

member of the American Society of Civil Engineers, the

American Water Works Association, and the Water

Environment Federation He is a consultant to the

governments of Taiwan and the United States and for

Water and Wastewater Calculations Manual

Shun Dar Lin

C C Lee

Editor of Handbook of Environmental Engineering Calculations

Second Edition

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INFORMA-or otherwise

DOI: 10.1036/0071476245

9 Determination of Reaeration Rate Constant K2 41

16 Velz Reaeration Curve (A Pragmatic Approach) 87

17 Stream DO Model (A Pragmatic Approach) 92

References 121

References 306

5 Urban Stormwater Management 549

20 Biological (Secondary) Treatment Systems 617

29 Sludge (Residuals) Treatment and Management 796

References 884

Appendix A Illinois Environmental Protection Agency’s

Appendix C Solubility Product Constants for Solution

Appendix D Freundlich Adsorption Isotherm Constants

Index 919

Contents vii

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This manual presents the basic principles and concepts relating towater/wastewater engineering and provides illustrative examples ofthe subject covered To the extent possible, examples rely on practicalfield data and regulatory requirements have been integrated into theenvironmental design process Each of the calculations provided herein

is solved step-by-step in a streamlined manner that is intended to itate understanding Examples (step-by-step solutions) range from cal-culations commonly used by operators to more complicated calculationsrequired for research or design For calculations provided herein usingthe US customary units, readers who use the International System mayapply the conversion factors listed in Appendix E Answers are also gen-erally given in SI units for most of problems solved by the US custom-ary units

facil-This book has been written for use by the following readers: studentstaking coursework relating to “Public Water Supply,” “Waste-WaterEngineering,” or “Stream Sanitation”; practicing environmental (sani-tary) engineers; regulatory officers responsible for the review andapproval of engineering project proposals; operators, engineers, andmanagers of water and/or wastewater treatment plants; and other pro-fessionals, such as chemists and biologists, who need some knowledge

of water/wastewater issues This work will benefit all operators andmanagers of public water supply and of wastewater treatment plants,environmental design engineers, military environmental engineers,undergraduate and graduate students, regulatory officers, local publicworks engineers, lake managers, and environmentalists

Advances and improvements in many fields are driven by competition

or the need for increased profits It may be fair to say, however, thatadvances and improvements in environmental engineering are driveninstead by regulation The US Environmental Protection Agency (US EPA)sets up maximum contaminant levels, which research and projectdesigns must reach as a goal The step-by-step solution examples pro-vided in this book are guided by the integration of rules and regulations

ix

Copyright © 2007, 2001 by The McGraw-Hill Companies, Inc Click here for terms of use

on every aspect of water and wastewater The author has performed anextensive survey of literature on surface water and groundwater per-taining to environmental engineering and compiled them in this book.Rules and regulations are described as simply as possible, and practicalexamples are given.

The text includes calculations for surface water, groundwater, ing water treatment, and wastewater engineering Chapter 1 comprisescalculations for river and stream waters Stream sanitation had beenstudied for nearly 100 years By mid-twentieth century, theoretical andempirical models for assessing waste-assimilating capacity of streamswere well developed Dissolved oxygen and biochemical oxygen demand

drink-in streams and rivers have been comprehensively illustrated drink-in thisbook Apportionment of stream users and pragmatic approaches forstream dissolved oxygen models also first appeared in this manual.From the 1950s through the 1980s, researchers focused extensively onwastewater treatment In the 1970s, rotating biological contactorsbecame a hot subject Design criteria and examples for all of these areincluded in this volume Some treatment and management technologiesare no longer suitable in the United States However, they are still ofsome use in developing countries

Chapter 2 is a compilation of adopted methods and documentedresearch In the early 1980s, the US EPA published Guidelines forDiagnostic and Feasibility Study of Public Owned Lakes (Clean LakesProgram, or CLP) This was intended to be as a guideline for lake man-agement CLP and its calculation (evaluation) methods are presentedfor the first time in this volume Hydrological, nutrient, and sedimentbudgets are presented for reservoir and lake waters Techniques forclassification of lake water quality and assessment of the lake trophicstate index and lake use support are also presented

