NH,-N

g
Fog yp ow EH

Giri 1. n 1 n L n
120

Fx)
la
8
8
8}
ia
iS

+UA16 7 67 +

117 20/ 67 + +
Sealer

3h 07 8900cn

T27-1e

RBC DATA

frevueee n 4 n L 1 1

130 120 HO too 30 80 70 60

DISTANCE IN MILES

Figure 10
Observed vs. Computed (solid line) - Nitrogen Profiles
November 1967 (after Anon, 1969)

NQ,-N

NQ,-N

2005

obese — sr a

130 120 ite} 100 90 80 7 60
DISTANCE IN MILES

DISTANCE IN MILES

55



zooplankton death rate resulting from all other causes is
given by a constant, the magnitude of which is to be de-
termined empirically. The severity of this assumption can
be tested by examining the sensitivity of the solutions of
the phytoplankton and zooplankton equations to the magnitude
of this constant. Hence, the resulting zooplankton death
rate is given by

D5 = KT + Ky (77)

where Ky is empirically determined.

With the growth and death rates given by Equations (76) and
(77), respectively, the source term for herbivorous zooplankton
biomass is given by Equation (75). The conservation of mass
equation which describes the behavior of Z, is given by Equa-
tion (53), with 2, as the dependent variables replacing Ps and
S55 replacing 855 as the source terms.

This completes the formulation of the equations which describe
the zooplankton system. The equations for the nutrient system
remain to be formulated.

The Nutrient System

The conservation of mass principle is applied to the nutrients
being considered in the same way as it has been previously ap-
plied to the phytoplankton and zooplankton biomass within a
volume segment. The number of mass conservation equations re-
quired is equal to the number of nutrients that are explicitly
included in the growth rate formulation for the phytoplankton.
For the sake of simplicity, the formulation for only one nutri-
ent is discussed below.

119


about 35 kg/ha-yr were reported for sites 2 km from a large
feedlot and 73 kg/ha-yr for sites 0.4 km from a large feedlot.
Based on these measurements the investigators conclude the in-
crease in nitrogen for a nearby lake could be as high as 0.6
mg/l.

Increases in nitrate concentration have been reported (Bormann,
1968) when forests of watershed systems are cut. Over the
period of one year, during which the land was clear cut and

a herbicide applied to prevent regrowth, the concentration

of nitrate in the drainage increased from about 1 mg/l to ap-
proximately 40 mg/l the following year and 50 mg/l the second
year. The runoff from the cleared area also increased over

the period. A mass balance of the nitrogen indicated an annual
loss of nitrogen from the drainage area of about 50 kg/ha.

The increase is accounted for by increase flow and drainage,
reduction of root surfaces and production of an environment

more favorable for bacterial mineralization.

The contribution of rainfall to the nitrogen balance may be
significant in larger bodies of water. Assuming a range of
0.5 to 1.0 mg/l in rainfall (Feth, 1966), the annual input
would be between 5 and 10 pounds per acre year for a rainfall

of about 50 inches per year.

The contribution of total nitrogen due to domestic wastes may
range from 5 to 50 pounds/capita year, depending on the eco-
nomic and social characteristics of the area. In urban and
suburban areas population densities may vary from 5 to 25 or
more people per acre. On an areal basis, therefore, nitrogen
from this source may run from 25 to 250 pounds/acre year.
This is obviously one of the most significant sources. The
concentration of organic nitrogen in untreated municipal sew-
age ranges from 5 to 35 mg/l with an average of about 20 mg/l

21


equation for organic nitrogen, Ny, is
d(Ny)
[v] at = [A] (N)) + {vl (s.1) +N (115)

where (N,y) is a vector of input sources of N)-

The generalized source - sink term is given by

a
1
Ss ,=—
nl ay Di2 + £(2,P,G,) + a,G(P)Ny - K33 + K34(T)N) (116)

where ay is the ratio of nitrogen to chlorophyll, ay is the
ratio of carbon to chlorophyll, K33 is the settling rate of
organic nitrogen K34 is the hydrolysis rate of organic
nitrogen - a function of water temperature and £(Z,P,G,) is
the excretion of organic nitrogen (see Eq. 78 ).

Ammonia nitrogen (System #4) is produced by the feed forward
hydrolysis reaction of the organic nitrogen and any direct
sources of ammonia due to waste inputs or river runoff. The
oxidation of the ammonia to nitrate forms an important sub-
system of the overall model. This nitrification effect is
used as a sink of dissolved oxygen. The mass balance equa-
tion for ammonia nitrogen No, is

a(Ny)
(vl qe = TAL (N,) + (V1 (Sy) + Oy) (17)

where (Wy ) is a vector of input sources of ammonia. The
source-sifk term for the ammonia is

Sn, = KqsNy - Kye (T)Ny + a,G,Pa (118)

where Kas is the rate of oxidation of ammonia due to nitrify-

ing bacteria (a function of temperature) and a represents an

150


tudinal segments using data on depth,cross-sectional areas

and segment volume given by earlier modeling work in the
Potomac and summarized in Jaworski and Clark (undated).

Figure 29 shows the location of these main estuary segments.

As discussed more fully below, it was found necessary to in-
clude the effects of the tidal bays on main channel quality.
The tidal flat areas have been cited as an important phenome-
non in the Potomac River as far back as 1916 (Phelps). Accord-
ingly, an additional fifteen spatial segments were incorporated
in the model to reflect lateral effects of the shallow water
areas. These tidal bay segments are also shown in Figure 29.

A total of thirty-eight spatial segments were therefore used

to represent the Potomac with primary emphasis on segments

#1 - #15, the Upper Estuary. Each segment is assumed com-
pletely mixed and therefore represents a finite difference
approximation to a continuous medium. In order to simplify

the analysis, waste loads were inputted into segments. #5, 6

and 7 which accounts for the major portion of direct discharge
load. No attempt was made in this preliminary model to input
urban or suburban runoff or overflows from combined sewers.

The model incorporates interactions between nine variables
which are space and time dependent. The variables in the model
are

1) Phytoplankton chlorophyll "a" - P

2) Zooplankton carbon - Z

3) Organic nitrogen - N,

4) Ammonia nitrogen - Ny

3
6) Organic phosphorous - Nou

5) Nitrate nitrogen - N.

7) Inorganic phosphorous - Noe

8) Carbonaceous biochemical oxygen demand - L
9) Dissolved oxygen - C

145


RESEARCH REPORTING SERIES

Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five bread
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technolcgy
transfer and a maximum interface in related
fields. The five series are:

1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research

4. Environmental Monitoring

5. Socioeconomic Environmental Studies

This report has been assigned to the ECOLOGICAL
RESEARCH series. This series describes research
on the effects of pollution on humans, plant = and
animal species, and materials. Problems are
assessed for their long- and short-term
influences. Investigations include formaticn,
transport, and pathway studies to determine the
fate of pollutants and their effects. This work
provides the technical basis for setting standards
to minimize undesirable changes in living
organisms in the aquatic, terrestrial and
atmospheric environments. .


average 0.18 day? (Mullin, 1967). For the Georges Bank
population, Riley used 0.08 day") (Riley, 1947) for the maxi-
mum zooplankton growth rate. For a value of the grazing
coefficient Cy of 0.5 liter/mg-dry wt-day and an assimilation
coefficient of 65%, the Michaelis constant for zooplankton
assimilation, Kap’ ranges between 0.25 and 0.55 mg-dry wt/
liter of phytoplankton biomass. However, these values should
only be taken as an indication of the order of magnitude of
K__. It is probable that its value can vary substantially in

mp
different situations.

The fact that the growth rate reaches a maximum or saturates
is an important feature of the formulation of the zooplankton
growth rate since in some cases the phytoplankton concentra-
tion during part of the year exceeds that which the zooplankton
can effectively metabolize. If the zooplankton growth rate

is not limited in some way and, instead, is assumed simply to
be proportional to the phytoplankton concentration, as propos-
ed in simpler models, the resulting zooplankton growth rate
during phytoplankton blooms can be very much larger than is
physiologically possible for zooplankton, an unrealistic re-
sult. The saturating growth rate also has implications in the
mathematical properties of the resulting equations. In parti-
cular, the behavior differs significantly from the classical
Volterra Preditor-Prey equations (Lotka, 1956). This is dis-
cussed further in a subsequent section.

The growth of the zooplankton population as a whole, of which
the herbivorous zooplankton are a part, is complicated by the
fact that some zooplankters are carnivorous or omniverous.
Thus, the nutrient for the total population should include

not only phytoplankton but also organic detritus as a food
source since this is also available to the grazing zooplankton.
However, for cases where the phytoplankton are abundant and

115


assurance; the steady-state DO profile is estimated to be
everywhere above 4.0 mg/l.

The 20% removal program is equivalent to a discharge load
about 95,000 lbs/day of oxidizable nitrogen which would be
allowable while a 50% removal program is equivalent to a
discharge load of about 60,000 lbs/day of oxidizable nitrogen.
Ultimately therefore waste removal programs must assign both
carbonaceous BOD loads and nitrogen loads both on a vounds/
day basis. General ungquantifiable factors that will tend to
further enhance the attainment of dissolved oxygen goals in-
clude algal utilization of ammonia with subseguent reductions
in the dissolved oxygen deficit due to nitrification, ammonia
oxidation at a slower rate than that assumed and specific en-
couragement of nitrogen removal. Factors that will tend to
mitigate against achievement of the objective include a faster
rate of ammonia oxidation which will intensify and shorten the
area of minimum dissolved oxygen or carbonaceous removal de-
signs that result in oxidizable nitrogen removal of less than
20%.

Application of First-Order Model
to the Potomac Estuary

The Potomac River discharges into Chesapeake Bay and extends
over 100 miles upstream to the head of the estuary at Little
Falls. The major waste source is the effluent of the Washington,
D.C. secondary waste treatment plant. Water quality problems
include low dissolved oxygen in the vicinity of the District

of Columbia discharge and generally high algal concentrations.

Hetling and O'Connell (1968) and Jaworski, et al (March, 1969;

May, 1969) summarized data pertaining to dissolved oxygen,
nutrients and chlorophyll. These results were obtained from

64


WASTE LOAD:

WASTE LOAD—————o

SYSTEM @ SYSTEM @
Corbonoceous Dissolved
Biochemical = |-—_—>} ‘Oxygen -————»
Oxygen Deficit
Demond
SYSTEM @ SYSTEM @ SYSTEM®
Organic Ammonia Nitrate
— Nitrogen Nitrogen Nitrogen
SYSTEM @ SYSTEM @)
Zooplankton [>———-®| _—Phytoplankion.
Corbon ‘Chlorophy!!
SYSTEM © SYSTEM @
Orgonic Inorganic
Phosphorous Phosphorous
WASTE LOAD

Fig.

30 Interactions of Nine Svstems Used in Preliminary

Phytoplankton Model

148


of mass in which the hydrodynamic transport of mass is assumed
to be insignificant. Let P(t) be the concentration of phyto-
plankton mass at time t in a suitably chosen region of water.
The principle of conservation of mass can be expressed as a
differential equation

where S is the net source or sink of phytoplankton mass within
the region. If hydrodynamic transport is not included, then
the rate at which P increased or decreases depends only on the
internal sources and sinks of phytoplankton in the region of
interest.

The form of the internal sources and sinks of phytoplankton
is dictated by the mechanisms which are assumed to govern the
growth and death of phytoplankton. Fleming (1939), as de-
scribed by Riley (1963), postulated that spring diatom flower-
ing in the English Channel is described by the equation,

ge = f[a- (b + ct)]P

where P is the phytoplankton concentration, a is a constant
growth rate, and (b + ct) is a death rate resulting from the
grazing of zooplankton. The zooplankton population, which
is increasing owing to its grazing, results in an increasing
death rate which is approximated by the linear increase of
the death rate as a function of time.

The less empirical model has been proposed by Riley (1963)
based on the equation

s- = [P, - R- GIP

75


SECTION Iv

NITRIFICATION AND ITS EFFECTS
ON DISSOLVED OXYGEN

The discharge of nitrogenous compounds into natural waters
produces a variety of changes in water quality. Changes
occur not only in the various forms of nitrogen, but also

in those substances with which the nitrogen may react. The
oxygen requirements for the ammonia and nitrite oxidation,
the utilization of ammonia and nitrate as a nutrient by phyto-
plankton, and the ammonia interference with chlorination, are
significant examples. The nitrogenous compounds, which may
be either organic or inorganic in nature, are found in urban
and agricultural runoff, domestic waste waters and industrial
effluents. A "background" concentration of nitrogen is
present in most water systems and is the result of a dynamic
equilibrium of the natural sources of nitrogen in rainfall,
from the land and within the ground. Although the man-made
effects are generally of greater significance in the pollu-
tion of natural waters, background nitrogen concentrations
may be present in amounts that must be considered in any
modeling effort.

The purpose of this section is to present several simplified
mathematical models of nitrification in rivers and estuaries
which are based on first order kinetics and the assumotion

of temporal steady state. Specific attention is directed to
the role of nitrification in the modeling of the dissolved
oxygen balance. The models presented here progress from rela-
tively simple nitrogen equivalent biochemical oxygen demand
models to more complex models which incorporate feedback
effects.

The broad aspects of the nitrogen cycle are reviewed and the
importance of nitrogenous waste loads in the dissolved oxygen

11


ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

Bormann, F.H. et al "Nutrient Loss Accelerated by Clear-
Cutting of a Forest Ecosystem" Science, Vol. 159,
Feb. 23, 1968, pp. 882-884

Burns, C.W., Rigler, F.H., "Comparison of Filtering Rates
of Daphnia in Lake Water and in Suspensions of Yeast"
Limnol. Oceanog., 1967, 12 (3) 492-502.

Burns, C.W. "Relation between Filtering Rate Temperature
and Body Size in Four Species of Daphnia" Limnol. Oceanog.
1969, 14(5), 693-700.

Buswell et al, "Study of the Nitrification Phase of the
BOD Test" Sewage & Ind. Wastes, 22, 4, 508, 1959.

Chen, C.W. “Concepts and Utilities of Ecologic Model"
J. Sanit. Engr. Div., Amer. Soc. Civil Eng., Oct. 1970,
96, 1085-97.

Cole, C.R. "A Look at Simulation through a Study on
Plankton Population Dynamics" Report BNWL-485, p. 1-19,
Battelle Northwest Lab, Richland, Wa., 1967.

Comita, G.W. "Oxygen Comsumption in Diaptomus" Limnol.
Oceanog., 1968, 13(1) 51-7.

Conover, R.J. "Oceanography of Long Island Sound, 1952-1954
VI Biology of Acartia Clausi and A. tonsa" Bull. Bingham
Oceanog. Coll, 1956, 15 156-233.

Conover, R.J. "Assimilation of Organic Matter by Zooplankton"
Limnol. Oceanog., 1966, 11, 338-45.

Davidson, R.S., Clymer, A.B. "The Desirability and Appli-
cability of Simulating Ecosystems" Ann. N.Y. Acad.Sci,
1966, 128 (3) 790-4

Davis, H.T. "Introduction to Nonlinear Differential and
Integral Equations" pp 99-109, Dover, N.Y. 1962.

Delwiche, C.C. "Biological Transformations of Nitrogen
Compounds" Ind. & Eng. Chem., Vol. 48, No. 9, Sept. 1956,
pp 1421-1427.

Di Toro, O'Connor, D.J. and Thomann, R.V. "A Dynamic
Model of the Phytoplankton Population in the Sacramento-
San Joaquin Delta" Adv. in Chemistry Series, No. 106,
Amer.Chem. Soc., 1971, pp 131-180.

194


Any additional sources and sinks that contribute can be
added to the source term as needed. With the source term
formulated, the conservation of nutrient mass equation is
given by Equation (53) with N, as the dependent variable

. replacing S_..

eplacing P. d $s
repla cer 32m nj pj

The Equations of the Model

In the previous sections, the equations for phytoplankton
and zooplankton biomass and nutrient concentration within
one volume element have been formulated. The resulting
equations are an attempt to describe the kinetics of the
growth and death of the phytoplankton and zooplankton popu-
lations and their interaction with the nutrients available.
The form of the equations for the volume Vy are as follows:

°
_ = ~(P.,N., - ,(Z., ~ + »(P.,2.,N., )
Pi = (G,,(P5,Ny,t) - Dis (25,t)] Py + S3(P.,25,N5,t) (80)
°
Z, = (G,,(P,,t) - Dj (t)] 2, + S,5(P,,25,t) (81)
°
N, = S.(P.,2.,t) (82)

where G_.and D_.are given by Equations (67) and (74), G and

i) Pj 23
D25 are given by Equations (76) and (77), and Sn5 by Equation
(79). The dependence of the growth and death rates on the
concentration of the three dependent variables and time is

made explicit in this notation.

These equations describe only the kinetics of the populations
in a single volume element V5: However, in a natural water
body there exists significant mass transport as well. The
mass transport mechanisms can be conveniently represented by
the matrix A with elements aay: If for particular segments

124


nentially, cannot grow quickly enough to terminate the phyto-
plankton growth by grazing, and the phytoplankton continue to
grow exponentially until P* is reached. Of course, in the
actual situation, for which G is not a constant, other
phenomena such as nutrient depletion and self-shading exert
their effect, and the growth may be stopped sooner. However,

if the growth rate of zooplankton at a phytoplankton concentra-
tion equal to the Michaelis constant Kup is only slightly larger
than their death rate Dor then the zooplankton alone do not
rapidly terminate the bloom.

This condition is an important dividing iine for the possible
behavior of the phytoplankton-zooplankton equations set forth
in the previous sections. In particular, it indicates what
must be true for a system wherein the zooplankton are a signi-
ficant feature in the resulting phytoplankton solution. How-
ever, if Equation (97) is satisfied, then the zooplankton are
not the dominant control of the phytoplankton population.

Application - San Joaquin River

As an example of the application of the equations proposed
herein, consider the phytoplankton and zooplankton population
observed at Mossdale Bridge on the San Joaquin River in
California during the two years 1966-1967. Mossdale is located
approximately 40 miles from the confluence of the San Joaquin
and the Sacramento Rivers. The data presented below have been
supplied to the investigators by the Dept of Water Resources,
State of Calif (Anon,1966), as part of an ongoing project to
assess the effects of propose’? nutrient lcads and flow di-
versions on the water quality of the San Francisco Bay Delta
(O'Connor et al, 1972).

130


treatment. For the nutrients, median concentrations as given
in Jaworski et al, 1971, were used. Kjehldahl nitrogen of
0.95 mg/l and nitrate nitrogen levels ranging from 0.1 to 0.9
mg/l were inputted as boundary values. Organic phosphorous

of 0.05 mg/l and an inorganic phosphorous concentration of
0.05 mg/l were also inputted as constant values throughout

the year into segment #1. The effect of incoming phytoplankton
chlorophyll was examined under median flows for two cases:

a) chlorophyll a concentrations of 1 ug/l entering the estuary
from up-river and b) chlorophyll a boundary concentration of
25 ug/l, the promulgated EPA criterion.

Figure 50 shows the temporal variation in chlorophyll for the
main channel segment #9 and its associated tidal embayment,
segment #28, Piscataway embayment, using median flows. For
the case of 1 ug/l chlorophyll entering the estuary (Figure
50a), it is seen that the effect of the reduced waste discharge
on the main channel is minor and results in a late summer in-
crease to about 14 wg/l chlorophyll. In the embayment section,
however, chlorophyll concentrations rise to about 55 ug/l
(twice the objective) for a period of about 120 days. It
should be noted however that the level of 50 ug/l is consider-
ably less than the concentrations of 150 - 200 ug/l before

any nutrient reduction. The incoming concentration of 1 ug/l
chlorophyll is unrealistic however and one would normally ex-
pect higher incoming concentrations.

Figure 50b shows the effect of constraining the boundary
chlorophyll concentration at a level equal to the objective
of 25 ug/l. As indicated, main channel chlorophyll levels
rise to over 50 ug/l in August and in the embayment, concen-
trations rise to over 70 ug/l again for a period of about

4 months.

185


Figure 41 shows the spatial chlorophyll profile for June 30 -
July 15, 1969 and the 1969 simulation with 90% reduction of
nutrient waste discharge. As indicated, even after waste
reduction, maximum concentration of phytoplankton chlorophyll
exceed 100 ug/l or four times the objective of 25 ug/l. The
effect of the removal program is significant however from

about Mile 25 to Mile 40. It should be recalled that no dis-
tributed sources of nutrient were included which would increase
the nutrients available for growth.

Figures 42 - 49 show space-time computer generated contour
plots from the 1969 simulations. As indicated in Figure 42
for almost two months, chlorophyll concentrations are greater
than 25 wg/l due in part to the incoming concentration as
given at segment #1. Figure 43, a plot of the computed

Michaelis expression N/K + Ny) as in Eq. (112) indicates

that phosphorous limits the growth and essentially results

in the decrease of the population from its peak value during
July. The flow transients create the "waves" in the contour
plot. As shown in Figure 43, the estuary from about segment
#12 on, is increasingly limited by phosphorous almost uni-
formly throughout the year. Figure 44 shows the accompanying
inorganic phosphorous levels and indicates the large area of
the estuary at which phosphorous is at the Michaelis constant
concentration (see Table 13). Note in Figure 44 the lack of
significant temporal dependence of the phosphorous. Also,
concentrations of inorganic phosphorous in the vicinity of
Washington D.C. range from .05 to .1 mg/l P significantly
above the Michaelis constant indicating that in that immediate
area, phosphorous is not limiting. This is also indicated in
Figure 43. It should also be recalled that other sources of
nutrients from urban and suburban runoff were not included.

Figure 45 is a contour plot of the saturated growth rate of

174


These coefficients represent the end result of manv solutions
of Eq. (51) which tested the effects of various interactions
and levels of coefficients. It can be noted that orqanic ni-
trogen is "settled out" of the svstem because of the difference
between the decav coefficient of organic nitrogen and the
conversion of organic nitrogen to ammonia nitrogen. This was
justified on the basis of bottom sampling which indicated
significant deposits in the vicinitv of the Washington D.C.
outfall. Ammonia nitrogen followed two vaths: a) utilization
in the algal nitrogen loop (K = 0.02 per day) and bh) oxidation
to nitrate (K = 0.28 per dav). This split allowed a vroner
spatial profile to be maintained. Nitrate was recvcled to
algal nitrogen all of which was allowed to decav to organic

nitrogen.

Figures 14 and 15 compare the observed data of the various
nitrogen forms to computed values generated by the model with
and without the feedback of ammonia and nitrate nitrogen to
organic nitrogen. For Figure 14 onlv Kjeldahl nitrogen ob-
served data were available. The effects of the feedback loop
are to increase all profiles in a non-linear spatial manner.
The relative downstream shift of the various nitrogen forms
is interesting and reflects the sequential nature of these
types of reactions. Steady state analyses such as shown in
Figures 14 and 15 can orovide a basis for estimating the ef-
fects of environmental changes on nitrogen distribution in

addition to the effects of nitrification on the oxvgen regimes,

This latter effect is shown in Figure 16 where the dissolved
oxygen deficit due to nitrification is given with and without
algal feedback of nitrogen. Therefore, if the nitrogen is
completely stored in the algae, the dissolved oxvgen deficit
is lower at greater distances downstream than if the algal
nitrogen is released and available for further nitrification.

70


the initial organisms and the substrate. Thus:

dM _
ae = KM (My - M) (15)
where Mn = M, + acy and
K, = Kn
2 ak
Ss

Equation (15) is referred to as an autocatalytic reaction in
which the rate is increased by the concentration of the end
products of the reaction. It is autocatalyzed by the micro-
organisms which increase as substrate is oxidized.

The interacting differential equations for the substrate c
and the microbial mass are given by

-K Mic
dc _ T
de Kite (16)
and
'
au! _ KMtc
dt Kote (17)

where all quantities are now expressed in terms of their
equivalent substrate concentrations. Thus, M' = M/a, the
microbial mass in substrate equivalents. Eq. (17) is the

same as Eq. (14) except in terms of the substrate concentra-
tion. In the interests of mathematical simplicity, the
microbial loss due to endogenous respiration has been neg-
lected in these equations. The solution to Eqs. (16) and (17) is:

d l+d,, - c/c ltd, - c/ce
_ s M M fe}
Kt = Taq log ait oe + log —r (18)
° M

where a. = Ko/cor a dimensionless "Michaelis Number" and

28


Dx 108 WG-P/ CELL-HR

(D PO, ABSO

0 2 4 6 3 1o 12

MEASURES

Fig. 21 Measured Phosphate Absorption Rate, after Ketchum (1939),
vs. Phosphate Absorption Rate Estimated Using uN,N J (Ky thy)
(K_,+N,) where N, and N, are the Nitrate and Phodp ate

coféenérations, hespecttvely

99


d*p dD.

N N
QO = E -uy-~- KD. + KL (27)
dx? ax aN N
where E = dispersion coefficient (usually included only for

estuaries), u = river velocity or net downstream velocity for

estuaries, x = distance, Ly = nitrogenous BOD, Dy = DO
deficit (due to nitrogenous BOD), Ky = rate of oxidation of
NBOD, K, = reaeration coefficient, Wy = waste discharges

of NBOD.

Eqs. (26) and (27) can be recognized as the same set of equa-
tions that are used to describe the DO deficit due to the
discharge of carbonaceous BOD (CBOD). For the constant coef-
ficient case, the solutions to Eqs. (26) and (27) are:

N= Luo exp (3X) (28a)
Dy = we [exp (JX) - exp (3X) ] (28b)

where for rivers where dispersive effects are small

K
+ _ ON
In OT
K
5 = 4
a u (29)
noe
No Q

and for estuaries where the tidal dispersion effect is im-

35


(a)

140
120]
100)
JwgCHLOROPHYLL
{60}

60}

40

20

fo) n 1 n 1 1 1 1 1 L ni 1
30 60 90 120 150 180 210 240 270 300 330 360

Y rr | May | June | July | Avs | Seot | Oct | Now | Dec | a0 | Feb |

DAYS
1968

(b)

fee) ZOOPLANKTON dio
| <— PHYTOPLANKTON /

80)

6 mg CARBON

60)
|

.g CHLOROPHYLL
poor

20

60 90 120 150 180 210 240 270 300 330 360
P moh aos May | sine | July | Avg | Sest | oct | Now | Dee | Jan | Feb |

DAYS
1968

Fig. 34 Effect of Zooplankton Grazing on Phvtoplankton
in Seqment #9 a) No Zooplankton Grazing
b) Zooplankton Grazing at 0.42 1/mq Carb-Dav.
1968 Flow Regime

169


09

OO DEFICIT ( mg/I)

JULY— AUG, 1964

DISTANCE IN MILES

ESTIMATED DO DEFICIT DUE TO NITRIFICATION (After Anon. ,1969)

FIGURE 12



88.

89.

90.

91.

Vollenweider, R.A. "Scientific Fundamentals of the Eu-
trophication of Lakes and Flowing Waters, with Particular
Reference to Nitrogen and Phosphorus as Factors in
Eutrophication" p. 117, Organization for Economic Co-
operation and Development Directorate for Scientific
Affairs, Paris, France, 1968.

Vollenweider, R.A. Ed., “Manual on Methods for Measuring
Primary Production in Aquatic Environments" Ch. 2, p.4-24,
Blackwell Scientific Publications, Oxford, England, 1969.

Wright, J.C. "The Limnology of Canyon Ferry Reservoir,
I. Phytoplankton-Zooplankton Relationships" Limnol.
Oceanog. , 1958, 3(2) 150-9.

Yentsch, C.S., Lew, R.W. "A Study of Photosynthetic Light
Reactions" J. Marine Res, 1966, 24(3).

200


waste reduction program is completed for the Potomac Estuary.
Primary emphasis here is on the efficacy of nutrient removal
programs especially in the Washington D. C. area and down-
stream.

The area of interest of the model is centered in the upper
forty-mile reach of the Potomac Estuary although as indicated
below,the model geographically extends over the entire 114-
mile length of the estuary from Little Falls to Chesapeake
Bay. The upper reach of 30 - 40 miles is generally unaffected
by the incursion of saltsfrom Chesapeake Bay. The Potomac is
tidal in the vicinity of Washington D.C. and is several
hundred feet wide with a shipping channel of minimum depth

of 24 feet maintained to Washington. The tidal portion
averages about 18 feet and is characterized by numerous

coves and embayments along its length with average depths
significantly less than the main estuary. Presently, the
Major waste source in the upper reach of the Potomac is the

effluent from the Washington D.C. secondary treatment plant.

Water quality problems include low values of dissolved oxygen
in the vicinity of the District of Columbia and high concen-
trations of phytoplankton especially of Anacystis, a blue
green form. Detailed reviews are given bv Jaworski et al
(March 1969), Jaworski (Nov 1969), Aalto et al (1970) and
Jaworski et al (1971).

Model Geometry and Kinetics

The basic model builds on previous efforts of phytoplankton
dynamics (Di Toro et al, 1971) and nitrification effects in
estuaries (Thomann et al, 1970).

