WATER QUALITY  MANAGEMENT PLANNING
                 Addendum  to
           Simplified  Mathematical
          Modeling of Water  Quality
  THIS REPORT SUPPORTS MATERIAL PRESENTED IN THE MAIN TEXT,
  "SIMPLIFIED MATHEMATICAL MODELING OF WATER QUALITY"
  AS THIS ADDENDUM EXPLAINS IN DETAIL SOME OF THE SIMLIFYING
  ASSUMPTIONS MADE IN THE MODEL DEVELOPMENT,  IT IS STRONGLY
  RECOMMENDED THAT USERS CAREFULLY REVIEW THIS DOCUMENT
  BEFORE ATTEMPTING TO APPLY THE MODELS PRESENTED IN THE MAIN TEXT.
        ENVIRONMENTAL PROTECTION AGENCY

              Washington,D.C. 20460
                      MAY 1972

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                           INTRODUCTION


This Addendum was prepared in response to suggestions from EPA
regional personnel on topics needing additional comment or clarification
in connection with the handbook,, "Simplified Mathematical Modeling
of ater Quality".  Topics covered in this Addendum are:

Section         Title                                    Page No.

I               Waste Treatment Efficiencies             E-l

II              Nitrification                            E-9

III             Determination of 0 from Observed
                Data                                     E-15

IV              Reaeration Over Dams                     E-25

V               Temperature Coefficients (for K^, Ka)    E-35

VI              Evaluation of the Dispersion
                Coefficient, E                           E-35

Sections I and II, which explain some of the simplifying assumptions
made during model development, indicate specific situations where
additional discretion and judgment may be required before using
the handbook.  When lagoon treatment is being considered, Figure E-1A
from Section I replaces Figure B-l (Chart B) and Figure E-l (Chart E)
from the main text.  If the possibility of high algal concentrations
in lagoon effluent exists, Section I recommends against the use of
this guide.  All handbook users faced with possible water quality
problems related to nitrogenous oxgen demand should read Section II
which discusses the assumptions made in the main text concerning
nitrification.  Sections III, V, and VI provide additional assistance
on the derivation and use of several model coefficients.  Section IV
presents new material on modeling reaeration over dams; this topic
is not discussed in the main text.

An errata sheet for the main text is included at the end of this
Addendum.

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                         APPENDIX E
                          ADDENDUM
I. Waste Treatment Efficiencies

          The purpose of this section is to further explain
the bases of Tables III-3 and III-4 of the main text  (see
pages 53-54), which deal with the estimated efficiencies of
treatment levels.  The basic assumptions used for these tables
were: (a) 125 gallons/capita-day;  (b) 0.174 pounds five-day
carbonaceous BOD  (CBODj.) /capita-day;  (c) 0.044 pounds oxidiz-
able nitrogen (organic nitrogen and ammonia)/capita-day; and
(d) 0.025 pounds PO./capita-day.  The ultimate to five-day
carbonaceous BOD ratio used was 1.43.  This term results from
a BOD bottle decay coefficient of 0.24/day, which is repre-
sentative of municipal wastes.  The ultimate nitrogenous oxy-
gen demand to nitrogen ratio used was 4.57.  This term results
from the stoichiometric balance of the conversion of ammonia
to nitrate, which shall be discussed in Section II of this
addendum.  Application of these ratios resulted in (a) 0.25
pounds ultimate carbonaceous BOD/capita-day and (b) 0.20
pounds ultimate nitrogenous BOD/capita-day.
                             E-l

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          With respect to carbonaceous BOD, the column labeled



"% Removal" in Table III-3 is in terms of estimated CBOD5 re-



moval efficiencies.  Since carbonaceous removal efficiencies



are usually reported in terms of CBOD,., the normal procedure



in a stream or estuary analysis is to calculate the pounds of



CBODj. inputed into the water body, and subsequently increase



this load by the ultimate to five-day carbonaceous BOD ratio.




In Table III-3, the calculations were initiated with the 0.25



pounds ultimate carbonaceous BOD/capita-day, and the CBODj-



efficiencies were applied to this value based on the assump-



tion that the ultimate to five-day carbonaceous BOD ratio does



not significantly change in the secondary effluent.  Under



this assumption, both procedures result in the same value when



reduced to the ultimate carbonaceous demand inputed into the



water body.  The ultimate to five-day carbonaceous BOD ratio



may increase with more advanced treatment.  However, the value



used is appropriate for interim planning.



          Figures B-l  (Chart B) and E-l  (Chart E) of the main



text are obtained directly from Table III-3.  The following



illustrative example problem on treatment efficiencies is




presented:
                             E-2

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Illustrative Example
          Existing Population - 25,500

          Proposed Treatment - High Rate Biological Treatment

          DO Standard - 4 mg/1

          River - Intermediate Channel,  depth 5-10 feet
                                                        2
          Drainage Area at Discharge Location - 1,700 mi
                                                  2
          Estimated Drought Flow Rate -  .05 cfs/mi

          Maximum Water Temperature - 30C
          1)  From knowledge of Area - moderate to high growth
              fl = 2.00 (Table III-l, p.  49)

              Design Popultion:  25,000 x  2.00 = 51,000

          2)  Chart B - Enter Figure B-l
              Design Popultion - 51,000 at High Rate Treat-
               ment gives  effluent load of 10,000 Ib/day
               ultimate BOD.
             Calculation:

