Asymptotic flux shape

In the effort to determine transient flux shapes certain interesting characteristics of the flux behavior become evident. If the power rise is not checked the flux eventually attains a stable or asymptotic shape. The method of harmonics also provides a means of determining the asymptotic flux shape and also the elapsed time in the attainment of this stable shape. After the asymptotic shape has been achieved there is, at least from a theoretical point of view, no further change in the flux shape and the kinetics problem becomes separable. 

How much different the asymptotic flux shape will be from the original steady state flux depends of course on the amount of the initial disturbance. 

Graphs are provided in Figures 7, 8, and 9 corresponding to three values of ASp. In each case there is exhibited the steady state flux, some of the transient fluxes, and the asymptotic flux, all normalized to equal areas. Also provided is the stabilization time for each case, that is, the total time elapsed in attainment of the asymptotic shape. The criterion used to specify a stabilization time is discussed at the end of this section. The step changes in reactivity exhibited in the three figures were computed by a conventional steady state two-group criticality calculation with the aid of the Wanda Code (IBM-704) [19]. For the calculations under discussion = A eff — 1-... 

It is also of interest that the coefficients in the asymptotic flux are almost identical in the two cases. Moreover, the stable periods are almost the same. Thus, not only are the asymptotic shapes identical, for all practical purposes,... 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62]... 

The neutron flux distribution is the shape function for the reactor that is running on an asymptotic period ... 

A key prediction of equation (6.3.11.5) is that is independent of d, a consequence of the radial divergent flux within a conical-shaped pore. For typical values of d obtained in preparing GNEs (-10°), the current (equation (6.3.11.1)) asymptotically approaches the depth-independent value (equation (6.3.11.5)) when the pore depth is at least 20X greater than the radius of the pore orifice. For instance, for a pore with a 20-nm radius orifice, any pore depth greater than -400 nm will yield a similar value of the steady-state limiting current. Experimental measurements of as a function of d confirm the behavior predicted by the above set of equations (2). 

In the film model, Nusselt and Sherwood numbers need to be determined. The analogy between heat and mass transfer suggests that Nu Sh. Extensive work has been devoted to the variation of these numbers with distance, and to their asymptotic values in very long channels of various shapes (Shah and London (1978)). Heck et al. (1976) showed that upstream the light-off point, the asymptotic wall heat flux Nusselt number must be used, whereas the asymptotic wall... 

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