Calculations for groundwater are given in Chapter 3 They includegroundwater hydrology, flow in aquifers, pumping and its influence zone,setback zone, and soil remediation Well setback zone is regulated by thestate EPA Determinations of setback zones are also included in thebook Well function for confined aquifers is presented in Appendix B.Hydraulics for environmental engineering is included in Chapter 4.This chapter covers fluid (water) properties and definitions, hydrostat-ics, fundamental concepts of water flow in pipes, weirs, orifices, and inopen channels, and flow measurements Pipe networks for water supplydistribution systems and hydraulics for water and wastewater treatmentplants are also included

Chapters 5 and 6 cover the unit process for drinking water and water treatment, respectively The US EPA developed design criteria andguidelines for almost all unit processes These two chapters depict theintegration of regulations (or standards) into water and wastewater

waste-x Preface

design procedures Drinking water regulations and membrane filtrationare updated in Chapter 5 In addition, three new sections on pellet soft-ening, disinfection by-products (DBP), and health risks, also are incor-porated in Chapter 5 The DBP section provides concise information fordrinking water professionals Although the pellet softening process isnot accepted in the United States, it has been successfully used in manyother countries It is believed that this is the first presentation of pelletsoftening in US environmental engineering books Another new section

of constructed wetlands is included in Chapter 6 These two chapters(5 and 6) are the heart of the book and provide the theoretical consid-erations of unit processes, traditional (or empirical) design concepts, andintegrated regulatory requirements Drinking water quality standards,wastewater effluent standards, and several new examples have alsobeen added

The current edition corrects certain computational, typographical,and grammatical errors found in the previous edition

Charles C C Lee initiated the project of Handbook of Environmental

Engineering Calculations Gita Raman of ITC (India) did excellent

editing of the final draft The author also wishes to acknowledge MeilingLin, for typing the corrected manuscript Ben Movahed, President ofWATEK Engineering, reviewed the section of membrane filtration.Alex Ya Ching Wu, Plant Manager of Cheng-Ching Lake AdvancedWater Purification Plant in Taiwan, provided the operational manualfor pellet softening Mike Henebry of Illinois EPA reviewed the section

of health risks Jessica Moorman, Editor of Water & Waste Digest,

pro-vided 2006 drinking water regulatory updates Thanks to Dr Chuan-juiLin, Dr C Eddie Tzeng, Nancy Simpson, Jau-hwan Tzeng, Heather Lin,Robert Greenlee, Luke Lin, Kevin Lin, and Lucy Lin for their assistance.Any reader suggestions and comments will be greatly appreciated

SHUNDARLIN

Peoria, Illinois

Preface xi

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5 Dissolved Oxygen and Water Temperature 5

6 Biochemical Oxygen Demand Analysis 10

7 Streeter–Phelps Oxygen Sag Formula 13

8.2 Determination of deoxygenation rate

9.4 Stationary field monitoring procedure 46

10.1 Relationship of sediment characteristics and SOD 51

13 Natural Self-Purification in Streams 55

16 Velz Reaeration Curve (a Pragmatic Approach) 87

por-at several sampling points along a stream reach Both approaches areuseful for developing or approving the design of wastewater treatmentfacilities that discharge into a stream

In addition, biological factors such as algae, indicator bacteria, sity index, and macroinvertebrate biotic index are also presented

diver-2 Point Source Dilution

Point source pollutants are commonly regulated by a deterministicmodel for an assumed design condition having a specific probability

of occurrence A simplistic dilution and/or balance equation can bewritten as

Cu⫽ constituent concentration of upstream flow, mg/L

Qe⫽flow of the effluent, cfs

Ce⫽constituent concentration of the effluent, mg/L

Cd5 QuCu1 QeCe

Qu1 Qe

2 Chapter 1

Under the worst case, a 7-day, 10-year low flow is generally used forstream flow condition, for design purposes.