               HR-BIO Treatment - Table III-3
                    85% removal of CBOD5

                    20% removal of oxidizable N
          Q>25      C - OOP x (   _  >            Ibs  C  -  UOD
                                  >       >
                  cap-day                       cap-day


          0.20  lbs N - UOD x (1 -  .20)  =  .160  lbs  N -  UOD
                 cap-day                        cap-day
          TOTAL                        =  .197  lbS  -  UOD
                                               cap-day
                            E-3

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.197 x 51,000 = 10,000 Ib UOD/day

             197
  Note:   25'+  2Q  (100) = 44% Remaining

                           56% Removal of UOD


Estimation of Municipal Loads


     Ibs C - UOD                lbs CBODc
0.25 lb      V
       cap-day   '  *-~   """  cap-day
         N - UOD   4>57 =       lbs oxid. N
 >                   >
       cap-day                     cap-day

 at 125 gallons/capita-day

        .174 lbs CBOD ./cap-day       6
    5     125 gal/cap-day        8.34 Ibs/gal

      =167 mg/1
  .,     .044 lbs Oxid.N/cap-day   106
uxia.N             ^25x 8T3T

      = 42 mg/1


3)  Alternate Calculation:

    Effluent CBOD5 reported in design as 25 mg/1

     F,   _ 51,000 x 125 gallons/cap-day
                  106 gal/MG
          = 6.375 MGD
      No nitrogen information.
      Assume 20% removal of Oxid. N

42 mg/l-H x (1 - .20,  x 4.57 *
 x 6.375 MGD x 8.34        = 8160 lbs N-UOD/day
                  E-4

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          25 mg/1 CBOD5 x 1.43 ibs C "600 X 6'375 x 8-34

            = 1900 Ibs C-UOD/day

          Total - 10,000 Ibs UOD/day
          4) See page 86 for solution

               Allowable load - 4,000 Ibs UOD/day
          5) Review of existing data indicates 20 mg/1
              Oxid N in effluent

               20 mg/l-N x 4.57 x 6.375 x 8.34 = 4860 Ibs N-UOD/day
                                                 1900 Ibs C-UOD/day
               TOTAL                           = 6760 Ibs   UOD/day


             Additional Treatment still required to meet
              DO standard in the future.
          6) The planner may investigate when additional
              treatment is required.  At the present pop-
              lation the load is 3380 Ibs UOD/day.  Since
              the allowable load is 4000 Ibs UOD/day, the
              DO standard will be met under the proposed
              treatment scheme for the present.  There-
              fore the planner could allow construction
              of the HR-BIO system now with provisions
              for the additional treatment needed as
              growth occurs.  A detailed planning study
              would be required after construction.


          Where sufficient land area is available lagoons have

been employed as a method of treatment.  Lagoons may be class-

ified as four types:
                             E-5

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    Type  I  - Aerobic Algae Ponds I _,.  , ., .   . .    ,
    Type II  - Facultative Ponds   J "Stabilization" basins
    Type III - Anaerobic Ponds     )( or simPle
    Type IV  - Aerated lagoons - mechanical aeration



          As a whole, lagoon treatment efficiencies average

about 80% CBOD5 removal, with a range of 50% to 90%.  In the

colder climates, the lower efficiencies will be encountered

during the winter, due to the sensitivity of this type of


treatment to ambient temperatures.  In mechanically aerated

lagoons, nitrogen  (organic nitrogen and ammonia) removals may

range from 10% to 20%.  However, for "stabilization" basins,

or simple lagoons which are not mechanically aerated, an over-

all average for nitrogen removals cannot be realistically de-

fined, due to the seasonal dependence of these forms of treat-

ment with the associated wide variation in nitrogen removals.

This situation is further complicated in that the effluents


from aerobic algae ponds and facultative ponds often have a

high content of organic nitrogen.  This nitrogen is associated

with both living cells and detritus.  The susceptability to

hydrolysis and the rate of oxidation will vary widely depend-

ing on the form of organic nitrogen in the effluent.

          For the purposes of the report, when lagoon treat-


ment is considered, Figure E-1A replaces Figure B-l  (Chart
                            E-6

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 100,000
en
en
Z

X
o


UJ
  10,000
   1,000
      1,000
10,000                100,000

    DESIGN  POPULATION
1,000,000
                            FIGURE E-IA

                 LAGOON  TREATMENT  EFFICIENCIES

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B), and Figure E-l  (Chart E).  The discharged loads shown in


Figure E-1A make use of the following table:






                          TABLE E-l


          ESTIMATED EFFICIENCY OF LAGOON TREATMENT
                   ULTIMATE OXYGEN DEMAND

                                                      f4
Treatment   % Removal   #/capita/UOD remaining   f    .
  Level      *      +    *       +               traction
            C      N    C       N        total     remaining
  Lagoon    80     15  .050    .170      .220          .49

 *
C  = carbonaceous BOD

N  = oxidizable nitrogen

f. = residual fraction after treatment
          The separation of algae and other suspended matter


from the effluents of aerobic algae ponds and facultative


ponds generally result in higher nitrogen removals than indi


cated in the foregoing table.  However, higher nitrogen re-


movals should only be applied when justified on the basis of


consistent historical data of removal efficiencies of "sta-


bilization" basins in the specific geographical area under


investigation.
                             E-8

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          When there is no separation, the algae in the ef-



fluents of aerobic algae ponds and facultative ponds may



cause significant diurnal variations in dissolved oxygen in



a stream.  In such a situation, the modeling procedures pre-



sented in this guide are inadequate.  In general, when the



possibility of high algal concentrations in an effluent exist,



the use of this guide is not recommended.  In addition the



possible high solids content and color of such effluents may



be of concern.






