Critical radius Sphere

It is interesting to note that Umax, when calculated by the above equations, may exceed the maximum surface or interfacial pressure to which the film can, in practice, be subjected. The absorbed film will then desorb or crumple, or it may rather suddenly be shed to the rear of the drop as a filament or, if the interfacial tension is very close to zero, as emulsion. The drop may thus, on suddenly losing its surface film, accelerate again, as has indeed been noted (without explanation) by Terjesen. For the same reason, drops larger than given by some critical radius, may have a calculated greater than the adsorbed film can maintain, and hence will rise or fall with virtually no retardation, though drops below this critical size will be retarded to the velocity expected for solid spheres (70). 

Calcns of the critical radius for Tetryl, based on the Merzhanov Friedman treatments are compared in Table 2. Agreement is quite good. However, note that, as expected, critical radii based on the Frank-Kamenetskii-Chambre treatment (steady-state conditions) are considerably smaller than those computed via the hot spot approach. For comparison, Eq 14 (based on Ref 7) gives a r = 2.76 x 10"3cm for a Tetryl sphere at 700 K, and acr = 2.37cm at 445°K, in close agreement with Merzhanov... 

Table 5 Comparison of the critical radius Rc (in au), n (/cm3) and critical pressure Pc (in atm) between the results obtained by using a Debye-Huckel model and an Ion-Sphere (IS) model. Reprinted with permission from [203] Copyright 2006, John Wiley Sons, Inc.

Derive the expression for the critical radius of insulation for a sphere. 

Heat Conduction in Cylinders and Spheres 150 Multilayered Cylinders and Spheres 152 3-5 Critical Radius of Insulation 156... 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

The discussions above can be repeated for a sphere, and it can be shown in a similar manner that the critical radius of insulation for a spherical shell is... 

A rough estimate of the critical radius of a homogeneous unreflected reactor may be obtained simply by estimating the neutron mean free path according to (14.6). Assuming metal with a density of 19 g cm and a fast fission cross section of 2 X crn, one obtains = 10 cm. A sphere with this radius weighs 80 ks. For an unreflected metal sphere containing 93.5% the correct value is 52 kg. Pu has the smallest unreflected critical size for Pu (5-phase, density 15.8 g cm ) it is 15.7 kg ( 6 kg reflected), and for 16.2 kg ( 6 kg reflected). 

FIGURE 40. Modelling class 1 (+) by a circle (d-dimensional hyper-sphere) around the centre of gravity c. Within a critical radius r almost only patterns of class 1 are found. 

Critical thiclDiess of slab or cube and critical radius of sphere or cylinder. 

Approximate Percent Pu Sphere WaU Thickness (cm) Plutonium. Cone (g/Uter) Uranium Cone (g/Uter) Acid Molarity Critical Radius (cm) Critical Volume (liter) Critical Mass (g/Pu)... 

The TART," ALICE, and MCN" codes were used to determine the critical radius of a sphere of UH, where the U enrichment varied from 20 to 100%. All codes derived their cross-section information from the same evaluated library." The TART and ALICE codes use continuous energy neutrons and the same 175-group cross-section structure, while MCN is continuous in both neutron energy and cross-section representation. The results" of these calculations are in Fig. 1. The ALICE. and TART codes were used to calculate 41 fast" and 4 slow experimental critical assemblies." " Forty of the fast assemblies have Keff between 0.984 and 1.014 for... 

In this case the liquid need only form a portion of the sphere in order to achieve the critical radius and the flee energy of formation of the nucleus will be multiplied by a shape factor, S(0y)> which will depend on how well the liquid wets the substrate. 

As appears from Equation 2.182, the critical radius R depends on the parameters of the disjoining pressure isotherm (a, and to) and the interface tension. Under otherwise equal conditions, a decrease in the interface tension reduces the values of R, thus limiting the region of the applicability of the theory of solid spheres. 

The calculations show that the values of R decrease as 0 increases. Thus, at small values of 0<, close to zero, R = 1(T cm at 0 = 5-6°, R = 10 cm at 0 = 20-30°, R = 5x 10 cm and at 0 = 90°, R = 5x 10- cm. In this manner, an increase in the contact angle, corresponding to the enhancement of the droplet interaction, causes a decrease in the critical radius R. Now, this means that in the case of a strong interparticle interaction, the approach of solid spheres becomes less applicable yet, in the case of weakly interacting droplets, the particular approach may be used even for relatively large-sized droplets. The solutions obtained allow the evaluation ofR and help to choose the corresponding equations for the calculation of the equilibrium shape of the droplets within the contact zone. 

CP-1 was assembled in an approximately spherical shape with the purest graphite in the center. About 6 tons of luanium metal fuel was used, in addition to approximately 40.5 tons of uranium oxide fuel. The lowest point of the reactor rested on the floor and the periphery was supported on a wooden structure. The whole pile was surrounded by a tent of mbberized balloon fabric so that neutron absorbing air could be evacuated. About 75 layers of 10.48-cm (4.125-in.) graphite bricks would have been required to complete the 790-cm diameter sphere. However, criticality was achieved at layer 56 without the need to evacuate the air, and assembly was discontinued at layer 57. The core then had an ellipsoidal cross section, with a polar radius of 209 cm and an equatorial radius of309 cm [20]. CP-1 was operated at low power (0.5 W) for several days. Fortuitously, it was found that the nuclear chain reaction could be controlled with cadmium strips which were inserted into the reactor to absorb neutrons and hence reduce the value of k to considerably less than 1. The pile was then disassembled and rebuilt at what is now the site of Argonne National Laboratory, U.S.A, with a concrete biological shield. Designated CP-2, the pile eventually reached a power level of 100 kW [22]. 

In [220] it was shown, for spheres with a radius tending to zero, that there must be a critical value of Bcr = 4 3 " 1ma3N in the system which is conductive to an infinite cluster of bonded sites (here a is site spacing N is volumetric concentration). 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

In studies of steric stabilizers too little attention is generally paid to the dispersion force attractions between particles and the critical separation distance (H ) needed to keep particles from flocculating. Adsorbed steric stabilizers can provide a certain film thickness on each particle but if the separation distance between colliding particles is less than H the particles will flocculate. The calculation of H is not cr difficult and measurements to prove or disprove such calculations are not difficult either. For equal-sized spheres of substance 1 with radius or in medium 2 the Hamaker equation for the dispersion force attractive energy (Uj2i) at close approach is (7) ... 

Applying Equations (5.21) to the adiabatic time corresponding to the critical Damkohler number, and realizing for a three-dimensional pile of effective radius, r,. A 3 (e.g. Sc = 3.32 for a sphere for Bi —> oo), then we estimate a typical ignition time at 6 = Sc 3 of... 

Stated physically, the critical condition for pyrophoricity under the proposed assumptions is that the heat release of the oxide coat formed on a nascent sphere at the ambient temperature must be sufficient to heat the metal to its vaporization point and supply enough heat to vaporize the remaining metal. In such an approach one must take into account the energy necessary to raise the metal from the ambient temperature to the vaporization temperature. If r is assumed to be the radius of the metal particle and 6 the thickness of the oxide coat [(r - <5) is the pure metal radius], then the critical heat balance for pyrophoricity contains three terms ... 

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

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