Flux Calculations

One of the most important reasons for using a small-volume light water— moderated core instead of the large-volume heavy water—moderated core described in MonP-108 and in pendix 1 is that the virgin neutron flux (fission neutrons which have suffered no collisions) and the y-ray flux are much higher. The virgin flux, is related to the number of fission neutrons 

The above equation is correct if the reactor dimensions are large compared to the mean free path, virgin neutrons. 

In the present reactor Q is given hy where 4 is the average slow 

U per liter of reactor. The mean free path of virgin neutrons in the core of light water plus aluminum, with 0.75, is about 5 cm thus 

The old proposed heavy water—moderated reactor had a core volume almost ten times greater than the present one, while the total power output, 3 10 kw, and virgin—neutron mean free path was the same. Its average virgin flux was thus smaller by a factor of 10. A comparison of the slow, resonance, and fast flux in these light water and heavy water reactors is given in Table 4. 2.A. 

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Global uranium flux calculations have typically been based on the following two assumptions (a) riverine-end member concentrations of dissolved uranium are relatively constant, and (b) no significant input or removal of uranium occurs in coastal environments. Other sources of uranium to the ocean may include mantle emanations, diffusion through pore waters of deep-sea sediments, leaching of river-borne sediments by seawater," and remobilization through reduction of a Fe-Mn carrier phase. However, there is still considerable debate... 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

Such an analysis indicates that the zero-sink assumption must be used with extreme caution if accurate flux calculations are required at the local root level. Potassium, for example, is close to the limiting value of A, for the zero sink assumption to be fulfilled, and simulations with larger roots or larger buffer powers could well lead to inaccurate simulation results. Any zero-sink model involving nitrate should be treated with some suspicion. The zero-sink assumption is also widely used in root architecture models (see later). 

Aerosol particles Table 3.13 shows the percentage change in the actinic flux calculated by Peterson (1976) and Demerjian et al. (1980) for two cases (1) a particle concentration of zero, corresponding to a very clean atmosphere, and (2) a total particle concentration doubled compared to the base case. The actinic flux is predicted to increase if the total particle concentration is zero and decrease if it doubles (note, however, as discussed later, the sensitivity to the vertical distribution of particles and the relative importance of light scattering compared to absorption). 

Fig. 3.1. The line shows correlation between bond valence and bond length for Ca-O bonds given by eqn (3.1). The circles represent bond fluxes calculated from the Madelung field (Preiser et al. 1999).

The similarity between eqns (2.7) and (3.3) (given the equality of g, and F,) is a necessary but not sufficient condition that bond fluxes, "hy, and bond valences, Sij, are the same. A theoretical proof of their equality is not possible, but it can be demonstrated by comparing the bond valences calculated using eqn (3.1) (the line in Fig. 3.1) with bond fluxes calculated from the Madelung fields in particular compounds (the points in Fig. 3.1). This figure shows that the flux and bond valence are essentially the same for Ca-0 bonds and similar agreement is found for other types of bond provided that electronic anisotropies of the kind discussed in Chapter 8 are not present (Preiser et al. 1999). 

Fig. 7.1. Bond flux-bond length correlation for H-O bonds. The points correspond to fluxes calculated from structures determined by neutron diffraction. The heavy line is drawn through these points. The thin line is a smooth interpolation between the two ends of the heavy line and represents the correlation that would be expected in the absence of 0-0 repulsion.

Fig. 8.3. Bond flux versus bond length in I2O5. The points are the Madelung fluxes (calculated without the dipole by Preiser et al. 1999). The line represents the empirical bond valence-bond length correlation which is expected to be close to the true flux.

Since initially the concentration of MCF in air, C,a, is zero, the evaporation flux calculated from Eq. 20-6 is (A = 0.3 m2, surface area of puddle) ... 

In Illustrative Example 19.2 we discussed the flux of trichloroethene (TCE) from a contaminated aquifer through the unsaturated zone into the atmosphere. The example was based on a real case of a polluted aquifer in New Jersey (Smith et al., 1996). These authors compared the diffusive fluxes, calculated from measured TCE vapor concentration gradients, with total fluxes measured with a vertical flux chamber. They found that the measured fluxes were often several orders of magnitude larger than the fluxes calculated from Fick s first law. In these situations the vapor profiles across the unsaturated zone were not always linear. The authors attributed this to the influence of advective transport through the unsaturated zone. In order to test this hypothesis you are asked to make the following checks ... 

Species fluxes calculated by either the multicomponent (Section 12.7.2) or the mixture-averaged (discussed subsequently in Section 12.7.4) formulations are obtained from the diffusion velocities V, which in turn depend explicitly on the concentration gradients of the species (as well as temperature and pressure gradients). Solving for the fluxes requires calculating either all j-k pairs of multicomponent diffusion coefficients Dy, or for the mixture-averaged diffusion coefficient D m for every species k. 