II. Nitrification






          In addition to the oxidation of carbonaceous ma-



terial in a water body, the oxidation of nitrogen by specific



bacteria also exerts an oxygen demand on the water body.  In




long term biochemical oxygen demand tests, this is usually



observed as a second stage.  This phenomenon is called nit-



rification.  The important forms of nitrogen in this pheno-



menon are:






           (a)  organic nitrogen (amines, proteins)



           (b)  ammonia



           (c)  nitrite



           (d)  nitrate
                             E-9

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          All forms may be discharged by municipal and indus-



trial waste sources.  Organic and ammonia nitrogen, however,



are the more common forms discharged by municipal sources.



A series of reactions takes place in the nitrification phen-



omenon which essentially convert organic and ammonia nitrogen



into nitrate.  In the process of this bacterial conversion or



oxidation, oxygen is drawn from available resources to allow




the reaction to proceed.  Figure II-l of the main text (see



page 20), shows this process.



          Organic nitrogen, when discharged into a water body,



undergoes an hydrolysis reaction with ammonia as one of the



end products.  The ammonia formed from organic nitrogen,  to-



gether with direct discharges of ammonia from waste sources,



is oxidized under aerobic conditions to nitrite by bacteria



of the genus Nitrosomonas, as, follows  ' '  :






  (NH4) + OH~ + 1.5 02 bacteria   H+ + NO2~ + 2H20 +59.4 Kcal.







Note that oxygen is utilized in this biochemical reaction.



Stoichiometrically, the reaction requires 3.43 pounds of oxy-



gen for one pound of nitrogen oxidized to nitrite.



          The nitrite formed is subsequently oxidized to nit-




rate by the Nitrobacter as follows   :
                             E-10

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         N02  + 0.5 O2 bacteria   NC>3  + 18 Kcal,
This reaction requires 1.14 pounds of oxygen for one pound



of nitrite nitrogen oxidized to nitrate.  The total oxygen



utilization in the entire forward nitrification process is



4.57 pounds of oxygen per pound of ammonia nitrogen.  By con-



trast, carbonaceous BOD,- has an ultimate oxygen demand of



about 1.43.  Thus, a pound of carbonaceous BOD5 requires about



1.43 pounds of oxygen, while a pound of organic nitrogen or



ammonia requires 4.57 pounds of oxygen.  Nitrification reac-



tions are, therefore, a potential large source of oxygen de-



pletion in natural waters.



          The forward sequential nitrification phenomenon is



illustrated in Figure E-2.  If the sequence of reactions is



allowed to proceed uninterrupted, the ammonia decays almost



exponentially, nitrite builds up, but is quickly oxidized to



nitrate.  Eventually, all of the original ammonia nitrogen is



converted bacteriologically to the nitrate form of nitrogen,



a bacterially stable end product.  In natural waters subjected



to large discharges of nitrogenous waste materials, the re-



action generally proceeds in the direction indicated in Fig-



ure E-2.  There are several factors which may modify the



sequence of the reactions.  Under conditions of low dissolved






                            E-ll

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LJ
                         DISTANCE
                         FIGURE E-2
            SEQUENTIAL REACTIONS IN NITRIFICATION

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oxygen, nitrification is inhibited, and at values approach-



ing zero, nitrification may be completely suppressed.  This



has been experimentally verified by the British in the Thames



Estuary work   , and earlier by Waksman   .  For all practical



purposes, nitrification may be assumed suppressed at dissolved



oxygen levels below 1.5 mg/1.  However, as a water body re-



covers, nitrification may resume and cause a secondary dis-



solved oxygen sag further downstream.  Since dissolved oxygen



concentrations of less than 1.5 mg/1 are intolerable in most



cases, the modeling of the secondary sag has not been included



in this guide.  This may, however, cause some difficulty in



verification analyses, as shall be discussed in Section III of



the addendum.



          At low water temperatures, less than 10C, nitrifi-




cation is inhibited   .  In application of this guide, it may



be assumed that nitrification is suppressed at water temper-



atures below 10C.  The suppression of nitrification can be



simulated by eliminating the nitrogenous ultimate oxygen de-



mand portion of the waste input in the modeling analyses.



This concept is important in the evaluation of seasonal treat-



ment schemes.



          If significant quantities of phytoplankton are pre-



sent,  ammonia will be used as a nutrient source.  As a result,
                            E-13

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the available ammonia for nitrification is depleted.  In this



situation, the diurnal variation in dissolved oxygen caused



by the phytoplankton may create problems of low dissolved oxy-



gen at times.



          Sawyer    indicates that the optimum pH range for



nitrification is 8.0 to 8.5.  An optimum pH range of 7.0 to


                                        (8)
9.0 was reported by Engle and Alexander   .  Nitrification



studies performed by Hydroscience, Inc., indicate that opti-



mum nitrification is not significantly affected within a pH



range of 7.0 to 8.5.  Since most natural waters are within



these ranges, pH should normally not hinder nitrification.



