CE5504 – Surface Water Quality Modeling
A Toxic Substance Model for a Well-Mixed Lake
To this point, we have dealt with what are often termed conventional pollutants, e.g. nitrogen,
phosphorus, organic carbon, etc. Now our attention will move to toxic substances, substances which
differ from conventional pollutants in several ways important to the modeler:
 natural versus alien: problems relating to conventional pollutants typically involve the natural
cycle of organic production and decomposition, most often over stimulation of natural processes.
Toxic substances do not occur naturally and the problem is one of interference with natural
processes.
 aesthetics versus health: a case could be made that mitigation of problems relating to
conventional pollutants largely seeks to eliminate aesthetic problems. Although such a case
would be overstated, it is clear that management of toxic substances focuses on health concerns,
e.g. drinking water and aquatic food stuffs.
 few versus many: conventional water quality management deals with on the order of ten
‘pollutants’. In contrast, there are tens of thousands of organic chemicals that may be introduced
to the environment.
 single versus multiple phase: conventional pollutants are largely modeled as single species,
although some partitioning is made between the dissolved and particulate phases. Such
partitioning is critical for modeling toxic substances, because they tend to exist in gaseous,
dissolved and particulate phases.
It is the liquid/solid partitioning which most significantly impacts the manner in which toxic
substances are modeled. A mechanistic treatment of partitioning is important because key
source/sink processes act selectively on either the dissolved or particulate form. For example,
volatilization acts solely on the dissolved form, settling only on the particulate form, and decay
reactions on both.
Mathematically, the total contaminant concentration (mg∙m-3) is given by,
c  cd  c p
where cd is the dissolved component and cp is the particulate component, each representing a fixed
fraction of the total concentration,
cd  Fd  c
and
c p  Fp  c
where Fd and Fp are the fractions of the total contaminant in the dissolved and particulate form and
are a function of the contaminant's partition coefficient and the lake's suspended solids concentration,
Fd 
1
1  Kd  m
Fp 
Kd  m
and
1  Kd  m
where Kd is the partition coefficient (m3∙gSS-1) and ‘m’ is the suspended solids concentration
(gSS∙m-3). The partition coefficient, Kd, describes the tendency of a contaminant to associate with
solid matter through sorption. The sum of the two fractions is 1, i.e. Fd + Fp =1.
Contaminant
Kd
Contaminant
Kd
210
Pb
10
DDT
0.01
PCB
0.1
Dieldrin
0.0003
137
0.02
90
0.0002
0.02
Chloride
0
Cs
239,240
Pu
Sr
For organic contaminants, Kd is known to vary with the organic content of the solids,
Kd  f oc  Koc
where Koc is the organic carbon partition coefficient with units of,
L
gChemical O
M
P
gOrganic
C
N
Q
gChemical O
L
M
Nm P
Q
3
and ƒoc is the weight fraction of organic carbon in the solid matter (gOrganicC∙gSS-1). Koc has been
empirically related to the octanol-water partition coefficient, Kow, which can be measured directly or
determined from tables for the contaminant of interest,
Koc  617
. x107  Kow
Thus, one could expect significant partitioning to the particulate phase for contaminants with high
Kow values and for lakes high in organic-rich suspended solids (e.g. algae).
The mass balance for a toxic organic substance in a well-mixed lake is developed by considering the
various sources and sinks for that contaminant; here, the source is a tributary loading and the sinks
are outflow, volatilization, settling, and reaction (e.g. photolysis or microbial degradation).
[T] Toxicant mass balance (Chapra, 1997, Figure 40.3)
Mathematically,
V
dC
 W  Q  C  V  k  C  v v  A  Fd  C  v s  A  Fp  C
dt
where V is the lake volume (m3), c is the toxicant concentration (mg∙m-3), W is the toxicant loading
(mg∙yr-1), Q is the inflow/outflow (m3∙yr-1), k is a first order reaction coefficient (yr-1), vv is a
volatilization velocity (m∙yr-1), vs is a settling velocity (m∙yr-1), and A is the lake surface area (m2).
The steady state solution to this equation is,
css 
W
Q  V  k  v v  A  Fd  v s  A  Fp
The expression for the transfer coefficient, B, is,

Q
Q  V  k  v v  A  Fd  v s  A  Fp
For values of  << 1, the lake tends to assimilate the pollutant; for values of 1, the lake tends to
pass the pollutant downstream.
The eigenvalue (sum of loss coefficients) is,
Q
v v  Fd v s  Fp
 k

V
H
H
And the 95% system response time is given by,
t 95 
3

Values for V and A are specified for the lake. The inflow/outflow (Q) and inflow concentration (ci)
are measured. Kinetic coefficients and the fraction in the particulate and dissolved phases are
estimated empirically or determined experimentally.
Thus, more ‘reactive’ pollutants reach a new steady state more rapidly, offering greater promise for
remediation. Note, however, that loss to sediments can result in long term pollutant feedback.
Parameterizing Toxicant Kinetics
1. Sorption
In modeling sorption, we consider toxicant in the dissolved phase (cd, mg∙m-3) and in the
particulate phase (, mg∙gSS-1). A plot of  versus cd illustrates the tendency of the solid phase
toxicant concentration to increase due to sorption as the liquid phase concentration increases and is
called an isotherm.
cd
Note that the solid phase toxicant concentration () reaches an asymptote (maximum, m) where all
of the sorption sites are saturated.
The sorption process actually describes the equilibrium between the rates of adsorption and
desorption:
R ad  R de
where the rate of adsorption (Rad, mg∙s-1) is given by:
R ad  k ad  M s  c d  (  m  )
where kad is the mass-specific volumetric rate of adsorption (m3∙mg-1∙s-1) and Ms is the mass of solids
(mgSS).
Rde  kde  M s 
where kde is a first-order desorption rate (s-1). Substituting to the original equilibrium expression
yields:
m  cd