Example: A power plant pumps 27 cfs from a stream, with a flow of 186 cfs The discharge of the plant’s ash-pond is 26 cfs The boron concentrations for upstream water and the effluent are 0.051 and 8.9 mg/L, respectively Compute the boron concentration in the stream after completely mixing.

solution: By Eq (1.1)

3 Discharge Measurement

Discharge (flow rate) measurement is very important to provide thebasic data required for river or stream water quality The total dis-charge for a stream can be estimated by float method with wind andother surface effects, by die study, or by actual subsection flow meas-urement, depending on cost, time, manpower, local conditions, etc Thedischarge in a stream cross section can be measured from a subsection

by the following formula:

Q ⫽ Sum (mean depth ⫻ width ⫻ mean velocity)

determined at any specific reach as the channel volume of the reachdivided by the flow as follows:

(1.3)

where t ⫽ time of travel at a stream reach, days

Q ⫽ average stream flow in the reach, ft3/s(cfs) or m3/s

86,400 ⫽ a factor, s/d

Example: The cross-sectional areas at river miles 62.5, 63.0, 63.5, 64.0, 64.5, and 64.8 are, respectively, 271, 265, 263, 259, 258, and 260 ft 2 at a sur- face water elevation The average flow in 34.8 cfs Find the time of travel for

a reach between river miles 62.5 and 64.8.

solution:

Step 1 Find average area in the reach

Step 2 Find volume

Step 3 Find t

5 Dissolved Oxygen and Water Temperature

Dissolved oxygen (DO) and water temperature are most commonly insitu monitored parameters for surface waters (rivers, streams, lakes,reservoirs, wetlands, oceans, etc.) DO concentration in milligrams per

5 262.7 ft 2 Average area 5 16s271 1 265 1 263 1 259 1 258 1 260d ft 2

86,400

Streams and Rivers 5

liter (mg/L) is a measurement of the amount of oxygen dissolved in water.

It can be determined with a DO meter or by a chemical titration method.The DO in water has an important impact on aquatic animals andplants Most aquatic animals, such as fish, require oxygen in the water

to survive The two major sources of oxygen in water are from diffusionfrom the atmosphere across the water surface and the photosyntheticoxygen production from aquatic plants such as algae and macrophytes.Important factors that affect DO in water (Fig 1.1) may include watertemperature, aquatic plant photosynthetic activity, wind and wave mix-ing, organic contents of the water, and sediment oxygen demand.Excessive growth of algae (bloom) or other aquatic plants may providevery high concentration of DO, so called supersaturation On the otherhand, oxygen deficiencies can occur when plant respiration depletesoxygen beyond the atmospheric diffusion rate This can occur especiallyduring the winter ice cover period and when intense decomposition oforganic matter in the lake bottom sediment occurs during the summer.These oxygen deficiencies will result in fish being killed

6 Chapter 1

Figure 1.1 Factors affecting dissolved oxygen concentration in

water.

5.1 Dissolved oxygen saturation

DO saturation (DOsat) values for various water temperatures can be puted using the American Society of Civil Engineers’ formula (AmericanSociety of Civil Engineering Committee on Sanitary EngineeringResearch, 1960):

com-DOsat⫽14.652 ⫺ 0.41022T ⫹ 0.0079910T2⫺0.000077774T3 (1.4)where DOsat⫽dissolved oxygen saturation concentration, mg/L

T ⫽ water temperature, ⬚C

This formula represents saturation values for distilled water (b ⫽ 1.0)

at sea level pressure Water impurities can increase the saturation level(b ⬎ 1.0) or decrease the saturation level (b ⬍ 1.0), depending on thesurfactant characteristics of the contaminant For most cases, b isassumed to be unity The DOsatvalues calculated from the above formulaare listed in Table 1.2 (example: DOsat⫽8.79 mg/L, when T ⫽ 21.3⬚C)

for water temperatures ranging from zero to 30⬚C (American Society ofEngineering Committee on Sanitary Engineering Research, 1960)

