## Important Factors In The Modeling Of Lakes Conceptual Model Development

5.5.1 Definitions of Terms:

In order to describe a lake system mathematically, we must first make a list of variables (mathematical symbols) for several terms:

Qi is the inlet flow from the main inlet to the lake (m3/yr).

Qe is the outlet, or effluent, flow rate from the lake (m3/yr) (We usually assume that Qi is equal to Qe and we represent both by simply Q).

Ci is the average pollutant concentration in the inlet to the lake (kg/m3) (this value is zero in many cases). Ce is the average pollutant concentration in the lake (kg/m3) and the concentration in the effluent from the lake.

k is the first-order rate constant for removal of pollutant from the lake (year-1).

W is the total mass flux of pollutant in the lake, which is equal to the sum

Note the units used in each of the terms and note that they are compatible with each other. This is important in using the simulator package Fate®. These terms are used to develop the mass balance of pollutant in the lake and develop equations for the individual components of the mass balance (inflow, outflow, sources, and sinks of pollutant).

### 5.5.2 Detention Times and Effective Mixing Volumes

In this section we will develop a more conceptual, rather than mathematical, derivation of the chemical and physical processes important in the governing equation for the fate and transport of pollutants in lake systems. For a slightly more mathematical approach, refer to the background section of the lake module in Fate®.

In lake systems, it is useful to know or estimate how long water will stay in the system, since this provides an estimate of the minimum time the pollutant will stay in the system. This parameter is called the detention or retention time, and this brings us to the weakest assumption of commonly used lake models. In order to keep the mathematics relatively simple, we must assume that the lake is completely mixed with respect to pollutant concentrations. In some cases (small to medium-sized lakes), this is a valid assumption, but for others (large lakes) it is a weak assumption at best. When we assume that the lake is completely mixed, we can estimate the hydraulic detention time (t0) by t0 = V/Q (5.1)

which is expressed in years, in accordance with the units specified in Section 5.5.1.

Mixing in lakes, an applied form of entropy, is one of the most difficult parameters to estimate. The predominate mixing force is wind blowing across the surface of the lake and is commonly referred to as wind-driven advection (mixing due to the movement of water). The exact extent of mixing can be determined by costly and long-term monitoring projects. One extreme approach would be to release a known mass of dye at the inlet of the lake. Usually a fluorescent dye is used to enable detection of extremely small concentrations. A few European studies have used the radioactive tracer tritium. Of course, this will not work for very large lakes, since the large volume in these systems will dilute the dye to nondetectable concentrations and large lakes can have detention times of decades. After the dye has been placed into the lake, the effluent stream of the lake is monitored with respect to the dye concentration. If the slow increase and subsequent decrease in dye concentration is analyzed, the effective mixing volume can be calculated. Thus, the effective mixing volume is the volume of water actually mixing with the pollutant as opposed to the entire volume of the lake. Of course, if the lake is large the monitoring dye program could take months to years, or even decades, to complete. Note that this process is further complicated when stratification of the lake occurs. So, the dye technique for determining mixing is only of use in small ponds and lakes. Usually historical data or "experience" is used to estimate effective mixing volumes for larger lakes.

When the effective mixing volume is determined or estimated—for example, 78% of the total volume—this value can easily be used in place of V in Eq. (5.1) to calculate a more accurate estimate of the detention time of water and pollutant in the system. This approach can also be used for stratified lakes, where the depth of the hypolimnion can be measured and the volume of the water body receiving the pollutant can be calculated.

### 5.5.3 Chemical Reactions

In Chapter 2, a variety of potential degradation schemes were presented, including photochemical, biological, abiotic (chemical), and nuclear reactions. All of these are possible transformation reactions in lake systems. Whatever the type or types of reaction(s), all of these are usually represented by first-order kinetics, and we can add the individual rate constants (k values) together to obtain one overall first-order rate constant. This component of the fate and transport model is of the form

where Ct is the pollutant concentration at time t, C0 is the initial pollutant concentration, e is the exponential function, k is the first-order rate constant, and t is time.

