Steady state box models

There is a bulk mass flux upwards in plumes. However, this is assumed to be small, so that the reservoir volume and concentrations are essentially unaffected, and a return flux can be ignored. 

The high He/ He and Ne/ Ne ratios found in Loihi are used to calculate He and Ne isotope concentrations (see Undepleted mantle section). 

The lower mantle Ar and Xe isotope compositions cannot be well constrained by available data and are taken as unknowns. Therefore, the isotopic compositions, and so nonradiogenic isotope abundances, are unknowns. However, these are calculated from the balance of fluxes into the upper mantle (see below). 

3) The upper mantle reservoir that is sampled by MORE. 

All concentrations and isotopic compositions of noble gases and parent elements are assumed to be in steady state (and uniform) over the last residence time. Note that a crustal reservoir is not included in the model, because neither the long-term depletion of parent element mantle concentrations, nor the long-term flux of daughter nuclides to the atmosphere, are considered. 

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Stutz et al. (1999) used a quasi-steady-state box model to explain the measurements of iodine oxides at Mace Head. Their model did not include the aerosol phase. They reproduced the measured 10 mixing ratios of 6 pmol mol and found that for these conditions, cycles involving the production of HOI by reaction of lO with HO2 and subsequent photolysis of HOI would lead to an O3 destruction rate of —12.5 nmol mol d , whereas the lO self-reaction would lead to an O3 destruction rate of —3.8 nmol mol d . ... 

Lamborg C. H., Fitzgerald W. F., O Donnell J., andTorgersenT. (2002) A non-steady state box model of global-scale mercury biogeochemistry with interhemispheric atmospheric gradients. Abstr. Pap. Am. Chem. Soc. 223 072-ENVR. 

Artola-Garicano et al. [27] compared their measured removals of AHTN and HHCB [24] to the predicted removal of these compounds by the wastewater treatment plant model Simple Treat 3.0. Simple Treat is a fugacity-based, nine-box model that breaks the treatment plant process into influent, primary settler, primary sludge, aeration tank, solid/liquid separator, effluent, and waste sludge and is a steady-state, nonequilibrium model [27]. The model inputs include information on the emission scenario of the FM, FM physical-chemical properties, and FM biodegradation rate in activated sludge. 

Models which assume uniform mixing throughout the volume of a three-dimensional box are useful for estimating concentrations, especially for first approximations. For steady-state emission and atmospheric conditions, with no upwind background concentrations, the concentration is given by... 

What is the steady-state concentrahon derived from the box model for a 10-km city with average emissions of 2 x 10" g m s when the mixing height is 500 m and the wind speed is 4 m s ... 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

Given that the long residence time of uranium should place limits on how much the marine value could change over Late Quaternary time scales, several workers have used models to determine what these limits should be (see Henderson and Anderson 2003). Chen et al. (1986) and Edwards (1988) used a simple one-box model and assumed steady state conditions. They showed that ... 

To understand the impact of individual processes on the compartmental distribution of DDT, model runs with a non-steady-state, zero-dimensional, multimedia mass balance box model (MPI-MBM) [Lammel (2004)] were conducted in addition to MPI-MCTM experiments. Parameterisations of intra- and intercompartmental mass exchange and conversion process in MPI-MBM are similar to those in MPI-MCTM. A detailed description of differences and a comparison of both models can be found in Lammel et al (2007). The DDT emissions were the global mean temporally varying DDT applications for the years 1950 to 1990. A repeating annual cycle around constant mean temperatures was simulated. Surface and air temperatures differ by 14 K constantly. 

For a chemical, C, whose concentration is in steady state, the rate of supply equals the rate of its loss. This can be represented by the simple box model shown in Figure 5.6 for a chemical, C, which is present in three reservoirs, and by the following chemical equation ... 

From the perspective of the surface box, the biolimiting elements are supplied via river runoff and from upweUing. The elements are removed via the sinking of biogenic particles and downwelling. Since this model considers only the transport of materials into and out of the ocean and between the two reservoirs, details as to what happens to the elements while they reside in the boxes are not needed other than that they are present in a steady state. In such a case, the input rate of a biolimiting element will equal its output rate. For the surface-water reservoir, the mass balance that describes this steady state is given by... 

Nutrients are carried back to the sea surface by the return flow of deep-water circulation. The degree of horizontal segregation exhibited by a biolimiting element is thus determined by the rates of water motion to and from the deep sea, the flux of biogenic particles, and the element s recycling efficiency (/and from the Broecker Box model). If a steady state exists, the deep-water concentration gradient must be the result of a balance between the rates of nutrient supply and removal via the physical return of water to the sea surface. 