Example 1: Calculate DO saturation concentration for a water temperature

for elevation above the mean sea level (MSL) The correction factor can be culated as follows:

cal-(1.5)

where f ⫽ correction factor for above MSL

A ⫽ air temperature, ⬚C

E ⫽ elevation of the site, feet above MSL

Example 2: Find the correction factor of DO sat value for water at 620 ft above the MSL and air temperature of 25⬚C What is DO sat at a water temperature

Step 1 Using Eq (1.5)

Step 2 Compute DO sat

From Example 1, at T ⫽ 20⬚C

DO sat⫽9.02 (mg/L) With an elevation correction factor of 0.977

DO sat⫽9.02 mg/L ⫻ 0.977 ⫽ 8.81 (mg/L)

5.2 Dissolved oxygen availability

Most regulatory agencies have standards for minimum DO tions in surface waters to support indigenous fish species in surfacewaters In Illinois, for example, the Illinois Pollution Control Board stip-ulate that dissolved oxygen shall not be less than 6.0 mg/L during at least

concentra-16 h of any 24-h period, nor less than 5.0 mg/L at any time (IEPA, 1999).The availability of dissolved oxygen in a flowing stream is highly vari-able due to several factors Daily and seasonal variations in DO levelshave been reported The diurnal variations in DO are primarily induced

by algal productivity Seasonal variations are attributable to changes intemperature that affect DO saturation values The ability of a stream

to absorb or reabsorb oxygen from the atmosphere is affected by flow tors such as water depth and turbulence, and it is expressed in terms

fac-of the reaeration coefficient Factors that may represent significantsources of oxygen use or oxygen depletion are biochemical oxygendemand (BOD) and sediment oxygen demand (SOD) BOD, includingcarbonaceous BOD (CBOD) and nitrogenous BOD (NBOD), may be theproduct of both naturally occurring oxygen use in the decomposition oforganic material and oxygen depletion in the stabilization of effluentsdischarged from wastewater treatment plants (WTPs) The significance

of any of these factors depends upon the specific stream conditions One

or all of these factors may be considered in the evaluation of oxygen useand availability

6 Biochemical Oxygen Demand Analysis

Laboratory analysis for organic matter in water and wastewaterincludes testing for biochemical oxygen demand, chemical oxygendemand (COD), total organic carbon (TOC), and total oxygen demand(TOD) The BOD test is a biochemical test involving the use of micro-organisms The COD test is a chemical test The TOC and TOD tests areinstrumental tests

The BOD determination is an empirical test that is widely used formeasuring waste (loading to and from wastewater treatment plants),evaluating the organic removal efficiency of treatment processes, andassessing stream assimilative capacity The BOD test measures: (1) themolecular oxygen consumed during a specific incubation period for thebiochemical degradation of organic matter (CBOD); (2) oxygen used tooxidize inorganic material such as sulfide and ferrous iron; and (3)reduced forms of nitrogen (NBOD) with an inhibitor (trichloromethylpyri-dine) If an inhibiting chemical is not used, the oxygen demand measured

is the sum of carbonaceous and nitrogenous demands, so-called totalBOD or ultimate BOD

The extent of oxidation of nitrogenous compounds during the 5-dayincubation period depends upon the type and concentration of micro-organisms that carry out biooxidation The nitrifying bacteria usuallyare not present in raw or settleable primary sewage These nitrifyingorganisms are present in sufficient numbers in biological (secondary)effluent A secondary effluent can be used as “seeding” material for anNBOD test of other samples Inhibition of nitrification is required for aCBOD test for secondary effluent samples, for samples seeded with sec-ondary effluent, and for samples of polluted waters

The result of the 5-day BOD test is recorded as carbonaceous chemical oxygen demand, CBOD5, when inhibiting nitrogenous oxygendemand When nitrification is not inhibited, the result is reported asBOD5(incubation at 15⬚C for 5 days)

bio-The BOD test procedures can be found in Standard Methods for the

Examination of Water and Wastewater (APHA, AWWA, and WEF, 1995).