Spatially, the estuary is divided into twenty-three longi-

144


growth of microorganisms, as given by Eq. (13) where nutri-
ents are not limiting, (kK, = 0 in Eq. (10)) and (b) Case II,
logistic growth of microorganisms, as given by Eq. (14)

where K, >> c. In terms of the substrate utilization and
utilizing the dimensionless notation, these two limits can be
summarized as shown in Fig. 5. As indicated, therein, for
various dy = M' feo the substrate utilization varies from a
linear decrease to exnonential to an autocatalytic form. Under
certain conditions, therefore, an appropriate exponential de-
crease (first order kinetics) may be assumed in mathematical
modeling under full recognition that other forms may prevail
depending on the "mix" of microbial population, nutrient con-
centration and Michaelis constant. In the logistic growth
region, for 0.1 < dy < 1.0 Fig. 5-shows that for comolex river
or estuarine systems it would be very difficult to detect de-
partures from exponential substrate decay. Some data are
available on laboratory studies to evaluate the range of M'o.
However, actual field estimates of microbial mass, and sub-
strate concentrations are not available. As a consequence,
laboratory studies generally use "large" initial substrate
concentrations (a, and ay small) leading to specific non-
exponential substrate utilization. Stratton and McCarty (1967)
for ammonia oxidation obtained values of My @ 20°C of 0.0033 mg
bacterial mass/l. At a yield coefficient, a, of 0.28 mg cells/
mg substrate, this is equivalent to M'y of about 0.01 mg/l NH,-N.
For this case however, K, = 2.6 mg/l and Co = 5.5 mg/l or

ds - 0.47. Stratton thus obtained ammonia curves similar to
those shown in Fig. 5 for low values of du intermediate between
exponential and logistic microbial growth. This then repre-
sents an initial low nitrifying population and “high" initial
substrate concentration. At 20°C, for K. # 1-3.0 mg/l Knowles
et al (1965) for normally low initial ammonia concentrations

in natural waters, (0.5 mg/l), a, for ammonia could be "large".
If the ratio of initial cell mass, My to initial concentrations
is from 0.1 - 1.0 one should expect to see appropriate exponenti-

31


dissolved oxygen deficit decreases from about 2.2 mg/l to
about 1.4 mg/l. The difference between the downstream dis-
solved oxygen deficit and upstream increase is due to the
increasing cross-sectional area of the estuary as one pro-
ceeds in the downstream direction. If a 50% removal of
oxidizable nitrogen were accomplished, the estimated dis-
xolved oxygen deficit profile is as shown in Figure 13. A
general decrease is noted with a maximum dissolved oxygen
deficit of about 1.3 mg/l in the area of mile 90. In order
to provide an overall estimate of the effect of this shift
in the dissolved oxygen deficit profile and projected water
quality goals, a preliminary analysis was made of the esti-
mated dissolved oxygen profile under existing waste removal
requirements.

The estuary proper has been divided by the Delaware River
Basin Commission into four (4) zones with ultimate carbon-
aceous BOD removal requirements ranging from 86% to 89%,
based on raw 1964 waste loads. These requirements will gen-
erally be met by various types of secondary treatment, in-
cluding for municipalities, biological waste reduction. It
is difficult at this stage to estimate the extent of nitrogen
reduction to be expected from this prejram. The particular
design practices will govern this factor. However, for esti-
mating purposes, a value of about 20% oxidizable nitrogen

reduction appears reasonable.

A dissolved oxygen analysis was therefore made, using the
same sectional breakdown as used in the nitrification model.
The analysis indicated that under a 20% nitrogen removal of
1964 loads the dissolved oxygen goal of the DRBC will proba-
bly be met. ‘The critical region is in the vicinity of mile
100-90. Under 50% nitrogen removal of 1964 loads, the DRBC
dissolved oxygen goals will be met with a greater degree of

63


CHLOROPHYLL -fg/1

Dp @
oO ce]
Td

&
1

MEDIAN FLOW SIMULATION

90% REMOVAL OF NITROGEN
90% REMOVAL OF PHOSPHOROUS
+

25 /dg/| CHLOROPHYLL ENTERING ESTUARY

noe - JULY 15

3
|

ec)

“OBJECTIVE
| | | al _| | |

°

10 15 20 25 30 35 40
MILES BELOW CHAIN BRIDGE

Pig. 50 Temporal Variation in Chloronvhvll a at
Segments #9 and #28, Median Flow Simulation.
a) luq/l Chlorovhvll Boundarv hb) 25yq/1 Chlorovhvll
Boundary

186


10.

SECTION VIII

REFERENCES

Aalto, J.A., Jaworski, N.A. and Lear, D.W. "Current Water
Quality Conditions and Investigations in the Upper Potomac
River Tidal System" Tech. Report No. 41, Ches. Tech.

Support Lab., FWQA, Annapolis, Md., May 1970, V. Chapters.

Adams, J.A., Steele, J.H. "Some Contemporary Studies in
Marine Science" Shipboard Experiments on the Feeding of
Calanus Finmarchicus p. 19-35, H. Barnes, Ed., G. Alien
and Unwin Ltd., London, 1966.

Allison, F.E. "The Enigma of Soil Nitrogen Balance Sheets"
Advances in Agronomy p. 213-250, A.G. Norman, Ed., Academic
Press Inc., N.Y. TOSS"

Anon, “Effects of Polluting Discharges on the Thames Estu-
ary" Water Pollution Res. Tech. Paper No. 11, Dept. of

Scien. & Ind. Res., Her Majesty's Stationary Office,
609 pp + xxvlll, 1964.

Anon, Department of Water Resources, State of California
private communication, 1966.

Anon, "Nitrification in the Delaware Estuary" Prepared by
Hydroscience Inc., Westwood, N.J. for Delaware River Basin
Comm., Trenton, N.J., June 1969, 43 pp + Append.

Anon, "Eutrophication of Surface Waters - Lake Tahoe"
Lake Tahoe Area Council, South Lake Tahoe, Calif. 95705,
Report for FWPCA, Dept. of Interior, WPD 48-02, May 1969b.

Anraku, M., Omori, M., "Preliminary Survey of the Rela-
tionship between the Feeding Habits and the Structure of
the Mouth Parts of Marine Copepods" Limnol. Oceanog.
1963, 8(1), 116-26.

Azad, H.S., Borchardt, J.A. "A Method for Predicting the
Effects of Light Intensity on Algal Growth and Phosphorus
Assimilation" J. Water Pollution Control Federation, 1969,
41 (11), part 2.

Bishop, J.W. “Respiration Rates of Migrating Zooplankton
in the Natural Habitat" Limnol. Oceanog., 1968, 13 (1)
58-62.

193


30.3 ug PO,-P/liter, with a saturated absorption rate of

15.1 x 1078 ug PO,-P/cell-hr. This approximation to the
growth rate behavior as a function of more than one nutri-
ent must be regarded as only a first approximation, however,
since the complex interaction reported between the nutrients
is neglected.

The result of the above investigation is the following growth
rate expression. For the case of one limiting nutrient, N,
with Michaelis constant Kae the growth expression for the
rate in the 3th segment is

/ 2.718£

Go. = K,T, ( gE8E fe
173 ( km,

Pa ? ( ej"y a

in which Equations (55) and (62) have been combined. This
is the functional form that is used subsequently in the ap-
plications of these equations to natural phytoplankton
populations.

Comparison with Other Growth Rate Expressions

The most extensive investigation of the relationship between
the growth rate of natural phytoplankton populations and the
significant environment variables, within the context of
phytoplankton models, is that of Riley et al (1949). The
expression which results from their work is

G
'p _ : _ _ 6573.8
log lee = 22.884 + log Vp log I, - = —s*(68)

° Pp

100


160-

140

T

soo, ane jie ll

S 20k OBSERVED DATA. il

1oor

CHLOROPHYLL 49/1
8.8

Oo 5 10 15 20 25 . 30 35
MILES BELOW CHAIN BRIDGE

Fig. 36 Sensitivitv Run - No Tidal Bav Segments.
Aug. 1968 Profile and 1968 Conditions

163


There is a difficulty, however, in using this term directly
as a source of nutrient. To illustrate this difficulty,
assume that the nutrient is inorganic nitrogen. A part of
the excreted nitrogen, however, is in organic form, and a
bacterial decomposition into the inorganic forms must pre-
cede utilization by the phytoplankton. The same is true for
the nutrient released by the death of phytoplankton, anpkoTPy

Vr

an zk3T45- where

ang is the nutrient-to-zooplankton biomass ratio. Therefore,

strictly speaking, a conservation of mass equation for the

and that released by the death of zooplankton,

organic form of the nutrient is required. The organic form

is then converted to the inorganic form. For the case of ni-
trogen, the kinetics of this conversion have been investigated
and applied to stream and estuarine situations (Thomann, 1963).
If the conversion rate is large by comparison with the other
rates in the phytoplankton and zooplankton equations, then the
direct inclusion of these sources is approximately correct.

The sources of nutrients arising from man-made inputs, such

as wastewater discharges and agricultural runoff, are included
explicitly into the source term since these sources are usually
the major control variables available to influence the biologi-
cal systems. An extensive review of the magnitude and relative
importance of these sources of nutrients, primarily nitrogen
and phosphorus, has recently been made (Vollenweider, 1968).

A useful distinction is made between diffuse sources such as
agricultural runoff loads and ground water infiltration, which
are difficult to measure and control, and point sources such as
wastewater discharges from municipal and industrial sources,
for which more information is available. The nitrogen and
phosphorus loads from agricultural runoff are quite variable
and depend on many variables such as soil type, fertilizer ap-

plication, rainfall, and irrigation practice. The nutrient

122


of the initial sources, C >C (for n reaction steps)

01 on

is:

c =

no t c

mas

- *
OL exp ( Kt )

and due to the intermediate steps is

; Ki-1°0,i-1
Cc. = go: [exp (K,_,t*)-exp-K,t*)
ni ger Rae Kyi i-l 1

the sequential nature of these solutions is shown in Figure 7
where the last nitrogen form, the nitrate nitrogen is assumed
conservative, i.e K, = 0.

For estuarine situations, the solutions essentially follow

the same form except that a series of unknown coefficients
are introduced by virtue of mass balances that are required.
The number of coefficients is equal to the number of equations
that can be written down so that explicit numerical determi-
nation of the coefficients is possible. For example, the
solution to Eq. (31) for point source loads is

Ny = By exp (1x) + Cy exp (V, x) + YBN (39)
_ f 2
where S) = oF (1 + 1+4 K,E/u )
= ._ f 2
Y= oF Q 1+ 4 K),E/u*)

B, and C, = constants to be evaluated from consideration of
boundary conditions
and YBN = particular integral of sources and sinks of Ny).

41


equation is too general to be solved analytically, and the
numerical techniques are used in its solution.

An effective approximation to Equation (52) is obtained by
segmenting the water body of interest into n volume elements
of volume vy and representing the derivatives in Equation (52)
by differences. Let V be the n x n diagonal matrix of volumes
yi A, the n x n matrix of dispersive and advective transport
terms; Sor the n vector of source terms $53" averaged over the
volume Vai and P, the n vector of concentrations P., which are
the concentrations in the volumes. Then the finite difference

equations can be expressed as a vector differential equation

fe}

VP = AP + vs, (53)
where the dot denotes a time derivative. The details of the
application of this version of the dispersion advection equa-
tion to natural bodies of water has been presented by Thomann
(1963) and reviewed by O'Connor et al (1966).

The main interest in this report is centered on the source
terms Sp5 for the particular application of these equations

to the phytoplankton population in natural water bodies. It
is convenient to express the source term of phytoplankton, $55"
as a difference between the growth rate, Saye of phytoplankton
and their death rate, Dos" in the volume V5: That is

_ = GL. - D..)P. 54
Sp} ( iz) pi) P5 (34)

pj and Da; have units (day). The subscript p identi-
fies the quantities as referring to phytoplankton; the subscript

where G

j refers to the volume element being considered. The balance
between the magnitude of the growth rate and death rate deter-
mines the rate at which phytoplankton mass is created or des-
troyed in the volume element V5: Thus, the form of the growth

83


SECTION I

CONCLUSIONS

Based on the results of the mathematical model formulations
and verifications presented in this report it is concluded
and the impact on water quality of the carbonaceous, ni-
trogenous, and phosphoric fractions of wastewaters discharged
into an estuarine environment can be assessed, to guide the
preliminary planning of remedial actions to improve water
quality. The water quality parameters that appear sus-
ceptible to such an analysis are the dissolved oxygen levels
and the phytoplankton biomass which result as a consequence
of natural and man-made inputs. The analyses and verifica-
tions presented for the Delaware and Potomac estuaries are
viewed as the foundation upon which a rational investigation
of wastewater treatment alternatives can be based for the
Management and control of estuarine water quality.


ABSTRACT

This report describes the formulation and initial verifica-
tion of two modeling frameworks. The first is directed
toward an analysis of the impact of the carbonaceous and
nitrogenous components and wastewater on the dissolved
oxygen resources of a natural water system. The second
modeling framework concentrates on the interactions between
the discharge of nutrient, both nitrogen and phosphorus, and
the biomass of the phytoplankton and zooplankton populations
which result, as well as incorporating the overall impact on
dissolved oxygen. The models are formulated in terms of
coupled differential equations which incorporate both the
effect of transport due to tidal motion and turbulence, and
the kinetics which describe the biological and chemical
transformations that can occur. The modeling frameworks are
applied to the Delaware and Potomac estuaries in order to
estimate the ability of such models to describe the water
quality effects of carbon, nitrogen, and phosphorous dis-
charges. The agreement achieved between observation and
calculation indicate that the major features of the impact
of wastewater components on eutrophication phenomena can be
successfully analyzed within the context of the models pre-
sented herein.

This report was submitted in fulfillment of Project Number R800369, by
Manhattan College, Bronx, New York, under the sponsorship of the
Environmental Protection Agency. Work was completed as of December 31,

1972,

rom
Be
pe


i) 5 10 1S
SCALE OF MILES
nN © =MAIN ESTUARY SEGMENTS
7 (J: TIDAL BAY SEGMENTS
VIRGINIA
MARYLAND

CHESAPEAKE
(bay

Fig. 29 Mav of Potomac Estuarv Showing Longitudinal
and Lateral Segments

146



Equations (31) through (34) permit the determination of the in-
dividual nitrogen components and therefore represent a more
realistic description of the nitrification process than
Eqs.(26) and (27). The forcing functions for the DO deficit
are 3.43 Ko3N5 and 1.14 K34N3; the first revresenting ammonia
oxidation and the second nitrite oxidation. The entire
scheme is sketched in a block diagram form in Figure 6.
The equation for DO deficit due to the oxidation of nitro-
gen forms is:

a’Dy aDy

-udx ~ *,?

+ 343K, Ny (x) + 1.14K3 4 N, (x) (35)

oO = E N

dx?
where N(x) and Ng (x) are given from solutions of Eqs. (31)
through (34).

Particular utilization of Eqs. (31) - (34) again depends on
the water system either a river svstem where E = O or an
estuarine system where tidal dispersion effects embodied in
the dispersion coefficient may be significant.

The nature of the solutions to Eqs. (31) - (34) can be de-
termined by considering the river case where advective forces
dominate and dispersion can be considered zero. Further, one

can assume that K;,; = K Liki i.e that all material is

i,i+
conserved in each system and none of the forms of nitrogen

are lost, for example, to the bottom sediments, (Ky > K, it1)°

1,
The rate equations for the advective stream are then:
any
aer = ~ KN, + Wy (x) (36a)
aN,
gee = KN, ~ KN, + Wy (x) (36b)

38


Kjeldahl
TOTAL-NITROGEN mg/l

CHLOROPHYLL pg/|

(a)

160-

'40r NUTRIENTS

CONSERVED——TM

BD
is}

3
is}

@
is}

@ NUTRIENT DECAY=0,05/day
@ NUTRIENT DECAY: 0.1 /day

20+ ®

40

n

fe 5 10 5 20 25 30 35
MILES BELOW CHAIN BRIDGE

(0) KNIT= 0.05/day
@® Knt=0.!/day

ce) 5 10 IS 20 25 30 35
MILES BELOW CHAIN BRIDGE

Fiq. 32 Comparison of Range of Observed Data and Model Outnut
August 1968 a) Chlorophvll a (uq/1l)
b) Total Kjheldahl Nitrogen (mq/1)

157



nN
o)

SAY “AUG. 1968

O = 2000 CFS

LEGEND:

@ compureo wit
ALGAL FEED BACK

® cowrvreo wrH
NO FEED BACK

np
°
T

ro.)
—T

NITRITE+NITRATE -N(mg/1)
o -
a o
T T

Hee} ) } }
iLES FROM. CHAIN? O 5 10 15 20 Fs
BRIDGE 11 letsalals|e! 7s] 9 |] ol un] * B |
250
NITROGEN -

Ie | || ee OF COSERVED DIA

seguran

69

a
°o
T

CHLOROPHYLL © (ug/!)
8

WILT

5O
fe) t 1
MILES FROM CHAIN O 5 lo is 20 25 30 3
BRIDGE (i fealslelsfe] 7a] osfolul eae | os] mw] oo |
SEGMENT NO.
Figure 15

Verification of Nitrite and Nitrate Nitrogen (upper plot)
and Chlorophyll "a" (lower plot) for Potomac Estuary


encountered in formulating simple theoretical models of phyto-
plankton-zooplankton population models were discussed by
Steele (1965).

Other models have been proposed which follow the outlines of
the equations already discussed. Equations with parameters
that vary as a function of temperature, sunlight, and nutrient
concentration have been presented by Davidson and Clymer (1966)
and simulated by Cole (1967). A set of equations which model
the population of phytoplankton, zooplankton, and a species of
fish in a large lake have been presented by Parker (1968).

The application of the techniques of phytoplankton modeling

to the problem of eutrophication in rivers and estuaries has
been proposed by Chen (1970). The interrelations between the
nitrogen cycle and the phytoplankton population in the Potomac
Estuary has been investigated using a feed-forward-feed-back
model of the dependent variables, which interact linearly fol-
lowing first order kinetics (Thomann, 1970).

The formulations and equations presented in the subsequent
sections are modifications and extensions of previously pre-
sented equations which incorporate some additional physiologi-
cal information on the behavior of phytoplankton and zoo-
plankton populations. In contrast to the majority of the
applications of phytoplankton models which have been made
previously, the equations presented in the subsequent sections
are applied to a relatively shallow reach of the San Joaquin
River and the estuary further downstream. The motivation for
this application is an investigation of the possibility of ex-
cessive phytoplankton growths as environmental conditions and
nutrient loadings are changed in this area. Thus, the primary
thrust of this investigation is to produce an engineering tool
which can be used in the solution of engineering problems to

protect the water quality of the region of interest.

78


EPA -660/3-73-009
August 1973

Ecological Research Series



depends on temperature and light intensity. The effect is to
lower the Michaelis constant at high temperatures and at high
light intensity levels, which seems to be a reasonable behavior
for a phytoplankton population.

More elementary growth rate formulations have been proposed
which do not span the range of conditions attempted in Equa-
tions (67) and (71). In particular, a common proposal is to
make the growth rate linearly proportional to the various
environmental variables. For example, Davidson and Clymer
(1966) assumed that the growth rate is proportional to phos-
phate concentration and photoperiod and a temperature factor
given by exp [-(T - 18)°/18). This temperature factor is

quite different from all others proposed and greatly magnifies
the effect of temperature on the growth rate. For example, at
T = 18°C, the factor equals 1.0, whereas at T = 9°C, the factor
drops to 0.01, a 100-fold decrease, compared with approximately
a 2-fold decrease predicted by Equations (67) and (71). This
exaggerated effect seems to be unrealistic.

A complete investigation of the environmental influences on

the growth rate is still to be made. In particular, it has
been emphasized that there is an interaction between nitrogen
and phosphorus limitations as well as other effects which in-
fluence the phytoplankton growth rate. Also, these effects

are different for differing species. The species-dependent
effects are important in the problem of the seasonal succession
of phytoplankton species.

For any particular application, it is advisable to investigate
the growth rate of the already-existing population, as the re-
sulting expression may differ significantly from the general
over-all behavior as described by Equations (67) and (71).
Also, in dealing with natural associations of species of phyto+

104


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3°Td AnojuOD uoTIeTNUTS G96T

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PHOSPHATE PHOSPHOROUS POY-P

p

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SIMULATION NO. 3

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4. 1g.
SEGMENT NUMBER

(MG/L)

11 CONTOUR LEVELS

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ton growth rate, of the phytoplankton and nutrient concen-
trations as well. However, for a situation with adequate
nutrients and low initial phytoplankton concentration, the
nonlinear interaction is small initially, and the time vari-
ation of S can be small during the summer months. In any
case, the analysis of this simplified situation is quite
instructive.

Although no analytical solution is available for these simpli-
fied equations, their properties are well understood (Davis,
1962). In particular, the equations have two sets of singular
points corresponding to the solution of the righthand side of
Equations (86) and (87) equated to zero: the trivial solutions
P* = 0, Z* = O, and

D, S, - Dy
pt = ze = PP (88)
co C
®2p'g g

A perturbation analysis of Equations (86) and (87) about this
singular point shows that the solutions whose initial condi-
tions are close to P*, Z*, oscillate sinusoidally about this
singular point. Hence, no constant solution is possible.

The prey and predator populations continually oscillate and
are out of phase with each other. When the predator predomin-
ates, the prey is reduced, which in turn causes the predator
to die for lack of food, which allows the prey to proliferate
for lack of predator, which then causes the predator to grow
because of the prey available as a food supply, and so on.
The interesting feature is that these oscillations continue

indefinitely.

The classical Lotka-Volterra equations assume an isolated
population with no mass transport into or out of the volume
being considered. To simulate the effect of mass transport
into the volume, assume that an additional source term of

127


SST

1 . 1 L L

!
'
i

1

4 1 1

30

25)
FLOW
(l00Ocfs) 20}
INTO
#1

vune July Aug Sep! Oct

{
|
|
|
|
T
1968 |
{
|
|
|
I
|
!
!

Nov Dec | Jan Feb

Apri

May

June

1969

1
July Aug

Sept

-K——————Data Period - 1969 ——————____>|

Oct Nov De

—

fe} rn L . m 1 1 . 1
April Moy June July Aug Sept Oct Nov Dec | Jon Feb April Moy June July Aug Sept Oct Nov Dec
1968 1969
Fiq. 31 Temoerature and Flow Regimes Used for 1968 and 1969 Verification Runs



EPA REVIEW NOTICE

This report has been reviewed by the Environmental Protection Agency
and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Environ-
mental Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation of use.

di


Figure 51 is a longitudinal profile for July 15 for the median
flow simulation and a 25 ug/l chlorophyll boundary condition.
For about a 25 mile region of the estuary, the concentration
of phytoplankton chlorophyll exceeds the objective and rises
to a maximum value of almost 60 wg/l. Although this is above
the objective, it is nevertheless a reduction of about 60%
from maximum chlorophyll concentrations before any nutrient
removals.

These simulations permit the following general observations.
Achievement of the objective of 25 ug/l chlorophyll in the
estuary may not be possible, primarily because of the effect
of discharges from the Upper Potomac into the estuary. Even
under a median flow regime (4,000 cfs during the growing
season), maximum concentrations of 50 ug/l chlorophyll are
calculated for the main channel and 75 wg/l in some embayments.
It should also be noted that additional sources of nutrients
such as local drainage and urban runoff have not been included
in the model. Therefore, even if reductions greater than the
90% used in these applications could be achieved, such reduc-
tions would be probably offset by other distributed nutrient
sources. On the basis of the preliminary model, then, state-
of-the-art nutrient reduction (90% or better) in the Potomac
may provide reductions in average chlorophyll levels by about
60% under median flow conditions. Again, the simulations
should be considered as indicative of general trends only and
not as a predictive certainty.

Summary

A non-linear dynamic model of phytoplankton chlorophyll in
the Upper Potomac estuary verifies the observed 1968 and 1969
phytoplankton and nutrient temporal and spatial trends and
approximately verifies the associated concentration levels.

187


plankton, the various constants which result from such an
investigation can be considered to be averages over the
population, and so they represent in some average way the
population behavior as a whole.

Phytoplankton Death Rate

Numerous mechanisms have been proposed which contribute to
the death rate of phytoplankton: endogenous respiration rate,
grazing by herbivorous zooplankton, a sinking rate, and
parasitization (Fogg, 1965). The first three mechanisms
have been included in previous models for phytoplankton dy-

namics, and they have been shown to be of general importance.

The endogenous respiration rate of phytoplankton is the rate
at which the phytoplankton oxidize their organic carbon to
carbon dioxide per unit weight of phytoplankton organic car-
bon. Respiration is the reverse of the photosynthesis process
and as such contributes to the death rate of the phytoplankton
population. If the respiration rate of the population as a
whole is greater than the photosynthesis or growth rate, there
is a net loss phytoplankton carbon, and the population biomass
is reduced in size. The respiration rate as a function of
temperature has been investigated, and some measurements are
presented in Figure 23 and Table 8. A straight line seems to
give an adequate fit of the data; that is, Respiration Rate =
K,T. For the respiration rate in days and T in °C, the
value of Ky is in the range 0.005 + 0.001. The lack of any
more precise data precludes exploring the respiration rate's
dependence on other environmental variables. However, an im-
portant interaction has been suggested by Lune (1965). During
nutrient-depleted conditions, "many algae pass into morphologi-
cal or physiological resting stages under such unfavorable
conditions. Resting stages are absent in Asterionella formosa,

105


GROWTH RATE- K (Yoay)

K

3 sUBSTRATE CONCENTRATION ~ C

Fig. 4 Variation of Growth Rate with. Substrate Concentrations

26


of biomass is present and illuminates all the cells, and
further that all the necessary nutrients are present in suf-
ficient quantity so that no nutrient is in short supply. For
this condition, the growth rate that is observed is called

the maximum or saturated growth rate, K'. Measurements of K'
(base e) as a function of temperature are shown in Figure 18
and listed in Table €. The experimental conditions under

which these data were collected appear to meet the require-
ments of optimum light intensity and sufficient nutrient supply.
The data presented are selected from larger groups of reported
values, and they represent the maximum of these reported growth
rates. The presumption is that these large values reflect the
maximum growth rates achievable. From an ecological point of
view, it is necessary to consider the species most able to
compete, and, in terms of growth rate, it is the species with
the largest growth rate which will predominate. A straight-
line fit to this data appears to be a crude but reasonable
approximation of the data relating saturated growth rate K'

to temperature, T

K' = KT (55)

1 1

1 °c’. This
coefficient indicates an approximate doubling of the saturated
growth rate for a temperature change from 10° to 20°C, in ac-

where K, has values in the range 0.10 + 0.025 day

cordance with the generally reported temperature-dependence of
biological growth rates. The optimum temperature for algal
growth appears to be in the range between 20° and 25°C, al-
through thermophilic strains are known to exist (Fogg, 1965).
At higher temperatures, there is usually a suppression of the
saturated growth rate, and the straight-line approximation is
no longer valid. It should also be noted that the scatter in
the data in Figure 18 is sufficiently large so that the linear

85


36.

37.

38.

39.

40.

41.

42.

43.

44,

45.

46.

47.

48.

Jaworski, N.A. et al "Nutrients in the Potomac River
Basin" Tech. Rept. No. 9, Ches. Tech. Sup. Lab., FWPCA,
Dept. of Int., May 1969, 40 pp.

Jaworski, N.A. "Water Quality and Wastewater Loadings,
Upper Potomac Estuary During 1969" Tech. Rept. No. 27,
Ches. Tech. Sup. Lab., FWQA, Annapolis, Md., Nov. 1969,
VII Chapters.

Jaworski, N.A., Clark, L.S. and Feigner, K.D. "A Water
Resource Water Supply Study of the Potomac Estuary"

Tech. Rept. No. 35, Ches. Tech. Sup. Lab, EPA, Annapolis,
Md., April 1971, XII Chapters.

Jaworski, N.A. and Clark, L.J. "Physical Data, Potomac
River Tidal System Including Mathematical Model Segmenta-
tion" Tech. Rept. No. 43, Ches. Tech. Sup. Lab., FWQA,
Annapolis, Md. undated, unpaginated.

Ketchum, B.H. "The Absorption of Phosphate and Nitrate
by Illuminated Cultures of Nitzschia Closterium" Am. J.
Botany, June 1939, 26.

Knowles, G., Downing, A.L. and Barrett, M.J. "Determina-
tion of Kinetic Constants for Nitrifying Bacteria in Mixed
Culture, with the Aid of an Electronic Computer" J. Gen.
Microbiol., Great Britain, 1965, 38, 263-278.

Kuentzel, L.E. "Bacteria, Carbon Dioxide and Algal Blooms"
J. Water Pol. Control Fed., Oct. 1969, 41 (10).

Lawrence, A.W. and McCarty, P.L. "Unified Basis for
Biological Treatment Design and Operation" J. SED, ASCE,
Vol. 96, No. SA3, June 1970, pp. 757-778.

Levy, H., Baggott, E.A. "Numerical Solutions of Differenti-
al Equations" p. 91-110, Dover, N.Y. 1950.

Lotka, A.J. "Elements of Mathematical Biology" vp. 88-94,
Dover, N.Y. 1956.

Lund, J.W.G. "The Ecology of the Freshwater Phytoplankton"
Biol. Rev, 1965, 40, 231-93.

MacIsaac, J.J., Dugsdale, R.C. "The Kinetics of Nitrate
and Ammonia Uptake by Natural Populations of Marine Phyto-
plankton" Deep Sea Res., 1969, 16, 415-22.

Martin, J.H. "Phytoplankton Zooplankton Relationships in
Narragansett Bay III" Limnol. Oceanog, 1968, 13 (1).

196


FLOW

| ZOOPLANKTON

PREY GRAZING

TEMPERATURE

= 7 PHYTOPLANKTON

NUTRIENT NUTRIENT
LIMITATION USE

Y
SOLAR
RADIATION ceo NUTRIENTS.

\

NX
1+

MAN MADE
INPUTS

I
Fig. 17 Interactions of Environmental Variables and the
Phytoplankton, Zooplankton and Nutrient Systems

80



function of position and time.

For complex, interacting situations, it is necessary to
simultaneously characterize the concentrations c, (x,t) of

a number of substances. In this case the kinetic interaction
terms are usually functions of all the concentrations,

Silc., Cor veer Cpe r,t) so that a set of n simultaneous

n
equations result of the form:

oc.

Et VG, = Sylepee-eecy eet) + Wy (r,t) 2 = l,..en (2)
where J; = - E(r,t)Ve, + Bir, t)e,, the mass flux, due to
transport.

It is also common to simplify these equations by analyzing
less than the three spatial dimensions. The simpliest situ-
ation occurs in a one-dimensional analysis for which the

set of equations (2) becomes

ac.
*)

ac
i123 3 _
se RK (EAs + (Qc,) = Sj (eye e eee xk, t) + W; (x,t) (3)

The procedure followed in the applications described herein
has been to numerically integrate these equations in order
to characterize the distribution of substances of concern.