Primary outputs are produced essentially by sedimentation and (to a much lower extent) by emissions in the atmosphere. The steady state models proposed for seawater are essentially of two types box models and tube models. In box models, oceans are visualized as neighboring interconnected boxes. Mass transfer between these boxes depends on the mean residence time in each box. The difference between mean residence times in two neighboring boxes determines the rate of flux of matter from one to the other. The box model is particularly efficient when the time of residence is derived through the chronological properties of first-order decay reactions in radiogenic isotopes. For instance, figure 8.39 shows the box model of Broecker et al. (1961), based on The ratio, normal-... 

Figure 8.39 Box model for steady state chemistry of seawater. Numbers in boxes ...

This equation can be very helpful for assessing the behavior of a substance in a natural system. If input and system concentration at steady-state are equal, ktot must be zero. In turn, for any reactive substance must be smaller than Cjn. A simple application of the one-box model is given in Illustrative Example 12.3. 

Figure 21.6 One-box models described by different production and reaction functions, p(C,) and r(C,), respectively (Eq. i 21-26). The steady-state(s) (Cf J, CF. ..) are given by the crossing points of the two functions (a) linear system (Eq. 21-4) with one steady-state (b) nonlinear system with two steady-states (Cy unstable) (c) nonlinear system (Eq. 21-28) with two steady-states (Cstable). The dashed r line shows the situation when there is no positive c7 ...

It can be shown that all the coefficients kxi of Eq. 1 are positive or zero, if the model equations result from a mass balance scheme. Then k, > 0, where, is the smaller of the two eigenvalues. By analogy to Eq. 4 of Box 12.1, the time to steady-state can be defined by ... 

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. 

As discussed below (Illustrative Example 21.6), the real advantage of the two-box model is the description of the transient behavior of the concentration. That is, such models allow us to examine the dynamic change from the initial value to the steady-state, rather than the computation of the steady-state itself. 

From a conceptual point of view, nothing new is needed to further extend the above approach. For instance, the one-box model with two variables shown in Fig. 21.7 can be combined with the two-box (epilimnion/ hypolimnion) model. This results in four coupled differential equations. Even if the equations are linear, it is fairly complicated to solve them analytically. Computers can deal more efficiently with such problems, thus we refrain from adding another example. But we should always remember that independently from how many equations we couple, the solutions of linear models always consist of the sum of a number of exponential terms which have exactly one steady-state, although it may be at infinity. In Section 21.4 we will discuss the general structure of linear differential equations. 

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... 

The question arose whether contaminants in the fairly dirty city air could pollute the drinking water by air-water exchange. You remember the two-box model shown in Fig. 21.9 and decide to make a first assessment by using the steady-state solution of this model. As an example you use the case of benzene, which can reach a partial pressure in air of up to p = 10 ppbv in polluted areas. You use a water temperature of 10°C and the corresponding Henry s law constant K, H = 3.1 L bar mol-1. The air-water exchange velocity of benzene under these conditions is estimated as vi a/w = 5 x 10 4 cm s 1. 

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 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. 

Figure 8.4. Main window of Gepasi. The main window of Gepasi consists of menus (File, Options, and Help), icons, and four tabs (Model definition, Tasks, Scan, and Time course). Activation of any of the tab opens an indexed page. At the start of Gepasi, the Model definition page is opened. Enter name of the metabolic pathway to the Title box. Click Reactions button to define enzymatic reactions (e.g., E + A+B = EAB for R1, EAB = EPQ for R2, and EPQ = E + P + Q for R3 shows 3 reactions and 7 metabolites), and then click Kinetics button to select kinetic type. Activate Tasks tab to assign Time course (end time, points, simufile.dyn), Steady state (simufile.ss) and Report request. Activate Scan tab to select scan parameters. Activate Time course tab to select data to be recorded and then initiate the time course run.

SimpleBox is a multimedia mass balance model of the so-called Mackay type. It represents the environment as a series of well-mixed boxes of air, water, sediment, soil, and vegetation (compartments). Calculations start with user-specified emission fluxes into the compartments. Intermedia mass transfer fluxes and degradation fluxes are calculated by the model on the basis of user-specified mass transfer coefficients and degradation rate constants. The model performs a simultaneous mass balance calculation for all the compartments, and produces steady-state concentrations in... 

When the kinetics (at least for the main reactions) and the type of the reactor are known, the performance of the control structure can be assessed. Firstly, by considering a steady-state model that incorporates kinetic reactor and black-box separation and performing sensitivity studies, it is possible to assess the feasibility... 

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