          Nitrifying bacteria use free carbon dioxide or bi-


               (4)
carbonate ions    as the primary sources of carbon for the



growth of new cells.  Generally, in natural waters, the con-



centration of inorganic carbon available in the above forms



is greatly in excess of requirements and nitrification pro-



ceeds.



          Under conditions of high organic carbon concentra-



tions, the heterotrophic bacteria (organic carbon oxidizers)



may predominate over the autotrophic bacteria  (nitrogen oxi-



dizers) which would result in a delay in nitrification.  This



is usually exemplified in the standard biochemical oxygen de-



mand test by the occurence of two stages.  With untreated and
                             E-14

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heavily polluted water, the two stages are usually distinct.
The first stage, which reflects the aerobic oxidation of the
organic material, is usually substantially completed before
the second stage, which reflects the oxidation of ammonia
through nitrite to nitrate, is significantly underway.  In
treated effluents and less polluted waters, the lag between
the two stages may be reduced.  As the first stage is reduced
with nitrifying organisms present, the two stages may occur
simultaneously.  Since this guide will be applied to effluents
which have at least secondary treatment, nitrification has
been assumed to occur simultaneously with the oxidation of
carbonaceous material in the modeling procedures.

III. Determination of $ from Observed Data

          The deoxygenation coefficient, K,, and the reaeration
coefficient, K , can be estimated from field data.  With both
              a.
stream area and flow as constants (an underlying assumption
used in the guide), Equation  (IV-2)  of the main text  (see page
69), yields:
                              _ V
                      L = LQe    U                             (E-l)
                             E-15

-------
          where:

          L    =    BOD concentration  (mg/1)

          L    =    BOD concentration at x =  0  (mg/1)

          U    =    velocity  (miles/day)

          K-,   =    deoxygenation coefficient (I/day)

          x    =    distance  (miles)


The field estimate at K, can be obtained from semi-logarithmic

plots of observed long term ultimate BOD stream data  (five

day BOD stream data may also be used) as a function of dis-

tance downstream.  Thus, in natural logs, Equation  (E-l)  is

written as:


                            K,x
                   In L =	|j-  + In LQ                       (E-la)



A semi-log plot of field data usually results in a straight

line, the slope of which is defined by:


                                  K,
                        Slope = 	J                           (E-lb)



This procedure provides a first estimate of the deoxygenation

coefficient.  An example of this procedure is shown in Figure

E-3 where a straight line has been fitted by  eye to the Mohawk
                            E-16

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 in
Q
O
CD
O
10.0

9.0

8-0

7.0

6.0


5.0


4.0



3.0
    2.0
           U = 4. MILES/DAY
           Kd= -SLOPE x U

            = 2.3 (-17
                     MILES

              Kd= 0.16/DAY
/4MILESx
\  DAY  I
                                       I
          0
         I
                          16        24

                      DISTANCE (MILES)
                      32
36
     INPUT
                      FIGURE  E-3

        ILLUSTRATION OF COMPUTATION OF  Kd
               FROM BOD  STREAM DATA

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River data.  Note that care must be taken to include the con-



version from base 10 logarithms as given in semi-log plots



to base e logarithms as required by Equation  (E-l) .



          The reaeration coefficient, K , may be estimated  from
                                       a


depth and velocity data by the following formula developed


            (9)

by O'Connor    from field studies:
(DO)17'

  L 3/2


   H
                                                               (E-2)
                                                        o
where DT is the oxygen diffusivity at 20C  (0.000081 ft /hour)
       j-i


U is the average stream velocity, and H is the average depth.



A more convenient from of this equation is:
                             1 9 QIT '

                        Ka -
                               ti
where:



          K    =    reaeration coefficient  (I/day)
           cl


          U    =    velocity  (fps)



          H    =    depth  (feet)






          The reaeration coefficient is a surface controlled



phenomenon and for all practical purposes,  is independent  of
                            E-18

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water quality.  In view of this, K, is usually the more var-



iable unknown coefficient in the term , it is necessary that the K  and K, terms be in-
                                      a      ci


vestigated separately as indicated.



          The estimation of the reaction coefficients K, and



K  from which  is obtained along with the estimation of the
 a


loading conditions from observed data is the first step in a



verification analysis.  The general procedure for a verifica-



tion analysis is presented in Section II-F of the main text



(see page 22).  Some typical dissolved oxygen verifications



accomplished by Hydroscience, Inc.   '     are shown in Fig-



ures E-4 and E-5.



          There are, however, possible difficulties or "pit-



falls" that may be encountered in the verification analysis



using the basic assumptions applied in this guide.