The water quality problems investigated center on the effects
of nitrogen and phosphorus discharges as well as the carbon-
aceous waste discharges. The analyses are directed toward
the effects on the dissolved oxygen distribution on the one
hand and the effects on the first two trophic levels of the
food chain, the phytoplankton and zooplankton, on the other.
A series of models are presented and verified in the subsequ-


SELECTED WATER 1 Revert Naf?
RESOURCES ABSTRACTS

INPUT TRANSACTION FORM Ww
S. AKeport Dae er
DYNAMIC WATER QUALITY FORECASTING AND MANAGEMENT be
8 2 storm: ~ Orga. zation
Repo-* No.
O'Connor, Donald J., Thomann, Robert V., and Di Toro, Dominic, ""
R800369
Manhattan College, Bronx, New York, Civil Engineering Dept. fiat
R800369

1 Typ: “Repo and

Period Covered
2S: asorir Qrgar’ tion yg Environmental Protection Agency Final Report

Environmental Protection Agency report number,
EPA-660/3-73-009, August 1973.

This report describes the formulation and initial verification of two modeling frame-
works. The first is directed toward an analysis of the impact of the carbonaceous and
nitrogenous components and wastewater on the dissolved oxygen resources of a natural
water system, The second modeling framework concentrates on the interactions between
the discharge of nutrient, both nitrogen and phosphorus, and the biomass of the phyto-
plankton and zooplankton populations which result, as well as incorporating the overall|
impact on dissolved oxygen. The models are formulated in terms of coupled differential
equations which incorporate both the effect of transport due to tidal motion and
turbulence, and the kinetics which describe the biological and chemical transformations
that can occur. The modeling frameworks are applied to the Delaware and Potomac
estuaries in order to estimate the ability of such models to describe the water quality
effects of carbon, nitrogen, and phosphorous discharges. The agreement achieved
between observation and calculation indicate that the major features of the impact of
wastewater components on eutrophication phenomena can be successfully analyzed within
the context of the models presented herein. (0'Connor-Manhattan College)

17a. Descriptors #Water quality, *Mathematical Models, *Computer Models, Water pollution,
Cycling nutrients, Eutrophication, Dispersion, Mass Transfer, Nutrients, Oxygen Demand,
Photosynthesis, Simulation Analysis

17b, Identifiers

We. CG RE Meld & Grong O5C, O5G, 06G

19. Security Class. | 21. No.of | send To:

‘Repo: } Pages
eyes , . WATER RESOURCES SCIENTIFIC INFORMATION CENTER
% Se rity os 2 Pas U.S. DEPARTMENT OF THE INT!
(Page) WASHINGTON. D.C. 20240,

+. Donald J. O'Connor I : Manhattan College, Bronx, New York

‘US, GOVERNMENT PRINTING OFFICE: 1973 546-311/98 1-3,


110 pounds of organic nitrogen/day/mile was inputted together
with an equal amount of ammonia nitrogen.

A continuous solution model was used to represent the organic,
ammonia, nitrite and nitrate forms. Only sequential bacterial
nitrification was included specifically in the model. Algal
and denitrification effects were included qualitatively. The
continuous solution model (see Eqs. 39-40) was applied to
seven reaches of the estuary each of which included its repre-
sentative geometry (area, depth) and sequential reaction coef-
ficient (Anon, 1969). This type of model in contrast to the
finite difference approximation models results in continuous
solutions. Thus, although the first segment in the Delaware
Estuary nitrogen model under discussion here is thirty miles
long, a continuous solution (for constant spatial parameters)
is obtained throughout this length. The cross-sectional area
changes in the estuary were therefore approximated by the seven
segments. Major point discharges were grouped according to
this segment breakdown. This is a simplification introduced

to avoid numerous segment junctions at each discharge location.

The verification procedure consisted of comparing calculated
profiles from the four system model to a set of observed data
representative of steady-state summer conditions. The reaction
rates obtained for each of the nitrogen components were then
used in the examination and verification of other profiles
under different flow and temperature conditions. A consistent
set of first-order reaction coefficients was therefore obtained
which provides a reasonable representation of the observed
phenomena.

Figure8 shows the results of the first verification analysis

of data collected by the Delaware Water Pollution Commission
during July - August, 1964. The reaction coefficients shown

50


-
io}
T

vl
a
T

JUNE 30

JULY 15
a

w
is)
T

}oompuTed

= - RP w
co ©
T T T T

ext

TOTAL PHOSPHOROUS - PO, mg/I
2
T

°

0 5 10 5 20 25 30 35 40
MILES BELOW CHAIN BRIDGE ’
#: JUNE 30,1969
+= JULY 8
X= JULY 15
25 JUNE 30
+ Ay YuLY Ig {COMPUTED
20) x

a

NITRATE-N- mg/I
o .
on fo}

Fig. 38 Spatial Profile Comnarison of Observed 1969 Data
and Computed Values for Total Phosphorous (top)
and Nitrate Nitrogen (bottom)

167


a]
H

a o<t<f (60)
co} f<t<l

I, (t)
I, (t)

a

where f is the daylight fraction of the day (i.e. the photo
period) and I, is the average incident solar radiation in-
tensity during the photo period.

Let TS be the reduction in growth rate attributed to nonoptimum
light conditions in volume V5 averaged over depth and time.
Then v5 is given by

H. [fe -k_.z -k
1 J 1 Te % -tre ©
a a
re = exp + 1} dt dz (61)
j Hy ° T ° Ty i t.
where T = 1 day, the time-averaging interval, Hy, = Hy = the

depth of segment Ve and Kes is the extinction coefficient

in V5: The result is

-O -O..
= 1 13 _, 03
mr 5 Ray e e (62
where
ae Lhe Sess (63)
1j qT,
ae
oj I

The integral given by Equation (61) is a form of an integral
used by Steeman Nielson (1952, Talling (1957), and Ryther (1952)
and Yentsch (1957), as described by Vollenweider (1958), and,
in particular, Steele (1965), for the purpose of relating an
instantaneous rate (e.g., growth, photosynthesis, etc ) to

an average day rate and an average depth rate.

92


T8T

3OTd ANOQUOD UOTIETNUTS 696T

‘bia

Ly

ME

nO
ao
ioe)

Tl

335 ._)

v
201.

134.

87

0.

“+

b

~

Le

TOTAL PHOSPHATE PHOS.

|

SIMULATION NO. 3

1

!

U
n

AY

|
LO.

T PO4-P

| |
4. 19.
SEGMENT NUMBER

(MG/L)

23.

B

11 CONTOUR LEVELS

xZecr TA 7M. 09 OD

.50000E-02
»55000E-01
»10500£+00
»15500£+00
.20500€+00
»255006+00
-30500£+60
»35500E+00
»405006+00
+4S5500£+00
»50500£+00


CHLOROPHYLL -pg/I

TOTAL KjeldahIN—mg/

Ss
fs)

o
is)

a
ie}

+
co}

nN
ce}

fe}

°

4.0

10 2 20 25 30 35 40
MILES BELOW CHAIN BRIDGE

© = JUNE 30,1969
+2 JULY 8
Ke JULY 15,

COMPUTED

° 1 1 1 1 n 1 1
ie) ile} 5 20 25 30 35 40.
MILES BELOW CHAIN BRIDGE
~ ones . — — a
Fig. 37 Svatial Profile Comparison of Observed 1969 Data

and Computed Values for Chloronhvll a (ton) and
Total Kjeldahl Nitrogen (bottom)

166


TL

Dissolved Oxygen Deficit (mg/!)

Due to Nitrification

ny

in

——— without feedback
with algal feedback

Miles Below Chain Bridge
Figure 16
Computed Dissolved Oxygen Deficit Due to Nitrification

Potomac Estuary, July - August, 1968


Kg>>C, large ds

Logistic

Kg-#0, small dg

Exponential

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.4
0.2

Kmt

Logistic and. Exponential Growth of Microorganisms and Accompanying Nutrient Utilization

5

Fig.


€b ‘bts

LLT
TIME

3OTd zNOZUOD UOTIeTNUTS 696T

SIMULATION NO. 3
PHOSPHOROUS REDUCTION FACTOR - XEMP2

EC

| t ‘ili
335. (| Yo
10 CONTOUR LEVELS
268 ._ LL A . 10000 +00
,20000E+00
as ,30D00E +00
.4OUDDE+00
201._ |_ .50000E+00
.60000E+00
.70000E+00
.@0000E+00
134. _ .90000£+00
. LODDDE+01
o {|
oO) J)
5.

|
| | |
lt. 10. 4. 19. 23.
SEGMENT NUMBER

x

|

cCetmoamocom



many of the features of the complete nitrogen cvcle if one
accepts the assumptions of linearity and first order kinetics.
Typical examples of feedback svstems include denitrification
and algal utilization. The denitrification phenomenon may
involve two feedback loops; NO, to NO, (with attendant evo-
lution of Ny gas) and NO, to NH. Both of these feedback
reactions occur under conditions of "low" dissolved oxygen.
Actually, a more complete model of the system would include

a non-linear interaction between DO and the feedback reaction
rates.

The utilization of ammonia and nitrate by phytoplankton also
introduces feedback loops. Organic nitrogen is formed as a
part of the complex living matter in algae cells which upon
death release the organic nitrogen in dissolved form thereby
completing the cycle. This is an obvious over-simplification
of the actual mechanism which is dynamic and includes non-
linear growth limiting terms. However, the problem addressed
here is to introduce the feedback aspect into the steady state
sequential models discussed in the previous section.

The general feedback model makes use of a finite difference
approximation to Eqs. (31) - (34) and incorporates the first
order feedback reaction coefficients. The differential equa-
tions which incorporate feedback are:

a?ny aN.
O=E 7? ~ Ugg 7 Ky, + Kg No +--+ Kgy Ny + W, (x)
a?N, any
O= Baye 0 ae * BraNi 7 Ka2No + Kg2g + gaa * Wo (x) (41)
2
a?n, an

ax? ~Uan Ky4Ny + Ky Ny +... + K34Ny + Wy (x)

44


where 6.5 and D_. have units day”+ and 2, is the concentration

of zooplankton garbon in the volume element V.. The magnitude
of the growth rate in comparison with the death rate determines
whether the net rate of zooplankton biomass production in v5
is positive, indicating net growth rate, or negative, indicat-

ing a net death rate.

As in the case of the phytoplankton population, the growth and
death rates, and in fact the whole life cycle of individual
zooplankters, are complicated affairs with many individual pecu-
liarities. The surveys by Hutchinson (1967) and Raymont (1963)
give detailed accounts of their complex biology. It is, how-
ever, beyond the scope of this report to summarize all the
differences and species-dependent attributes of the many zoo-
plankton species. The point of view adopted is macroscopic,
with the population characterized in units of biomass. The
resulting growth and death rates can be thought of as averages
over the many species present. These simplifications are made
in the interest of providing a model which is simple enough to
be manageable and yet representative of the over-all behavior
of the populations.

The grazing mechanism of the zooplankton provides the basis for
the growth rate of the herbivorous zooplankton, S25" For a
filtering rate Cyr the quantity of phytoplankton biomass in-
gested is CyP 5" where PS is the phytoplankton biomass concen-
tration in Vs. To convert this rate to a zooplankton growth
rate, a parameter which relates the phytoplankton biomass in-
gested to zooplankton biomass produced, a utilization effici-
ency, azp! is required. However, this utilization efficiency
or yield coefficient is not a constant. At high phytoplankton
concentrations, the zooplankton do not metabolize all the
phytoplankton that they graze, but rather they excrete a portion

of the phytoplankton in undigested or semidigested form (Riley,

113


Application of First-Order. Models
to the Delaware Estuary

This estuary extends for about 86 miles from Trenton, N.J. to
Delaware Bay. The river flows past the metropolitan Phila-
delphia area and enters Delaware Bay about 50 miles from the
Atlantic Ocean. The estuary receives large amounts of carbo-
naceous and nitrogenous wastes from municipalities and indus-
tries and is characteristic of a system where bacterial
nitrification leading to oxygen depletion appears to be pre-
dominant (Anon., 1969; O'Connor et al, 1969). The municival
and industrial nitrogenous components are indicated in

Table 2.

Table 2

Estimated Municipal and Industrial Nitrogen
Discharges to Delaware Estuary *

(pounds/day)
Organic Ammonia Nitrate
Nitrogen Nitrogen Nitrogen
Municipal 28,500 48,500 2,000
Industrial - 32,500 30,500
28,500 81,000 32,500
Delaware River
@ Trenton (3,000 cfs) 9,000 1,000 16,000
Total 37,500 82,000 48,500

*Anon (1969)

A total direct load of 109,500 lbs of oxidizable nitrogen
per day is discharged from the municipal and industrial
waste sources.

In order to incorporate the effect of local drainage along

the length of the estuary, a runoff load equivalent to about

49


SECTION IX
PUBLICATIONS

"The Management of Time Variable Stream and Estuarine
Systems" Robert V. Thomann, Donald J. O'Connor and
Dominic M. Di Toro, Chemical Engineering Progress

Symposium Series, Vol. 64, Nov. 1968.

"Photosynthesis and Oxygen Balance in Streams" Donald
J. O'Connor and Dominic M. Di Toro, Journal of the
Sanitary Engineering Division, ASCE, Vol. 96, No. SA2,
Proc. Paper 7240, April 1970, pp 547-571.

"Modeling of the Nitrogen and Algae Cycles in Estuaries"
Robert V. Thomann, Donald J. O'Connor and Dominic M.

Di Toro, Proc. of Fifth International Water Pollution
Conf., San Francisco, Cal., 1970.

"A Water Quality Model of Chlorides in Great Lakes",
Donald J. O'Connor and John A. Mueller, Journal of the
Sanitary Engineering Division, ASCE, Nol. 96, No. SA4,
Proc. Paper 7470, Aug. 1970, pp. 955-975.

"Recurrence Relations for First Order Sequential Reactions
in Natural Waters" Dominic M. Di Toro, Water Resources
Research, Vol. 8, No. 1, Feb. 1972.

"A Linear Ecologic Model - Nitrification in Estuaries”
Donald J. O'Connor, Robert V. Thomann, Dominic M.

Di Toro, Systems Approach to Water Management, American
Elsevier Co., New York, March 1972.

"Effect of Longitudinal Dispersion on Dynamic Water
Quality Response of Streams and Rivers" Robert V. Thomann,
Water Resources Research, April 1973.

201


are available for integrating a set of ordinary first order
differential equations. In particular, the method of Huen,
described by Stiefel (1966), is effective and stable. It

is self-starting and consists of a predictor and a corrector
step. Let y = f(t,y) be the vector differential equation and
let h be the step size. The predictor is that of Euler: with
Yo the initial condition vector at tor the predictor value of
y at ty thet

1 75

q

y,* Yo + hE(t,, yQ) (101)

the corrector value is simply

h
YF YtFT [E(t or¥g) + £(t)/¥)*)] (102)
That is, the corrector uses the predictor value at ty to esti-
mate the slope at ty which is averaged with the slope at ty
to provide the slope of the straight line approximation. A
variation of this method is discussed at some length by

Hamming (1962).
Another simple two-step method is that of Runge, described by
Levy (1950). The Euler predictor is used with a half-step

integration.

ye = ygt b £(ty, 5) (103)

This value of y is used to estimate the slope at the midpoint
of the interval, which is then used as the slope of the straight
line approximation

h
Yy = Yot he (ty +5 y*) (104)

134


Equations (41) indicate all possible feedforward reactions

K..
1)
separate inclusion of Ki allows for possible loss of material

(j>i) and all possible feedback reactions (i>}). The

from the system due for example to bottom deposition. In

general, to prevent creation of nitrogen it is true that

Ki 2 } Rij

(i 4 3)

If a finite difference approximation is made to these equa-
tions (or equivalently a mass balance is constructed around
a finite section) a series of n algebraic equations results
where n is the number of finite sections (Thomann, 1972).
It should be noted that this approach is not restricted to

one-dimensional systems.

A finite difference approximation to the first equation of
Eq. 41 is given for spatial segment k as:

= \ fo -
0 = PEs COGN + BgNz 5) + Ely My 57M, I}

~ MRT NF Vaan Na,k Fe Kar Mak

+ ik (k = 1,2...n) (42)

where corey is the net flow from section k to j (positive out-
ward); vy is the volume of the xth segment; re is a finite
difference weight = max (1/2, 1-E'/Q) chosen for solution

stability; Bas =l-a,.; E

3! is a bulk tidal dispersion co-

kj
efficient given by

. EARS
kj Ly + L,

45


largely subjective. Extensive programs of nutrient removal
have been called for, with little or no quantitative pre-
diction of the effects of such treatment programs. A
quantitative methodology is required to estimate the effect
of proposed treatment programs that are planned to restore
water quality or to predict the effects of expected future
nutrient discharges. This methodology should include a
model of the phytoplankton population which approximates

the behavior of the phytoplankton in the water body of inter-
est and, therefore, can be used to test the effects of the
various control procedures available. In this way, rational
planning and water quality management can be instituted with
at least some degree of confidence that the planned results
actually will be achieved.

This chapter presents a phytoplankton population model in
natural waters, constructed on the basis of the principle
of conservation of mass. This is an elementary physical law
which is satisfied by macroscopic natural systems. The use
of this principle is dictated primarily by the lack of any
more specific physical laws which can be applied to these
biological systems. An alternate conservation law, that of
conservation of energy, can also be used. However, the de-
tails of how mass is transferred from species to species
are better understood than the corresponding energy trans-
formations, The mass interactims are related, among other
factors, to the kinetics of the populations and it is this
that the bulk of this chapter is devoted to exploring.

Review of Previous Models

The initial attempts to model the dynamics of a phytoplankton
population were based on a version of the law of conservation

74


and death rates as functions of environmental parameters and
dependent variables is an important element in a successful

phytoplankton population model.

Phytoplankton Growth Rate

The growth rate of a population of phytoplankton in a natural
environment is a complicated function of the species of phyto-
plankton present and their differing reactions to solar radi-
ation, temperature, and the balance between nutrient availability
and phytoplankton requirements. The complex and often con-
flicting data pertinent to this problem have been reviewed
recently by Hutchinson (1967), Strickland (1965), Lund (1965)
and Raymont (1963). The available information is not suffici-
ently detailed to specify the growth kinetics for individual
phytoplankton species in natural environments. Hence, in order
to accomplish the task of constructing a growth rate function,
a simplified approach is followed. The problem of different
species and their associated nutrient and environmental re-
quirements is not addressed. Instead, the population is
characterized as a whole by a measurement of the biomass of
phytoplankton present. Typical quantities used are the chloro-
phyll concentration of the population, the number of organisms
per unit volume, or the dry weight of the phytoplankton per
unit volume (Vollenweider, 1969). With a choice of biomass
units established, the growth rate expresses the rate of pro-
duction of biomass as a function of the important environmental
variables. The environmental variables to be considered below
are light, temperature, and the various nutrients which are
necessary for phytoplankton growth.

Consider a population of phytoplankton, either a natural as-

sociation or a single species culture, and assume that the
optimum or saturating light intensity for maximum growth rate

84


6E

SYSTEM |!

ORGANIC
NIT SOURCE

—————-+]

IOISTRIBUTION

DISTRIBUTION
AND DECAY OF JORG N. ANO DECAY
ORGANIC. NIT OF NH3-N

NO2~ N NO3- N
source 3 ROE 4
ISTRIBUTION DISTRIBUTION
AND DECAY [NO2- K3q JANO DECAY [SO32N
OF NO2-N OF NO3- N
3A
0.0.
REAERATION a 20 OER
d NH3—NOZ

FIGURE 6

BLOCK DIAGRAM OF NITRIFICATION & DISSOLVED OXYGEN
UNUZATION



iand j the matrix element aay is nonzero, then the volume
segments vy and V5 interact, and mass is transported between
the two segments. Letting P, Z, and N be the vectors of

elements Por ase and N, and letting Sor Sar Ss, be the vectors
of elements S_., S_.

pj’ ~2j’
tions for the three systems including the mass transport and

and Sn" the conservation of mass equa-

kinetic interactions are

vP = AP + vs, (83)
° :

vVZ = AZ + VS, (84)
°

vN = AN + VS) (85)

where V is the diagonal matrix of the volumes of the segments.
These are the equations which form the basis for the phyto-
plankton population model. The detailed formulation and
evaluation of the mass transport matrix has been discussed
elsewhere (Thomann, 1963; O'Connor & Thomann, 1966; O'Connor
et al, 1971).

The form of Equations (83)-(85) makes explicit the linear and
nonlinear portions of the equations. In the equation for P,
the phytoplankton biomass, the concentration P,, in the volume
element Vye is linearly coupled to the other Pits through the
matrix multiplication by A. However, there is no nonlinear
interaction between Py and any other Pye k # j. The reason

is that the transport processes are described by linear equa-
tions. It is usually the case, however, that the A matrix is
a function of time, since at least the advective terms usually
vary in time. The nonlinear terms in the vector Sy involve Py
itself and the corresponding Za and N,- Hence, the P equation
is coupled to the Z and N equations through this term. Note,
however, that P. is not coupled to the Zee k # j, in any other
segment, so that the coupling takes place only within each
volume segment.

125


Table 9

Grazing Rates of Zooplankton

Grazing Rate

Liter/
Reported mg Dry
Reference Organism Grazing Rate Wt.-Day

ROTIFER

Hutchinson Brachionus calyciflorus 0.05-0.12% 0.6-1.5
1967

COPEPOD
Riley 1949b Calanus sp. 67-208 0.67-2.0
Adams, Calanus finmarchicus 27% 0.05
Steele 1966
Mullin, Rhincalamus nasutus 98-670° 0.3-2.2
Brooke 1967
Anraku Centropages hamatus 0.67-1.6
Omori,1963
CLADOCERA
Wright,1958 Daphnia sp. 1.06
Burns ,1969 Daphnia sp 0.2-1.6
Ryther,1954 Daphnia magna 81° 0.74
McMahon, Daphnia magna 57-82% 0.2-0.3
1956
NATURAL ASSOC.
Riley,1949b Georges Bank 80-110? 0.8-1.10

4y1/animal -day
Pu /mg wet wt-day

approximation. However, since this mathematical expression
does represent a physiological mechanism that has been investi-
gated and for which reported values of cy are available, this
approximation is a realistic first step. Also, it is difficult
to see, aside from refinements as to temperature and phyto

109


Table 10

Endogenous Respiration Rate of Zooplankton

Respiration
Rate,
Plotting Temp. Ml 0./Mg Dry

Reference Organism Symbol °c ft-Day
Bishop Cladocerans xX 18 14.2
1968 4 2.7
" Copepods + 18 12.2
4 3.8
" Copepods | 18 8.2
16 6.5
12 5.2
8 4.1
4 3.4
Riley Calanus A 20 4.2
1963 finmarchicus 15 2.3
10 1.4
4 1.3
Comita Diaptomus 4 25 12.1
1968 leptopus 20 7.4
15 5.3
10 2.8
5 2.5
" D. clavipes e 25 12.5
20 8.5
15 5.1
10 2.4
5 1.8

" D. siciloides Cl 25 21
20 13.5
15 7.8
10 5.5
5 4.8
" Diaptomus sp. Oo 25 4.3
20 3.0
15 2.1
10 1.7
5 1.1

118


sources from point loads can be estimated more directly. For
example, the nutrient load from biologically treated municipal
wastewater is in the order of 10 g/capita-day total nitrogen

and 2 g/capita-day total phosphorus. The ratio of per capita
phosphorus to physiologically-required phosphorus is approxi-
mately 2 to 3, the excess being primarily the result of de-
tergent use. Industrial loads can also be important, especially
effluents from food processing industries. If the required
loading rates are available, their loads should be included

in the nutrient mass balance equations. In particular, if the
investigation of the phytoplankton population is directed at

the probable effects of increasing or decreasing the nutrient
load, these loads must be explicitly identified and their
magnitude assessed.

Let Way be the rate of addition of the nutrient to the 3th
volume element. This source is then included as a component
in the nutrient source term in the mass balance equation.

An important additional source of inorganic nutrients which

may influence the availability of nutrients is the interaction
of the overlying water either with the underlying mineral

strata if exposed or with whatever sediment is present. These
interactions can complicate the source term but they should

be included if they add significantly to the available nutrient.

The source term which results from the inclusion of the phyto-
plankton utilization sink, the zooplankton excretion and the

mortality sources, and the man-made additions is

Ws : a, Kk

= 31 - - _220%np_\

Snj T. anp®p3P5 + Anp°g”3?5\ x FP. }
j mp * 3)

(79)

+ anpkg TP; + ay 2374,

123


Other variables, constructed from these primary variables
are also tracked through the estuary. These secondary vari-
ables include total nitrogen and total phosphorous as the
most important.

Figure 30 shows the general interaction scheme of the nine
primary system variables. The major interactions and phe-
nomena are a) predator-prey relationships between phyto-
plankton and zooplankton systems 1 and 2 (which is apparently
not an important phenomenon at present in the upper Potomac);
b) nitrification of oxidizable nitrogen, given by systems #3,
4, and 5; c) conversion of organic phosphorous to inorganic
phosphorous and da) dissolved oxygen depletions due to carbo-
naceous waste loads and effects of nitrification. Nutrient
interactions are shown with phytoplankton chlorophyll. The
basic kinetic equations are discussed below for each system.

The general matrix equation for the phytoplankton chlorophyll
system (Di Toro et al, 1971) under a finite difference ap-
proximation is

d(py) =
wv) SSP = (Alt) + Iv1(s,) (11)

where (P) is an n x 1 phytoplankton chlorophyll "a" vector,
[A] is an n x n matrix of transport and disversion effects,
{v] is an n x n diagonal matrix of segment volumes and (Sy)
is ann x 1 vector of source terms. Specifically for
segment j

S|. = Go. = +) Pa
PJ ( Pp) Pp3) 3

where S05 is the growth rate and Das is the death rate.
Turning first to the growth rate, the light and temperature
effects are as given in Eq. (67). The interaction with ni-
trogen and phosphorous is given by a product of Michaelis

147


assumed ammonia preference by phytoplankton and is qiven by
a = NL/(N, + Ky) (119)

The first term in Eq. (118) therefore represents the pro-
duction of ammonia due to hydrolysis of organic nitrogen.
The second term represents the utilization of ammonia by
phytoplankton and the oxidation of ammonia by nitrifying
bacteria. Note that at concentrations of ammonia greater
than about 0.5 mg/l Eq. (119) indicates a preference for
ammonia of about 95% ata Kan = 0.025 mg/l.

The primary source of nitrate nitrogen (System #5) is the
oxidation of ammonia by nitrifying bacteria. The primary
sink of nitrate is the utilization of nitrate by phytoplankton.
The governing equation used is

a(n;

vl ae = TAliN,) + Iv] (Sy) + (y,) (120)

3
where N3 is the nitrate nitrogen concentration. The nitrate
source-sink is

Ss = K

N 45 No 7 aS,P (1-0) (121)

The first term in Eq. (121) represents the production of ni-
trate by nitrification while the second term represents the
utilization by phytoplankton.

The organic phosphorous system is assumed to be generated
from the death of phytoplankton and zooplankton plus any ad-
ditions of organic phosphorous due to the discharge of wastes.
The organic phosphorous of this system therefore represents
the"hon-living" phosphorous. The sinks for System #6 are the

151


EPA~660/3-73-009
August 1973

DYNAMIC WATER QUALITY

FORECASTING AND MANAGEMENT

By

Donald J. O'Connor, Robert V. Thomann, and Dominic M. Di Toro

Project No. R800369
Program Element 1BA023

Project Officer

Dr, Walter M, Sanders LIT
Southeast Environmental Research Laboratory
National Environmental Research Center-Corvallis
U. S. Environmental Protection Agency
Athens, Georgia 30601

Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U. S, ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D. C, 20460

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402- Price $2.05


PHYTOPLANKTON GROWTH RATES
DAY"!

ZOOPLANKTON GROWTH
RATES - DAY!

x
-0.2 1 1 —_ =
ie) 60 120 180 240 300 360
TIME — DAYS
1966
0.2
Sz
OF 63-05
we i
0 ;
|p
-0.1 i 1 L 1 1 1
° 60 120 180 240 300 360

TIME - DAYS

Fig. 28 Theoretical Growth Rates for Phytoplankton
and Zooplankton Poprulations

141



where

22.9(T) - 336.4

Log, Ys (T) = T +273

(72)

and T is temperature in degrees centigrade. To compare this
expression with that proposed in the previous section, con~
sider first the nutrient saturated growth rate as a function
of solar radiation intensity and temperature. The equations
are compared in Figure 22a as a function of total daily solar
radiation for three temperatures. The dotted line is Equa-
tion (71), and the solid line is the product of Equations (55)
and (56). The rate expressions are comparable, although two
differences are apparent. In Riley's expression the effect of
temperature is less pronounced in the 15° to 25°C range, and
the effect of higher daily average solar radiation intensities
is opposite (i.e., tends to increase the rate) to that of
Equation (56) based on Steele's expression. The growth rate
equations averaged over depth are compared in Figure 22. The
depth average rate resulting from Riley's expression is

S, = Se in [1 + I,/ri(T)] (73)
which is compared with Equation (67). The differences are now
more pronounced. In particular, the higher growth rates at
lower light intensities given by Equation (67) are reflected
in the increased depth average growth rate. This is not unex-
pected since the majority of the population is exposed to
lower light levels at the lower depths. Also, the dependence
on temperature is quite di*ferent, being linear in the case
of Equation (67) but practicaliy suppressed in Eguation (73).

An interesting feature of Riley's Equation (71) is the multi-

plication of the Michaelis constant by an expression which

102


zo

DO. DEFICIT ( mg/l )

FAVORABLE WNITRIFICATION CONDITIONS (NUEITED MITRIFICATION
O% NIT REMOVAL Ml. 100-80

2k
IE

C ed

te U7 Ne

poe FAVORABLE NITRIFICATION CONDITIONS

50% NIT. REMOVAL =
Oo Ottis 1 i L LL 1 at
130 120 Te) 100 30 80 70 60 50

DISTANCE IN MILES

Figure 13
DO Deficits Under Different Nitrification Conditions (after Anon.,1969)



loops, the solution for the four variables in all spatial
segments is formally given by

z,-1

(N) = (AI (i) (48)

The DO deficit due to these reactions is now readily comvuted.
If (Nj) represents ammonia and (N3) nitrite then a similar
differencing procedure yields for the DO deficit

(B] (D)y 3.43 [VK53] (Ny) + 1.14 (VK, 4] (N5) (49)

where [B] is an n X n matrix identical to the form of the
{a;] matrices except that the reaeration rate, kK, appears

on the main diagonal instead of Kia and (D) y represents the
spatial distribution vector of DO deficit due to the nitrifi-
cation effect. Note that once Eas (45) have been programmed
Eg. (49) is a svecial case of that set of equations and does
not require a separate computer program.

