CE5504 – Surface Water Quality Modeling
Lab 8. 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, materials 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 overstimulation of natural processes. Toxic substances (excluding metals) 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 is most particularly characteristic of 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,
where cd is the dissolved component and cp is the particulate component, each representing a fixed fraction of the total concentration,
and
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,
and
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.
210Pb / 10 / DDT / 0.01
PCB / 0.1 / Dieldrin / 0.0003
Hg / 0.08 / 90Sr / 0.0002
137Cs / 0.02 / Chloride / 0
For organic contaminants, Kd is known to vary with the organic content of the solids,
where Koc is the organic carbon partition coefficient with units of,
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,
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). Note, then the potential need for coupled phytoplankton-toxicant models.
Toxicant Mass Balance
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).
Mathematically,
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,
The expression for the transfer function (Chapra, 1997, p. 55), β, is,
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; Chapra, 1997, p. 57-58), λ, is,
And the 95% system response time is given by,
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 do to the role of the ‘slow eigenvalue’ in retarding recovery.
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, an isotherm, illustrates the tendency of the solid phase toxicant concentration to increase due to sorption as the liquid phase concentration increases.
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:
where the rate of adsorption (Rad, mg∙s-1) is given by:
where kad is the mass-specific volumetric rate of adsorption (m3∙mg-1∙s-1) and Ms is the mass of solids (mgSS).
where kde is a first-order desorption rate (s-1).
Substituting to the original equilibrium expression yields:
which is a rectangular hyperbola or saturation function where increases with cd, approaching m 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 m. In this case, the rate of adsorption becomes:
and the equilibrium expression can be written as:
and defining a partition coefficient (Kd, m3∙gSS-1) as:
yields:
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:
and substituting
The total toxicant concentration is:
or
Defining the fraction dissolved (Fd) as:
yields the relationship originally introduced:
and by substitution to the total toxicant concentration (c):
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 may be described as:
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 interface (ci, mg∙m3) and the bulk water (cd, mg∙m3):
The concentration of the chemical at the interface (ci) is given by Henry's Law (Chapra, 1997, p. 360):
and
Most toxicants have very low concentrations in the air (but note the ‘grasshopper effect’) and the flux term reduces to:
The calculation of the transfer velocity (v) is based on the case where mass transport is governed by molecular diffusion within the laminar gas and liquid films at the air-water interface. Mathematically, the expression for transfer velocity develops from two-film theory (Chapra, 1997, p. 369-373):
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).
For gases that are very soluble in the liquid, the rate limiting step is the diffusion of the solute through the gas film. For gases that are poorly soluble in the liquid, such as oxygen in water, the rate limiting step is the diffusion of the solute gas through the liquid film.
Values for Kl are have been related to the oxygen transfer coefficient KL:
where M is the molecular weight of the toxicant and where KL is given by:
where Uw is the wind speed (m∙s-1). Values for Kg have been related to wind speed as well:
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:
where the settling velocity (s, m∙d-1) is determined through model calibration or by measuring the sediment flux with sediment traps:
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. For example, consider the microbially-mediated reductive dehalogenation of chlorinated solvents such as PCE and TCE. Here, the microbes replace chlorine with hydrogen, leading to:
PCE (4Cl) TCE (3Cl) DCE (2Cl) VC (1Cl) Ethene CO2
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.
And
Substituting the right-hand side of the biomass equation to the substrate equation yields:
Toxicant concentrations (~ppb) are significantly less than the half saturation constant for organic carbon (~ppm) and thus C can be dropped from the denominator.
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:
and
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:
and
and the units of the biotransformation constant are year-1.