Flat-flux assumption

The flat-flux assumption will lead to errors when... 

The theory of resonance absorption has been formulated in a more general form in [9] and [10], where two coupled integral equations for the flux in the absorber nd in the moderator are introduced. In case the flat-flux assumption is retained, this procedure gives identically the same results as the ones presented here. The task of obtaining better approximations would seem to be nearly equivalent to a full Monte Carlo treatment of the whole problem. 

Resonance absorption in closely packed assemblies. So far we have only treated the case in which the absorber lumps are so widely spaced that they do not interact with each other i.e., the distances between them are large compared to the moderator free path. An important generalization is the one to closely packed assemblies where this condition is not fulfilled. This case was first treated by Dancoff and Ginsburg [16]. The problem is solved in principle by a redefinition of the escape probability Pq, This quantity was defined as the probability that a neutron coming from a uniform source density in the absorber escapes from it, which means that for large separation of the absorber lumps it will make the next collision in the moderator. We now define an effective escape probability PJ for close assemblies, which means just the same, i.e., that a neutron bom with a flat distribution in an absorber makes its next collision in the moderator, excluding the cases in which, after traversal of one or more moderator sections, it collides inside another absorber lump. It is clear that under the flat-flux assumption, all previous formulas still hold when Pq is replaced by PJ, and it remains only to find appropriate expressions for the latter. 

When the requirements for the flat-flux assumption are well satisfied, the total number of neutrons slowing down in the cell Qo may be approximated by the asymptotic form thus we take... 

We have used here Eq. (10.131) for Pe., and Mr Vrpy When these expressions are compared with (10.38) and (10.39), it is seen that the NR approximation for leads rather easily to a statement in the standard form. Note, however, that because of the flat flux assumption the present derivation does not contain the resonance disadvantage factor fr. This quantity is customarily computed using a one-velocity model to represent the entire fast-neutron population. It is well recognized that this point of view is crude and somewhat unclear. On the one hand, when used with the NR approximation, it may be observed that only one collision is required to remove a resonance neutron from the vicinity of a resonance thus the spatial distribution would be very nearly uniform and isotropic. On the other hand, if the NRIA approximation is valid, then the absorptions are necessarily very strong and the use of a disadvantage factor based on diffusion theory is not well justified. For these reasons it has been omitted in this treatment as an unwarranted refinement not in keeping with the precision of the over-all model. ... 

The general problem is formulated in 2 for homogeneous geometries and in 3 for heterogeneous ones. In the latter case a special assumption, the flat-flux hypothesis, has to be made, which is further discussed in 8. In 4 it is shown that the slowing-down equations can be solved by a simple numerical procedure. 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

The simplest case for modeling particle dissolution is to assume that the particles are monodisperse. Under these conditions, only one initial radius is required in the derivation of the model. Further simplification is possible if the assumption is made that mass transport from a sphere can be approximated by a flat surface or a slab, as was the case for the derivation for the Hixson-Crowell cube root law [70], Using the Nernstian expression for uniaxial flux from a slab (ignoring radial geometry or mass balance), one can derive the expression... 

According to assumptions and concentration profiles, illustrated in Fig. 2.1 A, solute being extracted through a hydrophobic flat membrane can be described by the solute flux from the bulk aqueous phase to the bulk organic phase in terms of individual mass-transfer coefficients at steady state and additivity of a one-dimensional series of diffusion resistances. Overall mass-transfer coefficient, Ap/p ... 

To infer a dry deposition rate from an eddy correlation measurement, a nondivergent vertical species flux should exist. Nondivergence essentially stipulates that quasi-one-dimensional transport exists. The nondivergence assumption is, in fact, equivalent to the constant-flux-layer assumption of the surface layer in practical terms, nondivergence is best satisfied in relatively flat topography for which a substantial fetch over the terrain exists. 

The assumptions of similarity weaken as one moves from flat and uniform surfaces into hilly terrain and associated natural surface covers (Doran et al. 1989). Fluxes are likely to change substantially over rather short distances (1 km or less), and it may be extremely difficult to establish a representative flux value for more extended regions from measurements at a single site. Some measurements have been carried out over sloping... 

Another approach is to use a thermopile with a flat wavelength response over the scanned spectral range and measure the power with a well-calibrated power meter (laser power meters calibrated at specific wavelengths work well) and then convert power into photon flux at all other wavelengths. Errors due to assumption of a constant bandpass intensity at each wavelength are also typically not accounted for and could be significant when a bandpass >10 nm is being used. 

For the sake of simplicity, we wiU start explicitly with an approximation that is satisfactory for the majority of cases. (How corrections can be made wherever necessary will be discussed in 8.) This assumption, which makes the treatment of the heterogeneous case comparable with the homogeneous one, is that the source flux in the moderator remains flat even at resonance energies, e.g., that there is also spatial flux recovery. We further assume that this flux is the asymptotic flux (normalized to one source neutron per ceU) ... 

Many additional studies have been conducted with the boundary layer model by taking into account the variation of physical properties with composition (or temperature) and by relaxing the assumption that Vy = 0 at y = 0 when mass transfer is occurring. Under conditions of high mass transfer rates one finds that mass transfer to the plate decreases the thickness of the mass transfer boundary layer while a mass flux away from the wall increases the boundary layer thickness The analogous problem of uniform flux at the plate has also been solved. Skelland describes a number of additional mass transfer boundary layer problems such as developing hydrodynamic and mass transfin- profiles in the entrance region of parallel flat plates and round tubes. 

---