A general multi-dimensional first order kinetic steady state
water quality problem with any number of variables and system
configuration can thus be structured by solving an MN system
of equations (Eas. (45)) where M is the number of variabled
forms (for nitrogen, M = 4).

For one-dimensional estuaries, the form of the matrices [aj]

is tridiagonal which simplifies the computations. If no feed-
back loops are incorporated, Eqs. (45) reduce to the sequential
reaction models discussed in the previous section and [A] be-
comes upper triangular. In anv case, multidimensional systems
with interacting (first order) reactions can be readily and
quickly analyzed by simply solving a set of linear algebraic
equations. This of course by-passes the entire issue of how
one determines the Ki5 coefficients. This is discussed in the
succeeding sections on applications.

48


discrepency may be due to the simple preference structure
used in the model. A more detailed analysis of phytoplankton
preference for differing forms of nitrogen appears to be war-
ranted to provide better agreement.

In summary, the spatial agreement between observed and computed
data for 1969 conditions is good. The general shapes of the
spatial profiles are obtained and approximate quantitative
agreement is obtained. Several areas remain to be explored,
notably the model structure of the different nitrogen forms

where anomalous results were occasionally obtained.

Figures 39 and 40 show comparisons between the temporal vari-
ation in chlorophyll a at four stations. As shown in Figure
39, the observed data are scattered but with a general peak
in July 1969. The order and timing of this peak for both
Mile 12.1 and Mile 18.3 is properly duplicated by the model.
The rapid drop in chlorophyll concentration at the end of
July is attributed to an increase in fresh water inflow

(see Figure 31) at that time. The decrease is also success-~
fully duplicated by the model. The model approximates the
subsequent fall bloom of phytoplankton at Mile 18.3 but at
Mile 12.1 the model calculations are somewhat higher than
the observed data.

Figure 40 shows the comparison at two stations further down-
stream. At both stations a winter bloom of phytoplankton
occurred which was not modeled in this work. At Hallowing
Point, Mile 26.9, the spring growth pattern and subsequent
decrease is adequately modeled. In the fall of 1969 at
Hallowing Point, however, data indicated a significant in-
crease in phytoplankton (maximum levels of 445 g/l chloro-
phyll). This fall bloom was not duplicated by the vhyto-
plankton model as constructed. It is not clear from the

169


Reference
Thomas Chaetoceros gracilis
1968 (marine diatom)
L. Tahoe Scenedesmus gracile
A.C.1969
Riley Natural association
1965 Microcystis aeruginosa
(blue-green)
Gerloff Phaeodactylum tricornutum
1957
Eppley et Oceanic species
al 1969 Oceanic species
" Neritic diatoms
Neritic diatoms
" Neritic or littoral
Flagellates
MacIsaac Natural association
et al 1969 Oligotropic
" Natural association
Eutrophic
*Estimated.

Michaelis Constants
for Nitrogen and Phosphorus

Table 7

PO, 25
total N 150
total P 10
PO 6?
pot 10?

4
PO, 10
NO, 1.4-7.0
NHS 1.4-5.6
NO 6.3-28
NH 7.0-120
No, 8.4-130
NHS 7.0-77
No, 2.8
NHS 1.4-8.4
NO, 14
NES 18

97


and of ammonia ranges from 10 to 60 mg/l with an average of
about 25 mg/l.

Industrial waste waters may contain appreciable quantities
of nitrogen in various forms. Ammonia is one of the most
commonly used industrial chemicals. Nitrogen in its various
forms is used in manufacture of dye, glass, explosives, many
chemicals and synthetic products, and may therefore be found
in significant quantities in these waste waters. They are
not usually of importance in paper, tannery, textile, metal
and in some vegetable and fruit-processing wastes. On the
other hand, they are present in meat-packing, brewery, dairy
and coke plant wastes.

The nitrogen in plant and animal proteins is measured by the
Kjeldahl method, which converts all of the organic nitrogen

to ammonia, The suspended fraction includes the nitrogen
present in the living and dead plankton and in the animal
excreta, while the dissolved portion contains the nitrogen
from excreted materials, usually associated with the degrada-
tion of cells. The inorganic forms of nitrogen, ammonia, ni-
trite and nitrate may be measured directly. Thus the nitrogen
present in waste streams or natural waters is usually reported
in terms of the specific inorganic forms or as organic nitrogen,
which covers a range of compounds in various stages of degra-
dation.

The total oxidizable nitrogen may be measured indirectly by

the standard biochemical oxygen demand test. The first stage
reflects primarily aerobic oxidation of the organic material,
in which an end product is ammonia. Simultaneously the organic
nitrogen is hydrolyzed to release ammonia. The ammonia thus
formed from these two sources, in combination with the ammonia
present in waste waters undergoes oxidation through nitrite to

22


where they are usually found in high numbers. In rivers re-
ceiving waste waters, nitrifying bacteria are probably present
in varying degrees depending on the nature and the treatment
of waste waters. Their number in sewage is low (about 100/ml1)
but increases through biological treatment to an order of 1000/
ml. Natural habitats are found on biological aggregates or on sur-

faces where the environment appears appropriate for optimum
growth and acceleration metabolism. Surfaces such as these

are found in trickling filters and activated sludge treatment
plants as well as the rocky beds of shallow rivers.

Denitrification

Under conditions of low concentration of dissolved oxygen the
bacterial reduction of nitrate can occur. This reaction is
to be distinguished from the utilization of nitrate and sub-
sequent reduction by aquatic plants. A large variety of
facultative bacteria can reduce nitrate. E Coli are cited

as a common bacteria capable of reducing nitrate. (Hutchinson,
1957) The reduction by bacteria of nitrate is probably all

to NO, and then to nitrogen gas although complete reduction

2
to ammonia may also occur. The primary reactions seem to be:

0, + 12 wo, Bacteria + 12 nop + 6 co

C6H 2% + 6H,0 (6)

2 2
which represents the reduction of nitrate to nitrite and

C,H, ,0 + @ no, Pacteria > 4 y

6H 2% + 2CO, + 4CO, + 6 HAO (7)

2 2 3

for the reduction of nitrite to nitrogen gas. It should be
noted that these reactions serve in place of an oxygen source
for micro-organisms in the metabolic oxidation of organic
compounds in a water body, without drawing on the dissolved
oxygen resources of the stream.

18


where Ge is the growth rate (day 4), K' = 7.6, Ig = average
daily incident solar radiation (ly/min), T' - temperature in
°K, and Vp is the nutrient reduction factor for phosphate
concentration, Nor defined as

v. = 1.0 N, > 0.55 mg-at/m?
ve = (0.55) 2N N. < 0.55 mg-at/m* (69)
Pp Pp Pp .
where Gg is the growth rate (aay 4), K' = 7,6, I, = average
daily incident solar radiation (ly/min), T' = temperature

in °K, and Yp is the nutrient reduction factor for phosphate
concentration, Nor defined as

v
Pp

Yp

1.0 Ny > 0.55 mg-at/m>

-1 3
(0.55) *N, N, < 0.55 mg-at/m

"

In order to compare this expression with that in the previous
section, let the nutrient reduction factor be replaced by a
Michaelis-Menton expression

N

= Pp
v = 2 (70)
K_ +N
P mp * “p

where Kup is the Michaelis constant for phosphate. To be
comparable with Equation (67), Kap should equal approximately
0.20 mg-at/m? (6.2 mg p/m). Using Equation (70) for Yor the
growth rate expression becomes

ry, (T)

G = K'L
Pp oly (TY+I5 (71)
'

101


45r

40r

TOTAL PHOS. (PO4)-mg/I

35

3.0-

25-

20-

PHOSPHOROUS
CONSERVED:

OK phos.=0.05/day
@ K phos. = 0.1 /day

307
25-
2 ol O Knit, =0.05/day
2 : (Q) Knit =0.1/day t
NITROGEN CONSERVED
w 15h a |
< }
 1oF i
* {
O0.5- | I| 3
.
o ALLL!
° 5 10 IS 20 25 30 35 |
Fiq. 33 a) Total Phosnhorous - PO, (mq/1) Comparison

b) Nitrate Nitrogen Comparison, August 1968

158


(a)

30
2st
NITRATE ofl. NO ZOOPLANKTON
N
(mg/l) L
SEGMENT EXTENSIVE ZOOPLANKTON
#9 Lok GRAZING
ost
of 1 ree eee
0 30 60 90 120 150 180 210 240 270 300 330 360
DAYS
(b)
25
EXTENSIVE ZOOPLANKTON GRAZING
AMMONIA 2.0}
N
(mg/l) 1.5
SEGMENT
#6 ihe)

NO ZOOPLANKTON

n 4 n n n 4 1 n n mn m
Oo 3 60 90 120 150 180 210 240 270 300 230 360
DAYS

Fig 35 Effect of Zooplankton Grazing on Nitrate Nitrogen (tov)
in Segment #9 and Ammonia Nitrogen (bottom) in Segment #6,
1968 Flow Regime

161.


to verify independently the 1969 data under the flow and
temperature regime showed in Figure 31. As shown in Figure
31, fresh water inflow conditions were generally unsteady
throughout 1969. There was a gradual decrease in the flow
from April to mid-July at which point the flow increased
markedly.

Figures 37 and 38 show the comparison between the computed
output and some observed data for the period at the end of
the gradual decline in river inflow, from June 30, 1969 to
July 15, 1969. Three data surveys were conducted during
this period. It should be recalled that the computed output
for 1969 was generated directly using the parameters from
the 1968 runs; the parameters were not adjusted for the 1969
runs. Only flow temperature and incoming concentrations
into segment #1 based on observed 1969 data were used in

the 1969 computations. It should be noted, however, that
boundary conditions at segment #1 do influence results for
about the first 15 miles of the estuary. Incoming chloro-
phyll concentrations were not measured during 1969; the
first station for which data were available was at Key
Bridge, Mile 3.3. Boundary chlorophyll values were inputted
to approximately duplicate the order of magnitude of observed
data at that station. The values ranged from 4 wg/l in the
winter to a high of 55 ug/l in September 1969.

Figure 37 shows the comparison between chlorophyll values

as observed and the computed values for the beginning and

end of the survey period. The June 30 values varied mark-
edly especially in the vicinity of Mile 15 where a low value
of 30 ug/l was observed, an apparently abnormal value for
this location at that particular time of vear. The general
shape of the spatial profile as computed is good and approxi-
mately reproduces the spatial behavior of the observed data.

165


Case III - Large initial microbial mass, large Michaelis
constant

In this case, dy is considered large, i.e. there is a large

microbial mass relative to Co at t = 0. Under this condition,

M' is relatively constant since the additional microbes pro-

duced by metabolism of fo is small and M' = M'o- The appro-

priate dimensionless differential equation is then

d(c/c,) . “dy (c/c,) (21)
d (Ky) dite. co

whose solution is
ad, log c/o, + (c/e,-1) = Kydyt (23)

therefore, for large dey i.e small concentration of substrate
relative to the Michaelis constant, the substrate decays ex-
ponentially as

[en

¢ = ¢, exp (-Ky a t) (24)
s
Case IV - Large initial microbial mass, small Michaelis con-
stant

On the other hand, for small do. i.e K, << cy << Mtoe the

solution (Eq.20) is linear:

c = cy (1-K yd, ¢) (25)
until log (c/e,)as ~ (c/o, - 1) where the total solution

(Eq. 20) indicates an exponential "tail".

Intermediate solutions can be thought of as a “linear com-
bination" of two separate models: (a) Case I, an exponential

30


160

r ~ °
aes bin da
yaad Apa
140;— / i) yrN
Ul I 1 \ °
/ / HOLLOWING POINT
i2ol- 1 1 ! ‘> MILE 26.9
= . U oes
o T winter 7 v7
L 100;-— J BLOOM /
’ { I /
3 if
= ser ly >
x= \ | N
a N ~
io 60 \ 1 | .
S sy Toy
3 |
40}- f°
/ SEGMENT 12 °
(MAIN CHANNEL)
20 4
A Poti i | | yyy
0 30 60 90 120 150 180 210 240 270 300 330 360
TIME OF YEAR
Jon | Feb. | Mar. | Aprit | May |June | July ] Aug. | Sept. | Oct. | Nov. | Dec. |
160, TON
rary POSSUM POINT
8 4] MILE 38,0
140} t ! .
1° / .
1c 4 °
120 I 1
Ss / I
ao
.
* took / \ .
' WINTER
4 { 8LOOM \
a 1 ee
= 80 \° SEGMENT |5
z= [ ‘ ! °
a . 9
fe} NUS
«© - .
Oo 601-
J
x
G
40}—
20;--
4 4 | i i i ! i | | | |
19) Ea) 60 30 120 1SO 10 210 240° 270 300 330 360
TIME OF YEAR
van, Feb, | Mor. | Aprit | May |.June | July | Aug. | Sept. | Oct. | Nov. | Dec. |
Fig. 40 Temporal Comparison of Observed 1969 Data and

Computed Values for Stations at Miles 26.9 and 38.0°

171


plankton concentration dependence, what further improvements
could be made in the formulation so long as the phytoplankton
and zooplankton population are represented by a biomass
measurement which ignores the species present and their in-
dividual characteristics. For simplicity in this investiga-
tion, the grazing rate is assumed to be a constant. The
death rate of phytoplankton resulting from the grazing of
zooplankton is given by the expression Cy25" where 25 is the
concentration of herbivorous zooplankton biomass in the j
volume element.

For models of the phytoplankton populations in coastal oceanic
waters and in lakes, the sinking rate of phytoplankton cells

is an important contribution to the mortality of the popula-
tion. The cells have a net downward velocity, and they eventu-
ally sink out of the euphotic zone to the bottom of the water
body. This mechanism has been investigated and included in
phytoplankton population models (Riley, 1949; Chen, 1970).
However, for the application of these equations to a relatively
shallow vertically well mixed river or estuary, the degree of
vertical turbulence is sufficient to eliminate the effect of
sinking on the vertical distribution of phytoplankton.

Therefore, considering only the phytoplankton respiration and
the predation of zooplankton, the death rate of phytoplankton
is given by the equation

Do. = K,T+C 2. 74
Pj 2 g°j (74)

and for a zooplankton biomass concentration ase the mortality
rate can be calculated from this equation.

This completes the specification of the growth and death rates
of the phytoplankton population in terms of the physical vari-

lll


Further refinement of the model is therefore necessary before
any definitive statements can be made of the effects of nu-
trient reduction on phytoplankton chlorophyll in the Upper

Potomac Estuary.

190


1
120

!
240

9, SUNLVUS

l
120

120

DAYS

TiMe

1966

Sdo -AMO14

AWVO/X1 ~ NOLUMIGWY
uvOS Alivd NY3WN

a ee rs ad

1967

1966

Temperature, Flow, and Mean Daily Solar Radiation;

San Joaquin River, Mossdale,

26

Fig.

132


simulations discussed below are not to be considered as
quantitative projections. At best, the simulations in-
dicate general qualitative trends. Both simulations used

a 90% removal policy of present raw waste loads. For the
first simulation the 1969 flow regime was used; untreated
nitrogen, phosphorous and carbonaceous waste loads were re-
duced by 90%; and incoming boundary values into segment #1
were the same as those used in the 1969 verification analysis.
For the second simulation, median flow conditions were used
with reduced waste inputs as in the first simulation. In
addition several boundary conditions on chlorophyll were
explored in the median flow simulation. For both simula-
tions it was assumed that all organic and ammonia nitrogen
was converted to nitrate nitrogen at the treatment plants

and all phosphorous residual load is inorganic phosphorous.
The summary of the waste loads is given in Table 14.

Table 14
Direct Discharge Waste Loads

Waste Input Waste Input
for Verification for Simulation
lbs/day lbs/day
Total Nitrogen (N) 46,500 7,070
Total Phosphorous (P) 20,300 2,620
Carbonaceous BOD 151,200 10,000

As shown, the 90% removal of the untreated waste loads re-
sults in about a 15% residual discharge of the 1968-1969
discharged loads. Although the residual loads used in the
simulations are in excess of the allocations established for
the Potomac (Jaworski et al, 1971) the level is considered a
practical achievable level on a continued sustained basis.

173


oS

REACTION RATE K(I/DAY)

TTTTTTTT TT

weowe awe tL 1 1 1 1 1

Is 2 23) 30 35 40
WATER TEMPERATURE - °C

fe}
a
3

Figure 9
Estimated Temperature Dependence of Nitrification Reaction Rates (after Anom.1969)



next trophic level, the herbivorous zooplankton, is a complex
process for which only a first approximation can be given. A
basic mechanism by which zooplankters feed is by filtering the
surrounding water and clearing it of whatever phytoplankton

and detritus is present. Thus, the presence of zooplankton

prey on phytoplankton as a food source. The filtering or graz-
ing rate of some species of zooplankton have been measured and
are presented in Table 9. The grazing rate is sometimes reported
as a volume of water filtered per unit time per individual. In
order to be applicable to a natural zooplankton population con-
sisting of differing species, these grazing rates are converted
to a filtering rate per unit biomass of zooplankton and denoted
by Cy. A convenient biomass unit for zooplankton concentration
is their dry weight. As can be seen from Table 9, the resulting
values of cy vary considerably. This variation is not unex-
pected since the measurement of grazing rates of zooplankters

is a difficult procedure and subject to large variations in the
estimates.

Variations of the filtering rate with temperature change have
been reported (Anraku, 1963). Examples of this variation are
presented in Figure 24 for four species of genus Daphnia, a
small crustacean(Burns, 1969); two species of Acartia (Conover,
1956); and two species of Centropages (Anraku, 1963), both
copepods. The copepods show a marked grazing rate temperature-
dependence while the grazing rates of the Daphnia do not vary
as markedly. The filtering rate also varies as a function of
the size of the phytoplankton cell being ingested (Mullin,
1963), the concentration of phytoplankton (McMahon, 1965), and
the amount of particulate matter present (Burns, 1967). Se-
lective grazing of certain phytoplankton species has also been
reported (Burns, 1969). The complexity of this aspect of phyto-
plankton mortality is such that the use of one grazing coeffi-
cient to represent the process must be viewed as a first

108


The peak value of 3 mg/l DO deficit is significant from a
water quality management viewpoint and indicates the need
for nitrification of the principal waste discharges.

The nitrogen algal cycle in the Potomac estuary is obviously
more complex than given by this model. Non-linear kinetics,
algal growth dynamics and environmental influences of
temperature and light all affect the observed data. Asa
planning tool however the simplified model using first order
kinetics provides a rapid means for estimating order of
magnitude responses and points the direction for more complex

modeling efforts.

72


N
fe}
°
fe}
——

OXYGEN DEMAND
(M9/L)
o
Nitrogenous

—+-

Carbonaceous

0 2 4 #6 8 0 12 4 #6 BW 20. 22
INCUBATION TIME
(DAYS)

Fig. 3  Two-state BOD Curve - Mohawk River at St. Johnsville, N.Y.

24

BOD

BOD



These variations approximately follow the shape of Equation
(56) for low light intensities but differ for the region of
high light intensities, usually by not decreasing after some
optimum intensity is reached. In particular, Tamiya et al
(1964) have investigated the growth rate of Chlorella ellip-
soidea to various light and temperature regimes. The saturated
growth rates as a function of temperature are included in
Figure 18. The influence of varying light intensity fits the

function
F(t) = > (87)
T+ KT7a
where Rv is the saturated growth rate and @ = 0.45 day
kilolux -). However, since K' is a function of temperature,

the saturating light intensity for Equation (57) is also a
function of temperature. Similar data obtained by Sorokin
et al (1962) using a high-temperature strain of Chlorella
pyrenoidosa support the temperature-dependence of the satu-
rating light intensity for chlorella. Therefore, in using
Equation (56), a temperature-dependent light saturation in-
tensity may be warranted.

At this point in the analysis, the effect of the natural
environment on the light available to the phytoplankton must
be included. Equation (56) expresses the reduction in the
growth rate caused by nonoptimum light intensity. This ex-
pression can therefore be used to calculate the reduction

in growth rate to be expected at any intensity. However,
this is too detailed a description for conservation of mass
equations which deal with homogeneous volume elements, V.,

J
and the growth rate within these elements. What is required

is averages of the growth rate over the volume elements.

In order to calculate the light intensity which is present
in the volume, V5. the light penetration at the depth of

90


SECTION II

RECOMMENDATIONS

The modeling formulations which have been developed in this
study should be applied to other estuarine water bodies in an
attempt to further strengthen the verification and refine the
kinetic structures employed. In particular, the eutrophica-
tion analysis is an initial attempt to incorporate the varied
and complex interactions which characterize the growth and
decay of phytoplankton biomass in natural waters and the re-
lationships to nutrient concentrations. This analysis should
be extended to include effects of other microorganisms as well
as the effect of chemical and biological parameters not in-
cluded in the preliminary formulations.

In principle the models developed herein can be extended to
apply to other settings such as lakes and coastal waters.

The modifications and adjustments for such an attempt would

be a fruitful continuation. The need for further comparisons
between observed data and calculated concentrations cannot be
over emphasized. Only in this way can progress be made in the
understanding of the complex phenomena which control water
quality in natural bodies of water.

Finally, it is recommended that the models developed in this
work be applied in the planning and evaluation of water quality
enhancement programs. Although the models are in no way com-
pletely satisfactory from a scientific point of view, and may
in fact not incorporate certain effects which may prove to be
important, they have been shown to be realistic and capable

of reproducing the observed situation for the cases considered
in this report. The models, therefore, are worthy of consid-
eration in any attempt to rationally plan and execute water
juality enhancement programs.


SECTION V

A DYNAMIC MODEL OF
PHYTOPLANKTON POPULATIONS IN NATURAL WATERS

The quality of natural waters can be markedly influenced by
the growth and distribution of phytoplankton. Utilizing
radiant energy, these microscopic plants assimilate inor-
ganic chemicals and convert them to cell material which, in
turn, is comsumed by the various animal species in the next
tropic level. The phytoplankton, therefore, are the base
of the food chain in natural waters, and their existence is
essential to all aquatic life.

The quality of a body of water can be adversely affected if
the population of phytoplankton becomes so large as to inter-
fere with either water use or the higher forms of aquatic
life. In particular, high concentrations of algal biomass
cause large diurnal variations in dissolved oxygen which

can be fatal to fish life. Also, the growths can be nui-
sances in themselves, especially when they decay and either
settle to the bottom or accumulate in windrows on the shore-
line. Phytoplankton can cause taste and odor problems in
water supplies and, in addition, contribute to filter clog-
ging in the water treatment plant.

The development of large populations of phytoplankton and, in
some cases, larger aquatic plants can be accelerated by the
addition of nutrients which result from man's activities or
natural processes. The resulting fertilization provides more
than ample inorganic nutrients, with the resulting development
of excessive phytoplankton. This seguence of events is com-
monly referred to as eutrophication.

Generally, the management of water systems subjected to ac-
celerated eutrophication because of waste discharges has been

73


Phytoplankton System Interactions

The major obstacle to a rigorous quantitative theory of
phytoplankton population dynamics is the enormous complexity
of the biological and physical phenomena which influence the
population. It is necessary, therefore, to idealize and
simplify the conceptual model so that the result is a manage-
able set of dependent systems or variables and their inter-
relations. The model considered in the following sections is
formulated on the basis of three primary dependent systems:
the phytoplankton population, whose behavior is the object of
concern; the herbivorous zooplankton population, which are
the predators of the phytoplankton, utilizing the available
phytoplankton as a food supply; and the nutrient system, which
represents the nutrients, primarily inorganic substances, that
are required by the phytoplankton during growth. These three
systems are affected not only by their interactions, but also
by external environmental variables. The three principal
variables considered in this analysis are temperature, which
influences all biological and chemical reactions, dispersion
and advective flow, which are the primary mass transport
mechanisms in a natural body of water, and solar radiation,
the energy source for the photosynthetic growth of the phyto-
plankton.

In addition to these external variables, the effect of man's
activities on the system is felt predominately in the nutrient
system. Sources of the necessary nutrients may be the result
of, for example, inputs of wastewater from municipal and in-
dustrial discharges or agricultural runoff. The man-made
waste loads are in most cases the primary control variables
which are available to affect changes in the phytoplankton
and zooplankton systems. A schematic representation of these
systems and their interrelations is presented in Figure 17.

79


33

34

35

36

37

38

39

40

41
42
43
44
45
46
47

FIGURES

Page
Comparison of Range of Observed Data and Model
Output - August 1968. a) Chlorophyll a (g/1)
b) Total Kjheldahl Nitrogen (mg/1) 157
a) Total Phosphorous - PO, (mg/l) Comparison
b) Nitrate Nitrogen Comparison, August 1968 158

Effect of Zooplankton Grazing on Phytoplankton in
Segment #9 a) No Zooplankton Grazing b) Zooplankton
Grazing at 0.42 1/mg Carb-Day, 1968 Flow Regime 160

Effect of Zooplankton Grazing on Nitrate Nitrogen
in Segment #9 and Ammonia Nitrogen in Segment #6,
1968 Flow Regime 161

Sensitivity Run - No Tidal Bay Segments.
August 1968 Profile and 1968 Conditions 163

Spatial Profile Comparison of Observed 1969 Data
and Computed Values for Chlorophyll a and Total
Kjeldahl Nitrogen 166

Spatial Profile Comparison of Observed 1969 Data
and Computed Values for Total Phosphorous and
Nitrate Nitrogen 167

Temporal Comparison of Observed 1969 Data and 170
Computed Values for Stations at Miles 12.1 and 18.3

Temporal Comparison of Observed 1969 Data and Com-
puted Values for Stations at Miles 26.9 and 38.0 171

1969 Simulation of Chlorophyll, June 30 and July 15.172

1969 Simulation Contour Plot 173
1969 Simulation Contour Plot 174
1969 Simulation Contour Plot 175
1969 Simulation Contour Plot 176
1969 Simulation Contour Plot 177
1969 Simulation Contour Plot 178

viii


and this is why a mass death occurs in the nutrient-depleted
epilimnion after the vernal maximum." In terms of the respira-

tion rate, the resting stage corresponds to a lowering of the

Table 8
Endogenous Respiration Rates of Phytoplankton
(Riley, 1949)
Endogenous
Respiration
Temperature Rateise
Organism °c ase,
Nitzschia closterium 6 0.035
35 0.170
Nitzschia closterium 20 0.08
Coscinodiscus excentricus 16 0.075
16 0.121
Natural Association 2 0.03
18 0.12
2.0 0.024+0.012
17.9 0.110+0.007

respiration rate as the nutrient concentrations decrease. Thus,
a Michaelis-Menton expression for the respiration rate nutrient
dependence may also be required, and this dependence should be
investigated experimentally. This mechanism is quite signifi-
cant from a water quality point of view since the deaths of
algae after a bloom is of primary concern in protecting the
quality of natural bodies of water. The resulting mass of dead
algal cells becomes a sink of dissolved oxygen which can danger-
ously lower the available oxygen for fish and other aquatic

animals.

The interaction between the phytoplankton population and the

107


apply to the seasonal variation of the phytoplankton popula-
tion. Instead, he used two volume segments to represent the
upper and lower water levels and kept the time derivatives

in the equations. Thus, both temporal and spatial variations
were considered. In addition, the differential equations for
phytoplankton and zooplankton concentration were coupled so
that the interactions of the populations could be studied, as
well as the nutrient-phytoplankton dependence. The coeffici-
ents of the equations were not functions of time, however, so
that the effects of time-varying solar radiation intensity and
temperature were not included. The equations were numerically
integrated and the results compared with the observed distri-
bution. Steele (1964) applied similar equations to the verti-
cal distribution of chlorophyll in the Gulf of Mexico.

The models proposed by Riley et al and Steele are basically
similar. Each consider the primary dependent variables to be
the phytoplankton, zooplankton, and nutrient concentration.

A conservation of mass equation is written for each species,
and the spatial variation is incorporated by considering finite
volume elements which interact because of vertical eddy dif-
fusion and downward advective transport of the phytoplankton.
Their equations differ in some details (for example, the growth
coefficients that were used and the assumptions of steady
state) but the principle is the same. In addition, these
equations were applied by the authors to actual marine situ-
ations and their solutions compared with observed data. This
is a crucial part of any investigation discussion wherein the
assumptions that are made and the approximations that are used
are difficult to justify a priori.

The models of both Riley and Steele have been reviewed in

greater detail by Riley (1963) in a discussion of their appli-
cability and possible future development. The difficulties

77


the phytoplankton with the light limitation. At the beginning
and end of the year, growth rates are less than about .01/day
throughout the length of the estuary. During the peak grow-
ing season, growth rates reached maximum levels of .2 - .3/day.
The interaction of phytoplankton population and light penetra-
tion is evident in the vicinity of segments #7 - #10 during

the peak growth season. Figure 46 shows the net growth rate -
growth versus death and predation. As indicated, there is a
large region in space and time where the net growth rate is
positive but it should be recalled that the transport structure
will substantially modify the net growth rate due to "wash out"
effects.

Figure 47 displays the total phosphorous surface. As shown,
total phosphorous downstream of segment #10 is temporally con-
stant and uniformly decreases spatially. In the vicinity of
Washington D.C. total phosphorous concentrations range from

~3 ~ 0.5 mg/l PO,-P even with 90% removal at the plant.

4
Ammonia nitrogen and nitrate nitrogen are shown in Figures 48
and 49. The shape of these surfaces can be compared to the
phosphorous surfaces of Figures 44 and 47. The comparison
indicates the more complex interactions and structure of the
ammonia and nitrate nitrogen. Significant temporal and spatial
variations occur reflecting the recycling of the different ni-
trogen forms. Nitrogen in all regions is above limiting con-

centrations.

