Steady state model lakes

Results of model simulations of effects of coagulation (a = 0.1) and sedimentation at steady state in Lake Zurich during summer are presented in Fig. 7.16. Particle volume concentrations in the epilimnion and hypolimnion are plotted as functions of the particle production flux in the epilimnion. Biological degradation and chemical dissolution of particles are neglected in these calculations. Predicted particle... 

Simple steady-state models may be used in order to relate quantitatively the mean concentration in the lake water column and the residence time of metal ions to the removal rate by sedimentation (for a detailed treatment of lake models see Imboden and Schwarzenbach, 1985). In a simple steady-state model, the inputs to the lake equal the removal by sedimentation and by outflow the water column is considered as fully mixed mean concentrations and residence times in the water column can be derived from the measured sedimentation fluxes. The binding of metals to the particles is fast in comparison to the settling. 

Simple steady-state models can only predict mean concentrations. Seasonal variations and concentration depth profiles in the water column of lakes give further insight into the mechanisms governing the removal of metal ions. Data on depth concentration profiles of trace metals in lakes are however still scarce (Sigg, 1985 Sigg et al., 1983 Murray, 1987). In a similar way as in the oceans, it might be expected to observe in lakes different types of profiles for different elements, depending on the predominant removal mechanism (Murray, 1987 Whitfield and Turner, 1987). 

Steady-state modeling calculations were performed to examine how congener-specific properties (such as sediment-water partition coefficients, Henry s law constants, and molecular diffusion rates) affect the transport and fate of PCBs. A basic description of the model, along with modeling results, is presented here to further explain the importance of physiochemical weathering processes in controlling the fate and distribution of PCB congeners in Twelve Mile Creek and the upper portion of Lake Hartwell. 

Overall, the steady-state model for physiochemical weathering provided a good description of observed variations in congener distributions for sur-ficial sediments in the Twelve Mile Creek-Lake Hartwell system. In general,... 

Further studies examining time-variable behavior of PCBs in the Twelve Mile Creek-Lake Hartwell system and sensitivity of model calculations to various system parameters are presently being performed. The steady-state modeling results presented in this chapter, however, provide a reasonable base for an initial assessment of the fate of PCBs in the Twelve Mile Creek-Lake Hartwell system. The cumulative removals of PCBs from the system by volatilization and burial are shown as percents of the total PCB... 

Steady-state model calculations were performed to further examine physiochemical weathering behavior. Results were consistent with congener distributions in surficial sediments. In general, they showed that a preferential depletion of the lower chlorinated congeners occurred in the upper portion of Twelve Mile Creek, where volatilization was the only removal mechanism. In the lower portion of Twelve Mile Creek and the upper portion of Lake Hartwell, burial of PCBs in deeper sediments played a more important role. A preferential depletion of higher chlorinated congeners occurs when burial is the dominant process by physiochemical weathering. 

Simple steady-state models have been used to determine critical loads for lakes and forest soils. The basic principle is that primary mineral weathering in the watershed is the ultimate supplier of base cations, which are required elements for vegetation and lake water to ensure adequate acid-neutralizing... 

Figure 4. Simplified steady state model describing important steps in the limnological transformations of P in a lake...

In water and sediments, the time to chemical steady-states is controlled by the magnitude of transport mechanisms (diffusion, advection), transport distances, and reaction rates of chemical species. When advection (water flow, rate of sedimentation) is weak, diffusion controls the solute dispersal and, hence, the time to steady-state. Models of transient and stationary states include transport of conservative chemical species in two- and three-layer lakes, transport of salt between brine layers in the Dead Sea, oxygen and radium-226 in the oceanic water column, and reacting and conservative species in sediment. 

The relationships between particle flux, trace element flux and trace element concentration in sediment are more complicated in deep lakes. In a deep lake, there may be a significant proportion of dissolved element held in the water column. If the water column dissolved element inventory approaches the magnitude of the annual flux for that element, then a steady state model is invalid. Instead, the dynamic model outlined in Figure 7 must be used to allow for the time delay in the response of the sediment to changes in trace element supply rate. The disadvantage of this, compared with the steady state sitnation, is that an observed trace element concentration profile does not lead back to a nniqne trace element supply history. However, a trace element snpply history does lead to a definite trace element concentration profile, so it is possible to see if any particular supply history is compatible with the observed concentration data. A practical example of this from Lake Baikal is shown in Boyle et al. (1998), where the exceptional water depth makes this effect particularly strong. 