          For example, assume that a verification is to be



attempted in a stream into which a raw municipal effluent is



presently inputed.  Such an effluent probably has highly



settleable solids.  As a result, in the vicinity of the out-



fall, the removal of ultimate oxygen demand is accomplished



by the physical settling and the oxidation of the organic



matter,  simultaneously.  Oxidation of the ultimate oxygen
                            E-19

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   C O  B
er>
to
      B
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                                                                                    00
                                                                                    CO
                                                                                    oa

                                                                                    CO
LJ
o:
z>
O
                          !/3m  -  N33AXO Q3A1QSSIQ

-------
        a:
        LJ
          CO
          to
          OJ
        g^
   in
   06
   in
   O
g

Sc
y
a.
a:
WVQ
               
               a?
               o
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             O (/)
               _)
    II
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                                 UJ
    o o
              I
             I
                           10
                           C\J
             N-tom^    rocvi

             I/6UU-N39AXO QBAIOSSIQ
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                                                                 en

-------
demand requires dissolved oxygen, while removal of the ulti-
mate oxygen demand  (organic matter) by settling does not di-
rectly use dissolved oxygen.  As one proceeds downstream, the
physical removal is completed and only oxidative removal re-
mains.  In such a situation, a semilog plot similar to Figure
E-3 would probably yield two straight lines as shown in Figure
E-6.  The slopes of these lines are defined by:
                   Zone I:  slope = -K /U
                   Zone II: slope = -K,/U                     (E-3)
where :
          K    =    BOD removal coefficient (I/day)
          K,   =    deoxygentaiton coefficient (I/day)
          In this guide, a constant deoxygenation coefficient
has been assumed down the length of the water body.  This
assumption is considered valid, since the guide was meant to
be applied to municipal effluents with at least secondary
treatment.  As a result, verification with the modeling pro-
cedures presented in this guide for the hypothetical problem
presented above, may only be approximate.
          A constant background dissolved oxygen deficit of
1.0 mg/1 has been recommended in the guidelines.  However, if
                             E-22

-------
  Q
  O
  CD
  UJ
  CL
  h-
  C/)
                 SETTLING 8
                 'OXIDATION
OXIDATION
         ZONE I
                                                 STREAM
                                                 LENGTH
     WASTE
    DISCHARGE
                      FIGURE  E-6

ILLUSTRATION OF SPATIALLY  VARYING  BOD  REMOVAL RATE

-------
a verification analysis is being attempted where raw munici-



pal effluents are involved, a background level of 1.5 to 2.0



is recommended due to the probable high organic content of



the bottom due to settling.



          It has been pointed out that nitrification is sup-



pressed at low dissolved oxygen levels (<1.5 mg/1).   This re-



sults in a delay in nitrification and a secondary dissolved



oxygen sag is usually detected downstream.  The guide proce-



dures do not incorporate the calculation of such a secondary



sag.  However, if the primary dissolved oxygen sag is approx-



imately verified, the coefficientts used may be considered



valid.



          High phytoplankton populations or extensive growths




of rooted aquatic plants, will have a significant diurnal




effect on the variation of dissolved oxygen.  The modeling



procedures do not incorporate this phenomenon.



          Toxic substances  (copper, arsenic, zinc, mercury,



lead) will inhibit bacterial populations.  Therefore, high



concentrations of these toxicants will affect dissolved oxy-



gen in the water body.
                            E-24

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          The inclusion of the above phenomena is of the


                                              (12)
realm of a detailed planning effort.  Thomann     presents



more sophisticated modeling procedures, which incorporate



the modeling of these phenomena.



          In summary, the constituents of the term  may be



evaluated from observed data.  In situations where only dis-



solved oxygen data is available, a first approximation at



verification should begin with the appropriate average values



of K  and K,, or $ as presented in Figures A-2, A-3, and A-4
    cl      Ci


of Appendix A, respectively.  These figures were developed on



the basis of detailed planning efforts by Hydroscience and



others.  Final verification of observed data should be accom-



plished with coefficients within the ranges presented in these



figures.  If other phenomena above the assumptions used in



this guide are present, good engineering judgment will deter-



mine the degree of validity of the coefficients obtained.





IV. Reaeration Over Dams





          The reaeration occurring at dams is similar to the



natural reaeration phenomenon occurring in a stream, and al-



ways drives the dissolved oxygen concentration of the water



toward saturation.
                            E-25

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          Based on experimental data   ' and later verified

                (14 )
with field data   ', the British developed the following

equation for reaeration over dams:


                r = 1 + 0.11 ab(l + 0.046T)H                   (E-4 )


where :

                                    C  - C    D
          r    =    deficit ratio = ~ - ~ = =^
                                     s " Lb   Db

          C    =    dissolved oxygen concentration above
                    dam (mg/1)

          C,    =    dissolved oxygen concentration below
                    dam (mg/1)

          C    =    dissolved oxygen saturation (mg/1)
           o

          D    =    dissolved oxygen deficit above dam
          D,    =    dissolved oxygen deficit below dam
                     (mg/1)

          T    =    temperature  (C)

          H    =    height  (feet) through which the water
                    falls

          a    =    1.25 in clear to slightly polluted
                    water: 1.00 in polluted water: 0.80
                    in sewage effluents

          b    =    1.00 for weir with free fall: 1.3
                    for step weirs or cascades
                             E-26

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Rearrangement of the above equation to the appropriate form



to be used in this manual is as follows:
         Da - Db =  f1 - 1 + 0.11 abU + 0.046T)H  J Da         (E~4a)





          An alternate equation developed by Mastropietro



from field data on the Mohawk River and Barge Canal in New



York State is as follows:
                C.                 C

                - =  (1 - 0.037H) ^ + 0.037H                  (E-5)

                 s                 s
Rearrangement to the appropriate dissolved oxygen deficit



formulation is:
                     D  - D,= 0.037H D                        (E-5a)
                      a    jj           a
A plot of this formula which calculates the decrease in dis-



solved oxygen deficit over a dam, is presented in Figure E-7.



