Extrapolation distance

Figure 1. No-slip condition and slip condition with slip length, for one-dimensional shear flow. The slip length b is the extrapolation distance into the solid, to obtain the no-slip point. The slope of the linear velocity profile near the wall is the shear rate y.

In this equation we are assuming that the extrapolation distance g is the same for both plate surfaces. In general, the temperature jump will depend on the type of surface, and these extrapolation distances will not be equal unless the materials are identical. For different types of materials we should have... 

Vertical neutron distributions were studied also ,typical results.-will be found in Fig. A2.F.. -It will be recalled that no reflector was present at top or bottom of the experimental pile, and the solid lines are cosine curves fitted to. points near the center by a. least-squares method. The slight lateral shift of. the data from center is apparently due to the presence of a portion of a control rod in the top of the pile.- It is clear that the extrapolation distances for thermal and epithermal neutrons are the -same, within the experimental limits. 

A radial neutron distribution was measured for each lattice and the extrapolation distance was found to increase mcmotonically with increasing fuel concentration. Material buckling evaluation was made from the measured radial distribution in each case. 

The bucklings measured for the large fuel elements are lleted in Table I along with the corresponding volume ratios and extrapolation distances, k. 

The (ast-fisslon factor was determined from the ratio of the activities of a depleted and an enriched uranium foil, the resulting value being 1.0214 0.0021. Thebuclcllng was obtained from the critical size of the assemblies and the extrapolation distance. The latter, which was determined by least-squares fitting the flux distribution, was found to be 2.64 0.3 cm. The average buckling was 6.642 x 10 cm . The thermal-diffusion area, calculated from thermal cross sections, was 1,87 cm . The delayed-neutron age to thermal was calculated by O. G. Sullivan using a Monte Carlo moments-method calculation. The value was 15.6 cm . 

Because of such satisfactory agreement, our work with small, simple assemblies now concentrates on experimental discrepancies (e.g., uncertainties about the effectiveness of Pu-240, and absolute values of spectral indices), certain elusive parameters (such as the ratio of capture to fission in a thorium reflector, and effective extrapolation distance for shape conversion), and effects of dilution (as various metals mixed with U-235). Some recent results in these categories follow. 

For purposes of calculation the Plexiglas walls of the cylinders employed in the experiments were treated as HjO and were mixed in with the solution, giving a concentration of 349 g/U (92.6% U-23S) per liter. The homogenized units were approximated by spheres having a radius of 11.25 cm. The reflectors were all assumed to be HjO, and the effect of thickness was calculated from experimental data expressed as albedos. The material buckling of a unit was calculated to be 0.0306 cm" which, with bare and water reflected extrapolation distances of 2.6 and 5.8 cm, is consistent with experimental data for similar solutions. ... 

R was impossible to make most of the solution/ring mixtures critical and, in fact, the source neidron multiplication was often too low to allow sensible extrapolation. Addition of only a few inches of solution above a very sub- critical mixture established a critical system that served as a source of neutrons distributed over the mixture and had spectral characteristics that allowed a measure of the neutron relaxation length in the mixture. The material buckling estimated for these exponential-experiment data is somewhat uncertain because values of the radial extrapolation distance were unavailable. It has been possible to show, however, that the material buckling of some of the mixtures studied is negative. 

The material buckling for each exponential system may be calculated by assuming a value of the radial extrapolation distance. It was possible to estimate the radial extrapolation distance for reflected and unreflected mixtures of glass and the most concentrated solution as 5.0 and 2.9 In., respectively. R can be concluded that 1 for every mixture Of this solution at a concentration of 415 g ofU per liter and borosllicate glass, provided the glass occupies sd least 24.1% d the volume, contains a minimum of 3.3,wt% of twron and is uniformly distributed. 

L 2(xm F(0,E) Is the flux measured at the center of the slab, Xm is the blab half thickness, and e is the extrapolation distance. Spectrum measurements were made at 3.70, 3.07, 1 80 cm and at the center of the slab. A least- squares fit to a cosine function by iteration gives L for each value of neutron energy. The corresponding values of e are given in F ure 1. 

Lattice Spacing, in. Water-to-Uranium Volume Ratio Measured Extrapolation Distance, cm Material Buckling, 10 cm ... 