### 5.5.4 Sedimentation

In addition to washout of pollutants in the effluent from lakes, along with biological and chemical degradation, pollutants can be removed from a lake system by sorption to particles followed by subsequent settling to the lake bottom. This can be a significant removal mechanism for some pollutants, especially those that do not readily degrade through microbial or chemical means. In order to appreciate how pollutants can thus be removed from the water column of lakes, we will first look at the size of particles that can be present in aqueous systems. Table 5.5 shows the particle settling velocity as a function of particle size. As the particle size decreases, the surface area of the particle increases, and sorption processes become more important since more pollutant can sorb to the surface. In addition, smaller particles can contain more organic matter on the surface and be even more sorptive reactive. Hence, clay-sized particles can be very important in determining the fate of sorbed pollutants, and as you can see from the data in Table 5.5, they have the smallest settling velocities. This results in the particles and sorbed pollutants settling in the deepest and most quiescent (calmest) regions of the lake. This particle-settling veloc-

TABLE 5.5. Particle Settling Velocity as a Function of Particle Size (Lapple, 1961)

Classification Particle Diameter Range (mm) Settling Velocity in Water (cm/sec)

Fine sand 20-200 2 X 10-2 to 2

Coarse sand 200-2000 (0.2-2mm) 2-20

ity has been validated by monitoring results from lakes that find the highest concentration of polluted sediments in the deepest regions of lakes (referred to as pollutant focusing). In contrast, regions of higher energy flow and thorough mixing in lakes contain larger particles, which generally do not contain high levels of organic matter and therefore do not contain high levels of pollutants.

Settling velocities (w) given in Table 5.5 were obtained by a relatively simple calculation, defined as Stokes' law:

where g is the acceleration due to gravity (length/time2), ps is the density of the spherical particle (mass/length3), pf is the density of the fluid (mass/length3), r is the spherical particle radius (length), and h is kinematic viscosity of the fluid (length2/time). The kinematic viscosity is the ratio of the dyanamic viscosity of a the fluid to the density of the fluid. Note that Eq. (5.3) assumes a spherical particle, but average particle radius can be used.

While Eq. (5.3) describes the settling of a particle, it is of little use, since pollutant concentration is not present in this equation. You should recall from Chapters 2 and 3 that sorption behavior of a pollutant is described by the distribution coefficient (Kd) for metals and the partition coefficient (Kp) for hydrophobic pollutants. Thus, we need an expression that incorporates particle removal and pollutant concentration:

where rA is the rate of decrease in pollutant A concentration per unit volume of water (mass/length3-time), Kd is the distribution coefficient (or Kp is the partition coefficient), w is the particle settling velocity (length/time), S is the suspended solids concentration (mass/length3), H is the water depth (length), and C is the pollutant concentration in the water (mass/length3).

Thus, we can account for pollutant removal by sedimentation in calm water. Waters with rapid currents that mix the water and suspended material will result in slower settling rates. Also note the units of the settling rate constant, concentration per time. These are the units of a zero-order, rather than first-order, rate expression, whereas our fate and transport models will use first-order expressions. Thus, unfortunately, the rate of pollutant removal cannot be directly substituted into our fate and transport models; still, we can estimate the removal of pollutant through sedimentation. Also note that this is assumed to be a steady-state process, since the concentration of pollutant and the suspended solids concentration in the water are assumed to be relatively constant. This is usually a reasonable assumption, except during storm events.

It is also important to note the rate of sediment accumulation in the bottom of a lake, since the settling sediment can bury previously contaminated sediment. Baker (1994) reports a range from 50 to 600g of sediment per square meter per year for a variety of lake systems. This translates into an accumulation rate from millimeters to centimeters of sediment per year, which can result in significant deposits of sediment. While sediment accumulation and burial of contaminated sediment is important, it is also important to look at sediment resuspension rates. Sediment can be naturally resuspended through bioturbation (the mixing of sediment by bottom feeding fish and organisms living in the sediment) and by storm events. Wetzel (2001, p. 635) reports resuspension rates from 0.5 to 21 g/m2 ■ day, which are very significant.

Considering these sedimentation rates, it is understandable that contaminated sediments can be buried and therefore removed from the aquatic system. In a sense, burial of sediments is a form of natural remediation. An example of this is shown in Figure 5.6 for chromium. This sediment profile is from Upper Mystic Lake, which is the water basin for the Aberjona Watershed, north of Boston, Massachusetts. As you can see from the profile, chrome used in the local tanning industry started in ~1900 and declined after 1925. The spike in chromium in the sediment deposited around 1959 is from rendering operations that utilized chrome-tanned hides in the

Cr Concentration (mg/kg)

0 2000 4000 6000 8000

Cr Concentration (mg/kg)

0 2000 4000 6000 8000

process of making glue. Note that today the chromium pollution has been isolated and buried by the natural sedimentation of unpolluted, or less polluted, material in the lake. If the sediment cap remains intact then the chromium has been effectively (and inexpensively) removed from the lake system.

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