This formulation was developed specifically for the Mohawk



River and Barge Canal, and is valid for dams up to fifteen



feet high and critical temperatures in the range of 20 to



25C.  Successful application of this formula was accomplished



in a detailed planning study
                            E-27

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o
U-
LU
Q
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           Da : DEFICIT ABOVE DAM  (MG/L)
10
                   FIGURE E-7
             REAERATION  OVER DAMS

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          Under the same conditions, Equation  (E-5a) predicts

less reaeration than the British formulation and is, there-

fore, more conservative.  In view of this and  successful veri-

fication analyses in a detailed planning effort     , it is

recommended that Equation (E-5a) be used if conditions are

within the stated dam height and temperature constraints.

          To evaluate the effect of a dam, it  is necessary to

obtain the entire deficit profile even if there is  only one

waste source.  The usual procedure in a stream analysis is to

reevaluate the problem at the location of the  dam.  Reference

is made to Equation (IV-5) of the main text  (see page 72) , re-

peated here as :


                                V       V
                    K,L          U         U
               D =    _K   [e       - e       ]                (E-6)
                    a    d
This equation defines the deficit profile until the dam is

reached.  At this location, X = X , , the ultimate oxygen de-

mand not oxidized, L1 is calculated from Equation  (IV-2) of

the main text (see page 69), repeated here as:

                                  K,x
                                   d
                                   U~
                        L = LQe                                (E-7)
                            E-29

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The remaining ultimate oxygen demand is analogous to a waste



input at the dam.  The deficit above the dam, D  , calculated
                                               a


from Equation  (E-6) at X = X^ is applied to Equation  (E-5a) ,



which results in an estimate of the deficit, D, , after dam



reaeration.  With these parameters, the deficit profile  after



the dam is defined by:





                 K,               K                   K


       KdL'    ~ -U(x ~ V     - ~!J(X ~ V        " ~l(x  ~ xd)
 D =    d    [e   U      d  - e    U      d ] + D,e    U     d   (E-8)

     Ka " Kd                                     b




Due to the fact that these equations are linear, manipulation



of Equation  (E-8) results in the following equation:
                                                  a

        KqL         r:        p:                  fr(X  X.,)
        a    d
                                         - V
Note that the first part of this equation is the same as  if



there were no dam in the river .  Therefore , the procedures



outlined for multiple waste sources  (Section IV-D. , pp. 87),



can be applied as originally presented.  The analyst need



only subtract the second term of Equation (E-9) from the



total dissolved oxygen deficit response  (Table IV- 2, Column



8, pp.91) calculated.
                            E-30

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          Chart F shows the technique for computing the nega-



tive term of Equation  (E-9).  Figure F-l is entered with mile



point beginning at X = 0, at the dam location.  With knowledge



of K /U, this distance is converted to the dimensionless ratio
    a


of the negative deficit at the milepoint to the decrease in



deficit at the dam.  Since K,, (j>, and U have already been es-



timated, K /U is readily obtained.
          a


          Figure F-2 requires the decrease in deficit at the



dam, D  - D,, due to dam reaeration.  This value is obtained
      a    D


from Figure E-7, which is entered with the deficit previously



calculated above the dam, and the height of the dam.  The



deficit values obtained from Figure F-2 are subtracted from



column  (8) of Table IV-2 at the appropriate milepoints, and



the calculation of columns (9), (10), (11), and (12) proceed



as before.



          If one wishes to use the British formulation, the



decrease in deficit at the dam can be directly calculated from



Equation E-4a, and the same procedure as outlined above will



be followed, except for use of Figure E-7.





Illustrative Example





          The same problem as presented in Section IV-D-l-a



(see page 95), will be reevaluated with a ten foot dam at
                            E-31

-------
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-------
mile point 30.  Table E-2, which is a continuation of Table



IV-4 of the main text  (see page 97), indicates the numerical



analysis of the estimated dissolved oxygen profile due to the



four waste sources with the dam included.  Note that column



(8) is the same as that of Table IV-4, except that it is re-



labeled Subtotal DO, etc.  At milepoint 30, the Subtotal dis-



solved oxygen deficit response is 4.8 mg/1.  From Figure E-7,



with a dam height of 10 feet, the decrease in deficit at the



dam is 1.8 mg/1.  Column  (8A) was obtained from Chart F using



a K /U of 0.12/miles.  Column (8B) was obtained from the sub-
   a


traction of column  (8A) from column (8), and represents the



total dissolved oxygen deficit response.  The remaining columns



are obtained as previously outlined.  Column (10) shows the



estimated dissolved oxygen profile and indicates a new spatial



minimum of 3.2.  This may be compared to the spatial minimum



of 2.6 previously calculated without the dam.  As indicated,



the standards are still violated from about mile 25 to mile



60.
                            E-33

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-------
V. Temperature Coefficients





          The deoxygenation  (K,) and reaeration  (K ) coeffi-



cients presented throughout this manual are for water temper-



atures of 20C.  Conversion to other temperatures can be made



by applying the following formula:
                         = K20(6)T   2
where KT is the reaction coefficient at temperature, T  (C),



K~n is the reaction coefficient at 20C, and 0 is a constant.
 ^ u

                             (12)           (12)
The term Q is equal to 1.047V  ' and 1.0241  ' for the deoxy-



genation and reaeration coefficients, respectively.  Nitrifi-



cation is assumed to occur simultaneously, and at the same



rate as carbonaceous oxidation.  The 6 value for nitrification



is also assumed as 1.047.  However, at low temperatures  (<10C)



nitrification may be assumed suppressed.