The determination of the Isothermal temperature coefficients required the estimation of temperature effects on radial buckling, and on bottom extrapolation distance. The effect on was calculated with the SRL-HERESY code . The change in the bottom extrapolation distance was calculated with the FLOG code. These corrections are listed with the experimental results in Table I. 

Clark has calculated masses of critical spherical water lattices of U(5.00) metal rods for a variety of diameters and uranium concentrations (or lattice spacings). The results lor 1.52- and 1.02-cm-diam rods were interpolated to allow the comparison with the measurements with I.31-cm rods shown by curves A and B. of Fig. 2. Curves B and C were obtained from the data of Fig. I by equating buddings using an extrapolation distance of 6.0 cm. No calculations of the larger diameter rods are available for comparison with curve C. 

Lattice Spacing in. Vw/Vu Uranium Height in. No. of Tubes for Critical Extrapolation Distance cm Material Buckling m ... 

An additional comparison of the data io calculated critical masses was made in spherical geoketry. Radii of equivalent spheres were obtained by equating bucklings in spherical and the experimental rectanguto geometries in which appropriate values of the extrapolation distances and reflector savings were used. The masses derived... 

Measured reactivity contributions of the various materials provided the basis for minor corrections of composition, and for eliminating effects of control-rod perturbations.. Observed critical volumes were corrected to spherical shape for comparison with one-dimensional transport calculations using Hansen-Roach cross sections. The shape conversions made use of Stratton s empirical expression for extrapolation distance ... 

Earlier measurements of critical water-moderated and -reflected lattices of 4,89% U-enrlched uranium [U(4.89)] rods 1.31, 2.07, and 2.49 cm in diameter, which provided reference for calculation of the criticality of heterogeneous assemblies of uranium of this enrichment, have been extended to rods 0.76 cm in diameter. The influence of the lattice pitch on the critical dimensions was determined by constructing several lattices in both square and triangular patterns. The data were then transformed, by equating bucklings using an extrapolation distance of 6 cm, to equivalent spheres for comparison with Clark s calculations. The results are shown in Fig, 1 where critical masses are plotted as a function of the U concentration. Shown for comparison are the earlier data for the 1.31-cm-diam rods. It is noted that the calculated masses of the 0.76-cm-diam rods are less than the experimental values at all concentrations. 

Material buddings and extrapolation distances have been measured, for 25.2 wt% PuO,-U(Nat)0> fuel pins in ymter. The pins used in Hie experiments are a prototype design of driver fuel pins Hiat wiU be used in the Fast Flux Test Facility. This work is part of a criticalify measurements and calculational program to provide safe limits for the fabrication, storage, shipment, and reprocessing of FFTF fuels. Results are presented in Table I. 

Obtained from least-squares fit of "Ijaek-off inverse multiplication measurements. Extrapolation distances are with 0.25 cm. 

Few data cvirrently exist on the ellectlveness of boron and cadmium for criticality prevention In operations external to reactors that may involve fuel elements under conditions of water Immersion. Material bucklings and extrapolation distances have previously been measured and reported for 25.2 wt% Pu02-U(Nat)0a fuel pins in water. These experimental results have also teen compared with one-dimensional diffusion theory calculations using ENDF/B version n cross-section data. The previous measurements have now been extended to include criticality data On these same fuel pins positioned in lattices with boron- and cadmium-poisoned water. 

The light-water reflector necessitated the determination of an extrapolation distance lfor each of the arrays. This distance was taken to be tbat value of X which,- in conjunction with the measured , gave the same material buckling for each array size of the same pitch (Table II). The measured keif s were then obtained by using this derived X, the measured , and a calculated M. A 3.0-cm extrapolation distance was assumed for the air bmind-ary at the top of the array. 

Two other techniques were correlated with the same critical eiqieriments with water-reflected spheres. The ffrst of toese involved a 12-groiq > Bo calculation to Obtain the buckling and two-group parameters for a subsequent diffusion theory calculation of extrapolation distance. ... 

Then using one-group constants from ANL-5800, a value of Koo is calculated. From ANL-S800 a value of the extrapolation distance is obtained. This is used with the values of Em and sphere radius. obtained above to calculate an effective migration area. These constants are then used to calculate a value of Keff for the fuel plate holder. 

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