To illustrate the model a steady state solution is given which would apply to the lake after prolonged steady exposure to water emission of 10 mol/h and atmospheric input from air of 5.3 ng/m3. The solution is given in Figure 2B in the form of fugacities, concentrations and transport and transformation process rates. 

At first sight, the result is puzzling, and it becomes even more so if we calculate the corresponding steady-state concentration of the one-box model using the same lake parameters. From Eq. 7 of Box 12.1 ... 

Hence, the one-box and two-box models yield the same result. There is a simple reason for that. Since the only removal processes of PCE act at the lake surface, at steady-state the surface concentration in both models (C°°for the one-box model, ClE for the two-box model) must attain the same value to compensate for the input /, tot. Furthermore, since the hypolimnion has neither source nor sink, the net exchange flux across the thermocline must be zero, and this requires C(E= C,H. 

In Illustrative Example 21.5 we discussed the behavior of tetrachloroethene (PCE) in a stratified lake. As mentioned before, our conclusions suffer from the assumption that the concentrations of PCE in the lake reach a steady-state. Since in the moderate climate zones (most of Europe and North America) a lake usually oscillates between a state of stratification in the summer and of mixing in the winter, we must now address the question whether the system has enough time to reach a steady-state in either condition (mixed or stratified lake). To find an answer we need a tool like the recipe for one-dimensional models (Eq. 4, Box 12.1) to estimate the time to steady-state for multidimensional systems. 

Inhomogeneous systems. If Eq. 21-46 is an inhomogeneous system, that is, if at least one Ja is different from zero, then usually all eigenvalues are different from zero and negative, at least if the equations are built from mass balance considerations. Again, the eigenvalue with the smallest absolute size determines time to steady-state for the overall system, but some of the variables may reach steady-state earlier. In Illustrative Example 21.6 we continue the discussion on the behavior of tetrachloroethene (PCE) in a stratified lake (see also Illustrative Example 21.5). Problem 21.8 deals with a three-box model for which time to steady-state is different for each box. 

You have constructed a linear two-box model for tetrachloroethene (PCE) in a lake in which the only input of PCE is from the outlet of a sewage treatment plant. The atmospheric PCE concentration is assumed to be zero in your model. How will the steady-state of the model be altered if the PCE input from sewage is reduced by... 

There is still another point to be discussed, which may limit the calculations presented in Tables 23.4 and 23.5. In 1986, when the concentrations were measured, the lake may not have been at steady-state. In fact, the PCB input, which mainly occurred through the atmosphere, dropped by about a factor of 5 between 1965 and 1980. However, the response time of Lake Superior (time to steady-state, calculated according to Eq. 4 of Box 12.1 from the inverse sum of all removal rates listed in Table 23.4) for both congeners would be less than 3 years. This is quite short, especially if we use the model developed for an exponentially changing input (Chapter 21.2, Eq. 21-17) with a specific rate of change a = - 0.1 yr 1 (that is, the rate which... 

Application of the dynamic water/sediment model to the fate of the PCBs in Lake Michigan is summarized in Table 2 3.7. As it turns out, for both congeners the steady-state concentrations are virtually unchanged relative to the values calculated for the three-phase one-box model of Table 23.5. We also note that in the model the sorbed concentrations are still significantly smaller than the measured values. The same is... 

Application of steady-state solution of linear water-sediment model (Box 23.3) to two PCB congeners in Lake Superior. The steady-state is calculated from Box 21.6. 

In Part 2 of the PCB story, we introduced the exchange between the water column and the surface sediments in exactly the same way as we describe air/water exchange. That is, we used an exchange velocity, vsedex, or the corresponding exchange rate, ksedex (Table 23.6). Since at this stage the sediment concentration was treated as an external parameter (like the concentration in the air, Ca), this model refinement is not meant to produce new concentrations. Rather we wanted to find out how much the sediment-water interaction would contribute to the total elimination rate of the PCBs from the lake and how it would affect the time to steady-state of the system. As shown in Table 23.6, the contribution of sedex to the total rate is about 20% for both congeners. Furthermore, it turned out that diffusion between the lake and the sediment pore water was much more important than sediment resuspension and reequilibration, at least for the specific assumptions made to describe the physics and sorption equilibria at the sediment surface. 