VI. Evaluation of the Dispersion Coefficient, E





          The dispersion coefficient, E, for an estuary or



tidal river may be evaluated for a particular net advective



flow from the observed steady-state concentration profile of



salinity or chlorides, provided that vertical stratification
                            E-35

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of these constituents is not pronounced.  The underlying equa-

tion that may be utilized is Equation III-3 in the main text

(refer to page 38), repeated here as:
                     C = C  egx    X < 0
                                                               (E-ll)
                     C = C  e^x    X > 0
Considering salinity or chlorides as conservative, and begin-

ning the model at the furthest point downstream  (location of

maximum concentration), Equation  (E-ll) becomes:
                             UX
                              E
                     C = CQ e      X _< 0                       (E-12a)


                     C = CQ        X :> 0                       (E-12b)
where:


          C    =    salinity or chloride concentration  (mg/1)

          C    =    maximum concentration at X =  0  (mg/1)

          U    =    net advective velocity  (miles/day)

          E    =    dispersion coefficient  (miles2/day)

          X    =    distance upstream  (miles) - negative
                     (usually, X = 0 at mouth of estuary)


Taking the natural log of Equation  (E-12a) yields:
                             E-36

-------
                      In C =  . x + In C                        (E-13)
                             EI         O
Therefore, a semi-logarithmic plot of log salinity or log



chlorides versus distance upstream should yield a straight



line, the slope of which is U/E.  The dispersion coefficient,



E, may then be calculated directly from the net advective



velocity, which may be estimated from the freshwater inflow,



and cross-sectional area of the estuary.  Figure E-8 illus-



trates the computation for data collected by U.S. Geological



Survey     for a reach of the Hudson River Estuary.  Note



that care must be taken to include the conversion from base



10 logarithms as given in semi-log plots to base e logarithms,



as required by Equation (12a).



          If vertical stratification of salinity or chlorides



is encountered, it is suggested that the average over the



depth be used, which will result in a first cut evaluation of



the dispersion coefficient.



          Dye data may also be used to obtain an estimate of



dispersion coefficients.   Reference is made to O'Connor


              (18)
and Diachishin    ,  for the appropriate procedures which are



based on a more complicated analysis.
                            E-37

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10,000
 9,000
 8,000
 7,000
 6,000
 5,000
 4,000

 3,000
2,000
c/)
LU
Q
cr
3   1,000
g   900
   800
    700
    600
    500

    400
  300
  2001
     0
     20
                          HUDSON  ESTUARY (7)
                           SEPTEMBER 10,1962
                    Q
                    A
                    U
                    E
              5,000 CFS
              200,000 SQ.FT.
              0.41 MILES/DAY
                u      Q.4I
              SLOPE * 2.3(.02I4)
                                       = 8.3 MILES/DAY
-5   -10   -15   -20   -25   -30  -35 MODEL MILES
25   30   35    40   45    50   55 RIVER MILES  FROM BATTERY
                       FIGURE E-8
         ILLUSTRATION OF COMPUTATION  OF  E
                  FROM CHLORIDE DATA

-------
                         REFERENCES
(  '  Thomann,  R.V.,  O'Connor,  D.J.,  Di Toro,  D.M.,  "Effect of
    Nitrification  on the Dissolved  Oxygen of Streams and Es-
    tuaries"  Manhattan College,  New York, N.Y.,  1970.

(2)
    Sawyer, C.N.,  Chemistry for  Sanitary Engineers,  McGraw-
    Hill Book Company, New York, N.Y., 196TH

    Hutchinson,  G.E.,  A Treatise cm Limnology, Volume 1^ J.
    Wiley and Sons,  Inc. ,  New York~7 N.Y., 1957.

    Effects of Polluting Discharges on the Thames  Estuary,
    Water Pollution Research  Technical Paper No.11,  Department
    of  Scientific  and Industrial Research, Her Majesty's Sta-
    tionary Office,  1964.

^  '  Waksman,  Principles of Soil  Microbiology, First  Edition,
    London, 1964.

    Camp,  T.R.,  Water and  Its Impurities, Reinhold Publishing
    Corporation, New York, 1963.


'  '  Sawyer, C.N.,  Wild, H.E., and McMahon, T.C.,  "Factors
    Affecting Nitrification Kinetics", presented at  the 43rd
    Annual Conference of the  Water  Pollution Control Federa-
    tion,  Boston,  Massachusetts, October, 1970.

(8)
v  '  Engle, M.S., and Alexander,  M., "Growth  and Autotrophic
    Metabolism of  Nitrosomonas Europaea", J. Bact.,  76, 1958.

(9\
v  '  O'Connor, D.J.  and Dobbins,  W.E., "Mechanisms  of Reaera-
    tion in Natural  Streams,  Trans.A.S.C.E., Vol.  123,  1958.

*    "Water Qaulity  Analysis for  the Markland Pool  of the Ohio
    River", Malcolm Pirnie Engineers, Hydroscience,  Inc.,
    October,  1969.

-------
                         REFERENCES
                        (continued)


^   '"Water  Quality Analysis  of  the Mohawk River-Barge Canal"
    New York State Department of  Health.  O'Connor,  D.J.,  and
    Hydroscience,  Inc.,  July, T9~68.