k de
 cd
k ad
which is a rectangular hyperbola or saturation function where  increases with cd, approaching max
at high values of cd. Note that the slope of the linear portion of the curve is determined by the ratio
of kde to kad, i.e. when kad >>> kde, adsorption increases rapidly with cd.
Environmental concentrations (cd) of most toxicants are such that we are interested only in the linear
portion of the isotherm where  <<< max. In this case, the rate of adsorption becomes:
R ad  k ad  M s  c d   m
and the equilibrium expression can be written as:
k ad   m  c d

k de
and defining a partition coefficient (Kd, m3∙gSS-1) as:
kad  m
Kd 
kde
yields:
  Kd  cd
Thus the solid phase toxicant concentration becomes a linear function of the dissolved phase toxicant
concentration with a proportionality constant which accommodates the sorption capacity (max) and
the ratio of the rate of adsorption to desorption (kad/kde).
Remembering that the concentration of solid phase toxicant in the water is given by:
cp  m 
and substituting
c p  m  Kd  cd
The total toxicant concentration is:
c  cd  c p
or
c  cd  m  Kd  cd
Defining the fraction dissolved (Fd) as:
Fd 
cd
c
yields the relationship originally introduced:
Fd 
1
1  Kd  m
and by substitution to the total toxicant concentration (c):
Kd  m
Fp 
1  Kd  m
thus we see that the coefficients Fd and Fp have their basis in sorption kinetics, i.e. (1) the relative
rates of adsorption and desorption for a compound and the sorption capacity of the solids (as
manifested in Kd) and (2) the concentration of solids present.
2. Volatilization
The volatilization term in the toxicant mass balance is given as:
dc
V  J As
dt
where J is the net flux of the chemical across the air-water interface (mg∙m-2∙d-1), given as the
product of a transfer velocity (v, m∙d-1) and the concentration gradient of the chemical between the
air (cg, mg∙m3) and the water (cd, mg∙m3):
d i
J  v  cg  cd
The concentration of the chemical in air, from Henry's Law, is:
cg 
pg
He
substituting,
 pg

J  v  
 cd 
 He

Most toxicants have very low concentrations in the air and the flux term reduces to:
J   v  cd
and
dc
V    v  cd  As
dt
The calculation of the transfer velocity (v) is based on two-film theory where mass transport is
governed by molecular diffusion within laminar gas and liquid films at the air-water interface.
Mathematically, the transfer velocity is given by:
 v  Kl 
He
Kl
H e  R  Ta 
Kg
where Kl is the mass transfer velocity in the liquid laminar layer (m∙d-1) and Kg is the mass transfer
velocity in the gas laminar layer (m∙d-1). Values for Kl are have been related to the oxygen transfer
coefficient KL:
F
I
G
HJ
K
32
Kl  K L 
M
0.25
where M is the molecular weight of the toxicant and where KL is given by:
K L  0.864  U w
where Uw is the wind speed (m∙s-1). Values for Kg have been related to wind speed as well:
18 I
F
 GJ
HM K
0.25
K g  168  U w
In summary, the volatilization velocity has been related to characteristics of both the chemical
(molecular weight, Henry's constant) and the environment (temperature, wind speed). Values for the
Henry's constant and molecular weight have been compiled for many toxic chemicals.
3. Sedimentation
The sedimentation term in the mass balance is given as:
dc
V    s  A s  c p
dt
where the settling velocity (s, m∙d-1) is determined through model calibration or by measuring the
sediment flux with sediment traps:
J ss
s 
c ss
where Jss is the mass rate of suspended solids accumulation in the sediment traps (gSS∙m-2∙d-1 ) and
css is the suspended solids concentration of the overlying water (gSS∙m-3).
4. Reaction
Chapra (1997) indicates that toxic organic chemicals may be lost from the system through
photolysis (breakdown to simpler compounds by sunlight), hydrolysis (breakdown to simpler
compounds through chemical reactions), and biodegradation (breakdown to simpler compounds by
bacteria). Reaction processes yield decay products, potentially transforming the toxicant into carbon
dioxide, water, and inorganic chemicals.
We will examine biodegradation as an example reaction process here as it complements our study of
microbial population dynamics. The approach is based on Monod kinetics, relating growth rate and
substrate concentration.
dC
1 dX

dt
Y dt
and
dX
C
  max 
X
dt
Ks  C
Substituting the right-hand side of the biomass equation to the substrate equation yields:
dC

C
  max 
X
dt
Y Ks  C
Toxicant concentrations (~ppb) are significantly less than the half saturation constant for organic
carbon (~ppm) and thus C can be dropped from the denominator.
dC

  max  X  C
dt
Y  Ks
yielding a second order expression (it's a function of both C and X), with a biotransformation
constant, kb2, having units of m3∙cell-1∙year-1:
kb 2  
 max
Y  Ks
and
dC
  kb 2  X  C
dt
Typical values for bacterial population densities are given by Chapra (1997) together with values for
kb2 for selected toxic organics. Where bacterial populations are relatively constant, the expression
reduces to a first order relationship:
kb  kb 2  X 
 max  X
Y  Ks
and
dC
  kb  C
dt
and the units of the biotransformation constant are year-1.