In the last step (Part 3), the sedimentary compartment (the surface mixed sediment layer , SMSL) was treated as an independent box (Table 23.7). The steady-state solution of the combined sediment/water system explained another characteristic of the observed concentrations, which, as mentioned above, could not be resolved by the one-box model. As shown in Table 23.8, for both congeners the concentration measured on particles suspended in the lake is larger than on sediment particles. The two-box model explained this difference in terms of the different relative organic carbon content of epilimnetic and sedimentary particles. This model also gave a more realistic value for the response time of the combined lake/sediment system with respect to changes in external loading of PCBs. However, major differences between modeled and observed concentrations remained unexplained. 

For transport terms in eqs 1 and 3, the upper and lower reaches of Twelve Mile Creek were treated as one-dimensional advective (plug flow) systems. The upper basin of Lake Hartwell was modeled as a series of completely mixed reactors, with inflows into the first reactor from Twelve Mile Creek and the Keowee River. Analytical solutions for steady-state distributions of solids and PCB congeners in the water column and the active sediment layer were determined by using approaches outlined by O Connor (20-22). Because Kd values for a specific congener class were taken to be equivalent for the water column and active sediment layer, the solutions are independent of k[ values. 

The spring waters of the Sierra Nevada result from the attack of high C02 soil waters on typical igneous rocks and hence can be regarded as nearly ideal samples of a major water type. Their compositions are consistent with a model in which the primary rock-forming silicates are altered in a closed system to soil minerals plus a solution in steady-state equilibrium with these minerals. Isolation of Sierra waters from the solid alteration products followed by isothermal evaporation in equilibrium with the eartKs atmosphere should produce a highly alkaline Na-HCO.rCOA water a soda lake with calcium carbonate, magnesium hydroxy-silicate, and amorphous silica as precipitates. 

Landrum et al. (1992) developed a kinetic bioaccumulation model for PAHs in the amphipod Diporeia, employing first-order kinetic rate constants for uptake of dissolved chemical from the overlying water, uptake by ingestion of sediment, and elimination of chemical via the gills and feces. In this model, diet is restricted to sediment, and chemical metabolism is considered negligable. The model and its parameters, as Table 9.3 summarizes, treat steady-state and time-variable conditions. Empirically derived regression equations (Landrum and Poore, 1988 and Landrum, 1989) are used to estimate the uptake and elimination rate constants. A field study in Lake Michigan revealed substantial differences between predicted and observed concentrations of PAHs in the amphipod Diporeia. Until more robust kinetic rate constant data are available for a variety of benthic invertebrates and chemicals, this model is unlikely to provide accurate estimates of chemical concentrations in benthic invertebrates under field conditions. 

Less sophisticated but highly useful models have been developed to assess the relative importance of different sources of toxics to the lakes. These screening level models can be equilibrium, steady state or dynamic models. Examples of these are the fugacity-based models of Mackay.17... 

In a simple model of tetrabromoxylene (TBX) flow into and out of Lake Ontario (volume = 1.67 x 1012m3, average depth = 85.6 m), there are three inputs and three outputs. Unfortunately, only three inputs and two outputs have been characterized. TBX enters the lake through rain (concentration = 10ng/L), rivers (344 kg/year), and streams and creeks (102 kg/year). TBX leaves the lake through rivers (310 g/day), volatilization from the lake surface (251 kg/year), and sedimentation. Assume that TBX is at steady state in the lake and that its residence time is 7.2 years. 

One of the basic assumptions in the unstirred chemostat was that the turnover rate was so slow that any transport (perhaps induced by pumps operating the chemostat) was negligible. If one thinks of a model for a flowing stream, or a lake with circulation, this assumption is unwarranted. Hence a mathematical analysis of the case where transport has been added to the model equations would be an important contribution. The steady-state case (with equal diffusion) was considered in [JW], but the dynamical model is the important one. 

Kaste O., Henriksen A., and Posch A. (2002) Present and potential nitrogen outputs from Norwegian soft water lakes—an assessment made by applying steady-state first-order acidity balance (FAB) model. Hydrol. Earth Syst. Sci. 6, 101-112. 

A second type of culture described by Monod kinetics is the continuous culture, in which a chemical is constantly fed into a vessel and both microbial cells and the chemical are constantly lost from the vessel at a given rate. This culture is often called a chemostat when operated under steady-state conditions. Like the batch culture, a continuous culture may be a useful model of certain environmental systems, such as lakes receiving continuous discharges of pollutants. Continuous cultures are common in industrial processes as... 

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