(12)
v    Thomann, R.B.,  Systems Analysis  and Water Quality Manage-
    ment, Environmental  Research  and Applications,  In~c.,  New
    York, 1972.


*   'Gameson, A.L.,  Vandyke,  K.G.,  and Ogden,  C.G.,  "The  Effect
    of  Temperature on  Aeration  at Wiers", Water  and Water En-
    gineering,  London, November,  1958.

(14)
    Barrett, M.J.,  Gameson,  A.L.,  and Ogden,  C.G.,  "Aeration
    Studies of  Four Wier Systems", Water  and  Water  Engineering,
    London, September, 1960.


    Mastropietro,  M.A.,  "Effects  of  Dam Reaeration  on Waste
    Assimilation  Capacities  of  the Mohawk River", Proceedings
    of  the  23rd Industrial Waste  Conference,  Purdue University
    May,  1968.


*   'Giese,  G.L.,  and Barr, J.W.,  The Hudson River Estuary - A
    Preliminary Investigation of  Flow and Water  Quality  Char-
    acteristics,  U.S.G.S., Bulletin  61, State of New York,
    Conservation  Department,  Water Resources  Commission,  1967.


^   'Analysis of the Dye  Diffusion Data in the Delaware River
    Estuary - Evaluation of  Diffusion Coefficients, Regional
    Office, U.S.  Public  Health  Service, Philadelphia, O'Connor
    D.J., August,  1962.


*   'Diachishin, A.N.,  "Dye Dispersion Studies" Journal of
    Sanitary Engineering Division, A.S.C.E.,  Volume 89,  No.
    SA1,  January,  1963.

-------
                    ADDITIONAL REFERENCES
"Nitrification in the Delaware Estuary", Delaware River Basin
Commission, Trenton, New Jersey, Hydroscience, Inc., June,
1969.

"Nitrification in the Activated Sludge Process - City of Flint,
Michigan", Consoer, Townsend, and Associates, Hydroscience, Inc.,
July, 1971.

"Advanced Treatmnet Methods for the Roxbury, New Jersey, Treat-
ment Plant", Lee T. Purcell Associates, Hydroscience, Inc.,
December, 1971.

"Advanced Waste Treatment for Nitrogen and Phosphorus Removal"
Baldwin and Cornelius, Freeport, New York, Hydroscience, Inc.,
March, 1971.

"Studies to develop Solutions to Waste Treatment Problems",
Armour and Company, Pharmaceutical Division &_ Edible Oils
Division, Hydroscience, Inc., December, 19651

"Biological Waste Treatment", Hanmer Division Plant, American
Tobacco Company, Chester, Virginia, Hydroscience, Inc., April
1968.

"Comparison of Nutrient Nitrogen Sources on Anaerobic Lagoon
Performance", Allied Chemical Company, Hopewell, Virginia,
Hydroscience, Inc., March, 1966.

"Biological Treatment Studies - Treatment of Wastes from Tan-
nery Operations", Armour Leather Company, Bolivar, Tennessee,
Hydroscience, Inc., July,1970.

"Biological Waste Treament - Amcelle Plant", Celanese Corpor-
ation of_ America, Maryland, Virginia, Hydroscience, Inc.,
December, 1967.

"Lagoon Treatment for Fiber Wastes", Celanese Corporation of_
America, Houston, Texas, Hydroscience, Inc., November, 1966.

"Design Criteria for the Treatment of Organic Wastes in an
Aerated Lagoon", E.I. DuPont de_ Nemours and Company, Hydro-
science, Inc., November, 1963.

-------
                    ADDITIONAL REFERENCES
                          (continued)
"Treatment of Organic Wastes in Aerated Lagoons", Water Poll-
ution Control Board, New York State Department of_ Health,
W.W. Eckenfelder, Jr., and D.J. O'Connor, Sponsored by N.Y.S.
Department of Health, Water Pollution Control Board, Project
#C-12516, June, 1960.

"Preliminary Report on the Theory and Application of Lagoon-
ing to the Treatment of Organic Wastes", Bethlehem Steel Com-
pany, Bethlehem, Pennsylvania, W.W. Eckenfelder and D.J.
0'Connor, Hydroscience, Inc., February, 1960.

"Lagoon Treatment for Fiber Wastes", Celanese Corporation of
America, Houston, Texas, Hydroscience, Inc., November, T9~6F7

"Waste Treattment Alternatives for Swift de La Plata Slaugh-
terhouse and Packing Plant", Deltec International Limited,
Hydroscience, Inc., June, 1970.

And others cited in main text.

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ERRATA FOR SIMPLIFIED MATHEMATICAL MODELING OF WATER QUALITY
                     Hydroscience, Inc.
Page

 52
 53
Line
Correction
 54
 87

 99

113

124

125
Last line
Table III-3 Last
Column
Treatment Level 4
Treatment Level 5

Table III-4 Last
Column
Treatment Level 1
Treatment Level 2
Treatment Level 3

Line 7

Line 3

Line 5

Last Line

Line 7
e) .044 pounds nitrogen.
                               .05
                               .01
.040
.035
.035

30C, 5.9 mg/1...

...minimum of about  2.6 mg/1

(V-5)

(V-6_)

...(by Equation V-6)...
f, V S GOVERNMENT PRINTING OFFICE 1972484-486/291

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