Flows for the median flow simulation ranged from 13,000 cfs
during the winter to a high of 20,500 cfs in the spring, de-
creasing to 4,200 cfs in the later summer and then increasing
again to 9,500 cfs in December. Incoming boundary concentra-
tions as noted previously have an important relative impact

on the estuary especially after waste loads are reduced by

184


(Day)

SATURATED GROWTH RATE

0 10 20 “30
SOLAR RADIATION (LANGLEYS/MIN)

b 3.0

E .

a

x

a

S

cc ~20

Or

ae

uO

io)

qo

ae)

Ww S 1.0

xo
=

x

BK

jeu

(

a o¥

0 10 20 30
SOLAR RADIATION (LANGLEYS/MIN)

Fig. 22 Comparison of Phytoplankton Growth Rates as a
Function of Incident Solar Radiation Intensity
and Temperature

103


dependence on temperature and also the magnitude of K' can
vary considerably in particular situations.

In the natural environment, the light intensity to which the
phytoplankton are exposed is not uniformly at the optimum
value but it varies as a function of depth because of the
natural turbidity present and as a function of time over the
day. Thus, the phytoplankton in the lower layers are exposed
to intensities below the optimum and those at the surface may
be exposed to intensities above the o¢timum so that their
growth rate would be inhibited. Figure 19b,c,d from Ryther
(1956) are plots of the photosynthesis rate normalized by the
photosynthesis rate at the optimum or saturating light in-
tensity vs the light intensity, I, incident on the popula-
,tions. Figure 19a is a plot of function

F(I) = <= exp [- xt 1 | (56)
s s

for I, = 2000 ft-candles, proposed by Steele (1965) to des-
cribe the light-dependence of the growth rate of phytoplankton.

The similarity between this function and data from Ryther is
sufficient to warrant the use of this expression to express
the influence of nonoptimum light intensity on the growth rate
of phytoplankton. Other workers have suggested different forms
for this relationship (Shelef et al, 1970; Vollenweider, 1965).

83


9LT

2OTd ANOJZUOD UOTIETNUTS 696T Zh

‘bt

TIME

SIMULATION NO. 3
PHYTOPLANKTON CHLOROPHYLL A
| 1 | | |

w

w

o
=)

201.
Be
134.
67.
0.
7 | | | |
L. 5. 10. iy. 19.

SEGMENT NUMBER

(MCG/L)
i

|
23.

13 CONTOUR LEVELS
OQ.

Zromeer To wmoonwn

»10000£+02
. 200006 +02
+300006+02
-4O000E+02
»5000U0E +02
«§000UE +02
« 70000E +02
»S8000U0E +02
»90000E+02.
»100D0E+03
»LL0006+03
«12000E+03


6LT

3OTd ANoQuOD uoTReTNWTS G96T

Sp *btd

335.

20l.

134.

6).

SAT.

SIMULATION NO. 3
GROWTH RATE W LIGHT LIMIT.

| I
4. 1g.
SEGMENT NUMBER

GIT

10 CONTOUR LEVELS

A

B
c
0
E
F
&
H
1
d

+84 765E-02
s4“U296E-O1
‘GOL15E-21
»116593£+00
»151756+00
1Q757E+00
122339L +00
»25921E+00
»29503E+00
.330G5E+00


i= ®
ALL

0)
0 12 110 100 90 80 70 60 50 MILES
REACT. COEF. | 010 |

(1/DAY )

i

930 120 10 100 90 80 70 60 50 MLES
REACT. COEF. | Oll Q00! | 005 | Qu

130 120 Il0 100 90 80 70 60 50 MLES
REACT OOF | a3 | 002 Q05| a3

3
2k
ZS
a)
SP lbqw ee ;
opis L 1 es 1
O 12 0 100 90 80 70 60 50 MILES
REACT. COEF| a0 j 005 {| 09 |OBL . 6,
(1/DAY ) <7 2/67
Q=5600 cfs
T=2/°C
Figure ll
Observed vs. Computed (solid line) - Nitrogen Profiles

July 1967 (after Anon. ,1969)
56



for each of the species do not constitute a unique set of rate
coefficients but represent a trial and error fit of the ob-
served data. The coefficients are consistent however with
other verification analyses. This is explored further below.

Several other points should be noted in this verification
analysis. The direct discharge of organic nitrogen waste
loads does not alone account for the total of about 1/mg/1
that was observed. It was therefore hypothesized that 0.75
mg/l of organic :nitrogen was due to the presence of plankton
and does not enter into subsequent nitrification reactions.
This is equivalent to about 75 wg/l to 150 ug/l which is
within the range of observed chlorophyll measurements for
the Delaware.

It can also be noted that the ammonia profile was verified
by employing a reduced reaction rate from mile 100 to mile
85. This was justified because of the low dissolved oxygen
(< 2 mg/l and minimums of about 0.7 mg/l) in this reach.

The nitrate analysis indicates two areas of nitrate decay.
The first reach from mile 100-90 is attributed to denitri-
fication. Because feedback loops were not used in this in-
vestigation, it was not possible to recycle this nitrate re-
duction into the system. Nitrate decay at the lower end of
the estuary was assumed to be due to increased phytoplankton
utilization of nitrate.

Experimental information on the influence of temperature on
nitrification is meager. Work on the Thames estuary (Anon,
1964) indicated for the temperature dependence of the oxida-

tion of ammonia:

T-20

52


In addition to the removal of nitrate by bacterial reduction
it may also be used by the phytoplankton as a nutrient source.
The nitrate must be converted to ammonia by enzymatic reaction
before it is assimilated. The assimilated nitrogen becomes
part of the organic nitrogen in plants and subsequently in
animals. Excretion and decay of this material releases or-
ganic nitrogen, thereby completing the cycle. Ammonia, or
some form of nitrogen, is also required by the heterotrophic
bacteria in the oxidation of carbonaceous material; however,
this sink of nitrogen is relatively insignificant by contrast
to the oxidation process or algal usage.

In summary there are two broad areas of concern from the water
quality viewpoint with respect to the nitrogen cycle in na-
tural waters: 1. The oxidation and possible reduction of
various forms of nitrogen by bacteria and the associated
utilization of oxygen. 2. The assimilation of the inorganic
nitrogen and the release of organic nitrogen by phytoplankton
during growth and death respectively. In some cases, either
one or the other of these conditions dominate, the former in
areas of large sources of waste water with little or partial
treatment and the latter in water bodies receiving biologically
treated effluents or agricultural drainage.

This section is concerned with the first of these broad areas.
Its specific purpose is to present a mathematical model of the
nitrification reactions in streams and estuaries. The basic
theory of the nitrogen cycle is reviewed; several simplified
kinetic mathematical models of portions of the cycle are con-
structed and application of the simplified models to specific

situations is described.

14


An equation similar to (39)can also be written down for Ny
with the input from Ny appearing in the particular integral.
Thus,

Ny = B,exp (S5x) + Cyexp (Vox)
Kia (40)
+ g—— [B,exo(S,x) + C,exp(V,x)]
Roo Kay 1 1 1 1

Eq. (40) can be compared to Eq. (37). This procedure is
repeated for each nitrogen form resulting in four solution
equations (see also Eq. (38)) in eight coefficients. Appli-
cation of boundary conditions, mass balance and concentration
equality at each section provides the necessary equations to
evaluate these coefficients. Details are given in Anon., 1969
and the application of this model to the Delaware Estuary is
discussed below.

Another approach that can be used to model sequential reactions
is to replace the spatial derivatives with finite approxima-
tions. This results in a series of algebraic equations which
can be solved simultaneously to obtain the spatial distribu-
tion of each nitrogen form. A generalization of this approach
is explored in detail in the next section.

In summary, models of sequential reactions are readily structur-
ed utilizing the steady state and first order kinetic assump-
tions. For the nitrification effect, these models “track" the
spatial distribution of each of a series of forms allowing di-
rect computation of the DO deficit due to nitrogen oxidation.
For streams, the solutions can be obtained directly while for
estuaries some type of computer solution is often required.

Feedback models incorporate feedback loops between any of the
respective systems. In principle, this third model can include

43


rates. The dimensionality of the matrices and vectors repre-
sents the spatial distribution of the nitrogen species. Thus,
[A] is an X n matrix and (Ny) are n x 1 vectors.

The entire procedure used to obtain Eq. (44) for the first
nitrogen form is now repeated for the second through fourth
forms. The four matrix equations are then given by

(A (N) = Wy) + [VK,,) (N,) + [VK,,] (Nj) + [VK,)] (4)
[Ag] (Ny) = (Wy) + EVK, 9] (Ny) + (VK, 51 (N3) + [VKy9] (Ny) cas)
(A3,] (Nj) = (W3) + [VK) 3] (Ny) + [VK,3](N4) + [VKy3] (Ny)
TAQ] (Ng) = (Wg) + (Kg) + (VK, 41 (NQ) + [VK3 4] (5)

It should be recalled that in Eqs. (45), the [Aj] matrices
differ only in the reaction coefficient Kyi on the main di-
agonal (see Eq. 43a). There are 4 n algebraic equations in
Eqs. (45) which can be solved by a variety of block decompo-
sition and relaxation techniques. Some additional insight
can be obtained by continuing the matrix analysis. The set
of matrix equations (45) can be written as

PCy) = (Kay) = (Ky) = Kl} / OH) (0)
i i
~[VK, 5] [Aj] - [VK,5] - [VK,5] (Ny) (Wo) us)
~ (VK, 3] - [VK,3] [Aj] - [VK,3] (N3) (W3)
~(VK) 4] - (VKj4] - [VK3,] [Ag] | (Ny) (Wy)
4
or
{A] (N) = (Wi) (47)

where [A] is a 4n X 4n matrix and (N) and (W) are 4n X 1
vectors. Of course, in Eqs.(46) not all feedback or feed-
forward loops need be included. For a given sequence of

47


where PL is the photosynthetic growth rate, R is the endoge-
nous respiration rate of the phytoplankton, and G is the
death rate owing to zooplankton grazing. A major improvement
in Riley's equation is the attempt to relate the growth rate,
the respiration rate, and the grazing to more fundamental
environmental variables such as incident solar radiation,
temperature, extinction coefficient, and observed nutrient
and zooplankton concentration. As a consequence, the coef-
ficients of the equations are time-variable since the
environmental parameters vary throughout the year. This
precludes an analytical solution to the equation, and nu-
merical integration methods must be used. Three separate
applications (Riley, 1946, 1947, 1949) of these equations

to the near-shore ocean environment have been made, and the
resulting agreement with observed data is quite encouraging.

A complex set of equations, proposed by Riley, Stommel and
Bumpus (1949) first introduced the spatial variation of the
phytoplankton with respect to depth into the conservation

of mass equation. In addition, a conservation of mass equa-
tion for a nutrient (phosphate) was also introduced, as well
as simplified equations for the herbivorous and carnivorous
zooplankton concentrations. The phytoplankton and nutrient
equations were applied to 20 volume elements which extended
from the surface to well below the euphotic zone. In order
to simplify the calculations, a temporal steady-state was
assumed to exist in each volume element. Thus, the equations
apply to those periods of the year during which the dependent
variables are not changing significantly in time. Such con-
ditions usually prevail during the summer months. The results
of these calculations were compared with observed data, and
again the results were encouraging.

Steele (1956) found that the steady-state assumption did not

76


It is clear from both the data and the model results that a
classical predator-prey situation is observed in 1966: the
spring bloom of phytoplankton resulting from favorable tem-
perature and light intensity provides the food for zooplankton,
which then reduce thepopulation during the summer. The de-
crease of the zooplankton and the subsequent slight secondary
bloom of phytoplankton complete the cycle for the year. It

is not clear, however, from a casual inspection of the data,
whether the zooplankton population terminated the phytoplankton
growth, as in classical predator-prey situations, whether the
nutrient concentration dropped to a limiting value that reduced
the growth rate, or a combination of the two. This point is
elaborated in the next section.

The situation in 1967 is quite different. No significant
phytoplankton growth is observed until late in the year. The
controlling variable in this case is the large advective flow
during the spring and summer of 1967 (see Figure 26) which
effectively washes out the population in the region. Only
when the flow has sufficiently decreased so that a population
can develop do the phytoplankton show a slight increase.
However, the dropping temperature and light intensity level
terminate the growth for the year.

Growth Rate - Death Rate Interactions

The behavior of the equations which represent the phytoplankton,
zooplankton, and nutrient systems in one volume can be inter-
preted in terms of the growth and death rates of the phytoplank-
ton and zooplankton. The equations are as before

= - Q -
a OC«<=S (c, DIP + | (Py P) (105)

138


In order to simplify the spatial segmentation and the calcu-

lations, a one-volume segment is chosen for the region of the
San Joaquin for which Mossdale is representative. The volume
of this segment is, of course, somewhat arbitrary, and a more
representative spatial segmentation would remove this uncer-

tainty. However, it is instructive to consider the behavior

of the solution of this simplified model.

The nutrient data available indicate that phosphate, biocarbo-
nate, silicate, calcium, and magnesium are available at con-
centrations well above the levels for which it has been sug-
gested that these nutrients limit growth. Only the ammonia

and nitrate concentrations are low, and they both decrease
markedly during the 1966 spring bloom. Hence, these nutrients
must be considered explicitly. To simplify the computations,
the ammonia and nitrate nitrogen are combined, and the nutrient

considered is total inorganic nitrogen.

There is some uncertainty concerning the magnitude and the
temporal variation of the inorganic nitrogen load being dis-
charged to the system during the time interval of interest.

For lack of a better assumption, the inorganic nitrogen load

Wa being discharged into the volume is assumed to be a constant,
the magnitude of which is determined by comparison with the ob-
served inorganic nitrogen concentration data at Mossdale.

The variation of the environmental variables being considered

- namely, temperature, solar radiation, and advective flow in
the San Joaquin during the two~year period of interest - and

the straight line approximations that are used directly in the
numerical computation are shown in Figure 26. The influent
advective flow, which is assumed to have constant concentrations
of phytoplankton and zooplankton biomass and inorganic nitrogen,

131


ent chapters indicating the utility and power of these
methods. The evolution of the modeling structure is il-
lustrated in Figure 1 which presents the development and
increasing scope of the modeling frameworks with the latter
models encompassing a relatively broad range of environmental
variables associated with the eutrophication phenomena.


is routed through the volume. Since Mossdale is located above
the saline portion of the San Joaquin, no significant dis-
persive mass transfer is assumed to exist by comparison with
the advective mass transfer.

The equations which represent this one-segment model are

P= (G,-d)P+2 (Pp - P) (98)
Pp” Pp Vo

° Q

Z = (G6 - Di )Z+ | (2, - 2) (99)

° WwW Q

N= -a.Gp+ +5 (Ny - ND (100)

where Q = Q(t) is the advective flow entering and leaving the
volume; V is the volume of the segment; Por aor and Ny are
the phytoplankton, zooplankton, and inorganic nitrogen con-
centration of the flow entering the volume. The remaining
terms have been defined previously by Equations (67), (74),
(76), and (77). In the nutrient equation, only the direct
source of inorganic nitrogen, War has been included; the or-
ganic feedback terms representing excreted nitrogen, etc.,
Equation (79), have been dropped. Since the magnitude of Wa
is uncertain and is assigned by comparison with observed data
and computed model output, these feedback terms can be thought
of as being incorporated in the value obtained for Wie

The solution of Equations (98), (99), and (100) requires
numerical techniques. For such nonlinear equations, it is
usually wise to employ a simple numerical integration scheme
which is easily understood and pay the price of increased com-
putational time for execution rather than using a complex,
efficient, numerical integration scheme where unstable be-

havior is a distinct possibility. A variety of simple methods

133


20 =
se a K+ 010
_ . 8 8
S ° on .
oan 8 rn ry
: >. ;
° ° ° @ °
g Crd 7D ROKTON € ° °
00 1 1 oo = jo. — —_)
ao! 120 HO 100 90 80, 70 60 5
“TAY 30, BEA
10, 1964
30h (@-AUG Y, 1964
OWPC CATA
= ALL MWS.
x FLOW = 3000 CFS.
& 2 9 THHPea6-27°C
% 20
0
3
°
to} °
8
é 2
in ie L 1
00 % 30 ca 7 50
I 1 OSTANCE-MLES 1
ou or too | 95 | no | no core
Fig. 8 Observed vs. Computed (solid line)
Nitrogen Profile August 1964 (after anon.1969
20
—
>
€
z

NO3-N-mg/I

90
OSTANCE:

err
om pola! po

51


Section

It

IIt

Iv

vI

VII

VIII

IX

CONTENTS

Page
Conclusions 1
Recommendations 3
Introduction 5
Nitrification and Its Effects on Dissolved 11

Oxygen

A Dynamic Model of Phytoplankton Populations 73
in Natural Waters

A Preliminary Model of Phytoplankton Dynamics 143
in the Upper Potomac Estuary

Acknowledgments 191
References 193
Publications 201


CHLOROPHYLL a- pg/I

160-

40 WOODROW WILSON BRIDGE

Mi. 12.1
120- .

SEGMENT 7—~
‘SEGMENT 6

100;

80

60)

On 30 ren? ar 2g '20 ey 9 99 ay 210 a

160 1969

140-
. PISCATAWAY CREEK

. Mi. 18.3

120 .

ool COMPUTED «
SEGMENT 9

80;--

60

20-

fo} 1 1 l n n n n L n

1 sepi71 061390 novF9° acd GO

fo) 30 60 90 (20, 150 180 210 240 270 300 330
1 Jon. “I Feb 1 Mar I Apr 1 Moy 1 June | July | Aug. | Sspt. | Oct. | Nov, I Dec.

1969

Hl
360

Fig. 39 Temporal Comparison of Observed 1969 Data and
Computed Values for Stations at Miles 12.1 and 18.3

170


24,

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

Droop, M.R., “Physiology and Biochemistry of Algae"
Organic Micronutrients, p. 141-60, R.A. Lewin, Ed.,
Academic Press, New York, 1962.

Dugdale, R.C. "Nutrient Limitation in the Sea: Dynamics,
Identification and Significance" Limnol. Oceanog, 1967,
12(4) 685-95.

Eppley, R.W., Rogers, J.N., McCarthy, J.J. “Half Satura-
tion Constants for Uptake of Nitrate and Ammonium by
Marine Phytoplankton" Limnol. Oceanog., 1969, 14(6),
912-20.

Feth, J.H. "Nitrogen Compounds in Natural Water - A
Review" Water Res. Research, Vol. 2, No. 1, 1966, pp
41-58.

Fogg, G.E. "Algal Cultures and Phytoplankton Ecology"
p-. 20, University of Wisconsin ress, Madison, Wis. 1965.

Gerloff, Skoog, "Nitrogen as a Limiting Factor for the
Growth of Microcystis Aerugmosa in Southern Wisconsin
Lakes" Ecology, 1957, 38, 556-61.

Hamming, R.W. "Numerical Methods for Scientists and
Engineers” p. 194-210, McGraw Hill, N.Y. 1962.

Hetling, L.J. and O'Connell, R.L. "An O, Balance for the
Potomac Estuary" Unpublished Manuscript; Chesapeake
Field Sta., Ches. Bay-Sus. River Basin Project, FWPCA,
Feb. 1968.

Hutchinson, G.E. "A Treatise on Limnology, Vol. I" J.
Wiley & Sons, Inc., New York, N.Y. 1957, 1015 pp + xiv.

Hutchinson, G.E. "A Treatise on Limnology Vol. II.
Introduction to Lake Biology and the Limnoplankton"
p. 306-54, Wiley, New York, 1967.

Hutchinson, G.E. and Viets, F.G. "Nitrogen Enrichment of
Surface Water by Absorption of Ammonia Volatilized from
Cattle Feedlots" Science, Vol. 166, Oct. 24, 1969,

pp. 514-515. —

Jaworski, N.A., Lear, D.W. and Aalto, J.A. "A Technical
Assessment of Current Water Quality Conditions and Factors
Affecting Water Quality in the Upper Potomac Estuary" Tech.
Report No. 5 Ches. Tech. Sup. Lab, FWQA, Annapolis, March,
T1969.

195


in the Washington D. C. area down to about segment 15 are
Anacystis, a blue green form that is toxic to zooplankton.
Several runs were made of the model to show the sensitivity
of the various systems to perturbations on the zooplankton
system.

Figure 34(b) shows the effect of zooplankton grazing on the
phytoplankton chlorophyll "a" concentration in segment #9.
The time period extends from March to December 1968 with

an additional 60 days to show the general trend after
December 1968. As indicated in Figure 34(b) with extensive
zooplankton grazing, ohytoplankton populations are almost
completely depleted in September and maximum concentrations
of only 60 ug/l are computed as compared to observed values
of about 130 yg/l. It is interesting to note that the sharp
drop in zooplankton at day 90 is the effect of flow transient
at that time and as a consequence phytoplankton populations
remain high since the predators are flushed past the segment.

Figure 34(a) with no zooplankton grazing is considerably
closer to the observed data range as shown. The transient
drop around September 1968 is due to a flow increase as shown
in Figure 31.

The effect of zooplankton predation on the nutrient forms is
interesting and is shown in Figure 35. As shown, the ammonia
nitrogen is quite insensitive to the zooplankton effect under
the conditions run. The drop in ammonia concentration at day
90 and day 180 is due to flow transients during that time

(see Figure 31). With the cycling of organic nitrogen to
ammonia nitrogen there is a sufficient feed forward to the
ammonia system even when the phytoplankton population is high.

The effect on the nitrate concentration is marked however as
shown in the top of Figure 35. Under extensive zooplankton

159


89

TOTAL K-NIT-N

(mg/l)

40

JULY ~ AUG, 1968
asf Q= 2000 OFS
LEGENO:
or © ee i
@ COMPUTED WITH
25 NO FEED BACK
| | | | APPROX. RANGE,
OF CESERVED OAIA
20+ |
is | uu
tok HE ag
as pl 1 |
re) 1 1 L 4 L 1 1
(0) 5 fe) 5 20 25 30 35 MILES FROM CHAIN BR.

Lu lelslalsfel 7 [el o]olul w2zwitoatlwal os

SEGMENT NO,

Figure 14
Verification of Kjeldahl Nitrogen for Potomac Estuary


nitrate. This process of nitrification is often referred to

as the second stage of biochemical oxygen demand. In untreated
and heavily polluted water, the two stages do not usually occur
together, the first being substantially completed before the
second is significantly underway. In treated effluents and
less polluted waters, the lag between the two is reduced. As
the first stage is reduced and with nitrifying organisms
present, the two stages may occur simultaneously. An example
of this reaction in BOD samples is shown in Figure 3,
(O'Connor, 1968).

Kinetic Models

The nitrification process, in which ammonia is converted
through nitrite to nitrate is an autotrophic biochemical
reaction. The energy for the growth of the microorganisms
is obtained from the oxidation of ammonia or nitrite (see
Eqs. (4) and (5). In reactions of this type the rate is
generally assumed to be proportional to the concentration

of the substrate and also of the microorganisms. The sub-
strate concentration dependency, however, may be modified by
stipulating that the rate of the reaction is independent of
the concentration of substrate at high substrate concentra-
tion and becomes increasingly concentration dependent as the
concentration decreases. Substrate limiting kinetics have
been used in the analysis of biological waste treatment phe-
nomena (Lawrence and McCarty, 1970; Pearson, 1968). A
general review of the literature is given in Lawrence and
McCarty (1970). The growth equation for the microorganisms
may be written as:

dM
at KM (8)
where M = the microbial concentration
K = microbial growth coefficient.

23


utilizes the solution of the fourth equation.

A finite difference approximation was employed to solve

Eq. (50). The spatial segmentation and svstem parameters of
other work (Hetling and O'Connell, 1968: Jaworski et al,
March 1969 and May 1969) was used. A total of 23 spatial
segments was apolied to the reach of the Potomac from Little
Falls downstream, a distance of about 100 miles to the ap-
proximate entrance to Chesapeake Bay. In matrix form, the
equations to he solved are (see also Eq. (46).

(Ay] ° ° ~  (VK,yy] fo) (ay) (Wy)
- [VK 9] [ag] (0) fo) o (Ny) (Wy)
° [VK 3] [a3] ° fo) (N3) = 0 (51)
(o) ~ (VKy4] - [VK3,] (Ag] fe) (Ny) °
0 - [VK,.] 0 0 [As]] (BD) 0

where the (Aj) are 23 x 23 triadiagonal matrices incorporating
net advective flow and dispersion with the Ki appearing on

the main diagonal. Note that [Ag] has the reaeration rate K,
on the main diagonal, (VK; 51 are 23 x 23 diagonal matrices

(N;) and (D) and (W) are 23 x 1 vectors of the nitrogen forms,
DO deficit and input nitrogen loads, resvectively. The matrix
Equation (48) is therefore composed of a svstem of 115 algebraic
equations, the simultaneous solution of which provides the
steady state distribution of the four nitrogen forms and the

DO deficit in all 23 segments.

Data were available for the period July-August 1968 for veri-
fication purposes. These data included Kjeldahl nitrogen
(representing the sum of organic nitrogen from waste discharges,
ammonia nitrogen and algal nitrogen), nitrite and nitrate ni-
trogen and chlorovhyll a measurement. Major invut nitrogen

66


growth rate becomes positive when the phytoplankton popula-
tion is large enough to sustain the zooplankters. Then the
zooplankton grow until they have reduced the phytoplankton
population to a level where they are no longer numerous enough
to sustain the zooplankton. The net zooplankton growth rate
then becomes negative and the population diminishes in size.
This small zooplankton population no longer exerts a signifi-
cant effect on the death rate of the phytoplankton, Dor and
its value decreases, causing the net phytoplankton growth rate
to become positive again, and the smaller autumn bloom results.
The decreasing temperature and light intensity and the in-
creasing advective flow then effectively terminate the bloom

as the year ends.

142


10 - 15 miles downstream of the major waste sources reflect-
ing the inhibited nitrification upstream due to low dissolved
oxygen. It is also quite interesting to note a general back-
ground of 0.7 - 1.0 mg/l dissolved oxygen deficit due to the
nitrogenous discharges from tributaries and run-off as well

as municipal and industrial sources. The veak value of

2.5 mg/l dissolved oxygen deficit is somewhat lower than
previous estimates which placed the peak at about 3.0 mg/l
dissolved oxygen deficit, but at approximately the same lo-
cation. Part of this difference is attributable to the use

of the four system model rather than an approximation through
"nitrogenous BOD" with associated reduction in nitrogenous

BOD decay rate or simple translation of the input of nitroge-
nous BOD. In general, then, Figure 12 confirms previous work
which recognized a downstream shift of the satisfaction of

the nitrogenous oxygen demand. The results indicate that this
phenomenon is due to low upstream dissolved oxygen values which
together with the discharge of potentially toxic materials have
an inhibitory effect on the nitrifying bacteria. As the dis-
solved oxygen recovers, the nitrifying flora begins to develop
and bacterial nitrification proceeds at a relatively rapid
pace. This is then accompanied by an increasing utilization
of oxygen.

Projected effects of a nitrogen removal program can be esti-
mated using this model. Nitrogen removal from waste effluents
can be accomplished in several ways including biological ni-
trification, air stripping and ion exchange. Each of the
methods accomplishes varying degrees of removal of nitroge-
nous components and at widely varying costs.

Figure 12 showed the estimated dissolved oxygen deficit due
to nitrification for 1964 summer conditions. It should be
recalled that the peak dissolved oxygen deficit occurred at
about mile 75 because of assumed nitrification inhibition

59


The ammonia in turn is oxidized under aerobic conditions to
nitrite by bacteria of the genus Nitrosomonas as follows:
(Sawyer, 1960, Hutchinson, 1957, Stratton, 1967 and Delwiche,
1956)

bacteria

(NH,) + OH + 1.50, > H + NO> + 2H, + 59.4 K cal (4)

2
This reaction requires 3.43 gms of oxygen utilization for one
gram of nitrogen oxidized to nitrite. The nitrite thus formed
is subsequently oxidized to nitrate by bacteria of the genus
Nitrobacter as follows: (Hutchinson, 1957)

bacteria -

NO, + 0.50, > NO3+18 K cal (5)

2
This reaction requires 1.14 gms of oxygen utilization for one
gram of nitrite nitrogen oxidized to nitrate. The total oxygen
utilization in the entire forward nitrification process is
therefore 4.47 gms of oxygen per gm of ammonia nitrogen oxidized
to nitrate. The Nitrobacter bacteria process about three times
as much substrate as the Nitrosomonas bacteria to derive the
same amount of energy. Nitrite is therefore converted quite

rapidly to nitrate.

Essential factors for nitrification are oxygen, phosphates and
an alkaline environment to neutralize the resulting acids.
Nitrifying bacteria are very susceptible to action of toxic
substances (e.g. manganese): These bacteria are obligate auto-
trophs, which derive their energy from the oxidation of simple
inorganic compounds. The energy obtained from these reactions
is utilized for the assimilation of carbon from either carbon
dioxide or bicarbonate ion, but not from organic carbon.
However, it has been indicated that Nitrobacter can be grown
heterotrophically on an organic substrate, without losing its
ability to oxidize nitrite. Acetate was assimilated as cell

16


al behavior in the substrate. Therefore, for "dilute" systems
where substrate concentrations are "low" and for "well seeded"
systems where initial nitrifying populations are high relative
to initial substrate concentrations, first order decay of sub-
strate may be approximately justified. The actual application
of these conditions must of course be made with care but as a
first approximation, first order kinetics reactions for the
nitrification phenomenon is a meaningful step and can aid in
understanding the behavior of observed nitrogen forms in
natural water systems.

First Order Steady State Mathematical Model
of Streams and Estuaries

With an assumption of first order kinetic reactions and steady
state conditions, several levels of mathematical models can
be constructed each of which are useful in predictions of
long-term effects of nitrification on water quality. The
three levels of modeling are:

a) BOD equivalent models

b) Sequential reaction models

c) General feedback models

Each of the levels increases somewhat in the complexity of

the equations and provides greater detail in understanding the
phenomenon. The major reason for even considering several
problem levels of first order kinetic models is that all prob-
lem contexts do not necessarily require complex models for
their solution. Much can be learned from the first order
models although it is important to stress again that the actual
kinetic mechanisms are more complex than those which are con-
sidered herein, If it is obvious that non-linear kinetics
apply then one should use the models presented below only as

a gross approximation, if at all. For some vroblem contexts,

33


phosphorous from the organic form and Ki is the overall loss
coefficient of inorganic phosphorous from the water column.
Total organic phosphorous and total phosphorous are computed
from Systems #1, #2, #6 and #7 with appropriate conversion
factors.