When the dilution water is seeded, oxygen uptake (consumed) isassumed to be the same as the uptake in the seeded blank The differ-ence between the sample BOD and the blank BOD, corrected for theamount of seed used in the sample, is the true BOD Formulas for cal-culation of BOD are as follows (APHA, AWWA, and WEF, 1995):When dilution water is not seeded:

(1.6)

BOD, mg/L 5 D12P D2

10 Chapter 1

When dilution water is seeded:

(1.7)

where D1, Di⫽DO of diluted sample immediately after

preparation, mg/L

D2, De⫽ DO of diluted sample after incubation at 20⬚C, mg/L

P ⫽ decimal volumetric fraction of sample used; mL of

sample/300 mL for Eq (1.6)

Bi⫽DO of seed control before incubation, mg/L

Be⫽DO of seed control after incubation, mg/L

f ⫽ ratio of seed in diluted sample to seed in seed control

P ⫽ percent seed in diluted sample/percent seed in seed

control for Eq (1.7)

If seed material is added directly to the sample and to control bottles:

f ⫽ volume of seed in diluted sample/volume of seed in seed control

Example 1: For a BOD test, 75 mL of a river water sample is used in the

300 mL of BOD bottles without seeding with three duplications The initial

DO in three BOD bottles read 8.86, 8.88, and 8.83 mg/L, respectively The

DO levels after 5 days at 20⬚C incubation are 5.49, 5.65, and 5.53 mg/L, respectively Find the 5-day BOD (BOD 5 ) for the river water.

Example 2: The wastewater is diluted by a factor of 1/20 using seeded trol water DO levels in the sample and control bottles are measured at 1-day intervals The results are shown in Table 1.3 One milliliter of seed material

con-is added directly to diluted and to control bottles Find daily BOD values.

solution:

Step 1 Compute f and P

Step 2 Find BODs using Eq (1.7)

For other days, BOD can be determined in the same manner The results

of BODs are also presented in the above table It can be seen that BOD 5 for this wastewater is 102.4 mg/L.

7 Streeter–Phelps Oxygen Sag Formula

The method most widely used for assessing the oxygen resources instreams and rivers subjected to effluent discharges is the Streeter–Phelpsoxygen sag formula that was developed for the use on the Ohio River in

1914 The well-known formula is defined as follows (Streeter andPhelps, 1925):

(1.8a)or

(1.8b)

where D t⫽DO saturation deficit downstream, mg/L or lb

(DOsat– DOa) at time t

t ⫽ time of travel from upstream to downstream, days

Da⫽initial DO saturation deficit of upstream water,

mg/L or lb

La⫽ultimate upstream BOD, mg/L

e ⫽ base of natural logarithm, 2.7183

K1⫽deoxygenation coefficient to the base e, per day

K2⫽reoxygenation coefficient to the base e, per day

k1⫽deoxygenation coefficient to the base 10, per day

k2⫽reoxygenation coefficient to the base 10, per day

In the early days, K1or K2and k1or k2were used classically for values

based on e and 10, respectively Unfortunately, in recent years, many authors have mixed the usage of K and k Readers should be aware of this The logarithmic relationships between k and K are K1⫽2.3026k1

(1.10)where ⫽the net rate of change in the DO deficit, or the absolute

change of DO deficit (D) over an increment of time dt

due to stream waste assimilative capacity affected by

deoxygenation coefficient K1and due to an atmosphericexchange of oxygen at the air/water interface affected by

the reaeration coefficient K2

La⫽ultimate upstream BOD, mg/L

L t⫽ultimate downstream BOD at any time t, mg/L

Combining the above two differential equations and integrating between

the limits Da, the initial upstream sampling point, and t, any time of flow

below the initial point, yields the basic equation devised by Streeter andPhelps This stimulated intensive research on BOD, reaction rates, andstream sanitation

There are some shortcomings in the Streeter–Phelps equation The twoassumptions are likely to generate errors It is assumed that: (1) wastesdischarged to a receiving water are evenly distributed over the river’scross section; and (2) the wastes travel down the river as a plug flow with-out mixing along the axis of the river These assumptions only applywithin a reasonable distance downstream Effluent discharge generallytravels as a plume for some distance before mixing In addition, it isassumed that oxygen is removed by microbial oxidation of the organicmatter (BOD) and is replaced by reaeration from the surface Some fac-tors, such as the removal of BOD by sedimentation, conversion of sus-pended BOD to soluble BOD, sediment oxygen demand, and algalphotosynthesis and respiration are not included