The basinwide flux calculations from the seven lakes show that preindustrial Hg accumulation rates in the sediments ranged between 4.5 and 9.0 xg/m2 per year, and the modern rates range between 16 and 32 xg/m2 per year (Figure 9). More striking is the observation that the range in these rates is a function of the relative size of the terrestrial catchment surrounding each lake basin. Over 90% of the variation in modern Hg accumulation can be accounted for by the ratio of a lake s catchment area to its surface area (Ad A0). The correlation between preindustrial Hg accu-... 

Comparison of the depositional fluxes shows that diatoms were the most important particle component transporting P to the sediment surface, accounting for 50-55% of the flux (Table II). Terrigenous material and calcite were also important transport vectors. Deposition varied markedly with season, as shown by the time series plot of the major particle components (Figure 13). The total P flux calculated by using the particle components model agreed with the flux measured by sediment traps (157-227 versus 185 mg/m2). The close agreement indicated that the major particle vectors were represented and associated P concentrations were accurately quantified. 

A major factor governing diffusive fluxes of sulfate into sediments is lake sulfate concentration. A linear relationship exists between lake sulfate concentrations and diffusive fluxes calculated from pore-water profiles (Figure 5). The relationship extends over a range of 3 orders of magnitude in sulfate... 

The subsurface maximum in pore-water HgT (Figure 3) suggested that diffusion from the profundal sediments to the overlying water column could be important. Fickian diffusive flux calculations (eq 2) were used to estimate Hg loading from pore waters. Diffusion coefficients for mercury in pore waters were not available. However, free-water diffusion coefficients for monovalent anions (see Table I) averaged about 5 X 10"6 cm2/s (53, 55) and... 

Differential quantum yield. It is considered as a dynamic characteristics of the system and is defined as the ratio of the reaction rate to the photon flux (calculated horn the radiant flux, the intensities of the rays of the UV lamp, the transmittance of the optical window of the reactor, and the absorbance of TiC>2). A quantum yield of ca. 0.1 was found in the region where r varies linearly with . 

Lakes. Flux calculations based only on wet and dry particle deposition were close to measured sediment fluxes. PCA analysis confirmed that wet and dry particle deposition was much more important than dry vapor deposition, based on homolog patterns. 

Gas exchange studies of total toxaphene in Lakes Superior, Michigan and Ontario were conducted between 1996-2000 [46,50,67] using concurrently measured water and air concentrations. The air and water data from these studies are discussed in Sects. 2.2, 3.1.1 and 3.1.2, and are summarized in Tables 3, 5, and 6. An earlier mass budget for toxaphene [65] estimated gas exchange from measured water concentrations and historical air data from Hoff et al. [64], All studies based their fugacity and flux calculations on the temperature-dependent Henry s law constant for technical toxaphene [15]. 

The direction and magnitude of gas transfer of PAHs across the air-water interface can be calculated using a modified [69] two-layer resistance model. This model has been previously well described elsewhere [70] and is summarized here. The overall flux calculation is defined by... 

The rivers play a major role in the transfert of carbon and mineral nutrients from land to the sea and influence significantly the biogeochemical processes operating in coastal waters. Quantification of the material transport, both in the dissolved and particulate forms, has been attempted by several authors in the past (Clarke, 1924 Holeman, 1968 Garrels McKenzie, 1971 Martin et al., 1980 Meybeck, 1982 Milliman Meade, 1983). Depending on the type of sampling techniques and methods of calculations employed there are differences in the reported fluxes. A major problem in such calculations is the paucity of reliable data from some of the major rivers of the world especially of Asia (see e.g. Milliman Meade, 1983). Additionally the difficulty of obtaining representative samples from the rivers will adversely affect flux calculations. Most of the inferences drawn on the nature and transport of riverine materials rest on data collected randomly - at different points in time and space. Seasonal variations in the transport of materials are very common in some of the major world rivers, and in some cases more than 60 % of the material transport occurs within a very short period of time. Furthermore, available data are not always comparable since the analytical techniques used differ from river to river. 

Single-stage simulations reveal that intermolecular friction forces do not lead to reverse diffusion effects, and thus the molar fluxes calculated with the effective diffusion approach differ only slightly from those obtained via the Maxwell-Stefan equations without the consideration of generalized driving forces. This result is as expected for dilute solutions and allows one to reduce model complexity for the process studied (143). 

The ratio of total radioactivity between the growing and quiescent areas as estimated by scintillation counting (or electron flux calculated in part from counts per minute) was approximately 1.8 at 5 minutes, 2.3 at 1 hour, and 2.9 at 24 hours. 

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