In the carbonaceous BOD system only the direct sources of
biochemical oxygen demand (carbonaceous) are included. A
traditional first-order decay is assumed. The equation is

da(L)

[v] aE CO« [A] (L) + [v] (sy) + (Ww) (126)
where L is the carbonaceous BOD and Sy is simply
Ss, = - Kg (T)L

The DO deficit System #9 is classical in concept except that
the effect of nitrification is computed internally using the
output from System #4. The governing equation is
a(D) _
(Vv). —e- = [A] (D) + {v] (Sp) (127)
where D = DO deficit. The source-sink term for the deficit
is given by

s = 4.57 (K

D ) +k

45No 9h - K,D + £, (P) (128)
where Keg is the deoxygenation coefficient due to carbonaceous
BOD, kK, is the reaeration coefficient. The first term in

Eq. (128) represents the utilization of oxygen due to nitri-
fication while the last term represents the effect of phyto-

plankton on the dissolved oxygen.

153


with 6 = 1.017. Laboratory work however indicated @ = 1.10.
Others (Stratton and McCarty, 1967; Buswell et al, 1959)
have estimated 6 at about 1.08 while for nitrite oxidation
a value of 1.06 has been determined (Stratton and McCarty,
1967). After review of available results, the temperature
dependence of the reaction rates was hypothesized as shown
in Figure 9,

Data were available for the period November 1967 when water
temperatures were 7° - 10°c and river flows at Trenton were
about 8900 cfs. 1964 loads were used. Figure 10 summarizes
the results of the application of the model to these lower
temperature data. A reaction rate of 0.025 (@ temperature =
7.5°C) was used for the conversion of organic nitrogen, .01
for ammonia, .05 for nitrite and 0.0 for nitrate. At these
low rates, almost all forms behave as conservative variables.
It was not necessary in this case to make the assumption that
plankton had synthesized nitrogen into living tissue. At the
low estuary temperatures, most plankton activity would be
minimal. The ammonia plot in Figure10 distinctly shows the
effect of reduced nitrification. This can be especially seen
by comparing the Nov. 1967 data (Fig.19) to the August 1964
data (Fig. 8). The reduced nitrification effect at low
temperatures is also evident in the nitrite and nitrate pro-
files. This effect will also be reflected in reduced oxygen
utilization and has a significant impact on treatment vrograms
that may use nitrification.

The final step in the verification analyses was to use the pre-
ceding reaction coefficients temperature dependence and support-
ing assumption to "independently" verify other profiles.

Figure lL is an example. As indicated, agreement is good sup-
porting the assumption of a consistent set of coefficients.

The only change in the coefficients was to distribute the area

53


The dissolved oxygen conditions under which nitrate reduction
becomes significant are subject to some differences of opinion
resulting from the relatively small amount of work done in
this area. Under completely anaerobic conditions, nitrifi-
cation cannot occur since the nitrifying bacteria are strictly
aerobic. There is some evidence to indicate, however, that
nitrate reduction goes on constantly and is greatly acceler-

ated under low (0-2 mg/l) dissolved oxygen conditions.

Sources of Nitrogen

Agricultural sources of nitrogen may be significant with
respect to concentrations found in natural waters. The
nitrogen is present in the drainage water from these lands
and aside from that inherent in the soil itself, emanates
from the fertilizers, legumes and barnyard and silo effluents.
An attempt at a nitrogen balance for harvested crop areas of
the United States is given in Table l.

Table 1

Nitrogen Balance Sheet for the
Harvested Crop Area of the United States, 1930*

Nitrogen
lb/acre/year
Additions
Rain and irrigation 4.7
Seeds 1.0
Fertilizers 1.7
Manures 5.2
Symbiotic nitrogen fixation 9.2
Nonsymbiotic nitrogen fixation 6.0
27.8
Losses
Harvested crops 25.1
Erosion 24.2
Leaching 23.0
72.3
Net Annual Loss 44.5

*after Allison, 1955, quoted by Feth, 1966.

19


t Q at

= a -
P(t) = P(o) e "+P, oF le 1) (110)

For a negative, that is, for a negative net growth rate, the
solution tends to the steady state value Py Q/l|alv. However,
for a positive, the solution grows exponentially without limit.
Thus, for a negative but |a| small, or for a positive, the
solution becomes large; whereas for a negative but |a| large,
the solution stays small. Hence, the behavior of the solution
can be inferred from the plots of the net growth rates.

Figure 28a is a plot of the following terms from the 1966
Mossdale calculation: Ss, without the Michaelis-Menton multi-
plicative factor included - i.e., the growth rate at nutrient
saturation denoted by cy (1,T); S itself denoted by GS, (N,1I,T)
i.e., the growth rate considering the nutrient effects. The
net growth rate S, - (Dy, + Q/V) is also plotted. Similarly,

in Figure 28b the growth rate of zooplankton Ge the mortality
rate Do the flushing rate Q(t)/V, and the net growth rate

G, - (D, + Q/V) are plotted.

The analysis of the 1966 model calculations can now be made

by inspecting these figures. The net growth rate for the
phytoplankton &., - @, + Q/V) becomes positive at t = 85 days
owing to an increase in Gyr the result of rising temperature
and light intensity, and a decrease in Q/V as the advective
flow decreases. The positive net growth rate of the popula-
tion causes their numbers to increase exponentially fast until
the nutrient begins to be in short supply. This is evidenced
by the departure of the S, curve from the & at nutrient satura-
tion curve. At the same time, the Dy curve is showing a marked
increase because of the increased zooplankton population and
their grazing. The result is that the net growth rate becomes
zero and then negative as the zooplankton reduce the phyto-
plankton population by grazing. The growth of the zooplankton
can be analyzed in a similar fashion using Figure 28b. The net

140


SATURATED GROWTH RATE (DAYTM)

w&
°

id
°

Fig. 18

TEMPERATURE °C

Phytoplankton Saturated Growth Rate (Base e)
as a Function of Temperature

86


Peak values as computed during the period tend to be higher
than observed although the average of the two computed lines
of June 30 and July 15 is reasonably close to the average of
the observed data and differs in the peak region by about
20 ug/l and by 5 - 10 ug/l downstream of the peak region.

The total Kjeldahl nitrogen verification is also shown in
Figure 37 and agreement is very good with the exception of
the July 8 survey where there was no observed increase in
nitrogen in the vicinity of the major discharges. There

is no readily apparent explanation for this discrepency.

Figure 38 shows the comparison between the spatial profile
of total phosphorous (as PO,) as observed during the surveys
of June 30, July 8 and July 15, 1969 and the computed pro-
files for the time period bracketing those surveys. As
shown the agreement is quite good although in the downstream
direction, computed values decline more rapidly than ob-
served values. This is probably a consequence of the simple
first order adsorption kinetics that were used. A second
order assumption as used by Jaworski et al, 1971, would give
better results. Computed values in general are within 0.5 -
1.0 mg/l of observed values of PO,-

The nitrate profiles are also shown in Figure 38. The agree-
ment is good for the July 8 and July 15 surveys. Agreement
was not obtained for the June 30 survey. During that survey
the observed data showed a considerable downstream shift in
the observed nitrate data which was not duplicated by the
computed values. This could possibly be due to low dissolved
oxygen values which would delay the onset of nitrification
but DO data were not available to confirm this hypothesis.
There is a more rapid decline in computed values of nitrate
nitrogen in the vicinity of Mile 20 than was observed. This

168


aN,

sry = KAN, - K,N, + W(x) (36c)
dt 202 3°3 3
dn

4. - (36a)
ace = K3N3 K4Ny + Wy (x)

where t* = x/u, the time of travel.

Eqs. (36) represent a series of first order coupled equations.
The solution of Eq. (36a) is:

N

- *
fy Noy exp ( Kt )

where Noy is the initial value of Ny @ x = O given by mass
balance incorporating Wy (x).

Substitution of this solution into Eq. (36b) and integration

gives the solution for ammonia, Ny as:
ny = geOl (exp (k, t*) exp (-K,t*)) + Nasexp(-K,t*) (37)
2 7 Kay SOXP MG PICK, o2eXP (“Kp

Sequential substitution and integration gives the concentra-
tions of nitrite and nitrate. For example, for nitrite,

exp (~K, t*) -exp (-K,t*) exp (-Kyt*) -exp (-K,t*)

2
N, = pcx. (' = - = JN
3 Ky Ky K3 Ky K3 Ky 01
K,
“Kt*)-= -K.t*
+ rk [exp ( Kyt )-exp ( K,t I Noo (38)

+ K3Ny3exp (-K,t*)

Each of the preceding inputs is reflected in this equation.
In general, the total amount of end product formed bv virtue

40


In addition to the conceptual model which isolates the major
interacting systems, a further idealization is required which
sets the lower and upper limits of the temporal and spatial
scales being considered. Within the context of the problem

of eutrophication and its control, the seasonal distribution
of the phytoplankton is of major importance, so that the lower
limit of the temporal scale is on the order of days. The
spatial scale is set by the hydrodynamics of the water body
being considered. For example, in a tidal estuary, the spatial
scale is on the order of miles whereas in a small lake it is
likely a good deal smaller. The upper limits for the temporal
and spatial extent of the model are dictated primarily by
practical considerations such as the length of time for which
adequate information is available and the size of the computer
being used for the calculations.

These simplifying assumptions are made primarily on the basis
of an intuitive assessment of the important features of the
systems being considered and the experience gained by pre-
vious attempts to address these and related problems in natural
bodies of water. The basic principle to be applied to this
conceptual model, which can then be translated into mathe-

matical terms, is that of conservation of mass.

Conservation of Mass

The principle of conservation of mass is the basis upon which
the mathematical development is structured. Alternate formu-
lations, such as those based on the conservation of energy,
have been proposed. However, conservation of mass has proved
a useful starting point for many models of the natural en-
vironment.

The principle of conservation of mass simply states that the

81


data as presented whether the high values in the fall of
1969 at Mile 26.9 are a surface phenomenon and therefore
beyond the scope of the model formulation or whether the
high values extended throughout the water column.

For Mile Point 38, Possum Point, similar comments can be
made. The apparent winter bloom of phytoplankton is not
captured by the model since the growth parameters for the
phytoplankton biomass reflect a warm water population.
The model does reasonably well until about August 1969
after which a wide scatter of chlorophyll values was ob-
served. Peak values during August to October are not
obtained by the model although the model does approximate
conditions again during November and December.

In general, the phytoplankton model as formulated herein,
provided a reasonable approximation to 1969 conditions us-
ing a 1968 "tuning" of the model. As an independent check,
recognizing that no changes were made to the model from the
results of the 1968 analysis, the verification for 1969 pro-
vides an added measure of credibility to the overall model
structure. Overall spatial profiles during July 1969
verified well to about 40 miles downstream from Chain Bridge.
Dynamic variability in phytoplankton was verified well
throughout the first 20 miles and then only approximately
for the remaining 20 miles. Transient blooms in the late
winter and early fall of 1969 were not duplicated by the
model . :

Model Application
In order to illustrate the application of the preliminary

model, two simulations were prepared. It should be stressed
that this ‘model is largely explanatory in nature and the

172


76.

77.

78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

Spencer, C.P. "Studies on the Culture of a Marine Diatom"
J. Marine Biol. Assoc. U.K., 1954; quoted by Harvey, H.W.
"The Chemistry and Fertility of Sea Waters" p. 94,
Cambridge Univ. Press, 1966.

Steele, J.H. "Plant Production on Fladen Ground" J. Marine
Biol. Assoc. U.K., 1956, 35, 1-33. TO

Steele, J.H. "A Study of Production in the Gulf of Mexico"
Jd. Marine Res., 1964, 22, 211-22.

Steele, J.H. “Primary Production in Aquatic Environments"
Notes on Some Theoretical Problems in Production Ecology,
C.R. Goldman, Ed. Mem. Inst. Idrobiol. p. 383-98, 18
Suppl., Univ. of Calif. Press, Berkeley, 1965.

Stiefel, E.L. "An Introduction to Numerical Mathematics"
p. 163, Academic, New York, 1966.

Stratton, F.E. and McCarty, P.L. "Prediction of Nitrifi-~
cation Effects on the Dissolved Oxygen Balance of Streams"
Env. Sci. and Tech. Vol. 1, No. 5, May 1967, pp 405-410.

Strickland, J.D.H., "Chemical Oceanography" Production of
Organic Matter in the Primary Stages of the Marine Food
Chain, Vol. I, p. 503, J.P.Riley and G. Skivow, Eds.,
Academic, New York, 1965.

Tamiya, H., Hase, E., Shibata, K, et al "Algal Culture
from Laboratory to Pilot Plant" Kinetics of Growth of
Chlorella with Special Reference to Its Dependence on
Quantity of Available Light and on Temperature, pp 204-
234, J.S. Burlew, Ed., Publ. 600, Carnegie Inst. of
Washington D.C. 1964.

Thomann, R.V. "Mathematical Model for Dissolved Oxygen"
J. Sanit. Eng. Div., Proc. ASCE, Oct. 1963, 83, SA5, 1-30.

Thomann, R.V. "Systems Analysis and Water Quality
Management", ESSC Pub. Co., Stamford, Conn., 1972.

Thomas, W.H., Dodson, A.N. "Effects of Phosphate Concen-
tration on Cell Division Rates and Yield of a Tropical
Oceanic Diatom" Biol. Bull., 1968, 134 (1) 199-208.

Vollenweider, R.A. "Primary Production in Aquatic Envi-
ronments" Calculation Models of Photosynthesis - Depth
Curves and Some Implications Regarding Day Rate Estimates
in Primary Production Measurements, pp 425-57, C.R.
Goldman, Ed., Mem. Inst. Idrobiol, 18 Suppl, Univ. of
Calif. Press, Berkeley, 1965.

199


Table 3

SUMMARY OF REACTION COE#FICIENTS
DETERMINED [N VERLFiCALIGN ANSLYSiS OF NITROGEN
IN DELAWARE ESTUARY

First Order reaction Coefficient @ 20°C

Reach Org.to of to nee NO,
Survey No. NH3(K,>) NO,(K,,) NO,%K,,) (K44)
July-Aug. T O.7 v.1T 0.3 0.0
1964 2 0.001 0.002 0.05
Qe 3090 cfs 3 0.001 0.602 0.0
T = 26°C 4 0.05 0.110 0.0
5-6 ¥ Q.11 0.3 0.0
7 0.2 0.11 0.3 0.5
June-July 1 0.1 O.iL 0.3 0.0
1965 2 0.601 0.002 0.05
Q = 2020 cfs 3 0.001 0.002 0.0
T = 20°C 4 0.5 0.110
5 0.110 0.3 .
6 .
7 J | 0.05
8 0.1 0.110 0.3 0.05
July 1967 1 0.1 0.11 0.3 0.0
Q = 5690 cfs 2 0.001 0.002 0.05
T = 21°Cc 3 0.001 0.002 0.05
4 0.001 0.002 0.05
5 0.05 0.105 0.0
6 0.11 0.3 0.0
7 0.1 0.11 0.3 0.05
First Order Reaction Coefficient @ 7-10°C
Nov. 1967 1 0.025 0.01 0.05 0.0
Q= 8900 cfs 2
T = 7-10°C 3
4
5
6
7 0.025 0.01 0.05 0.0

58


mass of the substances being considered within an arbitrarily
selected volume must be accounted for by either mass transport
into and out of the volume or as mass produced or removed
within the volume. The transport of mass in a natural water
system arises primarily from two phenomena: dispersion, which

is caused by tidal action, density differences, turbulent dif-
fusion, wind action, etc.; and advection owing to a unidirec-
tional flow - for example, the fresh water flow in a river or.
estuary or the prevailing currents in a bay or a near-shore
environment. The distinction between the two phenomena is

that, over the time scale of interest, dispersive mass transport
mixes adjacent volumes of water so that a portion of the water
in adjacent volume elements is interchanged, and the mass trans-
port is proportional to the difference in concentrations of

mass in adjacent volumes. Advective transport, however, is
transport in the direction of the advective flow only. In ad-
dition to the mass transport phenomena, mass in the volume can
increase resulting from sources within the volume. These sources
represent the rate of addition or removal of mass per unit time
per unit volume by chemical and biological processes.

A mathematical expression of conservation of mass which includes
the terms:to describe the mass transport phenomena and the source
term is a partial differential equation of the following form

oP . _ .
Pred v EVP v oP + S, (52)

where P(x,y,z,t) is the concentration of the substance of in-
terest - e.g., phytoplankton biomass - as a function of position
and time; E is the diagonal matrix of dispersion coefficients;

Q is the advective flow rate vector; 85 is the vector whose
terms are the rate of mass addition by the sources and sinks;
and V is the gradient operator. This partial differential

82


t
<3
[oa
t
za f
om
oaned
i
a s
Ww >
xs
no
DE
oN
me)
oOo—
ot
a
za
ld
Fig.

is)

cS)

o

a

>

25

5 : 10 15 20
TEMPERATURE °C

Endogenous Respiration Rate of Zooplankton
vs. Temperature

117

25

0.40

0.30


SECTION VII

ACKNOWLEDGMENTS

The authors are pleased to acknowledge the guidance and
support of Walter Sanders of the Southeast Water Laboratory,
Environmental Protection Agency , in this research.

The Participation of John L. Mancini, Hydroscience, Inc.,
is also gratefully acknowledged.

The application of the nitrification model to the Delaware
Estuary was accomplished through a contractual arrangement
between the Delaware River Basin Commission and Hydroscience
Inc.

The assistance of Gerald Cox and Jack Hodges of the Department
of Water Resources and Harold Chadwick of the Department of
Fish and Game, State of California is also gratefully acknowl-
edges.

Richard Winfield, Research Engineer, Manhattan College, as-
sisted significantly in the verification and application of
the eutrophication model of the Potomac Estuary and his input
is specifically acknowledged.

Finally the authors are pleased to acknowledge the effect and

forebearance of Mrs. Berenice Maguire in the preparation of
this report.

191


The source term Sn5 in the conservation of mass equation for
the concentration of the nutrient Ny in the j volume seg-
ment v5 is the sum of all sources and sinks of the nutrient
within V5 The primary interaction between the nutrient
system and the phytoplankton system is the reduction or sink
of nutrient connected with phytoplankton growth. The rate

of increase of phytoplankton biomass is Go 4P4° To convert
this assimilation rate to the rate of utilization of the nu-
trient, the ratio of biomass productior to net nutrient as-
similated is required. Over a long time interval, this ratio
approximates the nutrient-to-biomass ratio of the phytoplankton
population. For example, if the nutrient being considered is
total inorganic nitrogen and the phytoplankton biomass is
characterized in terms of dry weight, then this ratio is the
nitrogen-to-dry-weight ratio of the population. For both
nitrogen and phosphorus, these ratios have been studied for

a number of phytoplankton species and natural associations.

An example of this information is presented in Table 1l, con-
densed from Strickland (1965). If ano is the nutrient-to-
phytoplankton biomass ratio of the population, then the sink
of the nutrient owing to phytoplankton growth is anpSp4?4°

A secondary interaction between the biological system and the
nutrient systems is the excretion of nutrients by the zoo-
plankton and the release of nutrients in an organic form by
the death of phytoplankton and zooplankton. The excretion
mechanism has been considered by Riley (1965) in a generaliza-
tion of the equations of Steele. The rate of phosphorus ex-
cretion has also been measured experimentally (Martin, 1968).
Using the formulation for zooplankton growth rate proposed
herein, the rate of nutrient excretion is the rate grazed,

aC P.Z., minus the rate metabolized, a ; that is,

G_.Z.
mp gj J np Z) J

120


in which a = stochiometric or yield constant (mg bacterial
mass per mg substrate)
Mo = initial bacterial mass

cy = initial substrate concentration

Substitution of Eq. (11)into Eq.(7oy gives upon integration

K Cc aK_+’M_+ ac
= ln (say7eqary) + HF
° fo}

M
Kat in g- (12)

~ M+ ac +a
° ° My So °

A detailed dimensionless analysis of the behavior of this
type of equation is given below in terms of the utilization
of the substrate.

Equation (10) indicates that at high concentrations of sub-
strate (c >> K.). the rate is independent of c and the rate
expression reduces to a first-order reaction in which the
growth rate is proportional to the concentration of micro-
organisms:

dM
ae ~ Kyu (13)

This equation indicates an unlimited exponential growth of
organisms. At low concentration of substrate (K, >> c), the

expression reduces to a second-order reaction in which the

rate is proportional to the product of the concentration as in

aM _ 14
dt K (24)

This equation can also be written entirely in terms of micro-
bial mass by considering the bacterial equivalent of the
initial substrate concentration and letting My be the sum of

27


conversion of organic phosphorous to inorganic dissolved
phosphorous forms and an assumed sink of organic phosphorous
out of the estuary, itself due to sorption of turbidity parti-
cles and subsequent settling.

The equation for System #6 is then

d (Ny)
tv] —gB— = TAN) + IVI By + (122 )
ol pl
where Nou is the organic phosphorous and the source sink term

is given by

a
)o= = D2 + £(Z,P,G,) + a,D.P (123)

(s
pl c PP

N
~ Ree

where a is the ratio of phosphorous to chlorophyll and Keg
is the overall decay of organic phosphorous.

The inorganic dissolved phosphorous system receives the output
from the conversion of organic nitrogen and any direct sources
of waste as input. The primary sinks of the dissolved phospho-
rous are the uptake by the phytoplankton and sorption on sus-
pended material. The System #7 equation is

tv] ote) ) (124)
v = [A] (N,,) + [vl(S, ) + [v](s +W
at p2 Noa No2 Nyo

where N is the dissolved inorganic phosphorous and

p2

s

N GP (125)

p2 = Ke7Npr 7 82 SpP 7 K77Nyo

where K is the rate of production of dissolved inorganic

67


1969 SIMULATION

90% REMOVAL OF NITROGEN
90% REMOVAL OF PHOSPHOROUS

Solid Line, July 15 Profile.

175

Compare to Fig. 37

180}— COMPUTED BEFORE
REMOVAL
160}—
S 140}—
iow
+ !20-—-
a
> 100;-—-
x
S 80-—
5
J 60;-— ‘\
x NN
40}
Se N
20-— oBJECTIVE a S
~
; l | | | | |
[e) 5 ie} 15 20 25 30 35 40
MILES BELOW CHAIN BRIDGE
—— = June 30 Computed
—-——= July 15 Computed
Fig. 41 1969 Simulation of Chlorophyll. Dashed Line, June 30:



TABLES

No. Page
1 Nitrogen Balance Sheet for the Harvested Crop 19
Area of the United States, 1930
2 Estimated Municipal and Industrial Nitrogen 49
Discharges to Delaware Estuary
3 Summary of Reaction Coefficients Determined in 58
Verification Analysis of Nitrogen in Delaware
Estuary
4 Estimated Significant Input Nitrogen Loads, 67
Potomac Estuary
5 First Order Reaction Coefficients, Potomac Estuary 67
6 Maximum Growth Rates as a Function of Temperature 87
7 Michaelis Constants for Nitrogen and Phosphorus 97
8 Endogenous Respiration Rates of Phytoplankton 107
9 Grazing Rates of Zooplankton 109
10 Endogenous Respiration Rate of Zooplankton 117
11 Dry Weight Percentage of Carbon, Nitrogen, and 121

Phosphorus in Phytoplankton
12 Parameter Values for the Mossdale Model 136

13 Parameters Used in Verification of 1968 and 1969 164
Potomac River Data - Preliminary Phytoplankton Model

14 Direct Discharge Waste Loads 173


Both of these methods are second order methods, being accurate
to terms of order at? in a comparison of Taylor series ex-
pansions of the exact and approximate values, and both methods
require two derivative evaluations per step. The method of
Runge has been used in the calculations presented below.

The equations themselves are programmed for solution using

a continuous simulation language and a digital computer. The
language, in this case CSMP/1130, is based on a block diagram,
analog computer, representation of the differential equations.
The flexibility of these languages which allow changes in the
equation structure to be made easily is an asset in modeling
complex systems.

The biomass variables used in the calculations are total cell
counts for the phytoplankton and rotifer counts for the zoo-
plankton. The rotifer population represented the large major-
ity of the zooplankton present on a weight basis as well. In
order to relate these variables to comparable units, a series
of conversion factors have been used. The phytoplankton count
-chlorophyll concentration ratio was measured. However, the
carbon-chlorophyll or dry weight~-chloroplyll conversions are
unknown. Hence, the conversion to an organic carbon basis

is made rather arbitrarily. However, the carbon-to-chlorophyll
ratio which results (see Table 12) is within the range reported
in the literature. The same problem exists with the rotifer
counts to rotifer carbon conversion: the value used is given

in Table 12.

The comparison of the model output and the observed data for
the two-year period for which data are available is shown in
Figure 27. The parameter values used in the equations are
listed in Table 12.

135


SECTION VI

A PRELIMINARY MODEL OF PHYTOPLANKTON DYNAMICS
IN THE UPPER POTOMAC ESTUARY

The Potomac Estuary has been the subject of investigation

and analysis over many decades and most intensively within
the past several years, This work has been motivated in large
measure by the location of Washington D.C. along its shores
and by a long history of poor water quality. Also, if water
quality is to be improved nationally, the river flowing
through the Nation's capital must serve as a model situation.
Several Enforcement Conferences have been held on the water
quality of the Potomac River with the aim of establishing
required effluent controls to achieve specified water quality
objectives. Important recommendations which evolved from
these efforts included waste load allocations which restrict
the mass discharge of oxygen demanding material, and nitrogen
and phosphorus residuals. These recommendations relied to
some degree on the application of detailed mathematical
modeling of key water quality constituents such as dissolved
oxygen and various nitrogen and phosphorous forms.

The purpose of this section is to present a preliminary
model of the dynamic behavior of phytoplankton in the Upper
Potomac Estuary. This research extends the previous model-
ing efforts on the Potomac to incorporate explicitly the
space-time variability of chlorophyll a as a water quality
parameter indicative of a eutrophied environment. The
research also extends previous work in this report (see
Section V to an estuarine situation dominated presently
by municipal waste discharges. This preliminary phyto-
plankton model is intended therefore to shed further light
on the water quality changes that can be expected when a

243


effects. Therefore,

Ny Ny
G.. = G,,(I,T,£,H,K) aN
pj It Kan* tn Kup Ib

(112)
where Noy is the total inorganic nitrogen with Kan 25 Michaelis
constant, Ny is orthophosphate concentration with Kup as the

Michaelis constant. represents the functional relation-

G
ships of the growth rate and the solar radiation, I; photo-
period, f; water temperature, T; depth, H and light extinction
coefficient, K. The latter coefficient is incorporated as a
non-linear relationship with chlorophyll as

K = KE + .0088 P + .054 p’o?

(113)
where KE is the extinction coefficient (l1/meter) estimated at
zero phytoplankton concentration. The death rate, Dos is as

given in Eq. (74 ) and incorporates phytoplankton respiration

as a function of temperature and grazing by phytoplankton.

The governing equation for the zooplankton carbon system is

wi $@) = fay(z) + (vite, - D1 (2) (114)
where Z is an x 1 column vector of zooplankton carbon con-
centration. The expressions for G4 and Dey are identical
to those used in the earlier work. Actually, this system
did not play a role in the verifications discussed below.
Some sensitivity runs were made however to show the effect

of different levels of zooplankton grazing.

The organic nitrogen system includes non-living forms of
organic nitrogen generated by the model and includes sources
due to death of zooplankton and phytoplankton and grazed but
unassimilated phytoplankton nitrogen. The mass balance

149


po .
Deficit
CNg-N4

N, | ] Ny | Ny | |
t
00
Deficit
CNo-N3) ff

Pig.7 Sequential Reactions in Nitrification
- First Order Kinetics - Stream System

42


No.
48
49
50

51

FIGURES

Page
1969 Simulation Contour Plot 179
1969 Simulation Contour Plot 183

Temporal Variation in Chlorophyll a at Segments #9
and #28, Median Flow Simulation. a) lug/l Chlorophyll

Boundary b) 25yg/l Chlorophyll Boundary 186
Median Flow Simulation Profile of Chlorophyll - 188
July 15

ix


portant

a TE
jy = ap tt 1+ ——
u2
aK_E
u
- pas/re
Ja Ze a? (30)
Wy
Lyo *
o 4/1 + SKE
u2

where for jy and jg the positive sign of the radical is as-
sociated with the negative x direction and the negative sign
is associated with the positive x direction. The DO deficit
profile due to the nitrogenous waste discharge is thus given
by Eq. (28) The DO deficit due to the carbonaceous BOD
must be added to this profile to determine the total DO
deficit due to waste discharges.

The disadvantage of this simple model is that usually some
judgment must be exercised in determining the spatial dis-

tribution of the reaction rate, K In some instances, the

field data may indicate that for various reasons (organic
nitrogen hydrolysis, insufficient nitrifiers) an apparent

lag exists in the exertion of the NBOD. The judgment incorpo-
ration of this effect becomes particularly significant when
projections are made of expected water quality under different

treatment schemes.

However, it is still somewhat surprising that even this simple
model is often not considered in DO balance studies. This ap-
pears to be especially true in the numerous analyses conducted
to estimate the effects of different levels of waste treatment

36


the growth rate saturates for the significant growing periods,
the simplification introduced by ignoring the detritus is
probably acceptable.

The death rate of herbivorous zooplankton is thought to be
caused primarily by the same mechanisms that cause the death
of the phytoplankton, namely, endogeneous respiration and
predation by higher trophic levels. The endogeneous respira-
tion rate of zooplankton populations has been measured and
the results of some of these experiments are presented in
Figure 25 and Table 10.

It is clear from these measurements that the respiration rate
of zooplankters is temperature-dependent. It is also depend-
ent on the weight of the zooplankter in question and its life
cycle stage (Comita, 1968). As a first approximation, a
straight line dependence is adequate, and the endogeneous
respiration rate is given by the equation: respiration rate =
KT where K3 = 0.140.005 (dav ec) 7}, The conversion from the
reported units to a death rate is made by assuming that 50%

of the zooplankton dry weight represents the carbon weight

and that carbohydrate (CH,0) is being oxidized. The data are
somewhat variable and serve only to establish a range of values
within which the respiration rate of a natural zooplankton as-

sociation might be expected.

