Mixing reservoir

The relation between tq and x is not as simple. Ta may be larger or less than tq depending on the shape of the age probability density function (as shown in Fig. 4-4). For a well-mixed reservoir, or one with a first-order removal process, x = tq (Fig. 4-4b). 

Let us define a two-box model for a steady-stafe ocean as shown in Fig. 10-22. The two well mixed reservoirs correspond to the surface ocean and deep oceans. We assume that rivers are the only source and sediments are the only sink. Elements are also removed from the surface box by biogenic particles (B). We also assume there is mixing between the two boxes that can be expressed as a velocity Vmix = 2 m/yr and that rivers input water to the surface box at a rate of Vnv = 0.1 m/yr. The resulting ratio of F mix/V riv is 20. 

Simple three-box models with the atmosphere assumed to be one well-mixed reservoir and the oceans described by a surface layer and a deep-sea reservoir have been used extensively. Keeling (1973) has discussed this type of model in detail. The two-box ocean model is refined by including a second surface box, simulating an "outcropping" (deep-water forming) polar sea (e.g.. Keeling and Bolin, 1967, 1968), and to include a better resolution of the main thermo-cline (e.g., Bjorkstrom, 1979). The terrestrial biota are included in a simple manner (e.g., Bolin and Eriksson, 1959) in some studies Fig. 11-18 shows a model used by Machta (1972) where the role of biota is simulated by one reservoir connected to the atmosphere with a time lag of 20 years. 

The inadequacy of the two-box model of the ocean led to the box-diffusion model (Oeschger et al, 1975). Instead of simulating the role of the deep sea with a well-mixed reservoir in exchange with the surface layer by first-order exchange processes, the transfer into the deep sea is maintained by vertical eddy diffusion. In... 

As an example we assume a well mixed reservoir with an average age of water T, i.e., the age distribution is... 

Equation 1.2 assumes that the concentration of C is constant throughout the ocean, i.e., that the rate of water mixing is much fester than the combined effects of any reaction rates. For chemicals that exhibit this behavior, the ocean can be treated as one well-mixed reservoir. This is generally only true for the six most abundant (major) ions in seawater. For the rest of the chemicals, the open ocean is better modeled as a two-reservoir system (surface and deep water) in which the rate of water exchange between these two boxes is explicitly accoimted for. 

The third transport coefficient that we address is the diffusion coefficient. The simplest case is diffusion of a dilute species (solute) into another fluid (solvent) that is present in great excess. Consider the experiment shown schematically in Fig 12.3. In the bottom portion is a large well-mixed reservoir containing a mixture of solute held at fixed concentration c = C in a solvent. The top portion is a similar well-mixed reservoir of the mixture with solute concentration held fixed at c = C + AC. A permeable thin film separates each reservoir from the center fluid mixture. As such, in the center fluid region the concentrations of solute at the upper and lower edges equal the concentrations just across the permeable films that is, they equal the reservoir concentrations. 

The atmosphere is a well-mixed reservoir with known concentrations of noble gases (Table 13.1). These atmospheric noble gases have characteristic isotopic abundances that are given in Table 13.2. The solubility of the noble gases is given in Fig. 13.1, expressed in cc STP noble gas/cc water. STP stands for standard temperature (0°C) and pressure (760mmHg=l atmosphere). What is the solubility of argon in distilled water at sea level at 15 °C The answer, from Fig. 13.1, is 3.5 x 10 4cc STP Ar/cc water. 

Example 4.13. Carbon-14 as a Tracer for Oceanic Mixing In a simplified two-box model of the ocean, the warm waters and the cold waters may tie subdivided into two well-mixed reservoirs—an upper one a few hundred meters in depth and a lower one of 3200 m depth. The Cj content of the upper and lower reservoirs (corresponding to the Pacific) are, respectively, 1.98 x 10 mol liter and 2.44 x 10 mol liter, whereas the C/C ratios for uppsr and lower reservoirs are, respectively, 0.92 x 10 and 0.77 x 10 mol/ mol. Estimate from this information the rate of vertical mixing and the residence time of the water in the deep sea (Broecker, 1974). 

Partly air-filled, mixed reservoir with calcite precipitation... 

For the well-mixed reservoirs, a conservation equation is written in which gain of 14C by inflow to the box (atmosphere or surface ocean) is balanced by the outflow to other boxes plus radioactive decay (see Section 2.8) of the tracer during its time in the reservoir. For the deep ocean, conservation is described by a partial differential advection-diffusion equation. The... 

Mass balance calculations suggest that this eclogitic reservoir is of considerable size. Kamber and Collerson (2000) estimated the mass of deeply subducted ocean crust over 4.3 Ga to be about 1.4 X 1026 g, that is, about 3% of the mass of the silicate Earth, greater than the mass of the continental crust, and equal to about 20% of the mass of oceanic crust subducted over time. Where this reservoir is located is not precisely known. However, Kamber and Collerson (2000) argued that the Nb and Ta contents of MORB worldwide are very well correlated, implying that, for these elements, the depleted mantle is a well-mixed reservoir, and unlikely therefore to contain high Nb/Ta slabs. For this reason they propose that the eclogitic slab reservoir is located in the lower mantle. 

Figure 4-3a might correspond to a lake with inlet and outlet on opposite sides of the lake. Most water molecules will then have a residence time in the lake roughly equal to the time it takes for the mean current to carry the water from the inlet to the outlet. Another example is a human population where most people live to a mature age. Figure 4-3b illustrates the common situation with exponential decay or with a well-mixed reservoir. A simple example could be the reservoir of aU on Earth. The half-life of this nudide is 4.5 x 10 years, implying that the content... 

Let us define a two-box model for a steady-state ocean as shown in Fig. 9-21. The two weU-mixed reservoirs correspond to the surface ocean and deep oceans. We assume that rivers are the only source... 

The batch adsorber problem will yield a finite steady-state concentration for both the individual particle and also within the well-mixed reservoir. This steady-state concentration can be readily obtained from Eqs. 11.174 as follows. 

So far we have analysed the two systems using the Maxwell-Stefan approach the steady state analysis of the Stefan tube and the transient analysis of the Loschmidt s tube, and they are conveniently used to study the diffusion characteristic of the system. Here, we consider another example which is also useful in the determination of diffusion characteristics. This system is the two bulb method, in which a small capillary tube or a bundle of capillaries is bounded by two well-mixed reservoirs as shown in Figure 7.2-3. 

---