The formula is a classic in sanitary engineering stream work Itsdetailed analyses can be found in almost all general environmental engi-neering texts Many modifications and adaptations of the basic equationhave been devised and have been reported in the literature Many

researches have been carried out on BOD, K1, and K2factors Illustrationsfor oxygen sag formulas will be presented in the latter sections

8 BOD Models and K1 Computation

Under aerobic conditions, organic matter and some inorganics can beused by bacteria to create new cells, energy, carbon dioxide, and residue.The oxygen used to oxidize total organic material and all forms of nitro-gen for 60 to 90 days is called the ultimate BOD (UBOD) It is commonthat measurements of oxygen consumed in a 5-day test period called

5-day BOD or BOD5are practiced The BOD progressive curve is shown

in Fig 1.2

8.1 First-order reaction

Phelps law states that the rate of biochemical oxidation of organic ter is proportional to the remaining concentration of unoxidized sub-stance Phelps law can be expressed in differential form as follows(monomolecular or unimolecular chemical reaction):

mat-(1.11)

by integration

oror

TIME, days

Figure 1.2 BOD progressive curve.

K1⫽deoxygenation rate, based on e, K1⫽2.303k1, per day

k1⫽deoxygenation rate, based on 10,

k1⫽0.4343K1(k1⫽0.1 at 20⬚C), per day

e ⫽ base of natural logarithm, 2.7183

Oxygen demand exerted up to time t, y, is a first-order reaction (see

Fig 1.2):

(1.13a)

or based on log10

(1.13b)When a delay occurs in oxygen uptake at the onset of a BOD test, a lag-

time factor t0should be included and Eqs (1.13a) and (1.13b) become

(1.14a)or

(1.14b)For the Upper Illinois Waterway study, many of the total and NBODcurves have an S-shaped configuration The BOD in waters from poolsoften consists primarily of high-profile second-stage or NBOD, andthe onset of the exertion of this NBOD is often delayed 1 or 2 days Thedelayed NBOD and the total BOD (TBOD) curves, dominated by theNBOD fraction, often exhibit an S-shaped configuration The generalmathematical formula used to simulate the S-shaped curve is (Butts

et al., 1975)

(1.15)

where m is a power factor, and the other terms are as previously defined.

Statistical results show that a power factor of 2.0 in Eq (1.15) bestrepresents the S-shaped BOD curve generated in the Lockport and

Brandon Road areas of the waterway Substituting m ⫽ 2 in Eq (1.15)

yields

(1.15a)

Example 1: Given K1⫽0.25 per day, BOD 5⫽6.85 mg/L, for a river water

sample Find Lawhen t0⫽0 days and t0⫽2 days.

solution: By Eq (1.13b)

When t ⫽ 0.25 days

Similar calculations can be performed as above The relationship between t

and is listed in Table 1.4.

8.2 Determination of deoxygenation rate

and ultimate BOD

Biological decomposition of organic matter is a complex phenomenon.Laboratory BOD results do not necessarily fit actual stream conditions.BOD reaction rate is influenced by immediate demand, stream or riverdynamic environment, nitrification, sludge deposit, and types and con-centrations of microbes in the water Therefore, laboratory BOD analy-ses and stream surveys are generally conducted for raw and treatedwastewaters and river water to determine BOD reaction rate

Many investigators have worked on developing and refining methods

and formulas for use in evaluating the deoxygenation (K1) and

reaera-tion (K2) constants and the ultimate BOD (La) There are several

meth-ods proposed to determine K1values Unfortunately, K1values determined

by different methods given by the same set of data have considerable

variations Reed–Theriault least-squares method published in 1927 (USPublic Health Service, 1927) gives the most consistent results, but it istime consuming and tedious Computation using a digital computer wasdeveloped by Gannon and Downs (1964).