The death rate attributed to predation by the higher trophic
levels, specifically the carnivorous zooplankton, has been
considered by previous models in a more or less empirical way.
The complication resulting from another equation and the un-
certainty as to the mechanisms involved are quite severe at
this trophic level. In particular, it is probable that an
equation for organic detritus is necessary to describe ade-
quately the available food. Hence, it is expedient to break
the causal chain at this point and assume that the herbivorous

116


Nitrification

Nitrogenous matter in waste waters consists of proteins, urea,
ammonia and, in some cases, nitrate. The intermediate decom-
position products of the proteins such as amino acids, amides
and amines are also present in varying degrees. The proteins
are broken down by hydrolysis in a series of steps into a
variety of amino acids. Both exocellular and endocellular
enzymes are involved in the process. Aeration and alkaline
conditions favor the production of the exo-enzyme. The amino
acids are very soluble in water and exert a strong buffering
action; the carboxyl group reacting with the hydrogen ion and
the amino group reacting with the hydroxyl. The decomposition
of the amino acids which can occur in a number of different
ways, is endocellular. Ammonia is released in this process

of deamination, which may be reductive, oxidative or hydro-
lytic reaction, depending on the nature and structure of the
amino acids. In any case, the significant end product is
ammonia.

Ammonia is also released in the aerobic decomposition of pro-
teins by heterotrophic bacteria. The ammonia, which is highly
soluble, combines with the hydrogen ion to form the ammonium
ion, thus tending to raise the pH. In the neutral pH zone,
all of the ammonia is present in this form and at the higher
pH it is evolved as a gas. The heterotrophic decomposition is
the typical reaction in the first stage of the biochemical de-
oxygenation of natural waters. Ammonia is an end product of
both this reaction and the reaction associated with the hydro-
lytic breakdown of proteins. The ammonia present in natural
waters is thus a result of either the direct discharge of the
material in waste waters or of the decomposition of organic

matter in various forms.

15


CARBONACEOUS BOD
DISSOLVED OXYGEN

Carbonaceous BOD
Nitrogenous BOD
Dissolved Oxygen

Carbonaceous BOD
Organic Nitrogen, Ammonia

Nitrate, Dissolved Oxygen

Phytoplankton Biomass
Zooplankton Biomass

Total Inorganic Nitrogen

Phytoplankton Biomass
Zooplankton Biomass
Organic Nitrogen, Ammonia, Nitrate
Organic Phosphorus, Orthophosphate
Corbonaceous BOD
Dissolved Oxygen

Linear
Steady-state

t— Dissolved Oxygen
Models

Non-linear
Unsteady-state
[— Eutrophication

Models

Fig. 1 Evolution of the Modeling Structures

9



10

11
12
13
14
15

16

17

FIGURES

Page
Evolution of the Modeling Structures 9
Major Features of the Nitrogen Cvcle 13
Two-State BOD Curve - Mohawk River at St.Johnsville, 24
New York
variation of Growth Rate with Substrate Concentration 26
Logistic and Exponential Growth of Microorganisms
and Accompanying Nutrient Utilization 32
Block Diagram of Nitrification and Dissolved Oxygen
Utilization 39
Sequential Reactions in Nitrification - First Order
Kinetics ~ Stream System 42
Observed vs. Computed Nitrogen Profile, August 1964 51
Estimated Temperature Dependence of Nitrification
Reaction Rates 54
Observed vs. Computed Nitrogen Profiles, 55
November 1967
Observed vs. Computed Nitrogen Profiles, July 1967 56
Estimated DO Deficit Due to Nitrification 60
DO Deficits Under Different Nitrification Conditions 62
Verification of Kjeldahl Nitrogen for Potomac Estuary 68
Verification of Nitrite and Nitrate Nitrogen and
Chlorophyll "a" for Potomac Estuary 69
Computed Dissolved Oxygen Deficit Due to Nitrification
in Potomac Estuary, July - August, 1968 71
Interactions of Environmental Variables and the
Phytoplankton, Zooplankton, and Nutrient Systems 80

vi


phytoplankton biomass exists and has constant magnitude Po
For this situation, the equations become

ap

= -p! -
ae = (G, D » P CyP2 + Py (89)
dz _

ae = «(0,2 + aap CPZ (90)

The nontrival singular point for these equations is

_p
ote D, ate arpPo . (G, - D',) (91)
a,c! D Cc
zpg Z g

The perturbation analysis about this singular point yields a
second order linear ordinary differential equation whose
characteristic equation has the roots Ay and ro where

fa, PC

2
zpo g! _ - -p
5 D, | ar gPo (G, D p 2 (92)

PC
Lee tepro’g
1,2 2

Since for Po? 0, these roots have negative real parts, this
singular point is a stable focus, and the steady state values
given by Equation (91) are approached either by a damped sinu-
soid or an exponential (Davis, 1962). Note that for Po = 0,
the classical case, the roots are purely imaginary, and the
oscillation persists indefinitely.

This analysis suggests that the effect of transport into the
system stabilizes the behavior of the equations and in parti-
cular allows the solutions to achieve a constant solution.
This is in marked contrast to the behavior of the classical
Lotka-Volterra equations.

Another modification, which has been introduced into the zoo-
plankton equations, changes the behavior of the proposed

128


Las
=<¢
S
uJ
-
4
a
z .10
Q
q
oe
a
u
uy
&
05
n
2
oO
Zz
i
oO
fo)
2
itv 1)

1) 5 10 1S 20 25 30

TEMPERATURE °C

Fig. 23 Endogenous Respiration Rate of Phytoplankton
vs. Temperature; after Riley (1949)

106


a series of sampling stations in the Potomac where data were
collected over a period of months during 1967, 1968 and again
during 1969. Waste load information from the point waste dis-
charge locations as well as from land run-off has also been
obtained.

A steady state feedback model which considers organic ammonia
and nitrate nitrogen form was constructed. The assumption

of first order kinetics prevailed throughout. A feedback
loop is incorporated which represents ammonia and nitrate
nitrogen utilization by algae with the subsequent nitrogen
release upon death recycled to the organic nitrogen form.

The nitrification phenomenon was also inputted to the dis-
solved oxygen deficit system. A total of five systems was
therefore modeled.

The equations are:

1a aN, 1
O= 5 ax | FA ag | 7 x A(ON,)/dx - Ky,N) + KyiNy
o=i | za m2 - £ A(0N,)/ax - Ky5Ny + Ky Ny
o= 5 ay | BA = ~ F A(ON3)/Ax ~ Ky3Nq + KyyNy (50)
O= k& EA nt - 2 a (ony) /ax ~ KggNq + KygNo + N34N3
OF t& EA eo - % 4 (QD) /ax - K,D + Kj5N)
where Ny is organic nitrogen, Ny is ammonia nitrogen, Ny is

nitrite plus nitrate nitrogen, Ny is algal nitrogen, D is

the dissolved oxygen deficit and Kos = 4.57 K As in-

22 > Kia
ii + “ij
appears as KyiNg- a source term in the first equation which

32°

dicated previously, K for all i. The feedback loop

65


The value for erosion is equivalent to an average annual value
of almost 8 mg/l of nitrate as nitrogen from a drainage area
of 1 square mile with a flow of 1 cubic foot per second. This
value appears to be high for most natural waters but may be
present in rivers draining harvested lands during certain peri-
ods of the year.

Nitrogen leached from drained soil on which alfalfa and blue-
grass are grown in Kentucky can yield up to 10 pounds/acre
year and corn fields in Wisconsin have yielded in the range
of 20 and 40 pounds per acre year. Irrigated lands with di-
versified forms drained between 2 and 25 pounds of nitrogen

per acre year.

Fertilizers are significant sources of nitrogen in harvested
lands. The fertilizers are in forms such as ammonium sulfate
and nitrate, anhydrous ammonia and in various forms of nitrates
and phosphates, all of which are highly soluble. During peri-
ods of excessive runoff or when irrigation is in excess of
plant requirements, nitrogen in various forms is carried off

to either the ground or surface water. Barnyard and silo
drainage, as well as wash waters and effluents from feedlot

and dairy operations, undoubtedly add nitrogen to water courses.

A recent report described the nitrogen enrichment of surface
waters by absorption of ammonia which was volatilized from
cattle feedlots (Hutchinson, 1969). Laboratory studies in-
dicate that as much as 90% of urine nitrogen may escape as
ammonia from feedyards. A significant quantity of this ni-
trogen may be absorbed by water surfaces within a few kilo-
meters from the feedlots (between 10 - 20 kq/ha-yr). A value
of 4 kg/ha-yr was measured at a site in which there were no
irrigated fields or feedlots within 3 km and no large lots or
cities within 15 km. On the other hand, absorption rates of

20


= - Q -
ae = (G,- D2 + % (2, - 2) (106)

where Po and Zo are the concentrations of phytoplankton and
zooplankton carbon in the influent flow, Q. A more useful
form for these equations is

dp

qe)
as)

= - Q Q
a = [c, , + PIL P+ ee, (107)
dZ _ . Q Q
ac {[G, (Dy + 9] 2+ % 4, (108)

A complete analysis of the properties of these equations is
quite difficult since the coefficients of P and Z are time
variables and also functions of P and 2. However, the be-
havior of the solution becomes more accessible if the
variation of these coefficients is studied as a function of
time. The expressions Ss - (D, + Q/V) and G, - (D, + Q/V)

can be considered the net growth rates for phytoplankton and
zooplankton. The advective or flushing rate, Q/V, is included
in these expressions since it acts as a death rate in one
segment system.

The sign and magnitude of the net growth rate controls the
behavior of the solution. For a linear equation, for which
the net growth rate is not a function of the dependent varia-
ble (i.e., P or Z), the type of solution obtained depends on
the sign and magnitude of the net growth rate. That is, for
the equation

= Q
ge 7 OP + GPS (109)

where a, Q, and V are constant, the solution is

139


SECTION III

INTRODUCTION

As the nation moves forward in its program for water pollution
control and water quality management, the need for both the
short and long term operation and management of water resource
systems becomes ever more important. Complex problems regard-
ing subtle interactions between waste treatment processes and
the environment must be considered. These problems can be
conceptualized and formulated using the techniques of dynamic
systems analyses. However the general theory must be specifi-
cally adapted to these problems. A whole expertise must be
developed which enables a translation of the problems of water
quality maintenance and prediction and water resources manage-
ment into the abstract formulations of dynamic systems analyses.

As is well known, a sequence of profound biological changes
occurs in a natural body of water receiving untreated waste
water. What is not so well known, however, is the chain of
events that is set into motion as a result of discharging
biologically treated, nutrient-rich, effluents. When the
waste is untreated, the bacterial populations predominate.
These are of the form which oxidize organic carbon in their
metabolic processes. When this material is removed in a bio-
logical treatment unit, the next component of the cycle, ni-~
trification, becomes more significant. If the treatment
processes are designed to allow the nitrifying bacteria to
develop in the plant, then an end product of relatively stable
nitrates results. If nitrification does not occur during
treatment, or is only partially completed, the remaining nitro~
gen is discharged. Both the residual carbonaceous material
and the nitrogenous material exert a separate and distinct
oxygen demand on the water bodies resources.


PHYTOPLANKTON

ZOOPLANKTON

TOTAL INORGANIC

100,000 ;

i
80,000 +
6Q,000F

40,0001

CELL /ML.

{
20,000;

2
Q
oO
{o)
T

NO/L
co
°
to}
Lo)

NITROGEN MG/L

h i
120 240 330 120
1966 TIME - CANS

Fig. 27 Phytoplankton, Zooplankton, and Total Inorganic Nitrogen;
Comparison of Theoretical Calculations and Observed Data;
San Joaquin River, Mossdale, 19666-1967

- 137


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In unlimited growth, the solution of Eq. (g) indicates that
the microbial population would grow at an exponential rate.
If however, a single substrate (say nitrogen) plays an im-
portant role in limiting the growth, one must also consider
the utilization of the substrate and consider that the growth
coefficient, K, in Eq. (8) is not a constant but depends on

a substrate concentration. Thus, let

Kae
K = Ke (9)
Ss

where Ky is the maximum growth rate (l/day), c is the concen-
tration of substrate (mg/l) and kK, (the so-called Michaelis
or half-saturation constant) is the concentration (mg/l) at
which the growth rate is one-half of the saturated rate.
Fig.4 is a sketch of this functional form.

Substitution of Eq. (9) into (g) gives:

K_M

= amt (10)
K_+¢c
s

al

This form of kinetic equation was first developed by Michaelis
and Menton to explain enzymatic reactions. It was later ap-
plied by Monod (1942) to systems involving growth of biological
organisms. This equation can be written in terms of micro-
organisms only by substituting for the substrate concentration
its equivalence in terms of microorganism i.e a unit increase
of organisms is equal to a unit decrease in substrate multi-
plied by the appropriate stochiometric or yield coefficient.
This is simply an expression of the conservation law, in which
the total mass, M, is equal to the sum of the initial organisms
present plus the substrate utilized to produce microbes, i.e:

M = My + a(c,-e) (11)

25


grazing, the utilization of nitrate is decreased and nitrate
approximately behaves as a conservative variable responding
only to changes in the flow regime while being fed by the ni-
trification system. As a consequence, No, values build up
to 3 mg/l under extensive zooplankton grazing - considerably
greater than observed (see Figure 33b). With no zooplankton
grazing, phytoplankton populations increase and therefore
utilize nitrate nitrogen and reduce the values to the 1 -

1.5 mg/l level. This is the range observed during 1968. All
of these runs indicate that even with minimal grazing rates
of the zooplankton on the phytoplankton, the phytoplankton
population is never computed to grow to the levels observed
in 1968. It was concluded therefore that the predatory effect
of zooplankton on the phytoplankton was minimal at least for
the upper end of the estuary below Washington D.C.

A sensitivity run was also made to indicate the effects of
including the tidal bay segments. The results are shown in
Figure 36. Observed chlorophyll concentration in the main
channel appear to be significantly influenced by the growth
of phytoplankton in the shallower side~-channel areas. With
average depths of about 5 feet in some of the tidal bay seg-
ments, significant populations are computed to grow in these
areas. These populations are then tidally exchanged with the
main channel flow and contribute to the population at that
location. It is hypothesized then that the growth of phyto-
plankton in the shallow areas is significant (even without
direct discharges to embayments) and can account for as much
as 40 ug/l chlorophyll in the observed main channel data.

Verification Analysis - 1969 Data

The parameters determined from the 1968 verification analysis
are listed in Table 13. This set of parameters was then used

162


il

Bh 02

z8T
SAIOAIS NOIINIIWd 3937709 NULIBHNEW tE9S

335.

WW
= 0.)
=

201.

E&

\

134.

87.

AMMONLA NITROGEN NH3-N

uu

SIMULATION NO. 3

i
10.

|
tu.

(MG/L)

I
Lg. 23.

SEGMENT NUMBER

Pig.

48

1969 Simulation Contour Plot

Rue ra 700oD

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Borchardt (1969)
h(P., = 0.17 P. 66
3) j (66)

for h in nt and Por the phytoplankton concentration is
mg/liter of dry weight. This expression gives values com-
parable with Equation (65) for a reasonable conversion factor
for the units involved.

To summarize the analysis to this point, the saturated growth
rate K' has been estimated from available data and its tem-
perature dependence established. The reduction to be ex-
pected from nonoptimum light intensities has been quantified
and used to calculate the reduction in growth rate, yr to
be expected in each volume element vy as a function of the
extinction coefficient and the depth of the segment. The
mechanism of self-shading has been included by specifying
the chlorophyll dependence of the extinction coefficient. It
remains to evaluate the effect of nutrients on the growth
rate.

The effects of various nutrient concentrations on the jrowti
of phytoplankton has been investigated and the results are
quite complex. As a first approximation to the effect of
nutrient corcentration on the growth rate, it is assumed
that the phytoplankton population in question follow Monod
growth kinetics with respect to the important nutrients.
That is, at an adequate level of substrate concentration,
the growth rate proceeds at the saturated rate for the tem-
perature and light conditions present. However, at low
substrate concentration, the growth rate becomes linearly
proportional to substrate concentration. Thus, for a nu-

trient with concentration Ny in the yuh segment, the factor

94


Table 12

Parameter Values for the Mossdale Model

Notation Description Parameter Value
K, Saturated growth rate 0.1 day! °c
of phytoplankton
I, Light saturation intensity 300 ly/day
for phytoplankton
Lan Extinction coefficient 4.0m?
H Depth 1.2m
Kh Michaelis constant for total 0.025 mg N/liter
inorganic nitrogen
£ Photoperiod 0.5+0.11 sin[0.0172(t-165) ]day
K, Endogenous respiration rate 0.005 day + oct
of phytoplankton
Cy Zooplankton grazing rate 0.13 liter/mg - C - day
Py Influent phytoplankton 5.0 ug Chl/liter
chlorophyll concentration
ay Zooplankton conversion 0.6 mg C/mg - C
P efficiency
Kn Phytoplankton Michaelis 60 upg Chl/liter
P constant
Dy Zooplankton death rate 0.075 day
2 Influent zooplankton carbon 0.05 mg C/liter
concentration
n Phytoplankton nitrogen- 0.17 mg N/mg - C
P carbon ratio
C/Cch1 Phytoplankton carbon to total 50 mg C/mg-Chl
chlorophyll ratio
No Influent total inorganic 0.1 mg N/liter
nitrogen concentration
Ww Direct discharge rate of 12500 lbs/day
nitrogen
v Segment volume 9.7 x 10° tv?
Phytoplankton total cell 100 cells/ml =
count/phytoplankton 1.75 ug Chl/liter
Zooplankton count/zooplankton 104 No./liter =
carbon ratio 1.30 mg C liter

136


Table 6

Maximum Growth Rates

As_a Function of Temperature

Saturated
Growth Rate,
K'
Temper- “1
Reference Organism ature (Base _,Da —)
Tamiya et Chlorella ellipsoidea 25 3.14
al, 1964 (green alga) 15 1.2
Yentsch Nannochloris atomus 20 2.16
1966 (marine flagellate) 10 1.54
Spencer Nitzschia closterium 27 1.75
1954 (marine diatom) 19 1.55
15.5 1.19
10 0.67
Riley Natural association 4 0.63
1949b 2.6 0.51
Myers Chlorella pyrenoidosa 25 1.96
1964
" Scenedesmus quadricauda 25 2.02
Sorokin Chlorella pyrenoidosa 25 2.15
1958 :
" Chlorella vulgaris 25 1.8
" Scenedesmus obliquus 25 1.52
" Chlamydomonas reinhardti 25 2.64
Sorokin Chlorella pyrenoidosa 10 0.2
1962 (synchronized culture) 15 1.1
(high-temperature strain) 20 2-4
25 9

87


The data for 1968 were used to tune the model and the 1969
data were then used for independent verifications using coef-
ficients from 1968 data and changing only flow, temperature
and incoming boundary conditions for the 1969 verification.
Late fall blooms of phytoplankton were not completely verified
and a late winter bloom was not verified. Phytoplankton
species discrimination is not incorporated in the model.

The transport through the estuary assumes an important role

in the phytoplankton dynamics of the upper Potomac Estuary.

The nutrients and phytoplankton associated with the incoming
river flow are particularly important. Unlike other problem
contexts such as dissolved oxygen, critical conditions for
phytoplankton growth may occur during non-drought flow con-
ditions. This is due primarily to the addition of nutrients
during high flows. The verification analyses indicate that,

at present, phosphorous and nitrogen are probably sorbed

from the estuary and enter the estuarine sediments. The
preliminary model does not incorporate any recycle of nutrients
from the sediments although the model does indicate that phos-
phorous and nitrogen are not conserved in the water column and
losses to the sediment can be considerable. These losses could
alter the effect of incoming river concentrations.

Simulations with the preliminary model indicate that under non-
drought flows and a 90% reduction of untreated raw nutrient
loads, chlorophyll concentrations in the main channel may rise
to about 50 ug/l with embayment values over 70 ug/l. These
concentrations are considerably above the objective of 25 ug/l
but are about 60% less than present levels. The simulations

are intended only to show possible trends under future loading
conditions and are not to be interpreted in any detailed quanti-
tative fashion. Models of the type presented herein require
additional evaluation and testing on a continuing basis.

189


Simultaneously, the photosynthetic and respiratory processes
of the algae are operative. If nitrogen is not removed in
treatment, the environment is more conducive to the growth
of algae, which then proliferate. When the algal blooms ex-
ceed the capacity of the system, mass mortality may occur.
Algal decay occurs and carbonaceous and nitrogenous demand
is returned to the water body thereby initiating the cycle
again. It is therefore necessary to proceed in a sequential
manner so that these important secondary ecological effects
are incorporated both in the analysis and planning phases.

Efficient dynamic regional management of these systems there-
fore requires a foundation of analytical tools and techniques
that will buttress the solution of the very complex problems
associated with short and long term water pollution control.

The basis for the methods presented herein is the principle
of conservation of mass. It can be expressed in mathematical
terms as a partial differential equation which related the
concentration of a substance c(r,t) at a position Y and time
t to the effects of mass transport, which are described by

a velocity vector field B(r,t) and a dispersion matrix E(r,t);
the effects of kinetic transformations, which are described
in terms of sources and sinks of the substance S(c,r,t); and
the effects of direct discharges to the water body W(r,t).

The requirement that the mass changes are accountable in these
terms requires that:

+ V4 [-Elr,t)Vo + U(r,t)c] = S(c,r,t) + W(r,t) (1)

ale

where V = 0/0x T+ 3/dy z + 9/dz . Given the mass transport
and kinetic descriptions and for specified direct discharges
and boundary conditions this equation is solvable,in principle,
for the resulting concentration distribution c(r,t) as a


The reduction factor ry is a function of the extinction co-
efficient Key of the volume V. However, the extinction
coefficient is a function of the phytoplankton concentration
present if their concentration is large. Thus, an important
feedback mechanism exists which can have a marked effect on
the growth rate of phytoplankton. As the concentration of
phytoplankton in a volume element increases, the extinction
coefficient, particularly at the green wavelengths, starts
to increase. This mechanism is called self-shading. The
most straightforward approach to including this effect into
the growth rate expression is to specify the extinction co-
efficient as a function of the phytoplankton concentration

key = Ka; + hips) (64)

where Ka is the extinction coefficient attributable to

other causes and key includes the phytoplankton's contribu-
tion. The function hips) has been investigated by Riley (1956),
who found that it can be approximated by

2/3

h(P.) = 0.0088 P. + 0.054 P.
(P;) = 00088 P, + 0.054 P. (65)

where Ps has the units yg/liter chlorophyll, concentration
and h has units mt, A more recent investigation (Small1,1968)
shows that this relationship applies to coastal waters of
Oregon for a range in chlorophyll, concentration of from 0

to 5.0 mg ch/m.

A theoretical basis for this relationship is the Beer-Lambert
law, which expresses the extinction coefficient in terms of
the concentration of light-absorbing material. For dense
algal cultures, this law has been experimentally verified
(Oswald, et al, 1953). A similar relationship based on this
law has been proposed by Chen (1970) from the data of Azad and

93


€8T

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67

3OTd ANOQUOD UOTRIeTNUTS 696T

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S(MULATION NO. 3
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SEGMENT NUMBER

(MG/L)

11 CONTOUR LEVELS

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.91000£+00
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rated growth rate concentration is in the range of 50-100
ug Si/liter (Stickland 1965).

There are a large number of trace inorganic elements which
have been implicated in the growth processes of algae, among
which are iron [for which a Michaelis constant of 5 ug/liter
for reactive iron has been reported by Lake Tahoe Area Council
(Anon, May 1969)],, manganese, calcium, magnesium, and po-
tassium (Lund, 1965). However, the significance of these
elements in the growth of phytoplankton in natural waters

is still unclear. Trace organic nutrients have also been
shown to be necessary for most species of algae: 80% of the
strains studied require vitamin Bio: 53% require thiamine,
and 10% require biotin (Droop, 1962). Presumably, these nu-
trients are available in sufficient quantities in natural
waters so that their concentration does not appreciably af-
fect the growth rate.

In the preceding discussion of nutrient influences on the
growth rate, it is tacitly assumed that only one nutrient is
in short supply and all the other nutrients are plentiful.
This is sometimes the case in a natural body of water. How-
ever, it is also possible that more than one nutrient is in
short supply. The most straightforward approach to formulat-
ing the growth rate reduction caused by a shortage of more
than one nutrient is to multiply the saturated growth rate

by the reduction factor for each nutrient. This approach has
also been suggested by Chen (1970). As an example, the data
from Ketchum (1939) for the rate of phosphate absorption as a
function of both phosphate and nitrate concentration can be
satisfactorily fit with a product of two Michaelis-Menton
expressions. The resulting fit, obtained by a multiple non-
linear regression analysis, is shown in Figure 21. The
Michaelis constants that result are 28.4 ug NO,-N/liter and

98


62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

Riley, G.A. "Factors Controlling Phytoplankton Populations
on Georges Bank" J. Marine Res., 1946, 6(1) 54-73.

Riley, G.A. "A Theoretical Analysis of the Zooplankton
Population of Georges Bank" J. Marine Res., 1947, 6(2),
104-25,

Riley, G.A., Von Arx, R. "Theoretical Analysis of Seasonal
Changes in the Phytoplankton of Husan Harbor, Korea,
J. Marine Res., 1949, 8 (1) 60-72.

Riley, G.A., Stommel, H., Bumpus, D.F. "Quantitative
Ecology of the Plankton of the Western North Atlantic"
Bull. Bingham Oceanog. Coll, 1949, 12(3) 1-169.

Riley, G.A. "Oceanography of Long Island Sound 1952-1954
II. Physical Oceanography" Bull. Bingham Oceanog. Coll.
1956, 15 15-46.

Riley, G.A. "The Sea" Theory of Food-Chain Relations in
the Ocean, p. 438-63, M. N. Hill, Ed., Interscience, N.Y.
1963.

Riley, G.A. "Mathematical Model of Regional Variations in
Plankton" Limnol. Oceanog, 1965, 10 (Suppl) R202-R215.

Ryther, J.H. "Inhibitory Effects of Phytoplankton upon the
Feeding of Daphnia Magna with Reference to Growth, Repro-
duction and Survival" Ecology, 1954, 35, 522-33.

Ryther, J.H. "Phytosynthesis in the Ocean as a Function of
Light Intensity" Limnol. Oceanog, 1956, 1, 61-70.

Sawyer, C.N. "Chemistry for Sanitary Engineers" McGraw-
Hill Bk. Co., New York, 1960, 367 pp + Viii.

Shelef, G., Oswald, W.J., McGauhey, P.H. “Algal Reactor
for Life Support Systems" J. SED, ASCE, No. SAl, Feb.1970.

Small, L.F., Curl, H. Jr. "The Relative Contribution of
Particulate Chlorophyll to the Extinction of Light off
the Coast of Oregon" Limnol. Oceanog., 1968, 13(1), 84.

Sorokin, C., Krauss, R.W. "The Effects of Light Intensity
on the Growth Rates of Green Algae" Plant Physiol. 1958,
33, 109-13.

Sorokin, C., Krauss,R.W. "Effects of Temperatures and Illu-

minance on Chlorella Growth Uncoupled from Cell Division"
Plant Physiol, 1962, 37, 37-42.

198


CHLOROPHYLL — yg/I

CHLOROPHYLL — #g/I

120};— (0)
|yg/l CHLOROPHYLL ENTERING ESTUARY
100 }-—
80 +—
Segment No.28
(Piscataway
60;— Embayment) —o-- ee
oa se
/ \
40}— / \
t XN
Objective t \
‘
20;—- 7 Segment No. \
7 \
0 =e a ae | oj J

i 60 90 120 150 180 210 240 270 300 330 360
van. ne Feb. | Mar. | Apr. | May | June | July | Aug. | Sept. | Oct. | Nov. | Dec. |

TIME OF YEAR

120}-— (b)
25u9/! CHLOROPHYLL ENTERING ESTUARY
100 -- Segment No.28
(Piscataway
Embayment)
80;
nen ~sL
— 7 N
60 a Segment No.9> _ ~~
“a \
40}— ”
a
----7 Objective
20
oL Lt tb i tt

fo 30 60 90 120 150 180 210 240 270 300 330 360
| Jan. | Feb.| Mar. | Apr. | May | vune | July | Aug. | Sept. | Oct. | Nov. | Dec. |

TIME OF YEAR

Da - re

Fig. 51 Median Flow Simulation Profile of Chlorophyll
July 15

188



1947). Thus, utilization efficiency is a function of the

phytoplankton concentration. A convenient choice for this
functional relationship is a,_K K_ + P.) so that the
u Pe zp imp’ ( ‘mp 3)

growth rate for the zooplankton population is

(76)

The resulting growth rate has the same form as that postulated
for the nutrient-phytoplankton relationship, namely, a Michaelis-
Menton expression with respect to phytoplankton biomass. In
fact, the argument which is used to justify its use in Equation
(67) can be repeated in this context. The difference is that

in this case the substrate or nutrient is phytoplankton biomass,
and the microbes are the zooplankton. The Michaelis constant

Kap is the phytoplankton biomass concentration at which the
growth rate S25 is one-half the maximum possible growth rate

aap gkmp* The fact that at high phytoplankton concentrations
the zooplankton growth rate saturates was incorporated by Riley
(1947) in the first model proposed for a zooplankton population.

The assimilation efficiency of the zooplankton at low phyto-
plankton concentrations, agp! which is the ratio of phyto-
plankton organic carbon utilized to zooplankton organic carbon
produced has been estimated by Conover (1966) for a mixed zoo-
plankton population. The results of 26 experiments gave an
average of 63% and a standard deviation of 20%. Other reported
values are within this range. Experimental values for Kap!
which in effect set the maximum growth rate of zooplankton,
are not available and would probably be highly species-
dependent. Perhaps a more effective way of estimating Knap
is first to estimate the maximum growth rate at saturating
phytoplankton concentrations, arp gkmp! and then calculate

Kp * Growth rates for copepods through their life cycle

114


from mile 100 - 85. This inhibition was ascribed to low dis-
solved oxygen conditions in that reach. If, following imple-
mentation of the waste control program, dissolved oxygen
conditions were at a higher level (say greater than 2 mg/l
everywhere), it is informative to explore the resulting effect
of nitrification on dissolved oxygen. Two possibilities exist
under improved dissolved oxygen:

a) ammonia oxidation will take place throughout the entire
length of the estuary at approximately a rate of
0.1/day. This will result in a shift of the maximum
dissolved oxygen deficit upstream.

b) because of generally improved water quality, algal
utilization of ammonia may now increase throughout
the length of the estuary. Since many algal species
utilize ammonia preferentially, the ammonia would be
tied up in organic form, and not contribute to the
deficit until some time later in the year. The rate
of this phenomenon is unknown.

Both effects will probably proceed simultaneously. However,
in order to provide a somewhat conservative estimate, it can
be assumed that all the ammonia will be oxidized and will
contribute to the dissolved oxygen deficit. Under this as-
sumption, model runs were made using ammonia oxidation rates
of 0.11/day everywhere, and the dissolved oxygen deficit was
computed. The results are shown in Figure j3,

As shown, under favorable nitrification conditions, with am-
monia oxidation proceeding uniformly, the maximum dissolved
oxygen deficit shifts upstream to about mile 90. There is a
decrease of about 0.2 mg/l in the maximum, and a general
spreading over a larger area. At mile 90, the deficit in-
creases from about 0.5 mg/l to about 2.2 mg/l under favorable
nitrification conditions. On the other hand, at mile 75, the

61


loads are summarized in Table 4 below.

Table 4
Estimated Significant Input Nitrogen Loads
Potomac Estuary
July-August, 1968

Ult.Carb.BOD org. N. NH3-N
lbs/dav lbs/day lbs/day

Arlington, Va. 5,900 1,300 2,100
Washington D.C. 132,000 20,000 20,000
Alexandria 11,700 1,300 2,300
Ft. Westgate 20,800 1,400 1,100

Figures 14 and 15 show the results of a verification analysis
of data collected during July-August, 1968. The first order
reaction coefficients for the verification analvses shown in
these figures are given in Table 5.

Table 5
First Order Reaction Coefficients
Potomac Estuary
July-August, 1968
Temp. = 28°C

Reaction
Reaction Step Symbol Coef. (1/dav)
Decay of Organic - NH Ky 0.2
Organic Nit. + NH -Nit. Kia 0.21
Decay of NH3 ~ Nit. K52 0.30
NH3 Nit. > NO - Nit. Ko3 6.28
NH Nit. + Algal Nit. Ko4 0.02
Decay of NO, - Nit. K33 0.10
NO, Nit. > Algal Nit. K54 0.10
Decay of Algal Nit. Ky4 0.12
Algal Nit. + Organic Nit. Kay 0.12

67


49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

McMahon, J.W., Rigler, F.H. "Feeding Rate of Daphnia
Magna Straus in Different Foods Labeled with Radioactive
Phosphorus" Limnol. Oceanog., 1965, 10 (1) 105-13.

Monod, J. "Recherches sur la Croissance des Cultures
Bacteriennes" Hermann, Paris, 1942.

Mullin, M.M. "Some Factors Affecting the Feeding of Marine
Copepods of the Genus Calanus" Limnol. Oceanog., 1963, 8
(2), 239-50.

Myers, J. “Algal Cuiture from Laboratory to Pilot Plant"
Growth Characteristics of Algae in Relation to the Problem
of Moss Culture, p. 37-54, J.S. Burlew, Ed., Carnegie
Inst. of Wash. D.C, Publ. 600, 1964.

O'Connor, D.J., Thomann, R.V., “Stream Modeling for Pollu-
tion Control" Proc. IBM Sci. Computing Symp. Environ. Sci.,
Nov. 14-16, 1966, p. 269.

O'Connor, D.J. "Water Quality Analysis of the Mohawk River-
Barge Canal" Prepared in coop. with Hydroscience, Inc, for
N.Y. State Dept. of Health, July 1968.

O'Connor, D.J., St. John, J.P. and Di Toro, D.M. "Water
Quality Analysis of the Delaware River Estuary, J.SED, ASCE,
No. SA6, Dec. 1969, pp 1225-1252.

O'Connor, D.J., Di Toro, D.M., Thomann, R.V., Mancini, J.L,
"Phytoplankton Population Model of the Sacramento - San
Joaquin Delta Bay" Tech. Rept. to Dept. of Water Resources,
State of Calif. by Hydroscience Inc., Westwood, N.J. 1971.

Oswald, W.J., Gotaas, H.B., Ludwig, H.F., Lynch, V.
“Photosynthetic Oxygenation" Sewage Ind. Wastes, 1953, 25
(6), 692.

Parker R.A. "Simulation of an Aquatic Ecosystem" Biometrics
1968, 24(4) 803-22.

Pearson, E.A. "Kinetics of Biological Treatment" Advances
in Water Quality Improvement, pp. 381-394, E. Gloyna and
W.W. Eckenfelder, Ed., Univ. of Texas Press, Austin, 1968.

Phelps, E.B., Stream Sanitation, pp. 148-49, J. Wiley & Sons,
New York, N.Y. 1944,

Raymont, J.E.G. "Plankton and Productivity in the Oceans"
pp 93-466, Pergamon, N.Y. 1963.

197


balance of natural wastes is stressed. This review is follow-
ed by a critical review of the reaction kinetics that actually
prevail in nitrification.

Figure 2 schematically outlines the major features of the ni-
trogen cycle that are of importance. It is appropriate to
initiate the cycle at the point where organic nitrogen
(amines, nitriles, proteins) and ammonia resulting from mu-
nicipal and industrial waters are discharged into a water
body. The organic nitrogen undergoes an hydrolytic reaction
producing ammonia as one of the end products, which in addi-
tion to the ammonia present in the waste waters, provides a
food source for the nitrifying bacteria. The oxidation pro-
cess proceeds sequentially from ammonia through nitrite to
nitrate. The conditions under which these reactions proceed
are comparatively restrictive, but, if present, they provide
an appropriate environment which may result in large deple-
tions of dissolved oxygen.

The forward sequential reactions of the nitrification process
often are the dominating features of the nitrogen cycle in
bodies of water receiving large discharges of nitrogenous
waste material. The reactions proceed in the forward di-~
rection, provided the concentration of dissolved oxygen is
sufficiently high to meet the requirements of the nitrifying
bacteria. However, under conditions of low concentration of
dissolved oxygen, bacterial reduction of nitrate to nitrite
can occur followed by the further reduction of nitrite pri-
marily to nitrogen gas, although a few species may reduce the
nitrite to ammonia. These reactions provide a source of oxygen
for the microorganisms in the stabilization of organic matter
without utilizing whatever dissolved oxygen is present in the
water.

12


Only summary plots of observed data were available for veri-
fication of the 1968 data. These plots are given in Jaworski
et al (1969). Figures 32 and 33 show a variety of results
from the validation analyses of the May to November 1968 data.
As indicated previously, the system equations are all time
variable; the results in Figures 32 and 33 show the average
spatial profile during August 1968, a period of low flow just
prior to the late summer flow transient that year.

The three curves of model output in Figures 32 and 33 repre~
sent three conditions on the nutrients; nutrients are totally
conserved in the model and nutrients are decayed out of the
system at rates of .05/day and 0.1/day. As shown, the agree-
ment is good when the organic nitrogen and total phosphorous
are allowed to decay at rates of .05/day or 0.1/day but there
is poor agreement when the nutrients are totally conserved

in the system. This illustrates one of the steps in the
"tuning" of the model using the 1968 data. It was obvious
from the early runs of the model where the nutrients were al-
lowed to recycle entirely within the model that the results
were not desirable below about Mile Point 15. The observed
data on all variables shown in Figures 32 and 33 decreased
more rapidly than the computations. Accordingly, the hypo-
thesis was adopted that incoming sediment load permitted
sorption of phosphorous and nitrogen with subsequent settling
to the bottom of the estuary and out of the domain of the
modeling framework. This hypothesis is consistent with that
given by Jaworski et al (1971) in determining phosphorous
balances in the estuary.

In addition, the determination of model parameters using the
1968 data indicated that zooplankton grazing of the phyto-
plankton was probably not a significant factor in the Potomac.
This is supported by the fact that the phytoplankton of concern

156


by which the saturated growth rate is in the 3th segment re-
duced is: Nj/ (K, + Nj). The constant, Kae which is called

the Michaelis or half saturation constant, is the nutrient
concentration at which the growth rate is half the saturated
growth rate. There exists an increasing body of experimental
evidence to support the use of this functional form for the
dependence of the growth rate on the concentration of either
phosphate (Dugdale, 1967), nitrate, or ammonia (Eppley, 1969)
if only one of these nutrients is in short supplv. An example
of this behavior, using the data from Ketchum (1939) is shown
in Figure 20a for the nitrate uptake rate as a function of ni-
trate concentration and in Figure 20b for the phosphate uptake
as a function of phosphate concentration. These results are
from batch experiments. Similar results from chemostat experi-
ments, which seem to be more suitable but more lengthy for this
type of analysis, have also been obtained. Table 7 is a list-
ing of measured and estimated Michaelis constants for ammonia,
nitrate, and phosphate. The estimates are obtained by taking
one-third the reported saturation concentration of the nutrients.
These measurements and estimates indicate that the Michaelis
constant for phosphorus is approximately 10 ug P/liter and for
inorganic nitrogen forms in the range from 1.0 to 100 ug N/
liter, depending on the species and its previous history.

The data on the effects of the concentration of other inorganic
nutrients on the growth rate is less complete. Since algae

use carbon dioxide as their carbon source during photosynthesis,
this is clearly a nutrient which can reduce the growth rate at
low concentrations (Kuentzel, 1969). Reported saturation con-
centration for Chlorella is < 0.1% atm (Myers, 1964).

The silicate concentration is a factor in the growth rate of
diatoms for which it is an essential requirement. The satu-

95


carbon, the optimum pH range being 8.5 - 9.5. The alkaline
environment is required to neutralize the acidic end-products.
Below pH of 6.0, which can occur in a poorly buffered system,
inhibition occurs. The presence of organic matter, particu-
larly amino compounds in excessive concentrations, inhibits
growth and respiration of the nitrifying bacteria. These
excessive concentrations in the order of thousands of mg/l
are seldom encountered in either waste water or natural waters.
However, even concentrations of organics in the order of hun-
dreds of mg/l which may be found in practical cases, appear
to still retard the nitrification process. At these concentra-
tions the heterotrophic bacteria probably predominate and
assimilate the ammonia in their metabolic processes. After
the death and lysis of these bacteria, nitrification takes
place. The experimental evidence in this regard is not con-
clusive and, in fact, is somewhat contradictory. While the
majority of reported work indicates some degree of inhibiting
effects, other reports show minimal or no retardation. At
lower concentrations of organic materials, as may be found

in natural water bodies, both heterotrophic and autotrophic
reactions may occur simultaneously.

Given the appropriate conditions, the concentration of the
organisms appears to be the significant factor controlling

the rate of nitrification, with the concentration of the re-
actant having a reduced effect. However, if there is an ample
supply of organisms, the rate appears to be controlled by the
concentration of the reactant (see also - Kinetic Models).

The number of nitrifiers is, of course, determined by the
generation time of organisms, which is in the order of one
day, by contrast to an order of a few hours for many hetero-
trophic bacteria.

The common source of the nitrifying organisms is rich soil,

17


where AS is the interfacial cross sectional area and Ly
and L, are segment lengths of segments k and j; and Wik
is the direct discharge of waste material, N)- The notation
Nk indicates the spatial distribution in all k seqments of
the water body of the first nitrogen form. If in Eq. (37)
all terms involving the dependent variables Ny are grouped
on the left hand side and the invout forcing function, Wik
and other nitrogen forms, Ni on the right hand side, one

obtains:

aN + t aM 5 = Ma + MaKor Nas te MKaa Nak (43)
=fT ’
where Akk ~ (55 + Bas) + YK x (43a)
- ep
a5 = 5 Bg - Eley (43b)

A total of n equations similar to Eq. (43) can be written
down for each of the spatial segments. The svstem of equa-

tions is then

SS Rs ies se RL SS Os A SL

= Ww. +V5K +...V5K

Fee 2%o1 ,2%2,2 2Ka1Nq,2

3o1%1 ,17822%) 2 an§i jn = “1,2

Any tn 2%) ot oe anny nm +

Want Mp¥21 n%2,n Va Nan

In matrix form, Eas.(44) are
{Ay} (Ny) = (Wy) + EVK9) 1 (N,) + [VK )70N,) + [VK 1 (Ny) (44)
where the subscripts refer to the nitrogen forms and (VK;

«J
J
is a diagonal matrix of the product of volumes and reaction

46


water where v5 is located must be evaluated. The rate at
which light is attenuated with respect to depth is given

by the extinction coefficient, k.: That is, at a depth z,
the intensity at that depth, I(z), is related to the surface
intensity, Igor by the formula

I(z) = I, exp (-k,2) (58)

where z = O is the water surface and z is vositive downward.
Thus, the reduction of the saturated growth rate at any depth
z resulting from the nonoptimum light intensity present is

given by Equation (58), substituted into Equation (56).

~k,z - ~k,2
I,e -I,e
F[I(z)] = T exp r +1 (59)
s s

To apply this equation to the finite volume elements, within
which it is assumed that the phytoplankton concentration is
uniform, it is necessary to average this reduction factor

throughout the depth of the volume element Vs- Let H and

45 be the depths of the surface and bottom, respectively,
of the volume element V;- For example, if the volume ele-
ment Y5 extends from the water surface to the bottom of the
water body, then Ho5 = O and hy is the water depth at the
location of V5- For the sake of simplicity, it is assumed
that this is the case. If 495 # O, a straight-forward gen-

eralization of the following average is required.

In addition to an average over depth, it is also expedient

to average the phytoplankton growth rate over a time interval.
Since the time scale within which this analysis is addressed
is the week-to-week change in the population over a year, a
daily average is appropriate. For simplicity, it is assumed
that the incident solar radiation as a function of time ov-

a day is given by the function

91


dM = M' /ege a dimensionless microbial-substrate
concentration ratio.
Equation (18) is simply a dimensionless reexpression of Eq. (12)
One can examine several special cases of Eqs. (16) and (17)
which will provide some understanding of the behavior of

these equations.

Case I - Small Michaelis constant, small initial microbial
mass

The limiting value for this case is K, = 0 which results in

the exponential growth of microbes as given from the solution

to Eq. (13) The substrate utilization is then

c/e, = 1l- dy exp (Kyt) (19)
In Eq. (18) the second term dominates for this case.

Case II - Large Michaelis constant, small initial microbial
mass

For this case, a, is considered large and one obtains the

logistic growth equations (Eq. (15). In terms of substrate

utilization, the solution is

(1+4y) exp (~ (Kyt) (14d) /d.)

c/eg = qd, + exp K€ (14a,)74,) (20)

for small dy (initial microbial mass relative to initial sub-
strate concentration)

exp (-Kyt/d.)

= = 1
C/o = ar expCK,t7a,) ~ Tea expU=Kyt7e,) (21)

29


Program . .

Name Description
KIT Saturated Growth Rate-Phyto.
Is Saturating Light Intensity
KMN Michaelis' Constant-Nitrogen
KMP Michaelis' " Phosphorous
KMPL Michaelis' " Phyto.Chlor
K2T Phyto. Endogeneous

Respiration Rate

CCHL Carbon-Chlorophyll Ratio

NCHL Nitrogen-Chlor. Ratio

PCHL Phos.~Chlor. Ratio

K34T Org.N-NH, Hydrolysis Rate

K33 Decay of Organic Nit.

K45T NH3-NO3 Nitrification

K67T Org P-Inorg.P Conversion
Rate

K77 Decay of Org. P.

K66 Decay of Inorg. P.

K11T BOD Decay Coef.

Table 13

Parameters Used in Verifications
of 1968 and 1969 Potomac River Data
Preliminary Phytoplankton Model

Unit Value
l/day- °C 0.1
ly/day 300.0
mg-N/1 .025
mg-P/1 .005
ug-Chlor/1 60.0
Rate 1/day-°C 005

mg Carb/ug Chlor 05
mg Nit/ug Chlor -01

mg P/ug Chlor 001
1/day-°c 007
1/day 0.10
1/day-°Cc 01
1/day-°C 007
1/day 0.10
1/aay 0.10
1/day-°C 0.01

164


equations in contrast to the Lotka-Volterra equations. It
has been argued that the zooplankton growth rate resulting
from grazing must approach its maximum value when the phyto-
plankton population becomes large since the zooplankters
cannot metabolize the continually increasing food that is
available. Thus, the growth rate ayplgh2 is replaced by
Cy2P K., p/ (P + Kn p) where Kap is a Michaelis constant for

®2p g
the weaction. The "equations then become

at (c, - D'y)P - CgP2 + Py (93)
C_2P
mp

The nonzero singular: points are

(95)

(96)

This solution reduces to the previous situation, Equation
(91), for large Kop’ This is expected since for Kup large
with respect to P, the expression Kp! (P + Kup) approaches
one. However, an interesting modification from classical
predator-prey behavior occurs if the following condition is
met

2 2p -gkmp = Do+e (97)

where € is a small positive number. For this condition, P*
is large and positive. What happens in this case is that the
zooplankton population, although it continues to grow expo-

129


eT

@TOAD usbOrZFN 8YR JO seanjeeg zolew 2 ‘Sta

NITRIFICATION
N,

MITROSOMONAS || MiTRITE
NITROGEN

'
{

MITROBACTER

WASTE
SOURCES

NITRATE
MTROGEN

WASTE WASTE
SOURCES SOURCES
ORGANIC | _HyOROLYSIs | AMMONIA
NITROGEN NITROGEN
"4
DEATH ANIMAL
NITROGEN

MTRATE REDUCTION

IN PHYTOPLANKTON

UTILIZATION.



of nitrification inhibition in accordance with low DO reaches.
The additional verifications were obtained with ease, once

the order of the coefficients had been established from the
August 1964 and November 1967 analyses. The consistency of
the reaction coefficients is illustrated in Table 3. While
the set of coefficients is certainly not unique, Table 3 shows
that the general order of magnitude of the coefficients is
consistent providing allowance is made for variable spatial
distributions of the coefficients due to low dissolved oxygen
values.

The four system model can be used to estimate the effects of
the nitrification reaction on the dissolved oxygen deficit,
as indicated in Figure 6.

The organic and ammonia waste sources are inputted into the
initial two systems. The output from the ammonia system is
then multiplied by the reaction rate Ko3ar which is given by
3.43 Ko3- This then represents the sink of dissolved oxygen
due to ammonia oxidation and is inputted into the third system
which now represents the dissolved oxygen system with its ac-
companying reaeration rate. The total of three systems are
therefore used and the output from the third system represents
the dissolved oxygen deficit due to NH, - NO, oxidation. A
similar procedure is followed for the NO, - No, oxidation,

= 1.14K,,.

where K34n 34

Figure 11 shows a typical result for the July - August 1964
condition. The reaction rates for nitrification shown in

Figure 8 were used in the computation together with a constant
spatial reaeration rate of 0.18/day at 20°c. The two components
of the nitrification are shown. A slight shift downstream of
the NO, - NO component relative to the NH - NO, component

can be noted. Also, the peak in the total deficit occurs some

57


Therefore, the nonlinearities provide the coupling between

the phytoplankton, zooplankton, and nutrient systems. This
coupling is accomplished within each volume and does not ex-
tend beyond the volume boundary. The coupling among the
volumes is accomplished by the linear transport interaction
represented by the matrix A. This matrix may be time-varying
but its elements are not functions of the phytoplankton, zoo-
plankton, or nutrient concentrations. Hence, in many ways
these equations behave linearly. In particular, their spatial
behavior is unaffected by the nonlinear source terms. However,
the temporal behavior and the relationships between each P.,

250 and Ny are distinctly nonlinear.

Comparison with Lotka-Volterra Equations

The classical theory of predator-prey interaction as formulated
by Volterra involves two equations which express the growth
rate of the prey and the predator (Lotka, 1956). Within the
context of phytoplankton and zooplankton population, the prey
is the phytoplankton and the predator _ zooplankton. In

the notation of the previous sections, for a one-volume system,
the Lotka-Volterra equations are:

S- = (G.-D') P- CPZ 86
at ( p " ) g (86)
dz _ _

ae = DAZ + app gPZ (87)

where all the coefficients, Soe Dor Cyr Do and Aap are assumed
to be constants and S, > Diy: This is a highly simplified
situation since, as indicated previously, the growth and death
rates are functions of time and, in the case of the phytoplank-

126


GRAZING RATE (1/mg-dry wt.- day)

a Daphnia Magna

2: D. Sch¢dleri

+ D. pulex

v D. galeaia

o Acartia clausi

o A. tonsa

a Centropages hamatus
@C. typicus

TEMPERATURE °C

Fig. 24 Grazing Rates of Zooplankton vs. Temperature

110


ables: light and temperature, the nutrient concentrations,
and the zooplankton present. With these variables known as

a function of time, it is possible to calculate the phyto-
plankton population resulting throughout the year. However,
the zooplankton population and the nutrient concentrations
are not known a priori since they depend on the phytoplankton
population which develops. That is, these systems are inter-
dependent and cannot be analyzed separately. It is therefore
necessary to characterize both the zooplankton population and
the nutrients in mathematical terms in order to predict the
phytoplankton population which would develop in a given set

of circumstances.

The Zooplankton System

As indicated in the previous section, the zooplankton popu-
lation exerts a considerable influence on the phytoplankton
death rate by its feeding on the phytoplankton. In some in-
stances, it has been suggested that this grazing is the
primary factor in the reduction of the concentration of phyto-
plankton after the spring bloom. In the earlier attempts to
model the phytoplankton system, the measured concentration of
zooplankton biomass was used to evaluate the phytoplankton
death rate resulting from grazing. However, it is clear that
the same arguments used to develop the equation for the con-
servation of phytoplankton biomass can be applied directly to
the zooplankton system. In particular, the source of zoo-
plankton biomass 835 within a volume element Y5 can be given
as the difference between a zooplankton growth rate S35 and

a zooplankton death rate Dose Thus, the equation for the
source of zooplankton biomass, which is analogous to Equa-
tion (54) is

S.. = (G.-D..) 2. (75)


No.

18

19

20

21

22

23

24

25

26

27

28

29

30

31

FIGURES

Page
Phytoplankton Saturated Growth Rate (Base e)
as a Function of Temperature 86
Normalized Rate of Photosynthesis vs Incident
Light Intensity 89

Nutrient Absorption Rate as a Function of Nutrient
Concentration: Comparison of Michaelis Menton
Theoretical Curve with Data from Ketchum 96

Measured Phosphate Absorption Rate, After Ketchum

vs Phosphate Absorption | Rate Estimated Using

UN,N. of Ker +N. (Ky ) Where N, and N, Are the

Ni rate aha phbsphitge cokeentratio Ss, Resfectively 99

Comparison of Phytoplankton Growth Rates as a
Function of Incident Solar Radiation Intensity and 13
Temperature

Endogenous Respiration Rate of Phytoplankton vs
Temperature after Riley (1949) 106

Grazing Rates of Zooplankton vs Temperature 110

Endogenous Respiration Rate of Zooplankton vs
Temperature 117

Temperature, Flow, and Mean Daily Solar Radiation;
San Joaquin River, Mossdale, 1966-1967 132

Phytoplankton, Zooplankton, and Total Inorganic
Nitrogen; Comparison of Theoretical Calculations

and Observed Data; San Joaquin River, Mossdale, 137
1966-1967

Theoretical Growth Rates for Phytoplankton and
Zooplankton Populations 141

Map of Potomac Estuary Showing Longitudinal and
Lateral Segments 146

Interactions of Nine Systems Used in Preliminary
Phytoplankton Model 148

Temperature and Flow Regimes Used for 1968 and
1969 Verification Runs 155

vii


NORMALIZED RATE OF PHOTOSYNTHESIS

1.0

(a) Plpg 0.5

1.0

(b) Plpg 0.5

y; Diatoms

© Plpg 05 ewe

LIGHT INTENSITY (FOOT CANDLES x 10%)

Fig. 19 Normalized Rate of Photosynthesis vs. Incident
Light Intensity: (a) Theoretical Curve after
Steele (1965) (b,c,d) Data after Ryther (1956)

89


the first order kinetics assumption may be justified on the
basis of the previous analysis. In anv case, first order
kinetics simplifies the structure of the models (linearity
is presumed) and provides for rapid numerical solutions.

BOD equivalent models are particularly appropriate when the
major portion of the nitrogenous demand is present as ammonia.
Either a direct measurement of the ammonia and its oxvgen
equivalence or the second stage BOD measurement mav be used.
Either is then inputted as a sink of dissolved oxvgen ina

DO model. This approach becomes more approximate as the con-
centration of organic nitrogen becomes more significant. The
organic nitrogen breaks down by hydrolysis to vield, among
other products, ammonia. The oxidation of ammonia and the
associate use of dissolved oxvgen may therefore be delayed

in accordance with the hydrolvtic rate of reaction. This ef-
fect, if appreciable, is taken into account in the oxygen
model by a reduction in the reaction coefficient of the ni-
trogenous BOD or by empirically introducing a lag in the
initiation of nitrification. In spite of its simplicity, this

model is quite adequate in a number of practical cases.

As indicated above, the nitrogenous oxygen demand, NBOD is
given approximately by:

NBOD = 4.57 (Org-N + NH,-N)

This demand can be considered as an oxvgen sink in the pair of
equations describing the distribution of NBOD and DO deficit.
For one dimensional steady state systems with constant coef-
ficients, these equations are:

a*L, dL

= = N -
Oo = E wae 7 Kyly + Wy (x) (26)

dx?

34


The nine system equations for each of 38 spatial segments re-
sults in a total of 342 simultaneous non-linear ordinary
differential equations. The entire set was programmed for
solution by a CDC 6600 computer. For a one-year simulation
at integration time steps of about 0.1/day, normal central
processing times were about 6 minutes.

Verification and Sensitivity Analysis
1968 Data

Two data periods were available for testing the validity and
degree of applicability of the preliminary model. The pro-
cedure followed in the verification was to “tune" the model
to the first data set collected in 1968. A forecasting
situation was then set up for 1969. The validated model for
1968 was used to independently estimate the 1969 data. The
only new input data used for 1969 was the river inflow with
associated water quality, water temperature and waste loads.
All coefficients determined from the 1968 validation were
used without modification for the 1969 verification. This
latter verification then provides the basis for determining
the degree to which the model will simulate future conditions

under different waste removal policies.

Figure 31 shows the data periods and the flow and temperature
regimes used for the 1968 and 1969 runs. During the 1968

data period, Figure 31 indicates a significant flow transient
in June 1968 with a lesser transient in September. The piece-
wise linear approximations to the time variable flow were de-
termined from the available data and do not account for the
day to day variability in inflow. The emphasis was rather on
describing the seasonal variation rather than on shorter term

fluctuations.


the excretion rate is

a, pK
Ane g?4P5 ( -x 2p mp ) (78)
mp 3

At high phytoplankton concentrations, almost all the grazed
phytoplankton is excreted since the bracketed term in Equa-
tion (78) approaches unity.

Table 11

Dry Weight Percentage* of Carbon, Nitrogen,

and Phosphorus in Phytoplankton”

% Carbon % Nitrogen % Phosphorus
Phyto- Aver- Aver- Aver-
plankter age Range age Range age Range
Myxo- _ _ _
phyceae 36 (28-45) 4.9 (4.5-5.8) 1.1 (0.8-1.4)
Chloro- _ _ .
phyceae 43 (35-48) 7.8 (6.6-9.1) 2.9 (2.4-3.3)
Dino- _ _ .
phyceae 43 (37-47) 4.4 (3.3-5.0) 1.0 (0.6-1.1)
Chryso- _ _ .
phyceae 40 (35-45) 8.4 (7.8-9.0) 2.1 (1.2-3.0)
Bacillario- _
phyceae 33 (19-50) 4.9 (2.7-5.9) 1.1 (0.4-2.0)

the units are (mg of carbon, nitrogen, or phos vhorus) / (mg dry
weight of phytoplankton) X 100%.
b

Condensed from Strickland (1965).

121


Ny
roy

xe
at
B=
ica)
rans
a= 0
z non
m
°
zo
0
0 50 wo 150 200 250 300
NOg (ug-N/1)

x
a .
wt
Da

oO
Ovo
BS
aq!

Oo
Sax
ao

0 10 20 30 40 50 60

PO4 (ug-P/I)

Fig. 20 Nutrient Absorption Rate as a Function of Nutrient
Concentration: Comparison of Michaelis Menten
Theoretical Curve with Data from Ketchum (1939)

96


where the nitrogen discharges are largely unaffected by
standard high rate biological treatment.

Sequential reaction models answer some of the disadvantages
described above and especially apply when organic as well
as ammonia nitrogen are present as inputs. Furthermore,
when it is desirable to trace the individual components of
the downstream nitrification process; organic, ammonia,
nitrite and nitrate, this approach is useful. The lag ef-
fect is incorporated into the kinetic expression by the
first step of a sequence of the coupled reactions. By con-
trast to the previous simplified approach, modeling with
consecutive reactions is slightly more complicated analvti-
cally, requires more data, time and experience, but provides

greater understanding and increased confidence in prediction.

The sequential reaction models follow each of the nitrogen
components individually. The ammonia and nitrite outputs
are then converted to oxygen demands and used as inputs into
the DO deficit equation. The general constant coefficient
equations are:

a?N, aN,
Oo = E Ge? -uge KN) + Wy (x) (31)
a?N, aN,
Oo = E Pa 7 UW gy - KyQNy + KyoN, + Wy (x) (32)
an
d?n 3 (33)
o = B— 2 -u de ~ ¥33Ngz + Ka3Nq + Wy Ox)
dx?
an
a?n 4 (34)
Oo =8 -u Ge 7 KagNg + KgqN3 + Wy Ge)
dx

where Ny, No, N3, and Ny are organic, ammonia, nitrite and

nitrate nitrogen respectively, K represents the first order

di
decay of substance i, Kis is the forward reaction coefficient

and W; is the discharge of substance it

37