In 1936, a simplified procedure, the so-called log-difference method

of estimating the constants of the first-stage BOD curve, was presented

by Fair (1936) The method is also mathematically sound, but is also ficult to solve

dif-Thomas (1937) followed Fair et al (1941a, 1941b) and developed the

“slope” method, which, for many years, was the most used procedure forcalculating the constants of the BOD curve Later, Thomas (1950) pre-sented a graphic method for BOD curve constants In the same year,

Moore et al (1950) developed the “moment” method that was simple,

reli-able, and accurate to analyze BOD data; this soon became the mostused technique for computing the BOD constants

Researchers found that K1varied considerably for different sources

of wastewaters and questioned the accepted postulate that the 5-dayBOD is proportional to the strength of the sewage Orford and Ingram(1953) discussed the monomolecular equation as being inaccurate andunscientific in its relation to BOD They proposed that the BOD curvecould be expressed as a logarithmic function

Tsivoglou (1958) proposed a “daily difference” method of BOD datasolved by a semigraphical solution A “rapid ratio” method can be solvedusing curves developed by Sheehy (1960) O’Connor (1966) modified theleast-squares method using BOD5

This book describes Thomas’s slope method, method of moments,

log-arithmic function, and rapid methods calculating K1(or k1) and La

con-stants via the least-squares treatment of the basic form of the order reaction equation or

first-(1.16)

where dy ⫽ increase in BOD per unit time at time t

K1⫽deoxygenation constant, per day

La⫽first stage ultimate BOD, mg/L

y ⫽ BOD exerted in time t, mg/L

This differential equation (Eq (1.16)) is linear between dy/dt and y Let

y⬘ ⫽ dy/dt to be the rate of change of BOD and n be the number of BOD

measurements minus one Two normal equations for finding K1and Laare

(1.18)

Solving Eqs (1.17) and (1.18) yields values of a and b, from which K1

and Lacan be determined directly by following relations:

and

The calculations include first determinations of y⬘, y⬘y, and y2 for

each value of y The summation of these gives the quantities of ⌺y⬘,

of the slopes are calculated from the given data of y and t as follows:

(1.21)

For the special case, when equal time increments t i⫹1 – t i⫽ t3– t2⫽

t2–t1⫽ ⌬t, y⬘ becomes

(1.21a)

A minimum of six observations (n ⬎ 6) of y and t are usually required

to give consistent results

Example 1: For equal time increments, BOD data at temperature of 20⬚C,

t and y, are shown in Table 1.5 Find K1and La

solution:

Step 1 Calculate y⬘, y⬘y, and y2

Step 2 Determine a and b

Writing normal equations (Eqs (1.17) and (1.18)), n ⫽ 9

a⌺y 1 b⌺y22 ⌺yyr 5 0

Streams and Rivers 21

TABLE 1.5 Calculations for y ⴕ, y ⴕy, and y2 Values

Example 2: For unequal time increments, observed BOD data, t and y are given in Table 1.6 Find K1and La

solution:

Step 1 Calculate ⌬t, ⌬y, y⬘, yy⬘, and y2 ; then complete Table 1.6

From Eq (1.21) (see Table 1.6)

Step 2 Compute a and b; while n ⫽ 9

(1) and

(2)

Eq (2) – Eq (1)

with Eq (2)

Step 3 Determine K1and La

be a series of regularly spaced time intervals Calculations are needed

for the sum of the BOD values, ⌺y, accumulated to the end of a series

of time intervals and the sum of the product of time and observed BOD

values, ⌺ty, accumulated to the end of the time series.

The rate constant K1 and the ultimate BOD La can then be easily

read from a prepared graph by entering values of ⌺y/⌺ty on the

appro-priate scale Treatments of BOD data with and without lag phase will

be different The authors (Moore et al., 1950) presented three graphs for

3-, 5-, and 7-day sequences (Figs 1.3, 1.4, and 1.5) with daily intervalsfor BOD value without lag phase There is another chart presented for

a 5-day sequence with lag phase (Fig 1.6)

Example 1: Use the BOD (without lag phase) on Example 1 of Thomas’

slope method, find K1and La